Group #14528152 — subgroups

Subgroups — #14528152

Order 2176.

156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 57 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	#11763188	C₂	1	1	yes	maximal, characteristic
2	#11763094	C₂	1	1	yes	maximal, characteristic
3	Q₈×#3622	C₂	1	1	yes	maximal, characteristic
4	#11762561	C₂	1	1	yes	maximal, characteristic
5	#11762555	C₂	1	1	yes	maximal, characteristic
6	#11762554	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#475	C₂	1	1	yes	maximal, characteristic
8	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×#63231	—	2	1	no	—
11	C₂×#63231	—	2	1	no	—
12	#10587555	C₂²	1	1	yes	characteristic
13	C₁₇×#106	—	2	1	no	—
14	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₁₇×#103	—	2	1	no	—
16	#10587506	—	2	1	no	—
17	#10587479	—	2	1	no	—
18	C₄⋊#3622	—	2	1	no	—
19	C₄⋊#3622	C₂²	1	1	yes	characteristic
20	C₄⋊#3622	C₂²	1	1	yes	characteristic
21	C₄×#3622	—	2	1	no	—
22	C₄×#3622	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₁₇	—	2	1	no	—
24	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂×Dic₃₄	—	2	1	no	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₄⋊C₄×C₁₇	—	2	1	no	—
28	C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
31	C₄²×C₁₇	—	2	1	no	—
32	C₄²×C₁₇	C₂³	1	1	yes	characteristic
33	#63231	—	4	2	no	—
34	#63231	—	4	2	no	—
35	C₄⋊Dic₁₇	—	2	1	no	—
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₂×#3622	—	2	1	no	—
38	C₂×#3622	—	2	1	no	—
39	C₂×#3622	D₄	1	1	yes	characteristic
40	C₂×#3622	D₄	1	1	yes	characteristic
41	Q₈×C₁₇	—	2	1	no	—
42	Q₈×C₁₇	—	4	1	no	—
43	Q₈×C₁₇	—	4	1	no	—
44	Q₈×C₁₇	Q₁₆	1	2	yes	complemented
45	C₂×C₄×C₁₇	—	2	1	no	—
46	C₂×C₄×C₁₇	—	2	1	no	—
47	C₂×C₄×C₁₇	—	4	1	no	—
48	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
51	C₂×Dic₁₇	—	4	1	no	—
52	Dic₃₄	—	4	1	no	—
53	Dic₃₄	—	4	1	no	—
54	C₈×C₁₇	—	4	1	no	—
55	#3622	—	2	1	no	—
56	#3622	—	4	1	no	—
57	#3622	—	4	1	no	—
58	#3622	SD₁₆	1	2	yes	—
59	#1670	—	17	1	no	maximal, Sylow
60	C₂²×C₁₇	#121	1	1	yes	characteristic
61	C₄×C₁₇	—	2	2	no	—
62	C₄×C₁₇	—	2	1	no	—
63	C₄×C₁₇	—	4	1	no	—
64	C₄×C₁₇	—	8	1	no	—
65	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₄×C₁₇	#135	1	1	yes	characteristic
68	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₁₇	—	8	1	no	—
70	#494	—	17	1	no	—
71	#475	—	17	1	no	—
72	#475	D₁₇	1	1	yes	characteristic
73	C₈×Q₈	—	17	1	no	—
74	#335	—	17	1	no	—
75	#332	—	17	1	no	—
76	#332	—	17	1	no	—
77	C₂×C₁₇	#460	1	1	yes	characteristic
78	C₂×C₁₇	#467	1	1	yes	characteristic
79	C₂×C₁₇	#473	1	1	yes	characteristic
80	C₂×Q₁₆	—	34	1	no	—
81	C₂×Q₁₆	—	34	1	no	—
82	C₄⋊Q₈	—	17	1	no	—
83	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
84	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
85	#106	—	2	1	no	—
86	#106	—	34	1	no	—
87	C₄⋊C₈	—	17	1	no	—
88	C₄⋊C₈	—	17	1	no	—
89	C₄⋊C₈	—	34	1	no	—
90	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
91	#103	—	2	1	no	—
92	#103	—	34	1	no	—
93	C₄×C₈	—	17	1	no	—
94	C₄×C₈	—	34	1	no	—
95	C₁₇	#1670	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	34	1	no	—
98	C₂×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
99	Q₁₆	—	68	2	no	—
100	Q₁₆	—	68	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	17	1	no	—
103	C₂×C₈	—	17	1	no	—
104	C₂×C₈	—	34	1	no	—
105	C₂×C₈	—	34	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	34	1	no	—
110	C₄⋊C₄	—	34	1	no	—
111	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂²×D₁₇	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	68	1	no	—
118	Q₈	—	68	1	no	—
119	Q₈	#63231	1	2	yes	complemented
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	68	1	no	—
124	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
125	C₂×C₄	#63254	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	17	2	no	—
129	C₈	—	34	1	no	—
130	C₈	—	68	1	no	—
131	C₈	—	68	1	no	—
132	C₂²	#10587625	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	8	1	no	—
137	C₄	—	136	1	no	—
138	C₄	SD₁₆×D₁₇	1	1	yes	characteristic
139	C₄	#10587600	1	1	yes	characteristic
140	C₄	C₂×#63231	1	1	yes	characteristic
141	C₄	#10587636	1	1	yes	characteristic
142	C₂	#11763007	1	1	yes	characteristic
143	C₂	#11763048	1	1	yes	characteristic
144	C₂	#11763082	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763188

C₂

yes

maximal, characteristic

#11763094

C₂

yes

maximal, characteristic

Q₈×#3622

C₂

yes

maximal, characteristic

#11762561

C₂

yes

maximal, characteristic

#11762555

C₂

yes

maximal, characteristic

#11762554

C₂

yes

maximal, characteristic

C₁₇×#475

C₂

yes

maximal, characteristic

C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₄×Q₈×C₁₇

C₂²

yes

—

—

C₂²

yes

characteristic

C₁₇×#106

—

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₁₇×#103

—

—

—

—

C₂²

yes

characteristic

C₄⋊#3622

C₂²

yes

characteristic

C₄×#3622

—

C₄×#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

D₄

yes

characteristic

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

#63231

—

#63231

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

Q₁₆

yes

complemented

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

—

—

—

SD₁₆

yes

—

—

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₇

C₂×Q₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

#494

—

#475

—

#475

D₁₇

yes

characteristic

C₈×Q₈

—

#335

—

#332

—

#332

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

C₄×Q₈

C₂×D₁₇

yes

characteristic

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#1670

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₇

yes

characteristic

Q₁₆

—

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²⋊D₁₇

yes

characteristic

112

C₄²

—

113

C₄²

C₂²×D₁₇

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

Q₈

#63231

yes

complemented

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

D₄×D₁₇

yes

characteristic

125

C₂×C₄

#63254

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

#10587625

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

136

—

138

C₄

SD₁₆×D₁₇

yes

characteristic

139

C₄

#10587600

yes

characteristic

140

C₄

C₂×#63231

yes

characteristic

141

C₄

#10587636

yes

characteristic

142

C₂

#11763007

yes

characteristic

143

C₂

#11763048

yes

characteristic

144

C₂

#11763082

yes

characteristic

	subgroup	G/H
center	C₂²	#10587625
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#475	C₂
socle, prime radical	C₂²×C₁₇	#121
hypercenter	#475	D₁₇
hypocenter	C₁₇	#1670
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#10587625

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#475

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
44. Q₈×C₁₇	Q₁₆	semidirect
95. C₁₇	#1670	semidirect
119. Q₈	#63231	semidirect

H ≅ G/N

type

44. Q₈×C₁₇

Q₁₆

semidirect

95. C₁₇

#1670

semidirect

119. Q₈

#63231

semidirect

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

subgroup	index
#11763188	2
#11763094	2
Q₈×#3622	2
#11762561	2
#11762555	2
#11762554	2
C₁₇×#475	2
#1670	17

subgroup

index

C₁₇×#475

Subgroups — #14528152

Conjugacy classes of subgroups

Complements

Maximal subgroups