Group #14527240 — subgroups

Subgroups — #14527240

Order 2176.

232 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 93 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	#11763366	C₂	1	1	yes	maximal, characteristic
2	#11762833	C₂	1	1	yes	maximal, characteristic
3	#11762541	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₁₇	—	2	1	no	—
6	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
7	#10587612	—	2	1	no	—
8	#10587610	—	2	1	no	—
9	#10587610	C₄	1	2	yes	—
10	C₂×#63223	—	2	1	no	—
11	#10587556	—	2	1	no	—
12	#10587555	C₄	1	1	yes	characteristic
13	#10587555	C₄	1	1	yes	characteristic
14	C₄×Dic₃₄	—	2	1	no	—
15	C₄×Dic₃₄	C₂²	1	2	yes	—
16	C₄×C₈×C₁₇	—	2	1	no	—
17	C₄×C₈×C₁₇	C₂²	1	2	yes	—
18	C₄⋊#3622	—	2	1	no	—
19	C₄⋊#3622	C₂²	1	2	yes	—
20	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
23	C₂×Dic₃₄	—	2	1	no	—
24	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₇	—	2	4	no	—
26	C₂×C₈×C₁₇	—	2	2	no	—
27	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₇	D₄	1	1	yes	characteristic
29	C₄²×C₁₇	D₄	1	1	yes	characteristic
30	C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	C₂²⋊Dic₁₇	—	2	2	no	—
32	C₄⋊Dic₁₇	—	2	1	no	—
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
36	#63225	—	2	2	no	—
37	#63225	—	2	2	no	—
38	C₄×Dic₁₇	—	2	2	no	—
39	#63223	—	4	2	no	—
40	C₂×#3622	—	2	2	no	—
41	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇	—	2	2	no	—
43	C₂×C₄×C₁₇	—	2	2	no	—
44	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₇	Q₁₆	1	2	yes	—
49	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₇	—	2	2	no	—
53	C₂×Dic₁₇	—	2	2	no	—
54	Dic₃₄	—	2	4	no	—
55	Dic₃₄	—	4	2	no	—
56	C₈×C₁₇	—	2	4	no	—
57	#3622	—	4	2	no	—
58	#1493	—	17	1	no	maximal, Sylow
59	C₂²×C₁₇	—	2	1	no	—
60	C₂²×C₁₇	—	2	1	no	—
61	C₂²×C₁₇	#103	1	2	yes	—
62	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₁₇	—	2	1	no	—
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	#103	1	2	yes	—
66	C₄×C₁₇	#104	1	2	yes	—
67	C₄×C₁₇	#131	1	2	yes	—
68	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
70	Dic₁₇	—	4	2	no	—
71	Dic₁₇	—	4	2	no	—
72	#533	—	17	1	no	—
73	#423	—	17	1	no	—
74	C₂×C₄×C₈	D₁₇	1	1	yes	characteristic
75	Q₈⋊C₈	—	17	4	no	—
76	C₂×C₁₇	—	2	2	no	—
77	C₂×C₁₇	#408	1	1	yes	characteristic
78	C₂×C₁₇	C₂×#103	1	1	yes	characteristic
79	C₂×C₁₇	C₂×#104	1	1	yes	characteristic
80	C₂×M₁₆	—	34	1	no	—
81	C₂²×C₈	—	2	1	no	—
82	C₄⋊Q₈	—	17	1	no	—
83	C₄⋊Q₈	—	17	1	no	—
84	#125	—	34	1	no	—
85	C₂²⋊Q₈	—	17	2	no	—
86	C₂²⋊Q₈	—	34	1	no	—
87	C₄×Q₈	—	17	2	no	—
88	C₄×Q₈	—	34	1	no	—
89	#117	—	34	1	no	—
90	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
91	C₄⋊C₈	—	17	2	no	—
92	C₄⋊C₈	—	34	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	C₂×D₁₇	1	2	yes	—
95	C₁₇	#1493	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	17	2	no	—
97	C₂×Q₈	—	34	1	no	—
98	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
99	C₂²×C₄	D₆₈	1	1	yes	characteristic
100	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
101	M₁₆	—	68	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	34	2	no	—
105	C₄⋊C₄	—	17	2	no	—
106	C₄⋊C₄	—	34	2	no	—
107	C₄⋊C₄	—	34	1	no	—
108	C₄⋊C₄	—	34	1	no	—
109	C₄⋊C₄	—	34	2	no	—
110	C₄⋊C₄	—	34	1	no	—
111	C₂²⋊C₄	—	34	2	no	—
112	C₄²	—	34	2	no	—
113	C₄²	C₄×D₁₇	1	1	yes	characteristic
114	C₄²	D₆₈	1	1	yes	characteristic
115	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
116	C₄²	C₂²×D₁₇	1	1	yes	characteristic
117	C₂³	#63227	1	1	yes	characteristic
118	Q₈	—	34	4	no	—
119	Q₈	—	68	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	34	2	no	—
123	C₂×C₄	—	34	2	no	—
124	C₂×C₄	#63219	1	2	yes	—
125	C₂×C₄	Dic₆₈	1	2	yes	—
126	C₂×C₄	#63227	1	1	yes	characteristic
127	C₂×C₄	#63227	1	1	yes	characteristic
128	C₂×C₄	#63227	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
130	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
132	C₈	—	2	4	no	—
133	C₈	—	68	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#10587487	1	2	yes	—
137	C₂²	C₂×#63227	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	68	2	no	—
141	C₄	—	68	2	no	—
142	C₄	#10587476	1	2	yes	—
143	C₄	#10587487	1	2	yes	—
144	C₄	#10587587	1	2	yes	—
145	C₄	C₂×#63219	1	1	yes	characteristic
146	C₄	C₂×Dic₆₈	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	#11762775	1	1	yes	characteristic
149	C₂	C₂×#10587487	1	1	yes	characteristic
150	C₂	C₂×#10587476	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763366

C₂

yes

maximal, characteristic

#11762833

C₂

yes

maximal, characteristic

#11762541

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₁₇

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

C₂²

yes

—

—

C₄

yes

—

C₂×#63223

—

#10587556

—

#10587555

C₄

yes

characteristic

#10587555

C₄

yes

characteristic

C₄×Dic₃₄

—

C₄×Dic₃₄

C₂²

yes

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂²

yes

—

C₄⋊#3622

—

C₄⋊#3622

C₂²

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

C₂×C₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

#63225

—

C₄×Dic₁₇

—

#63223

—

C₂×#3622

—

C₂³×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

SD₁₆

yes

—

C₂×C₄×C₁₇

Q₁₆

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

#3622

—

#1493

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#103

yes

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#103

yes

—

C₄×C₁₇

#104

yes

—

C₄×C₁₇

#131

yes

—

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₇

C₂×Q₁₆

yes

characteristic

Dic₁₇

—

Dic₁₇

—

#533

—

#423

—

C₂×C₄×C₈

D₁₇

yes

characteristic

Q₈⋊C₈

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

—

C₁₇

#1493

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₁₇

yes

characteristic

C₂²×C₄

D₆₈

yes

characteristic

100

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

C₄×D₁₇

yes

characteristic

114

C₄²

D₆₈

yes

characteristic

115

C₄²

C₂²⋊D₁₇

yes

characteristic

116

C₄²

C₂²×D₁₇

yes

characteristic

117

C₂³

#63227

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#63219

yes

—

125

C₂×C₄

Dic₆₈

yes

—

126

C₂×C₄

#63227

yes

characteristic

127

C₂×C₄

#63227

yes

characteristic

128

C₂×C₄

#63227

yes

characteristic

129

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

130

C₂×C₄

C₂×D₆₈

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

#10587487

yes

—

137

C₂²

C₂×#63227

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#10587476

yes

—

143

C₄

#10587487

yes

—

144

C₄

#10587587

yes

—

145

C₄

C₂×#63219

yes

characteristic

146

C₄

C₂×Dic₆₈

yes

characteristic

147

C₂

—

148

C₂

#11762775

yes

characteristic

149

C₂

C₂×#10587487

yes

characteristic

150

C₂

C₂×#10587476

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₇
Fitting	C₂×C₄×C₈×C₁₇	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₇	C₂²⋊C₄
cosocle	C₄²×C₁₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₇
hypocenter	C₁₇	#1493
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₆₈

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₇

Fitting

C₂×C₄×C₈×C₁₇

C₂

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₇

C₂²⋊C₄

cosocle

C₄²×C₁₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₇

hypocenter

C₁₇

#1493

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #11762541	C₂	semidirect
95. C₁₇	#1493	semidirect

H ≅ G/N

type

3. #11762541

C₂

semidirect

95. C₁₇

#1493

semidirect

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

subgroup	index
#11763366	2
#11762833	2
#11762541	2
C₂×C₄×C₈×C₁₇	2
#1493	17

subgroup

index

C₂×C₄×C₈×C₁₇

Subgroups — #14527240

Conjugacy classes of subgroups

Complements

Maximal subgroups