Group #17982817 — subgroups

Subgroups — #17982817

Order 4352.

158 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 77 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	#14545883	C₂	1	1	yes	maximal, characteristic
2	#14545883	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#10587528	C₂	1	1	yes	maximal, characteristic
4	#14545870	C₂	1	1	yes	maximal, characteristic
5	#14545869	C₂	1	1	yes	maximal, characteristic
6	#14545867	C₂	1	1	yes	maximal, characteristic
7	#14545198	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587528	C₂²	1	1	yes	characteristic
9	C₂×#10587528	C₂²	1	1	yes	characteristic
10	C₂×#10587528	C₂²	1	1	yes	characteristic
11	C₂×#10587528	C₂²	1	1	yes	characteristic
12	#11763730	C₄	1	1	yes	characteristic
13	#11763730	C₄	1	1	yes	characteristic
14	C₄⋊#63216	C₄	1	1	yes	characteristic
15	#11763695	C₂²	1	1	yes	characteristic
16	#11763695	C₂²	1	1	yes	characteristic
17	C₄×#63216	C₂²	1	1	yes	characteristic
18	#11762933	C₄	1	1	yes	characteristic
19	C₄×Dic₃₄	C₈	1	1	yes	characteristic
20	C₂×#63216	C₂×C₄	1	1	yes	characteristic
21	C₂×#63216	C₂×C₄	1	1	yes	characteristic
22	C₂×#63216	C₂×C₄	1	1	yes	characteristic
23	C₂×#63216	C₂³	1	1	yes	characteristic
24	#10587528	—	2	1	no	—
25	#10587528	—	2	1	no	—
26	#10587528	—	2	1	no	—
27	#10587528	Q₈	1	2	yes	complemented
28	C₁₇×C₄⋊C₈	C₈	1	1	yes	characteristic
29	#10587485	C₈	1	1	yes	characteristic
30	#10587485	C₈	1	1	yes	characteristic
31	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×#3622	C₂×C₄	1	1	yes	characteristic
34	C₂×Dic₃₄	C₁₆	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
38	C₄⋊Dic₁₇	C₁₆	1	1	yes	characteristic
39	#63225	C₁₆	1	1	yes	characteristic
40	#63225	C₁₆	1	1	yes	characteristic
41	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
42	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
43	C₂×#3622	C₂×C₈	1	1	yes	characteristic
44	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
45	#63216	—	2	1	no	—
46	#63216	—	2	1	no	—
47	#63216	—	2	1	no	—
48	#63216	C₂×Q₈	1	1	yes	characteristic
49	#63216	#41	1	1	yes	characteristic
50	#13660	—	17	1	no	maximal, Sylow
51	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
54	C₂×Dic₁₇	C₂×C₁₆	1	1	yes	characteristic
55	C₂×Dic₁₇	C₂×C₁₆	1	1	yes	characteristic
56	C₂×Dic₁₇	C₂×C₁₆	1	1	yes	characteristic
57	C₂×Dic₁₇	C₂×C₁₆	1	1	yes	characteristic
58	Dic₃₄	—	2	2	no	—
59	C₈×C₁₇	—	2	1	no	—
60	C₈×C₁₇	—	2	1	no	—
61	#3622	—	2	1	no	—
62	#3622	C₄×Q₈	1	1	yes	characteristic
63	#3622	#131	1	1	yes	characteristic
64	Q₈×C₁₆	—	17	1	no	—
65	C₄⋊C₃₂	—	17	1	no	—
66	C₄⋊C₃₂	—	17	1	no	—
67	C₄⋊C₃₂	—	17	1	no	—
68	#1397	—	17	1	no	—
69	#1397	—	17	1	no	—
70	C₄×C₃₂	—	17	1	no	—
71	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
72	C₄×C₁₇	—	2	1	no	—
73	C₄×C₁₇	—	2	1	no	—
74	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
75	C₄×C₁₇	#504	1	1	yes	characteristic
76	Dic₁₇	—	2	1	no	—
77	Dic₁₇	—	2	1	no	—
78	Dic₁₇	—	2	1	no	—
79	Dic₁₇	M₆₄	1	2	yes	—
80	C₈×Q₈	—	17	1	no	—
81	C₂×C₃₂	—	17	1	no	—
82	C₂×C₃₂	—	17	1	no	—
83	C₂×C₃₂	—	17	1	no	—
84	C₂×C₃₂	—	17	1	no	—
85	C₄⋊C₁₆	—	17	1	no	—
86	C₄⋊C₁₆	—	17	1	no	—
87	C₄⋊C₁₆	—	17	1	no	—
88	C₄×C₁₆	—	17	1	no	—
89	C₄×C₁₆	—	17	1	no	—
90	C₄×C₁₆	—	17	1	no	—
91	C₂×C₁₇	Q₈×C₁₆	1	1	yes	characteristic
92	C₂×C₁₇	C₂×M₆₄	1	1	yes	characteristic
93	C₂×C₁₇	#2258	1	1	yes	characteristic
94	C₄×Q₈	—	17	1	no	—
95	C₂×C₁₆	—	17	1	no	—
96	C₂×C₁₆	—	17	1	no	—
97	C₂×C₁₆	—	17	1	no	—
98	C₂×C₁₆	—	17	1	no	—
99	C₄⋊C₈	—	17	1	no	—
100	C₄⋊C₈	—	17	1	no	—
101	C₄⋊C₈	#3633	1	1	yes	characteristic
102	C₄×C₈	—	17	1	no	—
103	C₄×C₈	—	17	1	no	—
104	C₄×C₈	—	17	1	no	—
105	C₃₂	—	17	2	no	—
106	C₃₂	—	34	1	no	—
107	C₃₂	—	34	1	no	—
108	C₃₂	—	34	1	no	—
109	C₁₇	#13660	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₈	—	17	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	C₂×#3633	1	1	yes	characteristic
114	C₂×C₈	C₂×#3633	1	1	yes	characteristic
115	C₄⋊C₄	—	17	1	no	—
116	C₄⋊C₄	—	17	1	no	—
117	C₄⋊C₄	—	17	1	no	—
118	C₄²	—	17	1	no	—
119	C₄²	—	17	1	no	—
120	C₄²	C₂×#3633	1	1	yes	characteristic
121	C₁₆	—	17	1	no	—
122	C₁₆	—	17	1	no	—
123	C₁₆	—	34	1	no	—
124	C₁₆	—	34	1	no	—
125	C₁₆	—	34	1	no	—
126	Q₈	—	34	2	no	—
127	C₂×C₄	—	17	1	no	—
128	C₂×C₄	—	17	1	no	—
129	C₂×C₄	—	17	1	no	—
130	C₂×C₄	—	17	1	no	—
131	C₂×C₄	#10587665	1	1	yes	characteristic
132	C₂×C₄	#10587665	1	1	yes	characteristic
133	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	17	1	no	—
137	C₈	—	17	1	no	—
138	C₈	—	34	1	no	—
139	C₂²	C₂×#10587665	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	17	2	no	—
143	C₄	—	34	1	no	—
144	C₄	—	34	1	no	—
145	C₄	—	34	1	no	—
146	C₄	Q₈×#3633	1	1	yes	characteristic
147	C₄	#11763930	1	1	yes	characteristic
148	C₂	#14546008	1	1	yes	characteristic
149	C₂	#14546015	1	1	yes	characteristic
150	C₂	#14546018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545883

C₂

yes

maximal, characteristic

#14545883

C₂

yes

maximal, characteristic

C₄⋊#10587528

C₂

yes

maximal, characteristic

#14545870

C₂

yes

maximal, characteristic

#14545869

C₂

yes

maximal, characteristic

#14545867

C₂

yes

maximal, characteristic

#14545198

C₂

yes

maximal, characteristic

C₂×#10587528

C₂²

yes

characteristic

C₂×#10587528

C₂²

yes

characteristic

C₂×#10587528

C₂²

yes

characteristic

C₂×#10587528

C₂²

yes

characteristic

#11763730

C₄

yes

characteristic

#11763730

C₄

yes

characteristic

C₄⋊#63216

C₄

yes

characteristic

#11763695

C₂²

yes

characteristic

#11763695

C₂²

yes

characteristic

C₄×#63216

C₂²

yes

characteristic

#11762933

C₄

yes

characteristic

C₄×Dic₃₄

C₈

yes

characteristic

C₂×#63216

C₂×C₄

yes

characteristic

C₂×#63216

C₂×C₄

yes

characteristic

C₂×#63216

C₂×C₄

yes

characteristic

C₂×#63216

C₂³

yes

—

—

—

Q₈

yes

complemented

C₁₇×C₄⋊C₈

C₈

yes

characteristic

#10587485

C₈

yes

characteristic

#10587485

C₈

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₄×#3622

C₂×C₄

yes

characteristic

C₂×Dic₃₄

C₁₆

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄⋊Dic₁₇

C₁₆

yes

characteristic

#63225

C₁₆

yes

characteristic

#63225

C₁₆

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

characteristic

C₂×#3622

C₂×C₈

yes

characteristic

C₂×#3622

C₂²×C₄

yes

—

—

—

C₂×Q₈

yes

yes

—

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×Dic₁₇

C₂×C₁₆

yes

characteristic

C₂×Dic₁₇

C₂×C₁₆

yes

characteristic

C₂×Dic₁₇

C₂×C₁₆

yes

characteristic

C₂×Dic₁₇

C₂×C₁₆

yes

characteristic

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3622

C₄×Q₈

yes

yes

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1397

—

#1397

—

C₄×C₃₂

—

C₂²×C₁₇

C₂²×C₁₆

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₈×Q₈

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

M₆₄

yes

—

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂×C₁₇

Q₈×C₁₆

yes

characteristic

C₂×C₁₇

C₂×M₆₄

yes

characteristic

C₂×C₁₇

#2258

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

#3633

yes

characteristic

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₃₂

—

106

C₃₂

—

107

C₃₂

—

108

C₃₂

—

109

C₁₇

#13660

yes

characteristic, complemented, Sylow

110

C₂×Q₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂×#3633

yes

characteristic

114

C₂×C₈

C₂×#3633

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂×#3633

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#10587665

yes

characteristic

132

C₂×C₄

#10587665

yes

characteristic

133

C₂×C₄

C₂²×#3633

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₂²

C₂×#10587665

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

Q₈×#3633

yes

characteristic

147

C₄

#11763930

yes

characteristic

148

C₂

#14546008

yes

characteristic

149

C₂

#14546015

yes

characteristic

150

C₂

#14546018

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3633
commutator, socle, prime radical	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₁₇×C₄⋊C₈	C₈
cosocle	C₂×#63216	C₂³
hypercenter	C₄⋊C₈	#3633
hypocenter	C₁₇	#13660
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#3633

commutator, socle, prime radical

C₂²×C₁₇

C₂²×C₁₆

Fitting

C₁₇×C₄⋊C₈

C₈

C₂³

C₄⋊C₈

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
27. #10587528	Q₈	semidirect
109. C₁₇	#13660	semidirect

H ≅ G/N

type

27. #10587528

Q₈

semidirect

109. C₁₇

#13660

semidirect

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

subgroup	index
#14545883	2
C₄⋊#10587528	2
#14545870	2
#14545869	2
#14545867	2
#14545198	2
#13660	17

subgroup

index

Subgroups — #17982817

Conjugacy classes of subgroups

Complements

Maximal subgroups