Subgroups — C₃₁×#17763 #21474522

Order 7936.

908 conjugacy classes of subgroups.

The subgroup lattice has 504 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×#2989	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1561	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1559	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1552	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1550	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1550	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1546	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#332	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#557	—	2	1	no	—
14	C₃₁×#557	C₄	1	2	yes	—
15	C₃₁×#556	—	2	1	no	—
16	C₃₁×#556	C₄	1	2	yes	—
17	C₃₁×#552	—	2	2	no	—
18	C₃₁×#552	C₄	1	4	yes	—
19	C₃₁×#541	—	2	2	no	—
20	C₃₁×#536	—	2	1	no	—
21	C₃₁×#533	C₂²	1	1	yes	characteristic
22	C₃₁×#533	C₂²	1	2	yes	—
23	C₃₁×#533	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃₁×#519	—	2	1	no	—
26	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#446	—	2	1	no	—
28	C₃₁×#436	—	2	1	no	—
29	C₃₁×#435	—	2	2	no	—
30	C₃₁×#432	C₂²	1	2	yes	—
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#424	—	2	1	no	—
33	C₃₁×#424	—	2	2	no	—
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₃₁×#423	C₂²	1	2	yes	—
36	C₃₁×#423	C₂²	1	2	yes	—
37	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
40	C₃₁×#333	C₂²	1	1	yes	characteristic
41	C₃₁×#333	C₂²	1	1	yes	characteristic
42	C₃₁×#333	C₂²	1	2	yes	—
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	1	yes	characteristic
47	C₃₁×#332	C₂²	1	2	yes	—
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	C₃₁×#131	—	4	4	no	—
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₃₁×#126	—	2	4	no	—
67	C₃₁×#126	—	2	4	no	—
68	C₃₁×#125	—	2	4	no	—
69	C₃₁×#125	—	2	4	no	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#125	—	2	2	no	—
72	C₃₁×#125	C₂×C₄	1	4	yes	—
73	C₃₁×#124	—	2	2	no	—
74	C₃₁×#123	—	2	4	no	—
75	C₃₁×C₂²⋊Q₈	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	—	2	2	no	—
77	C₃₁×C₂²⋊Q₈	—	2	1	no	—
78	C₃₁×C₂²⋊Q₈	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
80	C₃₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
81	C₄×Q₈×C₃₁	—	2	2	no	—
82	C₄×Q₈×C₃₁	—	2	2	no	—
83	C₄×Q₈×C₃₁	—	2	2	no	—
84	C₄×Q₈×C₃₁	—	2	2	no	—
85	C₄×Q₈×C₃₁	—	2	2	no	—
86	C₄×Q₈×C₃₁	—	2	1	no	—
87	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
88	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
89	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
91	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
92	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₄×D₄×C₃₁	D₄	1	2	yes	—
94	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
95	C₃₁×#117	—	2	1	no	—
96	C₃₁×#117	—	2	2	no	—
97	C₃₁×#117	—	2	4	no	—
98	C₃₁×#117	C₂×C₄	1	4	yes	—
99	C₃₁×#117	D₄	1	1	yes	characteristic
100	C₃₁×#117	D₄	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
105	C₃₁×C₄⋊C₈	—	2	2	no	—
106	C₃₁×C₄⋊C₈	—	2	1	no	—
107	C₃₁×C₄⋊C₈	—	2	1	no	—
108	C₃₁×C₄⋊C₈	—	2	2	no	—
109	C₃₁×C₄⋊C₈	—	2	1	no	—
110	C₃₁×C₄⋊C₈	—	2	2	no	—
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₃₁×C₂²⋊C₈	—	2	1	no	—
118	C₃₁×C₂²⋊C₈	—	2	2	no	—
119	C₃₁×#97	—	2	2	no	—
120	C₄×C₈×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	2	no	—
122	#41×C₃₁	—	2	2	no	—
123	C₂×Q₈×C₃₁	—	2	2	no	—
124	C₂×Q₈×C₃₁	—	2	1	no	—
125	C₂×Q₈×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	—	4	2	no	—
127	C₂×Q₈×C₃₁	—	4	2	no	—
128	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
129	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
131	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂²×C₄	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	2	yes	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	M₁₆×C₃₁	—	4	4	no	—
140	M₁₆×C₃₁	—	4	2	no	—
141	M₁₆×C₃₁	—	8	2	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	4	4	no	—
149	C₂×C₈×C₃₁	—	4	2	no	—
150	C₂×C₈×C₃₁	—	8	1	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
164	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
165	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
166	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
167	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
168	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
169	C₂²⋊C₄×C₃₁	—	2	2	no	—
170	C₂²⋊C₄×C₃₁	—	2	4	no	—
171	C₂²⋊C₄×C₃₁	—	2	4	no	—
172	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
173	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
174	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
175	C₄²×C₃₁	—	2	4	no	—
176	C₄²×C₃₁	—	2	4	no	—
177	C₄²×C₃₁	—	2	2	no	—
178	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
179	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
180	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂×D₄	1	2	yes	—
182	C₄²×C₃₁	C₂×D₄	1	2	yes	—
183	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
185	#17763	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	Q₈×C₃₁	—	2	2	no	—
189	Q₈×C₃₁	—	4	2	no	—
190	Q₈×C₃₁	—	4	2	no	—
191	Q₈×C₃₁	—	4	2	no	—
192	Q₈×C₃₁	—	4	1	no	—
193	Q₈×C₃₁	—	4	1	no	—
194	Q₈×C₃₁	—	4	4	no	—
195	D₄×C₃₁	—	2	4	no	—
196	D₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	4	no	—
209	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
210	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
211	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
213	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
216	C₂×C₄×C₃₁	#137	1	2	yes	—
217	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₈×C₃₁	—	4	1	no	—
221	C₈×C₃₁	—	4	2	no	—
222	C₈×C₃₁	—	4	1	no	—
223	C₈×C₃₁	—	4	2	no	—
224	C₈×C₃₁	—	4	1	no	—
225	C₈×C₃₁	—	4	1	no	—
226	#3509	C₂×C₃₁	1	1	yes	characteristic
227	#2989	C₂×C₃₁	1	1	yes	characteristic
228	#2906	C₂×C₃₁	1	1	yes	characteristic
229	#1561	C₂×C₃₁	1	2	yes	complemented
230	#1559	C₂×C₃₁	1	2	yes	complemented
231	#1552	C₂×C₃₁	1	2	yes	complemented
232	#1550	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#1550	C₂×C₃₁	1	2	yes	complemented
234	#1546	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂×#332	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂²×C₃₁	—	2	1	no	—
238	C₂²×C₃₁	—	2	1	no	—
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	2	2	no	—
241	C₂²×C₃₁	—	2	2	no	—
242	C₂²×C₃₁	#417	1	2	yes	—
243	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
244	C₄×C₃₁	—	2	1	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	1	no	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	2	1	no	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	4	4	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	#417	1	2	yes	—
254	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
255	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
256	C₄×C₃₁	#577	1	1	yes	characteristic
257	C₄×C₃₁	#578	1	1	yes	characteristic
258	C₂×#131	—	2	1	no	—
259	#557	—	2	1	no	—
260	#557	C₄×C₃₁	1	2	yes	—
261	#556	—	2	1	no	—
262	#556	C₄×C₃₁	1	2	yes	—
263	#552	—	2	2	no	—
264	#552	C₄×C₃₁	1	4	yes	—
265	#541	—	2	2	no	—
266	#536	—	2	1	no	—
267	#533	C₂²×C₃₁	1	1	yes	characteristic
268	#533	C₂²×C₃₁	1	2	yes	—
269	#533	C₂²×C₃₁	1	2	yes	—
270	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	#519	—	2	1	no	—
272	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
273	#446	—	2	1	no	—
274	#436	—	2	1	no	—
275	#435	—	2	2	no	—
276	#432	C₂²×C₃₁	1	2	yes	—
277	#431	C₂²×C₃₁	1	1	yes	characteristic
278	#424	—	2	1	no	—
279	#424	—	2	2	no	—
280	#423	C₂²×C₃₁	1	1	yes	characteristic
281	#423	C₂²×C₃₁	1	2	yes	—
282	#423	C₂²×C₃₁	1	2	yes	—
283	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
286	#333	C₂²×C₃₁	1	1	yes	characteristic
287	#333	C₂²×C₃₁	1	1	yes	characteristic
288	#333	C₂²×C₃₁	1	2	yes	—
289	#332	C₂²×C₃₁	1	2	yes	—
290	#332	C₂²×C₃₁	1	2	yes	—
291	#332	C₂²×C₃₁	1	1	yes	characteristic
292	#332	C₂²×C₃₁	1	1	yes	characteristic
293	#332	C₂²×C₃₁	1	2	yes	—
294	#332	C₂²×C₃₁	1	2	yes	—
295	#332	C₂²×C₃₁	1	2	yes	—
296	C₂×C₃₁	—	2	2	no	—
297	C₂×C₃₁	—	4	2	no	—
298	C₂×C₃₁	#2889	1	1	yes	characteristic
299	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
300	C₂×C₃₁	#2898	1	1	yes	characteristic
301	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
302	C₂²×Q₈	—	2	1	no	—
303	#131	—	4	4	no	—
304	C₂×M₁₆	—	2	2	no	—
305	C₂×M₁₆	—	2	1	no	—
306	C₂×M₁₆	—	2	2	no	—
307	C₂²×C₈	—	2	2	no	—
308	C₂²×C₈	—	2	1	no	—
309	C₂²×C₈	—	2	1	no	—
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	—	2	1	no	—
312	C₄⋊Q₈	—	2	1	no	—
313	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
314	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
316	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
317	#126	—	2	4	no	—
318	#126	—	2	4	no	—
319	#125	—	2	4	no	—
320	#125	—	2	4	no	—
321	#125	—	2	1	no	—
322	#125	—	2	2	no	—
323	#125	C₂×C₄×C₃₁	1	4	yes	—
324	#124	—	2	2	no	—
325	#123	—	2	4	no	—
326	C₂²⋊Q₈	—	2	2	no	—
327	C₂²⋊Q₈	—	2	2	no	—
328	C₂²⋊Q₈	—	2	1	no	—
329	C₂²⋊Q₈	—	2	2	no	—
330	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
331	C₂²⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
332	C₄×Q₈	—	2	2	no	—
333	C₄×Q₈	—	2	2	no	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×Q₈	—	2	2	no	—
336	C₄×Q₈	—	2	2	no	—
337	C₄×Q₈	—	2	1	no	—
338	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
339	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
340	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
341	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
342	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
343	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
344	C₄×D₄	D₄×C₃₁	1	2	yes	—
345	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
346	#117	—	2	1	no	—
347	#117	—	2	2	no	—
348	#117	—	2	4	no	—
349	#117	C₂×C₄×C₃₁	1	4	yes	—
350	#117	D₄×C₃₁	1	1	yes	characteristic
351	#117	D₄×C₃₁	1	2	yes	—
352	C₂×C₄⋊C₄	—	2	1	no	—
353	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
354	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
356	C₄⋊C₈	—	2	2	no	—
357	C₄⋊C₈	—	2	1	no	—
358	C₄⋊C₈	—	2	1	no	—
359	C₄⋊C₈	—	2	2	no	—
360	C₄⋊C₈	—	2	1	no	—
361	C₄⋊C₈	—	2	2	no	—
362	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
363	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
364	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
365	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
366	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
367	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
368	C₂²⋊C₈	—	2	1	no	—
369	C₂²⋊C₈	—	2	2	no	—
370	#97	—	2	2	no	—
371	C₄×C₈	—	2	1	no	—
372	C₃₁	#17763	1	1	yes	characteristic, direct product factor, Sylow
373	#41	—	2	2	no	—
374	#41	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	1	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	—	4	2	no	—
379	C₂×Q₈	—	4	2	no	—
380	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
381	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
382	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
383	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	M₁₆	—	4	4	no	—
392	M₁₆	—	4	2	no	—
393	M₁₆	—	8	2	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	4	4	no	—
401	C₂×C₈	—	4	2	no	—
402	C₂×C₈	—	8	1	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
416	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
417	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
418	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
419	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
420	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
421	C₂²⋊C₄	—	2	2	no	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
425	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
426	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
427	C₄²	—	2	4	no	—
428	C₄²	—	2	4	no	—
429	C₄²	—	2	2	no	—
430	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
431	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
432	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₄²	C₂×D₄×C₃₁	1	2	yes	—
434	C₄²	C₂×D₄×C₃₁	1	2	yes	—
435	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
436	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
437	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
438	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
439	Q₈	—	2	2	no	—
440	Q₈	—	4	2	no	—
441	Q₈	—	4	4	no	—
442	Q₈	—	4	2	no	—
443	Q₈	—	4	2	no	—
444	Q₈	—	4	1	no	—
445	Q₈	—	4	1	no	—
446	D₄	—	2	4	no	—
447	D₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
461	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
462	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
464	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
465	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
467	C₂×C₄	C₃₁×#137	1	2	yes	—
468	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
471	C₈	—	4	1	no	—
472	C₈	—	4	2	no	—
473	C₈	—	4	1	no	—
474	C₈	—	4	2	no	—
475	C₈	—	4	1	no	—
476	C₈	—	4	1	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	C₃₁×#417	1	2	yes	—
483	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
484	C₄	—	2	1	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	1	no	—
487	C₄	—	2	2	no	—
488	C₄	—	2	1	no	—
489	C₄	—	2	2	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	4	no	—
492	C₄	—	4	2	no	—
493	C₄	C₃₁×#417	1	2	yes	—
494	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
495	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
496	C₄	C₃₁×#577	1	1	yes	characteristic
497	C₄	C₃₁×#578	1	1	yes	characteristic
498	C₂	—	2	2	no	—
499	C₂	—	4	2	no	—
500	C₂	C₃₁×#2889	1	1	yes	characteristic
501	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
502	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×#1561	C₂	semidirect
5. C₃₁×#1559	C₂	semidirect
6. C₃₁×#1552	C₂	semidirect
7. C₃₁×#1550	C₂	semidirect
8. C₃₁×#1550	C₂	semidirect
9. C₃₁×#1546	C₂	semidirect
10. C₃₁×#1542	C₂	semidirect
11. C₂×C₃₁×#332	C₂	semidirect
185. #17763	C₃₁	direct (N × H)
229. #1561	C₂×C₃₁	semidirect
230. #1559	C₂×C₃₁	semidirect
231. #1552	C₂×C₃₁	semidirect
232. #1550	C₂×C₃₁	semidirect
233. #1550	C₂×C₃₁	semidirect
234. #1546	C₂×C₃₁	semidirect
235. #1542	C₂×C₃₁	semidirect
236. C₂×#332	C₂×C₃₁	semidirect
372. C₃₁	#17763	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×#2989	2
C₃₁×#2906	2
C₃₁×#1561	2
C₃₁×#1559	2
C₃₁×#1552	2
C₃₁×#1550	2
C₃₁×#1546	2
C₃₁×#1542	2
C₂×C₃₁×#332	2
#17763	31

Subgroups — C31×#17763 #21474522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17763 #21474522