Subgroups — C₃₁×#17907 #21474666

Order 7936.

766 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3100	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1586	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1580	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1480	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#131	—	2	1	no	—
11	C₂³×C₈×C₃₁	—	2	1	no	—
12	C₃₁×#532	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#480	C₄	1	2	yes	—
15	C₃₁×#474	C₄	1	2	yes	—
16	C₃₁×#459	C₄	1	2	yes	—
17	C₃₁×#447	C₄	1	2	yes	—
18	C₄×D₈×C₃₁	C₂²	1	4	yes	—
19	C₃₁×#436	—	2	1	no	—
20	C₃₁×#436	C₂²	1	2	yes	complemented
21	C₈×D₄×C₃₁	—	2	1	no	—
22	C₈×D₄×C₃₁	—	2	1	no	—
23	C₈×D₄×C₃₁	—	2	1	no	—
24	C₈×D₄×C₃₁	C₂²	1	2	yes	—
25	C₈×D₄×C₃₁	C₂²	1	2	yes	—
26	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
27	C₃₁×#431	C₂²	1	1	yes	characteristic
28	C₃₁×#431	C₂²	1	1	yes	characteristic
29	C₃₁×#423	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₃₁×#408	—	2	1	no	—
32	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#334	C₂²	1	4	yes	complemented
35	C₃₁×D₄⋊C₈	C₂²	1	4	yes	—
36	C₃₁×D₄⋊C₈	C₂²	1	4	yes	complemented
37	C₃₁×#322	C₂²	1	2	yes	complemented
38	C₃₁×#322	C₂²	1	2	yes	complemented
39	C₂×#41×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	—	2	1	no	—
42	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
43	C₃₁×#131	—	4	4	no	—
44	C₂×M₁₆×C₃₁	—	2	1	no	—
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂²×C₈×C₃₁	—	2	1	no	—
47	C₂²×C₈×C₃₁	—	2	2	no	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	4	4	no	—
51	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
54	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
55	C₃₁×#124	—	2	1	no	—
56	C₃₁×#121	—	2	1	no	—
57	C₃₁×#121	C₂×C₄	1	2	yes	—
58	C₄×D₄×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	—	2	1	no	—
60	C₄×D₄×C₃₁	—	2	1	no	—
61	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₄×D₄×C₃₁	C₂³	1	2	yes	—
63	C₄×D₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#107	C₂×C₄	1	4	yes	—
68	C₃₁×C₄⋊C₈	—	2	1	no	—
69	C₃₁×C₄⋊C₈	—	2	1	no	—
70	C₃₁×C₄⋊C₈	—	2	1	no	—
71	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
72	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
73	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
74	C₃₁×#102	C₂×C₄	1	4	yes	—
75	C₃₁×#102	C₂×C₄	1	4	yes	—
76	C₃₁×C₂²⋊C₈	—	2	1	no	—
77	C₃₁×C₂²⋊C₈	—	2	2	no	—
78	C₃₁×C₂²⋊C₈	—	2	2	no	—
79	C₃₁×C₂²⋊C₈	—	2	2	no	—
80	C₃₁×C₂²⋊C₈	—	2	2	no	—
81	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
83	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
84	C₄×C₈×C₃₁	C₂³	1	2	yes	—
85	C₄×C₈×C₃₁	C₂³	1	2	yes	—
86	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂⁴×C₃₁	—	2	1	no	—
89	#41×C₃₁	—	4	2	no	—
90	#41×C₃₁	—	4	2	no	—
91	C₂×Q₈×C₃₁	—	2	1	no	—
92	C₂×D₄×C₃₁	—	2	4	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	1	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	—	2	1	no	—
97	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
98	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	4	2	no	—
104	C₂²×C₄×C₃₁	—	4	2	no	—
105	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
108	D₈×C₃₁	—	2	8	no	—
109	M₁₆×C₃₁	—	4	4	no	—
110	M₁₆×C₃₁	—	4	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	4	4	no	—
117	C₂×C₈×C₃₁	—	4	4	no	—
118	C₂×C₈×C₃₁	—	4	4	no	—
119	C₂×C₈×C₃₁	—	4	2	no	—
120	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
121	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
129	C₂²⋊C₄×C₃₁	—	2	2	no	—
130	C₂²⋊C₄×C₃₁	—	2	2	no	—
131	C₂²⋊C₄×C₃₁	—	2	2	no	—
132	C₂²⋊C₄×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
134	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₃₁	#41	1	1	yes	characteristic
136	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
137	#17907	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂³×C₃₁	—	2	4	no	—
139	C₂³×C₃₁	—	2	2	no	—
140	C₂³×C₃₁	—	2	2	no	—
141	C₂³×C₃₁	—	4	2	no	—
142	C₂³×C₃₁	—	4	2	no	—
143	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
144	Q₈×C₃₁	—	4	2	no	—
145	D₄×C₃₁	—	2	4	no	—
146	D₄×C₃₁	—	2	2	no	—
147	D₄×C₃₁	—	2	4	no	—
148	D₄×C₃₁	—	4	2	no	—
149	D₄×C₃₁	—	4	4	no	—
150	D₄×C₃₁	—	4	2	no	—
151	D₄×C₃₁	#131	1	4	yes	complemented
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	4	1	no	—
156	C₂×C₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	4	no	—
162	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
170	C₈×C₃₁	—	2	4	no	—
171	C₈×C₃₁	—	2	1	no	—
172	C₈×C₃₁	—	2	1	no	—
173	C₈×C₃₁	—	2	1	no	—
174	C₈×C₃₁	—	2	1	no	—
175	C₈×C₃₁	—	4	2	no	—
176	C₈×C₃₁	—	4	2	no	—
177	C₈×C₃₁	—	4	2	no	—
178	#3100	C₂×C₃₁	1	1	yes	characteristic
179	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
180	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#1586	C₂×C₃₁	1	4	yes	complemented
182	#1580	C₂×C₃₁	1	4	yes	complemented
183	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1486	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1480	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₂²×C₃₁	—	2	1	no	—
188	C₂²×C₃₁	—	2	1	no	—
189	C₂²×C₃₁	—	2	4	no	—
190	C₂²×C₃₁	—	4	8	no	—
191	C₂²×C₃₁	—	4	2	no	—
192	C₂²×C₃₁	—	4	2	no	—
193	C₂²×C₃₁	—	4	2	no	—
194	C₂²×C₃₁	—	4	2	no	—
195	C₂²×C₃₁	—	4	2	no	—
196	C₂²×C₃₁	#442	1	2	yes	—
197	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
198	C₄×C₃₁	—	2	1	no	—
199	C₄×C₃₁	—	2	1	no	—
200	C₄×C₃₁	—	2	1	no	—
201	C₄×C₃₁	—	4	2	no	—
202	C₄×C₃₁	—	4	2	no	—
203	C₄×C₃₁	#442	1	2	yes	—
204	C₄×C₃₁	C₂×#131	1	2	yes	—
205	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₂×#131	—	2	1	no	—
208	C₂³×C₈	—	2	1	no	—
209	#532	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
211	#480	C₄×C₃₁	1	2	yes	—
212	#474	C₄×C₃₁	1	2	yes	—
213	#459	C₄×C₃₁	1	2	yes	—
214	#447	C₄×C₃₁	1	2	yes	—
215	C₄×D₈	C₂²×C₃₁	1	4	yes	—
216	#436	—	2	1	no	—
217	#436	C₂²×C₃₁	1	2	yes	complemented
218	C₈×D₄	—	2	1	no	—
219	C₈×D₄	—	2	1	no	—
220	C₈×D₄	—	2	1	no	—
221	C₈×D₄	C₂²×C₃₁	1	2	yes	—
222	C₈×D₄	C₂²×C₃₁	1	2	yes	—
223	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
224	#431	C₂²×C₃₁	1	1	yes	characteristic
225	#431	C₂²×C₃₁	1	1	yes	characteristic
226	#423	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
228	#408	—	2	1	no	—
229	C₂×C₂²⋊C₈	—	2	1	no	—
230	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
231	#334	C₂²×C₃₁	1	4	yes	complemented
232	D₄⋊C₈	C₂²×C₃₁	1	4	yes	—
233	D₄⋊C₈	C₂²×C₃₁	1	4	yes	complemented
234	#322	C₂²×C₃₁	1	2	yes	complemented
235	#322	C₂²×C₃₁	1	2	yes	complemented
236	C₂×C₃₁	—	2	2	no	—
237	C₂×C₃₁	—	4	4	no	—
238	C₂×C₃₁	—	8	2	no	—
239	C₂×C₃₁	#2929	1	1	yes	characteristic
240	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
241	C₂×C₃₁	#2955	1	1	yes	characteristic
242	C₂×#41	—	2	1	no	—
243	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
244	C₂³×C₄	—	2	1	no	—
245	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
246	#131	—	4	4	no	—
247	C₂×M₁₆	—	2	1	no	—
248	C₂×M₁₆	—	2	2	no	—
249	C₂²×C₈	—	2	1	no	—
250	C₂²×C₈	—	2	2	no	—
251	C₂²×C₈	—	2	2	no	—
252	C₂²×C₈	—	2	2	no	—
253	C₂²×C₈	—	4	4	no	—
254	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
256	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
257	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
258	#124	—	2	1	no	—
259	#121	—	2	1	no	—
260	#121	C₂×C₄×C₃₁	1	2	yes	—
261	C₄×D₄	—	2	1	no	—
262	C₄×D₄	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
265	C₄×D₄	C₂³×C₃₁	1	2	yes	—
266	C₄×D₄	C₂³×C₃₁	1	2	yes	—
267	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
268	C₂×C₂²⋊C₄	—	2	1	no	—
269	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
270	#107	C₂×C₄×C₃₁	1	4	yes	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
275	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
276	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
277	#102	C₂×C₄×C₃₁	1	4	yes	—
278	#102	C₂×C₄×C₃₁	1	4	yes	—
279	C₂²⋊C₈	—	2	1	no	—
280	C₂²⋊C₈	—	2	2	no	—
281	C₂²⋊C₈	—	2	2	no	—
282	C₂²⋊C₈	—	2	2	no	—
283	C₂²⋊C₈	—	2	2	no	—
284	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
285	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
286	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
287	C₄×C₈	C₂³×C₃₁	1	2	yes	—
288	C₄×C₈	C₂³×C₃₁	1	2	yes	—
289	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₃₁	#17907	1	1	yes	characteristic, direct product factor, Sylow
292	C₂⁴	—	2	1	no	—
293	#41	—	4	2	no	—
294	#41	—	4	2	no	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×D₄	—	2	4	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂×D₄	—	2	1	no	—
300	C₂×D₄	—	2	1	no	—
301	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
302	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	4	2	no	—
309	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
312	D₈	—	2	8	no	—
313	M₁₆	—	4	4	no	—
314	M₁₆	—	4	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	4	4	no	—
321	C₂×C₈	—	4	4	no	—
322	C₂×C₈	—	4	4	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
325	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
332	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	2	2	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
338	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₄²	#41×C₃₁	1	1	yes	characteristic
340	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
341	C₂³	—	2	4	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	4	2	no	—
345	C₂³	—	4	2	no	—
346	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
347	Q₈	—	4	2	no	—
348	D₄	—	2	4	no	—
349	D₄	—	2	2	no	—
350	D₄	—	2	4	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	4	no	—
353	D₄	—	4	2	no	—
354	D₄	C₃₁×#131	1	4	yes	complemented
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	4	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
366	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
367	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
368	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
369	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
370	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
371	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
373	C₈	—	2	4	no	—
374	C₈	—	2	1	no	—
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	2	1	no	—
378	C₈	—	4	2	no	—
379	C₈	—	4	2	no	—
380	C₈	—	4	2	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	4	no	—
384	C₂²	—	4	8	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	C₃₁×#442	1	2	yes	—
391	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	4	2	no	—
396	C₄	—	4	2	no	—
397	C₄	C₃₁×#442	1	2	yes	—
398	C₄	C₂×C₃₁×#131	1	2	yes	—
399	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
400	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
401	C₂	—	2	2	no	—
402	C₂	—	4	4	no	—
403	C₂	—	8	2	no	—
404	C₂	C₃₁×#2929	1	1	yes	characteristic
405	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
406	C₂	C₃₁×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3100	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2931	C₂	semidirect
3. C₂×C₈×D₄×C₃₁	C₂	semidirect
4. C₃₁×#1586	C₂	semidirect
5. C₃₁×#1580	C₂	semidirect
6. C₃₁×#1570	C₂	semidirect
7. C₃₁×#1486	C₂	semidirect
8. C₃₁×#1480	C₂	semidirect
9. C₃₁×#1452	C₂	semidirect
20. C₃₁×#436	C₂²	semidirect
26. C₈×D₄×C₃₁	C₂²	semidirect
34. C₃₁×#334	C₂²	semidirect
36. C₃₁×D₄⋊C₈	C₂²	semidirect
37. C₃₁×#322	C₂²	semidirect
38. C₃₁×#322	C₂²	semidirect
137. #17907	C₃₁	direct (N × H)
151. D₄×C₃₁	#131	semidirect
179. #2931	C₂×C₃₁	semidirect
180. C₂×C₈×D₄	C₂×C₃₁	semidirect
181. #1586	C₂×C₃₁	semidirect
182. #1580	C₂×C₃₁	semidirect
183. #1570	C₂×C₃₁	semidirect
184. #1486	C₂×C₃₁	semidirect
185. #1480	C₂×C₃₁	semidirect
186. #1452	C₂×C₃₁	semidirect
217. #436	C₂²×C₃₁	semidirect
223. C₈×D₄	C₂²×C₃₁	semidirect
231. #334	C₂²×C₃₁	semidirect
233. D₄⋊C₈	C₂²×C₃₁	semidirect
234. #322	C₂²×C₃₁	semidirect
235. #322	C₂²×C₃₁	semidirect
291. C₃₁	#17907	direct (N × H)
354. D₄	C₃₁×#131	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3100	2
C₃₁×#2931	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#1586	2
C₃₁×#1580	2
C₃₁×#1570	2
C₃₁×#1486	2
C₃₁×#1480	2
C₃₁×#1452	2
#17907	31

Subgroups — C31×#17907 #21474666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17907 #21474666