Group #17958013 — subgroups

Subgroups — #17958013

Order 4352.

302 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 140 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	#14544985	C₂	1	1	yes	maximal, characteristic
2	#14544985	C₂	1	1	yes	maximal, characteristic
3	#14543702	C₂	1	4	yes	maximal, complemented
4	D₁₇×#322	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587529	C₄	1	2	yes	—
6	C₂×#10587529	C₄	1	2	yes	—
7	#11763698	C₄	1	2	yes	—
8	#11763698	C₄	1	2	yes	—
9	#11763697	C₂²	1	2	yes	—
10	#11763697	C₂²	1	2	yes	—
11	C₄×C₈×D₁₇	C₄	1	2	yes	—
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	#11762531	C₂²	1	2	yes	—
14	#11762530	C₄	1	1	yes	characteristic
15	C₁₇×#322	C₄	1	1	yes	characteristic
16	C₂×C₈×D₁₇	—	2	2	no	—
17	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
19	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×#63216	C₂×C₄	1	4	yes	—
21	C₂×#63216	C₂×C₄	1	4	yes	—
22	#10587529	—	2	4	no	—
23	#10587529	C₈	1	8	yes	complemented
24	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₁₇	—	2	2	no	—
27	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×#3622	C₂×C₄	1	2	yes	—
30	C₄×#3622	C₂³	1	1	yes	characteristic
31	C₂×C₄×D₁₇	C₄²	1	1	yes	characteristic
32	C₂×C₄×D₁₇	M₁₆	1	2	yes	—
33	C₂×C₈×C₁₇	—	2	2	no	—
34	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
36	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
38	C₄×Dic₁₇	M₁₆	1	2	yes	—
39	C₂×#3622	—	2	2	no	—
40	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
41	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
42	C₈×D₁₇	—	4	4	no	—
43	C₈×D₁₇	C₂×C₈	1	4	yes	—
44	C₈×D₁₇	M₁₆	1	4	yes	—
45	#63216	—	2	4	no	—
46	#63216	C₂×C₈	1	8	yes	complemented
47	C₂×#1313	—	17	1	no	maximal, Sylow
48	C₂²×D₁₇	#97	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
51	C₂×Dic₁₇	#97	1	2	yes	—
52	C₂×Dic₁₇	#97	1	1	yes	characteristic
53	C₄×D₁₇	—	2	4	no	—
54	C₄×D₁₇	C₄×C₈	1	2	yes	—
55	C₄×D₁₇	#97	1	2	yes	—
56	C₈×C₁₇	—	4	2	no	—
57	C₈×C₁₇	C₄×C₈	1	2	yes	—
58	C₈×C₁₇	#97	1	2	yes	—
59	#3622	—	4	2	no	—
60	#3622	C₂²×C₈	1	2	yes	—
61	#3622	C₂×M₁₆	1	2	yes	—
62	C₂×#346	—	17	1	no	—
63	C₂×#346	—	17	1	no	—
64	C₂×#322	—	17	1	no	—
65	#1313	—	17	4	no	—
66	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
67	C₂×D₁₇	#322	1	2	yes	—
68	C₂×D₁₇	#347	1	4	yes	—
69	C₄×C₁₇	—	2	2	no	—
70	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
71	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
72	Dic₁₇	—	2	2	no	—
73	Dic₁₇	#322	1	2	yes	—
74	C₂²×C₁₆	—	17	2	no	—
75	C₂²×C₁₆	—	17	2	no	—
76	C₂×C₄×C₈	—	17	1	no	—
77	C₂×C₄×C₈	—	17	2	no	—
78	#346	—	17	2	no	—
79	#346	—	17	2	no	—
80	#346	—	17	2	no	—
81	#346	—	17	2	no	—
82	#322	—	17	2	no	—
83	#322	—	17	1	no	—
84	#322	#595	1	1	yes	characteristic
85	C₂×C₁₇	C₂×#322	1	1	yes	characteristic
86	C₂×C₁₇	C₂×#347	1	2	yes	—
87	D₁₇	#1313	1	4	yes	complemented
88	C₂²×C₈	—	17	1	no	—
89	C₂²×C₈	—	17	1	no	—
90	C₂²×C₈	—	34	2	no	—
91	C₂×C₄²	—	17	1	no	—
92	C₂×C₁₆	—	17	4	no	—
93	C₂×C₁₆	—	17	8	no	—
94	C₂×C₁₆	—	17	4	no	—
95	C₂×C₁₆	—	34	4	no	—
96	C₄×C₈	—	17	1	no	—
97	C₄×C₈	—	17	1	no	—
98	C₄×C₈	—	17	1	no	—
99	C₄×C₈	—	17	2	no	—
100	C₄×C₈	—	34	2	no	—
101	C₄×C₈	C₂×#595	1	2	yes	—
102	C₄×C₈	C₂×#595	1	1	yes	characteristic
103	C₁₇	C₂×#1313	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₄	—	17	1	no	—
105	C₂²×C₄	—	17	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	17	1	no	—
108	C₂×C₈	—	17	1	no	—
109	C₂×C₈	—	17	4	no	—
110	C₂×C₈	—	17	4	no	—
111	C₂×C₈	—	34	2	no	—
112	C₂×C₈	—	68	4	no	—
113	C₂×C₈	C₄×#595	1	1	yes	characteristic
114	C₂×C₈	C₄×#595	1	1	yes	characteristic
115	C₄²	—	17	2	no	—
116	C₄²	—	17	1	no	—
117	C₄²	C₂²×#595	1	1	yes	characteristic
118	C₁₆	—	17	8	no	—
119	C₁₆	—	34	4	no	—
120	C₂³	—	17	1	no	—
121	C₂×C₄	—	17	1	no	—
122	C₂×C₄	—	17	2	no	—
123	C₂×C₄	—	17	2	no	—
124	C₂×C₄	—	17	2	no	—
125	C₂×C₄	—	34	4	no	—
126	C₂×C₄	D₁₇⋊M₁₆	1	2	yes	—
127	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₈	—	17	2	no	—
130	C₈	—	17	2	no	—
131	C₈	—	68	2	no	—
132	C₈	C₈×#595	1	2	yes	—
133	C₈	#10587532	1	2	yes	—
134	C₂²	—	17	4	no	—
135	C₂²	—	17	2	no	—
136	C₂²	#11763768	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	17	2	no	—
139	C₄	—	34	2	no	—
140	C₄	C₂×C₈×#595	1	1	yes	characteristic
141	C₄	C₂×#10587532	1	1	yes	characteristic
142	C₂	—	17	4	no	—
143	C₂	#14543980	1	1	yes	characteristic
144	C₂	#14545001	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544985

C₂

yes

maximal, characteristic

#14544985

C₂

yes

maximal, characteristic

#14543702

C₂

yes

maximal, complemented

D₁₇×#322

C₂

yes

maximal, characteristic

C₂×#10587529

C₄

yes

—

C₂×#10587529

C₄

yes

—

#11763698

C₄

yes

—

#11763698

C₄

yes

—

#11763697

C₂²

yes

—

#11763697

C₂²

yes

—

C₄×C₈×D₁₇

C₄

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762531

C₂²

yes

—

#11762530

C₄

yes

characteristic

C₁₇×#322

C₄

yes

characteristic

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₄²×D₁₇

C₂×C₄

yes

characteristic

C₂×#63216

C₂×C₄

yes

—

C₂×#63216

C₂×C₄

yes

—

#10587529

—

#10587529

C₈

yes

complemented

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

C₂³

yes

characteristic

C₂×C₄×D₁₇

C₄²

yes

characteristic

C₂×C₄×D₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

—

C₂×#3622

—

C₂×#3622

C₂²×C₄

yes

characteristic

C₂×#3622

C₂²×C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

C₂×C₈

yes

—

C₈×D₁₇

M₁₆

yes

—

#63216

—

#63216

C₂×C₈

yes

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

C₂×Dic₁₇

#97

yes

—

C₂×Dic₁₇

#97

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₄×C₈

yes

—

C₄×D₁₇

#97

yes

—

C₈×C₁₇

—

C₈×C₁₇

C₄×C₈

yes

—

C₈×C₁₇

#97

yes

—

#3622

—

#3622

C₂²×C₈

yes

—

#3622

C₂×M₁₆

yes

—

—

—

—

—

C₂²×C₁₇

C₂×#97

yes

characteristic

C₂×D₁₇

#322

yes

—

C₂×D₁₇

#347

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄×C₈

yes

characteristic

C₄×C₁₇

C₂×#97

yes

characteristic

Dic₁₇

—

Dic₁₇

#322

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

—

—

—

—

—

—

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

D₁₇

#1313

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂×#595

yes

—

102

C₄×C₈

C₂×#595

yes

characteristic

103

C₁₇

C₂×#1313

yes

characteristic, complemented, Sylow

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄×#595

yes

characteristic

114

C₂×C₈

C₄×#595

yes

characteristic

115

C₄²

—

116

C₄²

—

117

C₄²

C₂²×#595

yes

characteristic

118

C₁₆

—

119

C₁₆

—

120

C₂³

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₁₇⋊M₁₆

yes

—

127

C₂×C₄

C₂×C₄×#595

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

C₈×#595

yes

—

133

C₈

#10587532

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

#11763768

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

C₂×C₈×#595

yes

characteristic

141

C₄

C₂×#10587532

yes

characteristic

142

C₂

—

143

C₂

#14543980

yes

characteristic

144

C₂

#14545001

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#322	C₄
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂²×D₁₇	#97
cosocle	C₄×#3622	C₂³
hypercenter	#322	#595
hypocenter	C₁₇	C₂×#1313
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×C₄×#595

commutator

C₄×C₁₇

C₂×C₄×C₈

Frattini

C₄²

C₂²×#595

Fitting

C₁₇×#322

C₄

socle

C₂²×C₁₇

C₂×#97

C₂²×D₁₇

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ #322

N	H ≅ G/N	type
3. #14543702	C₂	semidirect
23. #10587529	C₈	semidirect
46. #63216	C₂×C₈	semidirect
87. D₁₇	#1313	semidirect
103. C₁₇	C₂×#1313	semidirect

H ≅ G/N

type

3. #14543702

C₂

semidirect

23. #10587529

C₈

semidirect

46. #63216

C₂×C₈

semidirect

87. D₁₇

#1313

semidirect

103. C₁₇

C₂×#1313

semidirect

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

subgroup	index
#14544985	2
#14543702	2
D₁₇×#322	2
C₂×#1313	17

subgroup

index

#14544985

#14543702

D₁₇×#322

C₂×#1313

Subgroups — #17958013

Conjugacy classes of subgroups

Complements

Maximal subgroups