Group #34032781 — subgroups

Subgroups — #34032781

Order 18688.

162 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 49 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	#22626995	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#11924040	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×#10596833	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924040	—	2	1	no	—
5	C₄×#11924040	—	2	1	no	—
6	D₄×#10596833	—	2	1	no	—
7	#17989450	C₄	1	1	yes	characteristic
8	D₄⋊#10596833	—	2	1	no	—
9	C₈×#10596833	C₂²	1	1	yes	characteristic
10	D₈×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924040	—	2	2	no	—
12	C₂×#11924040	—	4	1	no	—
13	C₂²×#10596833	—	4	1	no	—
14	C₄⋊#10596833	—	2	1	no	—
15	C₂²⋊#10596833	—	4	1	no	—
16	C₄×#10596833	D₄	1	1	yes	characteristic
17	C₄×#10596833	D₄	1	1	yes	characteristic
18	D₄×Dic₇₃	—	2	1	no	—
19	#14556525	C₈	1	1	yes	characteristic
20	#14556512	—	2	1	no	—
21	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×D₈×C₇₃	C₈	1	1	yes	characteristic
23	#11924040	—	2	4	no	—
24	#11924040	—	8	1	no	—
25	C₂×#10596833	—	2	1	no	—
26	C₂×#10596833	—	4	1	no	—
27	C₂×#10596833	—	8	1	no	—
28	C₂×#10596833	—	8	1	no	—
29	C₂×#10596833	D₈	1	1	yes	characteristic
30	C₂×#10596833	SD₁₆	1	1	yes	characteristic
31	C₂²×Dic₇₃	—	4	1	no	—
32	C₂×#10596831	M₁₆	1	1	yes	characteristic
33	C₄⋊Dic₇₃	—	2	1	no	—
34	C₂²⋊Dic₇₃	—	4	1	no	—
35	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×D₄×C₇₃	—	2	1	no	—
37	D₈×C₇₃	—	2	1	no	—
38	D₈×C₇₃	C₁₆	1	2	yes	complemented
39	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
40	C₂³×C₇₃	—	4	1	no	—
41	D₄×C₇₃	—	2	2	no	—
42	D₄×C₇₃	—	4	1	no	—
43	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×Dic₇₃	—	4	1	no	—
45	C₂×Dic₇₃	—	8	1	no	—
46	C₂×Dic₇₃	—	8	1	no	—
47	C₂×Dic₇₃	#102	1	1	yes	characteristic
48	C₂×Dic₇₃	#104	1	1	yes	characteristic
49	#10596833	—	2	2	no	—
50	#10596833	—	2	1	no	—
51	#10596833	—	8	1	no	—
52	#10596833	D₁₆	1	1	yes	characteristic
53	#10596833	SD₃₂	1	1	yes	characteristic
54	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
55	C₈×C₇₃	M₃₂	1	1	yes	characteristic
56	#10596831	—	2	1	no	—
57	C₂²×C₇₃	—	4	2	no	—
58	C₂²×C₇₃	—	8	1	no	—
59	C₂²×C₇₃	—	8	1	no	—
60	C₂²×C₇₃	D₄⋊C₈	1	1	yes	characteristic
61	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
62	C₄×C₇₃	#350	1	1	yes	characteristic
63	Dic₇₃	—	2	1	no	—
64	Dic₇₃	—	8	1	no	—
65	Dic₇₃	#357	1	1	yes	characteristic
66	Dic₇₃	#359	1	1	yes	characteristic
67	#7068	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	—	8	2	no	—
69	C₂×C₇₃	D₄⋊C₁₆	1	1	yes	characteristic
70	C₂×C₇₃	#1331	1	1	yes	characteristic
71	C₂×C₇₃	#1335	1	1	yes	characteristic
72	C₈×D₈	—	73	1	no	—
73	#1370	—	73	1	no	—
74	C₈×C₁₆	—	73	1	no	—
75	C₇₃	#7068	1	1	yes	characteristic, complemented, Sylow
76	C₄×D₈	—	73	1	no	—
77	C₈×D₄	—	146	1	no	—
78	C₄⋊C₁₆	—	146	1	no	—
79	C₄×C₁₆	—	146	1	no	—
80	#335	—	73	1	no	—
81	D₄⋊C₈	—	146	1	no	—
82	C₈²	—	73	1	no	—
83	C₂×D₈	#10596842	1	1	yes	characteristic
84	C₂²×C₈	—	292	1	no	—
85	C₄×D₄	—	146	1	no	—
86	C₂×C₁₆	—	292	1	no	—
87	C₂×C₁₆	—	146	2	no	—
88	#107	—	73	1	no	—
89	C₄⋊C₈	—	146	1	no	—
90	#102	—	146	1	no	—
91	C₂²⋊C₈	—	292	1	no	—
92	C₄×C₈	—	73	1	no	—
93	C₄×C₈	—	73	1	no	—
94	C₄×C₈	—	73	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂²×C₄	—	292	1	no	—
97	D₈	—	2	1	no	—
98	D₈	#11924040	1	2	yes	complemented
99	C₂×C₈	—	292	1	no	—
100	C₂×C₈	—	584	1	no	—
101	C₂×C₈	—	584	1	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	146	1	no	—
106	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
107	C₄⋊C₄	—	146	1	no	—
108	C₂²⋊C₄	—	292	1	no	—
109	C₄²	—	73	1	no	—
110	C₁₆	—	584	1	no	—
111	C₁₆	—	146	4	no	—
112	C₂³	—	4	1	no	—
113	D₄	—	2	2	no	—
114	D₄	—	4	1	no	—
115	C₂×C₄	—	292	1	no	—
116	C₂×C₄	—	584	1	no	—
117	C₂×C₄	—	584	1	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
121	C₈	—	584	1	no	—
122	C₈	—	73	1	no	—
123	C₈	—	73	1	no	—
124	C₈	—	146	1	no	—
125	C₈	—	146	2	no	—
126	C₈	—	146	1	no	—
127	C₈	C₂×#11924040	1	1	yes	characteristic
128	C₈	#14556687	1	1	yes	characteristic
129	C₂²	—	4	2	no	—
130	C₂²	—	8	1	no	—
131	C₂²	—	8	1	no	—
132	C₂²	D₄⋊#10596842	1	1	yes	characteristic
133	C₄	—	584	1	no	—
134	C₄	—	73	1	no	—
135	C₄	—	73	1	no	—
136	C₄	—	146	1	no	—
137	C₄	C₂²⋊#11924040	1	1	yes	characteristic
138	C₄	#17989687	1	1	yes	characteristic
139	C₂	—	8	2	no	—
140	C₂	D₄⋊#11924040	1	1	yes	characteristic
141	C₂	#22626977	1	1	yes	characteristic
142	C₂	#22626981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626995

C₂

yes

maximal, characteristic, complemented

C₈×#11924040

C₂

yes

maximal, characteristic, complemented

D₈×#10596833

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

characteristic

D₄⋊#10596833

—

C₈×#10596833

C₂²

yes

characteristic

D₈×Dic₇₃

C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

D₄×Dic₇₃

—

#14556525

C₈

yes

characteristic

#14556512

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₂×D₈×C₇₃

C₈

yes

—

—

—

—

—

—

D₈

yes

characteristic

C₂×#10596833

SD₁₆

yes

characteristic

C₂²×Dic₇₃

—

C₂×#10596831

M₁₆

yes

characteristic

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

C₁₆

yes

complemented

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

—

—

—

D₁₆

yes

characteristic

#10596833

SD₃₂

yes

characteristic

C₈×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

M₃₂

yes

characteristic

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

D₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₃

#350

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

D₄⋊C₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₈×D₈

—

#1370

—

C₈×C₁₆

—

C₇₃

#7068

yes

characteristic, complemented, Sylow

C₄×D₈

—

C₈×D₄

—

146

—

C₄⋊C₁₆

—

146

—

C₄×C₁₆

—

146

—

#335

—

D₄⋊C₈

—

146

—

C₈²

—

C₂×D₈

#10596842

yes

characteristic

C₂²×C₈

—

292

—

C₄×D₄

—

146

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

146

—

#107

—

C₄⋊C₈

—

146

—

#102

—

146

—

C₂²⋊C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

292

—

D₈

—

D₈

#11924040

yes

complemented

C₂×C₈

—

292

—

100

C₂×C₈

—

584

—

101

C₂×C₈

—

584

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

146

—

106

C₂×C₈

C₂×#10596842

yes

characteristic

107

C₄⋊C₄

—

146

—

108

C₂²⋊C₄

—

292

—

109

C₄²

—

110

C₁₆

—

584

—

111

C₁₆

—

146

—

112

C₂³

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

292

—

116

C₂×C₄

—

584

—

117

C₂×C₄

—

584

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²⋊#10596842

yes

characteristic

121

C₈

—

584

—

122

C₈

—

123

C₈

—

124

C₈

—

146

—

125

C₈

—

146

—

126

C₈

—

146

—

127

C₈

C₂×#11924040

yes

characteristic

128

C₈

#14556687

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

D₄⋊#10596842

yes

characteristic

133

C₄

—

584

—

134

C₄

—

135

C₄

—

136

C₄

—

146

—

137

C₄

C₂²⋊#11924040

yes

characteristic

138

C₄

#17989687

yes

characteristic

139

C₂

—

140

C₂

D₄⋊#11924040

yes

characteristic

141

C₂

#22626977

yes

characteristic

142

C₂

#22626981

yes

characteristic

	subgroup	G/H
center	C₂²	D₄⋊#10596842
commutator	C₈×C₇₃	C₂×C₁₆
Frattini	C₂×C₈	C₂×#10596842
Fitting, prime radical	C₂×D₈×C₇₃	C₈
socle	C₂²×C₇₃	D₄⋊C₈
cosocle	C₈×#10596833	C₂²
hypercenter	C₂×D₈	#10596842
hypocenter	C₇₃	#7068
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄⋊#10596842

commutator

C₈×C₇₃

C₂×C₁₆

Frattini

C₂×C₈

C₂×#10596842

Fitting, prime radical

C₂×D₈×C₇₃

C₈

socle

C₂²×C₇₃

D₄⋊C₈

C₂²

C₂×D₈

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

N	H ≅ G/N	type
1. #22626995	C₂	semidirect
2. C₈×#11924040	C₂	semidirect
38. D₈×C₇₃	C₁₆	semidirect
75. C₇₃	#7068	semidirect
98. D₈	#11924040	semidirect

H ≅ G/N

type

1. #22626995

C₂

semidirect

2. C₈×#11924040

C₂

semidirect

38. D₈×C₇₃

C₁₆

semidirect

75. C₇₃

#7068

semidirect

98. D₈

#11924040

semidirect

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

subgroup	index
#22626995	2
C₈×#11924040	2
D₈×#10596833	2
#7068	73

subgroup

index

Subgroups — #34032781

Conjugacy classes of subgroups

Complements

Maximal subgroups