Group #17986607 — subgroups

Subgroups — #17986607

Order 4544.

192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14556126	C₂	1	1	yes	maximal, characteristic, complemented
2	#14556123	C₂	1	2	yes	maximal
3	#14556122	C₂	1	2	yes	maximal
4	#14556120	C₂	1	2	yes	maximal, complemented
5	#14556119	C₂	1	1	yes	maximal, characteristic, complemented
6	#14556118	C₂	1	2	yes	maximal, complemented
7	#14556108	C₂	1	1	yes	maximal, characteristic
8	#14556104	C₂	1	1	yes	maximal, characteristic
9	C₄×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#14556085	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₁₄₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₇₁	C₂²	1	2	yes	—
16	C₄⋊Dic₇₁	C₂²	1	2	yes	—
17	Dic₇₁⋊C₄	C₂²	1	1	yes	characteristic
18	Dic₇₁⋊C₄	C₂²	1	1	yes	characteristic
19	Dic₇₁⋊C₄	C₂²	1	2	yes	—
20	Dic₇₁⋊C₄	C₂²	1	2	yes	—
21	Dic₇₁⋊C₄	C₂²	1	2	yes	—
22	Dic₇₁⋊C₄	C₂²	1	2	yes	—
23	Dic₇₁⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₇₁)⋊C₄	C₂²	1	1	yes	characteristic
25	(C₂×D₇₁)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₇₁)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₇₁)⋊C₄	C₂²	1	2	yes	—
28	C₄×Dic₇₁	C₂²	1	1	yes	characteristic
29	C₄×Dic₇₁	C₂²	1	2	yes	—
30	C₂×Q₈×C₇₁	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
33	C₄²×C₇₁	C₂²	1	1	yes	characteristic
34	C₄²×C₇₁	C₂²	1	2	yes	—
35	C₂²×D₇₁	C₂³	1	1	yes	characteristic
36	Q₈×C₇₁	—	2	2	no	—
37	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂³	1	2	yes	—
40	C₂×C₄×C₇₁	C₂³	1	2	yes	—
41	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₇₁	C₂³	1	1	yes	characteristic
43	C₂×Dic₇₁	C₂³	1	1	yes	characteristic
44	C₂×Dic₇₁	C₂³	1	2	yes	—
45	C₂×Dic₇₁	C₂³	1	2	yes	—
46	C₂×Dic₇₁	C₂³	1	1	yes	characteristic
47	C₄×D₇₁	—	2	2	no	—
48	Dic₁₄₂	—	2	2	no	—
49	C₂²×C₇₁	C₂⁴	1	1	yes	characteristic
50	C₂×D₇₁	—	2	1	no	—
51	C₂×D₇₁	—	2	1	no	—
52	C₂×D₇₁	—	2	1	no	—
53	C₄×C₇₁	—	2	2	no	—
54	C₄×C₇₁	—	2	1	no	—
55	C₄×C₇₁	—	2	2	no	—
56	C₄×C₇₁	—	2	1	no	—
57	C₄×C₇₁	#41	1	2	yes	—
58	Dic₇₁	—	2	2	no	—
59	Dic₇₁	—	2	1	no	—
60	Dic₇₁	—	2	1	no	—
61	Dic₇₁	—	2	1	no	—
62	Dic₇₁	—	2	2	no	—
63	C₂×C₇₁	C₂×#41	1	1	yes	characteristic
64	C₂×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
65	C₂×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
66	D₇₁	—	4	1	no	—
67	C₇₁	#541	1	1	yes	characteristic, complemented, Sylow
68	#541	—	71	1	no	maximal, Sylow
69	C₄⋊Q₈	—	71	1	no	—
70	#126	—	71	2	no	—
71	#126	—	71	2	no	—
72	#125	—	71	2	no	—
73	#125	—	71	1	no	—
74	#125	—	71	2	no	—
75	C₂²⋊Q₈	—	71	1	no	—
76	C₂²⋊Q₈	—	71	1	no	—
77	C₄×Q₈	—	71	1	no	—
78	C₄×Q₈	D₇₁	1	1	yes	characteristic, complemented
79	#117	—	71	1	no	—
80	C₂×Q₈	—	71	1	no	—
81	C₂×Q₈	C₂×D₇₁	1	1	yes	characteristic
82	C₂²×C₄	—	71	1	no	—
83	C₄⋊C₄	—	71	1	no	—
84	C₄⋊C₄	—	71	1	no	—
85	C₄⋊C₄	—	71	2	no	—
86	C₄⋊C₄	—	71	2	no	—
87	C₄⋊C₄	—	71	2	no	—
88	C₄⋊C₄	—	71	2	no	—
89	C₄⋊C₄	—	71	2	no	—
90	C₄⋊C₄	—	71	1	no	—
91	C₄⋊C₄	—	71	2	no	—
92	C₄⋊C₄	—	71	2	no	—
93	C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₇₁	1	2	yes	—
95	C₂²⋊C₄	—	71	1	no	—
96	C₂²⋊C₄	—	71	1	no	—
97	C₂²⋊C₄	—	71	2	no	—
98	C₂²⋊C₄	—	71	2	no	—
99	C₄²	—	71	1	no	—
100	C₄²	—	71	2	no	—
101	C₄²	C₂×D₇₁	1	1	yes	characteristic
102	C₄²	C₂×D₇₁	1	2	yes	—
103	C₂³	—	71	1	no	—
104	Q₈	—	2	2	no	—
105	Q₈	—	142	2	no	—
106	C₂×C₄	—	71	1	no	—
107	C₂×C₄	—	71	1	no	—
108	C₂×C₄	—	71	2	no	—
109	C₂×C₄	—	71	2	no	—
110	C₂×C₄	—	71	1	no	—
111	C₂×C₄	—	142	2	no	—
112	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₇₁	1	2	yes	—
115	C₂×C₄	C₂²×D₇₁	1	2	yes	—
116	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
117	C₂²	—	142	1	no	—
118	C₂²	—	142	1	no	—
119	C₂²	—	142	1	no	—
120	C₂²	C₂³×D₇₁	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	142	1	no	—
126	C₄	—	142	2	no	—
127	C₄	—	142	1	no	—
128	C₄	—	142	1	no	—
129	C₄	—	142	2	no	—
130	C₄	#11765865	1	2	yes	—
131	C₂	—	284	1	no	—
132	C₂	C₂×#11765865	1	1	yes	characteristic
133	C₂	#14556175	1	1	yes	characteristic
134	C₂	#14556176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14556126

C₂

yes

maximal, characteristic, complemented

#14556123

C₂

yes

maximal

#14556122

C₂

yes

maximal

#14556120

C₂

yes

maximal, complemented

#14556119

C₂

yes

maximal, characteristic, complemented

#14556118

C₂

yes

maximal, complemented

#14556108

C₂

yes

maximal, characteristic

#14556104

C₂

yes

maximal, characteristic

C₄×Dic₁₄₂

C₂

yes

maximal, characteristic, complemented

#14556085

C₂

yes

maximal, characteristic

C₄×Q₈×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₄₂

C₂²

yes

characteristic

C₂×C₄×D₇₁

C₂²

yes

characteristic

C₄⋊Dic₇₁

C₂²

yes

characteristic

C₄⋊Dic₇₁

C₂²

yes

—

C₄⋊Dic₇₁

C₂²

yes

—

Dic₇₁⋊C₄

C₂²

yes

characteristic

Dic₇₁⋊C₄

C₂²

yes

characteristic

Dic₇₁⋊C₄

C₂²

yes

—

Dic₇₁⋊C₄

C₂²

yes

—

Dic₇₁⋊C₄

C₂²

yes

—

Dic₇₁⋊C₄

C₂²

yes

—

Dic₇₁⋊C₄

C₂²

yes

—

(C₂×D₇₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₇₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₇₁)⋊C₄

C₂²

yes

—

(C₂×D₇₁)⋊C₄

C₂²

yes

—

C₄×Dic₇₁

C₂²

yes

characteristic

C₄×Dic₇₁

C₂²

yes

—

C₂×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₄⋊C₄×C₇₁

C₂²

yes

—

C₄²×C₇₁

C₂²

yes

characteristic

C₄²×C₇₁

C₂²

yes

—

C₂²×D₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₄×C₇₁

C₂³

yes

—

C₂×C₄×C₇₁

C₂³

yes

—

C₂×C₄×C₇₁

C₂³

yes

characteristic

C₂×Dic₇₁

C₂³

yes

characteristic

C₂×Dic₇₁

C₂³

yes

characteristic

C₂×Dic₇₁

C₂³

yes

—

C₂×Dic₇₁

C₂³

yes

—

C₂×Dic₇₁

C₂³

yes

characteristic

C₄×D₇₁

—

Dic₁₄₂

—

C₂²×C₇₁

C₂⁴

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#41

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

C₂×#41

yes

characteristic

C₂×C₇₁

2₋¹⁺⁴

yes

characteristic

C₂×C₇₁

2₋¹⁺⁴

yes

characteristic

D₇₁

—

C₇₁

#541

yes

characteristic, complemented, Sylow

#541

—

maximal, Sylow

C₄⋊Q₈

—

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

D₇₁

yes

characteristic, complemented

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₇₁

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇₁

yes

characteristic

C₄⋊C₄

C₂×D₇₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

C₂×D₇₁

yes

characteristic

102

C₄²

C₂×D₇₁

yes

—

103

C₂³

—

104

Q₈

—

105

Q₈

—

142

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

142

—

112

C₂×C₄

C₂²×D₇₁

yes

characteristic

113

C₂×C₄

C₂²×D₇₁

yes

characteristic

114

C₂×C₄

C₂²×D₇₁

yes

—

115

C₂×C₄

C₂²×D₇₁

yes

—

116

C₂×C₄

C₂²×D₇₁

yes

characteristic

117

C₂²

—

142

—

118

C₂²

—

142

—

119

C₂²

—

142

—

120

C₂²

C₂³×D₇₁

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

142

—

126

C₄

—

142

—

127

C₄

—

142

—

128

C₄

—

142

—

129

C₄

—

142

—

130

C₄

#11765865

yes

—

131

C₂

—

284

—

132

C₂

C₂×#11765865

yes

characteristic

133

C₂

#14556175

yes

characteristic

134

C₂

#14556176

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇₁
commutator, socle, cosocle	C₂²×C₇₁	C₂⁴
Fitting	C₄×Q₈×C₇₁	C₂
prime radical	C₂²×D₇₁	C₂³
hypercenter	C₄×Q₈	D₇₁
hypocenter	C₇₁	#541
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇₁

commutator, socle, cosocle

C₂²×C₇₁

C₂⁴

Fitting

C₄×Q₈×C₇₁

C₂

prime radical

C₂²×D₇₁

C₂³

hypercenter

C₄×Q₈

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #14556126	C₂	semidirect
4. #14556120	C₂	semidirect
5. #14556119	C₂	semidirect
6. #14556118	C₂	semidirect
9. C₄×Dic₁₄₂	C₂	semidirect
11. C₄×Q₈×C₇₁	C₂	semidirect
67. C₇₁	#541	semidirect
78. C₄×Q₈	D₇₁	semidirect

H ≅ G/N

type

1. #14556126

C₂

semidirect

4. #14556120

C₂

semidirect

5. #14556119

C₂

semidirect

6. #14556118

C₂

semidirect

9. C₄×Dic₁₄₂

C₂

semidirect

11. C₄×Q₈×C₇₁

C₂

semidirect

67. C₇₁

#541

semidirect

78. C₄×Q₈

D₇₁

semidirect

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

subgroup	index
#14556126	2
#14556123	2
#14556122	2
#14556120	2
#14556119	2
#14556118	2
#14556108	2
#14556104	2
C₄×Dic₁₄₂	2
#14556085	2
C₄×Q₈×C₇₁	2
#541	71

subgroup

index

C₄×Dic₁₄₂

C₄×Q₈×C₇₁

Subgroups — #17986607

Conjugacy classes of subgroups

Complements

Maximal subgroups