Subgroups — C₂×C₃₁×#3285 #21517804

Order 7936.

2794 conjugacy classes of subgroups.

The subgroup lattice has 596 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×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#548	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3285	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×#464	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#452	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#448	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₁₆×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	—
14	C₂³×Q₈×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	—	2	8	no	—
18	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
22	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
23	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
24	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#548	—	2	8	no	—
28	C₃₁×#548	C₂²	1	16	yes	complemented
29	C₂×C₃₁×#124	—	2	2	no	—
30	C₂×C₃₁×#124	C₂²	1	2	yes	—
31	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
32	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
33	C₂×C₃₁×#121	—	2	2	no	—
34	C₂×C₃₁×#121	C₂²	1	2	yes	—
35	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₃₁	—	2	2	no	—
37	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
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×#464	C₂²	1	16	yes	complemented
42	C₃₁×#452	C₂²	1	16	yes	complemented
43	C₃₁×#448	C₂²	1	16	yes	complemented
44	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
45	C₈×D₄×C₃₁	C₂²	1	8	yes	complemented
46	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
47	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#103	C₂²	1	2	yes	—
49	C₂×C₃₁×#103	C₂²	1	2	yes	—
50	C₂×C₃₁×#102	C₂²	1	2	yes	—
51	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
54	C₂×#41×C₃₁	—	2	4	no	—
55	C₂×#41×C₃₁	—	2	8	no	—
56	C₂×#41×C₃₁	—	4	4	no	—
57	C₂×#41×C₃₁	C₂³	1	8	yes	—
58	C₂²×Q₈×C₃₁	—	2	2	no	—
59	C₂²×Q₈×C₃₁	—	2	4	no	—
60	C₂²×Q₈×C₃₁	—	2	8	no	—
61	C₂²×Q₈×C₃₁	—	4	4	no	—
62	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
63	C₂²×Q₈×C₃₁	C₂³	1	8	yes	—
64	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₃₁	—	2	2	no	—
66	C₂²×D₄×C₃₁	—	2	2	no	—
67	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₃₁	—	2	2	no	—
70	C₂³×C₄×C₃₁	—	2	2	no	—
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#135	—	2	32	no	—
74	C₂×Q₁₆×C₃₁	—	2	8	no	—
75	C₂×Q₁₆×C₃₁	—	2	32	no	—
76	C₂×Q₁₆×C₃₁	—	2	4	no	—
77	C₂×Q₁₆×C₃₁	D₄	1	8	yes	complemented
78	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
79	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
80	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
81	C₂×SD₁₆×C₃₁	—	2	8	no	—
82	C₂×SD₁₆×C₃₁	—	4	4	no	—
83	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
84	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
85	C₂×D₈×C₃₁	—	2	4	no	—
86	C₂×D₈×C₃₁	C₂³	1	4	yes	—
87	C₂²×C₈×C₃₁	—	2	4	no	—
88	C₂²×C₈×C₃₁	—	2	4	no	—
89	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
93	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
94	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₃₁×#124	—	2	16	no	—
96	C₃₁×#124	—	2	4	no	—
97	C₃₁×#124	C₂³	1	8	yes	—
98	C₃₁×C₂²⋊Q₈	—	2	4	no	—
99	C₃₁×C₂²⋊Q₈	—	2	16	no	—
100	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
101	C₃₁×#121	—	2	16	no	—
102	C₃₁×#121	—	2	4	no	—
103	C₃₁×#121	C₂³	1	8	yes	—
104	C₄×Q₈×C₃₁	—	2	4	no	—
105	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
106	C₄×D₄×C₃₁	—	2	16	no	—
107	C₄×D₄×C₃₁	D₄	1	4	yes	—
108	C₄×D₄×C₃₁	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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	D₄	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	—	2	2	no	—
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₃₁×#107	C₂³	1	4	yes	—
122	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
123	C₃₁×#103	C₂³	1	8	yes	—
124	C₃₁×#103	C₂³	1	8	yes	—
125	C₃₁×#102	C₂³	1	8	yes	—
126	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
127	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
128	C₄×C₈×C₃₁	C₂³	1	4	yes	—
129	C₂⁴×C₃₁	—	2	2	no	—
130	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
132	#41×C₃₁	—	2	16	no	—
133	#41×C₃₁	—	4	8	no	—
134	#41×C₃₁	—	4	16	no	—
135	C₂×Q₈×C₃₁	—	2	2	no	—
136	C₂×Q₈×C₃₁	—	2	8	no	—
137	C₂×Q₈×C₃₁	—	2	2	no	—
138	C₂×Q₈×C₃₁	—	2	4	no	—
139	C₂×Q₈×C₃₁	—	2	16	no	—
140	C₂×Q₈×C₃₁	—	2	4	no	—
141	C₂×Q₈×C₃₁	—	2	4	no	—
142	C₂×Q₈×C₃₁	—	4	8	no	—
143	C₂×Q₈×C₃₁	—	4	8	no	—
144	C₂×Q₈×C₃₁	—	4	16	no	—
145	C₂×Q₈×C₃₁	C₂×D₄	1	8	yes	—
146	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
147	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
148	C₂×D₄×C₃₁	—	2	4	no	—
149	C₂×D₄×C₃₁	—	2	8	no	—
150	C₂×D₄×C₃₁	—	2	8	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	—	2	2	no	—
153	C₂×D₄×C₃₁	—	2	4	no	—
154	C₂×D₄×C₃₁	—	4	8	no	—
155	C₂×D₄×C₃₁	—	4	8	no	—
156	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
157	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₃₁	—	2	2	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	—	2	2	no	—
163	C₂²×C₄×C₃₁	—	2	8	no	—
164	C₂²×C₄×C₃₁	—	2	2	no	—
165	C₂²×C₄×C₃₁	—	2	2	no	—
166	C₂²×C₄×C₃₁	—	2	8	no	—
167	C₂²×C₄×C₃₁	—	4	8	no	—
168	C₂²×C₄×C₃₁	—	4	8	no	—
169	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	Q₁₆×C₃₁	—	2	16	no	—
179	Q₁₆×C₃₁	—	2	8	no	—
180	Q₁₆×C₃₁	C₂×D₄	1	16	yes	complemented
181	SD₁₆×C₃₁	—	2	16	no	—
182	SD₁₆×C₃₁	—	4	8	no	—
183	D₈×C₃₁	—	2	8	no	—
184	C₂×C₈×C₃₁	—	2	2	no	—
185	C₂×C₈×C₃₁	—	2	2	no	—
186	C₂×C₈×C₃₁	—	2	2	no	—
187	C₂×C₈×C₃₁	—	2	8	no	—
188	C₂×C₈×C₃₁	—	2	8	no	—
189	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
190	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
191	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
192	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
193	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
194	C₄⋊C₄×C₃₁	—	2	8	no	—
195	C₄⋊C₄×C₃₁	—	2	8	no	—
196	C₄⋊C₄×C₃₁	—	2	2	no	—
197	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
198	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
199	C₂²⋊C₄×C₃₁	—	2	8	no	—
200	C₂²⋊C₄×C₃₁	—	2	8	no	—
201	C₂²⋊C₄×C₃₁	—	2	8	no	—
202	C₂²⋊C₄×C₃₁	—	2	8	no	—
203	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
204	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
205	C₄²×C₃₁	—	2	8	no	—
206	C₄²×C₃₁	C₂×D₄	1	2	yes	—
207	C₄²×C₃₁	C₂⁴	1	2	yes	—
208	C₂×#3285	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂³×C₃₁	—	2	2	no	—
210	C₂³×C₃₁	—	2	2	no	—
211	C₂³×C₃₁	—	2	2	no	—
212	C₂³×C₃₁	—	2	2	no	—
213	C₂³×C₃₁	—	2	2	no	—
214	C₂³×C₃₁	—	2	4	no	—
215	C₂³×C₃₁	—	4	4	no	—
216	C₂³×C₃₁	—	4	4	no	—
217	C₂³×C₃₁	—	4	4	no	—
218	C₂³×C₃₁	#135	1	4	yes	complemented
219	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
220	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
221	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	Q₈×C₃₁	—	2	8	no	—
223	Q₈×C₃₁	—	2	4	no	—
224	Q₈×C₃₁	—	2	4	no	—
225	Q₈×C₃₁	—	2	4	no	—
226	Q₈×C₃₁	—	4	16	no	—
227	Q₈×C₃₁	—	4	4	no	—
228	Q₈×C₃₁	C₂²×D₄	1	8	yes	—
229	D₄×C₃₁	—	2	8	no	—
230	D₄×C₃₁	—	2	4	no	—
231	D₄×C₃₁	—	2	4	no	—
232	D₄×C₃₁	—	4	16	no	—
233	D₄×C₃₁	—	4	16	no	—
234	D₄×C₃₁	—	4	4	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	8	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	4	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	4	16	no	—
250	C₂×C₄×C₃₁	—	4	16	no	—
251	C₂×C₄×C₃₁	—	4	4	no	—
252	C₂×C₄×C₃₁	—	4	4	no	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
260	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
261	C₈×C₃₁	—	2	2	no	—
262	C₈×C₃₁	—	2	2	no	—
263	C₈×C₃₁	—	2	2	no	—
264	C₈×C₃₁	C₂²×D₄	1	4	yes	—
265	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
266	C₂³×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₂×#548	C₂×C₃₁	1	2	yes	complemented
268	#3285	C₂×C₃₁	1	16	yes	direct product factor
269	C₂×#495	C₂×C₃₁	1	1	yes	characteristic, complemented
270	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
271	C₂×#466	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂×#464	C₂×C₃₁	1	2	yes	complemented
273	C₂×#452	C₂×C₃₁	1	2	yes	complemented
274	C₂×#448	C₂×C₃₁	1	2	yes	complemented
275	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
276	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
277	C₂²×C₃₁	—	2	2	no	—
278	C₂²×C₃₁	—	2	8	no	—
279	C₂²×C₃₁	—	2	2	no	—
280	C₂²×C₃₁	—	2	2	no	—
281	C₂²×C₃₁	—	2	2	no	—
282	C₂²×C₃₁	—	2	2	no	—
283	C₂²×C₃₁	—	4	4	no	—
284	C₂²×C₃₁	—	4	4	no	—
285	C₂²×C₃₁	—	4	4	no	—
286	C₂²×C₃₁	—	4	4	no	—
287	C₂²×C₃₁	—	8	8	no	—
288	C₂²×C₃₁	D₄²	1	2	yes	—
289	C₂²×C₃₁	C₂×#135	1	2	yes	—
290	C₂²×C₃₁	C₂×#135	1	4	yes	complemented
291	C₂²×C₃₁	#578	1	2	yes	—
292	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	2	2	no	—
295	C₄×C₃₁	—	2	8	no	—
296	C₄×C₃₁	—	2	2	no	—
297	C₄×C₃₁	—	2	2	no	—
298	C₄×C₃₁	—	4	4	no	—
299	C₄×C₃₁	—	4	4	no	—
300	C₄×C₃₁	D₄²	1	2	yes	—
301	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
302	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
303	C₂²×#41	C₂²×C₃₁	1	2	yes	—
304	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
305	C₂×#135	—	2	8	no	—
306	C₂×#135	C₂²×C₃₁	1	8	yes	complemented
307	C₂²×Q₁₆	—	2	8	no	—
308	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
309	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
310	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
311	C₂²×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
312	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
313	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
314	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
315	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
316	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
317	#548	—	2	8	no	—
318	#548	C₂²×C₃₁	1	16	yes	complemented
319	C₂×#124	—	2	2	no	—
320	C₂×#124	C₂²×C₃₁	1	2	yes	—
321	C₂×C₂²⋊Q₈	—	2	2	no	—
322	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
323	C₂×#121	—	2	2	no	—
324	C₂×#121	C₂²×C₃₁	1	2	yes	—
325	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
326	C₂×C₄×D₄	—	2	2	no	—
327	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
328	#495	C₂²×C₃₁	1	8	yes	complemented
329	#467	C₂²×C₃₁	1	8	yes	complemented
330	#466	C₂²×C₃₁	1	8	yes	complemented
331	#464	C₂²×C₃₁	1	16	yes	complemented
332	#452	C₂²×C₃₁	1	16	yes	complemented
333	#448	C₂²×C₃₁	1	16	yes	complemented
334	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
335	C₈×D₄	C₂²×C₃₁	1	8	yes	complemented
336	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×#103	C₂²×C₃₁	1	2	yes	—
339	C₂×#103	C₂²×C₃₁	1	2	yes	—
340	C₂×#102	C₂²×C₃₁	1	2	yes	—
341	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
343	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×C₃₁	—	2	4	no	—
345	C₂×C₃₁	—	4	2	no	—
346	C₂×C₃₁	—	8	4	no	—
347	C₂×C₃₁	#3285	1	4	yes	direct product factor
348	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
349	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
350	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
351	C₂×#41	—	2	4	no	—
352	C₂×#41	—	2	8	no	—
353	C₂×#41	—	4	4	no	—
354	C₂×#41	C₂³×C₃₁	1	8	yes	—
355	C₂²×Q₈	—	2	2	no	—
356	C₂²×Q₈	—	2	4	no	—
357	C₂²×Q₈	—	2	8	no	—
358	C₂²×Q₈	—	4	4	no	—
359	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
360	C₂²×Q₈	C₂³×C₃₁	1	8	yes	—
361	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
362	C₂²×D₄	—	2	2	no	—
363	C₂²×D₄	—	2	2	no	—
364	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
365	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
366	C₂³×C₄	—	2	2	no	—
367	C₂³×C₄	—	2	2	no	—
368	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
370	#135	—	2	32	no	—
371	C₂×Q₁₆	—	2	8	no	—
372	C₂×Q₁₆	—	2	32	no	—
373	C₂×Q₁₆	—	2	4	no	—
374	C₂×Q₁₆	D₄×C₃₁	1	8	yes	complemented
375	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
376	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
377	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
378	C₂×SD₁₆	—	2	8	no	—
379	C₂×SD₁₆	—	4	4	no	—
380	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
381	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
382	C₂×D₈	—	2	4	no	—
383	C₂×D₈	C₂³×C₃₁	1	4	yes	—
384	C₂²×C₈	—	2	4	no	—
385	C₂²×C₈	—	2	4	no	—
386	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
387	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
388	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
389	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
390	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
391	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
392	#124	—	2	16	no	—
393	#124	—	2	4	no	—
394	#124	C₂³×C₃₁	1	8	yes	—
395	C₂²⋊Q₈	—	2	4	no	—
396	C₂²⋊Q₈	—	2	16	no	—
397	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
398	#121	—	2	16	no	—
399	#121	—	2	4	no	—
400	#121	C₂³×C₃₁	1	8	yes	—
401	C₄×Q₈	—	2	4	no	—
402	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
403	C₄×D₄	—	2	16	no	—
404	C₄×D₄	D₄×C₃₁	1	4	yes	—
405	C₄×D₄	C₂³×C₃₁	1	4	yes	—
406	C₂×C₄⋊C₄	—	2	2	no	—
407	C₂×C₄⋊C₄	—	2	2	no	—
408	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₂²⋊C₄	—	2	2	no	—
412	C₂×C₂²⋊C₄	—	2	2	no	—
413	C₂×C₂²⋊C₄	—	2	2	no	—
414	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
415	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄²	—	2	2	no	—
417	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
418	#107	C₂³×C₃₁	1	4	yes	—
419	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
420	#103	C₂³×C₃₁	1	8	yes	—
421	#103	C₂³×C₃₁	1	8	yes	—
422	#102	C₂³×C₃₁	1	8	yes	—
423	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
424	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
425	C₄×C₈	C₂³×C₃₁	1	4	yes	—
426	C₃₁	C₂×#3285	1	1	yes	characteristic, direct product factor, Sylow
427	C₂⁴	—	2	2	no	—
428	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
429	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
430	#41	—	2	16	no	—
431	#41	—	4	8	no	—
432	#41	—	4	16	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	2	8	no	—
435	C₂×Q₈	—	2	2	no	—
436	C₂×Q₈	—	2	4	no	—
437	C₂×Q₈	—	2	16	no	—
438	C₂×Q₈	—	2	4	no	—
439	C₂×Q₈	—	2	4	no	—
440	C₂×Q₈	—	4	8	no	—
441	C₂×Q₈	—	4	8	no	—
442	C₂×Q₈	—	4	16	no	—
443	C₂×Q₈	C₂×D₄×C₃₁	1	8	yes	—
444	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
445	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	8	no	—
448	C₂×D₄	—	2	8	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	4	8	no	—
453	C₂×D₄	—	4	8	no	—
454	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
455	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	8	no	—
464	C₂²×C₄	—	2	8	no	—
465	C₂²×C₄	—	4	8	no	—
466	C₂²×C₄	—	4	8	no	—
467	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
472	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
475	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
476	Q₁₆	—	2	16	no	—
477	Q₁₆	—	2	8	no	—
478	Q₁₆	C₂×D₄×C₃₁	1	16	yes	complemented
479	SD₁₆	—	2	16	no	—
480	SD₁₆	—	4	8	no	—
481	D₈	—	2	8	no	—
482	C₂×C₈	—	2	2	no	—
483	C₂×C₈	—	2	2	no	—
484	C₂×C₈	—	2	2	no	—
485	C₂×C₈	—	2	8	no	—
486	C₂×C₈	—	2	8	no	—
487	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
488	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
489	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
490	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
491	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
492	C₄⋊C₄	—	2	8	no	—
493	C₄⋊C₄	—	2	8	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
496	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	2	8	no	—
499	C₂²⋊C₄	—	2	8	no	—
500	C₂²⋊C₄	—	2	8	no	—
501	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
502	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
503	C₄²	—	2	8	no	—
504	C₄²	C₂×D₄×C₃₁	1	2	yes	—
505	C₄²	C₂⁴×C₃₁	1	2	yes	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	4	4	no	—
513	C₂³	—	4	4	no	—
514	C₂³	—	4	4	no	—
515	C₂³	C₃₁×#135	1	4	yes	complemented
516	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
517	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
518	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
519	Q₈	—	2	8	no	—
520	Q₈	—	2	4	no	—
521	Q₈	—	2	4	no	—
522	Q₈	—	2	4	no	—
523	Q₈	—	4	16	no	—
524	Q₈	—	4	4	no	—
525	Q₈	C₂²×D₄×C₃₁	1	8	yes	—
526	D₄	—	2	8	no	—
527	D₄	—	2	4	no	—
528	D₄	—	2	4	no	—
529	D₄	—	4	16	no	—
530	D₄	—	4	16	no	—
531	D₄	—	4	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	8	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	4	16	no	—
547	C₂×C₄	—	4	16	no	—
548	C₂×C₄	—	4	4	no	—
549	C₂×C₄	—	4	4	no	—
550	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
551	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
554	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
555	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
556	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
557	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
558	C₈	—	2	2	no	—
559	C₈	—	2	2	no	—
560	C₈	—	2	2	no	—
561	C₈	C₂²×D₄×C₃₁	1	4	yes	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	8	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	2	no	—
566	C₂²	—	2	2	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	8	8	no	—
573	C₂²	D₄²×C₃₁	1	2	yes	—
574	C₂²	C₂×C₃₁×#135	1	2	yes	—
575	C₂²	C₂×C₃₁×#135	1	4	yes	complemented
576	C₂²	C₃₁×#578	1	2	yes	—
577	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	2	8	no	—
581	C₄	—	2	2	no	—
582	C₄	—	2	2	no	—
583	C₄	—	4	4	no	—
584	C₄	—	4	4	no	—
585	C₄	D₄²×C₃₁	1	2	yes	—
586	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
587	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
588	C₂	—	2	4	no	—
589	C₂	—	4	2	no	—
590	C₂	—	8	4	no	—
591	C₂	C₃₁×#3285	1	4	yes	direct product factor
592	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic
593	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
594	C₂	C₂×C₃₁×#578	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×Q₁₆×C₃₁	C₂	semidirect
3. C₂×C₃₁×#548	C₂	semidirect
4. C₃₁×#3285	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×#464	C₂	semidirect
9. C₂×C₃₁×#452	C₂	semidirect
10. C₂×C₃₁×#448	C₂	semidirect
11. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
12. C₂×C₈×D₄×C₃₁	C₂	semidirect
16. C₂×C₃₁×#135	C₂²	semidirect
21. C₂²×Q₁₆×C₃₁	C₂²	semidirect
28. C₃₁×#548	C₂²	semidirect
38. C₃₁×#495	C₂²	semidirect
39. C₃₁×#467	C₂²	semidirect
40. C₃₁×#466	C₂²	semidirect
41. C₃₁×#464	C₂²	semidirect
42. C₃₁×#452	C₂²	semidirect
43. C₃₁×#448	C₂²	semidirect
44. C₄×Q₁₆×C₃₁	C₂²	semidirect
45. C₈×D₄×C₃₁	C₂²	semidirect
77. C₂×Q₁₆×C₃₁	D₄	semidirect
180. Q₁₆×C₃₁	C₂×D₄	semidirect
208. C₂×#3285	C₃₁	direct (N × H)
218. C₂³×C₃₁	#135	semidirect
265. C₂²×#135	C₂×C₃₁	semidirect
266. C₂³×Q₁₆	C₂×C₃₁	semidirect
267. C₂×#548	C₂×C₃₁	semidirect
268. #3285	C₂×C₃₁	direct (N × H)
269. C₂×#495	C₂×C₃₁	semidirect
270. C₂×#467	C₂×C₃₁	semidirect
271. C₂×#466	C₂×C₃₁	semidirect
272. C₂×#464	C₂×C₃₁	semidirect
273. C₂×#452	C₂×C₃₁	semidirect
274. C₂×#448	C₂×C₃₁	semidirect
275. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
276. C₂×C₈×D₄	C₂×C₃₁	semidirect
290. C₂²×C₃₁	C₂×#135	semidirect
306. C₂×#135	C₂²×C₃₁	semidirect
311. C₂²×Q₁₆	C₂²×C₃₁	semidirect
318. #548	C₂²×C₃₁	semidirect
328. #495	C₂²×C₃₁	semidirect
329. #467	C₂²×C₃₁	semidirect
330. #466	C₂²×C₃₁	semidirect
331. #464	C₂²×C₃₁	semidirect
332. #452	C₂²×C₃₁	semidirect
333. #448	C₂²×C₃₁	semidirect
334. C₄×Q₁₆	C₂²×C₃₁	semidirect
335. C₈×D₄	C₂²×C₃₁	semidirect
347. C₂×C₃₁	#3285	direct (N × H)
374. C₂×Q₁₆	D₄×C₃₁	semidirect
426. C₃₁	C₂×#3285	direct (N × H)
478. Q₁₆	C₂×D₄×C₃₁	semidirect
515. C₂³	C₃₁×#135	semidirect
575. C₂²	C₂×C₃₁×#135	semidirect
591. C₂	C₃₁×#3285	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×Q₁₆×C₃₁	2
C₂×C₃₁×#548	2
C₃₁×#3285	2
C₂×C₃₁×#495	2
C₂×C₃₁×#467	2
C₂×C₃₁×#466	2
C₂×C₃₁×#464	2
C₂×C₃₁×#452	2
C₂×C₃₁×#448	2
C₂×C₄×Q₁₆×C₃₁	2
C₂×C₈×D₄×C₃₁	2
C₂×#3285	31

Subgroups — C2×C31×#3285 #21517804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3285 #21517804