Group #14537855 — subgroups

Subgroups — #14537855

Order 2176.

394 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 197 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	#11762881	C₂	1	8	yes	maximal, complemented
2	#11762866	C₂	1	2	yes	maximal, complemented
3	#11762776	C₂	1	4	yes	maximal, complemented
4	C₁₇×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#41×C₁₇	C₂²	1	2	yes	—
6	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₁₇×#127	C₄	1	1	yes	characteristic
8	C₁₇×#124	C₄	1	2	yes	—
9	C₁₇×#121	C₄	1	4	yes	—
10	C₄×D₄×C₁₇	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×#63223	C₂²	1	4	yes	—
13	C₂²×#3622	C₂²	1	4	yes	—
14	C₂²⋊#3622	C₂²	1	4	yes	—
15	C₂²⋊#3622	C₂²	1	8	yes	—
16	C₄⋊#3622	C₂²	1	4	yes	—
17	#10587474	C₂²	1	4	yes	—
18	#41×C₁₇	—	4	2	no	—
19	#41×C₁₇	—	4	2	no	—
20	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×D₄×C₁₇	C₂×C₄	1	4	yes	—
23	C₂×D₄×C₁₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	C₂³	1	4	yes	—
27	C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
28	C₂²⋊C₄×C₁₇	C₂×C₄	1	8	yes	—
29	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₇	C₂³	1	2	yes	—
32	#63223	—	4	4	no	—
33	C₂×#3622	—	4	4	no	—
34	C₂×#3622	C₂³	1	8	yes	—
35	C₂³×C₁₇	C₂²×C₄	1	4	yes	—
36	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₇	—	4	2	no	—
38	D₄×C₁₇	—	4	4	no	—
39	D₄×C₁₇	—	4	2	no	—
40	D₄×C₁₇	—	4	4	no	—
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	—	2	4	no	—
43	C₂×C₄×C₁₇	—	2	4	no	—
44	C₂×C₄×C₁₇	—	4	2	no	—
45	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
50	#3622	—	2	8	no	—
51	#2980	—	17	1	no	maximal, Sylow
52	C₂²×C₁₇	—	2	1	no	—
53	C₂²×C₁₇	—	2	4	no	—
54	C₂²×C₁₇	—	2	2	no	—
55	C₂²×C₁₇	—	2	8	no	—
56	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
57	C₄×C₁₇	—	2	1	no	—
58	C₄×C₁₇	—	2	4	no	—
59	C₄×C₁₇	—	2	2	no	—
60	C₄×C₁₇	—	2	2	no	—
61	C₄×C₁₇	H(5,𝔽₂)	1	2	yes	—
62	#532	D₁₇	1	1	yes	characteristic
63	#435	—	17	8	no	—
64	#432	—	17	2	no	—
65	#408	—	17	4	no	—
66	C₂×C₁₇	—	4	1	no	—
67	C₂×C₁₇	—	4	4	no	—
68	C₂×C₁₇	#518	1	1	yes	characteristic
69	C₂×C₁₇	#568	1	2	yes	—
70	C₂×#41	C₂×D₁₇	1	2	yes	—
71	C₂×M₁₆	—	17	4	no	—
72	C₂²×C₈	—	17	4	no	—
73	C₄⋊Q₈	Dic₁₇	1	1	yes	characteristic
74	#127	Dic₁₇	1	1	yes	characteristic
75	#124	Dic₁₇	1	2	yes	—
76	#121	Dic₁₇	1	4	yes	—
77	C₄×D₄	C₂×D₁₇	1	4	yes	—
78	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
79	C₄⋊C₈	—	17	4	no	—
80	C₂²⋊C₈	—	17	4	no	—
81	C₂²⋊C₈	—	17	8	no	—
82	#97	—	17	4	no	—
83	C₁₇	#2980	1	1	yes	characteristic, complemented, Sylow
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	C₂×Dic₁₇	1	2	yes	—
87	C₂×D₄	C₂×Dic₁₇	1	4	yes	—
88	C₂×D₄	C₂×Dic₁₇	1	4	yes	—
89	C₂×D₄	C₂×Dic₁₇	1	2	yes	—
90	C₂²×C₄	C₂×Dic₁₇	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₁₇	1	4	yes	—
93	M₁₆	—	68	4	no	—
94	C₂×C₈	—	17	8	no	—
95	C₂×C₈	—	68	4	no	—
96	C₄⋊C₄	C₂×Dic₁₇	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₁₇	1	8	yes	—
98	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
99	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
100	C₄²	C₂²×D₁₇	1	2	yes	—
101	C₂³	C₂²×Dic₁₇	1	4	yes	—
102	C₂³	C₂²×Dic₁₇	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	4	no	—
106	D₄	—	4	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₁₇	1	2	yes	—
113	C₂×C₄	C₂²×Dic₁₇	1	2	yes	—
114	C₂×C₄	C₂²×Dic₁₇	1	4	yes	—
115	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
116	C₈	—	34	8	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	8	no	—
121	C₂²	C₂³×Dic₁₇	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	#10587691	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	—	4	4	no	—
129	C₂	#11763284	1	1	yes	characteristic
130	C₂	#11763576	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11762881

C₂

yes

maximal, complemented

#11762866

C₂

yes

maximal, complemented

#11762776

C₂

yes

maximal, complemented

C₁₇×#532

C₂

yes

maximal, characteristic

C₂×#41×C₁₇

C₂²

yes

—

C₁₇×C₄⋊Q₈

C₄

yes

characteristic

C₁₇×#127

C₄

yes

characteristic

C₁₇×#124

C₄

yes

—

C₁₇×#121

C₄

yes

—

C₄×D₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

—

C₂²×#3622

C₂²

yes

—

C₂²⋊#3622

C₂²

yes

—

C₂²⋊#3622

C₂²

yes

—

C₄⋊#3622

C₂²

yes

—

#10587474

C₂²

yes

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

C₂×C₄

yes

—

C₂×D₄×C₁₇

C₂×C₄

yes

—

C₂×D₄×C₁₇

C₂×C₄

yes

—

C₂×D₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂³

yes

—

#63223

—

C₂×#3622

—

C₂×#3622

C₂³

yes

—

C₂³×C₁₇

C₂²×C₄

yes

—

C₂³×C₁₇

C₂²×C₄

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂³×C₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

H(5,𝔽₂)

yes

—

#532

D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#518

yes

characteristic

C₂×C₁₇

#568

yes

—

C₂×#41

C₂×D₁₇

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₁₇

yes

characteristic

#127

Dic₁₇

yes

characteristic

#124

Dic₁₇

yes

—

#121

Dic₁₇

yes

—

C₄×D₄

C₂×D₁₇

yes

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₁₇

#2980

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

C₂×Dic₁₇

yes

—

C₂×D₄

C₂×Dic₁₇

yes

—

C₂×D₄

C₂×Dic₁₇

yes

—

C₂×D₄

C₂×Dic₁₇

yes

—

C₂²×C₄

C₂×Dic₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂²×C₄

C₂²×D₁₇

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₁₇

yes

—

C₂²⋊C₄

C₂×Dic₁₇

yes

—

C₄²

C₂×Dic₁₇

yes

characteristic

C₄²

C₂×Dic₁₇

yes

characteristic

100

C₄²

C₂²×D₁₇

yes

—

101

C₂³

C₂²×Dic₁₇

yes

—

102

C₂³

C₂²×Dic₁₇

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

112

C₂×C₄

C₂²×Dic₁₇

yes

—

113

C₂×C₄

C₂²×Dic₁₇

yes

—

114

C₂×C₄

C₂²×Dic₁₇

yes

—

115

C₂×C₄

C₂³×D₁₇

yes

characteristic

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×Dic₁₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#10587691

yes

—

127

C₂

—

128

C₂

—

129

C₂

#11763284

yes

characteristic

130

C₂

#11763576

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting, prime radical	C₁₇×#532	C₂
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#532	D₁₇
hypocenter	C₁₇	#2980
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₇

commutator, socle

C₂²×C₁₇

C₂³×C₄

Fitting, prime radical

C₁₇×#532

C₂

cosocle

C₂×C₄×C₁₇

C₂⁴

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #11762881	C₂	semidirect
2. #11762866	C₂	semidirect
3. #11762776	C₂	semidirect
83. C₁₇	#2980	semidirect

H ≅ G/N

type

1. #11762881

C₂

semidirect

2. #11762866

C₂

semidirect

3. #11762776

C₂

semidirect

83. C₁₇

#2980

semidirect

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

subgroup	index
#11762881	2
#11762866	2
#11762776	2
C₁₇×#532	2
#2980	17

subgroup

index

C₁₇×#532

Subgroups — #14537855

Conjugacy classes of subgroups

Complements

Maximal subgroups