Group #16950685 — subgroups

Subgroups — #16950685

Order 4352.

342 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 163 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	#14530540	C₂	1	2	yes	maximal
2	#14526866	C₂	1	4	yes	maximal
3	C₄²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587483	C₂²	1	4	yes	—
5	C₄⋊#63214	—	2	4	no	—
6	C₂×C₄×C₈×C₁₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₄³×C₁₇	C₄	1	1	yes	characteristic
11	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₁₇	—	2	2	no	—
15	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₄×C₈×C₁₇	—	2	4	no	—
18	C₄×C₈×C₁₇	—	2	4	no	—
19	C₄×C₈×C₁₇	C₈	1	2	yes	—
20	C₄×C₈×C₁₇	C₈	1	2	yes	—
21	C₄×C₈×C₁₇	D₄	1	2	yes	—
22	C₄×C₈×C₁₇	Q₈	1	2	yes	—
23	#10587483	—	2	8	no	—
24	C₂×#63214	—	2	4	no	—
25	C₂²×C₄×C₁₇	—	2	2	no	—
26	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
27	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₇	—	2	2	no	—
29	C₂×C₈×C₁₇	—	2	2	no	—
30	C₂×C₈×C₁₇	—	2	2	no	—
31	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₁₇	#41	1	4	yes	—
36	C₄²×C₁₇	—	2	8	no	—
37	C₄²×C₁₇	—	2	2	no	—
38	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
39	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₁₇	M₁₆	1	4	yes	—
42	#63214	—	4	4	no	—
43	#12883	—	17	1	no	maximal, Sylow
44	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
45	C₂×C₄×C₁₇	—	2	8	no	—
46	C₂×C₄×C₁₇	—	2	4	no	—
47	C₂×C₄×C₁₇	—	2	4	no	—
48	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
49	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
50	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#117	1	2	yes	—
52	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
54	C₈×C₁₇	—	2	2	no	—
55	C₈×C₁₇	—	2	4	no	—
56	C₈×C₁₇	#124	1	2	yes	—
57	C₈×C₁₇	#125	1	2	yes	—
58	#2150	—	17	2	no	—
59	C₄²×C₈	D₁₇	1	1	yes	characteristic
60	#1376	—	17	4	no	—
61	C₂²×C₁₇	—	2	2	no	—
62	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂²×C₁₇	#432	1	2	yes	—
64	C₄×C₁₇	—	2	8	no	—
65	C₄×C₁₇	—	2	2	no	—
66	C₄×C₁₇	#364	1	4	yes	—
67	C₄×C₁₇	#382	1	1	yes	characteristic
68	C₄×C₁₇	#423	1	1	yes	characteristic
69	C₄×C₁₇	#431	1	2	yes	—
70	C₂×M₃₂	—	17	4	no	—
71	C₂×C₄×C₈	Dic₁₇	1	2	yes	—
72	C₂×C₄×C₈	Dic₁₇	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
74	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
75	C₄³	Dic₁₇	1	1	yes	characteristic
76	C₄⋊C₁₆	—	34	4	no	—
77	C₂×C₁₇	—	2	2	no	—
78	C₂×C₁₇	#1831	1	1	yes	characteristic
79	C₂×C₁₇	C₂×#364	1	2	yes	—
80	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
81	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
82	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
83	C₂×C₄²	—	2	2	no	—
84	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
85	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
86	M₃₂	—	34	8	no	—
87	C₂×C₁₆	—	34	4	no	—
88	C₄×C₈	—	2	4	no	—
89	C₄×C₈	—	2	4	no	—
90	C₄×C₈	#3622	1	2	yes	—
91	C₄×C₈	#3622	1	2	yes	—
92	C₄×C₈	Dic₃₄	1	2	yes	—
93	C₄×C₈	D₆₈	1	2	yes	—
94	C₁₇	#12883	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#63223	1	2	yes	—
97	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂×#3622	1	4	yes	—
102	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
103	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
104	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
105	C₂×C₈	#63252	1	4	yes	—
106	C₄²	—	2	8	no	—
107	C₄²	—	2	2	no	—
108	C₄²	C₂×#3622	1	1	yes	characteristic
109	C₄²	C₂×#3622	1	1	yes	characteristic
110	C₄²	#63223	1	4	yes	—
111	C₄²	C₄⋊Dic₁₇	1	2	yes	—
112	C₁₆	—	68	4	no	—
113	C₂³	C₂×#63223	1	1	yes	characteristic
114	C₂×C₄	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₄⋊#3622	1	2	yes	—
118	C₂×C₄	C₄⋊#3622	1	2	yes	—
119	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
120	C₂×C₄	C₂×#63223	1	2	yes	—
121	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
122	C₂×C₄	#10587612	1	2	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	4	no	—
125	C₈	#10587556	1	2	yes	—
126	C₈	#10587561	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	C₂×C₄⋊#3622	1	1	yes	characteristic
129	C₂²	#11762866	1	2	yes	—
130	C₄	—	2	8	no	—
131	C₄	—	2	2	no	—
132	C₄	#11762630	1	4	yes	—
133	C₄	#11762675	1	1	yes	characteristic
134	C₄	#11762833	1	1	yes	characteristic
135	C₄	#11762862	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#14528901	1	1	yes	characteristic
138	C₂	C₂×#11762630	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14530540

C₂

yes

maximal

#14526866

C₂

yes

maximal

C₄²×C₈×C₁₇

C₂

yes

maximal, characteristic

C₂×#10587483

C₂²

yes

—

C₄⋊#63214

—

C₂×C₄×C₈×C₁₇

C₄

yes

—

C₂×C₄×C₈×C₁₇

C₄

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₄³×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₈

yes

—

C₄×C₈×C₁₇

C₈

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

#10587483

—

C₂×#63214

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

M₁₆

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

M₁₆

yes

—

#63214

—

#12883

—

maximal, Sylow

C₂³×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#124

yes

—

C₈×C₁₇

#125

yes

—

#2150

—

C₄²×C₈

D₁₇

yes

characteristic

#1376

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₇

#432

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#364

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₂×M₃₂

—

C₂×C₄×C₈

Dic₁₇

yes

—

C₂×C₄×C₈

Dic₁₇

yes

characteristic

C₂×C₄×C₈

C₂×D₁₇

yes

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

C₄³

Dic₁₇

yes

characteristic

C₄⋊C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

#1831

yes

characteristic

C₂×C₁₇

C₂×#364

yes

—

C₂²×C₈

C₂×Dic₁₇

yes

—

C₂²×C₈

C₂×Dic₁₇

yes

characteristic

C₂²×C₈

C₂²×D₁₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₁₇

yes

—

C₂×C₄²

C₂×Dic₁₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3622

yes

—

C₄×C₈

#3622

yes

—

C₄×C₈

Dic₃₄

yes

—

C₄×C₈

D₆₈

yes

—

C₁₇

#12883

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

#63223

yes

—

C₂²×C₄

C₂²×Dic₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×#3622

yes

—

102

C₂×C₈

C₄⋊Dic₁₇

yes

—

103

C₂×C₈

C₂×Dic₃₄

yes

characteristic

104

C₂×C₈

C₂×D₆₈

yes

characteristic

105

C₂×C₈

#63252

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₂×#3622

yes

characteristic

109

C₄²

C₂×#3622

yes

characteristic

110

C₄²

#63223

yes

—

111

C₄²

C₄⋊Dic₁₇

yes

—

112

C₁₆

—

113

C₂³

C₂×#63223

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₄⋊#3622

yes

—

118

C₂×C₄

C₄⋊#3622

yes

—

119

C₂×C₄

C₂²×#3622

yes

characteristic

120

C₂×C₄

C₂×#63223

yes

—

121

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

122

C₂×C₄

#10587612

yes

—

123

C₈

—

124

C₈

—

125

C₈

#10587556

yes

—

126

C₈

#10587561

yes

—

127

C₂²

—

128

C₂²

C₂×C₄⋊#3622

yes

characteristic

129

C₂²

#11762866

yes

—

130

C₄

—

131

C₄

—

132

C₄

#11762630

yes

—

133

C₄

#11762675

yes

characteristic

134

C₄

#11762833

yes

characteristic

135

C₄

#11762862

yes

—

136

C₂

—

137

C₂

#14528901

yes

characteristic

138

C₂

C₂×#11762630

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄²×C₈×C₁₇	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄²×C₈	D₁₇
hypocenter	C₁₇	#12883
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₂×C₈

C₂×D₆₈

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₄²×C₈×C₁₇

C₂

socle

C₂²×C₁₇

C₂×C₄⋊C₈

prime radical

C₂³×C₁₇

C₂×M₁₆

cosocle

C₂²×C₈×C₁₇

C₂³

hypercenter

C₄²×C₈

D₁₇

hypocenter

C₁₇

#12883

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
94. C₁₇	#12883	semidirect

H ≅ G/N

type

94. C₁₇

#12883

semidirect

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

subgroup	index
#14530540	2
#14526866	2
C₄²×C₈×C₁₇	2
#12883	17

subgroup

index

#14530540

#14526866

C₄²×C₈×C₁₇

#12883

Subgroups — #16950685

Conjugacy classes of subgroups

Complements

Maximal subgroups