Subgroups — C₃₁×#23716 #21480475

Order 7936.

816 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2089	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2078	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2063	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2051	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2045	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2018	C₂	1	1	yes	maximal, characteristic
10	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#100	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#442	—	2	2	no	—
17	C₃₁×#440	—	2	2	no	—
18	C₄×SD₁₆×C₃₁	—	2	2	no	—
19	C₄×D₈×C₃₁	—	2	2	no	—
20	C₄×D₈×C₃₁	—	2	2	no	—
21	C₄×D₈×C₃₁	—	2	2	no	—
22	C₃₁×#433	—	2	2	no	—
23	C₃₁×#431	—	2	2	no	—
24	C₂×C₃₁×#108	C₂²	1	2	yes	—
25	C₃₁×#428	C₂²	1	2	yes	—
26	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
27	C₃₁×#424	C₂²	1	1	yes	characteristic
28	C₃₁×#424	C₂²	1	1	yes	characteristic
29	C₃₁×#418	C₂²	1	2	yes	—
30	C₃₁×#417	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×#100	C₂²	1	2	yes	—
33	C₂×C₃₁×#100	C₂²	1	2	yes	—
34	C₃₁×#408	C₂²	1	1	yes	characteristic
35	C₃₁×#408	C₂²	1	1	yes	characteristic
36	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#343	C₂²	1	2	yes	—
40	C₃₁×#343	C₂²	1	2	yes	—
41	C₃₁×#341	C₂²	1	1	yes	characteristic
42	C₃₁×#341	C₂²	1	1	yes	characteristic
43	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂³×C₄×C₃₁	—	2	2	no	—
46	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
47	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	4	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
55	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
59	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
60	C₄×D₄×C₃₁	—	2	1	no	—
61	C₄×D₄×C₃₁	—	2	1	no	—
62	C₄×D₄×C₃₁	—	2	2	no	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
67	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
68	C₃₁×#117	—	2	1	no	—
69	C₃₁×#117	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#108	—	2	4	no	—
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#107	—	2	2	no	—
77	C₃₁×#106	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×#103	—	2	2	no	—
81	C₃₁×#102	—	2	2	no	—
82	C₃₁×#102	—	2	4	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#100	—	2	4	no	—
86	C₃₁×#100	—	2	4	no	—
87	C₃₁×C₂²⋊C₈	—	2	2	no	—
88	C₃₁×C₂²⋊C₈	—	2	2	no	—
89	C₃₁×#97	—	2	1	no	—
90	C₃₁×#97	—	2	2	no	—
91	C₄×C₈×C₃₁	—	2	1	no	—
92	C₄×C₈×C₃₁	—	2	2	no	—
93	C₄×C₈×C₃₁	D₄	1	2	yes	—
94	C₄×C₈×C₃₁	D₄	1	2	yes	—
95	C₂⁴×C₃₁	—	2	2	no	—
96	#41×C₃₁	—	2	2	no	—
97	#41×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×D₄×C₃₁	#41	1	4	yes	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	—	4	2	no	—
113	C₂²×C₄×C₃₁	—	8	4	no	—
114	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
117	SD₁₆×C₃₁	—	2	8	no	—
118	D₈×C₃₁	—	2	8	no	—
119	D₈×C₃₁	—	2	8	no	—
120	M₁₆×C₃₁	—	2	2	no	—
121	M₁₆×C₃₁	—	2	2	no	—
122	M₁₆×C₃₁	—	2	2	no	—
123	M₁₆×C₃₁	—	2	2	no	—
124	M₁₆×C₃₁	—	2	2	no	—
125	M₁₆×C₃₁	#41	1	4	yes	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²⋊C₄×C₃₁	—	2	1	no	—
141	C₂²⋊C₄×C₃₁	—	2	1	no	—
142	C₂²⋊C₄×C₃₁	—	2	1	no	—
143	C₂²⋊C₄×C₃₁	—	4	2	no	—
144	C₂²⋊C₄×C₃₁	—	4	2	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
152	#23716	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	4	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	8	2	no	—
158	C₂³×C₃₁	—	8	2	no	—
159	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
160	Q₈×C₃₁	—	2	2	no	—
161	D₄×C₃₁	—	2	4	no	—
162	D₄×C₃₁	—	2	2	no	—
163	D₄×C₃₁	—	2	4	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	8	4	no	—
178	C₂×C₄×C₃₁	—	8	4	no	—
179	C₂×C₄×C₃₁	—	8	2	no	—
180	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
181	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
182	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
183	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
187	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
188	C₈×C₃₁	—	2	2	no	—
189	C₈×C₃₁	—	2	4	no	—
190	C₈×C₃₁	—	2	2	no	—
191	C₈×C₃₁	—	4	1	no	—
192	C₈×C₃₁	—	4	2	no	—
193	C₈×C₃₁	—	4	1	no	—
194	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	2	yes	complemented
195	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#2089	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#2078	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#2063	C₂×C₃₁	1	2	yes	complemented
200	#2051	C₂×C₃₁	1	2	yes	complemented
201	#2045	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#2018	C₂×C₃₁	1	1	yes	characteristic
203	C₄×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₄×#100	C₂×C₃₁	1	2	yes	complemented
205	C₂²×C₃₁	—	2	1	no	—
206	C₂²×C₃₁	—	2	1	no	—
207	C₂²×C₃₁	—	4	1	no	—
208	C₂²×C₃₁	—	4	4	no	—
209	C₂²×C₃₁	—	4	1	no	—
210	C₂²×C₃₁	—	4	1	no	—
211	C₂²×C₃₁	—	8	4	no	—
212	C₂²×C₃₁	—	8	4	no	—
213	C₂²×C₃₁	—	8	2	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
216	C₂²×C₃₁	#529	1	2	yes	—
217	C₄×C₃₁	—	2	1	no	—
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	2	1	no	—
220	C₄×C₃₁	—	4	1	no	—
221	C₄×C₃₁	—	4	1	no	—
222	C₄×C₃₁	—	4	1	no	—
223	C₄×C₃₁	—	8	2	no	—
224	C₄×C₃₁	#472	1	2	yes	—
225	C₄×C₃₁	#525	1	1	yes	characteristic
226	C₄×C₃₁	#525	1	2	yes	—
227	C₄×C₃₁	#532	1	1	yes	characteristic
228	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
229	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
232	#442	—	2	2	no	—
233	#440	—	2	2	no	—
234	C₄×SD₁₆	—	2	2	no	—
235	C₄×D₈	—	2	2	no	—
236	C₄×D₈	—	2	2	no	—
237	C₄×D₈	—	2	2	no	—
238	#433	—	2	2	no	—
239	#431	—	2	2	no	—
240	C₂×#108	C₂²×C₃₁	1	2	yes	—
241	#428	C₂²×C₃₁	1	2	yes	—
242	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
243	#424	C₂²×C₃₁	1	1	yes	characteristic
244	#424	C₂²×C₃₁	1	1	yes	characteristic
245	#418	C₂²×C₃₁	1	2	yes	—
246	#417	C₂²×C₃₁	1	2	yes	—
247	C₂×#102	C₂²×C₃₁	1	2	yes	—
248	C₂×#100	C₂²×C₃₁	1	2	yes	—
249	C₂×#100	C₂²×C₃₁	1	2	yes	—
250	#408	C₂²×C₃₁	1	1	yes	characteristic
251	#408	C₂²×C₃₁	1	1	yes	characteristic
252	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
253	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
254	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
255	#343	C₂²×C₃₁	1	2	yes	—
256	#343	C₂²×C₃₁	1	2	yes	—
257	#341	C₂²×C₃₁	1	1	yes	characteristic
258	#341	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₃₁	—	2	2	no	—
260	C₂×C₃₁	—	8	1	no	—
261	C₂×C₃₁	—	8	4	no	—
262	C₂×C₃₁	#2392	1	1	yes	characteristic
263	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
264	C₂×C₃₁	#3067	1	1	yes	characteristic
265	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
266	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
267	C₂³×C₄	—	2	2	no	—
268	AGL(1,ℤ/8)	—	2	4	no	—
269	AGL(1,ℤ/8)	—	2	4	no	—
270	C₂×SD₁₆	—	2	2	no	—
271	C₂×D₈	—	2	2	no	—
272	C₂×D₈	—	2	4	no	—
273	C₂×D₈	—	2	2	no	—
274	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
275	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
276	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
277	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
278	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
281	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
282	C₄×D₄	—	2	1	no	—
283	C₄×D₄	—	2	1	no	—
284	C₄×D₄	—	2	2	no	—
285	C₄×D₄	—	2	2	no	—
286	C₄×D₄	—	2	2	no	—
287	C₄×D₄	—	2	2	no	—
288	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
289	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
290	#117	—	2	1	no	—
291	#117	C₂³×C₃₁	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
296	#108	—	2	4	no	—
297	#107	—	2	2	no	—
298	#107	—	2	2	no	—
299	#106	—	2	2	no	—
300	C₄⋊C₈	—	2	2	no	—
301	C₄⋊C₈	—	2	2	no	—
302	#103	—	2	2	no	—
303	#102	—	2	2	no	—
304	#102	—	2	4	no	—
305	#102	—	2	2	no	—
306	#102	—	2	2	no	—
307	#100	—	2	4	no	—
308	#100	—	2	4	no	—
309	C₂²⋊C₈	—	2	2	no	—
310	C₂²⋊C₈	—	2	2	no	—
311	#97	—	2	1	no	—
312	#97	—	2	2	no	—
313	C₄×C₈	—	2	1	no	—
314	C₄×C₈	—	2	2	no	—
315	C₄×C₈	D₄×C₃₁	1	2	yes	—
316	C₄×C₈	D₄×C₃₁	1	2	yes	—
317	C₃₁	#23716	1	1	yes	characteristic, direct product factor, Sylow
318	C₂⁴	—	2	2	no	—
319	#41	—	2	2	no	—
320	#41	—	2	2	no	—
321	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂×D₄	#41×C₃₁	1	4	yes	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	4	2	no	—
335	C₂²×C₄	—	4	2	no	—
336	C₂²×C₄	—	8	4	no	—
337	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
339	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
340	SD₁₆	—	2	8	no	—
341	D₈	—	2	8	no	—
342	D₈	—	2	8	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	2	2	no	—
345	M₁₆	—	2	2	no	—
346	M₁₆	—	2	2	no	—
347	M₁₆	—	2	2	no	—
348	M₁₆	#41×C₃₁	1	4	yes	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
356	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₂²⋊C₄	—	2	1	no	—
364	C₂²⋊C₄	—	2	1	no	—
365	C₂²⋊C₄	—	2	1	no	—
366	C₂²⋊C₄	—	4	2	no	—
367	C₂²⋊C₄	—	4	2	no	—
368	C₄²	—	2	1	no	—
369	C₄²	—	2	1	no	—
370	C₄²	—	2	1	no	—
371	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₂³	—	2	1	no	—
376	C₂³	—	2	4	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	4	2	no	—
379	C₂³	—	8	2	no	—
380	C₂³	—	8	2	no	—
381	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
382	Q₈	—	2	2	no	—
383	D₄	—	2	4	no	—
384	D₄	—	2	2	no	—
385	D₄	—	2	4	no	—
386	D₄	—	4	2	no	—
387	D₄	—	4	2	no	—
388	D₄	—	4	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	8	4	no	—
400	C₂×C₄	—	8	4	no	—
401	C₂×C₄	—	8	2	no	—
402	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
403	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
404	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
405	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
406	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
409	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
410	C₈	—	2	2	no	—
411	C₈	—	2	4	no	—
412	C₈	—	2	2	no	—
413	C₈	—	4	1	no	—
414	C₈	—	4	2	no	—
415	C₈	—	4	1	no	—
416	C₂²	—	2	1	no	—
417	C₂²	—	2	1	no	—
418	C₂²	—	4	1	no	—
419	C₂²	—	4	4	no	—
420	C₂²	—	4	1	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	8	4	no	—
423	C₂²	—	8	4	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	2	no	—
426	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
427	C₂²	C₃₁×#529	1	2	yes	—
428	C₄	—	2	1	no	—
429	C₄	—	2	1	no	—
430	C₄	—	2	1	no	—
431	C₄	—	4	1	no	—
432	C₄	—	4	1	no	—
433	C₄	—	4	1	no	—
434	C₄	—	8	2	no	—
435	C₄	C₃₁×#472	1	2	yes	—
436	C₄	C₃₁×#525	1	1	yes	characteristic
437	C₄	C₃₁×#525	1	2	yes	—
438	C₄	C₃₁×#532	1	1	yes	characteristic
439	C₂	—	2	2	no	—
440	C₂	—	8	1	no	—
441	C₂	—	8	4	no	—
442	C₂	C₃₁×#2392	1	1	yes	characteristic
443	C₂	C₂×C₃₁×#472	1	1	yes	characteristic
444	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#121
prime radical	C₃₁×#2018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₄×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2922	C₂	semidirect
4. C₃₁×#2089	C₂	semidirect
5. C₃₁×#2078	C₂	semidirect
6. C₃₁×#2063	C₂	semidirect
7. C₃₁×#2051	C₂	semidirect
8. C₃₁×#2045	C₂	semidirect
10. C₄×C₃₁×#108	C₂	semidirect
11. C₄×C₃₁×#100	C₂	semidirect
152. #23716	C₃₁	direct (N × H)
194. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
195. C₂×C₄×D₈	C₂×C₃₁	semidirect
196. #2922	C₂×C₃₁	semidirect
197. #2089	C₂×C₃₁	semidirect
198. #2078	C₂×C₃₁	semidirect
199. #2063	C₂×C₃₁	semidirect
200. #2051	C₂×C₃₁	semidirect
201. #2045	C₂×C₃₁	semidirect
203. C₄×#108	C₂×C₃₁	semidirect
204. C₄×#100	C₂×C₃₁	semidirect
317. C₃₁	#23716	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×AGL(1,ℤ/8)	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2922	2
C₃₁×#2089	2
C₃₁×#2078	2
C₃₁×#2063	2
C₃₁×#2051	2
C₃₁×#2045	2
C₃₁×#2018	2
C₄×C₃₁×#108	2
C₄×C₃₁×#100	2
#23716	31

Subgroups — C31×#23716 #21480475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23716 #21480475