Subgroups — C₃₁×#17971 #21474730

Order 7936.

600 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2981	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1592	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1590	C₂	1	2	yes	maximal, complemented
6	C₃₁×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1497	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1496	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#498	C₄	1	1	yes	characteristic
15	C₃₁×#492	C₄	1	1	yes	characteristic
16	C₃₁×#487	C₄	1	1	yes	characteristic
17	C₃₁×#486	C₄	1	1	yes	characteristic
18	C₃₁×#465	C₄	1	2	yes	—
19	C₃₁×#465	C₄	1	2	yes	—
20	C₃₁×#446	—	2	2	no	—
21	C₃₁×#446	C₂²	1	2	yes	—
22	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
23	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#436	C₂²	1	2	yes	—
26	C₃₁×#435	—	2	2	no	—
27	C₈×D₄×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#433	—	2	2	no	—
29	C₃₁×#432	C₂²	1	1	yes	characteristic
30	C₃₁×#432	C₂²	1	1	yes	characteristic
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#431	C₂²	1	1	yes	characteristic
33	C₃₁×#408	—	2	2	no	—
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#335	C₂²	1	2	yes	—
36	C₃₁×#334	C₂²	1	2	yes	—
37	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
39	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
40	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
41	C₃₁×#322	C₂²	1	1	yes	characteristic
42	C₃₁×#322	C₂²	1	1	yes	characteristic
43	C₈²×C₃₁	C₂²	1	1	yes	characteristic
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	1	no	—
46	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
54	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
55	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
56	C₃₁×#125	—	2	1	no	—
57	C₃₁×#124	—	2	1	no	—
58	C₃₁×C₂²⋊Q₈	—	2	1	no	—
59	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
60	C₃₁×#121	—	2	1	no	—
61	C₃₁×#121	C₂×C₄	1	2	yes	—
62	C₄×Q₈×C₃₁	—	2	1	no	—
63	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
64	C₄×D₄×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	C₂³	1	2	yes	—
66	C₃₁×#117	—	2	1	no	—
67	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#106	C₂×C₄	1	2	yes	—
69	C₃₁×#106	C₂×C₄	1	2	yes	—
70	C₃₁×C₄⋊C₈	—	2	2	no	—
71	C₃₁×C₄⋊C₈	—	2	2	no	—
72	C₃₁×C₄⋊C₈	—	2	1	no	—
73	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
74	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
75	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
76	C₃₁×#103	C₂×C₄	1	2	yes	—
77	C₃₁×#103	C₂×C₄	1	2	yes	—
78	C₃₁×#102	C₂×C₄	1	2	yes	—
79	C₃₁×#102	C₂×C₄	1	2	yes	—
80	C₃₁×C₂²⋊C₈	—	2	1	no	—
81	C₃₁×C₂²⋊C₈	—	2	1	no	—
82	C₃₁×C₂²⋊C₈	—	2	1	no	—
83	C₃₁×C₂²⋊C₈	—	2	2	no	—
84	C₃₁×C₂²⋊C₈	—	2	2	no	—
85	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
86	C₃₁×#97	—	2	2	no	—
87	C₃₁×#97	—	2	1	no	—
88	C₃₁×#97	—	2	1	no	—
89	C₄×C₈×C₃₁	—	2	2	no	—
90	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	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	1	yes	characteristic
98	#41×C₃₁	—	8	2	no	—
99	C₂×Q₈×C₃₁	—	2	1	no	—
100	C₂×Q₈×C₃₁	—	2	1	no	—
101	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
110	SD₁₆×C₃₁	—	2	4	no	—
111	SD₁₆×C₃₁	—	2	4	no	—
112	M₁₆×C₃₁	—	8	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	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	8	2	no	—
120	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	2	1	no	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
133	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
134	C₂²⋊C₄×C₃₁	—	2	2	no	—
135	C₂²⋊C₄×C₃₁	—	2	2	no	—
136	C₂²⋊C₄×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₃₁	#41	1	1	yes	characteristic
141	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
142	#17971	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	2	no	—
144	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
145	Q₈×C₃₁	—	2	2	no	—
146	Q₈×C₃₁	—	2	2	no	—
147	Q₈×C₃₁	—	8	1	no	—
148	Q₈×C₃₁	—	8	1	no	—
149	D₄×C₃₁	—	2	2	no	—
150	D₄×C₃₁	—	2	2	no	—
151	D₄×C₃₁	—	8	1	no	—
152	D₄×C₃₁	—	8	1	no	—
153	D₄×C₃₁	—	8	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	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
169	C₈×C₃₁	—	2	1	no	—
170	C₈×C₃₁	—	2	1	no	—
171	C₈×C₃₁	—	2	2	no	—
172	C₈×C₃₁	—	2	2	no	—
173	C₈×C₃₁	—	4	1	no	—
174	C₈×C₃₁	—	4	1	no	—
175	C₈×C₃₁	—	4	2	no	—
176	C₈×C₃₁	—	4	2	no	—
177	C₈×C₃₁	#131	1	2	yes	—
178	C₈×C₃₁	#131	1	2	yes	—
179	#3167	C₂×C₃₁	1	1	yes	characteristic
180	#2985	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#2981	C₂×C₃₁	1	1	yes	characteristic
182	#1592	C₂×C₃₁	1	2	yes	complemented
183	#1590	C₂×C₃₁	1	2	yes	complemented
184	SD₁₆⋊C₈	C₂×C₃₁	1	2	yes	complemented
185	C₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
186	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#1497	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#1496	C₂×C₃₁	1	1	yes	characteristic
189	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
190	C₂²×C₃₁	—	2	1	no	—
191	C₂²×C₃₁	—	2	1	no	—
192	C₂²×C₃₁	—	2	1	no	—
193	C₂²×C₃₁	—	4	2	no	—
194	C₂²×C₃₁	—	4	2	no	—
195	C₂²×C₃₁	—	4	2	no	—
196	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
197	C₄×C₃₁	—	2	1	no	—
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	—	4	2	no	—
204	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
205	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
206	C₄×C₃₁	#577	1	2	yes	—
207	#533	C₂²×C₃₁	1	1	yes	characteristic
208	#532	C₂²×C₃₁	1	1	yes	characteristic
209	#498	C₄×C₃₁	1	1	yes	characteristic
210	#492	C₄×C₃₁	1	1	yes	characteristic
211	#487	C₄×C₃₁	1	1	yes	characteristic
212	#486	C₄×C₃₁	1	1	yes	characteristic
213	#465	C₄×C₃₁	1	2	yes	—
214	#465	C₄×C₃₁	1	2	yes	—
215	#446	—	2	2	no	—
216	#446	C₂²×C₃₁	1	2	yes	—
217	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
218	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
219	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
220	#436	C₂²×C₃₁	1	2	yes	—
221	#435	—	2	2	no	—
222	C₈×D₄	C₂²×C₃₁	1	2	yes	—
223	#433	—	2	2	no	—
224	#432	C₂²×C₃₁	1	1	yes	characteristic
225	#432	C₂²×C₃₁	1	1	yes	characteristic
226	#431	C₂²×C₃₁	1	1	yes	characteristic
227	#431	C₂²×C₃₁	1	1	yes	characteristic
228	#408	—	2	2	no	—
229	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
230	#335	C₂²×C₃₁	1	2	yes	—
231	#334	C₂²×C₃₁	1	2	yes	—
232	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
233	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
234	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
235	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
236	#322	C₂²×C₃₁	1	1	yes	characteristic
237	#322	C₂²×C₃₁	1	1	yes	characteristic
238	C₈²	C₂²×C₃₁	1	1	yes	characteristic
239	C₈²	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×C₃₁	—	4	1	no	—
241	C₂×C₃₁	—	8	2	no	—
242	C₂×C₃₁	#2931	1	1	yes	characteristic
243	C₂×C₃₁	#2949	1	1	yes	characteristic
244	C₂×C₃₁	#2957	1	1	yes	characteristic
245	C₂×#41	—	2	1	no	—
246	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
247	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
248	C₂×M₁₆	—	2	2	no	—
249	C₂²×C₈	—	2	2	no	—
250	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
252	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
253	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
254	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
255	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
256	#125	—	2	1	no	—
257	#124	—	2	1	no	—
258	C₂²⋊Q₈	—	2	1	no	—
259	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
260	#121	—	2	1	no	—
261	#121	C₂×C₄×C₃₁	1	2	yes	—
262	C₄×Q₈	—	2	1	no	—
263	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	C₂³×C₃₁	1	2	yes	—
266	#117	—	2	1	no	—
267	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
268	#106	C₂×C₄×C₃₁	1	2	yes	—
269	#106	C₂×C₄×C₃₁	1	2	yes	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
274	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
275	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
276	#103	C₂×C₄×C₃₁	1	2	yes	—
277	#103	C₂×C₄×C₃₁	1	2	yes	—
278	#102	C₂×C₄×C₃₁	1	2	yes	—
279	#102	C₂×C₄×C₃₁	1	2	yes	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	2	2	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
286	#97	—	2	2	no	—
287	#97	—	2	1	no	—
288	#97	—	2	1	no	—
289	C₄×C₈	—	2	2	no	—
290	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
291	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
292	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
294	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₃₁	#17971	1	1	yes	characteristic, direct product factor, Sylow
299	#41	—	8	2	no	—
300	C₂×Q₈	—	2	1	no	—
301	C₂×Q₈	—	2	1	no	—
302	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
311	SD₁₆	—	2	4	no	—
312	SD₁₆	—	2	4	no	—
313	M₁₆	—	8	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	8	2	no	—
321	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
322	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
323	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
324	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
325	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
326	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
334	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
335	C₂²⋊C₄	—	2	2	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₄²	—	2	2	no	—
339	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
340	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₄²	#41×C₃₁	1	1	yes	characteristic
342	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
343	C₂³	—	2	2	no	—
344	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
345	Q₈	—	2	2	no	—
346	Q₈	—	2	2	no	—
347	Q₈	—	8	1	no	—
348	Q₈	—	8	1	no	—
349	D₄	—	2	2	no	—
350	D₄	—	2	2	no	—
351	D₄	—	8	1	no	—
352	D₄	—	8	1	no	—
353	D₄	—	8	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
364	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
365	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
366	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
367	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
368	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	2	no	—
372	C₈	—	2	2	no	—
373	C₈	—	4	1	no	—
374	C₈	—	4	1	no	—
375	C₈	—	4	2	no	—
376	C₈	—	4	2	no	—
377	C₈	C₃₁×#131	1	2	yes	—
378	C₈	C₃₁×#131	1	2	yes	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	4	2	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	2	no	—
393	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
394	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
395	C₄	C₃₁×#577	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	8	2	no	—
398	C₂	C₃₁×#2931	1	1	yes	characteristic
399	C₂	C₃₁×#2949	1	1	yes	characteristic
400	C₂	C₃₁×#2957	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×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2985	C₂	semidirect
4. C₃₁×#1592	C₂	semidirect
5. C₃₁×#1590	C₂	semidirect
6. C₃₁×SD₁₆⋊C₈	C₂	semidirect
7. C₈×SD₁₆×C₃₁	C₂	semidirect
8. C₃₁×#1568	C₂	semidirect
9. C₃₁×#1497	C₂	semidirect
11. C₃₁×#1450	C₂	semidirect
142. #17971	C₃₁	direct (N × H)
180. #2985	C₂×C₃₁	semidirect
182. #1592	C₂×C₃₁	semidirect
183. #1590	C₂×C₃₁	semidirect
184. SD₁₆⋊C₈	C₂×C₃₁	semidirect
185. C₈×SD₁₆	C₂×C₃₁	semidirect
186. #1568	C₂×C₃₁	semidirect
187. #1497	C₂×C₃₁	semidirect
189. #1450	C₂×C₃₁	semidirect
298. C₃₁	#17971	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×#3167	2
C₃₁×#2985	2
C₃₁×#2981	2
C₃₁×#1592	2
C₃₁×#1590	2
C₃₁×SD₁₆⋊C₈	2
C₈×SD₁₆×C₃₁	2
C₃₁×#1568	2
C₃₁×#1497	2
C₃₁×#1496	2
C₃₁×#1450	2
#17971	31

Subgroups — C31×#17971 #21474730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17971 #21474730