Subgroups — C₃₁×#7571 #21464330

Order 7936.

516 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊C₈	C₂	1	2	yes	maximal
3	C₃₁×#1272	C₂	1	4	yes	maximal
4	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
5	C₂×C₄×Q₈×C₃₁	C₄	1	2	yes	—
6	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
7	C₂×C₄×C₈×C₃₁	—	2	2	no	—
8	C₃₁×#391	C₂²	1	2	yes	—
9	C₃₁×#389	—	2	2	no	—
10	C₃₁×#384	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
12	C₄³×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#336	C₂²	1	4	yes	—
14	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₃₁	—	2	4	no	—
16	C₃₁×C₄⋊Q₈	—	2	2	no	—
17	C₃₁×C₄⋊Q₈	C₈	1	4	yes	complemented
18	C₄×Q₈×C₃₁	—	4	4	no	—
19	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
20	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
21	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₃₁	—	2	2	no	—
24	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₃₁	D₄	1	2	yes	—
27	C₂×C₄²×C₃₁	D₄	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
29	C₃₁×C₄⋊C₈	—	2	8	no	—
30	C₄×C₈×C₃₁	—	4	4	no	—
31	C₃₁×#95	—	2	4	no	—
32	C₂×Q₈×C₃₁	—	2	2	no	—
33	C₂×Q₈×C₃₁	—	4	2	no	—
34	C₂×Q₈×C₃₁	—	4	2	no	—
35	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₃₁	—	2	2	no	—
37	C₂²×C₄×C₃₁	—	2	2	no	—
38	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₃₁	—	2	8	no	—
44	C₂×C₈×C₃₁	—	4	4	no	—
45	C₂×C₈×C₃₁	—	4	4	no	—
46	C₄⋊C₄×C₃₁	—	2	8	no	—
47	C₄⋊C₄×C₃₁	—	4	4	no	—
48	C₄⋊C₄×C₃₁	—	4	2	no	—
49	C₄⋊C₄×C₃₁	—	4	4	no	—
50	C₄²×C₃₁	—	2	2	no	—
51	C₄²×C₃₁	—	2	4	no	—
52	C₄²×C₃₁	—	4	1	no	—
53	C₄²×C₃₁	—	4	4	no	—
54	C₄²×C₃₁	C₂×C₈	1	2	yes	—
55	C₄²×C₃₁	M₁₆	1	2	yes	—
56	C₄²×C₃₁	SD₁₆	1	4	yes	—
57	C₄²×C₃₁	Q₁₆	1	4	yes	—
58	#7571	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	Q₈×C₃₁	—	4	4	no	—
61	Q₈×C₃₁	—	4	4	no	—
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	1	no	—
64	C₂×C₄×C₃₁	—	2	1	no	—
65	C₂×C₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	—	2	4	no	—
67	C₂×C₄×C₃₁	—	2	1	no	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	4	2	no	—
70	C₂×C₄×C₃₁	—	4	2	no	—
71	C₂×C₄×C₃₁	—	4	2	no	—
72	C₂×C₄×C₃₁	—	4	4	no	—
73	C₂×C₄×C₃₁	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₃₁	#100	1	2	yes	—
75	C₂×C₄×C₃₁	#103	1	4	yes	—
76	C₂×C₄×C₃₁	#103	1	4	yes	—
77	C₂×C₄×C₃₁	#104	1	4	yes	—
78	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
82	C₈×C₃₁	—	4	8	no	—
83	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
84	C₄×C₄⋊C₈	C₂×C₃₁	1	2	yes	—
85	#1272	C₂×C₃₁	1	4	yes	—
86	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
87	C₂²×C₃₁	#410	1	1	yes	characteristic
88	C₂²×C₃₁	C₂×#100	1	1	yes	characteristic
89	C₂²×C₃₁	C₂×#103	1	2	yes	—
90	C₂²×C₃₁	C₂×#104	1	2	yes	—
91	C₄×C₃₁	—	2	1	no	—
92	C₄×C₃₁	—	2	1	no	—
93	C₄×C₃₁	—	2	2	no	—
94	C₄×C₃₁	—	2	2	no	—
95	C₄×C₃₁	—	4	4	no	—
96	C₄×C₃₁	—	4	1	no	—
97	C₄×C₃₁	—	8	2	no	—
98	C₄×C₃₁	Q₈⋊C₈	1	8	yes	—
99	C₄×C₃₁	#333	1	4	yes	—
100	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
101	C₂×C₄×Q₈	C₄×C₃₁	1	2	yes	—
102	C₂×C₄⋊C₈	—	2	2	no	—
103	C₂×C₄×C₈	—	2	2	no	—
104	#391	C₂²×C₃₁	1	2	yes	—
105	#389	—	2	2	no	—
106	#384	C₄×C₃₁	1	1	yes	characteristic
107	C₄×C₄⋊C₄	—	2	2	no	—
108	C₄³	C₂²×C₃₁	1	1	yes	characteristic
109	#336	C₂²×C₃₁	1	4	yes	—
110	C₂×C₃₁	#1460	1	1	yes	characteristic
111	C₂×C₃₁	C₂×Q₈⋊C₈	1	2	yes	—
112	C₂×C₃₁	#1503	1	2	yes	—
113	C₂×C₃₁	#1525	1	1	yes	characteristic
114	C₂×C₃₁	C₂×#333	1	1	yes	characteristic
115	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
116	C₂²×C₈	—	2	4	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	C₈×C₃₁	1	4	yes	complemented
119	C₄×Q₈	—	4	4	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
124	C₂×C₄²	—	2	2	no	—
125	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
126	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
127	C₂×C₄²	D₄×C₃₁	1	2	yes	—
128	C₂×C₄²	D₄×C₃₁	1	2	yes	—
129	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
130	C₄⋊C₈	—	2	8	no	—
131	C₄×C₈	—	4	4	no	—
132	#95	—	2	4	no	—
133	C₃₁	#7571	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
141	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
142	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	4	4	no	—
148	C₄⋊C₄	—	2	8	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	1	no	—
155	C₄²	—	4	4	no	—
156	C₄²	C₂×C₈×C₃₁	1	2	yes	—
157	C₄²	M₁₆×C₃₁	1	2	yes	—
158	C₄²	SD₁₆×C₃₁	1	4	yes	—
159	C₄²	Q₁₆×C₃₁	1	4	yes	—
160	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	Q₈	—	4	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	C₃₁×C₂²⋊C₈	1	4	yes	—
175	C₂×C₄	C₃₁×#100	1	2	yes	—
176	C₂×C₄	C₃₁×#103	1	4	yes	—
177	C₂×C₄	C₃₁×#103	1	4	yes	—
178	C₂×C₄	C₃₁×#104	1	4	yes	—
179	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
180	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
181	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
182	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
183	C₈	—	4	8	no	—
184	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
185	C₂²	C₃₁×#410	1	1	yes	characteristic
186	C₂²	C₂×C₃₁×#100	1	1	yes	characteristic
187	C₂²	C₂×C₃₁×#103	1	2	yes	—
188	C₂²	C₂×C₃₁×#104	1	2	yes	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	—	4	1	no	—
195	C₄	—	8	2	no	—
196	C₄	C₃₁×Q₈⋊C₈	1	8	yes	—
197	C₄	C₃₁×#333	1	4	yes	—
198	C₂	C₃₁×#1460	1	1	yes	characteristic
199	C₂	C₂×C₃₁×Q₈⋊C₈	1	2	yes	—
200	C₂	C₃₁×#1503	1	2	yes	—
201	C₂	C₃₁×#1525	1	1	yes	characteristic
202	C₂	C₂×C₃₁×#333	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
17. C₃₁×C₄⋊Q₈	C₈	semidirect
58. #7571	C₃₁	direct (N × H)
118. C₄⋊Q₈	C₈×C₃₁	semidirect
133. C₃₁	#7571	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×C₄⋊Q₈	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1272	2
#7571	31

Subgroups — C31×#7571 #21464330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7571 #21464330