Group #14537889 — subgroups

Subgroups — #14537889

Order 2176.

314 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 157 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	#11762938	C₂	1	4	yes	maximal, complemented
3	#11762937	C₂	1	4	yes	maximal, complemented
4	#11762867	C₂	1	4	yes	maximal
5	#11762866	C₂	1	1	yes	maximal, characteristic
6	C₁₇×#432	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	#10587612	C₂²	1	2	yes	—
9	#10587610	C₄	1	2	yes	—
10	#10587610	C₄	1	2	yes	—
11	#10587556	C₄	1	2	yes	—
12	#10587555	C₄	1	1	yes	characteristic
13	#10587555	C₄	1	1	yes	characteristic
14	C₄×Dic₃₄	C₂²	1	2	yes	—
15	C₄×Dic₃₄	C₂²	1	2	yes	—
16	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
17	C₁₇×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₁₇×#97	C₂²	1	2	yes	—
19	C₂²⋊#3622	C₂²	1	2	yes	—
20	#10587486	C₂²	1	4	yes	—
21	#10587485	C₂²	1	8	yes	—
22	C₈×Dic₁₇	C₂²	1	4	yes	—
23	C₄⋊#3622	C₂²	1	2	yes	—
24	#10587474	C₂²	1	2	yes	—
25	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
29	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₇	C₂³	1	4	yes	—
31	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	C₄²×C₁₇	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₁₇	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
40	#63225	C₂×C₄	1	4	yes	—
41	#63225	C₂×C₄	1	4	yes	—
42	C₄×Dic₁₇	C₂³	1	4	yes	—
43	C₂×#3622	C₂³	1	4	yes	—
44	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	—	2	2	no	—
46	C₂×C₄×C₁₇	—	4	1	no	—
47	C₂×C₄×C₁₇	—	4	1	no	—
48	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
55	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
56	Dic₃₄	—	4	2	no	—
57	Dic₃₄	—	4	2	no	—
58	C₈×C₁₇	—	2	4	no	—
59	#3622	—	2	4	no	—
60	#2987	—	17	1	no	maximal, Sylow
61	C₂²×C₁₇	—	2	1	no	—
62	C₂²×C₁₇	—	2	1	no	—
63	C₂²×C₁₇	—	2	1	no	—
64	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	—	2	1	no	—
67	C₄×C₁₇	—	2	1	no	—
68	C₄×C₁₇	—	2	1	no	—
69	C₄×C₁₇	—	2	1	no	—
70	C₄×C₁₇	2₋¹⁺⁴	1	2	yes	—
71	Dic₁₇	—	2	4	no	—
72	Dic₁₇	—	2	4	no	—
73	#533	—	17	1	no	—
74	#446	—	17	4	no	—
75	#446	—	17	4	no	—
76	#433	—	17	4	no	—
77	#432	—	17	1	no	—
78	#432	D₁₇	1	1	yes	characteristic
79	C₂×C₁₇	—	4	1	no	—
80	C₂×C₁₇	#519	1	1	yes	characteristic
81	C₂×C₁₇	#568	1	1	yes	characteristic
82	C₂×C₁₇	#568	1	1	yes	characteristic
83	C₄⋊Q₈	—	17	1	no	—
84	C₄⋊Q₈	—	17	1	no	—
85	#125	—	17	2	no	—
86	C₂²⋊Q₈	—	17	2	no	—
87	C₂²⋊Q₈	—	17	2	no	—
88	C₄×Q₈	—	17	2	no	—
89	C₄×Q₈	—	17	2	no	—
90	#117	—	17	2	no	—
91	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
92	C₄⋊C₈	—	17	8	no	—
93	C₄⋊C₈	—	17	2	no	—
94	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
95	C₂²⋊C₈	—	17	2	no	—
96	C₂²⋊C₈	C₂×D₁₇	1	2	yes	—
97	#97	—	17	4	no	—
98	#97	—	17	2	no	—
99	#97	C₂×D₁₇	1	2	yes	—
100	C₄×C₈	—	17	4	no	—
101	C₁₇	#2987	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	17	2	no	—
103	C₂×Q₈	—	17	2	no	—
104	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
105	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
107	C₂×C₈	—	17	4	no	—
108	C₂×C₈	C₂²×D₁₇	1	4	yes	—
109	C₄⋊C₄	—	17	2	no	—
110	C₄⋊C₄	—	17	2	no	—
111	C₄⋊C₄	—	17	4	no	—
112	C₄⋊C₄	—	17	4	no	—
113	C₄⋊C₄	—	17	2	no	—
114	C₄⋊C₄	—	17	2	no	—
115	C₂²⋊C₄	—	17	4	no	—
116	C₄²	—	17	4	no	—
117	C₄²	C₄×D₁₇	1	1	yes	characteristic
118	C₄²	C₄×D₁₇	1	1	yes	characteristic
119	C₄²	C₂²×D₁₇	1	1	yes	characteristic
120	C₄²	C₂²×D₁₇	1	1	yes	characteristic
121	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
122	Q₈	—	68	2	no	—
123	Q₈	—	68	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	17	4	no	—
128	C₂×C₄	—	17	4	no	—
129	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
135	C₈	—	2	4	no	—
136	C₈	—	34	4	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	34	4	no	—
147	C₄	—	34	4	no	—
148	C₄	#10587697	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#11763286	1	1	yes	characteristic
151	C₂	#11763575	1	1	yes	characteristic
152	C₂	#11763575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763366

C₂

yes

maximal, characteristic

#11762938

C₂

yes

maximal, complemented

#11762937

C₂

yes

maximal, complemented

#11762867

C₂

yes

maximal

#11762866

C₂

yes

maximal, characteristic

C₁₇×#432

C₂

yes

maximal, characteristic

C₂×C₄²×C₁₇

C₂²

yes

characteristic

#10587612

C₂²

yes

—

#10587610

C₄

yes

—

#10587610

C₄

yes

—

#10587556

C₄

yes

—

#10587555

C₄

yes

characteristic

#10587555

C₄

yes

characteristic

C₄×Dic₃₄

C₂²

yes

—

C₄×Dic₃₄

C₂²

yes

—

C₁₇×C₄⋊C₈

C₂²

yes

—

C₁₇×C₂²⋊C₈

C₂²

yes

—

C₁₇×#97

C₂²

yes

—

C₂²⋊#3622

C₂²

yes

—

#10587486

C₂²

yes

—

#10587485

C₂²

yes

—

C₈×Dic₁₇

C₂²

yes

—

C₄⋊#3622

C₂²

yes

—

#10587474

C₂²

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

C₂×C₄

yes

—

C₂×Dic₃₄

C₂×C₄

yes

—

C₂×C₈×C₁₇

C₂³

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

C₂²⋊Dic₁₇

C₂×C₄

yes

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

C₂×C₄

yes

—

#63225

C₂×C₄

yes

—

C₄×Dic₁₇

C₂³

yes

—

C₂×#3622

C₂³

yes

—

C₂³×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂×Dic₁₇

C₂²×C₄

yes

—

C₂×Dic₁₇

C₂²×C₄

yes

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

#3622

—

#2987

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂³×C₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

2₋¹⁺⁴

yes

—

Dic₁₇

—

Dic₁₇

—

—

—

—

—

—

D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

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

C₂×D₁₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₁₇

yes

—

#97

—

#97

—

#97

C₂×D₁₇

yes

—

100

C₄×C₈

—

101

C₁₇

#2987

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

C₄×D₁₇

yes

characteristic

105

C₂²×C₄

C₄×D₁₇

yes

characteristic

106

C₂²×C₄

C₂²×D₁₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₁₇

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

C₄×D₁₇

yes

characteristic

118

C₄²

C₄×D₁₇

yes

characteristic

119

C₄²

C₂²×D₁₇

yes

characteristic

120

C₄²

C₂²×D₁₇

yes

characteristic

121

C₂³

C₂×C₄×D₁₇

yes

characteristic

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

130

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

131

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

132

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

133

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

134

C₂×C₄

C₂³×D₁₇

yes

characteristic

135

C₈

—

136

C₈

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂²×C₄×D₁₇

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#10587697

yes

—

149

C₂

—

150

C₂

#11763286

yes

characteristic

151

C₂

#11763575

yes

characteristic

152

C₂

#11763575

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#432	C₂
prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#432	D₁₇
hypocenter	C₁₇	#2987
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₇

commutator, socle

C₂²×C₁₇

C₂³×C₄

Fitting

C₁₇×#432

C₂

prime radical

C₂³×C₁₇

C₂²×C₄

cosocle

C₂×C₄×C₁₇

C₂⁴

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #11762938	C₂	semidirect
3. #11762937	C₂	semidirect
101. C₁₇	#2987	semidirect

H ≅ G/N

type

2. #11762938

C₂

semidirect

3. #11762937

C₂

semidirect

101. C₁₇

#2987

semidirect

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

subgroup	index
#11763366	2
#11762938	2
#11762937	2
#11762867	2
#11762866	2
C₁₇×#432	2
#2987	17

subgroup

index

C₁₇×#432

Subgroups — #14537889

Conjugacy classes of subgroups

Complements

Maximal subgroups