Subgroups — C₃₁×#8267 #21465026

Order 7936.

390 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1564	C₂	1	2	yes	maximal
3	C₃₁×#1452	C₂	1	2	yes	maximal
4	C₃₁×#1276	C₂	1	2	yes	maximal
5	C₃₁×#431	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₄³×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#335	C₄	1	4	yes	complemented
12	C₃₁×#334	C₄	1	4	yes	complemented
13	C₃₁×#322	—	2	4	no	—
14	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₃₁	—	2	2	no	—
18	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
20	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
21	C₃₁×C₂²⋊C₈	—	2	4	no	—
22	C₄×C₈×C₃₁	—	2	4	no	—
23	C₄×C₈×C₃₁	—	2	4	no	—
24	C₄×C₈×C₃₁	—	2	4	no	—
25	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₃₁	D₄	1	2	yes	—
28	C₄×C₈×C₃₁	Q₈	1	2	yes	—
29	C₂²×C₄×C₃₁	—	2	2	no	—
30	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	#41	1	2	yes	—
32	C₂×C₈×C₃₁	—	2	2	no	—
33	C₂×C₈×C₃₁	—	2	2	no	—
34	C₂×C₈×C₃₁	—	2	2	no	—
35	C₂×C₈×C₃₁	—	2	8	no	—
36	C₂×C₈×C₃₁	C₄²	1	2	yes	—
37	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
40	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
41	C₄²×C₃₁	—	2	8	no	—
42	C₄²×C₃₁	—	2	2	no	—
43	C₄²×C₃₁	M₁₆	1	4	yes	—
44	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
45	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
47	#8267	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂³×C₃₁	#117	1	1	yes	characteristic
49	C₂×C₄×C₃₁	—	2	8	no	—
50	C₂×C₄×C₃₁	—	2	4	no	—
51	C₂×C₄×C₃₁	—	2	4	no	—
52	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
55	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
56	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
57	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
58	C₈×C₃₁	—	2	2	no	—
59	C₈×C₃₁	—	2	4	no	—
60	C₈×C₃₁	—	4	8	no	—
61	C₈×C₃₁	#97	1	2	yes	—
62	C₈×C₃₁	#97	1	2	yes	—
63	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
64	#1564	C₂×C₃₁	1	2	yes	—
65	#1452	C₂×C₃₁	1	2	yes	—
66	#1276	C₂×C₃₁	1	2	yes	—
67	C₂²×C₃₁	—	2	2	no	—
68	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₃₁	#432	1	2	yes	—
70	C₄×C₃₁	—	2	8	no	—
71	C₄×C₃₁	—	2	2	no	—
72	C₄×C₃₁	C₂×#97	1	1	yes	characteristic
73	C₄×C₃₁	C₄×M₁₆	1	1	yes	characteristic
74	C₄×C₃₁	#423	1	2	yes	—
75	C₄×C₃₁	#443	1	4	yes	—
76	#431	C₂²×C₃₁	1	4	yes	—
77	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
78	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
79	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
81	C₄³	C₂²×C₃₁	1	1	yes	characteristic
82	#335	C₄×C₃₁	1	4	yes	complemented
83	#334	C₄×C₃₁	1	4	yes	complemented
84	#322	—	2	4	no	—
85	C₂×C₃₁	—	2	2	no	—
86	C₂×C₃₁	#1767	1	1	yes	characteristic
87	C₂×C₃₁	#1772	1	2	yes	—
88	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
89	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
90	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
91	C₂×C₄²	—	2	2	no	—
92	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
93	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
94	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
95	C₂²⋊C₈	—	2	4	no	—
96	C₄×C₈	—	2	4	no	—
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	2	4	no	—
99	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
100	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
101	C₄×C₈	D₄×C₃₁	1	2	yes	—
102	C₄×C₈	Q₈×C₃₁	1	2	yes	—
103	C₃₁	#8267	1	1	yes	characteristic, direct product factor, Sylow
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
106	C₂²×C₄	#41×C₃₁	1	2	yes	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	C₄²×C₃₁	1	2	yes	—
112	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
113	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
114	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
115	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
116	C₄²	—	2	8	no	—
117	C₄²	—	2	2	no	—
118	C₄²	M₁₆×C₃₁	1	4	yes	—
119	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
120	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
121	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
122	C₂³	C₃₁×#117	1	1	yes	characteristic
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
128	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
129	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
130	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
131	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
132	C₈	—	2	2	no	—
133	C₈	—	2	4	no	—
134	C₈	—	4	8	no	—
135	C₈	C₃₁×#97	1	2	yes	—
136	C₈	C₃₁×#97	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
139	C₂²	C₃₁×#432	1	2	yes	—
140	C₄	—	2	8	no	—
141	C₄	—	2	2	no	—
142	C₄	C₂×C₃₁×#97	1	1	yes	characteristic
143	C₄	C₄×M₁₆×C₃₁	1	1	yes	characteristic
144	C₄	C₃₁×#423	1	2	yes	—
145	C₄	C₃₁×#443	1	4	yes	—
146	C₂	—	2	2	no	—
147	C₂	C₃₁×#1767	1	1	yes	characteristic
148	C₂	C₃₁×#1772	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₄⋊C₄
prime radical	C₂³×C₃₁	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₃₁×#335	C₄	semidirect
12. C₃₁×#334	C₄	semidirect
47. #8267	C₃₁	direct (N × H)
82. #335	C₄×C₃₁	semidirect
83. #334	C₄×C₃₁	semidirect
103. C₃₁	#8267	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₈×C₃₁	2
C₃₁×#1564	2
C₃₁×#1452	2
C₃₁×#1276	2
#8267	31

Subgroups — C31×#8267 #21465026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8267 #21465026