Subgroups — C₃₁×#36714 #21493473

Order 7936.

4170 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#534	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#120	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2846	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2697	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2679	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2678	C₂	1	8	yes	maximal, complemented
7	C₃₁×#2622	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2411	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2405	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2403	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2309	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁶×C₃₁	C₂²	1	1	yes	characteristic, complemented
13	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#534	—	2	8	no	—
17	C₂×C₃₁×#123	C₂²	1	2	yes	complemented
18	C₂×C₃₁×#123	C₂²	1	4	yes	complemented
19	C₂×C₃₁×#123	C₂²	1	2	yes	complemented
20	C₂×C₃₁×#121	C₂²	1	4	yes	complemented
21	C₂×C₃₁×#120	—	2	8	no	—
22	C₂×C₃₁×#120	C₂²	1	2	yes	—
23	C₂×C₃₁×#120	C₂²	1	4	yes	—
24	C₂×C₃₁×#120	C₂²	1	4	yes	—
25	C₂×C₃₁×#120	C₂²	1	4	yes	complemented
26	C₂×C₃₁×#120	C₂²	1	8	yes	complemented
27	C₂×C₃₁×#120	C₂²	1	2	yes	complemented
28	C₂×C₃₁×#120	C₂²	1	4	yes	complemented
29	C₂×C₃₁×#117	C₂²	1	2	yes	complemented
30	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
31	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
32	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#397	C₂²	1	8	yes	complemented
34	C₃₁×#396	C₂²	1	8	yes	complemented
35	C₃₁×#394	C₂²	1	8	yes	complemented
36	C₃₁×#394	C₂²	1	8	yes	complemented
37	C₃₁×#394	C₂²	1	8	yes	complemented
38	C₃₁×#392	C₂²	1	8	yes	complemented
39	C₃₁×#390	C₂²	1	4	yes	complemented
40	C₃₁×#390	C₂²	1	4	yes	complemented
41	C₃₁×#390	C₂²	1	8	yes	complemented
42	C₃₁×#388	C₂²	1	8	yes	complemented
43	C₃₁×#386	C₂²	1	4	yes	complemented
44	C₃₁×#386	C₂²	1	4	yes	complemented
45	C₃₁×#386	C₂²	1	4	yes	complemented
46	C₃₁×#385	C₂²	1	4	yes	complemented
47	C₃₁×#379	C₂²	1	2	yes	complemented
48	C₃₁×#379	C₂²	1	4	yes	complemented
49	C₃₁×#379	C₂²	1	8	yes	complemented
50	C₃₁×#379	C₂²	1	2	yes	complemented
51	C₂⁵×C₃₁	—	2	4	no	—
52	C₂⁵×C₃₁	—	4	8	no	—
53	C₂⁵×C₃₁	—	4	4	no	—
54	C₂⁵×C₃₁	C₂³	1	2	yes	—
55	C₂⁵×C₃₁	C₂³	1	4	yes	—
56	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
57	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
58	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic, complemented
59	C₂²×D₄×C₃₁	—	2	8	no	—
60	C₂²×D₄×C₃₁	—	2	4	no	—
61	C₂²×D₄×C₃₁	—	4	4	no	—
62	C₂²×D₄×C₃₁	—	4	8	no	—
63	C₂²×D₄×C₃₁	—	4	4	no	—
64	C₂²×D₄×C₃₁	D₄	1	8	yes	complemented
65	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
70	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
71	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
72	C₂²×D₄×C₃₁	C₂³	1	2	yes	complemented
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	2	yes	complemented
76	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic, complemented
77	C₃₁×#123	—	2	8	no	—
78	C₃₁×#123	—	2	8	no	—
79	C₃₁×#123	—	4	8	no	—
80	C₃₁×#121	—	2	16	no	—
81	C₃₁×#120	—	2	8	no	—
82	C₃₁×#120	—	2	16	no	—
83	C₃₁×#120	—	2	16	no	—
84	C₃₁×#120	—	2	8	no	—
85	C₃₁×#120	—	4	8	no	—
86	C₃₁×#120	—	4	16	no	—
87	C₃₁×#120	—	4	8	no	—
88	C₃₁×#117	—	4	4	no	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	complemented
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	complemented
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	complemented
92	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
94	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	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	2	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	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	2	yes	complemented
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	complemented
106	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	complemented
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	complemented
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	complemented
109	C₂×C₄²×C₃₁	C₂³	1	4	yes	complemented
110	C₃₁×#95	C₂³	1	8	yes	complemented
111	C₃₁×#95	C₂³	1	8	yes	complemented
112	C₂⁴×C₃₁	—	2	4	no	—
113	C₂⁴×C₃₁	—	2	8	no	—
114	C₂⁴×C₃₁	—	2	4	no	—
115	C₂⁴×C₃₁	—	2	4	no	—
116	C₂⁴×C₃₁	—	2	8	no	—
117	C₂⁴×C₃₁	—	2	16	no	—
118	C₂⁴×C₃₁	—	2	8	no	—
119	C₂⁴×C₃₁	—	2	2	no	—
120	C₂⁴×C₃₁	—	4	4	no	—
121	C₂⁴×C₃₁	—	4	4	no	—
122	C₂⁴×C₃₁	—	4	8	no	—
123	C₂⁴×C₃₁	—	4	4	no	—
124	C₂⁴×C₃₁	—	4	4	no	—
125	C₂⁴×C₃₁	—	4	4	no	—
126	C₂⁴×C₃₁	—	4	2	no	—
127	C₂⁴×C₃₁	—	4	4	no	—
128	C₂⁴×C₃₁	—	4	32	no	—
129	C₂⁴×C₃₁	—	4	16	no	—
130	C₂⁴×C₃₁	—	4	64	no	—
131	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
132	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
133	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
134	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
135	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
136	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
137	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
138	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
139	C₂×D₄×C₃₁	—	2	8	no	—
140	C₂×D₄×C₃₁	—	2	8	no	—
141	C₂×D₄×C₃₁	—	2	4	no	—
142	C₂×D₄×C₃₁	—	2	4	no	—
143	C₂×D₄×C₃₁	—	2	8	no	—
144	C₂×D₄×C₃₁	—	2	8	no	—
145	C₂×D₄×C₃₁	—	2	8	no	—
146	C₂×D₄×C₃₁	—	2	8	no	—
147	C₂×D₄×C₃₁	—	2	16	no	—
148	C₂×D₄×C₃₁	—	4	16	no	—
149	C₂×D₄×C₃₁	—	4	2	no	—
150	C₂×D₄×C₃₁	—	4	2	no	—
151	C₂×D₄×C₃₁	—	4	2	no	—
152	C₂×D₄×C₃₁	—	4	2	no	—
153	C₂×D₄×C₃₁	—	4	4	no	—
154	C₂×D₄×C₃₁	—	4	4	no	—
155	C₂×D₄×C₃₁	—	4	4	no	—
156	C₂×D₄×C₃₁	—	4	8	no	—
157	C₂×D₄×C₃₁	—	4	8	no	—
158	C₂×D₄×C₃₁	—	4	8	no	—
159	C₂×D₄×C₃₁	—	4	32	no	—
160	C₂×D₄×C₃₁	—	4	16	no	—
161	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	—
162	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	—
163	C₂²×C₄×C₃₁	—	2	4	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	2	no	—
166	C₂²×C₄×C₃₁	—	2	4	no	—
167	C₂²×C₄×C₃₁	—	4	4	no	—
168	C₂²×C₄×C₃₁	—	4	2	no	—
169	C₂²×C₄×C₃₁	—	4	4	no	—
170	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₃₁	—	2	8	no	—
177	C₄⋊C₄×C₃₁	—	4	4	no	—
178	C₄⋊C₄×C₃₁	—	4	8	no	—
179	C₂²⋊C₄×C₃₁	—	2	8	no	—
180	C₂²⋊C₄×C₃₁	—	2	4	no	—
181	C₂²⋊C₄×C₃₁	—	2	4	no	—
182	C₂²⋊C₄×C₃₁	—	2	8	no	—
183	C₂²⋊C₄×C₃₁	—	2	8	no	—
184	C₂²⋊C₄×C₃₁	—	2	16	no	—
185	C₂²⋊C₄×C₃₁	—	2	16	no	—
186	C₂²⋊C₄×C₃₁	—	2	8	no	—
187	C₂²⋊C₄×C₃₁	—	4	8	no	—
188	C₂²⋊C₄×C₃₁	—	4	8	no	—
189	C₂²⋊C₄×C₃₁	—	4	8	no	—
190	C₂²⋊C₄×C₃₁	—	4	4	no	—
191	C₂²⋊C₄×C₃₁	—	4	8	no	—
192	C₂²⋊C₄×C₃₁	—	4	4	no	—
193	C₄²×C₃₁	—	4	4	no	—
194	#36714	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂³×C₃₁	—	2	4	no	—
196	C₂³×C₃₁	—	2	16	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	8	no	—
203	C₂³×C₃₁	—	2	8	no	—
204	C₂³×C₃₁	—	2	2	no	—
205	C₂³×C₃₁	—	2	2	no	—
206	C₂³×C₃₁	—	2	2	no	—
207	C₂³×C₃₁	—	2	2	no	—
208	C₂³×C₃₁	—	2	4	no	—
209	C₂³×C₃₁	—	2	8	no	—
210	C₂³×C₃₁	—	2	2	no	—
211	C₂³×C₃₁	—	2	8	no	—
212	C₂³×C₃₁	—	2	4	no	—
213	C₂³×C₃₁	—	4	8	no	—
214	C₂³×C₃₁	—	4	8	no	—
215	C₂³×C₃₁	—	4	32	no	—
216	C₂³×C₃₁	—	4	32	no	—
217	C₂³×C₃₁	—	4	16	no	—
218	C₂³×C₃₁	—	4	8	no	—
219	C₂³×C₃₁	—	4	16	no	—
220	C₂³×C₃₁	—	4	16	no	—
221	C₂³×C₃₁	—	4	32	no	—
222	C₂³×C₃₁	—	4	4	no	—
223	C₂³×C₃₁	—	4	128	no	—
224	C₂³×C₃₁	—	4	16	no	—
225	C₂³×C₃₁	—	4	8	no	—
226	C₂³×C₃₁	—	8	8	no	—
227	C₂³×C₃₁	—	8	4	no	—
228	C₂³×C₃₁	—	8	8	no	—
229	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
230	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
231	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
232	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
233	C₂³×C₃₁	H(5,𝔽₂)	1	8	yes	—
234	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
235	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
236	D₄×C₃₁	—	4	8	no	—
237	D₄×C₃₁	—	4	8	no	—
238	D₄×C₃₁	—	4	16	no	—
239	D₄×C₃₁	—	4	16	no	—
240	D₄×C₃₁	—	4	8	no	—
241	D₄×C₃₁	—	4	8	no	—
242	D₄×C₃₁	—	8	8	no	—
243	D₄×C₃₁	—	8	16	no	—
244	C₂×C₄×C₃₁	—	2	8	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	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	8	no	—
251	C₂×C₄×C₃₁	—	2	8	no	—
252	C₂×C₄×C₃₁	—	2	8	no	—
253	C₂×C₄×C₃₁	—	2	8	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	4	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	4	4	no	—
258	C₂×C₄×C₃₁	—	4	8	no	—
259	C₂×C₄×C₃₁	—	8	4	no	—
260	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
261	C₂×#534	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂²×#120	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#2846	C₂×C₃₁	1	2	yes	complemented
264	#2697	C₂×C₃₁	1	4	yes	complemented
265	#2679	C₂×C₃₁	1	4	yes	complemented
266	#2678	C₂×C₃₁	1	8	yes	complemented
267	#2622	C₂×C₃₁	1	2	yes	complemented
268	#2411	C₂×C₃₁	1	4	yes	complemented
269	#2405	C₂×C₃₁	1	2	yes	complemented
270	#2403	C₂×C₃₁	1	2	yes	complemented
271	#2309	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂²×C₃₁	—	2	4	no	—
273	C₂²×C₃₁	—	2	8	no	—
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	2	2	no	—
276	C₂²×C₃₁	—	2	4	no	—
277	C₂²×C₃₁	—	2	4	no	—
278	C₂²×C₃₁	—	2	8	no	—
279	C₂²×C₃₁	—	2	4	no	—
280	C₂²×C₃₁	—	2	8	no	—
281	C₂²×C₃₁	—	2	16	no	—
282	C₂²×C₃₁	—	4	32	no	—
283	C₂²×C₃₁	—	4	16	no	—
284	C₂²×C₃₁	—	4	64	no	—
285	C₂²×C₃₁	—	4	4	no	—
286	C₂²×C₃₁	—	4	4	no	—
287	C₂²×C₃₁	—	4	4	no	—
288	C₂²×C₃₁	—	4	8	no	—
289	C₂²×C₃₁	—	4	4	no	—
290	C₂²×C₃₁	—	4	4	no	—
291	C₂²×C₃₁	—	4	4	no	—
292	C₂²×C₃₁	—	4	2	no	—
293	C₂²×C₃₁	—	4	4	no	—
294	C₂²×C₃₁	—	4	8	no	—
295	C₂²×C₃₁	—	8	8	no	—
296	C₂²×C₃₁	—	8	8	no	—
297	C₂²×C₃₁	#534	1	4	yes	—
298	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
299	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
300	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	4	yes	—
301	C₄×C₃₁	—	2	4	no	—
302	C₄×C₃₁	—	4	2	no	—
303	C₄×C₃₁	—	4	4	no	—
304	C₄×C₃₁	—	4	4	no	—
305	C₄×C₃₁	—	4	8	no	—
306	C₂⁶	C₂²×C₃₁	1	1	yes	characteristic, complemented
307	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
308	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
309	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
310	#534	—	2	8	no	—
311	C₂×#123	C₂²×C₃₁	1	2	yes	complemented
312	C₂×#123	C₂²×C₃₁	1	4	yes	complemented
313	C₂×#123	C₂²×C₃₁	1	2	yes	complemented
314	C₂×#121	C₂²×C₃₁	1	4	yes	complemented
315	C₂×#120	—	2	8	no	—
316	C₂×#120	C₂²×C₃₁	1	2	yes	—
317	C₂×#120	C₂²×C₃₁	1	4	yes	—
318	C₂×#120	C₂²×C₃₁	1	4	yes	—
319	C₂×#120	C₂²×C₃₁	1	4	yes	complemented
320	C₂×#120	C₂²×C₃₁	1	8	yes	complemented
321	C₂×#120	C₂²×C₃₁	1	2	yes	complemented
322	C₂×#120	C₂²×C₃₁	1	4	yes	complemented
323	C₂×#117	C₂²×C₃₁	1	2	yes	complemented
324	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
325	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
326	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
327	#397	C₂²×C₃₁	1	8	yes	complemented
328	#396	C₂²×C₃₁	1	8	yes	complemented
329	#394	C₂²×C₃₁	1	8	yes	complemented
330	#394	C₂²×C₃₁	1	8	yes	complemented
331	#394	C₂²×C₃₁	1	8	yes	complemented
332	#392	C₂²×C₃₁	1	8	yes	complemented
333	#390	C₂²×C₃₁	1	4	yes	complemented
334	#390	C₂²×C₃₁	1	4	yes	complemented
335	#390	C₂²×C₃₁	1	8	yes	complemented
336	#388	C₂²×C₃₁	1	8	yes	complemented
337	#386	C₂²×C₃₁	1	4	yes	complemented
338	#386	C₂²×C₃₁	1	4	yes	complemented
339	#386	C₂²×C₃₁	1	4	yes	complemented
340	#385	C₂²×C₃₁	1	4	yes	complemented
341	#379	C₂²×C₃₁	1	2	yes	complemented
342	#379	C₂²×C₃₁	1	4	yes	complemented
343	#379	C₂²×C₃₁	1	8	yes	complemented
344	#379	C₂²×C₃₁	1	2	yes	complemented
345	C₂×C₃₁	—	2	4	no	—
346	C₂×C₃₁	—	4	4	no	—
347	C₂×C₃₁	—	4	8	no	—
348	C₂×C₃₁	—	4	2	no	—
349	C₂×C₃₁	—	4	2	no	—
350	C₂×C₃₁	—	8	2	no	—
351	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
352	C₂×C₃₁	#3484	1	2	yes	—
353	C₂×C₃₁	#3532	1	4	yes	—
354	C₂⁵	—	2	4	no	—
355	C₂⁵	—	4	8	no	—
356	C₂⁵	—	4	4	no	—
357	C₂⁵	C₂³×C₃₁	1	2	yes	—
358	C₂⁵	C₂³×C₃₁	1	4	yes	—
359	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
360	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
361	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic, complemented
362	C₂²×D₄	—	2	8	no	—
363	C₂²×D₄	—	2	4	no	—
364	C₂²×D₄	—	4	4	no	—
365	C₂²×D₄	—	4	8	no	—
366	C₂²×D₄	—	4	4	no	—
367	C₂²×D₄	D₄×C₃₁	1	8	yes	complemented
368	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
369	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
370	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
371	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
372	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
373	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
374	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
375	C₂²×D₄	C₂³×C₃₁	1	2	yes	complemented
376	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
378	C₂³×C₄	C₂³×C₃₁	1	2	yes	complemented
379	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic, complemented
380	#123	—	2	8	no	—
381	#123	—	2	8	no	—
382	#123	—	4	8	no	—
383	#121	—	2	16	no	—
384	#120	—	2	8	no	—
385	#120	—	2	16	no	—
386	#120	—	2	16	no	—
387	#120	—	2	8	no	—
388	#120	—	4	8	no	—
389	#120	—	4	16	no	—
390	#120	—	4	8	no	—
391	#117	—	4	4	no	—
392	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	complemented
393	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	complemented
394	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	complemented
395	C₂×C₂²⋊C₄	—	2	4	no	—
396	C₂×C₂²⋊C₄	—	2	4	no	—
397	C₂×C₂²⋊C₄	—	2	8	no	—
398	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	complemented
408	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	complemented
409	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	complemented
410	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	complemented
411	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	complemented
412	C₂×C₄²	C₂³×C₃₁	1	4	yes	complemented
413	#95	C₂³×C₃₁	1	8	yes	complemented
414	#95	C₂³×C₃₁	1	8	yes	complemented
415	C₃₁	#36714	1	1	yes	characteristic, direct product factor, Sylow
416	C₂⁴	—	2	4	no	—
417	C₂⁴	—	2	8	no	—
418	C₂⁴	—	2	4	no	—
419	C₂⁴	—	2	4	no	—
420	C₂⁴	—	2	8	no	—
421	C₂⁴	—	2	16	no	—
422	C₂⁴	—	2	8	no	—
423	C₂⁴	—	2	2	no	—
424	C₂⁴	—	4	4	no	—
425	C₂⁴	—	4	4	no	—
426	C₂⁴	—	4	8	no	—
427	C₂⁴	—	4	4	no	—
428	C₂⁴	—	4	4	no	—
429	C₂⁴	—	4	4	no	—
430	C₂⁴	—	4	2	no	—
431	C₂⁴	—	4	4	no	—
432	C₂⁴	—	4	32	no	—
433	C₂⁴	—	4	16	no	—
434	C₂⁴	—	4	64	no	—
435	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
436	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
437	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
438	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
439	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
440	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
441	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
442	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	2	8	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	8	no	—
448	C₂×D₄	—	2	8	no	—
449	C₂×D₄	—	2	8	no	—
450	C₂×D₄	—	2	8	no	—
451	C₂×D₄	—	2	16	no	—
452	C₂×D₄	—	4	16	no	—
453	C₂×D₄	—	4	2	no	—
454	C₂×D₄	—	4	2	no	—
455	C₂×D₄	—	4	2	no	—
456	C₂×D₄	—	4	2	no	—
457	C₂×D₄	—	4	4	no	—
458	C₂×D₄	—	4	4	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	—	4	8	no	—
461	C₂×D₄	—	4	8	no	—
462	C₂×D₄	—	4	8	no	—
463	C₂×D₄	—	4	32	no	—
464	C₂×D₄	—	4	16	no	—
465	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	—
466	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	4	4	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	—	4	4	no	—
474	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
475	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
476	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
478	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
480	C₄⋊C₄	—	2	8	no	—
481	C₄⋊C₄	—	4	4	no	—
482	C₄⋊C₄	—	4	8	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	2	16	no	—
489	C₂²⋊C₄	—	2	16	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	—	4	8	no	—
492	C₂²⋊C₄	—	4	8	no	—
493	C₂²⋊C₄	—	4	8	no	—
494	C₂²⋊C₄	—	4	4	no	—
495	C₂²⋊C₄	—	4	8	no	—
496	C₂²⋊C₄	—	4	4	no	—
497	C₄²	—	4	4	no	—
498	C₂³	—	2	4	no	—
499	C₂³	—	2	16	no	—
500	C₂³	—	2	8	no	—
501	C₂³	—	2	4	no	—
502	C₂³	—	2	4	no	—
503	C₂³	—	2	4	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	2	8	no	—
506	C₂³	—	2	8	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	—	2	8	no	—
513	C₂³	—	2	2	no	—
514	C₂³	—	2	8	no	—
515	C₂³	—	2	4	no	—
516	C₂³	—	4	8	no	—
517	C₂³	—	4	8	no	—
518	C₂³	—	4	32	no	—
519	C₂³	—	4	32	no	—
520	C₂³	—	4	16	no	—
521	C₂³	—	4	8	no	—
522	C₂³	—	4	16	no	—
523	C₂³	—	4	16	no	—
524	C₂³	—	4	32	no	—
525	C₂³	—	4	4	no	—
526	C₂³	—	4	128	no	—
527	C₂³	—	4	16	no	—
528	C₂³	—	4	8	no	—
529	C₂³	—	8	8	no	—
530	C₂³	—	8	4	no	—
531	C₂³	—	8	8	no	—
532	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
533	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
534	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
535	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
536	C₂³	C₃₁×H(5,𝔽₂)	1	8	yes	—
537	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
538	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
539	D₄	—	4	8	no	—
540	D₄	—	4	8	no	—
541	D₄	—	4	16	no	—
542	D₄	—	4	16	no	—
543	D₄	—	4	8	no	—
544	D₄	—	4	8	no	—
545	D₄	—	8	8	no	—
546	D₄	—	8	16	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	8	no	—
554	C₂×C₄	—	2	8	no	—
555	C₂×C₄	—	2	8	no	—
556	C₂×C₄	—	2	8	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	8	no	—
562	C₂×C₄	—	8	4	no	—
563	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
564	C₂²	—	2	4	no	—
565	C₂²	—	2	8	no	—
566	C₂²	—	2	2	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	4	no	—
569	C₂²	—	2	4	no	—
570	C₂²	—	2	8	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	2	8	no	—
573	C₂²	—	2	16	no	—
574	C₂²	—	4	32	no	—
575	C₂²	—	4	16	no	—
576	C₂²	—	4	64	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	8	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	4	4	no	—
583	C₂²	—	4	4	no	—
584	C₂²	—	4	2	no	—
585	C₂²	—	4	4	no	—
586	C₂²	—	4	8	no	—
587	C₂²	—	8	8	no	—
588	C₂²	—	8	8	no	—
589	C₂²	C₃₁×#534	1	4	yes	—
590	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
591	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
592	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	4	yes	—
593	C₄	—	2	4	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	4	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	8	no	—
598	C₂	—	2	4	no	—
599	C₂	—	4	4	no	—
600	C₂	—	4	8	no	—
601	C₂	—	4	2	no	—
602	C₂	—	4	2	no	—
603	C₂	—	8	2	no	—
604	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
605	C₂	C₃₁×#3484	1	2	yes	—
606	C₂	C₃₁×#3532	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×#534	C₂	semidirect
2. C₂²×C₃₁×#120	C₂	semidirect
3. C₃₁×#2846	C₂	semidirect
4. C₃₁×#2697	C₂	semidirect
5. C₃₁×#2679	C₂	semidirect
6. C₃₁×#2678	C₂	semidirect
7. C₃₁×#2622	C₂	semidirect
8. C₃₁×#2411	C₂	semidirect
9. C₃₁×#2405	C₂	semidirect
10. C₃₁×#2403	C₂	semidirect
11. C₃₁×#2309	C₂	semidirect
12. C₂⁶×C₃₁	C₂²	semidirect
15. C₂³×D₄×C₃₁	C₂²	semidirect
17. C₂×C₃₁×#123	C₂²	semidirect
18. C₂×C₃₁×#123	C₂²	semidirect
19. C₂×C₃₁×#123	C₂²	semidirect
20. C₂×C₃₁×#121	C₂²	semidirect
25. C₂×C₃₁×#120	C₂²	semidirect
26. C₂×C₃₁×#120	C₂²	semidirect
27. C₂×C₃₁×#120	C₂²	semidirect
28. C₂×C₃₁×#120	C₂²	semidirect
29. C₂×C₃₁×#117	C₂²	semidirect
30. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
31. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
32. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
33. C₃₁×#397	C₂²	semidirect
34. C₃₁×#396	C₂²	semidirect
35. C₃₁×#394	C₂²	semidirect
36. C₃₁×#394	C₂²	semidirect
37. C₃₁×#394	C₂²	semidirect
38. C₃₁×#392	C₂²	semidirect
39. C₃₁×#390	C₂²	semidirect
40. C₃₁×#390	C₂²	semidirect
41. C₃₁×#390	C₂²	semidirect
42. C₃₁×#388	C₂²	semidirect
43. C₃₁×#386	C₂²	semidirect
44. C₃₁×#386	C₂²	semidirect
45. C₃₁×#386	C₂²	semidirect
46. C₃₁×#385	C₂²	semidirect
47. C₃₁×#379	C₂²	semidirect
48. C₃₁×#379	C₂²	semidirect
49. C₃₁×#379	C₂²	semidirect
50. C₃₁×#379	C₂²	semidirect
58. C₂⁵×C₃₁	C₂³	semidirect
64. C₂²×D₄×C₃₁	D₄	semidirect
72. C₂²×D₄×C₃₁	C₂³	semidirect
75. C₂³×C₄×C₃₁	C₂³	semidirect
76. C₂³×C₄×C₃₁	C₂³	semidirect
89. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
90. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
91. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
104. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
105. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
106. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
107. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
108. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
109. C₂×C₄²×C₃₁	C₂³	semidirect
110. C₃₁×#95	C₂³	semidirect
111. C₃₁×#95	C₂³	semidirect
194. #36714	C₃₁	direct (N × H)
261. C₂×#534	C₂×C₃₁	semidirect
262. C₂²×#120	C₂×C₃₁	semidirect
263. #2846	C₂×C₃₁	semidirect
264. #2697	C₂×C₃₁	semidirect
265. #2679	C₂×C₃₁	semidirect
266. #2678	C₂×C₃₁	semidirect
267. #2622	C₂×C₃₁	semidirect
268. #2411	C₂×C₃₁	semidirect
269. #2405	C₂×C₃₁	semidirect
270. #2403	C₂×C₃₁	semidirect
271. #2309	C₂×C₃₁	semidirect
306. C₂⁶	C₂²×C₃₁	semidirect
309. C₂³×D₄	C₂²×C₃₁	semidirect
311. C₂×#123	C₂²×C₃₁	semidirect
312. C₂×#123	C₂²×C₃₁	semidirect
313. C₂×#123	C₂²×C₃₁	semidirect
314. C₂×#121	C₂²×C₃₁	semidirect
319. C₂×#120	C₂²×C₃₁	semidirect
320. C₂×#120	C₂²×C₃₁	semidirect
321. C₂×#120	C₂²×C₃₁	semidirect
322. C₂×#120	C₂²×C₃₁	semidirect
323. C₂×#117	C₂²×C₃₁	semidirect
324. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
325. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
326. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
327. #397	C₂²×C₃₁	semidirect
328. #396	C₂²×C₃₁	semidirect
329. #394	C₂²×C₃₁	semidirect
330. #394	C₂²×C₃₁	semidirect
331. #394	C₂²×C₃₁	semidirect
332. #392	C₂²×C₃₁	semidirect
333. #390	C₂²×C₃₁	semidirect
334. #390	C₂²×C₃₁	semidirect
335. #390	C₂²×C₃₁	semidirect
336. #388	C₂²×C₃₁	semidirect
337. #386	C₂²×C₃₁	semidirect
338. #386	C₂²×C₃₁	semidirect
339. #386	C₂²×C₃₁	semidirect
340. #385	C₂²×C₃₁	semidirect
341. #379	C₂²×C₃₁	semidirect
342. #379	C₂²×C₃₁	semidirect
343. #379	C₂²×C₃₁	semidirect
344. #379	C₂²×C₃₁	semidirect
361. C₂⁵	C₂³×C₃₁	semidirect
367. C₂²×D₄	D₄×C₃₁	semidirect
375. C₂²×D₄	C₂³×C₃₁	semidirect
378. C₂³×C₄	C₂³×C₃₁	semidirect
379. C₂³×C₄	C₂³×C₃₁	semidirect
392. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
393. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
394. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
407. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
408. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
409. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
410. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
411. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
412. C₂×C₄²	C₂³×C₃₁	semidirect
413. #95	C₂³×C₃₁	semidirect
414. #95	C₂³×C₃₁	semidirect
415. C₃₁	#36714	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×#534	2
C₂²×C₃₁×#120	2
C₃₁×#2846	2
C₃₁×#2697	2
C₃₁×#2679	2
C₃₁×#2678	2
C₃₁×#2622	2
C₃₁×#2411	2
C₃₁×#2405	2
C₃₁×#2403	2
C₃₁×#2309	2
#36714	31

Subgroups — C31×#36714 #21493473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36714 #21493473