Subgroups — C₃₁×#34943 #21491702

Order 7936.

2036 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2572	C₂	1	4	yes	maximal, complemented
2	C₃₁×#2571	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2563	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2537	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2495	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2420	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2398	C₂	1	4	yes	maximal, complemented
8	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₃₁×#121	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2283	C₂	1	4	yes	maximal, complemented
11	C₄²×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#121	C₂²	1	4	yes	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
19	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
20	C₃₁×#400	C₂²	1	4	yes	—
21	C₃₁×#400	C₂²	1	4	yes	—
22	C₃₁×#397	C₂²	1	8	yes	—
23	C₃₁×#394	C₂²	1	4	yes	—
24	C₃₁×#394	C₂²	1	4	yes	—
25	C₃₁×#393	C₂²	1	8	yes	—
26	C₃₁×#391	C₂²	1	2	yes	—
27	C₃₁×#390	C₂²	1	4	yes	—
28	C₃₁×#389	C₂²	1	4	yes	—
29	C₃₁×#389	C₂²	1	4	yes	—
30	C₃₁×#388	C₂²	1	8	yes	—
31	C₃₁×#386	C₂²	1	8	yes	—
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#383	C₂²	1	8	yes	—
34	C₃₁×#382	C₂²	1	4	yes	—
35	C₃₁×#382	C₂²	1	4	yes	—
36	C₃₁×#381	C₂²	1	8	yes	—
37	C₃₁×#380	C₂²	1	4	yes	—
38	C₃₁×#380	C₂²	1	4	yes	—
39	C₃₁×#380	C₂²	1	4	yes	—
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
42	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
43	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
44	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
45	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
46	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
47	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
48	C₃₁×#376	C₂²	1	4	yes	—
49	C₄³×C₃₁	C₂²	1	1	yes	characteristic
50	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
52	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
56	C₃₁×C₄⋊Q₈	—	2	4	no	—
57	C₃₁×#121	—	2	16	no	—
58	C₄×Q₈×C₃₁	—	4	4	no	—
59	C₄×D₄×C₃₁	—	2	8	no	—
60	C₄×D₄×C₃₁	—	2	8	no	—
61	C₄×D₄×C₃₁	—	2	8	no	—
62	C₄×D₄×C₃₁	—	4	8	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
71	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
72	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
73	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
74	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
75	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
77	C₂×C₄²×C₃₁	—	2	16	no	—
78	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
79	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
80	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
81	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
82	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
87	C₃₁×#95	C₂³	1	4	yes	—
88	C₃₁×#95	C₂³	1	8	yes	—
89	C₃₁×#95	C₂³	1	8	yes	—
90	C₃₁×#95	C₂³	1	4	yes	—
91	C₃₁×#95	C₂³	1	4	yes	—
92	C₃₁×#95	C₂³	1	4	yes	—
93	C₃₁×#95	C₂³	1	8	yes	—
94	C₃₁×#95	C₂³	1	4	yes	—
95	C₃₁×#95	C₂³	1	8	yes	—
96	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
97	C₂×Q₈×C₃₁	—	2	4	no	—
98	C₂×Q₈×C₃₁	—	4	2	no	—
99	C₂×Q₈×C₃₁	—	4	2	no	—
100	C₂×D₄×C₃₁	—	2	16	no	—
101	C₂×D₄×C₃₁	—	2	4	no	—
102	C₂×D₄×C₃₁	—	2	4	no	—
103	C₂×D₄×C₃₁	—	4	4	no	—
104	C₂×D₄×C₃₁	#41	1	8	yes	—
105	C₂²×C₄×C₃₁	—	2	8	no	—
106	C₂²×C₄×C₃₁	—	2	8	no	—
107	C₂²×C₄×C₃₁	—	2	8	no	—
108	C₂²×C₄×C₃₁	—	2	8	no	—
109	C₂²×C₄×C₃₁	—	2	8	no	—
110	C₂²×C₄×C₃₁	—	2	8	no	—
111	C₂²×C₄×C₃₁	—	2	8	no	—
112	C₂²×C₄×C₃₁	—	2	8	no	—
113	C₂²×C₄×C₃₁	—	4	4	no	—
114	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
115	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
116	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
117	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
118	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
121	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	C₄⋊C₄×C₃₁	—	2	4	no	—
125	C₄⋊C₄×C₃₁	—	2	4	no	—
126	C₄⋊C₄×C₃₁	—	2	4	no	—
127	C₄⋊C₄×C₃₁	—	2	8	no	—
128	C₄⋊C₄×C₃₁	—	2	8	no	—
129	C₄⋊C₄×C₃₁	—	4	4	no	—
130	C₄⋊C₄×C₃₁	—	4	2	no	—
131	C₄⋊C₄×C₃₁	—	4	4	no	—
132	C₂²⋊C₄×C₃₁	—	2	8	no	—
133	C₂²⋊C₄×C₃₁	—	2	16	no	—
134	C₂²⋊C₄×C₃₁	—	2	8	no	—
135	C₂²⋊C₄×C₃₁	—	2	8	no	—
136	C₂²⋊C₄×C₃₁	—	4	8	no	—
137	C₂²⋊C₄×C₃₁	—	4	8	no	—
138	C₄²×C₃₁	—	2	8	no	—
139	C₄²×C₃₁	—	2	8	no	—
140	C₄²×C₃₁	—	2	8	no	—
141	C₄²×C₃₁	—	2	4	no	—
142	C₄²×C₃₁	—	2	8	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	4	1	no	—
145	C₄²×C₃₁	#41	1	8	yes	—
146	C₄²×C₃₁	#41	1	4	yes	—
147	#34943	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₃₁	—	2	4	no	—
149	C₂³×C₃₁	—	2	4	no	—
150	C₂³×C₃₁	—	2	4	no	—
151	C₂³×C₃₁	—	2	4	no	—
152	C₂³×C₃₁	—	2	4	no	—
153	C₂³×C₃₁	—	2	4	no	—
154	C₂³×C₃₁	C₂×#41	1	8	yes	—
155	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
156	Q₈×C₃₁	—	4	8	no	—
157	D₄×C₃₁	—	2	16	no	—
158	D₄×C₃₁	—	4	16	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	4	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	2	4	no	—
169	C₂×C₄×C₃₁	—	2	4	no	—
170	C₂×C₄×C₃₁	—	2	4	no	—
171	C₂×C₄×C₃₁	—	2	8	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	4	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	4	16	no	—
180	C₂×C₄×C₃₁	—	4	16	no	—
181	C₂×C₄×C₃₁	—	4	16	no	—
182	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
183	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
184	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
185	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
186	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
187	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
188	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
189	#2572	C₂×C₃₁	1	4	yes	complemented
190	#2571	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#2563	C₂×C₃₁	1	2	yes	complemented
192	#2537	C₂×C₃₁	1	1	yes	characteristic
193	#2495	C₂×C₃₁	1	4	yes	complemented
194	#2420	C₂×C₃₁	1	4	yes	complemented
195	#2398	C₂×C₃₁	1	4	yes	complemented
196	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
197	C₄×#121	C₂×C₃₁	1	4	yes	complemented
198	#2283	C₂×C₃₁	1	4	yes	complemented
199	C₄²×D₄	C₂×C₃₁	1	2	yes	complemented
200	C₂²×C₃₁	—	2	8	no	—
201	C₂²×C₃₁	—	2	8	no	—
202	C₂²×C₃₁	—	2	8	no	—
203	C₂²×C₃₁	—	4	4	no	—
204	C₂²×C₃₁	—	4	4	no	—
205	C₂²×C₃₁	—	4	4	no	—
206	C₂²×C₃₁	—	4	4	no	—
207	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
208	C₂²×C₃₁	C₂²×#41	1	2	yes	—
209	C₂²×C₃₁	C₂²×#41	1	2	yes	—
210	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
211	C₂²×C₃₁	#585	1	1	yes	characteristic
212	C₄×C₃₁	—	2	2	no	—
213	C₄×C₃₁	—	2	4	no	—
214	C₄×C₃₁	—	4	4	no	—
215	C₄×C₃₁	—	4	4	no	—
216	C₄×C₃₁	—	4	4	no	—
217	C₄×C₃₁	—	4	4	no	—
218	C₄×C₃₁	—	4	1	no	—
219	C₄×C₃₁	—	4	4	no	—
220	C₄×C₃₁	#529	1	8	yes	—
221	C₄×C₃₁	#555	1	4	yes	—
222	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#121	C₂²×C₃₁	1	4	yes	—
224	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
225	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
226	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
227	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
228	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
229	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
230	#400	C₂²×C₃₁	1	4	yes	—
231	#400	C₂²×C₃₁	1	4	yes	—
232	#397	C₂²×C₃₁	1	8	yes	—
233	#394	C₂²×C₃₁	1	4	yes	—
234	#394	C₂²×C₃₁	1	4	yes	—
235	#393	C₂²×C₃₁	1	8	yes	—
236	#391	C₂²×C₃₁	1	2	yes	—
237	#390	C₂²×C₃₁	1	4	yes	—
238	#389	C₂²×C₃₁	1	4	yes	—
239	#389	C₂²×C₃₁	1	4	yes	—
240	#388	C₂²×C₃₁	1	8	yes	—
241	#386	C₂²×C₃₁	1	8	yes	—
242	#384	C₂²×C₃₁	1	1	yes	characteristic
243	#383	C₂²×C₃₁	1	8	yes	—
244	#382	C₂²×C₃₁	1	4	yes	—
245	#382	C₂²×C₃₁	1	4	yes	—
246	#381	C₂²×C₃₁	1	8	yes	—
247	#380	C₂²×C₃₁	1	4	yes	—
248	#380	C₂²×C₃₁	1	4	yes	—
249	#380	C₂²×C₃₁	1	4	yes	—
250	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
251	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
252	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
253	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
254	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
255	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
256	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
257	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
258	#376	C₂²×C₃₁	1	4	yes	—
259	C₄³	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₃₁	—	4	8	no	—
261	C₂×C₃₁	C₂×#529	1	2	yes	—
262	C₂×C₃₁	#3460	1	1	yes	characteristic
263	C₂×C₃₁	C₂×#555	1	1	yes	characteristic
264	C₂×C₃₁	#3475	1	2	yes	—
265	C₂×C₃₁	#3481	1	1	yes	characteristic
266	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
267	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
268	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
269	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
270	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
271	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
272	C₄⋊Q₈	—	2	4	no	—
273	#121	—	2	16	no	—
274	C₄×Q₈	—	4	4	no	—
275	C₄×D₄	—	2	8	no	—
276	C₄×D₄	—	2	8	no	—
277	C₄×D₄	—	2	8	no	—
278	C₄×D₄	—	4	8	no	—
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
289	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
290	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
291	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
292	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
293	C₂×C₄²	—	2	16	no	—
294	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
295	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
298	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
299	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
300	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
301	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
302	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
303	#95	C₂³×C₃₁	1	4	yes	—
304	#95	C₂³×C₃₁	1	8	yes	—
305	#95	C₂³×C₃₁	1	4	yes	—
306	#95	C₂³×C₃₁	1	4	yes	—
307	#95	C₂³×C₃₁	1	4	yes	—
308	#95	C₂³×C₃₁	1	4	yes	—
309	#95	C₂³×C₃₁	1	8	yes	—
310	#95	C₂³×C₃₁	1	8	yes	—
311	#95	C₂³×C₃₁	1	8	yes	—
312	C₃₁	#34943	1	1	yes	characteristic, direct product factor, Sylow
313	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
314	C₂×Q₈	—	2	4	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×D₄	—	2	16	no	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	2	4	no	—
320	C₂×D₄	—	4	4	no	—
321	C₂×D₄	#41×C₃₁	1	8	yes	—
322	C₂²×C₄	—	2	8	no	—
323	C₂²×C₄	—	2	8	no	—
324	C₂²×C₄	—	2	8	no	—
325	C₂²×C₄	—	2	8	no	—
326	C₂²×C₄	—	2	8	no	—
327	C₂²×C₄	—	2	8	no	—
328	C₂²×C₄	—	2	8	no	—
329	C₂²×C₄	—	2	8	no	—
330	C₂²×C₄	—	4	4	no	—
331	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
332	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
333	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
334	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
335	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
336	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
338	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
339	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
341	C₄⋊C₄	—	2	4	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₄⋊C₄	—	2	4	no	—
344	C₄⋊C₄	—	2	8	no	—
345	C₄⋊C₄	—	2	8	no	—
346	C₄⋊C₄	—	4	4	no	—
347	C₄⋊C₄	—	4	2	no	—
348	C₄⋊C₄	—	4	4	no	—
349	C₂²⋊C₄	—	2	8	no	—
350	C₂²⋊C₄	—	2	16	no	—
351	C₂²⋊C₄	—	2	8	no	—
352	C₂²⋊C₄	—	2	8	no	—
353	C₂²⋊C₄	—	4	8	no	—
354	C₂²⋊C₄	—	4	8	no	—
355	C₄²	—	2	8	no	—
356	C₄²	—	2	8	no	—
357	C₄²	—	2	8	no	—
358	C₄²	—	2	4	no	—
359	C₄²	—	2	8	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	4	1	no	—
362	C₄²	#41×C₃₁	1	8	yes	—
363	C₄²	#41×C₃₁	1	4	yes	—
364	C₂³	—	2	4	no	—
365	C₂³	—	2	4	no	—
366	C₂³	—	2	4	no	—
367	C₂³	—	2	4	no	—
368	C₂³	—	2	4	no	—
369	C₂³	—	2	4	no	—
370	C₂³	C₂×#41×C₃₁	1	8	yes	—
371	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
372	Q₈	—	4	8	no	—
373	D₄	—	2	16	no	—
374	D₄	—	4	16	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	4	no	—
387	C₂×C₄	—	2	8	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	4	16	no	—
396	C₂×C₄	—	4	16	no	—
397	C₂×C₄	—	4	16	no	—
398	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
399	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
400	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
401	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
402	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
403	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
404	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
405	C₂²	—	2	8	no	—
406	C₂²	—	2	8	no	—
407	C₂²	—	2	8	no	—
408	C₂²	—	4	4	no	—
409	C₂²	—	4	4	no	—
410	C₂²	—	4	4	no	—
411	C₂²	—	4	4	no	—
412	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
413	C₂²	C₂²×#41×C₃₁	1	2	yes	—
414	C₂²	C₂²×#41×C₃₁	1	2	yes	—
415	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
416	C₂²	C₃₁×#585	1	1	yes	characteristic
417	C₄	—	2	2	no	—
418	C₄	—	2	4	no	—
419	C₄	—	4	4	no	—
420	C₄	—	4	4	no	—
421	C₄	—	4	4	no	—
422	C₄	—	4	4	no	—
423	C₄	—	4	4	no	—
424	C₄	—	4	1	no	—
425	C₄	C₃₁×#529	1	8	yes	—
426	C₄	C₃₁×#555	1	4	yes	—
427	C₂	—	4	8	no	—
428	C₂	C₂×C₃₁×#529	1	2	yes	—
429	C₂	C₃₁×#3460	1	1	yes	characteristic
430	C₂	C₂×C₃₁×#555	1	1	yes	characteristic
431	C₂	C₃₁×#3475	1	2	yes	—
432	C₂	C₃₁×#3481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₃₁×#2537	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2572	C₂	semidirect
2. C₃₁×#2571	C₂	semidirect
3. C₃₁×#2563	C₂	semidirect
5. C₃₁×#2495	C₂	semidirect
6. C₃₁×#2420	C₂	semidirect
7. C₃₁×#2398	C₂	semidirect
8. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
9. C₄×C₃₁×#121	C₂	semidirect
10. C₃₁×#2283	C₂	semidirect
11. C₄²×D₄×C₃₁	C₂	semidirect
147. #34943	C₃₁	direct (N × H)
189. #2572	C₂×C₃₁	semidirect
190. #2571	C₂×C₃₁	semidirect
191. #2563	C₂×C₃₁	semidirect
193. #2495	C₂×C₃₁	semidirect
194. #2420	C₂×C₃₁	semidirect
195. #2398	C₂×C₃₁	semidirect
196. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
197. C₄×#121	C₂×C₃₁	semidirect
198. #2283	C₂×C₃₁	semidirect
199. C₄²×D₄	C₂×C₃₁	semidirect
312. C₃₁	#34943	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2572	2
C₃₁×#2571	2
C₃₁×#2563	2
C₃₁×#2537	2
C₃₁×#2495	2
C₃₁×#2420	2
C₃₁×#2398	2
C₄×C₃₁×C₄⋊Q₈	2
C₄×C₃₁×#121	2
C₃₁×#2283	2
C₄²×D₄×C₃₁	2
#34943	31

Subgroups — C31×#34943 #21491702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34943 #21491702