Group #14530126 — subgroups

Subgroups — #14530126

Order 2176.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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	#11762852	C₂	1	2	yes	maximal
2	#11762678	C₂	1	1	yes	maximal, characteristic
3	#11762578	C₂	1	1	yes	maximal, characteristic
4	#11762568	C₂	1	2	yes	maximal
5	C₁₇×#544	C₂	1	1	yes	maximal, characteristic
6	C₁₇×C₄⋊Q₈	—	2	2	no	—
7	C₁₇×#125	—	2	2	no	—
8	C₁₇×C₂²⋊Q₈	—	2	2	no	—
9	C₁₇×#117	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
12	C₂×#63225	—	2	2	no	—
13	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₂×#63223	C₂²	1	2	yes	—
15	#10587479	—	2	2	no	—
16	#10587478	—	2	2	no	—
17	C₂×Q₈×C₁₇	—	2	2	no	—
18	C₂²×C₄×C₁₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₇	—	2	1	no	—
21	C₂²×Dic₁₇	—	2	1	no	—
22	C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₄⋊C₄×C₁₇	—	2	4	no	—
24	C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
27	C₂²⋊C₄×C₁₇	—	2	2	no	—
28	C₄²×C₁₇	—	2	2	no	—
29	C₄⋊Dic₁₇	—	2	2	no	—
30	#63225	—	4	4	no	—
31	C₄×Dic₁₇	—	2	1	no	—
32	C₄×Dic₁₇	Q₈	1	2	yes	complemented
33	#63223	—	2	2	no	—
34	#63223	Q₈	1	4	yes	complemented
35	C₂×#3622	—	2	2	no	—
36	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₁₇	—	8	2	no	—
38	C₂×C₄×C₁₇	—	2	2	no	—
39	C₂×C₄×C₁₇	—	2	2	no	—
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	—	4	2	no	—
42	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
46	C₂×Dic₁₇	—	2	2	no	—
47	C₂×Dic₁₇	—	2	2	no	—
48	C₂×Dic₁₇	—	2	2	no	—
49	C₂×Dic₁₇	—	4	2	no	—
50	C₂×Dic₁₇	—	4	1	no	—
51	#3622	—	2	4	no	—
52	#2060	—	17	1	no	maximal, Sylow
53	C₂²×C₁₇	—	2	1	no	—
54	C₂²×C₁₇	—	2	1	no	—
55	C₂²×C₁₇	#120	1	1	yes	characteristic
56	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂²×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×C₁₇	—	4	2	no	—
59	C₄×C₁₇	—	4	2	no	—
60	C₄×C₁₇	—	4	2	no	—
61	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
62	C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
63	Dic₁₇	—	2	4	no	—
64	Dic₁₇	—	8	1	no	—
65	#544	D₁₇	1	1	yes	characteristic
66	#428	—	17	2	no	—
67	#384	—	17	1	no	—
68	#343	—	17	1	no	—
69	#339	—	17	2	no	—
70	C₂×C₁₇	—	2	1	no	—
71	C₂×C₁₇	—	2	1	no	—
72	C₂×C₁₇	#395	1	1	yes	characteristic
73	C₂×C₁₇	#453	1	1	yes	characteristic
74	C₂×C₁₇	#456	1	1	yes	characteristic
75	C₂×M₁₆	—	17	2	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	#125	—	2	2	no	—
78	C₂²⋊Q₈	—	2	2	no	—
79	#117	C₂×D₁₇	1	2	yes	—
80	C₂×C₄⋊C₄	—	17	1	no	—
81	C₂×C₄⋊C₄	—	34	2	no	—
82	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
83	C₂×C₄²	—	17	1	no	—
84	#107	—	34	2	no	—
85	#106	—	34	2	no	—
86	C₁₇	#2060	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₈	—	2	2	no	—
88	C₂²×C₄	—	34	1	no	—
89	C₂²×C₄	—	34	1	no	—
90	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
92	M₁₆	—	17	4	no	—
93	M₁₆	—	34	2	no	—
94	C₂×C₈	—	34	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	34	2	no	—
99	C₄⋊C₄	—	68	4	no	—
100	C₄⋊C₄	C₂²⋊D₁₇	1	2	yes	—
101	C₄⋊C₄	C₂²⋊D₁₇	1	2	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	17	2	no	—
105	C₄²	—	34	1	no	—
106	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
107	Q₈	—	8	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	34	2	no	—
113	C₂×C₄	—	34	2	no	—
114	C₂×C₄	—	34	2	no	—
115	C₂×C₄	—	68	2	no	—
116	C₂×C₄	—	68	1	no	—
117	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
118	C₂×C₄	Q₈×D₁₇	1	2	yes	—
119	C₂×C₄	#63256	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
121	C₈	—	34	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#10587630	1	1	yes	characteristic
125	C₂²	#10587632	1	1	yes	characteristic
126	C₂²	#10587639	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	34	4	no	—
131	C₄	—	136	1	no	—
132	C₄	#10587630	1	2	yes	—
133	C₄	#10587632	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#11762728	1	1	yes	characteristic
137	C₂	#11762977	1	1	yes	characteristic
138	C₂	#11762992	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762852

C₂

yes

maximal

#11762678

C₂

yes

maximal, characteristic

#11762578

C₂

yes

maximal, characteristic

#11762568

C₂

yes

maximal

C₁₇×#544

C₂

yes

maximal, characteristic

C₁₇×C₄⋊Q₈

—

C₁₇×#125

—

C₁₇×C₂²⋊Q₈

—

C₁₇×#117

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂×#63225

—

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

—

#10587479

—

#10587478

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

Q₈

yes

complemented

#63223

—

#63223

Q₈

yes

complemented

C₂×#3622

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

#3622

—

#2060

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#120

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

C₄⋊Q₈

yes

—

Dic₁₇

—

Dic₁₇

—

#544

D₁₇

yes

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₁₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₂×C₄²

—

#107

—

#106

—

C₁₇

#2060

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

C₂²⋊D₁₇

yes

—

101

C₄⋊C₄

C₂²⋊D₁₇

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

Q₈×D₁₇

yes

characteristic

118

C₂×C₄

Q₈×D₁₇

yes

—

119

C₂×C₄

#63256

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

#10587630

yes

characteristic

125

C₂²

#10587632

yes

characteristic

126

C₂²

#10587639

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

136

—

132

C₄

#10587630

yes

—

133

C₄

#10587632

yes

—

134

C₂

—

135

C₂

—

136

C₂

#11762728

yes

characteristic

137

C₂

#11762977

yes

characteristic

138

C₂

#11762992

yes

characteristic

	subgroup	G/H
center	C₂²	#10587639
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#544	C₂
socle	C₂²×C₁₇	#120
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#544	D₁₇
hypocenter	C₁₇	#2060
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#10587639

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#544

C₂

socle

C₂²×C₁₇

#120

prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
32. C₄×Dic₁₇	Q₈	semidirect
34. #63223	Q₈	semidirect
86. C₁₇	#2060	semidirect

H ≅ G/N

type

32. C₄×Dic₁₇

Q₈

semidirect

34. #63223

Q₈

semidirect

86. C₁₇

#2060

semidirect

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

subgroup	index
#11762852	2
#11762678	2
#11762578	2
#11762568	2
C₁₇×#544	2
#2060	17

subgroup

index

C₁₇×#544

Subgroups — #14530126

Conjugacy classes of subgroups

Complements

Maximal subgroups