Group #34034017 — subgroups

Subgroups — #34034017

Order 18688.

236 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22627147	C₂	1	2	yes	maximal, complemented
2	#22626989	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#11924040	C₂	1	2	yes	maximal, complemented
4	#22626948	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×#10596833	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556687	—	2	2	no	—
7	C₄⋊#11924040	C₂²	1	2	yes	—
8	C₂²⋊#11924040	—	2	2	no	—
9	C₄×#11924040	C₂²	1	1	yes	characteristic
10	C₄×#11924040	C₂²	1	1	yes	characteristic
11	D₄×#10596833	C₂²	1	2	yes	—
12	#17989450	C₄	1	1	yes	characteristic
13	D₄⋊#10596833	C₄	1	2	yes	—
14	C₈×#10596833	C₂²	1	1	yes	characteristic
15	D₈×Dic₇₃	C₄	1	1	yes	characteristic
16	#14556687	—	4	4	no	—
17	C₂×#11924040	—	2	2	no	—
18	C₂×#11924040	—	2	1	no	—
19	C₂×#11924040	D₄	1	2	yes	—
20	C₂²×#10596833	—	2	2	no	—
21	C₄⋊#10596833	C₂×C₄	1	2	yes	—
22	C₂²⋊#10596833	—	2	2	no	—
23	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596833	C₂³	1	1	yes	characteristic
25	D₄×Dic₇₃	C₂×C₄	1	2	yes	—
26	#14556525	C₈	1	1	yes	characteristic
27	#14556512	C₈	1	2	yes	—
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×D₈×C₇₃	C₈	1	1	yes	characteristic
30	#11924040	—	2	2	no	—
31	#11924040	—	4	1	no	—
32	#11924040	—	4	2	no	—
33	C₂×#10596833	—	2	2	no	—
34	C₂×#10596833	—	2	1	no	—
35	C₂×#10596833	—	4	2	no	—
36	C₂×#10596833	—	4	2	no	—
37	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
38	C₂×#10596833	#41	1	1	yes	characteristic
39	C₂²×Dic₇₃	—	2	2	no	—
40	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
41	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
42	C₂²⋊Dic₇₃	—	2	2	no	—
43	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
45	D₈×C₇₃	—	2	2	no	—
46	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂³×C₇₃	—	2	2	no	—
48	D₄×C₇₃	—	2	2	no	—
49	D₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×Dic₇₃	—	2	2	no	—
52	C₂×Dic₇₃	—	4	2	no	—
53	C₂×Dic₇₃	—	4	2	no	—
54	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
55	C₂×Dic₇₃	#131	1	1	yes	characteristic
56	#10596833	—	2	1	no	—
57	#10596833	—	4	1	no	—
58	#10596833	—	4	2	no	—
59	#10596833	AGL(1,ℤ/8)	1	2	yes	—
60	C₈×C₇₃	M₃₂	1	2	yes	complemented
61	#10596831	—	2	1	no	—
62	C₂²×C₇₃	—	4	2	no	—
63	C₂²×C₇₃	—	4	2	no	—
64	C₂²×C₇₃	—	4	2	no	—
65	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
66	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
67	C₄×C₇₃	#504	1	1	yes	characteristic
68	Dic₇₃	—	2	1	no	—
69	Dic₇₃	—	4	2	no	—
70	Dic₇₃	#442	1	1	yes	characteristic
71	Dic₇₃	#444	1	1	yes	characteristic
72	#12064	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	—	8	2	no	—
74	C₂×C₇₃	#1580	1	1	yes	characteristic
75	C₂×C₇₃	#2169	1	1	yes	characteristic
76	C₂×C₇₃	#2171	1	1	yes	characteristic
77	#2169	—	73	2	no	—
78	C₈×D₈	—	73	1	no	—
79	#1367	—	73	1	no	—
80	D₄⋊C₁₆	—	73	2	no	—
81	#1312	—	73	1	no	—
82	C₇₃	#12064	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	146	2	no	—
84	C₄×D₈	—	73	1	no	—
85	C₈×D₄	—	73	2	no	—
86	C₄⋊C₁₆	—	73	2	no	—
87	C₂²⋊C₁₆	—	146	2	no	—
88	C₄×C₁₆	—	73	1	no	—
89	C₄×C₁₆	—	73	1	no	—
90	#335	—	73	1	no	—
91	D₄⋊C₈	—	73	2	no	—
92	C₈²	—	73	1	no	—
93	C₂×D₈	#10596842	1	1	yes	characteristic
94	C₂²×C₈	—	146	2	no	—
95	C₄×D₄	—	73	2	no	—
96	M₃₂	—	292	4	no	—
97	C₂×C₁₆	—	73	2	no	—
98	C₂×C₁₆	—	146	1	no	—
99	C₂×C₁₆	—	146	2	no	—
100	#107	—	73	1	no	—
101	C₄⋊C₈	—	73	2	no	—
102	#102	—	73	2	no	—
103	C₂²⋊C₈	—	146	2	no	—
104	C₄×C₈	—	73	1	no	—
105	C₄×C₈	—	73	1	no	—
106	C₄×C₈	—	73	1	no	—
107	C₂×D₄	C₂×#10596842	1	2	yes	—
108	C₂²×C₄	—	146	2	no	—
109	D₈	—	2	2	no	—
110	C₂×C₈	—	292	2	no	—
111	C₂×C₈	—	292	2	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	146	1	no	—
116	C₂×C₈	—	146	2	no	—
117	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
118	C₄⋊C₄	—	73	2	no	—
119	C₂²⋊C₄	—	146	2	no	—
120	C₄²	—	73	1	no	—
121	C₁₆	—	292	2	no	—
122	C₁₆	—	292	1	no	—
123	C₁₆	—	146	2	no	—
124	C₂³	—	2	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	2	2	no	—
127	C₂×C₄	—	292	2	no	—
128	C₂×C₄	—	292	2	no	—
129	C₂×C₄	—	73	1	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	146	2	no	—
132	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
133	C₈	—	292	2	no	—
134	C₈	—	292	1	no	—
135	C₈	—	73	2	no	—
136	C₈	—	146	1	no	—
137	C₈	—	146	1	no	—
138	C₈	#14556687	1	2	yes	complemented
139	C₂²	—	4	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	D₄×#10596842	1	1	yes	characteristic
143	C₄	—	292	2	no	—
144	C₄	—	73	1	no	—
145	C₄	—	73	1	no	—
146	C₄	—	146	1	no	—
147	C₄	C₂×#14556687	1	1	yes	characteristic
148	C₄	#17989708	1	1	yes	characteristic
149	C₂	—	8	2	no	—
150	C₂	#22627062	1	1	yes	characteristic
151	C₂	#22627147	1	1	yes	characteristic
152	C₂	#22627153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627147

C₂

yes

maximal, complemented

#22626989

C₂

yes

maximal, characteristic, complemented

D₄⋊#11924040

C₂

yes

maximal, complemented

#22626948

C₂

yes

maximal, characteristic, complemented

D₈×#10596833

C₂

yes

maximal, characteristic

C₂×#14556687

—

C₄⋊#11924040

C₂²

yes

—

C₂²⋊#11924040

—

C₄×#11924040

C₂²

yes

characteristic

C₄×#11924040

C₂²

yes

characteristic

D₄×#10596833

C₂²

yes

—

#17989450

C₄

yes

characteristic

D₄⋊#10596833

C₄

yes

—

C₈×#10596833

C₂²

yes

characteristic

D₈×Dic₇₃

C₄

yes

—

—

—

D₄

yes

—

C₂²×#10596833

—

C₄⋊#10596833

C₂×C₄

yes

—

C₂²⋊#10596833

—

C₄×#10596833

C₂×C₄

yes

characteristic

C₄×#10596833

C₂³

yes

characteristic

D₄×Dic₇₃

C₂×C₄

yes

—

#14556525

C₈

yes

characteristic

#14556512

C₈

yes

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₂×D₈×C₇₃

C₈

yes

—

—

—

—

—

—

—

C₂×D₄

yes

yes

C₂²×Dic₇₃

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₄⋊Dic₇₃

C₂×C₈

yes

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×D₄×C₇₃

C₂×C₈

yes

—

D₈×C₇₃

—

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

C₄×D₄

yes

C₂×Dic₇₃

yes

—

—

—

yes

—

C₈×C₇₃

M₃₂

yes

complemented

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

—

C₇₃

#12064

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

146

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

—

C₂×D₈

#10596842

yes

characteristic

C₂²×C₈

—

146

—

C₄×D₄

—

M₃₂

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

C₂²⋊C₈

—

146

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₂×D₄

C₂×#10596842

yes

—

108

C₂²×C₄

—

146

—

109

D₈

—

110

C₂×C₈

—

292

—

111

C₂×C₈

—

292

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

146

—

116

C₂×C₈

—

146

—

117

C₂×C₈

C₂×#10596842

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

146

—

120

C₄²

—

121

C₁₆

—

292

—

122

C₁₆

—

292

—

123

C₁₆

—

146

—

124

C₂³

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

292

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

146

—

132

C₂×C₄

C₂²×#10596842

yes

characteristic

133

C₈

—

292

—

134

C₈

—

292

—

135

C₈

—

136

C₈

—

146

—

137

C₈

—

146

—

138

C₈

#14556687

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×#10596842

yes

characteristic

143

C₄

—

292

—

144

C₄

—

145

C₄

—

146

C₄

—

146

—

147

C₄

C₂×#14556687

yes

characteristic

148

C₄

#17989708

yes

characteristic

149

C₂

—

150

C₂

#22627062

yes

characteristic

151

C₂

#22627147

yes

characteristic

152

C₂

#22627153

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#10596842
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting, prime radical	C₂×D₈×C₇₃	C₈
socle	C₂²×C₇₃	C₈×D₄
cosocle	C₄×#10596833	C₂³
hypercenter	C₂×D₈	#10596842
hypocenter	C₇₃	#12064
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×#10596842

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#10596842

Fitting, prime radical

C₂×D₈×C₇₃

C₈

socle

C₂²×C₇₃

C₈×D₄

C₂³

C₂×D₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. #22627147	C₂	semidirect
2. #22626989	C₂	semidirect
3. D₄⋊#11924040	C₂	semidirect
4. #22626948	C₂	semidirect
60. C₈×C₇₃	M₃₂	semidirect
82. C₇₃	#12064	semidirect
138. C₈	#14556687	semidirect

H ≅ G/N

type

1. #22627147

C₂

semidirect

2. #22626989

C₂

semidirect

3. D₄⋊#11924040

C₂

semidirect

4. #22626948

C₂

semidirect

60. C₈×C₇₃

M₃₂

semidirect

82. C₇₃

#12064

semidirect

138. C₈

#14556687

semidirect

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

subgroup	index
#22627147	2
#22626989	2
D₄⋊#11924040	2
#22626948	2
D₈×#10596833	2
#12064	73

subgroup

index

Subgroups — #34034017

Conjugacy classes of subgroups

Complements

Maximal subgroups