Group #14539440 — subgroups

Subgroups — #14539440

Order 2176.

338 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 117 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	#11763420	C₂	1	2	yes	maximal
2	#11763211	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763177	C₂	1	2	yes	maximal, complemented
4	#11763064	C₂	1	2	yes	maximal
5	#11763047	C₂	1	2	yes	maximal
6	#11762916	C₂	1	2	yes	maximal, complemented
7	Q₁₆×Dic₁₇	C₂	1	2	yes	maximal, complemented
8	#11762763	C₂	1	1	yes	maximal, characteristic
9	C₁₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₁₇	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₁₇	C₂²	1	2	yes	—
12	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
13	#10587632	C₂²	1	4	yes	—
14	Q₈×Dic₁₇	C₂²	1	4	yes	—
15	C₂×#63218	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	#10587584	—	2	4	no	—
18	#10587575	—	2	4	no	—
19	#10587558	C₂²	1	1	yes	characteristic
20	#10587556	C₂²	1	2	yes	—
21	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	#10587506	C₂²	1	4	yes	—
23	#10587506	C₂²	1	4	yes	—
24	#10587489	C₂²	1	2	yes	—
25	#10587488	C₂²	1	2	yes	—
26	#10587486	C₂²	1	1	yes	characteristic
27	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
28	#10587474	C₂²	1	1	yes	characteristic
29	C₂×Q₈×C₁₇	C₂³	1	4	yes	—
30	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
31	Q₁₆×C₁₇	—	2	4	no	—
32	Q₁₆×C₁₇	—	2	4	no	—
33	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₇	—	2	4	no	—
36	C₄²×C₁₇	C₂³	1	1	yes	characteristic
37	#63227	—	2	4	no	—
38	#63227	D₄	1	2	yes	—
39	C₄⋊Dic₁₇	—	2	4	no	—
40	C₄⋊Dic₁₇	C₂³	1	4	yes	—
41	#63225	—	2	4	no	—
42	#63225	—	2	4	no	—
43	#63225	D₄	1	2	yes	—
44	C₄×Dic₁₇	—	2	4	no	—
45	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
46	C₂×#3622	C₂³	1	1	yes	characteristic
47	C₂×#3622	C₂³	1	1	yes	characteristic
48	#63218	—	2	2	no	—
49	C₈×D₁₇	—	2	2	no	—
50	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₇	—	2	4	no	—
52	Q₈×C₁₇	—	2	4	no	—
53	C₂×C₄×C₁₇	—	2	4	no	—
54	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₁₇	—	2	4	no	—
57	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₁₇	C₂×D₄	1	2	yes	—
59	C₄×D₁₇	—	2	1	no	—
60	C₄×D₁₇	—	2	1	no	—
61	C₈×C₁₇	#41	1	2	yes	—
62	C₈×C₁₇	#41	1	2	yes	—
63	#3622	—	2	1	no	—
64	#3622	—	2	1	no	—
65	#3177	—	17	1	no	maximal, Sylow
66	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₁₇	—	2	1	no	—
68	C₂×D₁₇	—	2	2	no	—
69	C₄×C₁₇	—	2	2	no	—
70	C₄×C₁₇	—	4	4	no	—
71	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
72	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
73	Dic₁₇	—	2	1	no	—
74	Dic₁₇	—	2	2	no	—
75	Dic₁₇	—	4	4	no	—
76	#541	—	17	2	no	—
77	#499	—	17	1	no	—
78	#494	D₁₇	1	1	yes	characteristic, complemented
79	#491	—	17	2	no	—
80	#470	—	17	2	no	—
81	#467	—	17	2	no	—
82	#441	—	17	2	no	—
83	C₄×Q₁₆	—	17	2	no	—
84	#405	—	17	1	no	—
85	C₂×C₁₇	#532	1	1	yes	characteristic
86	C₂×C₁₇	#578	1	2	yes	—
87	D₁₇	—	4	1	no	—
88	C₂×Q₁₆	C₂×D₁₇	1	2	yes	—
89	C₂×Q₁₆	C₂×D₁₇	1	2	yes	—
90	C₂×M₁₆	—	17	1	no	—
91	C₂²×C₈	—	17	1	no	—
92	C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
93	#126	—	34	4	no	—
94	#125	—	17	2	no	—
95	#125	—	34	4	no	—
96	C₂²⋊Q₈	—	17	4	no	—
97	C₄×Q₈	—	17	4	no	—
98	#117	—	17	1	no	—
99	#107	—	17	2	no	—
100	#106	—	17	2	no	—
101	#103	—	17	4	no	—
102	#103	—	17	4	no	—
103	#97	—	17	1	no	—
104	#97	—	17	1	no	—
105	C₄×C₈	—	17	1	no	—
106	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
107	C₁₇	#3177	1	1	yes	characteristic, complemented, Sylow
108	C₂×Q₈	C₂²×D₁₇	1	4	yes	—
109	C₂²×C₄	—	17	1	no	—
110	Q₁₆	—	2	4	no	—
111	Q₁₆	—	2	4	no	—
112	M₁₆	—	34	2	no	—
113	C₂×C₈	—	17	1	no	—
114	C₂×C₈	—	17	1	no	—
115	C₂×C₈	—	34	2	no	—
116	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	17	2	no	—
120	C₄⋊C₄	—	17	4	no	—
121	C₄⋊C₄	—	34	4	no	—
122	C₄⋊C₄	—	34	4	no	—
123	C₄⋊C₄	—	34	4	no	—
124	C₂²⋊C₄	—	17	2	no	—
125	C₂²⋊C₄	—	34	4	no	—
126	C₄²	—	17	1	no	—
127	C₄²	—	34	4	no	—
128	C₄²	C₂²×D₁₇	1	1	yes	characteristic
129	C₂³	—	17	1	no	—
130	Q₈	—	2	4	no	—
131	Q₈	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	17	2	no	—
134	C₂×C₄	—	17	1	no	—
135	C₂×C₄	—	34	1	no	—
136	C₂×C₄	—	34	1	no	—
137	C₂×C₄	—	34	4	no	—
138	C₂×C₄	D₄×D₁₇	1	2	yes	—
139	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
140	C₈	—	34	1	no	—
141	C₈	—	34	1	no	—
142	C₈	#63254	1	2	yes	—
143	C₈	#63254	1	2	yes	—
144	C₂²	—	34	1	no	—
145	C₂²	—	34	2	no	—
146	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	34	1	no	—
150	C₄	—	34	2	no	—
151	C₄	—	68	4	no	—
152	C₄	C₂×#63254	1	1	yes	characteristic
153	C₄	C₂×#63254	1	1	yes	characteristic
154	C₂	—	68	1	no	—
155	C₂	#11763357	1	1	yes	characteristic
156	C₂	#11763622	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763420

C₂

yes

maximal

#11763211

C₂

yes

maximal, characteristic, complemented

#11763177

C₂

yes

maximal, complemented

#11763064

C₂

yes

maximal

#11763047

C₂

yes

maximal

#11762916

C₂

yes

maximal, complemented

Q₁₆×Dic₁₇

C₂

yes

maximal, complemented

#11762763

C₂

yes

maximal, characteristic

C₁₇×#494

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆×C₁₇

C₂²

yes

—

C₂×Q₁₆×C₁₇

C₂²

yes

—

C₁₇×C₄⋊Q₈

C₂²

yes

—

#10587632

C₂²

yes

—

Q₈×Dic₁₇

C₂²

yes

—

C₂×#63218

C₂²

yes

characteristic

C₂×C₈×D₁₇

C₂²

yes

—

—

C₂²

yes

characteristic

#10587556

C₂²

yes

—

C₄×C₈×C₁₇

C₂²

yes

characteristic

#10587506

C₂²

yes

—

#10587506

C₂²

yes

—

#10587489

C₂²

yes

—

#10587488

C₂²

yes

—

#10587486

C₂²

yes

characteristic

C₈×Dic₁₇

C₂²

yes

characteristic

#10587474

C₂²

yes

characteristic

C₂×Q₈×C₁₇

C₂³

yes

—

C₂×C₄×D₁₇

C₂³

yes

characteristic

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

#63227

—

#63227

D₄

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂³

yes

—

#63225

—

#63225

—

#63225

D₄

yes

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂³

yes

characteristic

C₂×#3622

C₂³

yes

characteristic

C₂×#3622

C₂³

yes

characteristic

#63218

—

C₈×D₁₇

—

C₂²×D₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

C₂×D₄

yes

—

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

#41

yes

—

C₈×C₁₇

#41

yes

—

—

—

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#41

yes

characteristic

C₄×C₁₇

C₂×#41

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

#541

—

#499

—

#494

D₁₇

yes

characteristic, complemented

—

—

—

—

C₄×Q₁₆

—

#405

—

C₂×C₁₇

#532

yes

characteristic

C₂×C₁₇

#578

yes

—

D₁₇

—

C₂×Q₁₆

C₂×D₁₇

yes

—

C₂×Q₁₆

C₂×D₁₇

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₁₇

yes

—

#126

—

#125

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#107

—

100

#106

—

101

#103

—

102

#103

—

103

#97

—

104

#97

—

105

C₄×C₈

—

106

C₄×C₈

C₂×D₁₇

yes

characteristic

107

C₁₇

#3177

yes

characteristic, complemented, Sylow

108

C₂×Q₈

C₂²×D₁₇

yes

—

109

C₂²×C₄

—

110

Q₁₆

—

111

Q₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂²×D₁₇

yes

characteristic

117

C₂×C₈

C₂²×D₁₇

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²×D₁₇

yes

characteristic

129

C₂³

—

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

D₄×D₁₇

yes

—

139

C₂×C₄

C₂³×D₁₇

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

#63254

yes

—

143

C₈

#63254

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×D₄×D₁₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₂×#63254

yes

characteristic

153

C₄

C₂×#63254

yes

characteristic

154

C₂

—

155

C₂

#11763357

yes

characteristic

156

C₂

#11763622

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#494	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂²×D₁₇	C₂×D₄
hypercenter	#494	D₁₇
hypocenter	C₁₇	#3177
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₁₇

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₁₇×#494

C₂

socle

C₂²×C₁₇

C₂²×D₄

prime radical

C₂²×D₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #11763211	C₂	semidirect
3. #11763177	C₂	semidirect
6. #11762916	C₂	semidirect
7. Q₁₆×Dic₁₇	C₂	semidirect
9. C₁₇×#494	C₂	semidirect
78. #494	D₁₇	semidirect
107. C₁₇	#3177	semidirect

H ≅ G/N

type

2. #11763211

C₂

semidirect

3. #11763177

C₂

semidirect

6. #11762916

C₂

semidirect

7. Q₁₆×Dic₁₇

C₂

semidirect

9. C₁₇×#494

C₂

semidirect

78. #494

D₁₇

semidirect

107. C₁₇

#3177

semidirect

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

subgroup	index
#11763420	2
#11763211	2
#11763177	2
#11763064	2
#11763047	2
#11762916	2
Q₁₆×Dic₁₇	2
#11762763	2
C₁₇×#494	2
#3177	17

subgroup

index

Q₁₆×Dic₁₇

C₁₇×#494

Subgroups — #14539440

Conjugacy classes of subgroups

Complements

Maximal subgroups