Subgroups — C₂×C₃₁×#3191 #21517492

Order 7936.

2394 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#525	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3191	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₁×#485	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#482	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#465	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#452	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#449	C₂	1	2	yes	maximal, complemented
10	C₂²×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₃₁	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
14	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
15	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#525	—	2	8	no	—
19	C₃₁×#525	C₂²	1	16	yes	complemented
20	C₂×C₃₁×#123	—	2	2	no	—
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#121	C₂²	1	2	yes	—
24	C₂×C₃₁×#121	C₂²	1	2	yes	—
25	C₂×C₃₁×#120	—	2	2	no	—
26	C₂×C₄×D₄×C₃₁	—	2	2	no	—
27	C₂×C₄×D₄×C₃₁	—	2	2	no	—
28	C₂×C₃₁×#117	C₂²	1	2	yes	—
29	C₃₁×#485	C₂²	1	16	yes	complemented
30	C₃₁×#482	C₂²	1	16	yes	complemented
31	C₃₁×#467	C₂²	1	8	yes	complemented
32	C₃₁×#466	C₂²	1	8	yes	complemented
33	C₃₁×#465	C₂²	1	16	yes	complemented
34	C₃₁×#452	C₂²	1	16	yes	complemented
35	C₃₁×#449	C₂²	1	16	yes	complemented
36	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
39	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
40	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
41	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
42	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
43	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
44	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
45	C₂×C₃₁×C₂²⋊C₈	C₂²	1	8	yes	complemented
46	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
47	C₂×#41×C₃₁	—	2	4	no	—
48	C₂×#41×C₃₁	—	4	4	no	—
49	C₂×#41×C₃₁	—	4	4	no	—
50	C₂×#41×C₃₁	C₂³	1	8	yes	—
51	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×D₄×C₃₁	—	2	2	no	—
53	C₂²×D₄×C₃₁	—	2	2	no	—
54	C₂²×D₄×C₃₁	—	2	2	no	—
55	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₃₁	—	2	2	no	—
57	C₂³×C₄×C₃₁	—	4	2	no	—
58	C₂³×C₄×C₃₁	—	4	2	no	—
59	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	D₄	1	2	yes	—
61	C₂³×C₄×C₃₁	D₄	1	4	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
65	C₂×Q₁₆×C₃₁	—	4	2	no	—
66	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	complemented
67	C₂×SD₁₆×C₃₁	—	4	4	no	—
68	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	complemented
69	C₂×D₈×C₃₁	—	4	2	no	—
70	C₂×D₈×C₃₁	C₂³	1	4	yes	complemented
71	C₂²×C₈×C₃₁	—	4	2	no	—
72	C₂²×C₈×C₃₁	—	4	2	no	—
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
77	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
78	C₃₁×#123	—	2	16	no	—
79	C₃₁×C₂²⋊Q₈	—	2	4	no	—
80	C₃₁×C₂²⋊Q₈	—	2	4	no	—
81	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
82	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
83	C₃₁×#121	—	2	4	no	—
84	C₃₁×#121	—	2	4	no	—
85	C₃₁×#121	C₂³	1	8	yes	—
86	C₃₁×#121	C₂³	1	8	yes	—
87	C₃₁×#120	—	2	16	no	—
88	C₄×D₄×C₃₁	—	2	16	no	—
89	C₄×D₄×C₃₁	—	2	16	no	—
90	C₃₁×#117	—	2	4	no	—
91	C₃₁×#117	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
100	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
101	C₂×C₄²×C₃₁	—	2	2	no	—
102	C₃₁×#107	C₂³	1	4	yes	complemented
103	C₃₁×#107	C₂³	1	4	yes	complemented
104	C₃₁×#106	C₂³	1	8	yes	complemented
105	C₃₁×#103	C₂³	1	8	yes	complemented
106	C₃₁×#103	C₂³	1	8	yes	complemented
107	C₃₁×#102	C₂³	1	8	yes	complemented
108	C₃₁×#102	C₂³	1	8	yes	complemented
109	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
110	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
111	C₂⁴×C₃₁	—	2	4	no	—
112	C₂⁴×C₃₁	—	2	2	no	—
113	C₂⁴×C₃₁	—	4	2	no	—
114	C₂⁴×C₃₁	—	4	2	no	—
115	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
116	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
118	#41×C₃₁	—	4	8	no	—
119	#41×C₃₁	—	4	8	no	—
120	#41×C₃₁	—	4	16	no	—
121	C₂×Q₈×C₃₁	—	2	2	no	—
122	C₂×Q₈×C₃₁	—	2	4	no	—
123	C₂×Q₈×C₃₁	—	2	4	no	—
124	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
125	C₂×D₄×C₃₁	—	2	8	no	—
126	C₂×D₄×C₃₁	—	2	8	no	—
127	C₂×D₄×C₃₁	—	2	8	no	—
128	C₂×D₄×C₃₁	—	2	2	no	—
129	C₂×D₄×C₃₁	—	2	4	no	—
130	C₂×D₄×C₃₁	—	2	4	no	—
131	C₂×D₄×C₃₁	—	4	8	no	—
132	C₂×D₄×C₃₁	—	4	8	no	—
133	C₂×D₄×C₃₁	—	4	8	no	—
134	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	2	2	no	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	—	2	8	no	—
143	C₂²×C₄×C₃₁	—	4	8	no	—
144	C₂²×C₄×C₃₁	—	4	4	no	—
145	C₂²×C₄×C₃₁	—	4	4	no	—
146	C₂²×C₄×C₃₁	—	4	2	no	—
147	C₂²×C₄×C₃₁	—	4	4	no	—
148	C₂²×C₄×C₃₁	—	4	8	no	—
149	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₃₁	—	4	4	no	—
159	SD₁₆×C₃₁	—	4	8	no	—
160	D₈×C₃₁	—	4	4	no	—
161	C₂×C₈×C₃₁	—	2	2	no	—
162	C₂×C₈×C₃₁	—	2	4	no	—
163	C₂×C₈×C₃₁	—	2	2	no	—
164	C₂×C₈×C₃₁	—	4	4	no	—
165	C₂×C₈×C₃₁	—	4	4	no	—
166	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
167	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
168	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
169	C₄⋊C₄×C₃₁	—	2	8	no	—
170	C₄⋊C₄×C₃₁	—	2	8	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
174	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
175	C₂²⋊C₄×C₃₁	—	2	8	no	—
176	C₂²⋊C₄×C₃₁	—	2	8	no	—
177	C₂²⋊C₄×C₃₁	—	2	8	no	—
178	C₂²⋊C₄×C₃₁	—	2	8	no	—
179	C₂²⋊C₄×C₃₁	—	2	8	no	—
180	C₄²×C₃₁	—	2	8	no	—
181	C₂×#3191	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂³×C₃₁	—	2	2	no	—
183	C₂³×C₃₁	—	2	2	no	—
184	C₂³×C₃₁	—	2	4	no	—
185	C₂³×C₃₁	—	2	8	no	—
186	C₂³×C₃₁	—	2	4	no	—
187	C₂³×C₃₁	—	2	4	no	—
188	C₂³×C₃₁	—	2	4	no	—
189	C₂³×C₃₁	—	4	4	no	—
190	C₂³×C₃₁	—	4	4	no	—
191	C₂³×C₃₁	—	4	16	no	—
192	C₂³×C₃₁	—	4	4	no	—
193	C₂³×C₃₁	—	4	4	no	—
194	C₂³×C₃₁	—	4	4	no	—
195	C₂³×C₃₁	#135	1	4	yes	—
196	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
197	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
198	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
199	Q₈×C₃₁	—	2	4	no	—
200	Q₈×C₃₁	—	2	4	no	—
201	Q₈×C₃₁	—	4	4	no	—
202	D₄×C₃₁	—	2	4	no	—
203	D₄×C₃₁	—	2	4	no	—
204	D₄×C₃₁	—	4	16	no	—
205	D₄×C₃₁	—	4	16	no	—
206	D₄×C₃₁	—	4	16	no	—
207	D₄×C₃₁	—	4	4	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	8	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	4	4	no	—
220	C₂×C₄×C₃₁	—	4	8	no	—
221	C₂×C₄×C₃₁	—	4	16	no	—
222	C₂×C₄×C₃₁	—	4	4	no	—
223	C₂×C₄×C₃₁	—	4	4	no	—
224	C₂×C₄×C₃₁	—	4	4	no	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
229	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
230	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
231	C₈×C₃₁	—	2	2	no	—
232	C₈×C₃₁	—	2	4	no	—
233	C₈×C₃₁	—	2	2	no	—
234	C₂×#525	C₂×C₃₁	1	2	yes	complemented
235	#3191	C₂×C₃₁	1	16	yes	direct product factor
236	C₂×#485	C₂×C₃₁	1	2	yes	complemented
237	C₂×#482	C₂×C₃₁	1	2	yes	complemented
238	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂×#466	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂×#465	C₂×C₃₁	1	2	yes	complemented
241	C₂×#452	C₂×C₃₁	1	2	yes	complemented
242	C₂×#449	C₂×C₃₁	1	2	yes	complemented
243	C₂²×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	2	8	no	—
246	C₂²×C₃₁	—	2	2	no	—
247	C₂²×C₃₁	—	2	4	no	—
248	C₂²×C₃₁	—	2	4	no	—
249	C₂²×C₃₁	—	2	4	no	—
250	C₂²×C₃₁	—	4	16	no	—
251	C₂²×C₃₁	—	4	8	no	—
252	C₂²×C₃₁	—	4	4	no	—
253	C₂²×C₃₁	—	4	4	no	—
254	C₂²×C₃₁	—	4	4	no	—
255	C₂²×C₃₁	—	8	8	no	—
256	C₂²×C₃₁	#534	1	2	yes	—
257	C₂²×C₃₁	C₂×#135	1	4	yes	—
258	C₂²×C₃₁	C₂×#135	1	2	yes	—
259	C₂²×C₃₁	#575	1	2	yes	—
260	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
261	C₄×C₃₁	—	2	4	no	—
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	4	4	no	—
264	C₄×C₃₁	—	4	4	no	—
265	C₄×C₃₁	—	4	4	no	—
266	C₄×C₃₁	#534	1	2	yes	—
267	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
268	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
269	C₂²×#41	C₂²×C₃₁	1	2	yes	—
270	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
271	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
272	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
273	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
274	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
276	#525	—	2	8	no	—
277	#525	C₂²×C₃₁	1	16	yes	complemented
278	C₂×#123	—	2	2	no	—
279	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
280	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
281	C₂×#121	C₂²×C₃₁	1	2	yes	—
282	C₂×#121	C₂²×C₃₁	1	2	yes	—
283	C₂×#120	—	2	2	no	—
284	C₂×C₄×D₄	—	2	2	no	—
285	C₂×C₄×D₄	—	2	2	no	—
286	C₂×#117	C₂²×C₃₁	1	2	yes	—
287	#485	C₂²×C₃₁	1	16	yes	complemented
288	#482	C₂²×C₃₁	1	16	yes	complemented
289	#467	C₂²×C₃₁	1	8	yes	complemented
290	#466	C₂²×C₃₁	1	8	yes	complemented
291	#465	C₂²×C₃₁	1	16	yes	complemented
292	#452	C₂²×C₃₁	1	16	yes	complemented
293	#449	C₂²×C₃₁	1	16	yes	complemented
294	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
295	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
296	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
297	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
298	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
299	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
300	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
301	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
302	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
303	C₂×C₂²⋊C₈	C₂²×C₃₁	1	8	yes	complemented
304	C₂×C₃₁	—	2	4	no	—
305	C₂×C₃₁	—	4	4	no	—
306	C₂×C₃₁	—	8	4	no	—
307	C₂×C₃₁	#3191	1	4	yes	direct product factor
308	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
309	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
310	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
311	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
312	C₂×#41	—	2	4	no	—
313	C₂×#41	—	4	4	no	—
314	C₂×#41	—	4	4	no	—
315	C₂×#41	C₂³×C₃₁	1	8	yes	—
316	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
317	C₂²×D₄	—	2	2	no	—
318	C₂²×D₄	—	2	2	no	—
319	C₂²×D₄	—	2	2	no	—
320	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
321	C₂³×C₄	—	2	2	no	—
322	C₂³×C₄	—	4	2	no	—
323	C₂³×C₄	—	4	2	no	—
324	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
325	C₂³×C₄	D₄×C₃₁	1	2	yes	—
326	C₂³×C₄	D₄×C₃₁	1	4	yes	—
327	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
328	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
330	C₂×Q₁₆	—	4	2	no	—
331	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	complemented
332	C₂×SD₁₆	—	4	4	no	—
333	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	complemented
334	C₂×D₈	—	4	2	no	—
335	C₂×D₈	C₂³×C₃₁	1	4	yes	complemented
336	C₂²×C₈	—	4	2	no	—
337	C₂²×C₈	—	4	2	no	—
338	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
339	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
340	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
341	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
342	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
343	#123	—	2	16	no	—
344	C₂²⋊Q₈	—	2	4	no	—
345	C₂²⋊Q₈	—	2	4	no	—
346	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
347	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
348	#121	—	2	4	no	—
349	#121	—	2	4	no	—
350	#121	C₂³×C₃₁	1	8	yes	—
351	#121	C₂³×C₃₁	1	8	yes	—
352	#120	—	2	16	no	—
353	C₄×D₄	—	2	16	no	—
354	C₄×D₄	—	2	16	no	—
355	#117	—	2	4	no	—
356	#117	C₂³×C₃₁	1	8	yes	—
357	C₂×C₄⋊C₄	—	2	2	no	—
358	C₂×C₄⋊C₄	—	2	2	no	—
359	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
360	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₂²⋊C₄	—	2	2	no	—
364	C₂×C₂²⋊C₄	—	2	2	no	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₄²	—	2	2	no	—
367	#107	C₂³×C₃₁	1	4	yes	complemented
368	#107	C₂³×C₃₁	1	4	yes	complemented
369	#106	C₂³×C₃₁	1	8	yes	complemented
370	#103	C₂³×C₃₁	1	8	yes	complemented
371	#103	C₂³×C₃₁	1	8	yes	complemented
372	#102	C₂³×C₃₁	1	8	yes	complemented
373	#102	C₂³×C₃₁	1	8	yes	complemented
374	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
375	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
376	C₃₁	C₂×#3191	1	1	yes	characteristic, direct product factor, Sylow
377	C₂⁴	—	2	4	no	—
378	C₂⁴	—	2	2	no	—
379	C₂⁴	—	4	2	no	—
380	C₂⁴	—	4	2	no	—
381	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
382	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
384	#41	—	4	8	no	—
385	#41	—	4	8	no	—
386	#41	—	4	16	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	—	2	4	no	—
390	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	—	2	8	no	—
393	C₂×D₄	—	2	8	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	4	no	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	—	4	8	no	—
398	C₂×D₄	—	4	8	no	—
399	C₂×D₄	—	4	8	no	—
400	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	4	8	no	—
410	C₂²×C₄	—	4	4	no	—
411	C₂²×C₄	—	4	4	no	—
412	C₂²×C₄	—	4	2	no	—
413	C₂²×C₄	—	4	4	no	—
414	C₂²×C₄	—	4	8	no	—
415	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
416	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
418	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
419	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
420	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
421	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
422	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
424	Q₁₆	—	4	4	no	—
425	SD₁₆	—	4	8	no	—
426	D₈	—	4	4	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	4	no	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	4	4	no	—
431	C₂×C₈	—	4	4	no	—
432	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
433	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
434	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
435	C₄⋊C₄	—	2	8	no	—
436	C₄⋊C₄	—	2	8	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
440	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
441	C₂²⋊C₄	—	2	8	no	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₂²⋊C₄	—	2	8	no	—
444	C₂²⋊C₄	—	2	8	no	—
445	C₂²⋊C₄	—	2	8	no	—
446	C₄²	—	2	8	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	2	8	no	—
451	C₂³	—	2	4	no	—
452	C₂³	—	2	4	no	—
453	C₂³	—	2	4	no	—
454	C₂³	—	4	4	no	—
455	C₂³	—	4	4	no	—
456	C₂³	—	4	16	no	—
457	C₂³	—	4	4	no	—
458	C₂³	—	4	4	no	—
459	C₂³	—	4	4	no	—
460	C₂³	C₃₁×#135	1	4	yes	—
461	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
462	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
463	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
464	Q₈	—	2	4	no	—
465	Q₈	—	2	4	no	—
466	Q₈	—	4	4	no	—
467	D₄	—	2	4	no	—
468	D₄	—	2	4	no	—
469	D₄	—	4	16	no	—
470	D₄	—	4	16	no	—
471	D₄	—	4	16	no	—
472	D₄	—	4	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	—	4	8	no	—
486	C₂×C₄	—	4	16	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	4	4	no	—
490	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
491	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
492	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
494	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
495	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
496	C₈	—	2	2	no	—
497	C₈	—	2	4	no	—
498	C₈	—	2	2	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	4	16	no	—
506	C₂²	—	4	8	no	—
507	C₂²	—	4	4	no	—
508	C₂²	—	4	4	no	—
509	C₂²	—	4	4	no	—
510	C₂²	—	8	8	no	—
511	C₂²	C₃₁×#534	1	2	yes	—
512	C₂²	C₂×C₃₁×#135	1	4	yes	—
513	C₂²	C₂×C₃₁×#135	1	2	yes	—
514	C₂²	C₃₁×#575	1	2	yes	—
515	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
516	C₄	—	2	4	no	—
517	C₄	—	2	2	no	—
518	C₄	—	4	4	no	—
519	C₄	—	4	4	no	—
520	C₄	—	4	4	no	—
521	C₄	C₃₁×#534	1	2	yes	—
522	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
523	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
524	C₂	—	2	4	no	—
525	C₂	—	4	4	no	—
526	C₂	—	8	4	no	—
527	C₂	C₃₁×#3191	1	4	yes	direct product factor
528	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
529	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
530	C₂	C₂×C₃₁×#575	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 ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#525	C₂	semidirect
2. C₃₁×#3191	C₂	direct (N × H)
3. C₂×C₃₁×#485	C₂	semidirect
4. C₂×C₃₁×#482	C₂	semidirect
5. C₂×C₃₁×#467	C₂	semidirect
6. C₂×C₃₁×#466	C₂	semidirect
7. C₂×C₃₁×#465	C₂	semidirect
8. C₂×C₃₁×#452	C₂	semidirect
9. C₂×C₃₁×#449	C₂	semidirect
10. C₂²×C₃₁×C₂²⋊C₈	C₂	semidirect
13. C₂²×Q₁₆×C₃₁	C₂²	semidirect
14. C₂²×SD₁₆×C₃₁	C₂²	semidirect
15. C₂²×D₈×C₃₁	C₂²	semidirect
19. C₃₁×#525	C₂²	semidirect
29. C₃₁×#485	C₂²	semidirect
30. C₃₁×#482	C₂²	semidirect
31. C₃₁×#467	C₂²	semidirect
32. C₃₁×#466	C₂²	semidirect
33. C₃₁×#465	C₂²	semidirect
34. C₃₁×#452	C₂²	semidirect
35. C₃₁×#449	C₂²	semidirect
36. C₂×C₃₁×#107	C₂²	semidirect
37. C₂×C₃₁×#107	C₂²	semidirect
38. C₂×C₃₁×#106	C₂²	semidirect
39. C₂×C₃₁×#103	C₂²	semidirect
40. C₂×C₃₁×#103	C₂²	semidirect
41. C₂×C₃₁×#102	C₂²	semidirect
42. C₂×C₃₁×#102	C₂²	semidirect
45. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
66. C₂×Q₁₆×C₃₁	C₂³	semidirect
68. C₂×SD₁₆×C₃₁	C₂³	semidirect
70. C₂×D₈×C₃₁	C₂³	semidirect
102. C₃₁×#107	C₂³	semidirect
103. C₃₁×#107	C₂³	semidirect
104. C₃₁×#106	C₂³	semidirect
105. C₃₁×#103	C₂³	semidirect
106. C₃₁×#103	C₂³	semidirect
107. C₃₁×#102	C₂³	semidirect
108. C₃₁×#102	C₂³	semidirect
181. C₂×#3191	C₃₁	direct (N × H)
234. C₂×#525	C₂×C₃₁	semidirect
235. #3191	C₂×C₃₁	direct (N × H)
236. C₂×#485	C₂×C₃₁	semidirect
237. C₂×#482	C₂×C₃₁	semidirect
238. C₂×#467	C₂×C₃₁	semidirect
239. C₂×#466	C₂×C₃₁	semidirect
240. C₂×#465	C₂×C₃₁	semidirect
241. C₂×#452	C₂×C₃₁	semidirect
242. C₂×#449	C₂×C₃₁	semidirect
243. C₂²×C₂²⋊C₈	C₂×C₃₁	semidirect
271. C₂²×Q₁₆	C₂²×C₃₁	semidirect
272. C₂²×SD₁₆	C₂²×C₃₁	semidirect
273. C₂²×D₈	C₂²×C₃₁	semidirect
277. #525	C₂²×C₃₁	semidirect
287. #485	C₂²×C₃₁	semidirect
288. #482	C₂²×C₃₁	semidirect
289. #467	C₂²×C₃₁	semidirect
290. #466	C₂²×C₃₁	semidirect
291. #465	C₂²×C₃₁	semidirect
292. #452	C₂²×C₃₁	semidirect
293. #449	C₂²×C₃₁	semidirect
294. C₂×#107	C₂²×C₃₁	semidirect
295. C₂×#107	C₂²×C₃₁	semidirect
296. C₂×#106	C₂²×C₃₁	semidirect
297. C₂×#103	C₂²×C₃₁	semidirect
298. C₂×#103	C₂²×C₃₁	semidirect
299. C₂×#102	C₂²×C₃₁	semidirect
300. C₂×#102	C₂²×C₃₁	semidirect
303. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
307. C₂×C₃₁	#3191	direct (N × H)
331. C₂×Q₁₆	C₂³×C₃₁	semidirect
333. C₂×SD₁₆	C₂³×C₃₁	semidirect
335. C₂×D₈	C₂³×C₃₁	semidirect
367. #107	C₂³×C₃₁	semidirect
368. #107	C₂³×C₃₁	semidirect
369. #106	C₂³×C₃₁	semidirect
370. #103	C₂³×C₃₁	semidirect
371. #103	C₂³×C₃₁	semidirect
372. #102	C₂³×C₃₁	semidirect
373. #102	C₂³×C₃₁	semidirect
376. C₃₁	C₂×#3191	direct (N × H)
527. C₂	C₃₁×#3191	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#525	2
C₃₁×#3191	2
C₂×C₃₁×#485	2
C₂×C₃₁×#482	2
C₂×C₃₁×#467	2
C₂×C₃₁×#466	2
C₂×C₃₁×#465	2
C₂×C₃₁×#452	2
C₂×C₃₁×#449	2
C₂²×C₃₁×C₂²⋊C₈	2
C₂×#3191	31

Subgroups — C2×C31×#3191 #21517492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3191 #21517492