Subgroups — C₃₁×#17762 #21474521

Order 7936.

1028 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3508	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2989	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1560	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1557	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1551	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1550	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1547	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#559	—	2	1	no	—
14	C₃₁×#559	C₄	1	2	yes	—
15	C₃₁×#550	—	2	1	no	—
16	C₃₁×#550	C₄	1	2	yes	—
17	C₃₁×#546	—	2	2	no	—
18	C₃₁×#546	C₄	1	4	yes	—
19	C₃₁×#540	—	2	2	no	—
20	C₃₁×#535	—	2	1	no	—
21	C₃₁×#533	C₂²	1	1	yes	characteristic
22	C₃₁×#532	C₂²	1	2	yes	—
23	C₃₁×#532	C₂²	1	2	yes	—
24	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
25	C₃₁×#518	—	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	complemented
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#424	—	2	2	no	—
33	C₃₁×#424	—	2	1	no	—
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₃₁×#423	C₂²	1	2	yes	complemented
36	C₃₁×#423	C₂²	1	2	yes	complemented
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	complemented
40	C₃₁×#333	C₂²	1	1	yes	characteristic, complemented
41	C₃₁×#333	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×#333	C₂²	1	2	yes	complemented
43	C₃₁×#332	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#332	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×#332	C₂²	1	2	yes	complemented
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₃₁×#331	C₂²	1	2	yes	—
48	C₃₁×#331	C₂²	1	2	yes	—
49	C₃₁×#331	C₂²	1	2	yes	—
50	C₂×#41×C₃₁	—	2	2	no	—
51	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×D₄×C₃₁	—	2	1	no	—
54	C₂²×D₄×C₃₁	—	2	1	no	—
55	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₃₁×#131	—	4	4	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×#127	—	2	2	no	—
66	C₃₁×#127	—	2	1	no	—
67	C₃₁×#127	—	2	1	no	—
68	C₃₁×#127	C₂×C₄	1	2	yes	—
69	C₃₁×#127	C₂³	1	2	yes	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#124	—	2	2	no	—
72	C₃₁×#124	—	2	2	no	—
73	C₃₁×#124	C₂×C₄	1	4	yes	—
74	C₃₁×#123	—	2	4	no	—
75	C₃₁×#123	—	2	4	no	—
76	C₃₁×C₂²⋊Q₈	—	2	1	no	—
77	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
78	C₃₁×#121	—	2	2	no	—
79	C₃₁×#121	—	2	2	no	—
80	C₃₁×#121	—	2	2	no	—
81	C₃₁×#121	—	2	4	no	—
82	C₃₁×#121	—	2	4	no	—
83	C₃₁×#121	C₂×C₄	1	4	yes	—
84	C₃₁×#120	—	2	4	no	—
85	C₄×Q₈×C₃₁	—	2	1	no	—
86	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
87	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
88	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	—	2	2	no	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	2	no	—
94	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
95	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
96	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
97	C₄×D₄×C₃₁	D₄	1	2	yes	—
98	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
99	C₃₁×#117	—	2	1	no	—
100	C₃₁×#117	D₄	1	1	yes	characteristic
101	C₃₁×#117	D₄	1	2	yes	—
102	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
105	C₃₁×C₄⋊C₈	—	2	1	no	—
106	C₃₁×C₄⋊C₈	—	2	1	no	—
107	C₃₁×C₄⋊C₈	—	2	2	no	—
108	C₃₁×C₄⋊C₈	—	2	1	no	—
109	C₃₁×C₄⋊C₈	—	2	2	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	C₂⁴×C₃₁	—	2	2	no	—
122	#41×C₃₁	—	2	2	no	—
123	#41×C₃₁	—	2	2	no	—
124	#41×C₃₁	—	8	2	no	—
125	#41×C₃₁	—	8	2	no	—
126	#41×C₃₁	—	8	4	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
129	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂×D₄×C₃₁	—	2	4	no	—
131	C₂×D₄×C₃₁	—	2	4	no	—
132	C₂×D₄×C₃₁	—	2	2	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	2	no	—
135	C₂×D₄×C₃₁	—	2	2	no	—
136	C₂×D₄×C₃₁	—	2	2	no	—
137	C₂×D₄×C₃₁	—	4	2	no	—
138	C₂×D₄×C₃₁	—	4	2	no	—
139	C₂×D₄×C₃₁	—	8	1	no	—
140	C₂×D₄×C₃₁	—	8	1	no	—
141	C₂×D₄×C₃₁	—	8	2	no	—
142	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
143	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
144	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	—	2	4	no	—
147	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
148	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	M₁₆×C₃₁	—	4	4	no	—
154	M₁₆×C₃₁	—	4	2	no	—
155	M₁₆×C₃₁	—	8	2	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	2	2	no	—
162	C₂×C₈×C₃₁	—	4	4	no	—
163	C₂×C₈×C₃₁	—	4	2	no	—
164	C₂×C₈×C₃₁	—	8	1	no	—
165	C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₄⋊C₄×C₃₁	—	2	2	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
171	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
172	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
173	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
174	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
175	C₂²⋊C₄×C₃₁	—	2	2	no	—
176	C₂²⋊C₄×C₃₁	—	2	4	no	—
177	C₂²⋊C₄×C₃₁	—	2	4	no	—
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	4	no	—
180	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
181	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
182	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
183	C₄²×C₃₁	—	2	2	no	—
184	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
185	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₃₁	C₂×D₄	1	2	yes	—
188	C₄²×C₃₁	C₂×D₄	1	2	yes	—
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
191	#17762	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂³×C₃₁	—	2	4	no	—
193	C₂³×C₃₁	—	4	4	no	—
194	C₂³×C₃₁	—	8	2	no	—
195	C₂³×C₃₁	—	8	2	no	—
196	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
197	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	Q₈×C₃₁	—	2	2	no	—
199	Q₈×C₃₁	—	4	2	no	—
200	Q₈×C₃₁	—	4	1	no	—
201	Q₈×C₃₁	—	4	1	no	—
202	D₄×C₃₁	—	2	4	no	—
203	D₄×C₃₁	—	2	2	no	—
204	D₄×C₃₁	—	4	4	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	4	2	no	—
208	D₄×C₃₁	—	4	2	no	—
209	D₄×C₃₁	—	4	4	no	—
210	D₄×C₃₁	—	8	2	no	—
211	D₄×C₃₁	—	8	2	no	—
212	D₄×C₃₁	—	8	4	no	—
213	D₄×C₃₁	—	8	4	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	8	4	no	—
225	C₂×C₄×C₃₁	—	8	4	no	—
226	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
228	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
230	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
233	C₂×C₄×C₃₁	#137	1	2	yes	—
234	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₈×C₃₁	—	4	1	no	—
238	C₈×C₃₁	—	4	2	no	—
239	C₈×C₃₁	—	4	1	no	—
240	C₈×C₃₁	—	4	2	no	—
241	C₈×C₃₁	—	4	1	no	—
242	C₈×C₃₁	—	4	1	no	—
243	#3508	C₂×C₃₁	1	1	yes	characteristic
244	#2989	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#1560	C₂×C₃₁	1	2	yes	complemented
247	#1557	C₂×C₃₁	1	2	yes	complemented
248	#1551	C₂×C₃₁	1	2	yes	complemented
249	#1550	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#1547	C₂×C₃₁	1	2	yes	complemented
251	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
253	C₂×#331	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₂²×C₃₁	—	2	1	no	—
255	C₂²×C₃₁	—	2	1	no	—
256	C₂²×C₃₁	—	2	2	no	—
257	C₂²×C₃₁	—	2	2	no	—
258	C₂²×C₃₁	—	2	2	no	—
259	C₂²×C₃₁	—	4	4	no	—
260	C₂²×C₃₁	—	4	4	no	—
261	C₂²×C₃₁	—	4	4	no	—
262	C₂²×C₃₁	—	16	4	no	—
263	C₂²×C₃₁	#417	1	2	yes	—
264	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
265	C₄×C₃₁	—	2	1	no	—
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	2	1	no	—
268	C₄×C₃₁	—	2	2	no	—
269	C₄×C₃₁	—	2	2	no	—
270	C₄×C₃₁	—	2	1	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	—	4	2	no	—
273	C₄×C₃₁	#417	1	2	yes	—
274	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
276	C₄×C₃₁	#576	1	1	yes	characteristic
277	C₄×C₃₁	#577	1	1	yes	characteristic
278	C₂×#131	—	2	1	no	—
279	#559	—	2	1	no	—
280	#559	C₄×C₃₁	1	2	yes	—
281	#550	—	2	1	no	—
282	#550	C₄×C₃₁	1	2	yes	—
283	#546	—	2	2	no	—
284	#546	C₄×C₃₁	1	4	yes	—
285	#540	—	2	2	no	—
286	#535	—	2	1	no	—
287	#533	C₂²×C₃₁	1	1	yes	characteristic
288	#532	C₂²×C₃₁	1	2	yes	—
289	#532	C₂²×C₃₁	1	2	yes	—
290	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
291	#518	—	2	1	no	—
292	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
293	#446	—	2	1	no	—
294	#436	—	2	1	no	—
295	#435	—	2	2	no	—
296	#432	C₂²×C₃₁	1	2	yes	complemented
297	#431	C₂²×C₃₁	1	1	yes	characteristic
298	#424	—	2	2	no	—
299	#424	—	2	1	no	—
300	#423	C₂²×C₃₁	1	1	yes	characteristic
301	#423	C₂²×C₃₁	1	2	yes	complemented
302	#423	C₂²×C₃₁	1	2	yes	complemented
303	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
306	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
307	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	#333	C₂²×C₃₁	1	2	yes	complemented
309	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
310	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
311	#332	C₂²×C₃₁	1	2	yes	complemented
312	#331	C₂²×C₃₁	1	2	yes	—
313	#331	C₂²×C₃₁	1	2	yes	—
314	#331	C₂²×C₃₁	1	2	yes	—
315	#331	C₂²×C₃₁	1	2	yes	—
316	C₂×C₃₁	—	2	2	no	—
317	C₂×C₃₁	—	4	2	no	—
318	C₂×C₃₁	—	8	4	no	—
319	C₂×C₃₁	#2888	1	1	yes	characteristic
320	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
321	C₂×C₃₁	#2897	1	1	yes	characteristic
322	C₂×#41	—	2	2	no	—
323	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
324	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
325	C₂²×D₄	—	2	1	no	—
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
328	#131	—	4	4	no	—
329	C₂×M₁₆	—	2	1	no	—
330	C₂×M₁₆	—	2	2	no	—
331	C₂×M₁₆	—	2	2	no	—
332	C₂²×C₈	—	2	1	no	—
333	C₂²×C₈	—	2	2	no	—
334	C₂²×C₈	—	2	1	no	—
335	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
336	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
337	#127	—	2	2	no	—
338	#127	—	2	1	no	—
339	#127	—	2	1	no	—
340	#127	C₂×C₄×C₃₁	1	2	yes	—
341	#127	C₂³×C₃₁	1	2	yes	—
342	#125	—	2	1	no	—
343	#124	—	2	2	no	—
344	#124	—	2	2	no	—
345	#124	C₂×C₄×C₃₁	1	4	yes	—
346	#123	—	2	4	no	—
347	#123	—	2	4	no	—
348	C₂²⋊Q₈	—	2	1	no	—
349	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#121	—	2	2	no	—
353	#121	—	2	4	no	—
354	#121	—	2	4	no	—
355	#121	C₂×C₄×C₃₁	1	4	yes	—
356	#120	—	2	4	no	—
357	C₄×Q₈	—	2	1	no	—
358	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
359	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
360	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
367	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
368	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
369	C₄×D₄	D₄×C₃₁	1	2	yes	—
370	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
371	#117	—	2	1	no	—
372	#117	D₄×C₃₁	1	1	yes	characteristic
373	#117	D₄×C₃₁	1	2	yes	—
374	C₂×C₂²⋊C₄	—	2	4	no	—
375	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
376	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	—	2	2	no	—
380	C₄⋊C₈	—	2	1	no	—
381	C₄⋊C₈	—	2	2	no	—
382	C₄⋊C₈	—	2	2	no	—
383	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
384	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
385	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
386	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
387	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
388	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
389	C₂²⋊C₈	—	2	1	no	—
390	C₂²⋊C₈	—	2	2	no	—
391	#97	—	2	2	no	—
392	C₄×C₈	—	2	1	no	—
393	C₃₁	#17762	1	1	yes	characteristic, direct product factor, Sylow
394	C₂⁴	—	2	2	no	—
395	#41	—	2	2	no	—
396	#41	—	2	2	no	—
397	#41	—	8	2	no	—
398	#41	—	8	2	no	—
399	#41	—	8	4	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
402	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
403	C₂×D₄	—	2	4	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	4	2	no	—
411	C₂×D₄	—	4	2	no	—
412	C₂×D₄	—	8	1	no	—
413	C₂×D₄	—	8	1	no	—
414	C₂×D₄	—	8	2	no	—
415	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
416	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
417	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
421	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
423	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
425	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
426	M₁₆	—	4	4	no	—
427	M₁₆	—	4	2	no	—
428	M₁₆	—	8	2	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	4	4	no	—
436	C₂×C₈	—	4	2	no	—
437	C₂×C₈	—	8	1	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
444	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
445	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
446	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
447	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
454	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
455	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
456	C₄²	—	2	2	no	—
457	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
458	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
459	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₄²	C₂×D₄×C₃₁	1	2	yes	—
461	C₄²	C₂×D₄×C₃₁	1	2	yes	—
462	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
464	C₂³	—	2	4	no	—
465	C₂³	—	4	4	no	—
466	C₂³	—	8	2	no	—
467	C₂³	—	8	2	no	—
468	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
469	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
470	Q₈	—	2	2	no	—
471	Q₈	—	4	2	no	—
472	Q₈	—	4	1	no	—
473	Q₈	—	4	1	no	—
474	D₄	—	2	4	no	—
475	D₄	—	2	2	no	—
476	D₄	—	4	4	no	—
477	D₄	—	4	4	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	8	2	no	—
483	D₄	—	8	2	no	—
484	D₄	—	8	4	no	—
485	D₄	—	8	4	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	8	4	no	—
497	C₂×C₄	—	8	4	no	—
498	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
499	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
500	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
501	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
502	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
503	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
504	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
505	C₂×C₄	C₃₁×#137	1	2	yes	—
506	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
508	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
509	C₈	—	4	1	no	—
510	C₈	—	4	2	no	—
511	C₈	—	4	1	no	—
512	C₈	—	4	2	no	—
513	C₈	—	4	1	no	—
514	C₈	—	4	1	no	—
515	C₂²	—	2	1	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	2	2	no	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	4	4	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	4	no	—
523	C₂²	—	16	4	no	—
524	C₂²	C₃₁×#417	1	2	yes	—
525	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
526	C₄	—	2	1	no	—
527	C₄	—	2	2	no	—
528	C₄	—	2	1	no	—
529	C₄	—	2	2	no	—
530	C₄	—	2	2	no	—
531	C₄	—	2	1	no	—
532	C₄	—	4	2	no	—
533	C₄	—	4	2	no	—
534	C₄	C₃₁×#417	1	2	yes	—
535	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
536	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
537	C₄	C₃₁×#576	1	1	yes	characteristic
538	C₄	C₃₁×#577	1	1	yes	characteristic
539	C₂	—	2	2	no	—
540	C₂	—	4	2	no	—
541	C₂	—	8	4	no	—
542	C₂	C₃₁×#2888	1	1	yes	characteristic
543	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
544	C₂	C₃₁×#2897	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	C₃₁×#3508	C₂
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
2. C₃₁×#2989	C₂	semidirect
3. C₃₁×#2906	C₂	semidirect
4. C₃₁×#1560	C₂	semidirect
5. C₃₁×#1557	C₂	semidirect
6. C₃₁×#1551	C₂	semidirect
7. C₃₁×#1550	C₂	semidirect
8. C₃₁×#1547	C₂	semidirect
9. C₃₁×#1545	C₂	semidirect
10. C₃₁×#1542	C₂	semidirect
11. C₂×C₃₁×#331	C₂	semidirect
30. C₃₁×#432	C₂²	semidirect
35. C₃₁×#423	C₂²	semidirect
36. C₃₁×#423	C₂²	semidirect
39. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
40. C₃₁×#333	C₂²	semidirect
41. C₃₁×#333	C₂²	semidirect
42. C₃₁×#333	C₂²	semidirect
43. C₃₁×#332	C₂²	semidirect
44. C₃₁×#332	C₂²	semidirect
45. C₃₁×#332	C₂²	semidirect
191. #17762	C₃₁	direct (N × H)
244. #2989	C₂×C₃₁	semidirect
245. #2906	C₂×C₃₁	semidirect
246. #1560	C₂×C₃₁	semidirect
247. #1557	C₂×C₃₁	semidirect
248. #1551	C₂×C₃₁	semidirect
249. #1550	C₂×C₃₁	semidirect
250. #1547	C₂×C₃₁	semidirect
251. #1545	C₂×C₃₁	semidirect
252. #1542	C₂×C₃₁	semidirect
253. C₂×#331	C₂×C₃₁	semidirect
296. #432	C₂²×C₃₁	semidirect
301. #423	C₂²×C₃₁	semidirect
302. #423	C₂²×C₃₁	semidirect
305. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
306. #333	C₂²×C₃₁	semidirect
307. #333	C₂²×C₃₁	semidirect
308. #333	C₂²×C₃₁	semidirect
309. #332	C₂²×C₃₁	semidirect
310. #332	C₂²×C₃₁	semidirect
311. #332	C₂²×C₃₁	semidirect
393. C₃₁	#17762	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×#3508	2
C₃₁×#2989	2
C₃₁×#2906	2
C₃₁×#1560	2
C₃₁×#1557	2
C₃₁×#1551	2
C₃₁×#1550	2
C₃₁×#1547	2
C₃₁×#1545	2
C₃₁×#1542	2
C₂×C₃₁×#331	2
#17762	31

Subgroups — C31×#17762 #21474521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17762 #21474521