Group #14537126 — subgroups

Subgroups — #14537126

Order 2176.

924 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 209 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×#10587624	C₂	1	6	yes	maximal, complemented
2	C₂×C₄²×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762712	C₂	1	6	yes	maximal, complemented
4	#11762679	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₇	C₂²	1	6	yes	—
7	C₂²×C₂²⋊D₁₇	C₂²	1	4	yes	—
8	C₂²×C₄×D₁₇	C₂²	1	3	yes	—
9	C₁₇×#127	—	2	4	no	—
10	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₁₇	C₂²	1	12	yes	—
12	#10587624	—	2	24	no	—
13	C₂×C₄⋊Dic₁₇	C₂²	1	6	yes	—
14	C₂×C₄×Dic₁₇	C₂²	1	3	yes	—
15	C₄²×D₁₇	—	2	4	no	—
16	C₂⁴×C₁₇	C₂³	1	4	yes	—
17	C₂³×D₁₇	C₂³	1	1	yes	characteristic
18	C₂×D₄×C₁₇	—	2	12	no	—
19	C₂×D₄×C₁₇	—	2	12	no	—
20	C₂×D₄×C₁₇	—	2	12	no	—
21	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
22	C₂×C₂²⋊D₁₇	—	2	24	no	—
23	C₂²×Dic₁₇	C₂³	1	4	yes	—
24	C₂²×Dic₁₇	C₂³	1	3	yes	—
25	C₂×C₄×D₁₇	—	2	6	no	—
26	C₂×C₄×D₁₇	—	2	6	no	—
27	C₂×C₄×D₁₇	D₄	1	12	yes	complemented
28	C₄²×C₁₇	D₄	1	4	yes	complemented
29	C₂²⋊Dic₁₇	—	2	24	no	—
30	C₄⋊Dic₁₇	—	2	12	no	—
31	C₄×Dic₁₇	—	2	6	no	—
32	C₂³×C₁₇	—	2	12	no	—
33	C₂³×C₁₇	—	2	4	no	—
34	C₂³×C₁₇	—	2	12	no	—
35	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
36	C₂²×D₁₇	—	2	1	no	—
37	C₂²×D₁₇	—	2	3	no	—
38	C₂²×D₁₇	C₂×D₄	1	6	yes	—
39	D₄×C₁₇	—	4	24	no	—
40	C₂×C₄×C₁₇	C₂×D₄	1	6	yes	—
41	C₂×C₄×C₁₇	C₂×D₄	1	6	yes	—
42	C₂×C₄×C₁₇	#41	1	6	yes	—
43	C₂²⋊D₁₇	—	4	16	no	—
44	C₂×Dic₁₇	—	2	3	no	—
45	C₂×Dic₁₇	—	2	3	no	—
46	C₂×Dic₁₇	—	2	12	no	—
47	C₂×Dic₁₇	C₂×D₄	1	6	yes	—
48	C₄×D₁₇	—	2	24	no	—
49	#2856	—	17	1	no	maximal, Sylow
50	C₂²×C₁₇	—	4	12	no	—
51	C₂²×C₁₇	—	4	4	no	—
52	C₂²×C₁₇	—	4	12	no	—
53	C₂²×C₁₇	C₂²×D₄	1	3	yes	—
54	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
55	C₂²×C₁₇	C₂×#41	1	3	yes	—
56	C₂×D₁₇	—	2	6	no	—
57	C₂×D₁₇	—	2	6	no	—
58	C₂×D₁₇	#127	1	4	yes	complemented
59	C₄×C₁₇	#121	1	12	yes	complemented
60	Dic₁₇	—	2	6	no	—
61	Dic₁₇	—	4	4	no	—
62	C₂×#127	D₁₇	1	1	yes	characteristic, complemented
63	C₂×#121	—	17	6	no	—
64	C₂²×C₄²	—	17	1	no	—
65	#390	—	17	6	no	—
66	#384	—	17	1	no	—
67	C₂×C₁₇	—	8	4	no	—
68	C₂×C₁₇	C₂×#121	1	3	yes	—
69	C₂×C₁₇	C₂×#127	1	1	yes	characteristic
70	C₂×C₁₇	#532	1	3	yes	—
71	D₁₇	—	2	4	no	—
72	C₂²×D₄	—	17	4	no	—
73	C₂²×D₄	C₂×D₁₇	1	6	yes	—
74	C₂³×C₄	—	17	3	no	—
75	#127	—	2	4	no	—
76	#121	—	34	24	no	—
77	C₂×C₄⋊C₄	—	17	6	no	—
78	C₂×C₂²⋊C₄	—	17	12	no	—
79	C₂×C₄²	—	17	3	no	—
80	C₂×C₄²	—	34	4	no	—
81	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
82	C₁₇	#2856	1	1	yes	characteristic, complemented, Sylow
83	C₂⁴	—	17	1	no	—
84	C₂⁴	C₂²×D₁₇	1	4	yes	—
85	C₂×D₄	—	2	12	no	—
86	C₂×D₄	—	2	12	no	—
87	C₂×D₄	—	2	12	no	—
88	C₂×D₄	—	34	24	no	—
89	C₂²×C₄	—	17	3	no	—
90	C₂²×C₄	—	17	12	no	—
91	C₂²×C₄	—	17	4	no	—
92	C₂²×C₄	—	34	6	no	—
93	C₂²×C₄	—	34	6	no	—
94	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
95	C₄⋊C₄	—	34	12	no	—
96	C₂²⋊C₄	—	34	24	no	—
97	C₄²	—	34	6	no	—
98	C₄²	C₂²⋊D₁₇	1	4	yes	complemented
99	C₂³	—	2	12	no	—
100	C₂³	—	2	12	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	17	6	no	—
103	C₂³	—	34	3	no	—
104	C₂³	—	34	1	no	—
105	C₂³	C₂³×D₁₇	1	1	yes	characteristic
106	D₄	—	4	24	no	—
107	D₄	—	68	16	no	—
108	C₂×C₄	—	17	6	no	—
109	C₂×C₄	—	34	3	no	—
110	C₂×C₄	—	34	3	no	—
111	C₂×C₄	—	34	24	no	—
112	C₂×C₄	—	34	12	no	—
113	C₂×C₄	D₄×D₁₇	1	6	yes	—
114	C₂×C₄	#63254	1	6	yes	—
115	C₂×C₄	C₂×C₂²⋊D₁₇	1	6	yes	—
116	C₂²	—	4	12	no	—
117	C₂²	—	4	4	no	—
118	C₂²	—	4	12	no	—
119	C₂²	—	17	4	no	—
120	C₂²	—	34	6	no	—
121	C₂²	—	34	6	no	—
122	C₂²	C₂×D₄×D₁₇	1	3	yes	—
123	C₂²	C₂×#63254	1	3	yes	—
124	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
125	C₄	—	34	6	no	—
126	C₄	—	68	4	no	—
127	C₄	#10587624	1	12	yes	complemented
128	C₂	—	8	4	no	—
129	C₂	—	34	4	no	—
130	C₂	C₂×#10587624	1	3	yes	—
131	C₂	#127×D₁₇	1	1	yes	characteristic
132	C₂	#11763357	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587624

C₂

yes

maximal, complemented

C₂×C₄²×D₁₇

C₂

yes

maximal, characteristic, complemented

#11762712

C₂

yes

maximal, complemented

#11762679

C₂

yes

maximal, characteristic, complemented

C₂×C₁₇×#127

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₇

C₂²

yes

—

C₂²×C₂²⋊D₁₇

C₂²

yes

—

C₂²×C₄×D₁₇

C₂²

yes

—

C₁₇×#127

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₇

C₂²

yes

—

#10587624

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

—

C₄²×D₁₇

—

C₂⁴×C₁₇

C₂³

yes

—

C₂³×D₁₇

C₂³

yes

characteristic

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂×C₂²⋊D₁₇

—

C₂²×Dic₁₇

C₂³

yes

—

C₂²×Dic₁₇

C₂³

yes

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

D₄

yes

complemented

C₄²×C₁₇

D₄

yes

complemented

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

C₂⁴

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

C₂×D₄

yes

—

D₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₄

yes

—

C₄×D₁₇

—

#2856

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²×D₄

yes

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₂²×C₁₇

C₂×#41

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂×#127

D₁₇

yes

characteristic, complemented

C₂×#121

—

C₂²×C₄²

—

#390

—

#384

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#121

yes

—

C₂×C₁₇

C₂×#127

yes

characteristic

C₂×C₁₇

#532

yes

—

D₁₇

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₇

yes

—

C₂³×C₄

—

#127

—

#121

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₁₇

#2856

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂⁴

C₂²×D₁₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊D₁₇

yes

complemented

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂³×D₁₇

yes

characteristic

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

D₄×D₁₇

yes

—

114

C₂×C₄

#63254

yes

—

115

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×D₄×D₁₇

yes

—

123

C₂²

C₂×#63254

yes

—

124

C₂²

C₂²×C₂²⋊D₁₇

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

#10587624

yes

complemented

128

C₂

—

129

C₂

—

130

C₂

C₂×#10587624

yes

—

131

C₂

#127×D₁₇

yes

characteristic

132

C₂

#11763357

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₂×C₁₇×#127	C₂
hypercenter	C₂×#127	D₁₇
hypocenter	C₁₇	#2856
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇

commutator, socle, cosocle

C₂³×C₁₇

C₂⁴

Fitting

C₂×C₁₇×#127

C₂

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#10587624	C₂	semidirect
2. C₂×C₄²×D₁₇	C₂	semidirect
3. #11762712	C₂	semidirect
4. #11762679	C₂	semidirect
5. C₂×C₁₇×#127	C₂	semidirect
27. C₂×C₄×D₁₇	D₄	semidirect
28. C₄²×C₁₇	D₄	semidirect
58. C₂×D₁₇	#127	semidirect
59. C₄×C₁₇	#121	semidirect
62. C₂×#127	D₁₇	semidirect
82. C₁₇	#2856	semidirect
98. C₄²	C₂²⋊D₁₇	semidirect
127. C₄	#10587624	semidirect

H ≅ G/N

type

1. C₂×#10587624

C₂

semidirect

2. C₂×C₄²×D₁₇

C₂

semidirect

3. #11762712

C₂

semidirect

4. #11762679

C₂

semidirect

5. C₂×C₁₇×#127

C₂

semidirect

27. C₂×C₄×D₁₇

D₄

semidirect

28. C₄²×C₁₇

D₄

semidirect

58. C₂×D₁₇

#127

semidirect

59. C₄×C₁₇

#121

semidirect

62. C₂×#127

D₁₇

semidirect

82. C₁₇

#2856

semidirect

98. C₄²

C₂²⋊D₁₇

semidirect

127. C₄

#10587624

semidirect

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

subgroup	index
C₂×#10587624	2
C₂×C₄²×D₁₇	2
#11762712	2
#11762679	2
C₂×C₁₇×#127	2
#2856	17

subgroup

index

C₂×C₄²×D₁₇

C₂×C₁₇×#127

Subgroups — #14537126

Conjugacy classes of subgroups

Complements

Maximal subgroups