Group #21490057 — subgroups

Subgroups — C₃₁×#33298 #21490057

Order 7936.

694 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 286 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2862	C₂	1	2	yes	maximal
2	C₃₁×#1993	C₂	1	8	yes	maximal
3	C₃₁×#1835	C₂	1	4	yes	maximal
4	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
6	C₂×C₃₁×#97	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
8	C₃₁×#391	—	2	8	no	—
9	C₃₁×#382	C₂²	1	4	yes	—
10	C₃₁×#382	C₂²	1	4	yes	—
11	C₄³×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#340	C₂²	1	16	yes	—
13	C₂²×Q₈×C₃₁	—	2	4	no	—
14	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
15	C₃₁×C₄⋊Q₈	—	2	16	no	—
16	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
17	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
18	C₂×C₄²×C₃₁	D₄	1	4	yes	—
19	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
21	C₃₁×#97	—	2	8	no	—
22	C₄×C₈×C₃₁	Q₈	1	8	yes	complemented
23	C₃₁×#95	—	2	8	no	—
24	C₃₁×#95	—	2	8	no	—
25	C₂×Q₈×C₃₁	—	4	16	no	—
26	C₂×Q₈×C₃₁	—	4	8	no	—
27	C₂²×C₄×C₃₁	—	2	8	no	—
28	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
29	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
30	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
31	C₂×C₈×C₃₁	—	2	8	no	—
32	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
33	C₄⋊C₄×C₃₁	—	2	16	no	—
34	C₄⋊C₄×C₃₁	—	4	8	no	—
35	C₄²×C₃₁	—	4	4	no	—
36	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
37	C₄²×C₃₁	#41	1	8	yes	—
38	#33298	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
40	Q₈×C₃₁	—	8	16	no	—
41	C₂×C₄×C₃₁	—	2	4	no	—
42	C₂×C₄×C₃₁	—	2	8	no	—
43	C₂×C₄×C₃₁	—	4	8	no	—
44	C₂×C₄×C₃₁	—	4	16	no	—
45	C₂×C₄×C₃₁	#124	1	8	yes	—
46	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
47	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
48	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
49	C₈×C₃₁	—	2	8	no	—
50	#2862	C₂×C₃₁	1	2	yes	—
51	#1993	C₂×C₃₁	1	8	yes	—
52	#1835	C₂×C₃₁	1	4	yes	—
53	C₄×#97	C₂×C₃₁	1	1	yes	characteristic
54	C₂²×C₃₁	C₂×#124	1	2	yes	—
55	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
56	C₂²×C₃₁	#575	1	4	yes	—
57	C₄×C₃₁	—	2	8	no	—
58	C₄×C₃₁	—	2	4	no	—
59	C₄×C₃₁	—	8	8	no	—
60	C₄×C₃₁	#533	1	4	yes	—
61	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
62	C₂×#97	C₂²×C₃₁	1	4	yes	—
63	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
64	#391	—	2	8	no	—
65	#382	C₂²×C₃₁	1	4	yes	—
66	#382	C₂²×C₃₁	1	4	yes	—
67	C₄³	C₂²×C₃₁	1	1	yes	characteristic
68	#340	C₂²×C₃₁	1	16	yes	—
69	C₂×C₃₁	#2862	1	1	yes	characteristic
70	C₂×C₃₁	#3140	1	4	yes	—
71	C₂×C₃₁	#3162	1	2	yes	—
72	C₂²×Q₈	—	2	4	no	—
73	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
74	C₄⋊Q₈	—	2	16	no	—
75	C₂×C₄⋊C₄	—	2	4	no	—
76	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
77	C₂×C₄²	D₄×C₃₁	1	4	yes	—
78	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
79	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
80	#97	—	2	8	no	—
81	C₄×C₈	Q₈×C₃₁	1	8	yes	complemented
82	#95	—	2	8	no	—
83	#95	—	2	8	no	—
84	C₃₁	#33298	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×Q₈	—	4	16	no	—
86	C₂×Q₈	—	4	8	no	—
87	C₂²×C₄	—	2	8	no	—
88	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
89	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
90	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
91	C₂×C₈	—	2	8	no	—
92	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
93	C₄⋊C₄	—	2	16	no	—
94	C₄⋊C₄	—	4	8	no	—
95	C₄²	—	4	4	no	—
96	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
97	C₄²	#41×C₃₁	1	8	yes	—
98	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
99	Q₈	—	8	16	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	8	no	—
102	C₂×C₄	—	4	8	no	—
103	C₂×C₄	—	4	16	no	—
104	C₂×C₄	C₃₁×#124	1	8	yes	—
105	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
106	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
107	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
108	C₈	—	2	8	no	—
109	C₂²	C₂×C₃₁×#124	1	2	yes	—
110	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
111	C₂²	C₃₁×#575	1	4	yes	—
112	C₄	—	2	8	no	—
113	C₄	—	2	4	no	—
114	C₄	—	8	8	no	—
115	C₄	C₃₁×#533	1	4	yes	—
116	C₂	C₃₁×#2862	1	1	yes	characteristic
117	C₂	C₃₁×#3140	1	4	yes	—
118	C₂	C₃₁×#3162	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2862

C₂

yes

maximal

C₃₁×#1993

C₂

yes

maximal

C₃₁×#1835

C₂

yes

maximal

C₄×C₃₁×#97

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂²

yes

—

C₂×C₃₁×#97

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₃₁×#391

—

C₃₁×#382

C₂²

yes

—

C₃₁×#382

C₂²

yes

—

C₄³×C₃₁

C₂²

yes

characteristic

C₃₁×#340

C₂²

yes

—

C₂²×Q₈×C₃₁

—

C₂²×C₈×C₃₁

C₂³

yes

—

C₃₁×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂³

yes

—

C₂×C₄²×C₃₁

D₄

yes

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₂×C₄²×C₃₁

C₂³

yes

—

C₃₁×#97

—

C₄×C₈×C₃₁

Q₈

yes

complemented

C₃₁×#95

—

C₃₁×#95

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×Q₈

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂×Q₈

yes

—

C₄²×C₃₁

#41

yes

—

#33298

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂²×D₄

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#124

yes

—

C₂×C₄×C₃₁

C₄⋊Q₈

yes

—

C₂×C₄×C₃₁

C₂²×Q₈

yes

—

C₂×C₄×C₃₁

C₂×#41

yes

—

C₈×C₃₁

—

#2862

C₂×C₃₁

yes

—

#1993

C₂×C₃₁

yes

—

#1835

C₂×C₃₁

yes

—

C₄×#97

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

C₂×#124

yes

—

C₂²×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₃₁

#575

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#533

yes

—

C₂×C₄⋊Q₈

C₂²×C₃₁

yes

—

C₂×#97

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

#391

—

#382

C₂²×C₃₁

yes

—

#382

C₂²×C₃₁

yes

—

C₄³

C₂²×C₃₁

yes

characteristic

#340

C₂²×C₃₁

yes

—

C₂×C₃₁

#2862

yes

characteristic

C₂×C₃₁

#3140

yes

—

C₂×C₃₁

#3162

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂³×C₃₁

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³×C₃₁

yes

—

C₂×C₄²

D₄×C₃₁

yes

—

C₂×C₄²

C₂³×C₃₁

yes

characteristic

C₂×C₄²

C₂³×C₃₁

yes

—

#97

—

C₄×C₈

Q₈×C₃₁

yes

complemented

#95

—

#95

—

C₃₁

#33298

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₃₁

yes

—

C₂²×C₄

C₂×D₄×C₃₁

yes

—

C₂²×C₄

C₂⁴×C₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×Q₈×C₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×Q₈×C₃₁

yes

—

C₄²

#41×C₃₁

yes

—

C₂³

C₂²×D₄×C₃₁

yes

characteristic

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₃₁×#124

yes

—

105

C₂×C₄

C₃₁×C₄⋊Q₈

yes

—

106

C₂×C₄

C₂²×Q₈×C₃₁

yes

—

107

C₂×C₄

C₂×#41×C₃₁

yes

—

108

C₈

—

109

C₂²

C₂×C₃₁×#124

yes

—

110

C₂²

C₂×C₃₁×C₄⋊Q₈

yes

characteristic

111

C₂²

C₃₁×#575

yes

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

C₃₁×#533

yes

—

116

C₂

C₃₁×#2862

yes

characteristic

117

C₂

C₃₁×#3140

yes

—

118

C₂

C₃₁×#3162

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₁

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₃₁

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

N	H ≅ G/N	type
22. C₄×C₈×C₃₁	Q₈	semidirect
38. #33298	C₃₁	direct (N × H)
81. C₄×C₈	Q₈×C₃₁	semidirect
84. C₃₁	#33298	direct (N × H)

H ≅ G/N

type

22. C₄×C₈×C₃₁

Q₈

semidirect

38. #33298

C₃₁

direct (N × H)

81. C₄×C₈

Q₈×C₃₁

semidirect

84. C₃₁

#33298

direct (N × H)

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

subgroup	index
C₃₁×#2862	2
C₃₁×#1993	2
C₃₁×#1835	2
C₄×C₃₁×#97	2
#33298	31

subgroup

index

C₃₁×#2862

C₃₁×#1993

C₃₁×#1835

C₄×C₃₁×#97

#33298

Subgroups — C₃₁×#33298 #21490057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#33298 #21490057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33298 #21490057