Group #14527464 — subgroups

Subgroups — #14527464

Order 2176.

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

normal subgroups only

This group has 240 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	C₂×#10587555	C₂	1	1	yes	maximal, characteristic
2	#11762833	C₂	1	1	yes	maximal, characteristic
3	#11762557	C₂	1	4	yes	maximal, complemented
4	C₁₇×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₄	C₄	1	2	yes	—
6	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₇	—	2	1	no	—
8	C₂×C₄⋊Dic₁₇	—	2	1	no	—
9	C₂×#63223	—	2	1	no	—
10	#10587555	—	2	1	no	—
11	#10587555	—	2	1	no	—
12	#10587555	C₄	1	2	yes	—
13	#10587555	C₂²	1	2	yes	—
14	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
15	C₁₇×C₂²⋊C₈	—	2	1	no	—
16	C₄×C₈×C₁₇	—	2	1	no	—
17	C₄⋊#3622	—	2	1	no	—
18	C₄⋊#3622	C₂²	1	2	yes	—
19	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×Dic₁₇	—	2	2	no	—
23	C₂×Dic₃₄	—	2	2	no	—
24	C₂×Dic₃₄	—	4	4	no	—
25	C₂×Dic₃₄	C₂×C₄	1	4	yes	—
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	C₄⋊Dic₁₇	—	2	2	no	—
32	C₄⋊Dic₁₇	—	2	2	no	—
33	C₄⋊Dic₁₇	—	2	2	no	—
34	C₄⋊Dic₁₇	—	2	2	no	—
35	#63223	—	4	2	no	—
36	C₂×#3622	—	2	2	no	—
37	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	—	2	2	no	—
39	C₂×C₄×C₁₇	—	2	2	no	—
40	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₇	SD₁₆	1	2	yes	—
44	C₂×C₄×C₁₇	Q₁₆	1	2	yes	—
45	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₇	—	2	4	no	—
49	C₂×Dic₁₇	—	4	4	no	—
50	Dic₃₄	—	4	4	no	—
51	Dic₃₄	—	4	4	no	—
52	C₈×C₁₇	—	4	2	no	—
53	#3622	—	4	2	no	—
54	#1548	—	17	1	no	maximal, Sylow
55	C₂²×C₁₇	—	2	1	no	—
56	C₂²×C₁₇	—	2	1	no	—
57	C₂²×C₁₇	#103	1	2	yes	—
58	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₁₇	—	2	1	no	—
60	C₄×C₁₇	—	2	1	no	—
61	C₄×C₁₇	#100	1	2	yes	—
62	C₄×C₁₇	#103	1	2	yes	—
63	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₁₇	#137	1	2	yes	—
66	Dic₁₇	—	4	4	no	—
67	C₂×C₄⋊Q₈	—	17	1	no	—
68	#431	D₁₇	1	1	yes	characteristic
69	#423	—	17	1	no	—
70	#333	—	17	4	no	—
71	C₂×C₁₇	—	2	2	no	—
72	C₂×C₁₇	C₂×#100	1	1	yes	characteristic
73	C₂×C₁₇	C₂×#103	1	1	yes	characteristic
74	C₂×C₁₇	#419	1	1	yes	characteristic
75	C₂²×Q₈	—	17	2	no	—
76	C₂×M₁₆	—	34	1	no	—
77	C₄⋊Q₈	—	17	2	no	—
78	C₄⋊Q₈	—	17	2	no	—
79	C₄⋊Q₈	—	34	1	no	—
80	C₄⋊Q₈	—	34	1	no	—
81	C₂×C₄⋊C₄	—	34	1	no	—
82	C₂×C₄⋊C₄	—	34	1	no	—
83	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
84	C₄⋊C₈	—	17	2	no	—
85	C₄⋊C₈	—	34	1	no	—
86	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
87	C₂²⋊C₈	—	2	1	no	—
88	C₄×C₈	—	2	1	no	—
89	C₁₇	#1548	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	17	4	no	—
91	C₂×Q₈	—	34	2	no	—
92	C₂×Q₈	—	68	4	no	—
93	C₂²×C₄	—	34	2	no	—
94	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
95	C₂²×C₄	D₆₈	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
97	M₁₆	—	68	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	34	2	no	—
100	C₄⋊C₄	—	34	2	no	—
101	C₄⋊C₄	—	34	2	no	—
102	C₄⋊C₄	—	34	2	no	—
103	C₄⋊C₄	—	34	2	no	—
104	C₄²	C₄×D₁₇	1	1	yes	characteristic
105	C₄²	D₆₈	1	1	yes	characteristic
106	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
107	C₄²	C₂²×D₁₇	1	1	yes	characteristic
108	C₂³	#63227	1	1	yes	characteristic
109	Q₈	—	68	4	no	—
110	Q₈	—	68	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	34	4	no	—
114	C₂×C₄	—	68	4	no	—
115	C₂×C₄	#63219	1	2	yes	—
116	C₂×C₄	Dic₆₈	1	2	yes	—
117	C₂×C₄	#63227	1	1	yes	characteristic
118	C₂×C₄	#63227	1	1	yes	characteristic
119	C₂×C₄	#63227	1	1	yes	characteristic
120	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
121	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	68	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#10587487	1	2	yes	—
128	C₂²	C₂×#63227	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	68	4	no	—
132	C₄	#10587487	1	2	yes	—
133	C₄	#10587494	1	2	yes	—
134	C₄	C₂×#63219	1	1	yes	characteristic
135	C₄	C₂×Dic₆₈	1	1	yes	characteristic
136	C₄	#10587637	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#10587494	1	1	yes	characteristic
139	C₂	C₂×#10587487	1	1	yes	characteristic
140	C₂	#11762817	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587555

C₂

yes

maximal, characteristic

#11762833

C₂

yes

maximal, characteristic

#11762557

C₂

yes

maximal, complemented

C₁₇×#431

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₄

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

—

—

—

C₄

yes

—

#10587555

C₂²

yes

—

C₁₇×C₄⋊C₈

C₂²

yes

—

C₁₇×C₂²⋊C₈

—

C₄×C₈×C₁₇

—

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

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

C₂×C₄

yes

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

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

—

#1548

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#103

yes

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#100

yes

—

C₄×C₁₇

#103

yes

—

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₇

C₂×Q₁₆

yes

characteristic

C₄×C₁₇

#137

yes

—

Dic₁₇

—

C₂×C₄⋊Q₈

—

#431

D₁₇

yes

characteristic

#423

—

#333

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₁₇

#1548

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₇

yes

characteristic

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₄×D₁₇

yes

characteristic

105

C₄²

D₆₈

yes

characteristic

106

C₄²

C₂²⋊D₁₇

yes

characteristic

107

C₄²

C₂²×D₁₇

yes

characteristic

108

C₂³

#63227

yes

characteristic

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#63219

yes

—

116

C₂×C₄

Dic₆₈

yes

—

117

C₂×C₄

#63227

yes

characteristic

118

C₂×C₄

#63227

yes

characteristic

119

C₂×C₄

#63227

yes

characteristic

120

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

121

C₂×C₄

C₂×D₆₈

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

123

C₈

—

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

#10587487

yes

—

128

C₂²

C₂×#63227

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#10587487

yes

—

133

C₄

#10587494

yes

—

134

C₄

C₂×#63219

yes

characteristic

135

C₄

C₂×Dic₆₈

yes

characteristic

136

C₄

#10587637

yes

—

137

C₂

—

138

C₂

C₂×#10587494

yes

characteristic

139

C₂

C₂×#10587487

yes

characteristic

140

C₂

#11762817

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#63227
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#431	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₇	C₂²⋊C₄
cosocle	C₄²×C₁₇	C₂³
hypercenter	#431	D₁₇
hypocenter	C₁₇	#1548
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×#63227

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#431

C₂

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₇

C₂²⋊C₄

cosocle

C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #11762557	C₂	semidirect
89. C₁₇	#1548	semidirect

H ≅ G/N

type

3. #11762557

C₂

semidirect

89. C₁₇

#1548

semidirect

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

subgroup	index
C₂×#10587555	2
#11762833	2
#11762557	2
C₁₇×#431	2
#1548	17

subgroup

index

C₁₇×#431

Subgroups — #14527464

Conjugacy classes of subgroups

Complements

Maximal subgroups