Group #17986708 — subgroups

Subgroups — #17986708

Order 4544.

228 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 113 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	#14556127	C₂	1	1	yes	maximal, characteristic, complemented
2	#14556120	C₂	1	1	yes	maximal, characteristic, complemented
3	#14556119	C₂	1	1	yes	maximal, characteristic, complemented
4	#14556107	C₂	1	2	yes	maximal, complemented
5	#14556105	C₂	1	2	yes	maximal, complemented
6	#14556105	C₂	1	2	yes	maximal, complemented
7	C₄×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×Dic₇₁	C₂	1	1	yes	maximal, characteristic
9	D₄×Dic₇₁	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₇₁⋊C₄	C₂	1	2	yes	maximal, complemented
11	C₄×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₁₄₂	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₄₂	C₂²	1	2	yes	—
14	C₂²×Dic₇₁	C₂²	1	2	yes	—
15	C₂²×Dic₇₁	C₂²	1	2	yes	—
16	C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₇₁	C₂²	1	2	yes	—
20	Dic₇₁⋊C₄	C₂²	1	1	yes	characteristic
21	Dic₇₁⋊C₄	C₂²	1	2	yes	—
22	Dic₇₁⋊C₄	C₂²	1	2	yes	—
23	Dic₇₁⋊C₄	C₂²	1	2	yes	—
24	Dic₇₁⋊C₄	C₂²	1	1	yes	characteristic
25	Dic₇₁⋊C₄	C₂²	1	2	yes	—
26	C₂²⋊Dic₇₁	C₂²	1	2	yes	—
27	C₂²⋊Dic₇₁	C₂²	1	2	yes	—
28	C₄×Dic₇₁	C₂²	1	1	yes	characteristic
29	C₄×Dic₇₁	C₂²	1	1	yes	characteristic
30	C₂×D₄×C₇₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₂²	1	2	yes	—
32	C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
33	C₂²⋊C₄×C₇₁	C₂²	1	2	yes	—
34	C₄²×C₇₁	C₂²	1	1	yes	characteristic
35	C₂³×C₇₁	C₂³	1	2	yes	—
36	D₄×C₇₁	Q₈	1	4	yes	complemented
37	C₂×C₄×C₇₁	—	4	2	no	—
38	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂³	1	2	yes	—
41	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₇₁	—	2	4	no	—
43	C₂×Dic₇₁	—	4	2	no	—
44	C₂×Dic₇₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₇₁	C₂³	1	1	yes	characteristic
46	C₂×Dic₇₁	C₂³	1	1	yes	characteristic
47	C₂×Dic₇₁	C₂³	1	2	yes	—
48	C₂×Dic₇₁	C₂³	1	2	yes	—
49	C₂×Dic₇₁	C₂³	1	1	yes	characteristic
50	Dic₁₄₂	—	2	2	no	—
51	Dic₁₄₂	—	4	2	no	—
52	C₂²×C₇₁	—	2	2	no	—
53	C₂²×C₇₁	—	2	2	no	—
54	C₂²×C₇₁	C₂×Q₈	1	4	yes	—
55	C₂²×C₇₁	C₂⁴	1	1	yes	characteristic
56	C₄×C₇₁	—	2	1	no	—
57	C₄×C₇₁	—	2	2	no	—
58	C₄×C₇₁	—	2	1	no	—
59	C₄×C₇₁	C₂×Q₈	1	2	yes	—
60	Dic₇₁	—	2	1	no	—
61	Dic₇₁	—	2	2	no	—
62	Dic₇₁	—	2	1	no	—
63	Dic₇₁	—	2	2	no	—
64	Dic₇₁	—	2	1	no	—
65	Dic₇₁	#41	1	2	yes	—
66	C₂×C₇₁	—	2	4	no	—
67	C₂×C₇₁	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₇₁	C₂×#41	1	1	yes	characteristic
69	C₂×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
70	C₇₁	#554	1	1	yes	characteristic, complemented, Sylow
71	#554	—	71	1	no	maximal, Sylow
72	C₄⋊Q₈	—	71	1	no	—
73	#125	—	71	1	no	—
74	#125	—	71	1	no	—
75	C₂²⋊Q₈	—	71	2	no	—
76	C₂²⋊Q₈	—	71	2	no	—
77	C₂²⋊Q₈	—	71	2	no	—
78	C₄×Q₈	—	71	1	no	—
79	C₄×D₄	—	71	1	no	—
80	C₄×D₄	—	71	1	no	—
81	C₄×D₄	D₇₁	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	71	2	no	—
83	C₂×Q₈	—	71	2	no	—
84	C₂×Q₈	—	71	1	no	—
85	C₂×D₄	C₂×D₇₁	1	1	yes	characteristic
86	C₂²×C₄	—	71	2	no	—
87	C₂²×C₄	—	71	2	no	—
88	C₂²×C₄	C₂×D₇₁	1	2	yes	—
89	C₄⋊C₄	—	71	1	no	—
90	C₄⋊C₄	—	71	2	no	—
91	C₄⋊C₄	—	71	2	no	—
92	C₄⋊C₄	—	71	2	no	—
93	C₄⋊C₄	—	71	1	no	—
94	C₄⋊C₄	—	71	2	no	—
95	C₄⋊C₄	—	71	1	no	—
96	C₄⋊C₄	—	71	1	no	—
97	C₄⋊C₄	—	71	1	no	—
98	C₄⋊C₄	—	71	2	no	—
99	C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
100	C₂²⋊C₄	—	71	2	no	—
101	C₂²⋊C₄	—	71	2	no	—
102	C₂²⋊C₄	C₂×D₇₁	1	2	yes	—
103	C₄²	—	71	1	no	—
104	C₄²	—	71	1	no	—
105	C₄²	C₂×D₇₁	1	1	yes	characteristic
106	C₂³	C₂²×D₇₁	1	2	yes	—
107	Q₈	—	284	2	no	—
108	Q₈	—	142	2	no	—
109	D₄	Dic₁₄₂	1	4	yes	complemented
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	284	2	no	—
112	C₂×C₄	—	71	1	no	—
113	C₂×C₄	—	71	1	no	—
114	C₂×C₄	—	71	1	no	—
115	C₂×C₄	—	71	1	no	—
116	C₂×C₄	—	71	2	no	—
117	C₂×C₄	—	71	2	no	—
118	C₂×C₄	—	142	4	no	—
119	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₇₁	1	2	yes	—
122	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	C₂×Dic₁₄₂	1	4	yes	—
126	C₂²	C₂³×D₇₁	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	71	2	no	—
131	C₄	—	142	2	no	—
132	C₄	—	142	1	no	—
133	C₄	—	142	1	no	—
134	C₄	—	142	2	no	—
135	C₄	—	142	1	no	—
136	C₄	C₂×Dic₁₄₂	1	2	yes	—
137	C₂	—	2	4	no	—
138	C₂	C₂²×Dic₁₄₂	1	1	yes	characteristic
139	C₂	#41×D₇₁	1	1	yes	characteristic
140	C₂	#14556173	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14556127

C₂

yes

maximal, characteristic, complemented

#14556120

C₂

yes

maximal, characteristic, complemented

#14556119

C₂

yes

maximal, characteristic, complemented

#14556107

C₂

yes

maximal, complemented

#14556105

C₂

yes

maximal, complemented

#14556105

C₂

yes

maximal, complemented

C₄×Dic₁₄₂

C₂

yes

maximal, characteristic, complemented

D₄×Dic₇₁

C₂

yes

maximal, characteristic

D₄×Dic₇₁

C₂

yes

maximal, characteristic

C₂×Dic₇₁⋊C₄

C₂

yes

maximal, complemented

C₄×D₄×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₁₄₂

C₂²

yes

characteristic

C₂×Dic₁₄₂

C₂²

yes

—

C₂²×Dic₇₁

C₂²

yes

—

C₂²×Dic₇₁

C₂²

yes

—

C₄⋊Dic₇₁

C₂²

yes

characteristic

C₄⋊Dic₇₁

C₂²

yes

characteristic

C₄⋊Dic₇₁

C₂²

yes

characteristic

C₄⋊Dic₇₁

C₂²

yes

—

Dic₇₁⋊C₄

C₂²

yes

characteristic

Dic₇₁⋊C₄

C₂²

yes

—

Dic₇₁⋊C₄

C₂²

yes

—

Dic₇₁⋊C₄

C₂²

yes

—

Dic₇₁⋊C₄

C₂²

yes

characteristic

Dic₇₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₇₁

C₂²

yes

—

C₂²⋊Dic₇₁

C₂²

yes

—

C₄×Dic₇₁

C₂²

yes

characteristic

C₄×Dic₇₁

C₂²

yes

characteristic

C₂×D₄×C₇₁

C₂²

yes

characteristic

C₂²×C₄×C₇₁

C₂²

yes

—

C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₂²⋊C₄×C₇₁

C₂²

yes

—

C₄²×C₇₁

C₂²

yes

characteristic

C₂³×C₇₁

C₂³

yes

—

D₄×C₇₁

Q₈

yes

complemented

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₄×C₇₁

C₂³

yes

—

C₂×C₄×C₇₁

C₂³

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

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

Dic₁₄₂

—

Dic₁₄₂

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×Q₈

yes

—

C₂²×C₇₁

C₂⁴

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×Q₈

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

#41

yes

—

C₂×C₇₁

—

C₂×C₇₁

C₂²×Q₈

yes

characteristic

C₂×C₇₁

C₂×#41

yes

C₂×C₇₁

yes

C₇₁

yes

characteristic, complemented, Sylow

#554

—

maximal, Sylow

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₇₁

yes

—

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

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

C₂×D₇₁

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×D₇₁

yes

characteristic

106

C₂³

C₂²×D₇₁

yes

—

107

Q₈

—

284

—

108

Q₈

—

142

—

109

D₄

Dic₁₄₂

yes

complemented

110

C₂×C₄

—

111

C₂×C₄

—

284

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

142

—

119

C₂×C₄

C₂²×D₇₁

yes

characteristic

120

C₂×C₄

C₂²×D₇₁

yes

characteristic

121

C₂×C₄

C₂²×D₇₁

yes

—

122

C₂×C₄

C₂²×D₇₁

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×Dic₁₄₂

yes

—

126

C₂²

C₂³×D₇₁

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

142

—

132

C₄

—

142

—

133

C₄

—

142

—

134

C₄

—

142

—

135

C₄

—

142

—

136

C₄

C₂×Dic₁₄₂

yes

—

137

C₂

—

138

C₂

C₂²×Dic₁₄₂

yes

characteristic

139

C₂

#41×D₇₁

yes

characteristic

140

C₂

#14556173

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇₁
commutator, socle, cosocle	C₂²×C₇₁	C₂⁴
Fitting	C₄×D₄×C₇₁	C₂
prime radical	C₂×D₄×C₇₁	C₂²
hypercenter	C₄×D₄	D₇₁
hypocenter	C₇₁	#554
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇₁

commutator, socle, cosocle

C₂²×C₇₁

C₂⁴

Fitting

C₄×D₄×C₇₁

C₂

prime radical

C₂×D₄×C₇₁

C₂²

hypercenter

C₄×D₄

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #14556127	C₂	semidirect
2. #14556120	C₂	semidirect
3. #14556119	C₂	semidirect
4. #14556107	C₂	semidirect
5. #14556105	C₂	semidirect
6. #14556105	C₂	semidirect
7. C₄×Dic₁₄₂	C₂	semidirect
10. C₂×Dic₇₁⋊C₄	C₂	semidirect
36. D₄×C₇₁	Q₈	semidirect
70. C₇₁	#554	semidirect
109. D₄	Dic₁₄₂	semidirect

H ≅ G/N

type

1. #14556127

C₂

semidirect

2. #14556120

C₂

semidirect

3. #14556119

C₂

semidirect

4. #14556107

C₂

semidirect

5. #14556105

C₂

semidirect

6. #14556105

C₂

semidirect

7. C₄×Dic₁₄₂

C₂

semidirect

10. C₂×Dic₇₁⋊C₄

C₂

semidirect

36. D₄×C₇₁

Q₈

semidirect

70. C₇₁

#554

semidirect

109. D₄

Dic₁₄₂

semidirect

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

subgroup	index
#14556127	2
#14556120	2
#14556119	2
#14556107	2
#14556105	2
C₄×Dic₁₄₂	2
D₄×Dic₇₁	2
C₂×Dic₇₁⋊C₄	2
C₄×D₄×C₇₁	2
#554	71

subgroup

index

C₄×Dic₁₄₂

D₄×Dic₇₁

C₂×Dic₇₁⋊C₄

C₄×D₄×C₇₁

#554

Subgroups — #17986708

Conjugacy classes of subgroups

Complements

Maximal subgroups