Group #90376 — subgroups

Subgroups — #90376

Order 448.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₇×#125	C₂	1	1	yes	maximal, characteristic
2	#6453	C₂	1	1	yes	maximal, characteristic
3	#6452	C₂	1	2	yes	maximal
4	#6452	C₂	1	2	yes	maximal
5	#6451	C₂	1	1	yes	maximal, characteristic
6	#6451	C₂	1	2	yes	maximal
7	#6450	C₂	1	1	yes	maximal, characteristic
8	#6450	C₂	1	2	yes	maximal
9	#6450	C₂	1	1	yes	maximal, characteristic
10	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₄	C₂²	1	2	yes	—
14	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
15	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₇	C₂²	1	2	yes	—
23	Dic₇⋊C₄	C₂²	1	1	yes	characteristic
24	Dic₇⋊C₄	C₂²	1	2	yes	—
25	Dic₇⋊C₄	C₂²	1	1	yes	characteristic
26	Dic₇⋊C₄	C₂²	1	2	yes	—
27	Dic₇⋊C₄	C₂²	1	2	yes	—
28	Dic₇⋊C₄	C₂²	1	2	yes	—
29	Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₄×Dic₇	C₂²	1	1	yes	characteristic
31	C₄×Dic₇	C₂²	1	2	yes	—
32	C₄×Dic₇	C₂²	1	1	yes	characteristic
33	C₄×Dic₇	C₂²	1	2	yes	—
34	C₄×Dic₇	C₂²	1	2	yes	—
35	#557	—	7	1	no	maximal, Sylow
36	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₇	C₂³	1	2	yes	—
38	C₂×C₄×C₇	C₂³	1	2	yes	—
39	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×Dic₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₇	C₂³	1	2	yes	—
44	C₂×Dic₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₇	C₂³	1	2	yes	—
46	C₂×Dic₇	C₂³	1	1	yes	characteristic
47	Dic₁₄	—	2	2	no	—
48	Dic₁₄	—	2	4	no	—
49	Dic₁₄	Q₈	1	4	yes	complemented
50	C₄⋊Q₈	—	7	2	no	—
51	C₄⋊Q₈	—	7	1	no	—
52	#125	—	7	2	no	—
53	#125	—	7	2	no	—
54	#125	—	7	1	no	—
55	#125	D₇	1	1	yes	characteristic
56	C₄×Q₈	—	7	1	no	—
57	C₄×Q₈	—	7	2	no	—
58	C₄×Q₈	—	7	1	no	—
59	C₄×Q₈	—	7	1	no	—
60	C₄×Q₈	—	7	1	no	—
61	C₂²×C₇	C₂⁴	1	1	yes	characteristic
62	C₄×C₇	—	2	2	no	—
63	C₄×C₇	—	2	1	no	—
64	C₄×C₇	—	2	1	no	—
65	C₄×C₇	—	2	2	no	—
66	C₄×C₇	C₂×Q₈	1	2	yes	—
67	Dic₇	—	2	1	no	—
68	Dic₇	—	2	2	no	—
69	Dic₇	—	2	1	no	—
70	Dic₇	—	2	1	no	—
71	Dic₇	C₂×Q₈	1	4	yes	—
72	Dic₇	#41	1	2	yes	—
73	C₂×Q₈	—	7	1	no	—
74	C₂×Q₈	—	7	2	no	—
75	C₂×Q₈	—	7	1	no	—
76	C₄⋊C₄	—	7	1	no	—
77	C₄⋊C₄	—	7	2	no	—
78	C₄⋊C₄	—	7	2	no	—
79	C₄⋊C₄	—	7	1	no	—
80	C₄⋊C₄	—	7	2	no	—
81	C₄⋊C₄	—	7	2	no	—
82	C₄⋊C₄	—	7	2	no	—
83	C₄⋊C₄	—	7	1	no	—
84	C₄⋊C₄	—	7	1	no	—
85	C₄⋊C₄	—	7	2	no	—
86	C₄⋊C₄	C₂×D₇	1	2	yes	—
87	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
88	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₇	1	2	yes	—
90	C₄²	—	7	2	no	—
91	C₄²	—	7	2	no	—
92	C₄²	—	7	1	no	—
93	C₄²	—	7	2	no	—
94	C₄²	—	7	1	no	—
95	C₄²	C₂×D₇	1	1	yes	characteristic
96	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₇	C₂×#41	1	1	yes	characteristic
98	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
99	Q₈	—	7	4	no	—
100	Q₈	—	14	4	no	—
101	Q₈	—	14	2	no	—
102	C₂×C₄	—	7	2	no	—
103	C₂×C₄	—	7	1	no	—
104	C₂×C₄	—	7	1	no	—
105	C₂×C₄	—	7	1	no	—
106	C₂×C₄	—	7	1	no	—
107	C₂×C₄	—	7	2	no	—
108	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
109	C₂×C₄	C₂²×D₇	1	2	yes	—
110	C₂×C₄	C₂²×D₇	1	2	yes	—
111	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
112	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
113	C₇	#557	1	1	yes	characteristic, complemented, Sylow
114	C₂²	C₂³×D₇	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	7	4	no	—
120	C₄	—	7	2	no	—
121	C₄	—	14	1	no	—
122	C₄	—	14	2	no	—
123	C₄	—	14	1	no	—
124	C₄	—	14	1	no	—
125	C₄	Q₈×D₇	1	2	yes	—
126	C₂	C₂×Q₈×D₇	1	1	yes	characteristic
127	C₂	D₇×#41	1	1	yes	characteristic
128	C₂	#6554	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#125

C₂

yes

maximal, characteristic

#6453

C₂

yes

maximal, characteristic

#6452

C₂

yes

maximal

#6452

C₂

yes

maximal

#6451

C₂

yes

maximal, characteristic

#6451

C₂

yes

maximal

#6450

C₂

yes

maximal, characteristic

#6450

C₂

yes

maximal

#6450

C₂

yes

maximal, characteristic

C₄×Dic₁₄

C₂

yes

maximal, characteristic

C₄×Dic₁₄

C₂

yes

maximal, characteristic

C₂×Dic₁₄

C₂²

yes

characteristic

C₂×Dic₁₄

C₂²

yes

—

C₂×Dic₁₄

C₂²

yes

characteristic

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₄⋊C₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₄⋊C₄

C₂²

yes

—

C₄²×C₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

characteristic

C₄×Dic₇

C₂²

yes

—

C₄×Dic₇

C₂²

yes

characteristic

C₄×Dic₇

C₂²

yes

—

C₄×Dic₇

C₂²

yes

—

#557

—

maximal, Sylow

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

characteristic

Dic₁₄

—

Dic₁₄

—

Dic₁₄

Q₈

yes

complemented

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

D₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂²×C₇

C₂⁴

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×Q₈

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

C₂×Q₈

yes

—

Dic₇

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

—

C₄⋊C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

C₂×D₇

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₇

yes

characteristic

C₂×C₇

C₂²×Q₈

yes

characteristic

C₂×C₇

C₂×#41

yes

characteristic

C₂×C₇

2₋¹⁺⁴

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×D₇

yes

characteristic

109

C₂×C₄

C₂²×D₇

yes

—

110

C₂×C₄

C₂²×D₇

yes

—

111

C₂×C₄

C₂²×D₇

yes

characteristic

112

C₂×C₄

C₂²×D₇

yes

characteristic

113

C₇

#557

yes

characteristic, complemented, Sylow

114

C₂²

C₂³×D₇

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

Q₈×D₇

yes

—

126

C₂

C₂×Q₈×D₇

yes

characteristic

127

C₂

D₇×#41

yes

characteristic

128

C₂

#6554

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#125	C₂
hypercenter	#125	D₇
hypocenter	C₇	#557
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, prime radical, cosocle

C₂²×C₇

C₂⁴

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
49. Dic₁₄	Q₈	semidirect
113. C₇	#557	semidirect

H ≅ G/N

type

49. Dic₁₄

Q₈

semidirect

113. C₇

#557

semidirect

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

subgroup	index
C₇×#125	2
#6453	2
#6452	2
#6451	2
#6450	2
C₄×Dic₁₄	2
#557	7

subgroup

index

C₄×Dic₁₄

Subgroups — #90376

Conjugacy classes of subgroups

Complements

Maximal subgroups