Subgroups — C₃₁×#17841 #21474600

Order 7936.

592 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2991	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2990	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1572	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1570	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1569	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1565	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#446	—	2	1	no	—
14	C₃₁×#446	C₄	1	1	yes	characteristic
15	C₃₁×#446	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₃₁	—	2	1	no	—
17	C₃₁×#436	—	2	1	no	—
18	C₃₁×#436	—	2	2	no	—
19	C₃₁×#436	C₄	1	1	yes	characteristic
20	C₃₁×#436	C₄	1	1	yes	characteristic
21	C₃₁×#435	—	2	2	no	—
22	C₃₁×#435	C₄	1	2	yes	—
23	C₃₁×#435	C₄	1	2	yes	—
24	C₈×D₄×C₃₁	—	2	1	no	—
25	C₃₁×#432	C₂²	1	2	yes	—
26	C₃₁×#431	C₂²	1	1	yes	characteristic
27	C₃₁×#431	C₂²	1	1	yes	characteristic
28	C₃₁×#431	C₂²	1	2	yes	—
29	C₃₁×#431	C₂²	1	2	yes	—
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₃₁×#423	C₂²	1	1	yes	characteristic
32	C₃₁×#423	C₂²	1	1	yes	characteristic
33	C₃₁×#423	C₂²	1	2	yes	—
34	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
35	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#335	C₂²	1	1	yes	characteristic
39	C₃₁×#335	C₂²	1	2	yes	—
40	C₃₁×#335	C₂²	1	2	yes	—
41	C₃₁×#334	C₂²	1	1	yes	characteristic
42	C₃₁×#334	C₂²	1	1	yes	characteristic
43	C₃₁×#334	C₂²	1	2	yes	—
44	C₃₁×#334	C₂²	1	2	yes	—
45	C₃₁×#334	C₂²	1	2	yes	—
46	C₃₁×#334	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	Q₈	1	1	yes	characteristic
48	C₂×M₁₆×C₃₁	—	2	1	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	1	no	—
51	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
58	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
59	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
61	C₄×D₄×C₃₁	D₄	1	2	yes	—
62	C₃₁×#117	Q₈	1	1	yes	characteristic
63	C₃₁×#117	Q₈	1	2	yes	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
66	C₃₁×C₄⋊C₈	—	2	2	no	—
67	C₃₁×C₄⋊C₈	—	2	1	no	—
68	C₃₁×C₄⋊C₈	—	2	1	no	—
69	C₃₁×C₄⋊C₈	—	2	2	no	—
70	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
71	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
72	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
73	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
74	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
75	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
76	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
77	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₂²⋊C₈	—	2	1	no	—
81	C₃₁×C₂²⋊C₈	—	2	2	no	—
82	C₃₁×C₂²⋊C₈	—	2	1	no	—
83	C₃₁×C₂²⋊C₈	—	2	2	no	—
84	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
85	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
86	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
87	C₃₁×#97	—	2	2	no	—
88	C₃₁×#97	C₂×C₄	1	2	yes	—
89	C₃₁×#97	C₂×C₄	1	2	yes	—
90	C₄×C₈×C₃₁	—	2	1	no	—
91	C₄×C₈×C₃₁	—	2	1	no	—
92	C₄×C₈×C₃₁	—	2	2	no	—
93	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₃₁	C₂³	1	2	yes	—
98	#41×C₃₁	—	4	2	no	—
99	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
100	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
101	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
108	M₁₆×C₃₁	—	2	4	no	—
109	M₁₆×C₃₁	—	4	2	no	—
110	M₁₆×C₃₁	—	8	1	no	—
111	M₁₆×C₃₁	—	8	2	no	—
112	M₁₆×C₃₁	—	8	1	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	4	1	no	—
121	C₂×C₈×C₃₁	—	8	2	no	—
122	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
129	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
130	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
131	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
132	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
133	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
134	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
135	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
137	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₃₁	C₂×D₄	1	2	yes	—
139	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
140	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
141	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
142	#17841	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
145	Q₈×C₃₁	—	2	2	no	—
146	D₄×C₃₁	—	2	2	no	—
147	D₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
163	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
167	C₈×C₃₁	—	2	1	no	—
168	C₈×C₃₁	—	2	2	no	—
169	C₈×C₃₁	—	2	1	no	—
170	C₈×C₃₁	—	2	1	no	—
171	C₈×C₃₁	—	2	2	no	—
172	C₈×C₃₁	—	4	1	no	—
173	C₈×C₃₁	—	4	1	no	—
174	C₈×C₃₁	—	4	1	no	—
175	C₈×C₃₁	—	4	2	no	—
176	C₈×C₃₁	—	4	1	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	#131	1	2	yes	—
179	#2991	C₂×C₃₁	1	1	yes	characteristic
180	#2990	C₂×C₃₁	1	1	yes	characteristic
181	#2965	C₂×C₃₁	1	1	yes	characteristic
182	#1574	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#1574	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1572	C₂×C₃₁	1	2	yes	complemented
185	#1570	C₂×C₃₁	1	2	yes	complemented
186	#1569	C₂×C₃₁	1	2	yes	complemented
187	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#1565	C₂×C₃₁	1	2	yes	complemented
189	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
190	C₂²×C₃₁	—	2	1	no	—
191	C₂²×C₃₁	—	2	2	no	—
192	C₂²×C₃₁	—	2	2	no	—
193	C₂²×C₃₁	—	2	1	no	—
194	C₂²×C₃₁	—	2	1	no	—
195	C₂²×C₃₁	—	2	2	no	—
196	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₄×C₃₁	—	2	1	no	—
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	2	2	no	—
200	C₄×C₃₁	—	2	1	no	—
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	#430	1	2	yes	—
204	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
205	C₄×C₃₁	#568	1	1	yes	characteristic
206	C₄×C₃₁	#577	1	1	yes	characteristic
207	C₄×C₃₁	#577	1	1	yes	characteristic
208	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
209	#446	—	2	1	no	—
210	#446	C₄×C₃₁	1	1	yes	characteristic
211	#446	C₄×C₃₁	1	1	yes	characteristic
212	C₈×Q₈	—	2	1	no	—
213	#436	—	2	1	no	—
214	#436	—	2	2	no	—
215	#436	C₄×C₃₁	1	1	yes	characteristic
216	#436	C₄×C₃₁	1	1	yes	characteristic
217	#435	—	2	2	no	—
218	#435	C₄×C₃₁	1	2	yes	—
219	#435	C₄×C₃₁	1	2	yes	—
220	C₈×D₄	—	2	1	no	—
221	#432	C₂²×C₃₁	1	2	yes	—
222	#431	C₂²×C₃₁	1	1	yes	characteristic
223	#431	C₂²×C₃₁	1	1	yes	characteristic
224	#431	C₂²×C₃₁	1	2	yes	—
225	#431	C₂²×C₃₁	1	2	yes	—
226	#431	C₂²×C₃₁	1	1	yes	characteristic
227	#423	C₂²×C₃₁	1	1	yes	characteristic
228	#423	C₂²×C₃₁	1	1	yes	characteristic
229	#423	C₂²×C₃₁	1	2	yes	—
230	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
231	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
232	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
233	#335	C₂²×C₃₁	1	1	yes	characteristic
234	#335	C₂²×C₃₁	1	1	yes	characteristic
235	#335	C₂²×C₃₁	1	2	yes	—
236	#335	C₂²×C₃₁	1	2	yes	—
237	#334	C₂²×C₃₁	1	1	yes	characteristic
238	#334	C₂²×C₃₁	1	1	yes	characteristic
239	#334	C₂²×C₃₁	1	2	yes	—
240	#334	C₂²×C₃₁	1	2	yes	—
241	#334	C₂²×C₃₁	1	2	yes	—
242	#334	C₂²×C₃₁	1	2	yes	—
243	C₂×C₃₁	—	4	1	no	—
244	C₂×C₃₁	—	4	2	no	—
245	C₂×C₃₁	#2906	1	1	yes	characteristic
246	C₂×C₃₁	#2912	1	1	yes	characteristic
247	C₂×C₃₁	C₂×#430	1	1	yes	characteristic
248	C₂×#41	Q₈×C₃₁	1	1	yes	characteristic
249	C₂×M₁₆	—	2	1	no	—
250	C₂×M₁₆	—	2	2	no	—
251	C₂×M₁₆	—	2	1	no	—
252	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
253	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
254	C₂²×C₈	—	2	2	no	—
255	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
257	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
258	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
259	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
260	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
261	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
262	C₄×D₄	D₄×C₃₁	1	2	yes	—
263	#117	Q₈×C₃₁	1	1	yes	characteristic
264	#117	Q₈×C₃₁	1	2	yes	—
265	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
267	C₄⋊C₈	—	2	2	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
272	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
273	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
274	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
275	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
277	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
278	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
279	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	2	2	no	—
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
286	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
287	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
288	#97	—	2	2	no	—
289	#97	C₂×C₄×C₃₁	1	2	yes	—
290	#97	C₂×C₄×C₃₁	1	2	yes	—
291	C₄×C₈	—	2	1	no	—
292	C₄×C₈	—	2	1	no	—
293	C₄×C₈	—	2	2	no	—
294	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
295	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
296	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₄×C₈	C₂³×C₃₁	1	2	yes	—
299	C₃₁	#17841	1	1	yes	characteristic, direct product factor, Sylow
300	#41	—	4	2	no	—
301	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
302	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
303	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
304	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
305	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
306	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
310	M₁₆	—	2	4	no	—
311	M₁₆	—	4	2	no	—
312	M₁₆	—	8	1	no	—
313	M₁₆	—	8	2	no	—
314	M₁₆	—	8	1	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	4	1	no	—
323	C₂×C₈	—	8	2	no	—
324	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
325	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
326	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
327	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
328	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
329	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
330	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
331	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
332	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
333	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
334	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
335	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
336	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
337	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
338	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
339	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₄²	C₂×D₄×C₃₁	1	2	yes	—
341	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
342	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
343	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
344	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
345	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
346	Q₈	—	2	2	no	—
347	D₄	—	2	2	no	—
348	D₄	—	4	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
360	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
361	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
362	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
364	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
365	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
367	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	2	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	2	no	—
373	C₈	—	4	1	no	—
374	C₈	—	4	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	4	1	no	—
378	C₈	—	4	2	no	—
379	C₈	C₃₁×#131	1	2	yes	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	2	no	—
386	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	2	2	no	—
393	C₄	C₃₁×#430	1	2	yes	—
394	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
395	C₄	C₃₁×#568	1	1	yes	characteristic
396	C₄	C₃₁×#577	1	1	yes	characteristic
397	C₄	C₃₁×#577	1	1	yes	characteristic
398	C₂	—	4	1	no	—
399	C₂	—	4	2	no	—
400	C₂	C₃₁×#2906	1	1	yes	characteristic
401	C₂	C₃₁×#2912	1	1	yes	characteristic
402	C₂	C₂×C₃₁×#430	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	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₃₁	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1574	C₂	semidirect
5. C₃₁×#1574	C₂	semidirect
6. C₃₁×#1572	C₂	semidirect
7. C₃₁×#1570	C₂	semidirect
8. C₃₁×#1569	C₂	semidirect
9. C₃₁×#1566	C₂	semidirect
10. C₃₁×#1565	C₂	semidirect
11. C₃₁×#1564	C₂	semidirect
142. #17841	C₃₁	direct (N × H)
182. #1574	C₂×C₃₁	semidirect
183. #1574	C₂×C₃₁	semidirect
184. #1572	C₂×C₃₁	semidirect
185. #1570	C₂×C₃₁	semidirect
186. #1569	C₂×C₃₁	semidirect
187. #1566	C₂×C₃₁	semidirect
188. #1565	C₂×C₃₁	semidirect
189. #1564	C₂×C₃₁	semidirect
299. C₃₁	#17841	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×#2991	2
C₃₁×#2990	2
C₃₁×#2965	2
C₃₁×#1574	2
C₃₁×#1572	2
C₃₁×#1570	2
C₃₁×#1569	2
C₃₁×#1566	2
C₃₁×#1565	2
C₃₁×#1564	2
#17841	31

Subgroups — C31×#17841 #21474600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17841 #21474600