Group #17982809 — subgroups

Subgroups — #17982809

Order 4352.

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

normal subgroups only

This group has 186 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	#14546018	C₂	1	2	yes	maximal, complemented
2	#14545884	C₂	1	1	yes	maximal, characteristic, complemented
3	#14545881	C₂	1	2	yes	maximal, complemented
4	#14545867	C₂	1	1	yes	maximal, characteristic, complemented
5	#14545173	C₂	1	1	yes	maximal, characteristic
6	#11763918	—	2	2	no	—
7	#11763917	C₂²	1	2	yes	—
8	#11763916	—	2	2	no	—
9	C₂×#10587528	C₂²	1	1	yes	characteristic
10	C₂×#10587528	C₂²	1	1	yes	characteristic
11	#11763868	C₂²	1	2	yes	—
12	#11763731	C₄	1	1	yes	characteristic
13	#11763708	C₄	1	2	yes	—
14	#11763695	C₂²	1	1	yes	characteristic
15	#11762925	C₄	1	1	yes	characteristic
16	#10587593	C₂×C₄	1	2	yes	—
17	#10587590	C₈	1	1	yes	characteristic
18	#10587538	C₂×C₄	1	2	yes	—
19	C₂×#63216	C₂×C₄	1	1	yes	characteristic
20	C₂×#63216	C₂³	1	1	yes	characteristic
21	#10587530	—	2	2	no	—
22	#10587529	—	2	2	no	—
23	#10587528	—	2	2	no	—
24	#10587528	—	2	1	no	—
25	#10587528	D₄	1	1	yes	characteristic
26	#10587528	D₄	1	1	yes	characteristic
27	C₁₇×#108	C₈	1	1	yes	characteristic
28	#10587496	C₈	1	2	yes	—
29	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
30	#63252	C₂×C₈	1	2	yes	—
31	M₁₆×C₁₇	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
35	Dic₆₈	C₁₆	1	1	yes	characteristic
36	D₁₃₆	C₁₆	1	1	yes	characteristic
37	#63219	C₁₆	1	2	yes	—
38	#63218	—	2	2	no	—
39	C₈×D₁₇	—	2	2	no	—
40	#63216	—	2	2	no	—
41	#63216	—	2	1	no	—
42	#63216	C₂×D₄	1	1	yes	characteristic
43	#63216	#41	1	1	yes	characteristic
44	#13647	—	17	1	no	maximal, Sylow
45	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
46	C₂²⋊D₁₇	—	2	2	no	—
47	C₂×Dic₁₇	—	2	1	no	—
48	D₆₈	C₂×C₁₆	1	2	yes	—
49	C₄×D₁₇	—	2	2	no	—
50	Dic₃₄	C₂×C₁₆	1	2	yes	—
51	C₈×C₁₇	—	2	2	no	—
52	C₈×C₁₇	C₂×C₁₆	1	1	yes	characteristic
53	C₈×C₁₇	C₂×C₁₆	1	1	yes	characteristic
54	#3622	C₄×D₄	1	1	yes	characteristic
55	#3622	#131	1	1	yes	characteristic
56	#2258	—	17	2	no	—
57	#2170	—	17	1	no	—
58	#1422	—	17	1	no	—
59	#1401	—	17	2	no	—
60	C₄×C₃₂	—	17	1	no	—
61	C₂²×C₁₇	#504	1	1	yes	characteristic
62	C₂×D₁₇	—	2	2	no	—
63	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
64	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
65	Dic₁₇	—	2	2	no	—
66	Dic₁₇	—	2	2	no	—
67	#504	—	17	2	no	—
68	#443	—	17	1	no	—
69	M₆₄	—	17	2	no	—
70	M₆₄	—	34	2	no	—
71	C₂×C₃₂	—	17	1	no	—
72	C₂×C₃₂	—	17	1	no	—
73	C₂×C₃₂	—	34	2	no	—
74	#364	—	17	1	no	—
75	#350	—	17	2	no	—
76	C₄×C₁₆	—	17	1	no	—
77	C₂×C₁₇	—	2	1	no	—
78	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
79	D₁₇	—	4	2	no	—
80	#135	—	17	1	no	—
81	#131	—	17	2	no	—
82	M₃₂	—	17	2	no	—
83	M₃₂	—	34	2	no	—
84	C₂×C₁₆	—	17	1	no	—
85	C₂×C₁₆	—	17	1	no	—
86	C₂×C₁₆	—	34	2	no	—
87	#108	#3633	1	1	yes	characteristic
88	#104	—	17	2	no	—
89	C₄×C₈	—	17	1	no	—
90	C₃₂	—	17	1	no	—
91	C₃₂	—	17	1	no	—
92	C₃₂	—	34	1	no	—
93	C₃₂	—	34	2	no	—
94	C₁₇	#13647	1	1	yes	characteristic, complemented, Sylow
95	#41	—	17	2	no	—
96	Q₁₆	—	17	1	no	—
97	SD₁₆	—	17	2	no	—
98	D₈	—	17	1	no	—
99	M₁₆	—	34	2	no	—
100	M₁₆	C₂×#3633	1	2	yes	—
101	C₂×C₈	—	17	1	no	—
102	C₂×C₈	—	34	2	no	—
103	C₂×C₈	C₂×#3633	1	1	yes	characteristic
104	C₄²	—	17	1	no	—
105	C₁₆	—	17	1	no	—
106	C₁₆	—	17	1	no	—
107	C₁₆	—	34	1	no	—
108	C₁₆	—	34	2	no	—
109	Q₈	—	17	2	no	—
110	D₄	—	17	2	no	—
111	D₄	—	34	2	no	—
112	C₂×C₄	—	34	1	no	—
113	C₂×C₄	—	34	2	no	—
114	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
115	C₈	—	2	2	no	—
116	C₈	—	17	1	no	—
117	C₈	—	17	1	no	—
118	C₈	#10587665	1	1	yes	characteristic
119	C₈	#10587665	1	1	yes	characteristic
120	C₂²	—	34	2	no	—
121	C₂²	#11763931	1	1	yes	characteristic
122	C₄	—	34	2	no	—
123	C₄	—	34	2	no	—
124	C₄	D₄×#3633	1	1	yes	characteristic
125	C₄	C₂×#10587665	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	68	2	no	—
128	C₂	#14545993	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14546018

C₂

yes

maximal, complemented

#14545884

C₂

yes

maximal, characteristic, complemented

#14545881

C₂

yes

maximal, complemented

#14545867

C₂

yes

maximal, characteristic, complemented

#14545173

C₂

yes

maximal, characteristic

#11763918

—

#11763917

C₂²

yes

—

#11763916

—

C₂×#10587528

C₂²

yes

characteristic

C₂×#10587528

C₂²

yes

characteristic

#11763868

C₂²

yes

—

#11763731

C₄

yes

characteristic

#11763708

C₄

yes

—

#11763695

C₂²

yes

characteristic

#11762925

C₄

yes

characteristic

#10587593

C₂×C₄

yes

—

#10587590

C₈

yes

characteristic

#10587538

C₂×C₄

yes

—

C₂×#63216

C₂×C₄

yes

characteristic

C₂×#63216

C₂³

yes

—

—

—

—

D₄

yes

characteristic

#10587528

D₄

yes

characteristic

C₁₇×#108

C₈

yes

characteristic

#10587496

C₈

yes

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

#63252

C₂×C₈

yes

—

M₁₆×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

characteristic

C₂×#3622

C₂²×C₄

yes

characteristic

Dic₆₈

C₁₆

yes

characteristic

D₁₃₆

C₁₆

yes

characteristic

#63219

C₁₆

yes

—

#63218

—

C₈×D₁₇

—

#63216

—

#63216

—

#63216

C₂×D₄

yes

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

D₆₈

C₂×C₁₆

yes

—

C₄×D₁₇

—

Dic₃₄

C₂×C₁₆

yes

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₁₆

yes

characteristic

C₈×C₁₇

C₂×C₁₆

yes

characteristic

#3622

C₄×D₄

yes

yes

—

—

—

—

C₄×C₃₂

—

C₂²×C₁₇

#504

yes

characteristic

C₂×D₁₇

—

C₄×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

C₂²×C₁₆

yes

characteristic

Dic₁₇

—

Dic₁₇

—

#504

—

#443

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#364

—

#350

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

D₄×C₁₆

yes

characteristic

D₁₇

—

#135

—

#131

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

yes

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

—

C₃₂

—

C₁₇

#13647

yes

characteristic, complemented, Sylow

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

100

M₁₆

C₂×#3633

yes

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₂×#3633

yes

characteristic

104

C₄²

—

105

C₁₆

—

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

Q₈

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×#3633

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

#10587665

yes

characteristic

119

C₈

#10587665

yes

characteristic

120

C₂²

—

121

C₂²

#11763931

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

D₄×#3633

yes

characteristic

125

C₄

C₂×#10587665

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#14545993

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#3633
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#3633
Fitting	C₁₇×#108	C₈
socle	C₂×C₁₇	D₄×C₁₆
prime radical	#10587590	C₈
cosocle	C₂×#63216	C₂³
hypercenter	#108	#3633
hypocenter	C₁₇	#13647
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₄

D₄×#3633

commutator

C₄×C₁₇

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#3633

Fitting

C₁₇×#108

C₈

socle

C₂×C₁₇

D₄×C₁₆

C₈

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
1. #14546018	C₂	semidirect
2. #14545884	C₂	semidirect
3. #14545881	C₂	semidirect
4. #14545867	C₂	semidirect
94. C₁₇	#13647	semidirect

H ≅ G/N

type

1. #14546018

C₂

semidirect

2. #14545884

C₂

semidirect

3. #14545881

C₂

semidirect

4. #14545867

C₂

semidirect

94. C₁₇

#13647

semidirect

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

subgroup	index
#14546018	2
#14545884	2
#14545881	2
#14545867	2
#14545173	2
#13647	17

subgroup

index

Subgroups — #17982809

Conjugacy classes of subgroups

Complements

Maximal subgroups