Subgroups — C₃₁×#17725 #21474484

Order 7936.

884 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1554	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1552	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1550	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1546	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1544	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1542	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#332	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#557	—	2	1	no	—
13	C₃₁×#557	C₄	1	2	yes	—
14	C₃₁×#556	—	2	1	no	—
15	C₃₁×#556	C₄	1	2	yes	—
16	C₃₁×#552	—	2	2	no	—
17	C₃₁×#552	C₄	1	4	yes	—
18	C₃₁×#541	—	2	2	no	—
19	C₃₁×#536	—	2	1	no	—
20	C₃₁×#533	C₂²	1	1	yes	characteristic
21	C₃₁×#533	C₂²	1	2	yes	—
22	C₃₁×#533	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₃₁×#519	—	2	1	no	—
25	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₃₁	—	2	1	no	—
27	C₈×D₄×C₃₁	—	2	2	no	—
28	C₈×D₄×C₃₁	—	2	1	no	—
29	C₃₁×#433	—	2	2	no	—
30	C₃₁×#432	C₂²	1	2	yes	—
31	C₃₁×#431	C₂²	1	2	yes	—
32	C₃₁×#431	C₂²	1	1	yes	characteristic
33	C₃₁×#431	C₂²	1	2	yes	—
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
38	C₃₁×#408	—	2	1	no	—
39	C₃₁×#405	—	2	1	no	—
40	C₃₁×#333	C₂²	1	1	yes	characteristic
41	C₃₁×#333	C₂²	1	2	yes	—
42	C₃₁×#333	C₂²	1	1	yes	characteristic
43	C₃₁×#332	C₂²	1	2	yes	—
44	C₃₁×#332	C₂²	1	2	yes	—
45	C₃₁×#332	C₂²	1	1	yes	characteristic
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₃₁×#332	C₂²	1	1	yes	characteristic
48	C₃₁×#332	C₂²	1	2	yes	—
49	C₃₁×#332	C₂²	1	2	yes	—
50	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×Q₈×C₃₁	—	2	1	no	—
52	C₂×M₁₆×C₃₁	—	2	1	no	—
53	C₂²×C₈×C₃₁	—	2	1	no	—
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
60	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×#126	—	2	4	no	—
64	C₃₁×#126	—	2	4	no	—
65	C₃₁×#125	—	2	4	no	—
66	C₃₁×#125	—	2	4	no	—
67	C₃₁×#125	—	2	1	no	—
68	C₃₁×#125	—	2	2	no	—
69	C₃₁×#125	C₂×C₄	1	4	yes	—
70	C₃₁×#124	—	2	2	no	—
71	C₃₁×#123	—	2	4	no	—
72	C₃₁×C₂²⋊Q₈	—	2	2	no	—
73	C₃₁×C₂²⋊Q₈	—	2	2	no	—
74	C₃₁×C₂²⋊Q₈	—	2	1	no	—
75	C₃₁×C₂²⋊Q₈	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
77	C₃₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
78	C₄×Q₈×C₃₁	—	2	2	no	—
79	C₄×Q₈×C₃₁	—	2	2	no	—
80	C₄×Q₈×C₃₁	—	2	2	no	—
81	C₄×Q₈×C₃₁	—	2	2	no	—
82	C₄×Q₈×C₃₁	—	2	2	no	—
83	C₄×Q₈×C₃₁	—	2	1	no	—
84	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
85	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
86	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
88	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
89	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄×C₃₁	D₄	1	2	yes	—
91	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
92	C₃₁×#117	—	2	1	no	—
93	C₃₁×#117	—	2	2	no	—
94	C₃₁×#117	—	2	4	no	—
95	C₃₁×#117	C₂×C₄	1	4	yes	—
96	C₃₁×#117	D₄	1	1	yes	characteristic
97	C₃₁×#117	D₄	1	2	yes	—
98	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
102	C₃₁×C₄⋊C₈	—	2	1	no	—
103	C₃₁×C₄⋊C₈	—	2	2	no	—
104	C₃₁×C₄⋊C₈	—	2	1	no	—
105	C₃₁×C₄⋊C₈	—	2	1	no	—
106	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
107	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
108	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
109	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₂²⋊C₈	—	2	2	no	—
113	C₃₁×C₂²⋊C₈	—	2	2	no	—
114	C₃₁×C₂²⋊C₈	—	2	1	no	—
115	C₃₁×C₂²⋊C₈	—	2	2	no	—
116	C₃₁×#97	—	2	2	no	—
117	C₄×C₈×C₃₁	—	2	2	no	—
118	C₄×C₈×C₃₁	—	2	2	no	—
119	C₄×C₈×C₃₁	—	2	1	no	—
120	#41×C₃₁	—	2	2	no	—
121	#41×C₃₁	—	2	2	no	—
122	C₂×Q₈×C₃₁	—	2	2	no	—
123	C₂×Q₈×C₃₁	—	2	1	no	—
124	C₂×Q₈×C₃₁	—	2	2	no	—
125	C₂×Q₈×C₃₁	—	4	2	no	—
126	C₂×Q₈×C₃₁	—	4	2	no	—
127	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
128	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
130	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	M₁₆×C₃₁	—	4	2	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	4	2	no	—
146	C₂×C₈×C₃₁	—	8	1	no	—
147	C₂×C₈×C₃₁	—	8	2	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	4	no	—
151	C₄⋊C₄×C₃₁	—	2	4	no	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
161	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
162	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
163	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
164	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
165	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
166	C₂²⋊C₄×C₃₁	—	2	2	no	—
167	C₂²⋊C₄×C₃₁	—	2	4	no	—
168	C₂²⋊C₄×C₃₁	—	2	4	no	—
169	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
170	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
171	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
172	C₄²×C₃₁	—	2	4	no	—
173	C₄²×C₃₁	—	2	4	no	—
174	C₄²×C₃₁	—	2	2	no	—
175	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
176	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
177	C₄²×C₃₁	C₂×D₄	1	2	yes	—
178	C₄²×C₃₁	C₂×D₄	1	2	yes	—
179	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
182	#17725	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	Q₈×C₃₁	—	2	2	no	—
186	Q₈×C₃₁	—	4	2	no	—
187	Q₈×C₃₁	—	4	2	no	—
188	Q₈×C₃₁	—	4	2	no	—
189	Q₈×C₃₁	—	4	1	no	—
190	Q₈×C₃₁	—	4	1	no	—
191	Q₈×C₃₁	—	4	4	no	—
192	D₄×C₃₁	—	2	4	no	—
193	D₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	4	no	—
206	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
207	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
210	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	#135	1	2	yes	—
213	C₂×C₄×C₃₁	#135	1	2	yes	—
214	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
217	C₈×C₃₁	—	4	1	no	—
218	C₈×C₃₁	—	4	1	no	—
219	C₈×C₃₁	—	4	2	no	—
220	C₈×C₃₁	—	4	2	no	—
221	C₈×C₃₁	—	4	1	no	—
222	C₈×C₃₁	—	4	1	no	—
223	#3509	C₂×C₃₁	1	1	yes	characteristic
224	#2988	C₂×C₃₁	1	1	yes	characteristic
225	#2925	C₂×C₃₁	1	1	yes	characteristic
226	#1554	C₂×C₃₁	1	2	yes	complemented
227	#1552	C₂×C₃₁	1	2	yes	complemented
228	#1550	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#1546	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1544	C₂×C₃₁	1	2	yes	complemented
231	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#1542	C₂×C₃₁	1	2	yes	complemented
233	C₂×#332	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂²×C₃₁	—	2	1	no	—
235	C₂²×C₃₁	—	2	1	no	—
236	C₂²×C₃₁	—	2	2	no	—
237	C₂²×C₃₁	—	2	2	no	—
238	C₂²×C₃₁	—	2	2	no	—
239	C₂²×C₃₁	#416	1	2	yes	—
240	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
241	C₄×C₃₁	—	2	1	no	—
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	2	1	no	—
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	1	no	—
247	C₄×C₃₁	—	4	2	no	—
248	C₄×C₃₁	—	4	4	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	#416	1	2	yes	—
251	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
252	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
253	C₄×C₃₁	#577	1	1	yes	characteristic
254	C₄×C₃₁	#578	1	1	yes	characteristic
255	#557	—	2	1	no	—
256	#557	C₄×C₃₁	1	2	yes	—
257	#556	—	2	1	no	—
258	#556	C₄×C₃₁	1	2	yes	—
259	#552	—	2	2	no	—
260	#552	C₄×C₃₁	1	4	yes	—
261	#541	—	2	2	no	—
262	#536	—	2	1	no	—
263	#533	C₂²×C₃₁	1	1	yes	characteristic
264	#533	C₂²×C₃₁	1	2	yes	—
265	#533	C₂²×C₃₁	1	2	yes	—
266	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
267	#519	—	2	1	no	—
268	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
269	C₈×Q₈	—	2	1	no	—
270	C₈×D₄	—	2	2	no	—
271	C₈×D₄	—	2	1	no	—
272	#433	—	2	2	no	—
273	#432	C₂²×C₃₁	1	2	yes	—
274	#431	C₂²×C₃₁	1	2	yes	—
275	#431	C₂²×C₃₁	1	1	yes	characteristic
276	#431	C₂²×C₃₁	1	2	yes	—
277	#423	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
281	#408	—	2	1	no	—
282	#405	—	2	1	no	—
283	#333	C₂²×C₃₁	1	1	yes	characteristic
284	#333	C₂²×C₃₁	1	2	yes	—
285	#333	C₂²×C₃₁	1	1	yes	characteristic
286	#332	C₂²×C₃₁	1	2	yes	—
287	#332	C₂²×C₃₁	1	2	yes	—
288	#332	C₂²×C₃₁	1	1	yes	characteristic
289	#332	C₂²×C₃₁	1	2	yes	—
290	#332	C₂²×C₃₁	1	1	yes	characteristic
291	#332	C₂²×C₃₁	1	2	yes	—
292	#332	C₂²×C₃₁	1	2	yes	—
293	C₂×C₃₁	—	2	2	no	—
294	C₂×C₃₁	—	4	2	no	—
295	C₂×C₃₁	#2889	1	1	yes	characteristic
296	C₂×C₃₁	C₂×#416	1	1	yes	characteristic
297	C₂×C₃₁	#2898	1	1	yes	characteristic
298	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
299	C₂²×Q₈	—	2	1	no	—
300	C₂×M₁₆	—	2	1	no	—
301	C₂²×C₈	—	2	1	no	—
302	C₂²×C₈	—	2	1	no	—
303	C₂²×C₈	—	2	2	no	—
304	C₄⋊Q₈	—	2	2	no	—
305	C₄⋊Q₈	—	2	1	no	—
306	C₄⋊Q₈	—	2	1	no	—
307	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
308	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
310	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
311	#126	—	2	4	no	—
312	#126	—	2	4	no	—
313	#125	—	2	4	no	—
314	#125	—	2	4	no	—
315	#125	—	2	1	no	—
316	#125	—	2	2	no	—
317	#125	C₂×C₄×C₃₁	1	4	yes	—
318	#124	—	2	2	no	—
319	#123	—	2	4	no	—
320	C₂²⋊Q₈	—	2	2	no	—
321	C₂²⋊Q₈	—	2	2	no	—
322	C₂²⋊Q₈	—	2	1	no	—
323	C₂²⋊Q₈	—	2	2	no	—
324	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
325	C₂²⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
326	C₄×Q₈	—	2	2	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	—	2	2	no	—
331	C₄×Q₈	—	2	1	no	—
332	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
333	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
334	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
335	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
336	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
337	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
338	C₄×D₄	D₄×C₃₁	1	2	yes	—
339	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
340	#117	—	2	1	no	—
341	#117	—	2	2	no	—
342	#117	—	2	4	no	—
343	#117	C₂×C₄×C₃₁	1	4	yes	—
344	#117	D₄×C₃₁	1	1	yes	characteristic
345	#117	D₄×C₃₁	1	2	yes	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
348	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
350	C₄⋊C₈	—	2	1	no	—
351	C₄⋊C₈	—	2	2	no	—
352	C₄⋊C₈	—	2	1	no	—
353	C₄⋊C₈	—	2	1	no	—
354	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
355	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
356	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
357	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
358	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
359	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
360	C₂²⋊C₈	—	2	2	no	—
361	C₂²⋊C₈	—	2	2	no	—
362	C₂²⋊C₈	—	2	1	no	—
363	C₂²⋊C₈	—	2	2	no	—
364	#97	—	2	2	no	—
365	C₄×C₈	—	2	2	no	—
366	C₄×C₈	—	2	2	no	—
367	C₄×C₈	—	2	1	no	—
368	C₃₁	#17725	1	1	yes	characteristic, direct product factor, Sylow
369	#41	—	2	2	no	—
370	#41	—	2	2	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	4	2	no	—
375	C₂×Q₈	—	4	2	no	—
376	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
377	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
378	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
379	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
383	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
384	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
387	M₁₆	—	4	2	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	4	2	no	—
395	C₂×C₈	—	8	1	no	—
396	C₂×C₈	—	8	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
410	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
411	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
412	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
413	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
414	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
419	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
420	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
421	C₄²	—	2	4	no	—
422	C₄²	—	2	4	no	—
423	C₄²	—	2	2	no	—
424	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
425	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
426	C₄²	C₂×D₄×C₃₁	1	2	yes	—
427	C₄²	C₂×D₄×C₃₁	1	2	yes	—
428	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
429	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
431	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
432	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
433	Q₈	—	2	2	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	4	4	no	—
436	Q₈	—	4	2	no	—
437	Q₈	—	4	2	no	—
438	Q₈	—	4	1	no	—
439	Q₈	—	4	1	no	—
440	D₄	—	2	4	no	—
441	D₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
455	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
456	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
458	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
459	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₃₁×#135	1	2	yes	—
461	C₂×C₄	C₃₁×#135	1	2	yes	—
462	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
465	C₈	—	4	1	no	—
466	C₈	—	4	1	no	—
467	C₈	—	4	2	no	—
468	C₈	—	4	2	no	—
469	C₈	—	4	1	no	—
470	C₈	—	4	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	1	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	C₃₁×#416	1	2	yes	—
477	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
478	C₄	—	2	1	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	2	no	—
482	C₄	—	2	2	no	—
483	C₄	—	2	1	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	2	no	—
487	C₄	C₃₁×#416	1	2	yes	—
488	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
489	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
490	C₄	C₃₁×#577	1	1	yes	characteristic
491	C₄	C₃₁×#578	1	1	yes	characteristic
492	C₂	—	2	2	no	—
493	C₂	—	4	2	no	—
494	C₂	C₃₁×#2889	1	1	yes	characteristic
495	C₂	C₂×C₃₁×#416	1	1	yes	characteristic
496	C₂	C₃₁×#2898	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1554	C₂	semidirect
5. C₃₁×#1552	C₂	semidirect
6. C₃₁×#1550	C₂	semidirect
7. C₃₁×#1546	C₂	semidirect
8. C₃₁×#1544	C₂	semidirect
9. C₃₁×#1542	C₂	semidirect
10. C₃₁×#1542	C₂	semidirect
11. C₂×C₃₁×#332	C₂	semidirect
182. #17725	C₃₁	direct (N × H)
226. #1554	C₂×C₃₁	semidirect
227. #1552	C₂×C₃₁	semidirect
228. #1550	C₂×C₃₁	semidirect
229. #1546	C₂×C₃₁	semidirect
230. #1544	C₂×C₃₁	semidirect
231. #1542	C₂×C₃₁	semidirect
232. #1542	C₂×C₃₁	semidirect
233. C₂×#332	C₂×C₃₁	semidirect
368. C₃₁	#17725	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×#3509	2
C₃₁×#2988	2
C₃₁×#2925	2
C₃₁×#1554	2
C₃₁×#1552	2
C₃₁×#1550	2
C₃₁×#1546	2
C₃₁×#1544	2
C₃₁×#1542	2
C₂×C₃₁×#332	2
#17725	31

Subgroups — C31×#17725 #21474484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17725 #21474484