Subgroups — C₃₁×#7919 #21464678

Order 7936.

626 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2920	C₂	1	2	yes	maximal
2	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1281	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₃₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#432	—	2	2	no	—
7	C₃₁×#432	C₄	1	4	yes	—
8	C₃₁×#431	—	2	2	no	—
9	C₃₁×#431	C₂²	1	4	yes	—
10	C₃₁×#423	—	2	2	no	—
11	C₃₁×#407	C₄	1	4	yes	—
12	C₄×M₁₆×C₃₁	—	2	2	no	—
13	C₂×C₃₁×#97	—	2	2	no	—
14	C₂×C₃₁×#97	—	2	2	no	—
15	C₂×C₃₁×#97	C₄	1	2	yes	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₃₁	—	2	4	no	—
20	C₂²×C₈×C₃₁	—	2	4	no	—
21	C₂²×C₈×C₃₁	—	2	8	no	—
22	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₃₁	D₄	1	2	yes	—
26	C₂×C₄²×C₃₁	D₄	1	2	yes	—
27	C₂×C₄²×C₃₁	D₄	1	2	yes	—
28	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
30	C₃₁×C₄⋊C₈	—	2	4	no	—
31	C₃₁×C₄⋊C₈	—	2	4	no	—
32	C₃₁×C₂²⋊C₈	—	2	4	no	—
33	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
34	C₃₁×#97	—	2	2	no	—
35	C₃₁×#97	—	2	2	no	—
36	C₃₁×#97	—	2	4	no	—
37	C₃₁×#97	—	2	4	no	—
38	C₄×C₈×C₃₁	—	2	4	no	—
39	C₂⁴×C₃₁	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	—	2	4	no	—
41	C₂²×C₄×C₃₁	—	4	4	no	—
42	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₃₁	—	4	8	no	—
53	C₂×C₈×C₃₁	—	2	16	no	—
54	C₂×C₈×C₃₁	—	2	4	no	—
55	C₂×C₈×C₃₁	—	2	4	no	—
56	C₂×C₈×C₃₁	—	2	8	no	—
57	C₂×C₈×C₃₁	M₁₆	1	8	yes	complemented
58	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
63	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
64	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
65	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₃₁	C₂×D₄	1	2	yes	—
67	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
68	#7919	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₃₁	—	2	4	no	—
70	C₂³×C₃₁	—	2	2	no	—
71	C₂³×C₃₁	#95	1	2	yes	—
72	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₃₁	—	2	4	no	—
74	C₂×C₄×C₃₁	—	2	4	no	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	2	4	no	—
77	C₂×C₄×C₃₁	—	2	4	no	—
78	C₂×C₄×C₃₁	—	4	2	no	—
79	C₂×C₄×C₃₁	—	4	2	no	—
80	C₂×C₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	#95	1	2	yes	—
83	C₂×C₄×C₃₁	#95	1	2	yes	—
84	C₂×C₄×C₃₁	#95	1	2	yes	—
85	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
90	C₈×C₃₁	—	2	8	no	—
91	C₈×C₃₁	—	4	8	no	—
92	#2920	C₂×C₃₁	1	2	yes	—
93	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic
94	#1281	C₂×C₃₁	1	4	yes	complemented
95	C₂²×C₃₁	—	2	8	no	—
96	C₂²×C₃₁	—	2	2	no	—
97	C₂²×C₃₁	—	2	2	no	—
98	C₂²×C₃₁	—	2	2	no	—
99	C₂²×C₃₁	#344	1	4	yes	—
100	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
101	C₂²×C₃₁	C₄×M₁₆	1	2	yes	—
102	C₄×C₃₁	—	2	2	no	—
103	C₄×C₃₁	—	2	2	no	—
104	C₄×C₃₁	—	2	2	no	—
105	C₄×C₃₁	—	2	2	no	—
106	C₄×C₃₁	—	2	2	no	—
107	C₄×C₃₁	#407	1	2	yes	—
108	C₄×C₃₁	#423	1	2	yes	—
109	C₂³×C₈	C₄×C₃₁	1	2	yes	—
110	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
111	#432	—	2	2	no	—
112	#432	C₄×C₃₁	1	4	yes	—
113	#431	—	2	2	no	—
114	#431	C₂²×C₃₁	1	4	yes	—
115	#423	—	2	2	no	—
116	#407	C₄×C₃₁	1	4	yes	—
117	C₄×M₁₆	—	2	2	no	—
118	C₂×#97	—	2	2	no	—
119	C₂×#97	—	2	2	no	—
120	C₂×#97	C₄×C₃₁	1	2	yes	—
121	C₂×#97	C₂²×C₃₁	1	2	yes	—
122	C₂×C₃₁	—	2	4	no	—
123	C₂×C₃₁	—	2	2	no	—
124	C₂×C₃₁	#1728	1	1	yes	characteristic
125	C₂×C₃₁	C₂×#344	1	2	yes	—
126	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
127	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
128	C₂×M₁₆	—	2	4	no	—
129	C₂²×C₈	—	2	4	no	—
130	C₂²×C₈	—	2	8	no	—
131	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
132	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
133	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
134	C₂×C₄²	D₄×C₃₁	1	2	yes	—
135	C₂×C₄²	D₄×C₃₁	1	2	yes	—
136	C₂×C₄²	D₄×C₃₁	1	2	yes	—
137	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
138	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
139	C₄⋊C₈	—	2	4	no	—
140	C₄⋊C₈	—	2	4	no	—
141	C₂²⋊C₈	—	2	4	no	—
142	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
143	#97	—	2	2	no	—
144	#97	—	2	2	no	—
145	#97	—	2	4	no	—
146	#97	—	2	4	no	—
147	C₄×C₈	—	2	4	no	—
148	C₃₁	#7919	1	1	yes	characteristic, direct product factor, Sylow
149	C₂⁴	C₄²×C₃₁	1	1	yes	characteristic
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	4	4	no	—
152	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
153	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
154	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
155	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
156	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
157	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
158	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
159	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
161	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
162	M₁₆	—	4	8	no	—
163	C₂×C₈	—	2	16	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	M₁₆×C₃₁	1	8	yes	complemented
168	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
169	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
170	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
171	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
172	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
173	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
174	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
175	C₄²	C₂×D₄×C₃₁	1	2	yes	—
176	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
177	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
178	C₂³	—	2	2	no	—
179	C₂³	—	2	4	no	—
180	C₂³	C₃₁×#95	1	2	yes	—
181	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₃₁×#95	1	2	yes	—
192	C₂×C₄	C₃₁×#95	1	2	yes	—
193	C₂×C₄	C₃₁×#95	1	2	yes	—
194	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
195	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
197	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
199	C₈	—	2	8	no	—
200	C₈	—	4	8	no	—
201	C₂²	—	2	8	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	C₃₁×#344	1	4	yes	—
206	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
207	C₂²	C₄×M₁₆×C₃₁	1	2	yes	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	C₃₁×#407	1	2	yes	—
214	C₄	C₃₁×#423	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	2	4	no	—
217	C₂	C₃₁×#1728	1	1	yes	characteristic
218	C₂	C₂×C₃₁×#344	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#95
prime radical	C₂⁴×C₃₁	C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1281	C₂	semidirect
57. C₂×C₈×C₃₁	M₁₆	semidirect
68. #7919	C₃₁	direct (N × H)
94. #1281	C₂×C₃₁	semidirect
148. C₃₁	#7919	direct (N × H)
167. C₂×C₈	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2920	2
C₂²×C₃₁×#97	2
C₃₁×#1281	2
#7919	31

Subgroups — C31×#7919 #21464678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7919 #21464678