Group #42119777 — subgroups

Subgroups — #42119777

Order 27584.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	#28352188	C₂	1	1	yes	maximal, characteristic, complemented
2	#28352183	C₂	1	2	yes	maximal, complemented
3	#28352182	C₂	1	2	yes	maximal, complemented
4	#28352175	C₂	1	2	yes	maximal
5	#28352169	C₂	1	1	yes	maximal, characteristic, complemented
6	#28352168	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₈₆₂	C₂	1	2	yes	maximal, complemented
8	#28352154	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊D₄₃₁	C₂	1	1	yes	maximal, characteristic
10	#28352147	C₂	1	1	yes	maximal, characteristic, complemented
11	#117×C₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₈₆₂	C₂²	1	1	yes	characteristic
13	C₂×Dic₈₆₂	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₄₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄₃₁	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₄₃₁	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₄₃₁	C₂²	1	2	yes	—
18	Dic₄₃₁⋊C₄	C₂²	1	1	yes	characteristic
19	Dic₄₃₁⋊C₄	C₂²	1	2	yes	—
20	Dic₄₃₁⋊C₄	C₂²	1	2	yes	—
21	Dic₄₃₁⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₄₃₁	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₄₃₁	C₂²	1	2	yes	—
24	(C₂×D₄₃₁)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₄₃₁)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₄₃₁)⋊C₄	C₂²	1	2	yes	—
27	C₄×Dic₄₃₁	C₂²	1	2	yes	—
28	C₄×Dic₄₃₁	C₂²	1	1	yes	characteristic
29	C₄×Dic₄₃₁	C₂²	1	2	yes	—
30	C₂²×C₄×C₄₃₁	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₄₃₁	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₄₃₁	C₂²	1	1	yes	characteristic
33	C₂²⋊C₄×C₄₃₁	C₂²	1	2	yes	—
34	C₄²×C₄₃₁	C₂²	1	2	yes	—
35	C₂²×D₄₃₁	C₂³	1	1	yes	characteristic
36	Dic₈₆₂	—	2	2	no	—
37	Dic₈₆₂	—	2	4	no	—
38	C₂²⋊D₄₃₁	—	2	2	no	—
39	C₂×Dic₄₃₁	C₂³	1	1	yes	characteristic
40	C₂×Dic₄₃₁	C₂³	1	1	yes	characteristic
41	C₂×Dic₄₃₁	C₂³	1	2	yes	—
42	C₂×Dic₄₃₁	C₂³	1	2	yes	—
43	C₂×Dic₄₃₁	C₂³	1	1	yes	characteristic
44	C₄×D₄₃₁	—	2	2	no	—
45	C₂³×C₄₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₄₃₁	—	2	2	no	—
47	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₄₃₁	C₂³	1	2	yes	—
49	C₂×C₄×C₄₃₁	C₂³	1	2	yes	—
50	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄₃₁	—	2	1	no	—
52	C₂²×C₄₃₁	—	2	1	no	—
53	C₂²×C₄₃₁	—	2	1	no	—
54	C₂²×C₄₃₁	C₂⁴	1	1	yes	characteristic
55	C₂×D₄₃₁	—	2	1	no	—
56	C₂×D₄₃₁	—	2	1	no	—
57	C₂×D₄₃₁	—	2	1	no	—
58	C₄×C₄₃₁	—	2	1	no	—
59	C₄×C₄₃₁	—	2	2	no	—
60	C₄×C₄₃₁	—	2	2	no	—
61	C₄×C₄₃₁	#41	1	2	yes	—
62	Dic₄₃₁	—	2	1	no	—
63	Dic₄₃₁	—	2	2	no	—
64	Dic₄₃₁	—	2	2	no	—
65	Dic₄₃₁	—	2	1	no	—
66	Dic₄₃₁	#41	1	2	yes	—
67	C₂×C₄₃₁	—	4	1	no	—
68	C₂×C₄₃₁	C₂×#41	1	1	yes	characteristic
69	C₂×C₄₃₁	C₂×#41	1	1	yes	characteristic
70	C₂×C₄₃₁	2₋¹⁺⁴	1	1	yes	characteristic
71	D₄₃₁	—	4	1	no	—
72	C₄₃₁	#556	1	1	yes	characteristic, complemented, Sylow
73	#556	—	431	1	no	maximal, Sylow
74	C₄⋊Q₈	—	431	1	no	—
75	#126	—	431	2	no	—
76	#126	—	431	2	no	—
77	#124	—	431	2	no	—
78	C₂²⋊Q₈	—	431	1	no	—
79	C₂²⋊Q₈	—	431	1	no	—
80	C₄×Q₈	—	431	1	no	—
81	C₄×Q₈	—	431	2	no	—
82	C₄×D₄	—	431	1	no	—
83	#117	—	431	1	no	—
84	#117	D₄₃₁	1	1	yes	characteristic, complemented
85	C₂×Q₈	—	431	1	no	—
86	C₂×Q₈	—	431	2	no	—
87	C₂×D₄	—	431	1	no	—
88	C₂²×C₄	—	431	1	no	—
89	C₂²×C₄	C₂×D₄₃₁	1	1	yes	characteristic
90	C₄⋊C₄	—	431	1	no	—
91	C₄⋊C₄	—	431	1	no	—
92	C₄⋊C₄	—	431	2	no	—
93	C₄⋊C₄	—	431	2	no	—
94	C₄⋊C₄	—	431	2	no	—
95	C₄⋊C₄	—	431	2	no	—
96	C₄⋊C₄	C₂×D₄₃₁	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₄₃₁	1	1	yes	characteristic
98	C₂²⋊C₄	—	431	1	no	—
99	C₂²⋊C₄	—	431	2	no	—
100	C₂²⋊C₄	—	431	2	no	—
101	C₂²⋊C₄	—	431	1	no	—
102	C₂²⋊C₄	—	431	2	no	—
103	C₂²⋊C₄	C₂×D₄₃₁	1	2	yes	—
104	C₄²	—	431	2	no	—
105	C₄²	—	431	1	no	—
106	C₄²	—	431	2	no	—
107	C₄²	C₂×D₄₃₁	1	2	yes	—
108	C₂³	—	431	1	no	—
109	C₂³	C₂²×D₄₃₁	1	1	yes	characteristic
110	Q₈	—	862	4	no	—
111	Q₈	—	862	2	no	—
112	D₄	—	862	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	862	2	no	—
115	C₂×C₄	—	431	1	no	—
116	C₂×C₄	—	431	1	no	—
117	C₂×C₄	—	431	2	no	—
118	C₂×C₄	—	431	2	no	—
119	C₂×C₄	—	431	1	no	—
120	C₂×C₄	C₂²×D₄₃₁	1	2	yes	—
121	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₄₃₁	1	2	yes	—
123	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	862	1	no	—
128	C₂²	—	862	1	no	—
129	C₂²	—	862	1	no	—
130	C₂²	C₂³×D₄₃₁	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	862	2	no	—
135	C₄	—	862	1	no	—
136	C₄	—	862	2	no	—
137	C₄	—	862	1	no	—
138	C₄	—	431	2	no	—
139	C₄	#20293800	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	1724	1	no	—
142	C₂	#41×D₄₃₁	1	1	yes	characteristic
143	C₂	C₂×#20293800	1	1	yes	characteristic
144	C₂	#28352236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28352188

C₂

yes

maximal, characteristic, complemented

#28352183

C₂

yes

maximal, complemented

#28352182

C₂

yes

maximal, complemented

#28352175

C₂

yes

maximal

#28352169

C₂

yes

maximal, characteristic, complemented

#28352168

C₂

yes

maximal, characteristic, complemented

C₄×Dic₈₆₂

C₂

yes

maximal, complemented

#28352154

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₄₃₁

C₂

yes

maximal, characteristic

#28352147

C₂

yes

maximal, characteristic, complemented

#117×C₄₃₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₈₆₂

C₂²

yes

characteristic

C₂×Dic₈₆₂

C₂²

yes

—

C₂×C₂²⋊D₄₃₁

C₂²

yes

characteristic

C₂×C₄×D₄₃₁

C₂²

yes

characteristic

C₄⋊Dic₄₃₁

C₂²

yes

characteristic

C₄⋊Dic₄₃₁

C₂²

yes

—

Dic₄₃₁⋊C₄

C₂²

yes

characteristic

Dic₄₃₁⋊C₄

C₂²

yes

—

Dic₄₃₁⋊C₄

C₂²

yes

—

Dic₄₃₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₄₃₁

C₂²

yes

characteristic

C₂²⋊Dic₄₃₁

C₂²

yes

—

(C₂×D₄₃₁)⋊C₄

C₂²

yes

—

(C₂×D₄₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₄₃₁)⋊C₄

C₂²

yes

—

C₄×Dic₄₃₁

C₂²

yes

—

C₄×Dic₄₃₁

C₂²

yes

characteristic

C₄×Dic₄₃₁

C₂²

yes

—

C₂²×C₄×C₄₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₄₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₄₃₁

C₂²

yes

characteristic

C₂²⋊C₄×C₄₃₁

C₂²

yes

—

C₄²×C₄₃₁

C₂²

yes

—

C₂²×D₄₃₁

C₂³

yes

characteristic

Dic₈₆₂

—

Dic₈₆₂

—

C₂²⋊D₄₃₁

—

C₂×Dic₄₃₁

C₂³

yes

characteristic

C₂×Dic₄₃₁

C₂³

yes

characteristic

C₂×Dic₄₃₁

C₂³

yes

—

C₂×Dic₄₃₁

C₂³

yes

—

C₂×Dic₄₃₁

C₂³

yes

characteristic

C₄×D₄₃₁

—

C₂³×C₄₃₁

C₂³

yes

characteristic

C₂×C₄×C₄₃₁

—

C₂×C₄×C₄₃₁

C₂³

yes

characteristic

C₂×C₄×C₄₃₁

C₂³

yes

—

C₂×C₄×C₄₃₁

C₂³

yes

—

C₂×C₄×C₄₃₁

C₂³

yes

characteristic

C₂²×C₄₃₁

—

C₂²×C₄₃₁

—

C₂²×C₄₃₁

—

C₂²×C₄₃₁

C₂⁴

yes

characteristic

C₂×D₄₃₁

—

C₂×D₄₃₁

—

C₂×D₄₃₁

—

C₄×C₄₃₁

—

C₄×C₄₃₁

—

C₄×C₄₃₁

—

C₄×C₄₃₁

#41

yes

—

Dic₄₃₁

—

Dic₄₃₁

—

Dic₄₃₁

—

Dic₄₃₁

—

Dic₄₃₁

#41

yes

—

C₂×C₄₃₁

—

C₂×C₄₃₁

C₂×#41

yes

characteristic

C₂×C₄₃₁

C₂×#41

yes

characteristic

C₂×C₄₃₁

2₋¹⁺⁴

yes

characteristic

D₄₃₁

—

C₄₃₁

#556

yes

characteristic, complemented, Sylow

#556

—

431

maximal, Sylow

C₄⋊Q₈

—

431

—

#126

—

431

—

#126

—

431

—

#124

—

431

—

C₂²⋊Q₈

—

431

—

C₂²⋊Q₈

—

431

—

C₄×Q₈

—

431

—

C₄×Q₈

—

431

—

C₄×D₄

—

431

—

#117

—

431

—

#117

D₄₃₁

yes

characteristic, complemented

C₂×Q₈

—

431

—

C₂×Q₈

—

431

—

C₂×D₄

—

431

—

C₂²×C₄

—

431

—

C₂²×C₄

C₂×D₄₃₁

yes

characteristic

C₄⋊C₄

—

431

—

C₄⋊C₄

—

431

—

C₄⋊C₄

—

431

—

C₄⋊C₄

—

431

—

C₄⋊C₄

—

431

—

C₄⋊C₄

—

431

—

C₄⋊C₄

C₂×D₄₃₁

yes

characteristic

C₄⋊C₄

C₂×D₄₃₁

yes

characteristic

C₂²⋊C₄

—

431

—

C₂²⋊C₄

—

431

—

100

C₂²⋊C₄

—

431

—

101

C₂²⋊C₄

—

431

—

102

C₂²⋊C₄

—

431

—

103

C₂²⋊C₄

C₂×D₄₃₁

yes

—

104

C₄²

—

431

—

105

C₄²

—

431

—

106

C₄²

—

431

—

107

C₄²

C₂×D₄₃₁

yes

—

108

C₂³

—

431

—

109

C₂³

C₂²×D₄₃₁

yes

characteristic

110

Q₈

—

862

—

111

Q₈

—

862

—

112

D₄

—

862

—

113

C₂×C₄

—

114

C₂×C₄

—

862

—

115

C₂×C₄

—

431

—

116

C₂×C₄

—

431

—

117

C₂×C₄

—

431

—

118

C₂×C₄

—

431

—

119

C₂×C₄

—

431

—

120

C₂×C₄

C₂²×D₄₃₁

yes

—

121

C₂×C₄

C₂²×D₄₃₁

yes

characteristic

122

C₂×C₄

C₂²×D₄₃₁

yes

—

123

C₂×C₄

C₂²×D₄₃₁

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

862

—

128

C₂²

—

862

—

129

C₂²

—

862

—

130

C₂²

C₂³×D₄₃₁

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

862

—

135

C₄

—

862

—

136

C₄

—

862

—

137

C₄

—

862

—

138

C₄

—

431

—

139

C₄

#20293800

yes

—

140

C₂

—

141

C₂

—

1724

—

142

C₂

#41×D₄₃₁

yes

characteristic

143

C₂

C₂×#20293800

yes

characteristic

144

C₂

#28352236

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃₁
commutator, socle, cosocle	C₂²×C₄₃₁	C₂⁴
Fitting	#117×C₄₃₁	C₂
prime radical	C₂×C₂²⋊D₄₃₁	C₂²
hypercenter	#117	D₄₃₁
hypocenter	C₄₃₁	#556
LCS	G ▷ C₂²×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃₁

commutator, socle, cosocle

C₂²×C₄₃₁

C₂⁴

Fitting

#117×C₄₃₁

C₂

prime radical

C₂×C₂²⋊D₄₃₁

C₂²

D₄₃₁

C₄₃₁

G ▷ C₂²×C₄₃₁ ▷ C₄₃₁

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #28352188	C₂	semidirect
2. #28352183	C₂	semidirect
3. #28352182	C₂	semidirect
5. #28352169	C₂	semidirect
6. #28352168	C₂	semidirect
7. C₄×Dic₈₆₂	C₂	semidirect
8. #28352154	C₂	semidirect
10. #28352147	C₂	semidirect
11. #117×C₄₃₁	C₂	semidirect
72. C₄₃₁	#556	semidirect
84. #117	D₄₃₁	semidirect

H ≅ G/N

type

1. #28352188

C₂

semidirect

2. #28352183

C₂

semidirect

3. #28352182

C₂

semidirect

5. #28352169

C₂

semidirect

6. #28352168

C₂

semidirect

7. C₄×Dic₈₆₂

C₂

semidirect

8. #28352154

C₂

semidirect

10. #28352147

C₂

semidirect

11. #117×C₄₃₁

C₂

semidirect

72. C₄₃₁

#556

semidirect

84. #117

D₄₃₁

semidirect

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

subgroup	index
#28352188	2
#28352183	2
#28352182	2
#28352175	2
#28352169	2
#28352168	2
C₄×Dic₈₆₂	2
#28352154	2
C₄×C₂²⋊D₄₃₁	2
#28352147	2
#117×C₄₃₁	2
#556	431

subgroup

index

C₄×Dic₈₆₂

C₄×C₂²⋊D₄₃₁

#117×C₄₃₁

431

Subgroups — #42119777

Conjugacy classes of subgroups

Complements

Maximal subgroups