Subgroups — C₃₁×#17724 #21474483

Order 7936.

1004 conjugacy classes of subgroups.

The subgroup lattice has 540 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×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1553	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1551	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1550	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1543	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1540	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#559	—	2	1	no	—
13	C₃₁×#559	C₄	1	2	yes	—
14	C₃₁×#550	—	2	1	no	—
15	C₃₁×#550	C₄	1	2	yes	—
16	C₃₁×#546	—	2	2	no	—
17	C₃₁×#546	C₄	1	4	yes	—
18	C₃₁×#540	—	2	2	no	—
19	C₃₁×#535	—	2	1	no	—
20	C₃₁×#533	C₂²	1	1	yes	characteristic
21	C₃₁×#532	C₂²	1	2	yes	—
22	C₃₁×#532	C₂²	1	2	yes	—
23	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
24	C₃₁×#518	—	2	1	no	—
25	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₃₁	—	2	1	no	—
27	C₈×D₄×C₃₁	—	2	1	no	—
28	C₈×D₄×C₃₁	—	2	2	no	—
29	C₃₁×#433	—	2	2	no	—
30	C₃₁×#432	C₂²	1	2	yes	complemented
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#431	C₂²	1	2	yes	complemented
33	C₃₁×#431	C₂²	1	2	yes	complemented
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	complemented
38	C₃₁×#408	—	2	1	no	—
39	C₃₁×#405	—	2	1	no	—
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	2	yes	complemented
45	C₃₁×#332	C₂²	1	1	yes	characteristic, 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	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×#127	—	2	2	no	—
63	C₃₁×#127	—	2	1	no	—
64	C₃₁×#127	—	2	1	no	—
65	C₃₁×#127	C₂×C₄	1	2	yes	—
66	C₃₁×#127	C₂³	1	2	yes	—
67	C₃₁×#125	—	2	1	no	—
68	C₃₁×#124	—	2	2	no	—
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	C₂×C₄	1	4	yes	—
71	C₃₁×#123	—	2	4	no	—
72	C₃₁×#123	—	2	4	no	—
73	C₃₁×C₂²⋊Q₈	—	2	1	no	—
74	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
75	C₃₁×#121	—	2	2	no	—
76	C₃₁×#121	—	2	2	no	—
77	C₃₁×#121	—	2	2	no	—
78	C₃₁×#121	—	2	4	no	—
79	C₃₁×#121	—	2	4	no	—
80	C₃₁×#121	C₂×C₄	1	4	yes	—
81	C₃₁×#120	—	2	4	no	—
82	C₄×Q₈×C₃₁	—	2	1	no	—
83	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
84	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
92	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
93	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₃₁	D₄	1	2	yes	—
95	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
96	C₃₁×#117	—	2	1	no	—
97	C₃₁×#117	D₄	1	1	yes	characteristic
98	C₃₁×#117	D₄	1	2	yes	—
99	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
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	1	no	—
104	C₃₁×C₄⋊C₈	—	2	2	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	1	no	—
119	C₄×C₈×C₃₁	—	2	2	no	—
120	C₂⁴×C₃₁	—	2	2	no	—
121	#41×C₃₁	—	2	2	no	—
122	#41×C₃₁	—	2	2	no	—
123	#41×C₃₁	—	8	2	no	—
124	#41×C₃₁	—	8	2	no	—
125	#41×C₃₁	—	8	4	no	—
126	C₂×Q₈×C₃₁	—	2	1	no	—
127	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
128	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₃₁	—	2	4	no	—
130	C₂×D₄×C₃₁	—	2	4	no	—
131	C₂×D₄×C₃₁	—	2	2	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	—	4	2	no	—
137	C₂×D₄×C₃₁	—	4	2	no	—
138	C₂×D₄×C₃₁	—	8	1	no	—
139	C₂×D₄×C₃₁	—	8	1	no	—
140	C₂×D₄×C₃₁	—	8	2	no	—
141	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
142	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
143	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
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	1	yes	characteristic
152	M₁₆×C₃₁	—	4	2	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	1	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	4	2	no	—
160	C₂×C₈×C₃₁	—	8	1	no	—
161	C₂×C₈×C₃₁	—	8	2	no	—
162	C₄⋊C₄×C₃₁	—	2	2	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
168	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
169	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
170	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
171	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
172	C₂²⋊C₄×C₃₁	—	2	2	no	—
173	C₂²⋊C₄×C₃₁	—	2	4	no	—
174	C₂²⋊C₄×C₃₁	—	2	4	no	—
175	C₂²⋊C₄×C₃₁	—	2	4	no	—
176	C₂²⋊C₄×C₃₁	—	2	4	no	—
177	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
178	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
179	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
180	C₄²×C₃₁	—	2	2	no	—
181	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
182	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
183	C₄²×C₃₁	C₂×D₄	1	2	yes	—
184	C₄²×C₃₁	C₂×D₄	1	2	yes	—
185	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
188	#17724	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₃₁	—	2	4	no	—
190	C₂³×C₃₁	—	4	4	no	—
191	C₂³×C₃₁	—	8	2	no	—
192	C₂³×C₃₁	—	8	2	no	—
193	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
194	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	Q₈×C₃₁	—	2	2	no	—
196	Q₈×C₃₁	—	4	2	no	—
197	Q₈×C₃₁	—	4	1	no	—
198	Q₈×C₃₁	—	4	1	no	—
199	D₄×C₃₁	—	2	4	no	—
200	D₄×C₃₁	—	2	2	no	—
201	D₄×C₃₁	—	4	4	no	—
202	D₄×C₃₁	—	4	2	no	—
203	D₄×C₃₁	—	4	2	no	—
204	D₄×C₃₁	—	4	2	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	4	4	no	—
207	D₄×C₃₁	—	8	2	no	—
208	D₄×C₃₁	—	8	2	no	—
209	D₄×C₃₁	—	8	4	no	—
210	D₄×C₃₁	—	8	4	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	8	4	no	—
222	C₂×C₄×C₃₁	—	8	4	no	—
223	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
224	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
225	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	#135	1	2	yes	—
230	C₂×C₄×C₃₁	#135	1	2	yes	—
231	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
234	C₈×C₃₁	—	4	1	no	—
235	C₈×C₃₁	—	4	1	no	—
236	C₈×C₃₁	—	4	2	no	—
237	C₈×C₃₁	—	4	1	no	—
238	C₈×C₃₁	—	4	2	no	—
239	C₈×C₃₁	—	4	1	no	—
240	#3508	C₂×C₃₁	1	1	yes	characteristic
241	#2988	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#2925	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#1553	C₂×C₃₁	1	2	yes	complemented
244	#1551	C₂×C₃₁	1	2	yes	complemented
245	#1550	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1543	C₂×C₃₁	1	2	yes	complemented
248	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1540	C₂×C₃₁	1	2	yes	complemented
250	C₂×#331	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂²×C₃₁	—	2	1	no	—
252	C₂²×C₃₁	—	2	1	no	—
253	C₂²×C₃₁	—	2	2	no	—
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	2	no	—
256	C₂²×C₃₁	—	4	4	no	—
257	C₂²×C₃₁	—	4	4	no	—
258	C₂²×C₃₁	—	4	4	no	—
259	C₂²×C₃₁	—	16	4	no	—
260	C₂²×C₃₁	#416	1	2	yes	—
261	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	2	2	no	—
264	C₄×C₃₁	—	2	1	no	—
265	C₄×C₃₁	—	2	2	no	—
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	2	1	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	#416	1	2	yes	—
271	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
272	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
273	C₄×C₃₁	#576	1	1	yes	characteristic
274	C₄×C₃₁	#577	1	1	yes	characteristic
275	#559	—	2	1	no	—
276	#559	C₄×C₃₁	1	2	yes	—
277	#550	—	2	1	no	—
278	#550	C₄×C₃₁	1	2	yes	—
279	#546	—	2	2	no	—
280	#546	C₄×C₃₁	1	4	yes	—
281	#540	—	2	2	no	—
282	#535	—	2	1	no	—
283	#533	C₂²×C₃₁	1	1	yes	characteristic
284	#532	C₂²×C₃₁	1	2	yes	—
285	#532	C₂²×C₃₁	1	2	yes	—
286	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
287	#518	—	2	1	no	—
288	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
289	C₈×Q₈	—	2	1	no	—
290	C₈×D₄	—	2	1	no	—
291	C₈×D₄	—	2	2	no	—
292	#433	—	2	2	no	—
293	#432	C₂²×C₃₁	1	2	yes	complemented
294	#431	C₂²×C₃₁	1	1	yes	characteristic
295	#431	C₂²×C₃₁	1	2	yes	complemented
296	#431	C₂²×C₃₁	1	2	yes	complemented
297	#423	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
301	#408	—	2	1	no	—
302	#405	—	2	1	no	—
303	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
304	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
305	#333	C₂²×C₃₁	1	2	yes	complemented
306	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
307	#332	C₂²×C₃₁	1	2	yes	complemented
308	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
309	#331	C₂²×C₃₁	1	2	yes	—
310	#331	C₂²×C₃₁	1	2	yes	—
311	#331	C₂²×C₃₁	1	2	yes	—
312	#331	C₂²×C₃₁	1	2	yes	—
313	C₂×C₃₁	—	2	2	no	—
314	C₂×C₃₁	—	4	2	no	—
315	C₂×C₃₁	—	8	4	no	—
316	C₂×C₃₁	#2888	1	1	yes	characteristic
317	C₂×C₃₁	C₂×#416	1	1	yes	characteristic
318	C₂×C₃₁	#2897	1	1	yes	characteristic
319	C₂×#41	—	2	2	no	—
320	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
321	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
322	C₂²×D₄	—	2	1	no	—
323	C₂²×D₄	—	2	1	no	—
324	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
325	C₂×M₁₆	—	2	1	no	—
326	C₂²×C₈	—	2	1	no	—
327	C₂²×C₈	—	2	1	no	—
328	C₂²×C₈	—	2	2	no	—
329	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
330	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
331	#127	—	2	2	no	—
332	#127	—	2	1	no	—
333	#127	—	2	1	no	—
334	#127	C₂×C₄×C₃₁	1	2	yes	—
335	#127	C₂³×C₃₁	1	2	yes	—
336	#125	—	2	1	no	—
337	#124	—	2	2	no	—
338	#124	—	2	2	no	—
339	#124	C₂×C₄×C₃₁	1	4	yes	—
340	#123	—	2	4	no	—
341	#123	—	2	4	no	—
342	C₂²⋊Q₈	—	2	1	no	—
343	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
344	#121	—	2	2	no	—
345	#121	—	2	2	no	—
346	#121	—	2	2	no	—
347	#121	—	2	4	no	—
348	#121	—	2	4	no	—
349	#121	C₂×C₄×C₃₁	1	4	yes	—
350	#120	—	2	4	no	—
351	C₄×Q₈	—	2	1	no	—
352	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
353	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
354	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
355	C₄×D₄	—	2	2	no	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
361	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
362	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
363	C₄×D₄	D₄×C₃₁	1	2	yes	—
364	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
365	#117	—	2	1	no	—
366	#117	D₄×C₃₁	1	1	yes	characteristic
367	#117	D₄×C₃₁	1	2	yes	—
368	C₂×C₂²⋊C₄	—	2	4	no	—
369	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
371	C₄⋊C₈	—	2	1	no	—
372	C₄⋊C₈	—	2	1	no	—
373	C₄⋊C₈	—	2	2	no	—
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
376	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
377	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
380	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
381	C₂²⋊C₈	—	2	2	no	—
382	C₂²⋊C₈	—	2	2	no	—
383	C₂²⋊C₈	—	2	1	no	—
384	C₂²⋊C₈	—	2	2	no	—
385	#97	—	2	2	no	—
386	C₄×C₈	—	2	2	no	—
387	C₄×C₈	—	2	1	no	—
388	C₄×C₈	—	2	2	no	—
389	C₃₁	#17724	1	1	yes	characteristic, direct product factor, Sylow
390	C₂⁴	—	2	2	no	—
391	#41	—	2	2	no	—
392	#41	—	2	2	no	—
393	#41	—	8	2	no	—
394	#41	—	8	2	no	—
395	#41	—	8	4	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
398	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
399	C₂×D₄	—	2	4	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	4	2	no	—
407	C₂×D₄	—	4	2	no	—
408	C₂×D₄	—	8	1	no	—
409	C₂×D₄	—	8	1	no	—
410	C₂×D₄	—	8	2	no	—
411	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
412	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
413	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
417	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
418	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
419	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
420	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
422	M₁₆	—	4	2	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	4	2	no	—
430	C₂×C₈	—	8	1	no	—
431	C₂×C₈	—	8	2	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
438	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
439	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
440	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
441	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
448	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
449	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
450	C₄²	—	2	2	no	—
451	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
452	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
453	C₄²	C₂×D₄×C₃₁	1	2	yes	—
454	C₄²	C₂×D₄×C₃₁	1	2	yes	—
455	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
458	C₂³	—	2	4	no	—
459	C₂³	—	4	4	no	—
460	C₂³	—	8	2	no	—
461	C₂³	—	8	2	no	—
462	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
463	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
464	Q₈	—	2	2	no	—
465	Q₈	—	4	2	no	—
466	Q₈	—	4	1	no	—
467	Q₈	—	4	1	no	—
468	D₄	—	2	4	no	—
469	D₄	—	2	2	no	—
470	D₄	—	4	4	no	—
471	D₄	—	4	4	no	—
472	D₄	—	4	2	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	8	2	no	—
477	D₄	—	8	2	no	—
478	D₄	—	8	4	no	—
479	D₄	—	8	4	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	8	4	no	—
491	C₂×C₄	—	8	4	no	—
492	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
493	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
494	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
496	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
497	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
498	C₂×C₄	C₃₁×#135	1	2	yes	—
499	C₂×C₄	C₃₁×#135	1	2	yes	—
500	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
503	C₈	—	4	1	no	—
504	C₈	—	4	1	no	—
505	C₈	—	4	2	no	—
506	C₈	—	4	1	no	—
507	C₈	—	4	2	no	—
508	C₈	—	4	1	no	—
509	C₂²	—	2	1	no	—
510	C₂²	—	2	1	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	2	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	16	4	no	—
518	C₂²	C₃₁×#416	1	2	yes	—
519	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
520	C₄	—	2	1	no	—
521	C₄	—	2	2	no	—
522	C₄	—	2	1	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	1	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	2	no	—
528	C₄	C₃₁×#416	1	2	yes	—
529	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
530	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
531	C₄	C₃₁×#576	1	1	yes	characteristic
532	C₄	C₃₁×#577	1	1	yes	characteristic
533	C₂	—	2	2	no	—
534	C₂	—	4	2	no	—
535	C₂	—	8	4	no	—
536	C₂	C₃₁×#2888	1	1	yes	characteristic
537	C₂	C₂×C₃₁×#416	1	1	yes	characteristic
538	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×#2988	C₂	semidirect
3. C₃₁×#2925	C₂	semidirect
4. C₃₁×#1553	C₂	semidirect
5. C₃₁×#1551	C₂	semidirect
6. C₃₁×#1550	C₂	semidirect
7. C₃₁×#1545	C₂	semidirect
8. C₃₁×#1543	C₂	semidirect
9. C₃₁×#1542	C₂	semidirect
10. C₃₁×#1540	C₂	semidirect
11. C₂×C₃₁×#331	C₂	semidirect
30. C₃₁×#432	C₂²	semidirect
32. C₃₁×#431	C₂²	semidirect
33. C₃₁×#431	C₂²	semidirect
37. 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
188. #17724	C₃₁	direct (N × H)
241. #2988	C₂×C₃₁	semidirect
242. #2925	C₂×C₃₁	semidirect
243. #1553	C₂×C₃₁	semidirect
244. #1551	C₂×C₃₁	semidirect
245. #1550	C₂×C₃₁	semidirect
246. #1545	C₂×C₃₁	semidirect
247. #1543	C₂×C₃₁	semidirect
248. #1542	C₂×C₃₁	semidirect
249. #1540	C₂×C₃₁	semidirect
250. C₂×#331	C₂×C₃₁	semidirect
293. #432	C₂²×C₃₁	semidirect
295. #431	C₂²×C₃₁	semidirect
296. #431	C₂²×C₃₁	semidirect
300. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
303. #333	C₂²×C₃₁	semidirect
304. #333	C₂²×C₃₁	semidirect
305. #333	C₂²×C₃₁	semidirect
306. #332	C₂²×C₃₁	semidirect
307. #332	C₂²×C₃₁	semidirect
308. #332	C₂²×C₃₁	semidirect
389. C₃₁	#17724	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×#2988	2
C₃₁×#2925	2
C₃₁×#1553	2
C₃₁×#1551	2
C₃₁×#1550	2
C₃₁×#1545	2
C₃₁×#1543	2
C₃₁×#1542	2
C₃₁×#1540	2
C₂×C₃₁×#331	2
#17724	31

Subgroups — C31×#17724 #21474483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17724 #21474483