Group #17982806 — subgroups

Subgroups — #17982806

Order 4352.

202 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₂×#11763918	C₂	1	2	yes	maximal, complemented
2	C₄⋊#10587528	C₂	1	1	yes	maximal, characteristic, complemented
3	#14545877	C₂	1	2	yes	maximal, complemented
4	C₄×#10587528	C₂	1	1	yes	maximal, characteristic, complemented
5	#14545163	C₂	1	1	yes	maximal, characteristic
6	#11763918	—	2	4	no	—
7	C₂×#10587528	C₂²	1	2	yes	—
8	C₂×#10587528	C₂²	1	1	yes	characteristic
9	C₂×#10587528	C₂²	1	1	yes	characteristic
10	C₂×#10587529	C₂²	1	2	yes	—
11	C₄⋊#63216	C₄	1	1	yes	characteristic
12	#11763704	C₄	1	2	yes	—
13	C₄×#63216	C₂²	1	1	yes	characteristic
14	#11762874	C₄	1	1	yes	characteristic
15	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
16	C₄×D₆₈	C₈	1	1	yes	characteristic
17	C₂×#63216	C₂×C₄	1	2	yes	—
18	C₂×#63216	C₂×C₄	1	1	yes	characteristic
19	C₂×#63216	C₂³	1	1	yes	characteristic
20	#10587529	—	2	2	no	—
21	#10587529	—	2	2	no	—
22	#10587528	—	2	2	no	—
23	#10587528	—	2	1	no	—
24	#10587528	D₄	1	2	yes	complemented
25	C₁₇×C₄⋊C₈	C₈	1	1	yes	characteristic
26	#10587491	C₈	1	2	yes	—
27	C₄×#3622	C₂×C₄	1	1	yes	characteristic
28	C₂×D₆₈	C₁₆	1	1	yes	characteristic
29	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
31	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
32	#63227	C₁₆	1	2	yes	—
33	C₄⋊Dic₁₇	C₁₆	1	1	yes	characteristic
34	C₂×#3622	C₂×C₈	1	1	yes	characteristic
35	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
36	C₈×D₁₇	—	2	2	no	—
37	C₈×D₁₇	—	2	2	no	—
38	#63216	—	2	2	no	—
39	#63216	—	2	1	no	—
40	#63216	C₂×D₄	1	1	yes	characteristic
41	#63216	#41	1	1	yes	characteristic
42	#13646	—	17	1	no	maximal, Sylow
43	C₂²×D₁₇	C₂×C₁₆	1	2	yes	—
44	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
47	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
48	D₆₈	—	2	2	no	—
49	C₄×D₁₇	—	2	2	no	—
50	C₄×D₁₇	—	2	2	no	—
51	C₈×C₁₇	—	2	2	no	—
52	#3622	—	2	1	no	—
53	#3622	C₄×D₄	1	1	yes	characteristic
54	#3622	#131	1	1	yes	characteristic
55	C₂×M₆₄	—	17	2	no	—
56	D₄×C₁₆	—	17	1	no	—
57	C₄⋊C₃₂	—	17	1	no	—
58	C₂²⋊C₃₂	—	17	2	no	—
59	C₄×C₃₂	—	17	1	no	—
60	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
61	C₂×D₁₇	—	2	2	no	—
62	C₂×D₁₇	—	2	2	no	—
63	C₂×D₁₇	—	2	2	no	—
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	M₆₄	1	2	yes	complemented
66	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
67	C₄×C₁₇	#504	1	1	yes	characteristic
68	Dic₁₇	—	2	2	no	—
69	C₂²×C₁₆	—	17	2	no	—
70	C₈×D₄	—	17	1	no	—
71	M₆₄	—	34	4	no	—
72	C₂×C₃₂	—	17	1	no	—
73	C₂×C₃₂	—	17	1	no	—
74	C₂×C₃₂	—	17	2	no	—
75	C₄⋊C₁₆	—	17	1	no	—
76	C₂²⋊C₁₆	—	17	2	no	—
77	C₄×C₁₆	—	17	1	no	—
78	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
79	C₂×C₁₇	C₂×M₆₄	1	1	yes	characteristic
80	C₂×C₁₇	#2258	1	1	yes	characteristic
81	D₁₇	—	4	2	no	—
82	C₂²×C₈	—	17	2	no	—
83	C₄×D₄	—	17	1	no	—
84	C₂×C₁₆	—	17	1	no	—
85	C₂×C₁₆	—	17	1	no	—
86	C₂×C₁₆	—	17	2	no	—
87	C₂×C₁₆	—	34	2	no	—
88	C₂×C₁₆	—	34	2	no	—
89	C₄⋊C₈	#3633	1	1	yes	characteristic
90	C₂²⋊C₈	—	17	2	no	—
91	C₄×C₈	—	17	1	no	—
92	C₃₂	—	17	2	no	—
93	C₃₂	—	34	2	no	—
94	C₃₂	—	34	1	no	—
95	C₁₇	#13646	1	1	yes	characteristic, complemented, Sylow
96	C₂×D₄	—	17	1	no	—
97	C₂²×C₄	—	17	2	no	—
98	C₂×C₈	—	17	1	no	—
99	C₂×C₈	—	17	1	no	—
100	C₂×C₈	—	34	2	no	—
101	C₂×C₈	—	34	2	no	—
102	C₂×C₈	C₂×#3633	1	2	yes	—
103	C₄⋊C₄	—	17	1	no	—
104	C₂²⋊C₄	—	17	2	no	—
105	C₄²	C₂×#3633	1	1	yes	characteristic
106	C₁₆	—	17	1	no	—
107	C₁₆	—	17	1	no	—
108	C₁₆	—	34	2	no	—
109	C₁₆	—	34	1	no	—
110	C₂³	—	17	2	no	—
111	D₄	—	34	2	no	—
112	C₂×C₄	—	17	2	no	—
113	C₂×C₄	—	34	2	no	—
114	C₂×C₄	—	34	2	no	—
115	C₂×C₄	#10587665	1	1	yes	characteristic
116	C₂×C₄	#10587665	1	1	yes	characteristic
117	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	17	1	no	—
120	C₈	—	17	1	no	—
121	C₈	—	34	1	no	—
122	C₂²	—	34	2	no	—
123	C₂²	—	34	2	no	—
124	C₂²	—	34	2	no	—
125	C₂²	C₂×#10587665	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	34	2	no	—
128	C₄	#11763918	1	2	yes	complemented
129	C₄	D₄×#3633	1	1	yes	characteristic
130	C₄	#11763931	1	1	yes	characteristic
131	C₂	—	68	2	no	—
132	C₂	#14545993	1	1	yes	characteristic
133	C₂	C₂×#11763918	1	1	yes	characteristic
134	C₂	#14546018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763918

C₂

yes

maximal, complemented

C₄⋊#10587528

C₂

yes

maximal, characteristic, complemented

#14545877

C₂

yes

maximal, complemented

C₄×#10587528

C₂

yes

maximal, characteristic, complemented

#14545163

C₂

yes

maximal, characteristic

#11763918

—

C₂×#10587528

C₂²

yes

—

C₂×#10587528

C₂²

yes

characteristic

C₂×#10587528

C₂²

yes

characteristic

C₂×#10587529

C₂²

yes

—

C₄⋊#63216

C₄

yes

characteristic

#11763704

C₄

yes

—

C₄×#63216

C₂²

yes

characteristic

#11762874

C₄

yes

characteristic

C₂×C₈×D₁₇

C₂×C₄

yes

—

C₄×D₆₈

C₈

yes

characteristic

C₂×#63216

C₂×C₄

yes

—

C₂×#63216

C₂×C₄

yes

characteristic

C₂×#63216

C₂³

yes

—

—

—

—

D₄

yes

complemented

C₁₇×C₄⋊C₈

C₈

yes

characteristic

#10587491

C₈

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂×D₆₈

C₁₆

yes

characteristic

C₂×C₄×D₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

#63227

C₁₆

yes

—

C₄⋊Dic₁₇

C₁₆

yes

characteristic

C₂×#3622

C₂×C₈

yes

characteristic

C₂×#3622

C₂²×C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

#63216

—

#63216

—

#63216

C₂×D₄

yes

yes

—

C₂²×D₁₇

C₂×C₁₆

yes

—

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×Dic₁₇

C₂×C₁₆

yes

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

—

#3622

—

#3622

C₄×D₄

yes

yes

C₂×M₆₄

—

D₄×C₁₆

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

C₄×C₃₂

—

C₂²×C₁₇

C₂²×C₁₆

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

M₆₄

yes

complemented

C₄×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

Dic₁₇

—

C₂²×C₁₆

—

C₈×D₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂×C₁₇

D₄×C₁₆

yes

characteristic

C₂×C₁₇

C₂×M₆₄

yes

characteristic

C₂×C₁₇

#2258

yes

characteristic

D₁₇

—

C₂²×C₈

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

#3633

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

—

C₁₇

#13646

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×#3633

yes

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

C₂×#3633

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

—

110

C₂³

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#10587665

yes

characteristic

116

C₂×C₄

#10587665

yes

characteristic

117

C₂×C₄

C₂²×#3633

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×#10587665

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

#11763918

yes

complemented

129

C₄

D₄×#3633

yes

characteristic

130

C₄

#11763931

yes

characteristic

131

C₂

—

132

C₂

#14545993

yes

characteristic

133

C₂

C₂×#11763918

yes

characteristic

134

C₂

#14546018

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3633
commutator, socle	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₁₇×C₄⋊C₈	C₈
prime radical	C₂×D₆₈	C₁₆
cosocle	C₂×#63216	C₂³
hypercenter	C₄⋊C₈	#3633
hypocenter	C₁₇	#13646
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#3633

commutator, socle

C₂²×C₁₇

C₂²×C₁₆

Fitting

C₁₇×C₄⋊C₈

C₈

prime radical

C₂×D₆₈

C₁₆

C₂³

C₄⋊C₈

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
1. C₂×#11763918	C₂	semidirect
2. C₄⋊#10587528	C₂	semidirect
3. #14545877	C₂	semidirect
4. C₄×#10587528	C₂	semidirect
24. #10587528	D₄	semidirect
65. C₄×C₁₇	M₆₄	semidirect
95. C₁₇	#13646	semidirect
128. C₄	#11763918	semidirect

H ≅ G/N

type

1. C₂×#11763918

C₂

semidirect

2. C₄⋊#10587528

C₂

semidirect

3. #14545877

C₂

semidirect

4. C₄×#10587528

C₂

semidirect

24. #10587528

D₄

semidirect

65. C₄×C₁₇

M₆₄

semidirect

95. C₁₇

#13646

semidirect

128. C₄

#11763918

semidirect

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

subgroup	index
C₂×#11763918	2
C₄⋊#10587528	2
#14545877	2
C₄×#10587528	2
#14545163	2
#13646	17

subgroup

index

Subgroups — #17982806

Conjugacy classes of subgroups

Complements

Maximal subgroups