Subgroups — C₃₁×#20650 #21477409

Order 7936.

1602 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#518	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#418	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1804	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1802	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1736	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1735	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#518	—	2	4	no	—
14	C₃₁×#518	C₄	1	8	yes	complemented
15	C₂×C₄×D₄×C₃₁	—	2	2	no	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
22	C₃₁×#424	—	2	4	no	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	complemented
24	C₃₁×#418	—	2	4	no	—
25	C₂×C₃₁×#102	—	2	4	no	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	4	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₃₁×#389	—	2	2	no	—
30	C₃₁×#385	—	2	2	no	—
31	C₃₁×#384	C₂²	1	2	yes	complemented
32	C₃₁×#382	C₂²	1	2	yes	complemented
33	C₃₁×#340	C₂²	1	2	yes	complemented
34	C₃₁×#340	C₂²	1	4	yes	complemented
35	C₃₁×#340	C₂²	1	2	yes	complemented
36	C₂⁵×C₃₁	—	2	1	no	—
37	C₂²×D₄×C₃₁	—	2	1	no	—
38	C₂²×D₄×C₃₁	—	2	2	no	—
39	C₂²×D₄×C₃₁	—	2	2	no	—
40	C₂²×D₄×C₃₁	—	2	2	no	—
41	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
42	C₂²×D₄×C₃₁	D₄	1	4	yes	—
43	C₂²×D₄×C₃₁	Q₈	1	4	yes	—
44	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂³×C₄×C₃₁	—	2	4	no	—
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	—	2	2	no	—
52	C₂²×C₈×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₄×D₄×C₃₁	—	2	16	no	—
57	C₄×D₄×C₃₁	—	2	8	no	—
58	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
59	C₃₁×#117	—	2	2	no	—
60	C₃₁×#117	—	2	2	no	—
61	C₃₁×#117	—	2	2	no	—
62	C₃₁×#117	C₂×C₄	1	4	yes	—
63	C₃₁×#117	C₂×C₄	1	4	yes	—
64	C₃₁×#117	D₄	1	4	yes	—
65	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
69	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
79	C₂×C₄²×C₃₁	—	2	1	no	—
80	C₂×C₄²×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
82	C₃₁×C₄⋊C₈	—	2	8	no	—
83	C₃₁×#102	—	2	8	no	—
84	C₃₁×#102	—	2	4	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#95	—	2	4	no	—
87	C₂⁴×C₃₁	—	2	2	no	—
88	C₂⁴×C₃₁	—	2	4	no	—
89	C₂⁴×C₃₁	—	2	4	no	—
90	C₂⁴×C₃₁	—	2	4	no	—
91	C₂⁴×C₃₁	—	8	2	no	—
92	C₂⁴×C₃₁	—	8	2	no	—
93	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×D₄×C₃₁	—	2	4	no	—
95	C₂×D₄×C₃₁	—	2	8	no	—
96	C₂×D₄×C₃₁	—	2	8	no	—
97	C₂×D₄×C₃₁	—	2	4	no	—
98	C₂×D₄×C₃₁	—	2	8	no	—
99	C₂×D₄×C₃₁	—	2	4	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	C₄⋊C₄	1	16	yes	complemented
104	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
105	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
106	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	16	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	4	2	no	—
113	C₂²×C₄×C₃₁	—	8	4	no	—
114	C₂²×C₄×C₃₁	—	8	4	no	—
115	C₂²×C₄×C₃₁	—	8	2	no	—
116	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
122	C₂²×C₄×C₃₁	#41	1	2	yes	—
123	C₂²×C₄×C₃₁	#41	1	2	yes	—
124	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	M₁₆×C₃₁	—	4	4	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	4	4	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	—	8	4	no	—
142	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
143	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
144	C₂²⋊C₄×C₃₁	—	2	4	no	—
145	C₂²⋊C₄×C₃₁	—	2	16	no	—
146	C₂²⋊C₄×C₃₁	—	4	2	no	—
147	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
148	C₄²×C₃₁	—	2	4	no	—
149	C₄²×C₃₁	—	4	2	no	—
150	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
151	C₄²×C₃₁	C₂×D₄	1	4	yes	—
152	#20650	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	—	2	4	no	—
156	C₂³×C₃₁	—	2	4	no	—
157	C₂³×C₃₁	—	2	4	no	—
158	C₂³×C₃₁	—	2	16	no	—
159	C₂³×C₃₁	—	4	4	no	—
160	C₂³×C₃₁	—	4	4	no	—
161	C₂³×C₃₁	—	8	8	no	—
162	C₂³×C₃₁	—	8	8	no	—
163	C₂³×C₃₁	—	8	4	no	—
164	C₂³×C₃₁	—	8	4	no	—
165	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂³×C₃₁	#121	1	2	yes	—
167	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
168	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
169	D₄×C₃₁	—	2	4	no	—
170	D₄×C₃₁	—	2	4	no	—
171	D₄×C₃₁	—	2	4	no	—
172	D₄×C₃₁	—	2	16	no	—
173	D₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	4	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	4	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	8	16	no	—
186	C₂×C₄×C₃₁	—	8	4	no	—
187	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
188	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
189	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
190	C₂×C₄×C₃₁	#117	1	4	yes	—
191	C₂×C₄×C₃₁	#121	1	4	yes	—
192	C₂×C₄×C₃₁	#121	1	2	yes	—
193	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
194	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
195	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
201	C₈×C₃₁	—	4	2	no	—
202	C₈×C₃₁	—	4	2	no	—
203	C₂×#518	C₂×C₃₁	1	1	yes	characteristic
204	C₂×#424	C₂×C₃₁	1	1	yes	characteristic, complemented
205	C₂×#418	C₂×C₃₁	1	1	yes	characteristic
206	C₂²×#102	C₂×C₃₁	1	1	yes	characteristic
207	#2317	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1804	C₂×C₃₁	1	4	yes	complemented
209	#1802	C₂×C₃₁	1	4	yes	complemented
210	#1736	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1735	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₂²×C₃₁	—	2	4	no	—
213	C₂²×C₃₁	—	2	4	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	2	8	no	—
217	C₂²×C₃₁	—	4	8	no	—
218	C₂²×C₃₁	—	4	8	no	—
219	C₂²×C₃₁	—	4	4	no	—
220	C₂²×C₃₁	—	4	4	no	—
221	C₂²×C₃₁	—	8	16	no	—
222	C₂²×C₃₁	#380	1	4	yes	—
223	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
224	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
225	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
226	C₂²×C₃₁	#576	1	2	yes	—
227	C₂²×C₃₁	#577	1	2	yes	—
228	C₄×C₃₁	—	2	8	no	—
229	C₄×C₃₁	—	4	4	no	—
230	C₄×C₃₁	—	8	1	no	—
231	C₄×C₃₁	—	8	2	no	—
232	C₄×C₃₁	—	8	1	no	—
233	C₄×C₃₁	#380	1	4	yes	—
234	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
235	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
236	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
237	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
238	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
239	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
240	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
241	#518	—	2	4	no	—
242	#518	C₄×C₃₁	1	8	yes	complemented
243	C₂×C₄×D₄	—	2	2	no	—
244	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
245	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
248	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
249	C₂²×C₂²⋊C₄	—	2	2	no	—
250	#424	—	2	4	no	—
251	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	complemented
252	#418	—	2	4	no	—
253	C₂×#102	—	2	4	no	—
254	C₂×#102	C₂²×C₃₁	1	2	yes	—
255	C₂×#102	C₂²×C₃₁	1	4	yes	—
256	C₂×#102	C₂²×C₃₁	1	2	yes	—
257	#389	—	2	2	no	—
258	#385	—	2	2	no	—
259	#384	C₂²×C₃₁	1	2	yes	complemented
260	#382	C₂²×C₃₁	1	2	yes	complemented
261	#340	C₂²×C₃₁	1	2	yes	complemented
262	#340	C₂²×C₃₁	1	4	yes	complemented
263	#340	C₂²×C₃₁	1	2	yes	complemented
264	C₂×C₃₁	—	2	4	no	—
265	C₂×C₃₁	—	4	8	no	—
266	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
267	C₂×C₃₁	#2897	1	2	yes	—
268	C₂×C₃₁	#3041	1	1	yes	characteristic
269	C₂×C₃₁	#3042	1	1	yes	characteristic
270	C₂×C₃₁	#3081	1	2	yes	—
271	C₂⁵	—	2	1	no	—
272	C₂²×D₄	—	2	1	no	—
273	C₂²×D₄	—	2	2	no	—
274	C₂²×D₄	—	2	2	no	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
277	C₂²×D₄	D₄×C₃₁	1	4	yes	—
278	C₂²×D₄	Q₈×C₃₁	1	4	yes	—
279	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
280	C₂³×C₄	—	2	4	no	—
281	C₂³×C₄	—	2	1	no	—
282	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×M₁₆	—	2	2	no	—
284	C₂×M₁₆	—	2	2	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂²×C₈	—	2	2	no	—
287	C₂²×C₈	—	2	2	no	—
288	C₂²×C₈	—	2	2	no	—
289	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
290	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
291	C₄×D₄	—	2	16	no	—
292	C₄×D₄	—	2	8	no	—
293	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
294	#117	—	2	2	no	—
295	#117	—	2	2	no	—
296	#117	—	2	2	no	—
297	#117	C₂×C₄×C₃₁	1	4	yes	—
298	#117	C₂×C₄×C₃₁	1	4	yes	—
299	#117	D₄×C₃₁	1	4	yes	—
300	C₂×C₄⋊C₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	4	no	—
304	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
309	C₂×C₂²⋊C₄	—	2	1	no	—
310	C₂×C₂²⋊C₄	—	2	4	no	—
311	C₂×C₂²⋊C₄	—	2	16	no	—
312	C₂×C₂²⋊C₄	—	2	1	no	—
313	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	—	2	1	no	—
316	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
317	C₄⋊C₈	—	2	8	no	—
318	#102	—	2	8	no	—
319	#102	—	2	4	no	—
320	#102	—	2	4	no	—
321	#95	—	2	4	no	—
322	C₃₁	#20650	1	1	yes	characteristic, direct product factor, Sylow
323	C₂⁴	—	2	2	no	—
324	C₂⁴	—	2	4	no	—
325	C₂⁴	—	2	4	no	—
326	C₂⁴	—	2	4	no	—
327	C₂⁴	—	8	2	no	—
328	C₂⁴	—	8	2	no	—
329	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂×D₄	—	2	4	no	—
331	C₂×D₄	—	2	8	no	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	2	4	no	—
334	C₂×D₄	—	2	8	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	C₄⋊C₄×C₃₁	1	16	yes	complemented
340	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
341	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
342	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	16	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	—	8	4	no	—
350	C₂²×C₄	—	8	4	no	—
351	C₂²×C₄	—	8	2	no	—
352	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
357	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
358	C₂²×C₄	#41×C₃₁	1	2	yes	—
359	C₂²×C₄	#41×C₃₁	1	2	yes	—
360	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
361	M₁₆	—	4	4	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	4	4	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	—	8	4	no	—
378	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
379	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
380	C₂²⋊C₄	—	2	4	no	—
381	C₂²⋊C₄	—	2	16	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
384	C₄²	—	2	4	no	—
385	C₄²	—	4	2	no	—
386	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
387	C₄²	C₂×D₄×C₃₁	1	4	yes	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	2	4	no	—
391	C₂³	—	2	4	no	—
392	C₂³	—	2	4	no	—
393	C₂³	—	2	16	no	—
394	C₂³	—	4	4	no	—
395	C₂³	—	4	4	no	—
396	C₂³	—	8	8	no	—
397	C₂³	—	8	8	no	—
398	C₂³	—	8	4	no	—
399	C₂³	—	8	4	no	—
400	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
401	C₂³	C₃₁×#121	1	2	yes	—
402	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
403	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
404	D₄	—	2	4	no	—
405	D₄	—	2	4	no	—
406	D₄	—	2	4	no	—
407	D₄	—	2	16	no	—
408	D₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	4	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	8	16	no	—
421	C₂×C₄	—	8	4	no	—
422	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
423	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
424	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
425	C₂×C₄	C₃₁×#117	1	4	yes	—
426	C₂×C₄	C₃₁×#121	1	4	yes	—
427	C₂×C₄	C₃₁×#121	1	2	yes	—
428	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
429	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
430	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
436	C₈	—	4	2	no	—
437	C₈	—	4	2	no	—
438	C₂²	—	2	4	no	—
439	C₂²	—	2	4	no	—
440	C₂²	—	2	2	no	—
441	C₂²	—	2	2	no	—
442	C₂²	—	2	8	no	—
443	C₂²	—	4	8	no	—
444	C₂²	—	4	8	no	—
445	C₂²	—	4	4	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	8	16	no	—
448	C₂²	C₃₁×#380	1	4	yes	—
449	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
450	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
451	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
452	C₂²	C₃₁×#576	1	2	yes	—
453	C₂²	C₃₁×#577	1	2	yes	—
454	C₄	—	2	8	no	—
455	C₄	—	4	4	no	—
456	C₄	—	8	2	no	—
457	C₄	—	8	1	no	—
458	C₄	—	8	1	no	—
459	C₄	C₃₁×#380	1	4	yes	—
460	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
461	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
462	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
463	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
464	C₂	—	2	4	no	—
465	C₂	—	4	8	no	—
466	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
467	C₂	C₃₁×#2897	1	2	yes	—
468	C₂	C₃₁×#3041	1	1	yes	characteristic
469	C₂	C₃₁×#3042	1	1	yes	characteristic
470	C₂	C₃₁×#3081	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#424	C₂	semidirect
5. C₃₁×#2317	C₂	semidirect
6. C₃₁×#1804	C₂	semidirect
7. C₃₁×#1802	C₂	semidirect
8. C₃₁×#1736	C₂	semidirect
9. C₃₁×#1735	C₂	semidirect
14. C₃₁×#518	C₄	semidirect
23. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
31. C₃₁×#384	C₂²	semidirect
32. C₃₁×#382	C₂²	semidirect
33. C₃₁×#340	C₂²	semidirect
34. C₃₁×#340	C₂²	semidirect
35. C₃₁×#340	C₂²	semidirect
58. C₄×D₄×C₃₁	C₂×C₄	semidirect
103. C₂×D₄×C₃₁	C₄⋊C₄	semidirect
152. #20650	C₃₁	direct (N × H)
204. C₂×#424	C₂×C₃₁	semidirect
207. #2317	C₂×C₃₁	semidirect
208. #1804	C₂×C₃₁	semidirect
209. #1802	C₂×C₃₁	semidirect
210. #1736	C₂×C₃₁	semidirect
211. #1735	C₂×C₃₁	semidirect
242. #518	C₄×C₃₁	semidirect
251. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
259. #384	C₂²×C₃₁	semidirect
260. #382	C₂²×C₃₁	semidirect
261. #340	C₂²×C₃₁	semidirect
262. #340	C₂²×C₃₁	semidirect
263. #340	C₂²×C₃₁	semidirect
293. C₄×D₄	C₂×C₄×C₃₁	semidirect
322. C₃₁	#20650	direct (N × H)
339. C₂×D₄	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#518	2
C₂×C₃₁×#424	2
C₂×C₃₁×#418	2
C₂²×C₃₁×#102	2
C₃₁×#2317	2
C₃₁×#1804	2
C₃₁×#1802	2
C₃₁×#1736	2
C₃₁×#1735	2
#20650	31

Subgroups — C31×#20650 #21477409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20650 #21477409