Subgroups — C₃₁×#29148 #21485907

Order 7936.

830 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2024	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2023	C₂	1	2	yes	maximal
5	C₃₁×#1959	C₂	1	2	yes	maximal
6	C₃₁×#1836	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1803	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1802	C₂	1	2	yes	maximal
9	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×#103	C₂²	1	4	yes	—
18	C₂×C₃₁×#102	C₂²	1	4	yes	—
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₃₁×#391	C₂²	1	1	yes	characteristic
21	C₃₁×#390	C₂²	1	1	yes	characteristic
22	C₃₁×#388	—	2	1	no	—
23	C₃₁×#388	—	2	1	no	—
24	C₃₁×#384	—	2	1	no	—
25	C₃₁×#384	—	2	1	no	—
26	C₃₁×#384	C₂²	1	2	yes	—
27	C₃₁×#384	C₂²	1	2	yes	—
28	C₃₁×#382	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
30	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
31	C₄³×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#340	C₂²	1	4	yes	—
33	C₃₁×#340	C₂²	1	4	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₁	—	2	2	no	—
36	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	—	2	1	no	—
39	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
40	C₃₁×C₄⋊Q₈	—	4	4	no	—
41	C₃₁×C₄⋊Q₈	—	4	4	no	—
42	C₃₁×C₄⋊Q₈	—	4	4	no	—
43	C₃₁×#124	—	2	2	no	—
44	C₃₁×#124	Q₈	1	4	yes	complemented
45	C₄×Q₈×C₃₁	—	2	4	no	—
46	C₄×D₄×C₃₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄²×C₃₁	—	2	1	no	—
62	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×C₄⋊C₈	—	2	4	no	—
70	C₃₁×#103	—	4	4	no	—
71	C₃₁×#102	—	4	4	no	—
72	C₃₁×#97	—	2	4	no	—
73	C₃₁×#95	—	2	1	no	—
74	C₃₁×#95	—	2	1	no	—
75	C₃₁×#95	—	2	1	no	—
76	C₃₁×#95	—	2	1	no	—
77	C₂⁴×C₃₁	—	2	1	no	—
78	C₂×Q₈×C₃₁	—	2	1	no	—
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	2	2	no	—
81	C₂×Q₈×C₃₁	—	4	4	no	—
82	C₂×Q₈×C₃₁	—	4	4	no	—
83	C₂×Q₈×C₃₁	—	4	4	no	—
84	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	1	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₃₁	—	2	1	no	—
90	C₂²×C₄×C₃₁	—	2	1	no	—
91	C₂²×C₄×C₃₁	—	2	1	no	—
92	C₂²×C₄×C₃₁	—	2	4	no	—
93	C₂²×C₄×C₃₁	—	4	2	no	—
94	C₂²×C₄×C₃₁	—	4	2	no	—
95	C₂²×C₄×C₃₁	—	4	2	no	—
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₂×C₈×C₃₁	—	2	4	no	—
104	C₂×C₈×C₃₁	—	2	4	no	—
105	C₂×C₈×C₃₁	—	2	4	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	4	no	—
109	C₄⋊C₄×C₃₁	—	2	4	no	—
110	C₄⋊C₄×C₃₁	—	4	4	no	—
111	C₄⋊C₄×C₃₁	—	4	4	no	—
112	C₄⋊C₄×C₃₁	—	4	4	no	—
113	C₄⋊C₄×C₃₁	—	4	4	no	—
114	C₄⋊C₄×C₃₁	—	4	2	no	—
115	C₄⋊C₄×C₃₁	—	4	4	no	—
116	C₂²⋊C₄×C₃₁	—	2	4	no	—
117	C₂²⋊C₄×C₃₁	—	2	4	no	—
118	C₂²⋊C₄×C₃₁	—	4	2	no	—
119	C₂²⋊C₄×C₃₁	—	4	2	no	—
120	C₂²⋊C₄×C₃₁	—	4	2	no	—
121	C₄²×C₃₁	—	2	2	no	—
122	C₄²×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	—	4	2	no	—
127	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
128	C₄²×C₃₁	#41	1	2	yes	—
129	C₄²×C₃₁	#41	1	4	yes	—
130	#29148	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂³×C₃₁	—	2	2	no	—
132	C₂³×C₃₁	—	4	1	no	—
133	C₂³×C₃₁	—	4	1	no	—
134	C₂³×C₃₁	—	4	1	no	—
135	C₂³×C₃₁	—	4	1	no	—
136	C₂³×C₃₁	—	4	1	no	—
137	C₂³×C₃₁	—	4	1	no	—
138	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	Q₈×C₃₁	—	2	4	no	—
140	Q₈×C₃₁	—	4	2	no	—
141	Q₈×C₃₁	—	8	8	no	—
142	D₄×C₃₁	—	2	4	no	—
143	D₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	4	1	no	—
156	C₂×C₄×C₃₁	—	4	1	no	—
157	C₂×C₄×C₃₁	—	4	1	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	4	4	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	4	no	—
162	C₂×C₄×C₃₁	—	4	4	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	8	4	no	—
167	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
168	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
169	C₂×C₄×C₃₁	#124	1	2	yes	—
170	C₂×C₄×C₃₁	#124	1	2	yes	—
171	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
172	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
173	C₂×C₄×C₃₁	#137	1	2	yes	—
174	C₂×C₄×C₃₁	#137	1	2	yes	—
175	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
177	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
179	C₈×C₃₁	—	4	4	no	—
180	#2868	C₂×C₃₁	1	1	yes	characteristic, complemented
181	C₄×#124	C₂×C₃₁	1	1	yes	characteristic
182	#2024	C₂×C₃₁	1	2	yes	complemented
183	#2023	C₂×C₃₁	1	2	yes	—
184	#1959	C₂×C₃₁	1	2	yes	—
185	#1836	C₂×C₃₁	1	2	yes	complemented
186	#1803	C₂×C₃₁	1	2	yes	complemented
187	#1802	C₂×C₃₁	1	2	yes	—
188	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
189	C₂²×C₃₁	—	4	2	no	—
190	C₂²×C₃₁	—	4	2	no	—
191	C₂²×C₃₁	—	4	2	no	—
192	C₂²×C₃₁	—	8	1	no	—
193	C₂²×C₃₁	—	8	1	no	—
194	C₂²×C₃₁	—	8	1	no	—
195	C₂²×C₃₁	—	8	1	no	—
196	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
197	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
198	C₂²×C₃₁	#535	1	1	yes	characteristic
199	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
202	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
203	C₄×C₃₁	—	2	2	no	—
204	C₄×C₃₁	—	2	2	no	—
205	C₄×C₃₁	—	2	2	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	2	2	no	—
208	C₄×C₃₁	—	4	2	no	—
209	C₄×C₃₁	—	8	1	no	—
210	C₄×C₃₁	—	8	1	no	—
211	C₄×C₃₁	—	8	4	no	—
212	C₄×C₃₁	#489	1	4	yes	—
213	C₄×C₃₁	#554	1	2	yes	—
214	C₄×C₃₁	#555	1	2	yes	—
215	C₂×C₄⋊Q₈	—	2	1	no	—
216	C₂×C₄⋊Q₈	—	2	1	no	—
217	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
218	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
222	C₂×#103	C₂²×C₃₁	1	4	yes	—
223	C₂×#102	C₂²×C₃₁	1	4	yes	—
224	C₂×#97	C₂²×C₃₁	1	2	yes	—
225	#391	C₂²×C₃₁	1	1	yes	characteristic
226	#390	C₂²×C₃₁	1	1	yes	characteristic
227	#388	—	2	1	no	—
228	#388	—	2	1	no	—
229	#384	—	2	1	no	—
230	#384	—	2	1	no	—
231	#384	C₂²×C₃₁	1	2	yes	—
232	#384	C₂²×C₃₁	1	2	yes	—
233	#382	C₂²×C₃₁	1	1	yes	characteristic
234	C₄×C₂²⋊C₄	—	2	1	no	—
235	C₄×C₂²⋊C₄	—	2	1	no	—
236	C₄³	C₂²×C₃₁	1	1	yes	characteristic
237	#340	C₂²×C₃₁	1	4	yes	—
238	#340	C₂²×C₃₁	1	4	yes	—
239	#336	C₂²×C₃₁	1	2	yes	—
240	C₂×C₃₁	—	8	2	no	—
241	C₂×C₃₁	#2563	1	1	yes	characteristic
242	C₂×C₃₁	#3079	1	1	yes	characteristic
243	C₂×C₃₁	#3079	1	1	yes	characteristic
244	C₂×C₃₁	C₂×#489	1	1	yes	characteristic
245	C₂×C₃₁	#3141	1	1	yes	characteristic
246	C₂×C₃₁	#3217	1	1	yes	characteristic
247	C₂×C₃₁	#3218	1	1	yes	characteristic
248	C₂²×Q₈	—	2	2	no	—
249	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
251	C₂³×C₄	—	2	1	no	—
252	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
253	C₄⋊Q₈	—	4	4	no	—
254	C₄⋊Q₈	—	4	4	no	—
255	C₄⋊Q₈	—	4	4	no	—
256	#124	—	2	2	no	—
257	#124	Q₈×C₃₁	1	4	yes	complemented
258	C₄×Q₈	—	2	4	no	—
259	C₄×D₄	—	2	4	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
268	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
269	C₂×C₂²⋊C₄	—	2	1	no	—
270	C₂×C₂²⋊C₄	—	2	1	no	—
271	C₂×C₂²⋊C₄	—	2	1	no	—
272	C₂×C₂²⋊C₄	—	2	1	no	—
273	C₂×C₂²⋊C₄	—	2	1	no	—
274	C₂×C₄²	—	2	1	no	—
275	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
278	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
281	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
282	C₄⋊C₈	—	2	4	no	—
283	#103	—	4	4	no	—
284	#102	—	4	4	no	—
285	#97	—	2	4	no	—
286	#95	—	2	1	no	—
287	#95	—	2	1	no	—
288	#95	—	2	1	no	—
289	#95	—	2	1	no	—
290	C₃₁	#29148	1	1	yes	characteristic, direct product factor, Sylow
291	C₂⁴	—	2	1	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	4	no	—
296	C₂×Q₈	—	4	4	no	—
297	C₂×Q₈	—	4	4	no	—
298	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	1	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	—	2	4	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	4	2	no	—
309	C₂²×C₄	—	4	2	no	—
310	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
317	C₂×C₈	—	2	4	no	—
318	C₂×C₈	—	2	4	no	—
319	C₂×C₈	—	2	4	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	—	4	4	no	—
325	C₄⋊C₄	—	4	4	no	—
326	C₄⋊C₄	—	4	4	no	—
327	C₄⋊C₄	—	4	4	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₄⋊C₄	—	4	4	no	—
330	C₂²⋊C₄	—	2	4	no	—
331	C₂²⋊C₄	—	2	4	no	—
332	C₂²⋊C₄	—	4	2	no	—
333	C₂²⋊C₄	—	4	2	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	2	no	—
339	C₄²	—	2	2	no	—
340	C₄²	—	4	2	no	—
341	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
342	C₄²	#41×C₃₁	1	2	yes	—
343	C₄²	#41×C₃₁	1	4	yes	—
344	C₂³	—	2	2	no	—
345	C₂³	—	4	1	no	—
346	C₂³	—	4	1	no	—
347	C₂³	—	4	1	no	—
348	C₂³	—	4	1	no	—
349	C₂³	—	4	1	no	—
350	C₂³	—	4	1	no	—
351	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
352	Q₈	—	2	4	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	8	8	no	—
355	D₄	—	2	4	no	—
356	D₄	—	4	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	4	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	4	no	—
375	C₂×C₄	—	4	4	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	8	4	no	—
380	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
381	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
382	C₂×C₄	C₃₁×#124	1	2	yes	—
383	C₂×C₄	C₃₁×#124	1	2	yes	—
384	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
385	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
386	C₂×C₄	C₃₁×#137	1	2	yes	—
387	C₂×C₄	C₃₁×#137	1	2	yes	—
388	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
389	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
390	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
392	C₈	—	4	4	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	4	2	no	—
396	C₂²	—	8	1	no	—
397	C₂²	—	8	1	no	—
398	C₂²	—	8	1	no	—
399	C₂²	—	8	1	no	—
400	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
401	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
402	C₂²	C₃₁×#535	1	1	yes	characteristic
403	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
404	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
405	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
406	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
407	C₄	—	2	2	no	—
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	2	2	no	—
411	C₄	—	2	2	no	—
412	C₄	—	4	2	no	—
413	C₄	—	8	1	no	—
414	C₄	—	8	1	no	—
415	C₄	—	8	4	no	—
416	C₄	C₃₁×#489	1	4	yes	—
417	C₄	C₃₁×#554	1	2	yes	—
418	C₄	C₃₁×#555	1	2	yes	—
419	C₂	—	8	2	no	—
420	C₂	C₃₁×#2563	1	1	yes	characteristic
421	C₂	C₃₁×#3079	1	1	yes	characteristic
422	C₂	C₃₁×#3079	1	1	yes	characteristic
423	C₂	C₂×C₃₁×#489	1	1	yes	characteristic
424	C₂	C₃₁×#3141	1	1	yes	characteristic
425	C₂	C₃₁×#3217	1	1	yes	characteristic
426	C₂	C₃₁×#3218	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2868	C₂	semidirect
3. C₃₁×#2024	C₂	semidirect
6. C₃₁×#1836	C₂	semidirect
7. C₃₁×#1803	C₂	semidirect
9. C₃₁×#1767	C₂	semidirect
44. C₃₁×#124	Q₈	semidirect
130. #29148	C₃₁	direct (N × H)
180. #2868	C₂×C₃₁	semidirect
182. #2024	C₂×C₃₁	semidirect
185. #1836	C₂×C₃₁	semidirect
186. #1803	C₂×C₃₁	semidirect
188. #1767	C₂×C₃₁	semidirect
257. #124	Q₈×C₃₁	semidirect
290. C₃₁	#29148	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×#2868	2
C₄×C₃₁×#124	2
C₃₁×#2024	2
C₃₁×#2023	2
C₃₁×#1959	2
C₃₁×#1836	2
C₃₁×#1803	2
C₃₁×#1802	2
C₃₁×#1767	2
#29148	31

Subgroups — C31×#29148 #21485907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29148 #21485907