Subgroups — C₃₁×#36710 #21493469

Order 7936.

2910 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#537	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#123	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2847	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2696	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2673	C₂	1	8	yes	maximal, complemented
6	C₃₁×#2617	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2591	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2433	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2404	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2404	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2313	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#537	—	2	8	no	—
16	C₂×C₃₁×#127	C₂²	1	4	yes	—
17	C₂×C₃₁×#123	—	2	8	no	—
18	C₂×C₃₁×#123	C₂²	1	8	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#121	C₂²	1	2	yes	—
21	C₂×C₃₁×#121	C₂²	1	2	yes	—
22	C₂×C₃₁×#121	C₂²	1	4	yes	complemented
23	C₂×C₃₁×#121	C₂²	1	4	yes	complemented
24	C₂×C₃₁×#120	C₂²	1	4	yes	—
25	C₂×C₃₁×#120	C₂²	1	4	yes	—
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#400	C₂²	1	4	yes	complemented
32	C₃₁×#397	C₂²	1	4	yes	complemented
33	C₃₁×#394	C₂²	1	4	yes	complemented
34	C₃₁×#394	C₂²	1	8	yes	complemented
35	C₃₁×#393	C₂²	1	4	yes	complemented
36	C₃₁×#393	C₂²	1	4	yes	complemented
37	C₃₁×#392	C₂²	1	8	yes	—
38	C₃₁×#392	C₂²	1	4	yes	complemented
39	C₃₁×#392	C₂²	1	4	yes	complemented
40	C₃₁×#392	C₂²	1	4	yes	complemented
41	C₃₁×#390	C₂²	1	4	yes	—
42	C₃₁×#388	C₂²	1	8	yes	complemented
43	C₃₁×#387	C₂²	1	8	yes	complemented
44	C₃₁×#386	C₂²	1	4	yes	—
45	C₃₁×#386	C₂²	1	4	yes	—
46	C₃₁×#386	C₂²	1	8	yes	complemented
47	C₃₁×#386	C₂²	1	8	yes	complemented
48	C₃₁×#386	C₂²	1	8	yes	complemented
49	C₃₁×#381	C₂²	1	2	yes	—
50	C₃₁×#380	C₂²	1	2	yes	—
51	C₃₁×#380	C₂²	1	4	yes	complemented
52	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
53	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
54	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×#41×C₃₁	—	2	8	no	—
56	C₂×#41×C₃₁	—	4	8	no	—
57	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
58	C₂²×D₄×C₃₁	—	2	4	no	—
59	C₂²×D₄×C₃₁	—	4	4	no	—
60	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
61	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
62	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
63	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
64	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
67	C₂³×C₄×C₃₁	—	2	4	no	—
68	C₂³×C₄×C₃₁	—	4	4	no	—
69	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#127	—	4	8	no	—
77	C₃₁×#123	—	2	32	no	—
78	C₃₁×C₂²⋊Q₈	—	2	16	no	—
79	C₃₁×#121	—	2	8	no	—
80	C₃₁×#121	—	2	8	no	—
81	C₃₁×#121	—	2	16	no	—
82	C₃₁×#121	—	4	8	no	—
83	C₃₁×#120	—	4	8	no	—
84	C₃₁×#120	—	4	8	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	complemented
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
94	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
105	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
106	C₃₁×#95	C₂³	1	8	yes	—
107	C₃₁×#95	C₂³	1	4	yes	—
108	C₃₁×#95	C₂³	1	4	yes	—
109	C₃₁×#95	C₂³	1	8	yes	—
110	C₃₁×#95	C₂³	1	4	yes	—
111	C₃₁×#95	C₂³	1	4	yes	—
112	C₂⁴×C₃₁	—	2	2	no	—
113	C₂⁴×C₃₁	—	2	4	no	—
114	C₂⁴×C₃₁	—	2	2	no	—
115	C₂⁴×C₃₁	—	2	4	no	—
116	C₂⁴×C₃₁	—	2	4	no	—
117	C₂⁴×C₃₁	—	4	2	no	—
118	C₂⁴×C₃₁	—	4	2	no	—
119	C₂⁴×C₃₁	—	4	2	no	—
120	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
121	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
122	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
123	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
124	#41×C₃₁	—	4	32	no	—
125	C₂×Q₈×C₃₁	—	2	4	no	—
126	C₂×Q₈×C₃₁	—	4	4	no	—
127	C₂×D₄×C₃₁	—	2	8	no	—
128	C₂×D₄×C₃₁	—	2	8	no	—
129	C₂×D₄×C₃₁	—	2	8	no	—
130	C₂×D₄×C₃₁	—	2	8	no	—
131	C₂×D₄×C₃₁	—	2	16	no	—
132	C₂×D₄×C₃₁	—	2	4	no	—
133	C₂×D₄×C₃₁	—	2	8	no	—
134	C₂×D₄×C₃₁	—	2	4	no	—
135	C₂×D₄×C₃₁	—	2	4	no	—
136	C₂×D₄×C₃₁	—	4	4	no	—
137	C₂×D₄×C₃₁	—	4	4	no	—
138	C₂×D₄×C₃₁	—	4	8	no	—
139	C₂×D₄×C₃₁	—	4	8	no	—
140	C₂×D₄×C₃₁	—	4	4	no	—
141	C₂×D₄×C₃₁	—	4	4	no	—
142	C₂×D₄×C₃₁	—	4	8	no	—
143	C₂×D₄×C₃₁	—	4	8	no	—
144	C₂×D₄×C₃₁	—	4	8	no	—
145	C₂×D₄×C₃₁	—	4	16	no	—
146	C₂²×C₄×C₃₁	—	2	4	no	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	8	no	—
150	C₂²×C₄×C₃₁	—	2	8	no	—
151	C₂²×C₄×C₃₁	—	2	8	no	—
152	C₂²×C₄×C₃₁	—	4	2	no	—
153	C₂²×C₄×C₃₁	—	4	4	no	—
154	C₂²×C₄×C₃₁	—	4	2	no	—
155	C₂²×C₄×C₃₁	—	4	2	no	—
156	C₂²×C₄×C₃₁	—	4	2	no	—
157	C₂²×C₄×C₃₁	—	4	4	no	—
158	C₂²×C₄×C₃₁	—	4	4	no	—
159	C₂²×C₄×C₃₁	—	4	16	no	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
170	C₄⋊C₄×C₃₁	—	2	8	no	—
171	C₄⋊C₄×C₃₁	—	2	4	no	—
172	C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	4	4	no	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	16	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	C₂²⋊C₄×C₃₁	—	2	8	no	—
181	C₂²⋊C₄×C₃₁	—	4	8	no	—
182	C₂²⋊C₄×C₃₁	—	4	8	no	—
183	C₂²⋊C₄×C₃₁	—	4	8	no	—
184	C₂²⋊C₄×C₃₁	—	4	8	no	—
185	C₄²×C₃₁	—	4	4	no	—
186	#36710	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂³×C₃₁	—	2	2	no	—
188	C₂³×C₃₁	—	2	2	no	—
189	C₂³×C₃₁	—	2	8	no	—
190	C₂³×C₃₁	—	2	2	no	—
191	C₂³×C₃₁	—	2	4	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	4	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	—	2	4	no	—
196	C₂³×C₃₁	—	2	4	no	—
197	C₂³×C₃₁	—	2	8	no	—
198	C₂³×C₃₁	—	2	4	no	—
199	C₂³×C₃₁	—	2	4	no	—
200	C₂³×C₃₁	—	2	4	no	—
201	C₂³×C₃₁	—	2	8	no	—
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	—	4	16	no	—
204	C₂³×C₃₁	—	4	4	no	—
205	C₂³×C₃₁	—	4	4	no	—
206	C₂³×C₃₁	—	4	4	no	—
207	C₂³×C₃₁	—	4	4	no	—
208	C₂³×C₃₁	—	4	4	no	—
209	C₂³×C₃₁	—	8	8	no	—
210	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
211	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
212	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
213	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
214	Q₈×C₃₁	—	4	8	no	—
215	D₄×C₃₁	—	4	16	no	—
216	D₄×C₃₁	—	4	8	no	—
217	D₄×C₃₁	—	4	16	no	—
218	D₄×C₃₁	—	8	4	no	—
219	D₄×C₃₁	—	8	4	no	—
220	D₄×C₃₁	—	8	16	no	—
221	D₄×C₃₁	—	8	16	no	—
222	C₂×C₄×C₃₁	—	2	8	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	8	no	—
230	C₂×C₄×C₃₁	—	2	8	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	2	4	no	—
233	C₂×C₄×C₃₁	—	2	4	no	—
234	C₂×C₄×C₃₁	—	2	4	no	—
235	C₂×C₄×C₃₁	—	2	8	no	—
236	C₂×C₄×C₃₁	—	4	4	no	—
237	C₂×C₄×C₃₁	—	4	4	no	—
238	C₂×C₄×C₃₁	—	4	16	no	—
239	C₂×C₄×C₃₁	—	4	16	no	—
240	C₂×C₄×C₃₁	—	4	4	no	—
241	C₂×C₄×C₃₁	—	4	4	no	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
243	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
244	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
245	C₂×#537	C₂×C₃₁	1	1	yes	characteristic, complemented
246	C₂²×#123	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#2847	C₂×C₃₁	1	2	yes	complemented
248	#2696	C₂×C₃₁	1	8	yes	complemented
249	#2673	C₂×C₃₁	1	8	yes	complemented
250	#2617	C₂×C₃₁	1	2	yes	complemented
251	#2591	C₂×C₃₁	1	2	yes	complemented
252	#2433	C₂×C₃₁	1	2	yes	complemented
253	#2404	C₂×C₃₁	1	2	yes	complemented
254	#2404	C₂×C₃₁	1	2	yes	complemented
255	#2313	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₂²×C₃₁	—	2	2	no	—
257	C₂²×C₃₁	—	2	2	no	—
258	C₂²×C₃₁	—	2	8	no	—
259	C₂²×C₃₁	—	2	4	no	—
260	C₂²×C₃₁	—	2	8	no	—
261	C₂²×C₃₁	—	4	16	no	—
262	C₂²×C₃₁	—	4	2	no	—
263	C₂²×C₃₁	—	4	2	no	—
264	C₂²×C₃₁	—	4	4	no	—
265	C₂²×C₃₁	—	4	4	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	4	2	no	—
268	C₂²×C₃₁	—	4	4	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	—	4	2	no	—
271	C₂²×C₃₁	—	4	8	no	—
272	C₂²×C₃₁	—	4	8	no	—
273	C₂²×C₃₁	—	4	8	no	—
274	C₂²×C₃₁	—	8	8	no	—
275	C₂²×C₃₁	#535	1	4	yes	—
276	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
277	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
278	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
279	C₂²×C₃₁	#585	1	2	yes	—
280	C₄×C₃₁	—	2	4	no	—
281	C₄×C₃₁	—	2	4	no	—
282	C₄×C₃₁	—	2	4	no	—
283	C₄×C₃₁	—	4	4	no	—
284	C₄×C₃₁	—	4	2	no	—
285	C₄×C₃₁	—	4	2	no	—
286	C₄×C₃₁	—	4	8	no	—
287	C₂²×#41	C₂²×C₃₁	1	2	yes	—
288	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
289	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
290	#537	—	2	8	no	—
291	C₂×#127	C₂²×C₃₁	1	4	yes	—
292	C₂×#123	—	2	8	no	—
293	C₂×#123	C₂²×C₃₁	1	8	yes	—
294	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
295	C₂×#121	C₂²×C₃₁	1	2	yes	—
296	C₂×#121	C₂²×C₃₁	1	2	yes	—
297	C₂×#121	C₂²×C₃₁	1	4	yes	complemented
298	C₂×#121	C₂²×C₃₁	1	4	yes	complemented
299	C₂×#120	C₂²×C₃₁	1	4	yes	—
300	C₂×#120	C₂²×C₃₁	1	4	yes	—
301	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
302	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
303	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
304	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
305	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
306	#400	C₂²×C₃₁	1	4	yes	complemented
307	#397	C₂²×C₃₁	1	4	yes	complemented
308	#394	C₂²×C₃₁	1	4	yes	complemented
309	#394	C₂²×C₃₁	1	8	yes	complemented
310	#393	C₂²×C₃₁	1	4	yes	complemented
311	#393	C₂²×C₃₁	1	4	yes	complemented
312	#392	C₂²×C₃₁	1	8	yes	—
313	#392	C₂²×C₃₁	1	4	yes	complemented
314	#392	C₂²×C₃₁	1	4	yes	complemented
315	#392	C₂²×C₃₁	1	4	yes	complemented
316	#390	C₂²×C₃₁	1	4	yes	—
317	#388	C₂²×C₃₁	1	8	yes	complemented
318	#387	C₂²×C₃₁	1	8	yes	complemented
319	#386	C₂²×C₃₁	1	4	yes	—
320	#386	C₂²×C₃₁	1	4	yes	—
321	#386	C₂²×C₃₁	1	8	yes	complemented
322	#386	C₂²×C₃₁	1	8	yes	complemented
323	#386	C₂²×C₃₁	1	8	yes	complemented
324	#381	C₂²×C₃₁	1	2	yes	—
325	#380	C₂²×C₃₁	1	2	yes	—
326	#380	C₂²×C₃₁	1	4	yes	complemented
327	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
328	C₂×C₃₁	—	2	4	no	—
329	C₂×C₃₁	—	4	4	no	—
330	C₂×C₃₁	—	8	4	no	—
331	C₂×C₃₁	—	8	2	no	—
332	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
333	C₂×C₃₁	#3484	1	1	yes	characteristic
334	C₂×C₃₁	#3487	1	1	yes	characteristic
335	C₂×C₃₁	#3534	1	4	yes	—
336	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
337	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×#41	—	2	8	no	—
339	C₂×#41	—	4	8	no	—
340	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
341	C₂²×D₄	—	2	4	no	—
342	C₂²×D₄	—	4	4	no	—
343	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
344	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
345	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
346	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
347	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
348	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
349	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
350	C₂³×C₄	—	2	4	no	—
351	C₂³×C₄	—	4	4	no	—
352	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
353	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
357	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
359	#127	—	4	8	no	—
360	#123	—	2	32	no	—
361	C₂²⋊Q₈	—	2	16	no	—
362	#121	—	2	8	no	—
363	#121	—	2	8	no	—
364	#121	—	2	16	no	—
365	#121	—	4	8	no	—
366	#120	—	4	8	no	—
367	#120	—	4	8	no	—
368	C₂×C₄⋊C₄	—	2	4	no	—
369	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	complemented
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
375	C₂×C₂²⋊C₄	—	2	4	no	—
376	C₂×C₂²⋊C₄	—	2	4	no	—
377	C₂×C₂²⋊C₄	—	2	4	no	—
378	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
388	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
389	#95	C₂³×C₃₁	1	4	yes	—
390	#95	C₂³×C₃₁	1	4	yes	—
391	#95	C₂³×C₃₁	1	8	yes	—
392	#95	C₂³×C₃₁	1	4	yes	—
393	#95	C₂³×C₃₁	1	4	yes	—
394	#95	C₂³×C₃₁	1	8	yes	—
395	C₃₁	#36710	1	1	yes	characteristic, direct product factor, Sylow
396	C₂⁴	—	2	2	no	—
397	C₂⁴	—	2	4	no	—
398	C₂⁴	—	2	2	no	—
399	C₂⁴	—	2	4	no	—
400	C₂⁴	—	2	4	no	—
401	C₂⁴	—	4	2	no	—
402	C₂⁴	—	4	2	no	—
403	C₂⁴	—	4	2	no	—
404	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
405	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
406	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
407	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
408	#41	—	4	32	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	—	4	4	no	—
411	C₂×D₄	—	2	8	no	—
412	C₂×D₄	—	2	8	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	2	16	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	4	4	no	—
421	C₂×D₄	—	4	4	no	—
422	C₂×D₄	—	4	8	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	4	no	—
426	C₂×D₄	—	4	8	no	—
427	C₂×D₄	—	4	8	no	—
428	C₂×D₄	—	4	8	no	—
429	C₂×D₄	—	4	16	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	8	no	—
434	C₂²×C₄	—	2	8	no	—
435	C₂²×C₄	—	2	8	no	—
436	C₂²×C₄	—	4	2	no	—
437	C₂²×C₄	—	4	4	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	4	2	no	—
440	C₂²×C₄	—	4	2	no	—
441	C₂²×C₄	—	4	4	no	—
442	C₂²×C₄	—	4	4	no	—
443	C₂²×C₄	—	4	16	no	—
444	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
445	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
446	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
447	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
448	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
450	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
451	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
452	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	4	4	no	—
458	C₄⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
459	C₂²⋊C₄	—	2	8	no	—
460	C₂²⋊C₄	—	2	8	no	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	8	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	—	4	8	no	—
468	C₂²⋊C₄	—	4	8	no	—
469	C₄²	—	4	4	no	—
470	C₂³	—	2	2	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	8	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	8	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	8	no	—
485	C₂³	—	2	4	no	—
486	C₂³	—	4	16	no	—
487	C₂³	—	4	4	no	—
488	C₂³	—	4	4	no	—
489	C₂³	—	4	4	no	—
490	C₂³	—	4	4	no	—
491	C₂³	—	4	4	no	—
492	C₂³	—	8	8	no	—
493	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
494	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
495	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
496	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
497	Q₈	—	4	8	no	—
498	D₄	—	4	16	no	—
499	D₄	—	4	8	no	—
500	D₄	—	4	16	no	—
501	D₄	—	8	4	no	—
502	D₄	—	8	4	no	—
503	D₄	—	8	16	no	—
504	D₄	—	8	16	no	—
505	C₂×C₄	—	2	8	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	8	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	8	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	4	no	—
521	C₂×C₄	—	4	16	no	—
522	C₂×C₄	—	4	16	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
526	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
527	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
528	C₂²	—	2	2	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	2	8	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	2	8	no	—
533	C₂²	—	4	16	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	4	4	no	—
538	C₂²	—	4	2	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	4	2	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	8	no	—
544	C₂²	—	4	8	no	—
545	C₂²	—	4	8	no	—
546	C₂²	—	8	8	no	—
547	C₂²	C₃₁×#535	1	4	yes	—
548	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
549	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
550	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
551	C₂²	C₃₁×#585	1	2	yes	—
552	C₄	—	2	4	no	—
553	C₄	—	2	4	no	—
554	C₄	—	2	4	no	—
555	C₄	—	4	4	no	—
556	C₄	—	4	2	no	—
557	C₄	—	4	2	no	—
558	C₄	—	4	8	no	—
559	C₂	—	2	4	no	—
560	C₂	—	4	4	no	—
561	C₂	—	8	2	no	—
562	C₂	—	8	4	no	—
563	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
564	C₂	C₃₁×#3484	1	1	yes	characteristic
565	C₂	C₃₁×#3487	1	1	yes	characteristic
566	C₂	C₃₁×#3534	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#537	C₂	semidirect
2. C₂²×C₃₁×#123	C₂	semidirect
3. C₃₁×#2847	C₂	semidirect
4. C₃₁×#2696	C₂	semidirect
5. C₃₁×#2673	C₂	semidirect
6. C₃₁×#2617	C₂	semidirect
7. C₃₁×#2591	C₂	semidirect
8. C₃₁×#2433	C₂	semidirect
9. C₃₁×#2404	C₂	semidirect
10. C₃₁×#2404	C₂	semidirect
11. C₃₁×#2313	C₂	semidirect
19. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
22. C₂×C₃₁×#121	C₂²	semidirect
23. C₂×C₃₁×#121	C₂²	semidirect
31. C₃₁×#400	C₂²	semidirect
32. C₃₁×#397	C₂²	semidirect
33. C₃₁×#394	C₂²	semidirect
34. C₃₁×#394	C₂²	semidirect
35. C₃₁×#393	C₂²	semidirect
36. C₃₁×#393	C₂²	semidirect
38. C₃₁×#392	C₂²	semidirect
39. C₃₁×#392	C₂²	semidirect
40. C₃₁×#392	C₂²	semidirect
42. C₃₁×#388	C₂²	semidirect
43. C₃₁×#387	C₂²	semidirect
46. C₃₁×#386	C₂²	semidirect
47. C₃₁×#386	C₂²	semidirect
48. C₃₁×#386	C₂²	semidirect
51. C₃₁×#380	C₂²	semidirect
86. C₂×C₄⋊C₄×C₃₁	D₄	semidirect
186. #36710	C₃₁	direct (N × H)
245. C₂×#537	C₂×C₃₁	semidirect
246. C₂²×#123	C₂×C₃₁	semidirect
247. #2847	C₂×C₃₁	semidirect
248. #2696	C₂×C₃₁	semidirect
249. #2673	C₂×C₃₁	semidirect
250. #2617	C₂×C₃₁	semidirect
251. #2591	C₂×C₃₁	semidirect
252. #2433	C₂×C₃₁	semidirect
253. #2404	C₂×C₃₁	semidirect
254. #2404	C₂×C₃₁	semidirect
255. #2313	C₂×C₃₁	semidirect
294. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
297. C₂×#121	C₂²×C₃₁	semidirect
298. C₂×#121	C₂²×C₃₁	semidirect
306. #400	C₂²×C₃₁	semidirect
307. #397	C₂²×C₃₁	semidirect
308. #394	C₂²×C₃₁	semidirect
309. #394	C₂²×C₃₁	semidirect
310. #393	C₂²×C₃₁	semidirect
311. #393	C₂²×C₃₁	semidirect
313. #392	C₂²×C₃₁	semidirect
314. #392	C₂²×C₃₁	semidirect
315. #392	C₂²×C₃₁	semidirect
317. #388	C₂²×C₃₁	semidirect
318. #387	C₂²×C₃₁	semidirect
321. #386	C₂²×C₃₁	semidirect
322. #386	C₂²×C₃₁	semidirect
323. #386	C₂²×C₃₁	semidirect
326. #380	C₂²×C₃₁	semidirect
369. C₂×C₄⋊C₄	D₄×C₃₁	semidirect
395. C₃₁	#36710	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×#537	2
C₂²×C₃₁×#123	2
C₃₁×#2847	2
C₃₁×#2696	2
C₃₁×#2673	2
C₃₁×#2617	2
C₃₁×#2591	2
C₃₁×#2433	2
C₃₁×#2404	2
C₃₁×#2313	2
#36710	31

Subgroups — C31×#36710 #21493469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36710 #21493469