Subgroups — C₂×C₃₁×#3280 #21517799

Order 7936.

3074 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#546	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3280	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#463	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#449	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#450	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	2	yes	complemented
14	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#135	—	2	8	no	—
16	C₂×C₃₁×#135	C₂²	1	8	yes	complemented
17	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
18	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
19	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
20	C₂²×D₈×C₃₁	—	2	8	no	—
21	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
23	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×D₈×C₃₁	C₂²	1	8	yes	complemented
25	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#546	—	2	8	no	—
28	C₃₁×#546	C₂²	1	16	yes	complemented
29	C₂×C₃₁×#124	C₂²	1	2	yes	complemented
30	C₂×C₃₁×#123	—	2	2	no	—
31	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
32	C₂×C₃₁×#121	—	2	2	no	—
33	C₂×C₃₁×#121	C₂²	1	2	yes	complemented
34	C₂×C₃₁×#120	—	2	2	no	—
35	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
37	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
38	C₃₁×#495	C₂²	1	8	yes	complemented
39	C₃₁×#467	C₂²	1	8	yes	complemented
40	C₃₁×#466	C₂²	1	8	yes	complemented
41	C₃₁×#463	C₂²	1	16	yes	complemented
42	C₃₁×#450	C₂²	1	16	yes	complemented
43	C₃₁×#449	C₂²	1	16	yes	complemented
44	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
45	C₈×D₄×C₃₁	C₂²	1	8	yes	complemented
46	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
49	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
50	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
51	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
53	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
54	C₂⁵×C₃₁	—	2	2	no	—
55	C₂×#41×C₃₁	—	2	4	no	—
56	C₂×#41×C₃₁	—	2	8	no	—
57	C₂×#41×C₃₁	—	4	4	no	—
58	C₂×#41×C₃₁	C₂³	1	8	yes	complemented
59	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×D₄×C₃₁	—	2	8	no	—
61	C₂²×D₄×C₃₁	—	2	2	no	—
62	C₂²×D₄×C₃₁	—	2	2	no	—
63	C₂²×D₄×C₃₁	—	2	2	no	—
64	C₂²×D₄×C₃₁	—	2	4	no	—
65	C₂²×D₄×C₃₁	—	4	4	no	—
66	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
67	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₃₁	—	2	2	no	—
71	C₂³×C₄×C₃₁	—	2	2	no	—
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#135	—	2	32	no	—
75	C₂×Q₁₆×C₃₁	—	2	4	no	—
76	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	complemented
77	C₂×SD₁₆×C₃₁	—	2	8	no	—
78	C₂×SD₁₆×C₃₁	—	4	4	no	—
79	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	complemented
80	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	complemented
81	C₂×D₈×C₃₁	—	2	8	no	—
82	C₂×D₈×C₃₁	—	2	32	no	—
83	C₂×D₈×C₃₁	—	2	4	no	—
84	C₂×D₈×C₃₁	D₄	1	8	yes	complemented
85	C₂×D₈×C₃₁	C₂³	1	8	yes	—
86	C₂×D₈×C₃₁	C₂³	1	4	yes	—
87	C₂×D₈×C₃₁	C₂³	1	4	yes	—
88	C₂²×C₈×C₃₁	—	2	4	no	—
89	C₂²×C₈×C₃₁	—	2	4	no	—
90	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
94	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₃₁	C₂³	1	4	yes	complemented
96	C₃₁×#124	—	2	4	no	—
97	C₃₁×#124	C₂³	1	8	yes	complemented
98	C₃₁×#123	—	2	16	no	—
99	C₃₁×C₂²⋊Q₈	—	2	4	no	—
100	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
101	C₃₁×#121	—	2	4	no	—
102	C₃₁×#121	—	2	16	no	—
103	C₃₁×#121	C₂³	1	8	yes	complemented
104	C₃₁×#120	—	2	16	no	—
105	C₄×D₄×C₃₁	—	2	4	no	—
106	C₄×D₄×C₃₁	D₄	1	4	yes	—
107	C₄×D₄×C₃₁	C₂³	1	4	yes	—
108	C₄×D₄×C₃₁	C₂³	1	8	yes	complemented
109	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
118	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₃₁×#107	C₂³	1	4	yes	complemented
122	C₃₁×C₄⋊C₈	C₂³	1	4	yes	complemented
123	C₃₁×#103	C₂³	1	8	yes	complemented
124	C₃₁×#102	C₂³	1	8	yes	complemented
125	C₃₁×#102	C₂³	1	8	yes	complemented
126	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
127	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	complemented
128	C₄×C₈×C₃₁	C₂³	1	4	yes	complemented
129	C₂⁴×C₃₁	—	2	2	no	—
130	C₂⁴×C₃₁	—	2	2	no	—
131	C₂⁴×C₃₁	—	2	8	no	—
132	C₂⁴×C₃₁	—	4	8	no	—
133	C₂⁴×C₃₁	—	4	8	no	—
134	C₂⁴×C₃₁	—	8	4	no	—
135	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
137	#41×C₃₁	—	2	16	no	—
138	#41×C₃₁	—	4	8	no	—
139	#41×C₃₁	—	4	16	no	—
140	C₂×Q₈×C₃₁	—	2	2	no	—
141	C₂×Q₈×C₃₁	—	2	4	no	—
142	C₂×Q₈×C₃₁	—	2	4	no	—
143	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
144	C₂×D₄×C₃₁	—	2	4	no	—
145	C₂×D₄×C₃₁	—	2	16	no	—
146	C₂×D₄×C₃₁	—	2	8	no	—
147	C₂×D₄×C₃₁	—	2	8	no	—
148	C₂×D₄×C₃₁	—	2	2	no	—
149	C₂×D₄×C₃₁	—	2	4	no	—
150	C₂×D₄×C₃₁	—	2	8	no	—
151	C₂×D₄×C₃₁	—	2	2	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	4	no	—
154	C₂×D₄×C₃₁	—	4	8	no	—
155	C₂×D₄×C₃₁	—	4	16	no	—
156	C₂×D₄×C₃₁	—	4	8	no	—
157	C₂×D₄×C₃₁	—	4	8	no	—
158	C₂×D₄×C₃₁	—	8	4	no	—
159	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
160	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	—
161	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
162	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	2	no	—
165	C₂²×C₄×C₃₁	—	2	2	no	—
166	C₂²×C₄×C₃₁	—	2	2	no	—
167	C₂²×C₄×C₃₁	—	2	2	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	8	no	—
170	C₂²×C₄×C₃₁	—	2	8	no	—
171	C₂²×C₄×C₃₁	—	4	8	no	—
172	C₂²×C₄×C₃₁	—	8	4	no	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	Q₁₆×C₃₁	—	2	8	no	—
183	SD₁₆×C₃₁	—	2	16	no	—
184	SD₁₆×C₃₁	—	4	8	no	—
185	D₈×C₃₁	—	2	16	no	—
186	D₈×C₃₁	—	2	8	no	—
187	D₈×C₃₁	C₂×D₄	1	16	yes	complemented
188	C₂×C₈×C₃₁	—	2	2	no	—
189	C₂×C₈×C₃₁	—	2	2	no	—
190	C₂×C₈×C₃₁	—	2	2	no	—
191	C₂×C₈×C₃₁	—	2	8	no	—
192	C₂×C₈×C₃₁	—	2	8	no	—
193	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
194	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
195	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
196	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
197	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
198	C₄⋊C₄×C₃₁	—	2	8	no	—
199	C₄⋊C₄×C₃₁	—	2	2	no	—
200	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
201	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
202	C₂²⋊C₄×C₃₁	—	2	8	no	—
203	C₂²⋊C₄×C₃₁	—	2	8	no	—
204	C₂²⋊C₄×C₃₁	—	2	8	no	—
205	C₂²⋊C₄×C₃₁	—	2	8	no	—
206	C₂²⋊C₄×C₃₁	—	2	8	no	—
207	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
208	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
209	C₄²×C₃₁	C₂×D₄	1	2	yes	—
210	C₄²×C₃₁	C₂⁴	1	2	yes	—
211	C₂×#3280	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₃₁	—	2	2	no	—
213	C₂³×C₃₁	—	2	2	no	—
214	C₂³×C₃₁	—	2	2	no	—
215	C₂³×C₃₁	—	2	8	no	—
216	C₂³×C₃₁	—	2	2	no	—
217	C₂³×C₃₁	—	2	2	no	—
218	C₂³×C₃₁	—	2	4	no	—
219	C₂³×C₃₁	—	2	4	no	—
220	C₂³×C₃₁	—	4	16	no	—
221	C₂³×C₃₁	—	4	4	no	—
222	C₂³×C₃₁	—	4	4	no	—
223	C₂³×C₃₁	—	4	4	no	—
224	C₂³×C₃₁	—	4	4	no	—
225	C₂³×C₃₁	—	4	16	no	—
226	C₂³×C₃₁	—	4	4	no	—
227	C₂³×C₃₁	—	8	8	no	—
228	C₂³×C₃₁	—	8	32	no	—
229	C₂³×C₃₁	#135	1	4	yes	complemented
230	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
231	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
232	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	Q₈×C₃₁	—	2	4	no	—
234	Q₈×C₃₁	—	2	4	no	—
235	Q₈×C₃₁	—	4	4	no	—
236	D₄×C₃₁	—	2	8	no	—
237	D₄×C₃₁	—	2	8	no	—
238	D₄×C₃₁	—	2	4	no	—
239	D₄×C₃₁	—	2	4	no	—
240	D₄×C₃₁	—	2	4	no	—
241	D₄×C₃₁	—	4	16	no	—
242	D₄×C₃₁	—	4	16	no	—
243	D₄×C₃₁	—	4	4	no	—
244	D₄×C₃₁	—	8	8	no	—
245	D₄×C₃₁	C₂²×D₄	1	8	yes	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	2	4	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	2	4	no	—
258	C₂×C₄×C₃₁	—	4	16	no	—
259	C₂×C₄×C₃₁	—	4	4	no	—
260	C₂×C₄×C₃₁	—	4	4	no	—
261	C₂×C₄×C₃₁	—	8	8	no	—
262	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
269	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
270	C₈×C₃₁	—	2	2	no	—
271	C₈×C₃₁	—	2	2	no	—
272	C₈×C₃₁	—	2	2	no	—
273	C₈×C₃₁	C₂²×D₄	1	4	yes	—
274	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
275	C₂³×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
276	C₂×#546	C₂×C₃₁	1	2	yes	complemented
277	#3280	C₂×C₃₁	1	16	yes	direct product factor
278	C₂×#495	C₂×C₃₁	1	1	yes	characteristic, complemented
279	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
280	C₂×#466	C₂×C₃₁	1	1	yes	characteristic, complemented
281	C₂×#463	C₂×C₃₁	1	2	yes	complemented
282	C₂×#449	C₂×C₃₁	1	2	yes	complemented
283	C₂×#450	C₂×C₃₁	1	2	yes	complemented
284	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
285	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	8	no	—
288	C₂²×C₃₁	—	2	2	no	—
289	C₂²×C₃₁	—	2	2	no	—
290	C₂²×C₃₁	—	2	8	no	—
291	C₂²×C₃₁	—	2	2	no	—
292	C₂²×C₃₁	—	2	2	no	—
293	C₂²×C₃₁	—	4	4	no	—
294	C₂²×C₃₁	—	4	16	no	—
295	C₂²×C₃₁	—	4	4	no	—
296	C₂²×C₃₁	—	4	4	no	—
297	C₂²×C₃₁	—	4	4	no	—
298	C₂²×C₃₁	—	4	4	no	—
299	C₂²×C₃₁	—	4	4	no	—
300	C₂²×C₃₁	—	8	32	no	—
301	C₂²×C₃₁	—	8	8	no	—
302	C₂²×C₃₁	D₄²	1	2	yes	—
303	C₂²×C₃₁	C₂×#135	1	2	yes	—
304	C₂²×C₃₁	C₂×#135	1	4	yes	complemented
305	C₂²×C₃₁	#576	1	2	yes	—
306	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
307	C₄×C₃₁	—	2	2	no	—
308	C₄×C₃₁	—	2	2	no	—
309	C₄×C₃₁	—	2	2	no	—
310	C₄×C₃₁	—	2	2	no	—
311	C₄×C₃₁	—	4	4	no	—
312	C₄×C₃₁	—	4	4	no	—
313	C₄×C₃₁	D₄²	1	2	yes	—
314	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
315	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
316	C₂²×#41	C₂²×C₃₁	1	2	yes	complemented
317	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
318	C₂×#135	—	2	8	no	—
319	C₂×#135	C₂²×C₃₁	1	8	yes	complemented
320	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
321	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
322	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
323	C₂²×D₈	—	2	8	no	—
324	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
325	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
326	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
327	C₂²×D₈	C₂²×C₃₁	1	8	yes	complemented
328	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
329	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
330	#546	—	2	8	no	—
331	#546	C₂²×C₃₁	1	16	yes	complemented
332	C₂×#124	C₂²×C₃₁	1	2	yes	complemented
333	C₂×#123	—	2	2	no	—
334	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
335	C₂×#121	—	2	2	no	—
336	C₂×#121	C₂²×C₃₁	1	2	yes	complemented
337	C₂×#120	—	2	2	no	—
338	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
340	C₂²×C₂²⋊C₄	—	2	2	no	—
341	#495	C₂²×C₃₁	1	8	yes	complemented
342	#467	C₂²×C₃₁	1	8	yes	complemented
343	#466	C₂²×C₃₁	1	8	yes	complemented
344	#463	C₂²×C₃₁	1	16	yes	complemented
345	#450	C₂²×C₃₁	1	16	yes	complemented
346	#449	C₂²×C₃₁	1	16	yes	complemented
347	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
348	C₈×D₄	C₂²×C₃₁	1	8	yes	complemented
349	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
350	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
351	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
352	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
353	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
354	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
355	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
356	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
357	C₂×C₃₁	—	2	4	no	—
358	C₂×C₃₁	—	4	2	no	—
359	C₂×C₃₁	—	4	8	no	—
360	C₂×C₃₁	—	8	4	no	—
361	C₂×C₃₁	#3280	1	4	yes	direct product factor
362	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
363	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
364	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
365	C₂⁵	—	2	2	no	—
366	C₂×#41	—	2	4	no	—
367	C₂×#41	—	2	8	no	—
368	C₂×#41	—	4	4	no	—
369	C₂×#41	C₂³×C₃₁	1	8	yes	complemented
370	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
371	C₂²×D₄	—	2	8	no	—
372	C₂²×D₄	—	2	2	no	—
373	C₂²×D₄	—	2	2	no	—
374	C₂²×D₄	—	2	2	no	—
375	C₂²×D₄	—	2	4	no	—
376	C₂²×D₄	—	4	4	no	—
377	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
378	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
379	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
380	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
381	C₂³×C₄	—	2	2	no	—
382	C₂³×C₄	—	2	2	no	—
383	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
384	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
385	#135	—	2	32	no	—
386	C₂×Q₁₆	—	2	4	no	—
387	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	complemented
388	C₂×SD₁₆	—	2	8	no	—
389	C₂×SD₁₆	—	4	4	no	—
390	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	complemented
391	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	complemented
392	C₂×D₈	—	2	8	no	—
393	C₂×D₈	—	2	32	no	—
394	C₂×D₈	—	2	4	no	—
395	C₂×D₈	D₄×C₃₁	1	8	yes	complemented
396	C₂×D₈	C₂³×C₃₁	1	8	yes	—
397	C₂×D₈	C₂³×C₃₁	1	4	yes	—
398	C₂×D₈	C₂³×C₃₁	1	4	yes	—
399	C₂²×C₈	—	2	4	no	—
400	C₂²×C₈	—	2	4	no	—
401	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
402	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
403	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
404	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
405	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
406	C₂²×C₈	C₂³×C₃₁	1	4	yes	complemented
407	#124	—	2	4	no	—
408	#124	C₂³×C₃₁	1	8	yes	complemented
409	#123	—	2	16	no	—
410	C₂²⋊Q₈	—	2	4	no	—
411	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
412	#121	—	2	4	no	—
413	#121	—	2	16	no	—
414	#121	C₂³×C₃₁	1	8	yes	complemented
415	#120	—	2	16	no	—
416	C₄×D₄	—	2	4	no	—
417	C₄×D₄	D₄×C₃₁	1	4	yes	—
418	C₄×D₄	C₂³×C₃₁	1	4	yes	—
419	C₄×D₄	C₂³×C₃₁	1	8	yes	complemented
420	C₂×C₄⋊C₄	—	2	2	no	—
421	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
423	C₂×C₂²⋊C₄	—	2	2	no	—
424	C₂×C₂²⋊C₄	—	2	2	no	—
425	C₂×C₂²⋊C₄	—	2	2	no	—
426	C₂×C₂²⋊C₄	—	2	2	no	—
427	C₂×C₂²⋊C₄	—	2	2	no	—
428	C₂×C₂²⋊C₄	—	2	16	no	—
429	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
432	#107	C₂³×C₃₁	1	4	yes	complemented
433	C₄⋊C₈	C₂³×C₃₁	1	4	yes	complemented
434	#103	C₂³×C₃₁	1	8	yes	complemented
435	#102	C₂³×C₃₁	1	8	yes	complemented
436	#102	C₂³×C₃₁	1	8	yes	complemented
437	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
438	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	complemented
439	C₄×C₈	C₂³×C₃₁	1	4	yes	complemented
440	C₃₁	C₂×#3280	1	1	yes	characteristic, direct product factor, Sylow
441	C₂⁴	—	2	2	no	—
442	C₂⁴	—	2	2	no	—
443	C₂⁴	—	2	8	no	—
444	C₂⁴	—	4	8	no	—
445	C₂⁴	—	4	8	no	—
446	C₂⁴	—	8	4	no	—
447	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
449	#41	—	2	16	no	—
450	#41	—	4	8	no	—
451	#41	—	4	16	no	—
452	C₂×Q₈	—	2	2	no	—
453	C₂×Q₈	—	2	4	no	—
454	C₂×Q₈	—	2	4	no	—
455	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	16	no	—
458	C₂×D₄	—	2	8	no	—
459	C₂×D₄	—	2	8	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	2	8	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	4	8	no	—
467	C₂×D₄	—	4	16	no	—
468	C₂×D₄	—	4	8	no	—
469	C₂×D₄	—	4	8	no	—
470	C₂×D₄	—	8	4	no	—
471	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
472	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	—
473	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
474	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	8	no	—
482	C₂²×C₄	—	2	8	no	—
483	C₂²×C₄	—	4	8	no	—
484	C₂²×C₄	—	8	4	no	—
485	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
490	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
491	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
493	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
494	Q₁₆	—	2	8	no	—
495	SD₁₆	—	2	16	no	—
496	SD₁₆	—	4	8	no	—
497	D₈	—	2	16	no	—
498	D₈	—	2	8	no	—
499	D₈	C₂×D₄×C₃₁	1	16	yes	complemented
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	2	8	no	—
504	C₂×C₈	—	2	8	no	—
505	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
506	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
507	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
508	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
509	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
510	C₄⋊C₄	—	2	8	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
513	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
514	C₂²⋊C₄	—	2	8	no	—
515	C₂²⋊C₄	—	2	8	no	—
516	C₂²⋊C₄	—	2	8	no	—
517	C₂²⋊C₄	—	2	8	no	—
518	C₂²⋊C₄	—	2	8	no	—
519	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
520	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
521	C₄²	C₂×D₄×C₃₁	1	2	yes	—
522	C₄²	C₂⁴×C₃₁	1	2	yes	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	8	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	2	4	no	—
531	C₂³	—	4	16	no	—
532	C₂³	—	4	4	no	—
533	C₂³	—	4	4	no	—
534	C₂³	—	4	4	no	—
535	C₂³	—	4	4	no	—
536	C₂³	—	4	16	no	—
537	C₂³	—	4	4	no	—
538	C₂³	—	8	8	no	—
539	C₂³	—	8	32	no	—
540	C₂³	C₃₁×#135	1	4	yes	complemented
541	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
542	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
543	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
544	Q₈	—	2	4	no	—
545	Q₈	—	2	4	no	—
546	Q₈	—	4	4	no	—
547	D₄	—	2	8	no	—
548	D₄	—	2	8	no	—
549	D₄	—	2	4	no	—
550	D₄	—	2	4	no	—
551	D₄	—	2	4	no	—
552	D₄	—	4	16	no	—
553	D₄	—	4	16	no	—
554	D₄	—	4	4	no	—
555	D₄	—	8	8	no	—
556	D₄	C₂²×D₄×C₃₁	1	8	yes	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	4	16	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	8	8	no	—
573	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
574	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
575	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
576	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
577	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
578	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
579	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
580	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
581	C₈	—	2	2	no	—
582	C₈	—	2	2	no	—
583	C₈	—	2	2	no	—
584	C₈	C₂²×D₄×C₃₁	1	4	yes	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	8	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	8	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	16	no	—
594	C₂²	—	4	4	no	—
595	C₂²	—	4	4	no	—
596	C₂²	—	4	4	no	—
597	C₂²	—	4	4	no	—
598	C₂²	—	4	4	no	—
599	C₂²	—	8	32	no	—
600	C₂²	—	8	8	no	—
601	C₂²	D₄²×C₃₁	1	2	yes	—
602	C₂²	C₂×C₃₁×#135	1	2	yes	—
603	C₂²	C₂×C₃₁×#135	1	4	yes	complemented
604	C₂²	C₃₁×#576	1	2	yes	—
605	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	4	4	no	—
611	C₄	—	4	4	no	—
612	C₄	D₄²×C₃₁	1	2	yes	—
613	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
614	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
615	C₂	—	2	4	no	—
616	C₂	—	4	2	no	—
617	C₂	—	4	8	no	—
618	C₂	—	8	4	no	—
619	C₂	C₃₁×#3280	1	4	yes	direct product factor
620	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic
621	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
622	C₂	C₂×C₃₁×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₃₁×#135	C₂	semidirect
2. C₂³×D₈×C₃₁	C₂	semidirect
3. C₂×C₃₁×#546	C₂	semidirect
4. C₃₁×#3280	C₂	direct (N × H)
5. C₂×C₃₁×#495	C₂	semidirect
6. C₂×C₃₁×#467	C₂	semidirect
7. C₂×C₃₁×#466	C₂	semidirect
8. C₂×C₃₁×#463	C₂	semidirect
9. C₂×C₃₁×#449	C₂	semidirect
10. C₂×C₃₁×#450	C₂	semidirect
11. C₂×C₄×D₈×C₃₁	C₂	semidirect
12. C₂×C₈×D₄×C₃₁	C₂	semidirect
13. C₂²×#41×C₃₁	C₂²	semidirect
16. C₂×C₃₁×#135	C₂²	semidirect
17. C₂²×Q₁₆×C₃₁	C₂²	semidirect
18. C₂²×SD₁₆×C₃₁	C₂²	semidirect
19. C₂²×SD₁₆×C₃₁	C₂²	semidirect
24. C₂²×D₈×C₃₁	C₂²	semidirect
26. C₂³×C₈×C₃₁	C₂²	semidirect
28. C₃₁×#546	C₂²	semidirect
29. C₂×C₃₁×#124	C₂²	semidirect
33. C₂×C₃₁×#121	C₂²	semidirect
36. C₂×C₄×D₄×C₃₁	C₂²	semidirect
38. C₃₁×#495	C₂²	semidirect
39. C₃₁×#467	C₂²	semidirect
40. C₃₁×#466	C₂²	semidirect
41. C₃₁×#463	C₂²	semidirect
42. C₃₁×#450	C₂²	semidirect
43. C₃₁×#449	C₂²	semidirect
44. C₄×D₈×C₃₁	C₂²	semidirect
45. C₈×D₄×C₃₁	C₂²	semidirect
46. C₂×C₃₁×#107	C₂²	semidirect
47. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
48. C₂×C₃₁×#103	C₂²	semidirect
49. C₂×C₃₁×#102	C₂²	semidirect
50. C₂×C₃₁×#102	C₂²	semidirect
52. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
53. C₂×C₄×C₈×C₃₁	C₂²	semidirect
58. C₂×#41×C₃₁	C₂³	semidirect
76. C₂×Q₁₆×C₃₁	C₂³	semidirect
79. C₂×SD₁₆×C₃₁	C₂³	semidirect
80. C₂×SD₁₆×C₃₁	C₂³	semidirect
84. C₂×D₈×C₃₁	D₄	semidirect
95. C₂²×C₈×C₃₁	C₂³	semidirect
97. C₃₁×#124	C₂³	semidirect
103. C₃₁×#121	C₂³	semidirect
108. C₄×D₄×C₃₁	C₂³	semidirect
121. C₃₁×#107	C₂³	semidirect
122. C₃₁×C₄⋊C₈	C₂³	semidirect
123. C₃₁×#103	C₂³	semidirect
124. C₃₁×#102	C₂³	semidirect
125. C₃₁×#102	C₂³	semidirect
127. C₃₁×C₂²⋊C₈	C₂³	semidirect
128. C₄×C₈×C₃₁	C₂³	semidirect
187. D₈×C₃₁	C₂×D₄	semidirect
211. C₂×#3280	C₃₁	direct (N × H)
229. C₂³×C₃₁	#135	semidirect
274. C₂²×#135	C₂×C₃₁	semidirect
275. C₂³×D₈	C₂×C₃₁	semidirect
276. C₂×#546	C₂×C₃₁	semidirect
277. #3280	C₂×C₃₁	direct (N × H)
278. C₂×#495	C₂×C₃₁	semidirect
279. C₂×#467	C₂×C₃₁	semidirect
280. C₂×#466	C₂×C₃₁	semidirect
281. C₂×#463	C₂×C₃₁	semidirect
282. C₂×#449	C₂×C₃₁	semidirect
283. C₂×#450	C₂×C₃₁	semidirect
284. C₂×C₄×D₈	C₂×C₃₁	semidirect
285. C₂×C₈×D₄	C₂×C₃₁	semidirect
304. C₂²×C₃₁	C₂×#135	semidirect
316. C₂²×#41	C₂²×C₃₁	semidirect
319. C₂×#135	C₂²×C₃₁	semidirect
320. C₂²×Q₁₆	C₂²×C₃₁	semidirect
321. C₂²×SD₁₆	C₂²×C₃₁	semidirect
322. C₂²×SD₁₆	C₂²×C₃₁	semidirect
327. C₂²×D₈	C₂²×C₃₁	semidirect
329. C₂³×C₈	C₂²×C₃₁	semidirect
331. #546	C₂²×C₃₁	semidirect
332. C₂×#124	C₂²×C₃₁	semidirect
336. C₂×#121	C₂²×C₃₁	semidirect
339. C₂×C₄×D₄	C₂²×C₃₁	semidirect
341. #495	C₂²×C₃₁	semidirect
342. #467	C₂²×C₃₁	semidirect
343. #466	C₂²×C₃₁	semidirect
344. #463	C₂²×C₃₁	semidirect
345. #450	C₂²×C₃₁	semidirect
346. #449	C₂²×C₃₁	semidirect
347. C₄×D₈	C₂²×C₃₁	semidirect
348. C₈×D₄	C₂²×C₃₁	semidirect
349. C₂×#107	C₂²×C₃₁	semidirect
350. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
351. C₂×#103	C₂²×C₃₁	semidirect
352. C₂×#102	C₂²×C₃₁	semidirect
353. C₂×#102	C₂²×C₃₁	semidirect
355. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
356. C₂×C₄×C₈	C₂²×C₃₁	semidirect
361. C₂×C₃₁	#3280	direct (N × H)
369. C₂×#41	C₂³×C₃₁	semidirect
387. C₂×Q₁₆	C₂³×C₃₁	semidirect
390. C₂×SD₁₆	C₂³×C₃₁	semidirect
391. C₂×SD₁₆	C₂³×C₃₁	semidirect
395. C₂×D₈	D₄×C₃₁	semidirect
406. C₂²×C₈	C₂³×C₃₁	semidirect
408. #124	C₂³×C₃₁	semidirect
414. #121	C₂³×C₃₁	semidirect
419. C₄×D₄	C₂³×C₃₁	semidirect
432. #107	C₂³×C₃₁	semidirect
433. C₄⋊C₈	C₂³×C₃₁	semidirect
434. #103	C₂³×C₃₁	semidirect
435. #102	C₂³×C₃₁	semidirect
436. #102	C₂³×C₃₁	semidirect
438. C₂²⋊C₈	C₂³×C₃₁	semidirect
439. C₄×C₈	C₂³×C₃₁	semidirect
440. C₃₁	C₂×#3280	direct (N × H)
499. D₈	C₂×D₄×C₃₁	semidirect
540. C₂³	C₃₁×#135	semidirect
603. C₂²	C₂×C₃₁×#135	semidirect
619. C₂	C₃₁×#3280	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₁×#135	2
C₂³×D₈×C₃₁	2
C₂×C₃₁×#546	2
C₃₁×#3280	2
C₂×C₃₁×#495	2
C₂×C₃₁×#467	2
C₂×C₃₁×#466	2
C₂×C₃₁×#463	2
C₂×C₃₁×#449	2
C₂×C₃₁×#450	2
C₂×C₄×D₈×C₃₁	2
C₂×C₈×D₄×C₃₁	2
C₂×#3280	31

Subgroups — C2×C31×#3280 #21517799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3280 #21517799