Subgroups — C₃₁×#9644 #21466403

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×D₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#127	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₃₁	—	2	2	no	—
7	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#390	—	2	2	no	—
12	C₃₁×#390	C₄	1	2	yes	—
13	C₃₁×#384	C₄	1	1	yes	characteristic
14	C₄³×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#336	—	2	2	no	—
16	C₃₁×D₄⋊C₈	—	2	8	no	—
17	C₂²×D₄×C₃₁	—	2	2	no	—
18	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₃₁	—	2	2	no	—
20	C₂²×C₈×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	—	2	2	no	—
22	C₃₁×#127	—	2	4	no	—
23	C₄×D₄×C₃₁	—	2	4	no	—
24	C₄×D₄×C₃₁	—	2	4	no	—
25	C₄×D₄×C₃₁	—	4	4	no	—
26	C₄×D₄×C₃₁	—	4	4	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
30	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	D₄	1	2	yes	—
34	C₂×C₄²×C₃₁	D₄	1	2	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×C₄⋊C₈	—	2	4	no	—
37	C₃₁×C₄⋊C₈	—	4	4	no	—
38	C₄×C₈×C₃₁	—	2	2	no	—
39	C₄×C₈×C₃₁	D₄	1	4	yes	complemented
40	C₂⁴×C₃₁	—	2	2	no	—
41	C₂×D₄×C₃₁	—	2	8	no	—
42	C₂×D₄×C₃₁	—	2	4	no	—
43	C₂×D₄×C₃₁	—	2	2	no	—
44	C₂×D₄×C₃₁	—	2	4	no	—
45	C₂×D₄×C₃₁	—	4	4	no	—
46	C₂×D₄×C₃₁	—	4	4	no	—
47	C₂×D₄×C₃₁	M₁₆	1	4	yes	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	—	4	2	no	—
51	C₂²×C₄×C₃₁	—	4	4	no	—
52	C₂²×C₄×C₃₁	—	4	4	no	—
53	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	—	4	4	no	—
60	C₂×C₈×C₃₁	—	4	2	no	—
61	C₂×C₈×C₃₁	—	4	4	no	—
62	C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₄⋊C₄×C₃₁	—	4	4	no	—
64	C₂²⋊C₄×C₃₁	—	4	8	no	—
65	C₂²⋊C₄×C₃₁	—	4	4	no	—
66	C₄²×C₃₁	—	2	2	no	—
67	C₄²×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	4	2	no	—
69	C₄²×C₃₁	D₈	1	4	yes	—
70	C₄²×C₃₁	SD₁₆	1	4	yes	—
71	C₄²×C₃₁	C₂×D₄	1	2	yes	—
72	C₄²×C₃₁	#41	1	2	yes	—
73	#9644	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₃₁	—	2	4	no	—
75	C₂³×C₃₁	—	4	2	no	—
76	C₂³×C₃₁	—	4	2	no	—
77	C₂³×C₃₁	—	4	4	no	—
78	C₂³×C₃₁	—	4	4	no	—
79	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	D₄×C₃₁	—	2	8	no	—
81	D₄×C₃₁	—	4	16	no	—
82	D₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	4	2	no	—
89	C₂×C₄×C₃₁	—	4	4	no	—
90	C₂×C₄×C₃₁	—	8	4	no	—
91	C₂×C₄×C₃₁	—	8	2	no	—
92	C₂×C₄×C₃₁	—	8	2	no	—
93	C₂×C₄×C₃₁	#102	1	4	yes	—
94	C₂×C₄×C₃₁	#102	1	4	yes	—
95	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₃₁	#121	1	2	yes	—
97	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
98	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₃₁	#131	1	2	yes	—
101	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
103	C₈×C₃₁	—	4	4	no	—
104	C₈×C₃₁	—	8	2	no	—
105	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
106	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
107	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂×D₄⋊C₈	C₂×C₃₁	1	4	yes	complemented
109	C₂²×C₃₁	—	4	4	no	—
110	C₂²×C₃₁	—	4	8	no	—
111	C₂²×C₃₁	—	8	2	no	—
112	C₂²×C₃₁	—	8	2	no	—
113	C₂²×C₃₁	—	8	4	no	—
114	C₂²×C₃₁	#390	1	1	yes	characteristic
115	C₂²×C₃₁	#407	1	1	yes	characteristic
116	C₂²×C₃₁	#408	1	1	yes	characteristic
117	C₂²×C₃₁	C₂×#102	1	2	yes	—
118	C₂²×C₃₁	#421	1	2	yes	—
119	C₄×C₃₁	—	2	2	no	—
120	C₄×C₃₁	—	2	4	no	—
121	C₄×C₃₁	—	2	4	no	—
122	C₄×C₃₁	—	8	2	no	—
123	C₄×C₃₁	#435	1	4	yes	—
124	C₄×C₃₁	#486	1	2	yes	—
125	C₄×C₃₁	#492	1	1	yes	characteristic
126	C₄×C₃₁	#493	1	1	yes	characteristic
127	C₂×#127	C₄×C₃₁	1	1	yes	characteristic
128	C₂×C₄×D₄	—	2	2	no	—
129	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
130	C₂×C₄⋊C₈	—	2	2	no	—
131	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
133	#390	—	2	2	no	—
134	#390	C₄×C₃₁	1	2	yes	—
135	#384	C₄×C₃₁	1	1	yes	characteristic
136	C₄³	C₂²×C₃₁	1	1	yes	characteristic
137	#336	—	2	2	no	—
138	D₄⋊C₈	—	2	8	no	—
139	C₂×C₃₁	—	8	4	no	—
140	C₂×C₃₁	#1480	1	2	yes	—
141	C₂×C₃₁	#1486	1	2	yes	—
142	C₂×C₃₁	#1955	1	1	yes	characteristic
143	C₂×C₃₁	#1957	1	1	yes	characteristic
144	C₂×C₃₁	#1963	1	1	yes	characteristic
145	C₂²×D₄	—	2	2	no	—
146	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
147	C₂³×C₄	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	#127	—	2	4	no	—
151	C₄×D₄	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₄×D₄	—	4	4	no	—
154	C₄×D₄	—	4	4	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
157	C₂×C₂²⋊C₄	—	2	4	no	—
158	C₂×C₂²⋊C₄	—	2	2	no	—
159	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
161	C₂×C₄²	D₄×C₃₁	1	2	yes	—
162	C₂×C₄²	D₄×C₃₁	1	2	yes	—
163	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	4	4	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	D₄×C₃₁	1	4	yes	complemented
168	C₃₁	#9644	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁴	—	2	2	no	—
170	C₂×D₄	—	2	8	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	M₁₆×C₃₁	1	4	yes	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
183	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
184	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₂²⋊C₄	—	4	8	no	—
194	C₂²⋊C₄	—	4	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	D₈×C₃₁	1	4	yes	—
199	C₄²	SD₁₆×C₃₁	1	4	yes	—
200	C₄²	C₂×D₄×C₃₁	1	2	yes	—
201	C₄²	#41×C₃₁	1	2	yes	—
202	C₂³	—	2	4	no	—
203	C₂³	—	4	2	no	—
204	C₂³	—	4	4	no	—
205	C₂³	—	4	4	no	—
206	C₂³	—	4	2	no	—
207	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
208	D₄	—	2	8	no	—
209	D₄	—	4	16	no	—
210	D₄	—	4	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₃₁×#102	1	4	yes	—
222	C₂×C₄	C₃₁×#102	1	4	yes	—
223	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
224	C₂×C₄	C₃₁×#121	1	2	yes	—
225	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
226	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
227	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
228	C₂×C₄	C₃₁×#131	1	2	yes	—
229	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
230	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
231	C₈	—	4	4	no	—
232	C₈	—	8	2	no	—
233	C₂²	—	4	4	no	—
234	C₂²	—	4	8	no	—
235	C₂²	—	8	2	no	—
236	C₂²	—	8	2	no	—
237	C₂²	—	8	4	no	—
238	C₂²	C₃₁×#390	1	1	yes	characteristic
239	C₂²	C₃₁×#407	1	1	yes	characteristic
240	C₂²	C₃₁×#408	1	1	yes	characteristic
241	C₂²	C₂×C₃₁×#102	1	2	yes	—
242	C₂²	C₃₁×#421	1	2	yes	—
243	C₄	—	2	2	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	4	no	—
246	C₄	—	8	2	no	—
247	C₄	C₃₁×#435	1	4	yes	—
248	C₄	C₃₁×#486	1	2	yes	—
249	C₄	C₃₁×#492	1	1	yes	characteristic
250	C₄	C₃₁×#493	1	1	yes	characteristic
251	C₂	—	8	4	no	—
252	C₂	C₃₁×#1480	1	2	yes	—
253	C₂	C₃₁×#1486	1	2	yes	—
254	C₂	C₃₁×#1955	1	1	yes	characteristic
255	C₂	C₃₁×#1957	1	1	yes	characteristic
256	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	C₂×C₃₁×#127	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×#1842	C₂	semidirect
3. C₃₁×#1744	C₂	semidirect
4. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
39. C₄×C₈×C₃₁	D₄	semidirect
73. #9644	C₃₁	direct (N × H)
106. #1842	C₂×C₃₁	semidirect
107. #1744	C₂×C₃₁	semidirect
108. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
167. C₄×C₈	D₄×C₃₁	semidirect
168. C₃₁	#9644	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×#127	2
C₃₁×#1842	2
C₃₁×#1744	2
C₂×C₃₁×D₄⋊C₈	2
#9644	31

Subgroups — C31×#9644 #21466403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9644 #21466403