Group #16848431 — subgroups

Subgroups — #16848431

Order 3968.

516 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 237 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	C₂×Q₈×Dic₃₁	C₂	1	2	yes	maximal
2	#14520956	C₂	1	2	yes	maximal
3	#14520935	C₂	1	8	yes	maximal
4	#14520915	C₂	1	1	yes	maximal, characteristic
5	#14520895	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
8	C₃₁×C₄⋊Q₈	C₄	1	8	yes	complemented
9	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	Q₈×Dic₃₁	—	2	8	no	—
12	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
14	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
16	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
17	C₃₁⋊#95	C₂²	1	4	yes	—
18	C₃₁⋊#95	C₂²	1	4	yes	—
19	C₃₁⋊#95	C₂²	1	8	yes	—
20	C₂×Q₈×C₃₁	—	2	4	no	—
21	C₂×Q₈×C₃₁	—	2	4	no	—
22	C₂×Q₈×C₃₁	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
24	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₁	C₂³	1	2	yes	—
27	C₂²×Dic₃₁	C₂³	1	4	yes	—
28	C₂²×Dic₃₁	C₂³	1	2	yes	—
29	C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
30	C₄²×C₃₁	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₃₁	—	2	4	no	—
32	C₄⋊Dic₃₁	—	2	8	no	—
33	C₄×Dic₃₁	—	2	2	no	—
34	C₄×Dic₃₁	—	2	8	no	—
35	C₄×Dic₃₁	D₄	1	4	yes	—
36	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
37	Q₈×C₃₁	—	2	16	no	—
38	C₂×C₄×C₃₁	—	2	4	no	—
39	C₂×C₄×C₃₁	—	2	4	no	—
40	C₂×C₄×C₃₁	—	2	4	no	—
41	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
42	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
43	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₃₁	#41	1	2	yes	—
46	C₂×Dic₃₁	—	2	4	no	—
47	C₂×Dic₃₁	—	2	4	no	—
48	C₂×Dic₃₁	—	2	2	no	—
49	C₂×Dic₃₁	—	2	8	no	—
50	C₂×Dic₃₁	—	2	4	no	—
51	C₂×Dic₃₁	C₂×D₄	1	4	yes	—
52	#2327	—	31	1	no	maximal, Sylow
53	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
54	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
55	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
56	C₂²×C₃₁	2₋¹⁺⁴	1	4	yes	—
57	C₄×C₃₁	—	2	8	no	—
58	C₄×C₃₁	—	2	4	no	—
59	C₄×C₃₁	C₄×D₄	1	4	yes	—
60	Dic₃₁	—	2	4	no	—
61	Dic₃₁	—	4	4	no	—
62	Dic₃₁	—	4	2	no	—
63	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
64	C₂×C₄×Q₈	—	31	2	no	—
65	#391	—	31	2	no	—
66	#387	—	31	8	no	—
67	#382	—	31	1	no	—
68	#376	—	31	1	no	—
69	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
70	C₂×C₃₁	#519	1	2	yes	—
71	C₂×C₃₁	#536	1	2	yes	—
72	C₂×C₃₁	#541	1	2	yes	—
73	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
74	C₄⋊Q₈	Dic₃₁	1	8	yes	complemented
75	C₄×Q₈	—	62	8	no	—
76	C₂×C₄⋊C₄	—	31	2	no	—
77	C₂×C₄⋊C₄	—	31	4	no	—
78	C₂×C₄⋊C₄	C₂×D₃₁	1	4	yes	—
79	C₂×C₄²	—	31	1	no	—
80	C₂×C₄²	—	31	4	no	—
81	C₂×C₄²	—	31	1	no	—
82	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
83	#95	—	31	4	no	—
84	#95	—	31	4	no	—
85	#95	—	31	8	no	—
86	C₃₁	#2327	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₈	—	2	4	no	—
88	C₂×Q₈	—	2	4	no	—
89	C₂×Q₈	C₂×Dic₃₁	1	8	yes	—
90	C₂²×C₄	—	31	2	no	—
91	C₂²×C₄	—	31	2	no	—
92	C₂²×C₄	—	31	4	no	—
93	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
94	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
95	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
96	C₄⋊C₄	—	62	8	no	—
97	C₄⋊C₄	—	62	4	no	—
98	C₄⋊C₄	C₂×Dic₃₁	1	16	yes	—
99	C₄²	—	31	4	no	—
100	C₄²	—	62	8	no	—
101	C₄²	—	62	2	no	—
102	C₄²	C₂×Dic₃₁	1	4	yes	—
103	C₂³	C₂³×D₃₁	1	1	yes	characteristic
104	Q₈	—	2	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	31	4	no	—
109	C₂×C₄	—	62	4	no	—
110	C₂×C₄	—	62	4	no	—
111	C₂×C₄	—	62	2	no	—
112	C₂×C₄	—	62	8	no	—
113	C₂×C₄	—	62	4	no	—
114	C₂×C₄	D₄×D₃₁	1	2	yes	—
115	C₂×C₄	#92497	1	2	yes	—
116	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
117	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
118	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
119	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
120	C₂²	C₂×#92497	1	1	yes	characteristic
121	C₂²	#11758564	1	4	yes	—
122	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
123	C₄	—	2	8	no	—
124	C₄	—	2	4	no	—
125	C₄	—	62	4	no	—
126	C₄	—	124	4	no	—
127	C₄	—	124	2	no	—
128	C₄	D₄×Dic₃₁	1	4	yes	—
129	C₂	C₂×D₄×Dic₃₁	1	1	yes	characteristic
130	C₂	#14521527	1	2	yes	—
131	C₂	#14521624	1	2	yes	—
132	C₂	#14521660	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₃₁

C₂

yes

maximal

#14520956

C₂

yes

maximal

#14520935

C₂

yes

maximal

#14520915

C₂

yes

maximal, characteristic

#14520895

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

—

C₃₁×C₄⋊Q₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

Q₈×Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

—

C₂³×C₃₁

C₂⁴

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×D₄

yes

—

#2327

—

maximal, Sylow

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂²×C₃₁

C₂×#41

yes

characteristic

C₂²×C₃₁

2₋¹⁺⁴

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊Q₈

D₃₁

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

C₂×C₃₁

C₂×C₄×D₄

yes

characteristic

C₂×C₃₁

#519

yes

—

C₂×C₃₁

#536

yes

—

C₂×C₃₁

#541

yes

—

C₂²×Q₈

C₂×D₃₁

yes

—

C₄⋊Q₈

Dic₃₁

yes

complemented

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

C₃₁

#2327

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂×Dic₃₁

yes

—

103

C₂³

C₂³×D₃₁

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

D₄×D₃₁

yes

—

115

C₂×C₄

#92497

yes

—

116

C₂×C₄

C₂²×Dic₃₁

yes

—

117

C₂×C₄

C₂²×Dic₃₁

yes

—

118

C₂×C₄

C₂²×Dic₃₁

yes

—

119

C₂²

C₂×D₄×D₃₁

yes

characteristic

120

C₂²

C₂×#92497

yes

characteristic

121

C₂²

#11758564

yes

—

122

C₂²

C₂³×Dic₃₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

124

—

127

C₄

—

124

—

128

C₄

D₄×Dic₃₁

yes

—

129

C₂

C₂×D₄×Dic₃₁

yes

characteristic

130

C₂

#14521527

yes

—

131

C₂

#14521624

yes

—

132

C₂

#14521660

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#2327
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator

C₂²×C₃₁

C₂³×C₄

Fitting

C₂×C₃₁×C₄⋊Q₈

C₂

socle, prime radical, cosocle

C₂³×C₃₁

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
8. C₃₁×C₄⋊Q₈	C₄	semidirect
74. C₄⋊Q₈	Dic₃₁	semidirect
86. C₃₁	#2327	semidirect

H ≅ G/N

type

8. C₃₁×C₄⋊Q₈

C₄

semidirect

74. C₄⋊Q₈

Dic₃₁

semidirect

86. C₃₁

#2327

semidirect

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

subgroup	index
C₂×Q₈×Dic₃₁	2
#14520956	2
#14520935	2
#14520915	2
#14520895	2
C₂×C₃₁×C₄⋊Q₈	2
#2327	31

subgroup

index

C₂×Q₈×Dic₃₁

C₂×C₃₁×C₄⋊Q₈

Subgroups — #16848431

Conjugacy classes of subgroups

Complements

Maximal subgroups