Group #14539132 — subgroups

Subgroups — #14539132

Order 2176.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#11763359	C₂	1	1	yes	maximal, characteristic
2	#11763179	C₂	1	2	yes	maximal, complemented
3	#11763175	C₂	1	2	yes	maximal, complemented
4	#11763171	C₂	1	4	yes	maximal, complemented
5	#11762803	C₂	1	4	yes	maximal, complemented
6	C₂×#10587474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#63252	C₂²	1	2	yes	—
9	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₁₇×#125	C₂²	1	2	yes	—
11	C₁₇×C₂²⋊Q₈	—	2	2	no	—
12	C₄×Q₈×C₁₇	—	2	2	no	—
13	C₁₇×#117	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
15	#10587616	—	2	2	no	—
16	C₂²×#3622	C₂²	1	2	yes	—
17	#10587561	C₂²	1	2	yes	—
18	#10587560	C₂²	1	1	yes	characteristic
19	C₄×D₆₈	—	2	2	no	—
20	#10587555	C₂²	1	1	yes	characteristic
21	#10587481	C₂²	1	8	yes	—
22	#10587480	C₂²	1	8	yes	—
23	#10587474	C₂²	1	2	yes	—
24	#10587474	C₂²	1	2	yes	—
25	C₂×Q₈×C₁₇	—	2	2	no	—
26	C₂²×C₄×C₁₇	D₄	1	2	yes	—
27	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₁₇	—	2	2	no	—
29	#63252	—	2	8	no	—
30	C₂×D₆₈	—	2	2	no	—
31	C₂×D₆₈	C₂³	1	2	yes	—
32	C₂×C₄×D₁₇	—	2	2	no	—
33	C₂×Dic₃₄	C₂³	1	2	yes	—
34	C₄⋊C₄×C₁₇	—	2	4	no	—
35	C₄⋊C₄×C₁₇	—	2	2	no	—
36	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₁₇	—	2	2	no	—
38	C₄²×C₁₇	—	2	2	no	—
39	C₄²×C₁₇	D₄	1	2	yes	—
40	C₄²×C₁₇	C₂³	1	1	yes	characteristic
41	C₄²×C₁₇	C₂³	1	1	yes	characteristic
42	#63227	—	2	4	no	—
43	C₄⋊Dic₁₇	—	2	2	no	—
44	C₂×#3622	—	2	4	no	—
45	C₂×#3622	C₂³	1	4	yes	—
46	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₁₇	—	2	2	no	—
48	Q₈×C₁₇	—	4	4	no	—
49	C₂×C₄×C₁₇	—	2	4	no	—
50	C₂×C₄×C₁₇	—	4	2	no	—
51	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#41	1	4	yes	—
55	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
56	C₂²⋊D₁₇	—	4	4	no	—
57	C₂×Dic₁₇	—	2	2	no	—
58	D₆₈	—	2	4	no	—
59	D₆₈	—	4	4	no	—
60	C₄×D₁₇	—	4	4	no	—
61	Dic₃₄	—	2	4	no	—
62	#3622	—	2	4	no	—
63	#3135	—	17	1	no	maximal, Sylow
64	C₂²×C₁₇	—	2	2	no	—
65	C₂²×C₁₇	#124	1	2	yes	—
66	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₁₇	—	4	2	no	—
68	C₂×D₁₇	—	4	4	no	—
69	C₄×C₁₇	—	2	2	no	—
70	C₄×C₁₇	—	2	2	no	—
71	C₄×C₁₇	—	4	4	no	—
72	C₄×C₁₇	#124	1	2	yes	—
73	C₄×C₁₇	C₂×#41	1	2	yes	—
74	Dic₁₇	—	4	2	no	—
75	#533	D₁₇	1	1	yes	characteristic, complemented
76	#532	—	17	1	no	—
77	#491	—	17	2	no	—
78	#490	—	17	2	no	—
79	#489	—	17	4	no	—
80	#416	—	17	4	no	—
81	C₂×#97	—	17	1	no	—
82	C₂×C₁₇	—	2	2	no	—
83	C₂×C₁₇	C₂×#124	1	1	yes	characteristic
84	C₂×C₁₇	#575	1	2	yes	—
85	D₁₇	—	8	2	no	—
86	C₂×#41	—	17	2	no	—
87	C₂²×C₈	—	17	2	no	—
88	C₄⋊Q₈	—	17	1	no	—
89	C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
90	#127	—	17	1	no	—
91	#125	C₂×D₁₇	1	2	yes	—
92	#124	—	17	2	no	—
93	C₂²⋊Q₈	—	2	2	no	—
94	#121	—	34	2	no	—
95	C₄×Q₈	—	2	2	no	—
96	C₄×D₄	—	34	2	no	—
97	#117	C₂×D₁₇	1	2	yes	—
98	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
99	#103	—	17	8	no	—
100	#102	—	17	8	no	—
101	#97	—	17	2	no	—
102	#97	—	17	2	no	—
103	C₁₇	#3135	1	1	yes	characteristic, complemented, Sylow
104	#41	—	34	8	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	17	2	no	—
107	C₂×D₄	—	17	2	no	—
108	C₂×D₄	—	34	2	no	—
109	C₂×D₄	—	34	2	no	—
110	C₂²×C₄	—	34	2	no	—
111	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
113	C₂×C₈	—	17	4	no	—
114	C₂×C₈	—	34	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	34	2	no	—
118	C₄⋊C₄	C₂²×D₁₇	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	34	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂²⋊D₁₇	1	2	yes	—
123	C₄²	C₂²×D₁₇	1	1	yes	characteristic
124	C₄²	C₂²×D₁₇	1	1	yes	characteristic
125	C₂³	—	34	2	no	—
126	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	Q₈	—	34	4	no	—
129	D₄	—	34	4	no	—
130	D₄	—	68	4	no	—
131	D₄	—	68	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	34	2	no	—
135	C₂×C₄	—	68	4	no	—
136	C₂×C₄	#63256	1	4	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
140	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
141	C₈	—	34	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	68	2	no	—
144	C₂²	—	68	4	no	—
145	C₂²	#10587633	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	68	2	no	—
151	C₄	#10587633	1	2	yes	—
152	C₄	C₂×#63256	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	136	2	no	—
155	C₂	C₂×#10587633	1	1	yes	characteristic
156	C₂	#11763607	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763359

C₂

yes

maximal, characteristic

#11763179

C₂

yes

maximal, complemented

#11763175

C₂

yes

maximal, complemented

#11763171

C₂

yes

maximal, complemented

#11762803

C₂

yes

maximal, complemented

C₂×#10587474

C₂

yes

maximal, characteristic, complemented

C₁₇×#533

C₂

yes

maximal, characteristic, complemented

C₂×#63252

C₂²

yes

—

C₁₇×C₄⋊Q₈

C₂²

yes

—

C₁₇×#125

C₂²

yes

—

C₁₇×C₂²⋊Q₈

—

C₄×Q₈×C₁₇

—

C₁₇×#117

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

#10587616

—

C₂²×#3622

C₂²

yes

—

#10587561

C₂²

yes

—

#10587560

C₂²

yes

characteristic

C₄×D₆₈

—

#10587555

C₂²

yes

characteristic

#10587481

C₂²

yes

—

#10587480

C₂²

yes

—

#10587474

C₂²

yes

—

#10587474

C₂²

yes

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₂²⋊D₁₇

—

#63252

—

C₂×D₆₈

—

C₂×D₆₈

C₂³

yes

—

C₂×C₄×D₁₇

—

C₂×Dic₃₄

C₂³

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂³

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

D₄

yes

—

C₄²×C₁₇

C₂³

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

#63227

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

C₂³

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂²×D₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

Dic₃₄

—

#3622

—

#3135

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

#124

yes

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#124

yes

—

C₄×C₁₇

C₂×#41

yes

—

Dic₁₇

—

#533

D₁₇

yes

characteristic, complemented

—

—

—

—

—

C₂×#97

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#124

yes

characteristic

C₂×C₁₇

#575

yes

—

D₁₇

—

C₂×#41

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

—

#127

—

#125

C₂×D₁₇

yes

—

#124

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

C₂×D₁₇

yes

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#103

—

100

#102

—

101

#97

—

102

#97

—

103

C₁₇

#3135

yes

characteristic, complemented, Sylow

104

#41

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₁₇

yes

—

112

C₂²×C₄

C₂²×D₁₇

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂²×D₁₇

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

C₂²⋊D₁₇

yes

—

123

C₄²

C₂²×D₁₇

yes

characteristic

124

C₄²

C₂²×D₁₇

yes

characteristic

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#63256

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

139

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

140

C₂×C₄

C₂³×D₁₇

yes

characteristic

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#10587633

yes

—

146

C₂²

C₂²×C₂²⋊D₁₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#10587633

yes

—

152

C₄

C₂×#63256

yes

—

153

C₂

—

154

C₂

—

136

—

155

C₂

C₂×#10587633

yes

characteristic

156

C₂

#11763607

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#533	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763359	C₂
hypercenter	#533	D₁₇
hypocenter	C₁₇	#3135
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₇

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₁₇×#533

C₂

socle

C₂²×C₁₇

C₂²×D₄

C₂

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #11763179	C₂	semidirect
3. #11763175	C₂	semidirect
4. #11763171	C₂	semidirect
5. #11762803	C₂	semidirect
6. C₂×#10587474	C₂	semidirect
7. C₁₇×#533	C₂	semidirect
75. #533	D₁₇	semidirect
103. C₁₇	#3135	semidirect

H ≅ G/N

type

2. #11763179

C₂

semidirect

3. #11763175

C₂

semidirect

4. #11763171

C₂

semidirect

5. #11762803

C₂

semidirect

6. C₂×#10587474

C₂

semidirect

7. C₁₇×#533

C₂

semidirect

75. #533

D₁₇

semidirect

103. C₁₇

#3135

semidirect

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

subgroup	index
#11763359	2
#11763179	2
#11763175	2
#11763171	2
#11762803	2
C₂×#10587474	2
C₁₇×#533	2
#3135	17

subgroup

index

C₁₇×#533

Subgroups — #14539132

Conjugacy classes of subgroups

Complements

Maximal subgroups