Subgroups — C₃₁×#19355 #21476114

Order 7936.

678 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3251	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1690	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1689	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1688	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1687	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1666	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1665	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1557	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₂×C₃₁×#135	—	2	1	no	—
14	C₃₁×#532	C₂²	1	2	yes	—
15	C₃₁×#529	C₂²	1	1	yes	characteristic
16	C₃₁×#495	—	2	1	no	—
17	C₃₁×#495	—	2	1	no	—
18	C₃₁×#494	C₂²	1	1	yes	characteristic
19	C₃₁×#493	C₂²	1	1	yes	characteristic
20	C₃₁×#492	C₂²	1	2	yes	—
21	C₃₁×#478	—	2	2	no	—
22	C₃₁×#475	C₂²	1	2	yes	—
23	C₃₁×#474	C₂²	1	2	yes	—
24	C₃₁×#469	—	2	2	no	—
25	C₃₁×#468	—	2	2	no	—
26	C₃₁×#463	—	2	2	no	—
27	C₃₁×#460	C₂²	1	2	yes	—
28	C₃₁×#459	C₂²	1	2	yes	—
29	C₃₁×#446	C₂²	1	1	yes	characteristic
30	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#436	C₂²	1	1	yes	characteristic
32	C₃₁×#435	—	2	1	no	—
33	C₈×D₄×C₃₁	—	2	1	no	—
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#433	—	2	1	no	—
36	C₃₁×#432	C₂²	1	1	yes	characteristic
37	C₃₁×#423	C₂²	1	2	yes	—
38	C₃₁×#405	—	2	1	no	—
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#334	C₂²	1	2	yes	—
41	C₃₁×#334	C₂²	1	1	yes	characteristic
42	C₃₁×#334	C₂²	1	1	yes	characteristic
43	C₃₁×#332	C₂²	1	2	yes	—
44	C₃₁×#332	C₂²	1	2	yes	—
45	C₃₁×#331	C₂²	1	2	yes	—
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	—	2	2	no	—
48	C₃₁×#135	—	4	4	no	—
49	C₃₁×#135	—	4	4	no	—
50	C₂×Q₁₆×C₃₁	—	2	1	no	—
51	C₂×Q₁₆×C₃₁	—	2	1	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×D₈×C₃₁	—	2	2	no	—
56	C₂×D₈×C₃₁	—	2	1	no	—
57	C₂×D₈×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂×M₁₆×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
64	C₃₁×#127	C₂³	1	2	yes	—
65	C₃₁×#126	—	2	1	no	—
66	C₃₁×#125	D₄	1	1	yes	characteristic
67	C₃₁×#124	—	2	2	no	—
68	C₃₁×#124	D₄	1	1	yes	characteristic
69	C₃₁×#123	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
71	C₃₁×#121	—	2	2	no	—
72	C₃₁×#121	—	2	2	no	—
73	C₃₁×#121	D₄	1	1	yes	characteristic
74	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	2	no	—
77	C₄×D₄×C₃₁	—	2	2	no	—
78	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#117	—	2	1	no	—
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#107	—	2	2	no	—
82	C₃₁×#106	—	2	2	no	—
83	C₃₁×C₄⋊C₈	—	2	1	no	—
84	C₃₁×C₄⋊C₈	—	2	1	no	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
87	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
88	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
89	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×C₂²⋊C₈	—	2	1	no	—
95	C₃₁×C₂²⋊C₈	—	2	1	no	—
96	C₃₁×C₂²⋊C₈	—	2	1	no	—
97	C₃₁×#97	—	2	1	no	—
98	C₃₁×#97	—	2	1	no	—
99	C₄×C₈×C₃₁	—	2	1	no	—
100	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
101	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
102	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
103	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
104	#41×C₃₁	—	4	4	no	—
105	#41×C₃₁	—	4	4	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	4	2	no	—
113	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	4	2	no	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
119	Q₁₆×C₃₁	—	4	2	no	—
120	Q₁₆×C₃₁	—	4	2	no	—
121	SD₁₆×C₃₁	—	4	4	no	—
122	SD₁₆×C₃₁	—	4	4	no	—
123	SD₁₆×C₃₁	—	8	2	no	—
124	D₈×C₃₁	—	4	2	no	—
125	D₈×C₃₁	—	4	2	no	—
126	D₈×C₃₁	—	8	2	no	—
127	M₁₆×C₃₁	—	4	2	no	—
128	M₁₆×C₃₁	—	8	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	4	2	no	—
136	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₂²⋊C₄×C₃₁	—	2	1	no	—
149	C₂²⋊C₄×C₃₁	—	2	1	no	—
150	C₂²⋊C₄×C₃₁	—	2	1	no	—
151	C₂²⋊C₄×C₃₁	—	2	1	no	—
152	C₂²⋊C₄×C₃₁	—	2	1	no	—
153	C₂²⋊C₄×C₃₁	—	4	2	no	—
154	C₂²⋊C₄×C₃₁	—	4	2	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₃₁	#41	1	1	yes	characteristic
159	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
160	#19355	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	1	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	#121	1	1	yes	characteristic
164	Q₈×C₃₁	—	2	2	no	—
165	Q₈×C₃₁	—	4	2	no	—
166	Q₈×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	2	2	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	4	2	no	—
170	D₄×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	8	2	no	—
174	D₄×C₃₁	—	8	4	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	8	4	no	—
186	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
187	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
188	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
192	C₈×C₃₁	—	2	1	no	—
193	C₈×C₃₁	—	2	1	no	—
194	C₈×C₃₁	—	4	1	no	—
195	C₈×C₃₁	—	4	1	no	—
196	C₈×C₃₁	—	4	2	no	—
197	C₈×C₃₁	—	4	2	no	—
198	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
199	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
200	#3251	C₂×C₃₁	1	2	yes	complemented
201	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#2966	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1690	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1689	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1688	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1687	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#1666	C₂×C₃₁	1	2	yes	complemented
208	#1665	C₂×C₃₁	1	2	yes	complemented
209	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1557	C₂×C₃₁	1	2	yes	complemented
211	C₂²×C₃₁	—	2	1	no	—
212	C₂²×C₃₁	—	2	1	no	—
213	C₂²×C₃₁	—	2	1	no	—
214	C₂²×C₃₁	—	4	1	no	—
215	C₂²×C₃₁	—	4	1	no	—
216	C₂²×C₃₁	—	4	1	no	—
217	C₂²×C₃₁	—	8	2	no	—
218	C₂²×C₃₁	—	8	2	no	—
219	C₂²×C₃₁	—	8	2	no	—
220	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
221	C₄×C₃₁	—	2	1	no	—
222	C₄×C₃₁	—	2	1	no	—
223	C₄×C₃₁	—	2	1	no	—
224	C₄×C₃₁	—	2	1	no	—
225	C₄×C₃₁	—	4	1	no	—
226	C₄×C₃₁	—	4	1	no	—
227	C₄×C₃₁	—	4	1	no	—
228	C₄×C₃₁	—	8	2	no	—
229	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
230	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₄×C₃₁	#577	1	2	yes	—
232	C₂×#135	—	2	1	no	—
233	C₂×#135	—	2	1	no	—
234	#532	C₂²×C₃₁	1	2	yes	—
235	#529	C₂²×C₃₁	1	1	yes	characteristic
236	#495	—	2	1	no	—
237	#495	—	2	1	no	—
238	#494	C₂²×C₃₁	1	1	yes	characteristic
239	#493	C₂²×C₃₁	1	1	yes	characteristic
240	#492	C₂²×C₃₁	1	2	yes	—
241	#478	—	2	2	no	—
242	#475	C₂²×C₃₁	1	2	yes	—
243	#474	C₂²×C₃₁	1	2	yes	—
244	#469	—	2	2	no	—
245	#468	—	2	2	no	—
246	#463	—	2	2	no	—
247	#460	C₂²×C₃₁	1	2	yes	—
248	#459	C₂²×C₃₁	1	2	yes	—
249	#446	C₂²×C₃₁	1	1	yes	characteristic
250	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
251	#436	C₂²×C₃₁	1	1	yes	characteristic
252	#435	—	2	1	no	—
253	C₈×D₄	—	2	1	no	—
254	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
255	#433	—	2	1	no	—
256	#432	C₂²×C₃₁	1	1	yes	characteristic
257	#423	C₂²×C₃₁	1	2	yes	—
258	#405	—	2	1	no	—
259	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	#334	C₂²×C₃₁	1	2	yes	—
261	#334	C₂²×C₃₁	1	1	yes	characteristic
262	#334	C₂²×C₃₁	1	1	yes	characteristic
263	#332	C₂²×C₃₁	1	2	yes	—
264	#332	C₂²×C₃₁	1	2	yes	—
265	#331	C₂²×C₃₁	1	2	yes	—
266	#331	C₂²×C₃₁	1	2	yes	—
267	C₂×C₃₁	—	4	1	no	—
268	C₂×C₃₁	—	8	1	no	—
269	C₂×C₃₁	—	16	2	no	—
270	C₂×C₃₁	#3038	1	1	yes	characteristic
271	C₂×C₃₁	#3060	1	1	yes	characteristic
272	C₂×C₃₁	#3067	1	1	yes	characteristic
273	C₂×#41	—	2	2	no	—
274	#135	—	4	4	no	—
275	#135	—	4	4	no	—
276	C₂×Q₁₆	—	2	1	no	—
277	C₂×Q₁₆	—	2	1	no	—
278	C₂×SD₁₆	—	2	2	no	—
279	C₂×SD₁₆	—	2	2	no	—
280	C₂×SD₁₆	—	2	2	no	—
281	C₂×D₈	—	2	2	no	—
282	C₂×D₈	—	2	1	no	—
283	C₂×D₈	—	2	1	no	—
284	C₂×M₁₆	—	2	2	no	—
285	C₂×M₁₆	—	2	1	no	—
286	C₂²×C₈	—	2	1	no	—
287	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
288	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
289	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
290	#127	C₂³×C₃₁	1	2	yes	—
291	#126	—	2	1	no	—
292	#125	D₄×C₃₁	1	1	yes	characteristic
293	#124	—	2	2	no	—
294	#124	D₄×C₃₁	1	1	yes	characteristic
295	#123	—	2	1	no	—
296	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
297	#121	—	2	2	no	—
298	#121	—	2	2	no	—
299	#121	D₄×C₃₁	1	1	yes	characteristic
300	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
301	C₄×D₄	—	2	1	no	—
302	C₄×D₄	—	2	2	no	—
303	C₄×D₄	—	2	2	no	—
304	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
305	#117	—	2	1	no	—
306	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
307	#107	—	2	2	no	—
308	#106	—	2	2	no	—
309	C₄⋊C₈	—	2	1	no	—
310	C₄⋊C₈	—	2	1	no	—
311	C₄⋊C₈	—	2	2	no	—
312	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
315	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
316	#103	—	2	2	no	—
317	#102	—	2	2	no	—
318	#102	—	2	2	no	—
319	#102	—	2	2	no	—
320	C₂²⋊C₈	—	2	1	no	—
321	C₂²⋊C₈	—	2	1	no	—
322	C₂²⋊C₈	—	2	1	no	—
323	#97	—	2	1	no	—
324	#97	—	2	1	no	—
325	C₄×C₈	—	2	1	no	—
326	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
327	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
328	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
329	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
330	C₃₁	#19355	1	1	yes	characteristic, direct product factor, Sylow
331	#41	—	4	4	no	—
332	#41	—	4	4	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
346	Q₁₆	—	4	2	no	—
347	Q₁₆	—	4	2	no	—
348	SD₁₆	—	4	4	no	—
349	SD₁₆	—	4	4	no	—
350	SD₁₆	—	8	2	no	—
351	D₈	—	4	2	no	—
352	D₈	—	4	2	no	—
353	D₈	—	8	2	no	—
354	M₁₆	—	4	2	no	—
355	M₁₆	—	8	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	4	2	no	—
363	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₂²⋊C₄	—	2	1	no	—
376	C₂²⋊C₄	—	2	1	no	—
377	C₂²⋊C₄	—	2	1	no	—
378	C₂²⋊C₄	—	2	1	no	—
379	C₂²⋊C₄	—	2	1	no	—
380	C₂²⋊C₄	—	4	2	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₄²	—	2	1	no	—
383	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₄²	#41×C₃₁	1	1	yes	characteristic
386	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
387	C₂³	—	2	1	no	—
388	C₂³	—	4	2	no	—
389	C₂³	C₃₁×#121	1	1	yes	characteristic
390	Q₈	—	2	2	no	—
391	Q₈	—	4	2	no	—
392	Q₈	—	4	2	no	—
393	D₄	—	2	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	8	2	no	—
400	D₄	—	8	4	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	8	4	no	—
412	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
413	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
414	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
415	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
418	C₈	—	2	1	no	—
419	C₈	—	2	1	no	—
420	C₈	—	4	1	no	—
421	C₈	—	4	1	no	—
422	C₈	—	4	2	no	—
423	C₈	—	4	2	no	—
424	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
425	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
426	C₂²	—	2	1	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	2	1	no	—
429	C₂²	—	4	1	no	—
430	C₂²	—	4	1	no	—
431	C₂²	—	4	1	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	2	no	—
434	C₂²	—	8	2	no	—
435	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	2	1	no	—
439	C₄	—	2	1	no	—
440	C₄	—	4	1	no	—
441	C₄	—	4	1	no	—
442	C₄	—	4	1	no	—
443	C₄	—	8	2	no	—
444	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
445	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
446	C₄	C₃₁×#577	1	2	yes	—
447	C₂	—	4	1	no	—
448	C₂	—	8	1	no	—
449	C₂	—	16	2	no	—
450	C₂	C₃₁×#3038	1	1	yes	characteristic
451	C₂	C₃₁×#3060	1	1	yes	characteristic
452	C₂	C₃₁×#3067	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3251	C₂	semidirect
2. C₃₁×#3147	C₂	semidirect
3. C₃₁×#2966	C₂	semidirect
4. C₃₁×#1690	C₂	semidirect
5. C₃₁×#1689	C₂	semidirect
6. C₃₁×#1688	C₂	semidirect
7. C₃₁×#1687	C₂	semidirect
8. C₃₁×#1666	C₂	semidirect
9. C₃₁×#1665	C₂	semidirect
10. C₃₁×#1566	C₂	semidirect
11. C₃₁×#1557	C₂	semidirect
160. #19355	C₃₁	direct (N × H)
200. #3251	C₂×C₃₁	semidirect
201. #3147	C₂×C₃₁	semidirect
202. #2966	C₂×C₃₁	semidirect
203. #1690	C₂×C₃₁	semidirect
204. #1689	C₂×C₃₁	semidirect
205. #1688	C₂×C₃₁	semidirect
206. #1687	C₂×C₃₁	semidirect
207. #1666	C₂×C₃₁	semidirect
208. #1665	C₂×C₃₁	semidirect
209. #1566	C₂×C₃₁	semidirect
210. #1557	C₂×C₃₁	semidirect
330. C₃₁	#19355	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×#3251	2
C₃₁×#3147	2
C₃₁×#2966	2
C₃₁×#1690	2
C₃₁×#1689	2
C₃₁×#1688	2
C₃₁×#1687	2
C₃₁×#1666	2
C₃₁×#1665	2
C₃₁×#1566	2
C₃₁×#1557	2
#19355	31

Subgroups — C31×#19355 #21476114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19355 #21476114