Group #14537867 — subgroups

Subgroups — #14537867

Order 2176.

356 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 177 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×C₄×Dic₃₄	C₂	1	1	yes	maximal, characteristic
2	#11762939	C₂	1	4	yes	maximal, complemented
3	#11762936	C₂	1	4	yes	maximal, complemented
4	#11762863	C₂	1	4	yes	maximal
5	#11762862	C₂	1	1	yes	maximal, characteristic
6	C₁₇×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₄	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇	C₄	1	1	yes	characteristic
10	C₂×#63225	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
12	C₄×Dic₃₄	C₄	1	2	yes	—
13	C₄×Dic₃₄	C₄	1	2	yes	—
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	C₄×C₈×C₁₇	C₂²	1	2	yes	—
19	C₂²⋊#3622	C₂²	1	2	yes	—
20	#10587486	C₂²	1	8	yes	—
21	#10587485	C₂²	1	8	yes	—
22	C₄⋊#3622	C₂²	1	2	yes	—
23	C₄×#3622	C₂²	1	2	yes	—
24	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
28	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₃₄	—	2	4	no	—
30	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
31	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₇	C₂³	1	4	yes	—
33	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₇	C₂³	1	1	yes	characteristic
36	C₄²×C₁₇	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
39	#63225	C₂×C₄	1	4	yes	—
40	#63225	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₇	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	M₁₆	1	4	yes	—
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	—	2	4	no	—
55	C₂×Dic₁₇	—	4	2	no	—
56	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
57	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
58	Dic₃₄	—	2	4	no	—
59	Dic₃₄	—	2	4	no	—
60	C₈×C₁₇	—	2	4	no	—
61	#3622	—	2	4	no	—
62	#2983	—	17	1	no	maximal, Sylow
63	C₂²×C₁₇	—	2	1	no	—
64	C₂²×C₁₇	—	2	1	no	—
65	C₂²×C₁₇	C₂×M₁₆	1	2	yes	—
66	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₁₇	—	2	1	no	—
68	C₄×C₁₇	—	2	1	no	—
69	C₄×C₁₇	—	2	1	no	—
70	C₄×C₁₇	C₂×M₁₆	1	2	yes	—
71	C₄×C₁₇	C₂×M₁₆	1	2	yes	—
72	C₄×C₁₇	2₋¹⁺⁴	1	2	yes	—
73	Dic₁₇	—	2	4	no	—
74	Dic₁₇	—	2	4	no	—
75	C₂×C₄×Q₈	—	17	1	no	—
76	#446	—	17	4	no	—
77	#446	—	17	4	no	—
78	#432	—	17	4	no	—
79	#431	—	17	1	no	—
80	#431	D₁₇	1	1	yes	characteristic
81	C₂×C₁₇	—	2	2	no	—
82	C₂×C₁₇	#519	1	1	yes	characteristic
83	C₂×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
84	C₂×C₁₇	#568	1	1	yes	characteristic
85	C₂²×Q₈	—	17	1	no	—
86	C₄×Q₈	—	17	2	no	—
87	C₄×Q₈	—	17	2	no	—
88	C₄×Q₈	—	17	2	no	—
89	C₄×Q₈	—	17	2	no	—
90	C₂×C₄⋊C₄	—	17	2	no	—
91	C₂×C₄⋊C₄	—	17	1	no	—
92	C₂×C₄²	—	17	2	no	—
93	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
94	C₄⋊C₈	—	17	8	no	—
95	C₄⋊C₈	—	17	2	no	—
96	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
97	C₂²⋊C₈	—	17	2	no	—
98	C₂²⋊C₈	C₂×D₁₇	1	2	yes	—
99	#97	—	17	8	no	—
100	C₄×C₈	—	17	2	no	—
101	C₄×C₈	C₂×D₁₇	1	2	yes	—
102	C₁₇	#2983	1	1	yes	characteristic, complemented, Sylow
103	C₂×Q₈	—	17	2	no	—
104	C₂×Q₈	—	17	2	no	—
105	C₂×Q₈	—	34	4	no	—
106	C₂²×C₄	—	17	2	no	—
107	C₂²×C₄	—	17	2	no	—
108	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
111	C₂×C₈	—	17	4	no	—
112	C₂×C₈	C₂²×D₁₇	1	4	yes	—
113	C₄⋊C₄	—	17	4	no	—
114	C₄⋊C₄	—	17	4	no	—
115	C₄⋊C₄	—	17	2	no	—
116	C₄⋊C₄	—	17	2	no	—
117	C₄²	—	17	4	no	—
118	C₄²	—	17	4	no	—
119	C₄²	C₄×D₁₇	1	1	yes	characteristic
120	C₄²	C₄×D₁₇	1	1	yes	characteristic
121	C₄²	C₂²×D₁₇	1	1	yes	characteristic
122	C₄²	C₂²×D₁₇	1	1	yes	characteristic
123	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
124	Q₈	—	34	4	no	—
125	Q₈	—	34	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	17	4	no	—
129	C₂×C₄	—	17	4	no	—
130	C₂×C₄	—	34	4	no	—
131	C₂×C₄	—	68	2	no	—
132	C₂×C₄	#63218	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	34	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#63218	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	34	4	no	—
149	C₄	—	34	4	no	—
150	C₄	C₂×#63218	1	2	yes	—
151	C₄	C₂×#63218	1	2	yes	—
152	C₄	#10587697	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#11763286	1	1	yes	characteristic
155	C₂	C₂²×#63218	1	1	yes	characteristic
156	C₂	#11763575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₃₄

C₂

yes

maximal, characteristic

#11762939

C₂

yes

maximal, complemented

#11762936

C₂

yes

maximal, complemented

#11762863

C₂

yes

maximal

#11762862

C₂

yes

maximal, characteristic

C₁₇×#431

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₄

yes

characteristic

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₄

yes

characteristic

C₂×#63225

C₄

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

—

C₄×Dic₃₄

C₄

yes

—

C₄×Dic₃₄

C₄

yes

—

C₄×Dic₃₄

C₂²

yes

—

C₄×Dic₃₄

C₂²

yes

—

C₁₇×C₄⋊C₈

C₂²

yes

—

C₁₇×C₂²⋊C₈

C₂²

yes

—

C₄×C₈×C₁₇

C₂²

yes

—

C₂²⋊#3622

C₂²

yes

—

#10587486

C₂²

yes

—

#10587485

C₂²

yes

—

C₄⋊#3622

C₂²

yes

—

C₄×#3622

C₂²

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

C₂×C₄

yes

—

C₂²×Dic₁₇

C₂×C₄

yes

—

C₂×Dic₃₄

—

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

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

C₂×C₄

yes

—

#63225

C₂×C₄

yes

—

C₄×Dic₁₇

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

M₁₆

yes

—

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

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂²×C₄

yes

—

C₂×Dic₁₇

C₂²×C₄

yes

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

#3622

—

#2983

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×M₁₆

yes

—

C₂²×C₁₇

C₂³×C₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×M₁₆

yes

—

C₄×C₁₇

C₂×M₁₆

yes

—

C₄×C₁₇

2₋¹⁺⁴

yes

—

Dic₁₇

—

Dic₁₇

—

C₂×C₄×Q₈

—

—

—

—

—

D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#519

yes

characteristic

C₂×C₁₇

C₂²×M₁₆

yes

characteristic

C₂×C₁₇

#568

yes

characteristic

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₁₇

yes

—

#97

—

100

C₄×C₈

—

101

C₄×C₈

C₂×D₁₇

yes

—

102

C₁₇

#2983

yes

characteristic, complemented, Sylow

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₁₇

yes

characteristic

109

C₂²×C₄

C₄×D₁₇

yes

characteristic

110

C₂²×C₄

C₂²×D₁₇

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₁₇

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄×D₁₇

yes

characteristic

120

C₄²

C₄×D₁₇

yes

characteristic

121

C₄²

C₂²×D₁₇

yes

characteristic

122

C₄²

C₂²×D₁₇

yes

characteristic

123

C₂³

C₂×C₄×D₁₇

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#63218

yes

—

133

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

134

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

135

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

136

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

137

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

138

C₂×C₄

C₂³×D₁₇

yes

characteristic

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#63218

yes

—

144

C₂²

C₂²×C₄×D₁₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₂×#63218

yes

—

151

C₄

C₂×#63218

yes

—

152

C₄

#10587697

yes

—

153

C₂

—

154

C₂

#11763286

yes

characteristic

155

C₂

C₂²×#63218

yes

characteristic

156

C₂

#11763575

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#431	C₂
prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#431	D₁₇
hypocenter	C₁₇	#2983
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₇

commutator, socle

C₂²×C₁₇

C₂³×C₄

Fitting

C₁₇×#431

C₂

prime radical

C₂³×C₁₇

C₂²×C₄

cosocle

C₂×C₄×C₁₇

C₂⁴

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #11762939	C₂	semidirect
3. #11762936	C₂	semidirect
102. C₁₇	#2983	semidirect

H ≅ G/N

type

2. #11762939

C₂

semidirect

3. #11762936

C₂

semidirect

102. C₁₇

#2983

semidirect

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

subgroup	index
C₂×C₄×Dic₃₄	2
#11762939	2
#11762936	2
#11762863	2
#11762862	2
C₁₇×#431	2
#2983	17

subgroup

index

C₂×C₄×Dic₃₄

C₁₇×#431

Subgroups — #14537867

Conjugacy classes of subgroups

Complements

Maximal subgroups