Group #14530292 — subgroups

Subgroups — #14530292

Order 2176.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#11762852	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587478	C₂	1	1	yes	maximal, characteristic
3	#11762836	C₂	1	1	yes	maximal, characteristic
4	#11762578	C₂	1	1	yes	maximal, characteristic
5	#11762577	C₂	1	1	yes	maximal, characteristic
6	#11762573	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#428	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#117	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
12	C₂×#63223	C₂²	1	1	yes	characteristic
13	C₂×#63223	C₂²	1	1	yes	characteristic
14	C₂²×#3622	C₂²	1	1	yes	characteristic
15	C₁₇×#107	—	2	1	no	—
16	C₁₇×#106	—	2	1	no	—
17	#10587479	—	2	1	no	—
18	#10587478	—	2	2	no	—
19	#10587478	—	2	1	no	—
20	C₄⋊#3622	—	2	2	no	—
21	C₂²×C₄×C₁₇	—	2	1	no	—
22	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₇	—	2	1	no	—
24	M₁₆×C₁₇	—	2	1	no	—
25	M₁₆×C₁₇	Q₈	1	2	yes	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₄⋊C₄×C₁₇	—	2	1	no	—
28	C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
31	C₂²⋊C₄×C₁₇	—	2	1	no	—
32	C₄²×C₁₇	—	2	1	no	—
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	#63223	—	2	1	no	—
36	#63223	—	2	1	no	—
37	#63223	—	2	1	no	—
38	#63223	Q₈	1	2	yes	—
39	C₂×#3622	—	2	1	no	—
40	C₂×#3622	—	2	1	no	—
41	C₂×#3622	—	2	1	no	—
42	C₂×#3622	—	2	1	no	—
43	C₂×#3622	D₄	1	2	yes	—
44	C₂³×C₁₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₇	—	2	1	no	—
46	C₂×C₄×C₁₇	—	2	1	no	—
47	C₂×C₄×C₁₇	—	4	1	no	—
48	C₂×C₄×C₁₇	—	8	1	no	—
49	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
55	C₂×Dic₁₇	—	4	1	no	—
56	C₂×Dic₁₇	—	8	1	no	—
57	C₈×C₁₇	—	2	2	no	—
58	#3622	—	2	2	no	—
59	#3622	—	2	2	no	—
60	#3622	—	4	1	no	—
61	#2089	—	17	1	no	maximal, Sylow
62	C₂²×C₁₇	—	2	1	no	—
63	C₂²×C₁₇	—	2	1	no	—
64	C₂²×C₁₇	#121	1	1	yes	characteristic
65	C₂²×C₁₇	#125	1	1	yes	characteristic
66	C₂²×C₁₇	#125	1	1	yes	characteristic
67	C₄×C₁₇	—	4	1	no	—
68	C₄×C₁₇	—	4	1	no	—
69	C₄×C₁₇	—	8	1	no	—
70	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₄×C₁₇	#123	1	1	yes	characteristic
73	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
74	Dic₁₇	—	8	1	no	—
75	#428	—	17	1	no	—
76	#428	D₁₇	1	1	yes	characteristic
77	C₂×#107	—	17	1	no	—
78	#424	—	17	1	no	—
79	#343	—	17	1	no	—
80	#343	—	17	1	no	—
81	#341	—	17	1	no	—
82	C₂×C₁₇	—	2	1	no	—
83	C₂×C₁₇	—	2	1	no	—
84	C₂×C₁₇	#398	1	1	yes	characteristic
85	C₂×C₁₇	#472	1	1	yes	characteristic
86	C₂×C₁₇	#473	1	1	yes	characteristic
87	C₂×M₁₆	—	17	1	no	—
88	C₂×M₁₆	—	17	1	no	—
89	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
90	C₂²×C₈	—	17	1	no	—
91	#117	C₂×D₁₇	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	17	1	no	—
93	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
94	#107	—	2	1	no	—
95	#107	—	34	2	no	—
96	#107	—	34	1	no	—
97	#106	—	2	1	no	—
98	#106	—	34	1	no	—
99	C₄⋊C₈	—	34	2	no	—
100	C₁₇	#2089	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	34	1	no	—
103	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
104	M₁₆	—	2	1	no	—
105	M₁₆	—	17	2	no	—
106	M₁₆	—	34	1	no	—
107	M₁₆	—	34	1	no	—
108	M₁₆	—	34	1	no	—
109	M₁₆	Dic₃₄	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	17	2	no	—
112	C₂×C₈	—	34	1	no	—
113	C₂×C₈	—	34	1	no	—
114	C₂×C₈	—	34	1	no	—
115	C₂×C₈	—	34	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	34	1	no	—
119	C₄⋊C₄	—	34	1	no	—
120	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
121	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₂³	#63254	1	1	yes	characteristic
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	68	1	no	—
130	C₂×C₄	—	136	1	no	—
131	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
132	C₂×C₄	#63252	1	1	yes	characteristic
133	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
135	C₂×C₄	#63256	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	34	2	no	—
139	C₈	—	34	2	no	—
140	C₈	—	68	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#10587575	1	1	yes	characteristic
144	C₂²	#10587576	1	1	yes	characteristic
145	C₂²	#10587625	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	8	1	no	—
149	C₄	—	136	1	no	—
150	C₄	#10587574	1	1	yes	characteristic
151	C₄	#10587580	1	1	yes	characteristic
152	C₄	#10587610	1	1	yes	characteristic
153	C₄	#10587632	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#11762742	1	1	yes	characteristic
157	C₂	#11763077	1	1	yes	characteristic
158	C₂	#11763082	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762852

C₂

yes

maximal, characteristic

C₂×#10587478

C₂

yes

maximal, characteristic

#11762836

C₂

yes

maximal, characteristic

#11762578

C₂

yes

maximal, characteristic

#11762577

C₂

yes

maximal, characteristic

#11762573

C₂

yes

maximal, characteristic

C₁₇×#428

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

characteristic

C₁₇×#107

—

C₁₇×#106

—

—

—

—

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

Q₈

yes

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

characteristic

C₄⋊C₄×C₁₇

D₄

yes

characteristic

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

—

—

Q₈

yes

—

—

—

—

—

D₄

yes

—

C₂³×C₁₇

#41

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₇

#123

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

Dic₁₇

—

#428

—

#428

D₁₇

yes

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

—

#117

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

—

—

—

—

C₄⋊C₈

—

100

C₁₇

#2089

yes

characteristic, complemented, Sylow

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×D₁₇

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

Dic₃₄

yes

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

C₂²⋊D₁₇

yes

characteristic

121

C₄⋊C₄

C₂²⋊D₁₇

yes

characteristic

122

C₂²⋊C₄

—

123

C₄²

—

124

C₂³

#63254

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

136

—

131

C₂×C₄

C₂×Dic₃₄

yes

characteristic

132

C₂×C₄

#63252

yes

characteristic

133

C₂×C₄

D₄×D₁₇

yes

characteristic

134

C₂×C₄

Q₈×D₁₇

yes

characteristic

135

C₂×C₄

#63256

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

#10587575

yes

characteristic

144

C₂²

#10587576

yes

characteristic

145

C₂²

#10587625

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

136

—

150

C₄

#10587574

yes

characteristic

151

C₄

#10587580

yes

characteristic

152

C₄

#10587610

yes

characteristic

153

C₄

#10587632

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#11762742

yes

characteristic

157

C₂

#11763077

yes

characteristic

158

C₂

#11763082

yes

characteristic

	subgroup	G/H
center	C₂²	#10587625
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#428	C₂
socle	C₂²×C₁₇	#121
prime radical	C₂³×C₁₇	#41
hypercenter	#428	D₁₇
hypocenter	C₁₇	#2089
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

subgroup

G/H

center

C₂²

#10587625

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#428

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

N	H ≅ G/N	type
100. C₁₇	#2089	semidirect

H ≅ G/N

type

100. C₁₇

#2089

semidirect

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

subgroup	index
#11762852	2
C₂×#10587478	2
#11762836	2
#11762578	2
#11762577	2
#11762573	2
C₁₇×#428	2
#2089	17

subgroup

index

C₁₇×#428

Subgroups — #14530292

Conjugacy classes of subgroups

Complements

Maximal subgroups