Subgroups — C₃₁×#9645 #21466404

Order 7936.

558 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×D₄⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₃₁×#124	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#391	C₄	1	1	yes	characteristic
12	C₃₁×#390	C₄	1	1	yes	characteristic
13	C₃₁×#388	—	2	2	no	—
14	C₃₁×#382	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
16	C₄³×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#336	—	2	2	no	—
18	C₃₁×#336	—	2	2	no	—
19	C₃₁×Q₈⋊C₈	—	2	4	no	—
20	C₃₁×D₄⋊C₈	—	2	4	no	—
21	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₁	—	2	1	no	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₃₁×#124	—	2	2	no	—
27	C₃₁×#124	—	2	2	no	—
28	C₄×Q₈×C₃₁	—	2	2	no	—
29	C₄×Q₈×C₃₁	—	2	2	no	—
30	C₄×D₄×C₃₁	—	2	2	no	—
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
38	C₂×C₄²×C₃₁	—	2	1	no	—
39	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	D₄	1	2	yes	—
42	C₂×C₄²×C₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	—	2	2	no	—
46	C₄×C₈×C₃₁	—	2	2	no	—
47	C₄×C₈×C₃₁	D₄	1	4	yes	—
48	C₃₁×#95	—	2	2	no	—
49	C₃₁×#95	—	2	2	no	—
50	C₂⁴×C₃₁	—	2	1	no	—
51	C₂×Q₈×C₃₁	—	2	1	no	—
52	C₂×Q₈×C₃₁	—	2	2	no	—
53	C₂×Q₈×C₃₁	—	2	2	no	—
54	C₂×Q₈×C₃₁	M₁₆	1	2	yes	—
55	C₂×D₄×C₃₁	—	2	2	no	—
56	C₂×D₄×C₃₁	—	2	1	no	—
57	C₂×D₄×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	M₁₆	1	2	yes	—
59	C₂²×C₄×C₃₁	—	2	1	no	—
60	C₂²×C₄×C₃₁	—	2	1	no	—
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	—	4	1	no	—
64	C₂²×C₄×C₃₁	—	4	4	no	—
65	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	4	4	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	—	4	4	no	—
74	C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₄⋊C₄×C₃₁	—	4	2	no	—
77	C₂²⋊C₄×C₃₁	—	2	8	no	—
78	C₂²⋊C₄×C₃₁	—	4	4	no	—
79	C₂²⋊C₄×C₃₁	—	4	2	no	—
80	C₄²×C₃₁	—	2	1	no	—
81	C₄²×C₃₁	—	2	1	no	—
82	C₄²×C₃₁	—	2	2	no	—
83	C₄²×C₃₁	—	2	4	no	—
84	C₄²×C₃₁	—	4	2	no	—
85	C₄²×C₃₁	—	4	1	no	—
86	C₄²×C₃₁	—	4	1	no	—
87	C₄²×C₃₁	C₂×D₄	1	2	yes	—
88	C₄²×C₃₁	#41	1	2	yes	—
89	#9645	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	2	no	—
91	C₂³×C₃₁	—	4	1	no	—
92	C₂³×C₃₁	—	4	1	no	—
93	C₂³×C₃₁	—	4	2	no	—
94	C₂³×C₃₁	—	4	2	no	—
95	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	Q₈×C₃₁	—	2	4	no	—
97	Q₈×C₃₁	—	4	2	no	—
98	D₄×C₃₁	—	2	4	no	—
99	D₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	4	1	no	—
109	C₂×C₄×C₃₁	—	4	1	no	—
110	C₂×C₄×C₃₁	—	4	1	no	—
111	C₂×C₄×C₃₁	—	4	1	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	8	2	no	—
115	C₂×C₄×C₃₁	—	8	2	no	—
116	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₃₁	#121	1	2	yes	—
118	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
119	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#131	1	2	yes	—
123	C₂×C₄×C₃₁	#135	1	4	yes	—
124	C₈×C₃₁	—	4	4	no	—
125	C₈×C₃₁	—	8	2	no	—
126	C₄×#124	C₂×C₃₁	1	1	yes	characteristic
127	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
129	C₂×Q₈⋊C₈	C₂×C₃₁	1	2	yes	complemented
130	C₂×D₄⋊C₈	C₂×C₃₁	1	2	yes	—
131	C₂²×C₃₁	—	4	2	no	—
132	C₂²×C₃₁	—	4	4	no	—
133	C₂²×C₃₁	—	8	1	no	—
134	C₂²×C₃₁	—	8	1	no	—
135	C₂²×C₃₁	—	8	2	no	—
136	C₂²×C₃₁	#390	1	1	yes	characteristic
137	C₂²×C₃₁	#407	1	1	yes	characteristic
138	C₂²×C₃₁	#408	1	1	yes	characteristic
139	C₂²×C₃₁	#416	1	2	yes	—
140	C₂²×C₃₁	#421	1	2	yes	—
141	C₄×C₃₁	—	2	2	no	—
142	C₄×C₃₁	—	4	2	no	—
143	C₄×C₃₁	—	4	2	no	—
144	C₄×C₃₁	—	4	2	no	—
145	C₄×C₃₁	—	8	1	no	—
146	C₄×C₃₁	—	8	1	no	—
147	C₄×C₃₁	#435	1	2	yes	—
148	C₄×C₃₁	#435	1	2	yes	—
149	C₄×C₃₁	#488	1	2	yes	—
150	C₄×C₃₁	#495	1	2	yes	—
151	C₂×#124	C₄×C₃₁	1	1	yes	characteristic
152	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
156	#391	C₄×C₃₁	1	1	yes	characteristic
157	#390	C₄×C₃₁	1	1	yes	characteristic
158	#388	—	2	2	no	—
159	#382	C₄×C₃₁	1	1	yes	characteristic
160	C₄×C₂²⋊C₄	—	2	2	no	—
161	C₄³	C₂²×C₃₁	1	1	yes	characteristic
162	#336	—	2	2	no	—
163	#336	—	2	2	no	—
164	Q₈⋊C₈	—	2	4	no	—
165	D₄⋊C₈	—	2	4	no	—
166	C₂×C₃₁	—	8	2	no	—
167	C₂×C₃₁	#1482	1	2	yes	—
168	C₂×C₃₁	#1488	1	2	yes	—
169	C₂×C₃₁	#1955	1	1	yes	characteristic
170	C₂×C₃₁	#1958	1	1	yes	characteristic
171	C₂×C₃₁	#1963	1	1	yes	characteristic
172	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
173	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₂³×C₄	—	2	1	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	#124	—	2	2	no	—
178	#124	—	2	2	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×Q₈	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	1	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₃₁	1	2	yes	—
193	C₂×C₄²	D₄×C₃₁	1	2	yes	—
194	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	D₄×C₃₁	1	4	yes	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	C₃₁	#9645	1	1	yes	characteristic, direct product factor, Sylow
202	C₂⁴	—	2	1	no	—
203	C₂×Q₈	—	2	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	M₁₆×C₃₁	1	2	yes	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	M₁₆×C₃₁	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
218	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	4	4	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	4	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	2	8	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂×D₄×C₃₁	1	2	yes	—
240	C₄²	#41×C₃₁	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	4	1	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	1	no	—
246	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	D₄	—	2	4	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
268	C₂×C₄	C₃₁×#121	1	2	yes	—
269	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
270	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
271	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
273	C₂×C₄	C₃₁×#131	1	2	yes	—
274	C₂×C₄	C₃₁×#135	1	4	yes	—
275	C₈	—	4	4	no	—
276	C₈	—	8	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	8	1	no	—
280	C₂²	—	8	1	no	—
281	C₂²	—	8	2	no	—
282	C₂²	C₃₁×#390	1	1	yes	characteristic
283	C₂²	C₃₁×#407	1	1	yes	characteristic
284	C₂²	C₃₁×#408	1	1	yes	characteristic
285	C₂²	C₃₁×#416	1	2	yes	—
286	C₂²	C₃₁×#421	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	2	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	C₃₁×#435	1	2	yes	—
294	C₄	C₃₁×#435	1	2	yes	—
295	C₄	C₃₁×#488	1	2	yes	—
296	C₄	C₃₁×#495	1	2	yes	—
297	C₂	—	8	2	no	—
298	C₂	C₃₁×#1482	1	2	yes	—
299	C₂	C₃₁×#1488	1	2	yes	—
300	C₂	C₃₁×#1955	1	1	yes	characteristic
301	C₂	C₃₁×#1958	1	1	yes	characteristic
302	C₂	C₃₁×#1963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1831	C₂	semidirect
3. C₃₁×#1744	C₂	semidirect
4. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
89. #9645	C₃₁	direct (N × H)
127. #1831	C₂×C₃₁	semidirect
128. #1744	C₂×C₃₁	semidirect
129. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
201. C₃₁	#9645	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#124	2
C₃₁×#1831	2
C₃₁×#1744	2
C₂×C₃₁×Q₈⋊C₈	2
C₂×C₃₁×D₄⋊C₈	2
#9645	31

Subgroups — C31×#9645 #21466404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9645 #21466404