Group #17981217 — subgroups

Subgroups — #17981217

Order 4352.

210 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14545988	C₂	1	2	yes	maximal
2	#14543978	C₂	1	1	yes	maximal, characteristic
3	#11763912	—	2	2	no	—
4	#11763906	—	2	2	no	—
5	#11763903	C₄	1	4	yes	—
6	C₄×#63242	C₄	1	1	yes	characteristic
7	C₄×#63242	C₂²	1	1	yes	characteristic
8	#11763650	—	2	2	no	—
9	C₄×C₈×D₁₇	C₄	1	1	yes	characteristic
10	C₂×#63242	C₈	1	2	yes	—
11	C₂×#63242	D₄	1	1	yes	characteristic
12	C₂×#63242	Q₈	1	1	yes	characteristic
13	C₂×#63241	—	2	4	no	—
14	C₂×C₈×D₁₇	—	2	1	no	—
15	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
16	C₄×#3624	—	2	1	no	—
17	C₄×#3624	—	2	1	no	—
18	C₄×#3624	C₂×C₄	1	2	yes	—
19	C₄×#3624	D₄	1	2	yes	—
20	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
21	C₈×Dic₁₇	—	2	1	no	—
22	C₄×#3622	C₈	1	1	yes	characteristic
23	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
26	C₂×#3624	—	2	2	no	—
27	C₂×#3624	C₂²⋊C₄	1	2	yes	—
28	C₂×#3624	C₄⋊C₄	1	2	yes	—
29	#63242	—	4	1	no	—
30	#63242	—	4	1	no	—
31	#63242	—	4	2	no	—
32	#63241	—	4	4	no	—
33	C₂×C₈×C₁₇	—	2	1	no	—
34	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
36	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
38	C₂×#3622	—	2	1	no	—
39	C₈×D₁₇	—	2	4	no	—
40	#7146	—	17	1	no	maximal, Sylow
41	C₂²×D₁₇	C₄⋊C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
45	C₂×Dic₁₇	#95	1	1	yes	characteristic
46	C₂×Dic₁₇	#97	1	1	yes	characteristic
47	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
48	C₄×D₁₇	—	2	2	no	—
49	C₄×D₁₇	—	2	1	no	—
50	C₄×D₁₇	—	2	1	no	—
51	C₄×D₁₇	—	2	2	no	—
52	#3624	—	2	2	no	—
53	#3624	—	2	2	no	—
54	#3624	—	4	2	no	—
55	C₈×C₁₇	—	2	2	no	—
56	#3622	—	2	2	no	—
57	#2163	—	17	2	no	—
58	C₂×#322	—	17	1	no	—
59	C₂²×C₁₇	#336	1	1	yes	characteristic
60	C₂×D₁₇	—	2	1	no	—
61	C₂×D₁₇	—	2	1	no	—
62	C₂×D₁₇	#364	1	2	yes	—
63	C₄×C₁₇	—	2	1	no	—
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	—	2	1	no	—
66	Dic₁₇	—	2	1	no	—
67	Dic₁₇	H(3,ℤ/4)	1	1	yes	characteristic
68	Dic₁₇	#338	1	1	yes	characteristic
69	Dic₁₇	#349	1	2	yes	—
70	C₂×C₄×C₈	—	17	1	no	—
71	C₂×C₄×C₈	—	17	1	no	—
72	C₂×C₄×C₈	—	17	1	no	—
73	C₄⋊C₁₆	—	34	2	no	—
74	C₂²⋊C₁₆	—	34	2	no	—
75	#346	—	17	4	no	—
76	#322	—	34	2	no	—
77	C₂×C₁₇	#1275	1	1	yes	characteristic
78	C₂×C₁₇	#1376	1	1	yes	characteristic
79	C₂×C₁₇	#1377	1	1	yes	characteristic
80	D₁₇	—	2	2	no	—
81	C₂²×C₈	—	17	1	no	—
82	C₂²×C₈	—	17	1	no	—
83	C₂²×C₈	—	17	2	no	—
84	C₂²×C₈	—	34	1	no	—
85	C₂×C₄²	—	17	1	no	—
86	C₂×C₁₆	—	34	4	no	—
87	C₄×C₈	—	17	2	no	—
88	C₄×C₈	—	17	2	no	—
89	C₄×C₈	—	17	1	no	—
90	C₄×C₈	—	34	1	no	—
91	C₄×C₈	—	34	1	no	—
92	C₄×C₈	—	34	1	no	—
93	C₄×C₈	#3633	1	1	yes	characteristic
94	C₁₇	#7146	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄	—	17	1	no	—
96	C₂²×C₄	—	17	1	no	—
97	C₂²×C₄	—	17	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	17	2	no	—
100	C₂×C₈	—	17	2	no	—
101	C₂×C₈	—	34	4	no	—
102	C₂×C₈	—	34	1	no	—
103	C₂×C₈	—	34	2	no	—
104	C₂×C₈	—	68	1	no	—
105	C₂×C₈	—	68	1	no	—
106	C₂×C₈	—	68	2	no	—
107	C₄²	—	17	1	no	—
108	C₄²	—	17	1	no	—
109	C₄²	—	17	1	no	—
110	C₄²	C₂×#3633	1	1	yes	characteristic
111	C₁₆	—	68	4	no	—
112	C₂³	—	17	1	no	—
113	C₂×C₄	—	17	1	no	—
114	C₂×C₄	—	17	1	no	—
115	C₂×C₄	—	17	1	no	—
116	C₂×C₄	—	34	1	no	—
117	C₂×C₄	—	34	1	no	—
118	C₂×C₄	—	34	2	no	—
119	C₂×C₄	—	34	2	no	—
120	C₂×C₄	C₄×#3633	1	1	yes	characteristic
121	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊#3633	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	34	2	no	—
125	C₈	—	34	2	no	—
126	C₈	—	34	2	no	—
127	C₈	—	68	2	no	—
128	C₂²	—	17	2	no	—
129	C₂²	—	34	1	no	—
130	C₂²	—	34	1	no	—
131	C₂²	(C₂×#3633)⋊C₄	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	17	1	no	—
136	C₄	—	17	1	no	—
137	C₄	—	17	2	no	—
138	C₄	—	34	1	no	—
139	C₂	—	34	2	no	—
140	C₂	#14545788	1	1	yes	characteristic
141	C₂	#14545858	1	1	yes	characteristic
142	C₂	#14545861	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545988

C₂

yes

maximal

#14543978

C₂

yes

maximal, characteristic

#11763912

—

#11763906

—

#11763903

C₄

yes

—

C₄×#63242

C₄

yes

characteristic

C₄×#63242

C₂²

yes

characteristic

#11763650

—

C₄×C₈×D₁₇

C₄

yes

characteristic

C₂×#63242

C₈

yes

—

C₂×#63242

D₄

yes

characteristic

C₂×#63242

Q₈

yes

characteristic

C₂×#63241

—

C₂×C₈×D₁₇

—

C₄²×D₁₇

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×#3624

D₄

yes

—

C₄×C₈×C₁₇

C₈

yes

characteristic

C₈×Dic₁₇

—

C₄×#3622

C₈

yes

characteristic

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×C₈

yes

characteristic

C₂×C₄×D₁₇

M₁₆

yes

characteristic

C₂×#3624

—

C₂×#3624

C₂²⋊C₄

yes

—

C₂×#3624

C₄⋊C₄

yes

—

—

—

—

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

characteristic

C₂×#3622

—

C₈×D₁₇

—

#7146

—

maximal, Sylow

C₂²×D₁₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

C₂²⋊C₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

#3624

—

#3624

—

C₈×C₁₇

—

#3622

—

#2163

—

C₂×#322

—

C₂²×C₁₇

#336

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#364

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

Dic₁₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3633

yes

characteristic

C₁₇

#7146

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₂×#3633

yes

characteristic

111

C₁₆

—

112

C₂³

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₄×#3633

yes

characteristic

121

C₂×C₄

C₄⋊#3633

yes

characteristic

122

C₂×C₄

C₂²⋊#3633

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

(C₂×#3633)⋊C₄

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₂

—

140

C₂

#14545788

yes

characteristic

141

C₂

#14545858

yes

characteristic

142

C₂

#14545861

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#3633)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₄²	C₂×#3633
Fitting	C₄×C₈×C₁₇	C₈
socle	C₂²×C₁₇	#336
prime radical	C₂²×D₁₇	C₄⋊C₈
cosocle	C₄×#63242	C₂²
hypercenter	C₄×C₈	#3633
hypocenter	C₁₇	#7146
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

(C₂×#3633)⋊C₄

commutator

C₂×C₄×C₁₇

C₄×C₈

Frattini

C₄²

C₂×#3633

Fitting

C₄×C₈×C₁₇

C₈

socle

C₂²×C₁₇

#336

prime radical

C₂²×D₁₇

C₄⋊C₈

C₂²

C₄×C₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
94. C₁₇	#7146	semidirect

H ≅ G/N

type

94. C₁₇

#7146

semidirect

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

subgroup	index
#14545988	2
#14543978	2
#7146	17

subgroup

index

#14545988

#14543978

#7146

Subgroups — #17981217

Conjugacy classes of subgroups

Complements

Maximal subgroups