Subgroups — C₃₁×#19410 #21476169

Order 7936.

758 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3035	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1682	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1681	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1668	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1667	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1571	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
13	C₂×C₃₁×#135	—	2	1	no	—
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#493	C₂²	1	2	yes	—
18	C₃₁×#486	C₂²	1	2	yes	—
19	C₃₁×#464	—	2	1	no	—
20	C₃₁×#463	—	2	1	no	—
21	C₃₁×#462	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#461	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#460	—	2	1	no	—
24	C₃₁×#460	C₂²	1	2	yes	—
25	C₃₁×#460	C₂²	1	1	yes	characteristic, complemented
26	C₃₁×#459	—	2	1	no	—
27	C₃₁×#459	—	2	1	no	—
28	C₃₁×#459	—	2	1	no	—
29	C₃₁×#459	C₂²	1	2	yes	—
30	C₃₁×#459	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#447	—	2	2	no	—
32	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
33	C₈×D₄×C₃₁	—	2	2	no	—
34	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
35	C₃₁×#431	—	2	1	no	—
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#423	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×#418	—	2	1	no	—
42	C₃₁×#416	—	2	1	no	—
43	C₃₁×#335	C₂²	1	2	yes	complemented
44	C₃₁×#335	C₂²	1	2	yes	complemented
45	C₃₁×#332	C₂²	1	2	yes	complemented
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₃₁×#331	C₂²	1	2	yes	—
48	C₃₁×#331	C₂²	1	2	yes	complemented
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	—	2	1	no	—
54	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
55	C₃₁×#135	—	4	4	no	—
56	C₂×Q₁₆×C₃₁	—	2	1	no	—
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	—	2	1	no	—
59	C₂×SD₁₆×C₃₁	—	2	1	no	—
60	C₂×D₈×C₃₁	—	2	2	no	—
61	C₂×D₈×C₃₁	—	2	1	no	—
62	C₂×D₈×C₃₁	—	2	2	no	—
63	C₂×D₈×C₃₁	—	2	2	no	—
64	C₂×M₁₆×C₃₁	—	2	1	no	—
65	C₂²×C₈×C₃₁	—	2	1	no	—
66	C₂²×C₈×C₃₁	—	2	1	no	—
67	C₂²×C₈×C₃₁	—	2	2	no	—
68	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₃₁×#127	—	2	1	no	—
70	C₃₁×#127	—	2	1	no	—
71	C₃₁×#127	—	2	1	no	—
72	C₃₁×#127	C₂³	1	2	yes	—
73	C₃₁×#127	C₂³	1	1	yes	characteristic
74	C₃₁×#124	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₃₁×#121	—	2	1	no	—
77	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
78	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	1	no	—
81	C₄×D₄×C₃₁	—	2	1	no	—
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
84	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#117	—	2	1	no	—
86	C₃₁×#117	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₃₁	—	2	1	no	—
88	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
89	C₃₁×C₄⋊C₈	—	2	2	no	—
90	C₃₁×C₄⋊C₈	—	2	1	no	—
91	C₃₁×C₄⋊C₈	—	2	1	no	—
92	C₃₁×C₄⋊C₈	—	2	1	no	—
93	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
94	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
95	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
96	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
97	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
98	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
99	C₃₁×#103	—	2	1	no	—
100	C₃₁×#103	—	2	1	no	—
101	C₃₁×#102	—	2	2	no	—
102	C₃₁×#102	—	2	2	no	—
103	C₃₁×#102	—	2	1	no	—
104	C₃₁×#102	—	2	1	no	—
105	C₃₁×#102	—	2	2	no	—
106	C₃₁×#102	—	2	2	no	—
107	C₃₁×C₂²⋊C₈	—	2	2	no	—
108	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
109	C₄×C₈×C₃₁	—	2	2	no	—
110	C₄×C₈×C₃₁	C₂³	1	2	yes	—
111	C₂⁴×C₃₁	—	4	1	no	—
112	#41×C₃₁	—	2	2	no	—
113	#41×C₃₁	—	4	2	no	—
114	#41×C₃₁	—	4	2	no	—
115	#41×C₃₁	D₈	1	4	yes	complemented
116	C₂×Q₈×C₃₁	—	2	1	no	—
117	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	2	1	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	4	1	no	—
126	C₂×D₄×C₃₁	—	4	1	no	—
127	C₂×D₄×C₃₁	—	4	1	no	—
128	C₂×D₄×C₃₁	—	4	1	no	—
129	C₂×D₄×C₃₁	—	4	2	no	—
130	C₂×D₄×C₃₁	—	4	2	no	—
131	C₂×D₄×C₃₁	—	8	1	no	—
132	C₂×D₄×C₃₁	—	8	1	no	—
133	C₂×D₄×C₃₁	—	8	2	no	—
134	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	2	1	no	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	4	1	no	—
139	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
142	Q₁₆×C₃₁	—	4	2	no	—
143	SD₁₆×C₃₁	—	4	2	no	—
144	SD₁₆×C₃₁	—	4	4	no	—
145	SD₁₆×C₃₁	—	4	2	no	—
146	D₈×C₃₁	—	4	2	no	—
147	D₈×C₃₁	—	4	4	no	—
148	D₈×C₃₁	—	8	2	no	—
149	D₈×C₃₁	—	8	2	no	—
150	M₁₆×C₃₁	—	4	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	1	no	—
155	C₂×C₈×C₃₁	—	4	2	no	—
156	C₂×C₈×C₃₁	—	8	2	no	—
157	C₂×C₈×C₃₁	—	8	1	no	—
158	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²⋊C₄×C₃₁	—	2	1	no	—
167	C₂²⋊C₄×C₃₁	—	2	1	no	—
168	C₂²⋊C₄×C₃₁	—	2	1	no	—
169	C₂²⋊C₄×C₃₁	—	4	1	no	—
170	C₂²⋊C₄×C₃₁	—	4	1	no	—
171	C₄²×C₃₁	—	2	1	no	—
172	C₄²×C₃₁	—	2	1	no	—
173	C₄²×C₃₁	—	2	1	no	—
174	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₃₁	#41	1	1	yes	characteristic
177	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
178	#19410	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
179	C₂³×C₃₁	—	2	1	no	—
180	C₂³×C₃₁	—	4	1	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	8	2	no	—
183	C₂³×C₃₁	—	8	2	no	—
184	C₂³×C₃₁	—	8	2	no	—
185	C₂³×C₃₁	#121	1	1	yes	characteristic
186	Q₈×C₃₁	—	2	1	no	—
187	Q₈×C₃₁	—	4	1	no	—
188	Q₈×C₃₁	—	4	1	no	—
189	Q₈×C₃₁	C₂×D₈	1	2	yes	—
190	D₄×C₃₁	—	2	4	no	—
191	D₄×C₃₁	—	2	1	no	—
192	D₄×C₃₁	—	4	2	no	—
193	D₄×C₃₁	—	4	1	no	—
194	D₄×C₃₁	—	4	2	no	—
195	D₄×C₃₁	—	4	1	no	—
196	D₄×C₃₁	—	4	1	no	—
197	D₄×C₃₁	—	4	2	no	—
198	D₄×C₃₁	—	4	1	no	—
199	D₄×C₃₁	—	4	2	no	—
200	D₄×C₃₁	—	8	1	no	—
201	D₄×C₃₁	—	8	2	no	—
202	D₄×C₃₁	—	8	2	no	—
203	D₄×C₃₁	—	8	2	no	—
204	D₄×C₃₁	—	8	2	no	—
205	D₄×C₃₁	—	8	2	no	—
206	D₄×C₃₁	C₂×D₈	1	2	yes	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	8	1	no	—
218	C₂×C₄×C₃₁	—	8	2	no	—
219	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
220	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
221	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
223	C₂×C₄×C₃₁	#135	1	2	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
227	C₈×C₃₁	—	2	2	no	—
228	C₈×C₃₁	—	2	2	no	—
229	C₈×C₃₁	—	4	1	no	—
230	C₈×C₃₁	—	4	1	no	—
231	C₈×C₃₁	—	4	2	no	—
232	C₈×C₃₁	—	4	2	no	—
233	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#3038	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#3035	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#2988	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1682	C₂×C₃₁	1	2	yes	complemented
238	#1681	C₂×C₃₁	1	2	yes	complemented
239	#1668	C₂×C₃₁	1	2	yes	complemented
240	#1667	C₂×C₃₁	1	2	yes	complemented
241	#1571	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂×#331	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	1	no	—
246	C₂²×C₃₁	—	2	2	no	—
247	C₂²×C₃₁	—	4	1	no	—
248	C₂²×C₃₁	—	4	1	no	—
249	C₂²×C₃₁	—	8	2	no	—
250	C₂²×C₃₁	—	8	1	no	—
251	C₂²×C₃₁	—	8	1	no	—
252	C₂²×C₃₁	—	8	2	no	—
253	C₂²×C₃₁	—	8	2	no	—
254	C₂²×C₃₁	—	8	1	no	—
255	C₂²×C₃₁	—	16	4	no	—
256	C₂²×C₃₁	#466	1	2	yes	—
257	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	2	2	no	—
260	C₄×C₃₁	—	2	1	no	—
261	C₄×C₃₁	—	2	1	no	—
262	C₄×C₃₁	—	4	1	no	—
263	C₄×C₃₁	—	4	1	no	—
264	C₄×C₃₁	—	8	1	no	—
265	C₄×C₃₁	#466	1	2	yes	—
266	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
267	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
268	C₄×C₃₁	#576	1	1	yes	characteristic
269	C₄×C₃₁	#577	1	1	yes	characteristic
270	C₂×AGL(1,ℤ/8)	—	2	1	no	—
271	C₂×#135	—	2	1	no	—
272	#532	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
274	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
275	#493	C₂²×C₃₁	1	2	yes	—
276	#486	C₂²×C₃₁	1	2	yes	—
277	#464	—	2	1	no	—
278	#463	—	2	1	no	—
279	#462	C₂²×C₃₁	1	1	yes	characteristic, complemented
280	#461	C₂²×C₃₁	1	1	yes	characteristic, complemented
281	#460	—	2	1	no	—
282	#460	C₂²×C₃₁	1	2	yes	—
283	#460	C₂²×C₃₁	1	1	yes	characteristic, complemented
284	#459	—	2	1	no	—
285	#459	—	2	1	no	—
286	#459	—	2	1	no	—
287	#459	C₂²×C₃₁	1	2	yes	—
288	#459	C₂²×C₃₁	1	1	yes	characteristic, complemented
289	#447	—	2	2	no	—
290	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
291	C₈×D₄	—	2	2	no	—
292	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
293	#431	—	2	1	no	—
294	#431	C₂²×C₃₁	1	1	yes	characteristic
295	#423	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₄⋊C₈	—	2	1	no	—
297	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
299	#418	—	2	1	no	—
300	#416	—	2	1	no	—
301	#335	C₂²×C₃₁	1	2	yes	complemented
302	#335	C₂²×C₃₁	1	2	yes	complemented
303	#332	C₂²×C₃₁	1	2	yes	complemented
304	#331	C₂²×C₃₁	1	2	yes	—
305	#331	C₂²×C₃₁	1	2	yes	—
306	#331	C₂²×C₃₁	1	2	yes	complemented
307	C₂×C₃₁	—	2	2	no	—
308	C₂×C₃₁	—	4	2	no	—
309	C₂×C₃₁	—	16	1	no	—
310	C₂×C₃₁	—	16	2	no	—
311	C₂×C₃₁	#3045	1	1	yes	characteristic
312	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
313	C₂×C₃₁	#3068	1	1	yes	characteristic
314	C₂×#41	—	2	1	no	—
315	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	—	2	1	no	—
318	C₂²×D₄	—	2	1	no	—
319	AGL(1,ℤ/8)	—	4	4	no	—
320	#135	—	4	4	no	—
321	C₂×Q₁₆	—	2	1	no	—
322	C₂×SD₁₆	—	2	1	no	—
323	C₂×SD₁₆	—	2	2	no	—
324	C₂×SD₁₆	—	2	1	no	—
325	C₂×D₈	—	2	1	no	—
326	C₂×D₈	—	2	2	no	—
327	C₂×D₈	—	2	2	no	—
328	C₂×D₈	—	2	2	no	—
329	C₂×M₁₆	—	2	1	no	—
330	C₂²×C₈	—	2	1	no	—
331	C₂²×C₈	—	2	1	no	—
332	C₂²×C₈	—	2	2	no	—
333	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
334	#127	—	2	1	no	—
335	#127	—	2	1	no	—
336	#127	—	2	1	no	—
337	#127	C₂³×C₃₁	1	2	yes	—
338	#127	C₂³×C₃₁	1	1	yes	characteristic
339	#124	—	2	1	no	—
340	#121	—	2	1	no	—
341	#121	—	2	1	no	—
342	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
343	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₄×D₄	—	2	1	no	—
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	—	2	1	no	—
347	C₄×D₄	—	2	1	no	—
348	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
349	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
350	#117	—	2	1	no	—
351	#117	D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₄²	—	2	1	no	—
353	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
354	C₄⋊C₈	—	2	2	no	—
355	C₄⋊C₈	—	2	1	no	—
356	C₄⋊C₈	—	2	1	no	—
357	C₄⋊C₈	—	2	1	no	—
358	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
359	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
360	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
361	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
362	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
363	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
364	#103	—	2	1	no	—
365	#103	—	2	1	no	—
366	#102	—	2	2	no	—
367	#102	—	2	2	no	—
368	#102	—	2	1	no	—
369	#102	—	2	1	no	—
370	#102	—	2	2	no	—
371	#102	—	2	2	no	—
372	C₂²⋊C₈	—	2	2	no	—
373	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
374	C₄×C₈	—	2	2	no	—
375	C₄×C₈	C₂³×C₃₁	1	2	yes	—
376	C₃₁	#19410	1	1	yes	characteristic, direct product factor, Sylow
377	C₂⁴	—	4	1	no	—
378	#41	—	2	2	no	—
379	#41	—	4	2	no	—
380	#41	—	4	2	no	—
381	#41	D₈×C₃₁	1	4	yes	complemented
382	C₂×Q₈	—	2	1	no	—
383	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	4	1	no	—
392	C₂×D₄	—	4	1	no	—
393	C₂×D₄	—	4	1	no	—
394	C₂×D₄	—	4	1	no	—
395	C₂×D₄	—	4	2	no	—
396	C₂×D₄	—	4	2	no	—
397	C₂×D₄	—	8	1	no	—
398	C₂×D₄	—	8	1	no	—
399	C₂×D₄	—	8	2	no	—
400	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	4	1	no	—
405	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
407	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
408	Q₁₆	—	4	2	no	—
409	SD₁₆	—	4	2	no	—
410	SD₁₆	—	4	4	no	—
411	SD₁₆	—	4	2	no	—
412	D₈	—	4	2	no	—
413	D₈	—	4	4	no	—
414	D₈	—	8	2	no	—
415	D₈	—	8	2	no	—
416	M₁₆	—	4	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	4	2	no	—
422	C₂×C₈	—	8	2	no	—
423	C₂×C₈	—	8	1	no	—
424	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
425	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₂²⋊C₄	—	2	1	no	—
433	C₂²⋊C₄	—	2	1	no	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	4	1	no	—
436	C₂²⋊C₄	—	4	1	no	—
437	C₄²	—	2	1	no	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	1	no	—
440	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₄²	#41×C₃₁	1	1	yes	characteristic
443	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
444	C₂³	—	2	1	no	—
445	C₂³	—	4	1	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	8	2	no	—
448	C₂³	—	8	2	no	—
449	C₂³	—	8	2	no	—
450	C₂³	C₃₁×#121	1	1	yes	characteristic
451	Q₈	—	2	1	no	—
452	Q₈	—	4	1	no	—
453	Q₈	—	4	1	no	—
454	Q₈	C₂×D₈×C₃₁	1	2	yes	—
455	D₄	—	2	4	no	—
456	D₄	—	2	1	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	1	no	—
459	D₄	—	4	2	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	1	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	1	no	—
464	D₄	—	4	2	no	—
465	D₄	—	8	1	no	—
466	D₄	—	8	2	no	—
467	D₄	—	8	2	no	—
468	D₄	—	8	2	no	—
469	D₄	—	8	2	no	—
470	D₄	—	8	2	no	—
471	D₄	C₂×D₈×C₃₁	1	2	yes	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	8	1	no	—
483	C₂×C₄	—	8	2	no	—
484	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
485	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
486	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
487	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
488	C₂×C₄	C₃₁×#135	1	2	yes	—
489	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
490	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
491	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
492	C₈	—	2	2	no	—
493	C₈	—	2	2	no	—
494	C₈	—	4	1	no	—
495	C₈	—	4	1	no	—
496	C₈	—	4	2	no	—
497	C₈	—	4	2	no	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	8	2	no	—
504	C₂²	—	8	1	no	—
505	C₂²	—	8	1	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	1	no	—
509	C₂²	—	16	4	no	—
510	C₂²	C₃₁×#466	1	2	yes	—
511	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
512	C₄	—	2	1	no	—
513	C₄	—	2	2	no	—
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	4	1	no	—
517	C₄	—	4	1	no	—
518	C₄	—	8	1	no	—
519	C₄	C₃₁×#466	1	2	yes	—
520	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
521	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
522	C₄	C₃₁×#576	1	1	yes	characteristic
523	C₄	C₃₁×#577	1	1	yes	characteristic
524	C₂	—	2	2	no	—
525	C₂	—	4	2	no	—
526	C₂	—	16	2	no	—
527	C₂	—	16	1	no	—
528	C₂	C₃₁×#3045	1	1	yes	characteristic
529	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
530	C₂	C₃₁×#3068	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×#3205	C₂	semidirect
2. C₃₁×#3038	C₂	semidirect
3. C₃₁×#3035	C₂	semidirect
4. C₃₁×#2988	C₂	semidirect
5. C₃₁×#1682	C₂	semidirect
6. C₃₁×#1681	C₂	semidirect
7. C₃₁×#1668	C₂	semidirect
8. C₃₁×#1667	C₂	semidirect
9. C₃₁×#1571	C₂	semidirect
10. C₃₁×#1547	C₂	semidirect
11. C₂×C₃₁×#331	C₂	semidirect
21. C₃₁×#462	C₂²	semidirect
22. C₃₁×#461	C₂²	semidirect
25. C₃₁×#460	C₂²	semidirect
30. C₃₁×#459	C₂²	semidirect
32. C₈×Q₈×C₃₁	C₂²	semidirect
34. C₈×D₄×C₃₁	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#335	C₂²	semidirect
45. C₃₁×#332	C₂²	semidirect
48. C₃₁×#331	C₂²	semidirect
115. #41×C₃₁	D₈	semidirect
178. #19410	C₃₁	direct (N × H)
233. #3205	C₂×C₃₁	semidirect
234. #3038	C₂×C₃₁	semidirect
235. #3035	C₂×C₃₁	semidirect
236. #2988	C₂×C₃₁	semidirect
237. #1682	C₂×C₃₁	semidirect
238. #1681	C₂×C₃₁	semidirect
239. #1668	C₂×C₃₁	semidirect
240. #1667	C₂×C₃₁	semidirect
241. #1571	C₂×C₃₁	semidirect
242. #1547	C₂×C₃₁	semidirect
243. C₂×#331	C₂×C₃₁	semidirect
279. #462	C₂²×C₃₁	semidirect
280. #461	C₂²×C₃₁	semidirect
283. #460	C₂²×C₃₁	semidirect
288. #459	C₂²×C₃₁	semidirect
290. C₈×Q₈	C₂²×C₃₁	semidirect
292. C₈×D₄	C₂²×C₃₁	semidirect
301. #335	C₂²×C₃₁	semidirect
302. #335	C₂²×C₃₁	semidirect
303. #332	C₂²×C₃₁	semidirect
306. #331	C₂²×C₃₁	semidirect
376. C₃₁	#19410	direct (N × H)
381. #41	D₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3205	2
C₃₁×#3038	2
C₃₁×#3035	2
C₃₁×#2988	2
C₃₁×#1682	2
C₃₁×#1681	2
C₃₁×#1668	2
C₃₁×#1667	2
C₃₁×#1571	2
C₃₁×#1547	2
C₂×C₃₁×#331	2
#19410	31

Subgroups — C31×#19410 #21476169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19410 #21476169