Subgroups — C₃₁×#22708 #21479467

Order 7936.

910 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2354	C₂	1	2	yes	maximal
3	C₃₁×#1948	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1854	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1807	C₂	1	2	yes	maximal, complemented
7	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1735	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#117	C₂²	1	4	yes	—
12	C₈×D₄×C₃₁	C₄	1	8	yes	complemented
13	C₂×C₃₁×#107	C₂²	1	2	yes	—
14	C₂×C₃₁×#106	C₂²	1	4	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#390	—	2	2	no	—
19	C₃₁×#388	—	2	4	no	—
20	C₃₁×#384	C₂²	1	2	yes	—
21	C₃₁×#380	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#340	C₂²	1	1	yes	characteristic
25	C₃₁×#340	C₂²	1	2	yes	—
26	C₃₁×#340	C₂²	1	2	yes	—
27	C₃₁×#340	C₂²	1	4	yes	—
28	C₃₁×#340	C₂²	1	1	yes	characteristic
29	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₃₁	—	4	4	no	—
32	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
37	C₄×D₄×C₃₁	Q₈	1	4	yes	—
38	C₃₁×#117	—	2	8	no	—
39	C₃₁×#117	—	4	4	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
47	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
48	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
49	C₂×C₄²×C₃₁	—	2	4	no	—
50	C₂×C₄²×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	—	2	4	no	—
53	C₃₁×#106	—	4	4	no	—
54	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
55	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
57	C₃₁×#95	—	2	4	no	—
58	C₃₁×#95	—	2	2	no	—
59	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×D₄×C₃₁	—	2	2	no	—
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₃₁	—	2	4	no	—
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	4	no	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	4	8	no	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	4	4	no	—
87	C₄⋊C₄×C₃₁	—	8	2	no	—
88	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
89	C₄⋊C₄×C₃₁	#41	1	4	yes	—
90	C₂²⋊C₄×C₃₁	—	4	8	no	—
91	C₂²⋊C₄×C₃₁	—	8	4	no	—
92	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
93	C₄²×C₃₁	—	4	8	no	—
94	C₄²×C₃₁	—	4	4	no	—
95	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
96	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
97	#22708	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	2	4	no	—
99	C₂³×C₃₁	—	2	2	no	—
100	C₂³×C₃₁	—	2	2	no	—
101	C₂³×C₃₁	—	2	4	no	—
102	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
103	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₃₁	—	2	8	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	2	8	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	4	no	—
119	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₃₁	#135	1	2	yes	—
123	C₂×C₄×C₃₁	#135	1	2	yes	—
124	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
127	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
128	C₈×C₃₁	—	2	2	no	—
129	C₈×C₃₁	—	2	2	no	—
130	C₈×C₃₁	—	2	4	no	—
131	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
132	#2354	C₂×C₃₁	1	2	yes	—
133	#1948	C₂×C₃₁	1	2	yes	complemented
134	#1854	C₂×C₃₁	1	4	yes	complemented
135	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#1807	C₂×C₃₁	1	2	yes	complemented
137	C₄×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#1735	C₂×C₃₁	1	2	yes	complemented
139	C₂²×C₃₁	—	2	4	no	—
140	C₂²×C₃₁	—	2	4	no	—
141	C₂²×C₃₁	—	2	4	no	—
142	C₂²×C₃₁	—	4	4	no	—
143	C₂²×C₃₁	—	4	4	no	—
144	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₃₁	#546	1	2	yes	—
146	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
147	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
148	C₂²×C₃₁	#577	1	2	yes	—
149	C₄×C₃₁	—	2	4	no	—
150	C₄×C₃₁	—	2	2	no	—
151	C₄×C₃₁	—	4	4	no	—
152	C₄×C₃₁	—	8	2	no	—
153	C₄×C₃₁	—	8	4	no	—
154	C₄×C₃₁	#427	1	4	yes	—
155	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
156	C₄×C₃₁	#518	1	1	yes	characteristic
157	C₄×C₃₁	#554	1	2	yes	—
158	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
159	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×#117	C₂²×C₃₁	1	4	yes	—
161	C₈×D₄	C₄×C₃₁	1	8	yes	complemented
162	C₂×#107	C₂²×C₃₁	1	2	yes	—
163	C₂×#106	C₂²×C₃₁	1	4	yes	—
164	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
167	#390	—	2	2	no	—
168	#388	—	2	4	no	—
169	#384	C₂²×C₃₁	1	2	yes	—
170	#380	C₂²×C₃₁	1	4	yes	—
171	C₄×C₄⋊C₄	—	2	2	no	—
172	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
173	#340	C₂²×C₃₁	1	1	yes	characteristic
174	#340	C₂²×C₃₁	1	2	yes	—
175	#340	C₂²×C₃₁	1	2	yes	—
176	#340	C₂²×C₃₁	1	4	yes	—
177	#340	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×C₃₁	—	4	4	no	—
179	C₂×C₃₁	#2349	1	1	yes	characteristic
180	C₂×C₃₁	C₂×#427	1	1	yes	characteristic
181	C₂×C₃₁	#2912	1	1	yes	characteristic
182	C₂×C₃₁	#3303	1	2	yes	—
183	C₂×C₃₁	#3304	1	2	yes	—
184	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
186	C₂²×C₈	—	4	4	no	—
187	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
188	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
190	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
191	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
192	C₄×D₄	Q₈×C₃₁	1	4	yes	—
193	#117	—	2	8	no	—
194	#117	—	4	4	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
198	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
199	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
201	C₂×C₂²⋊C₄	—	2	4	no	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
207	#107	—	2	4	no	—
208	#106	—	4	4	no	—
209	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
210	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
211	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
212	#95	—	2	4	no	—
213	#95	—	2	2	no	—
214	C₃₁	#22708	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
225	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
229	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	8	no	—
234	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
235	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	8	2	no	—
244	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
245	C₄⋊C₄	#41×C₃₁	1	4	yes	—
246	C₂²⋊C₄	—	4	8	no	—
247	C₂²⋊C₄	—	8	4	no	—
248	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
249	C₄²	—	4	8	no	—
250	C₄²	—	4	4	no	—
251	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
252	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
253	C₂³	—	2	4	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	4	no	—
257	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
258	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
259	D₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
276	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
277	C₂×C₄	C₃₁×#135	1	2	yes	—
278	C₂×C₄	C₃₁×#135	1	2	yes	—
279	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
282	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
292	C₂²	C₃₁×#546	1	2	yes	—
293	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
294	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
295	C₂²	C₃₁×#577	1	2	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	4	no	—
301	C₄	C₃₁×#427	1	4	yes	—
302	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
303	C₄	C₃₁×#518	1	1	yes	characteristic
304	C₄	C₃₁×#554	1	2	yes	—
305	C₂	—	4	4	no	—
306	C₂	C₃₁×#2349	1	1	yes	characteristic
307	C₂	C₂×C₃₁×#427	1	1	yes	characteristic
308	C₂	C₃₁×#2912	1	1	yes	characteristic
309	C₂	C₃₁×#3303	1	2	yes	—
310	C₂	C₃₁×#3304	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	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1948	C₂	semidirect
4. C₃₁×#1854	C₂	semidirect
5. C₃₁×#1834	C₂	semidirect
6. C₃₁×#1807	C₂	semidirect
7. C₄×C₃₁×#107	C₂	semidirect
8. C₃₁×#1735	C₂	semidirect
12. C₈×D₄×C₃₁	C₄	semidirect
97. #22708	C₃₁	direct (N × H)
133. #1948	C₂×C₃₁	semidirect
134. #1854	C₂×C₃₁	semidirect
135. #1834	C₂×C₃₁	semidirect
136. #1807	C₂×C₃₁	semidirect
137. C₄×#107	C₂×C₃₁	semidirect
138. #1735	C₂×C₃₁	semidirect
161. C₈×D₄	C₄×C₃₁	semidirect
214. C₃₁	#22708	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₃₁	2
C₃₁×#2354	2
C₃₁×#1948	2
C₃₁×#1854	2
C₃₁×#1834	2
C₃₁×#1807	2
C₄×C₃₁×#107	2
C₃₁×#1735	2
#22708	31

Subgroups — C31×#22708 #21479467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22708 #21479467