Group #14544598 — subgroups

Subgroups — #14544598

Order 2176.

272 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 100 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	C₂×#10587534	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587533	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊#595	C₂	1	2	yes	maximal
4	#11763683	C₂	1	2	yes	maximal
5	#97×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#595	C₂²	1	2	yes	—
7	C₂×C₄⋊#595	C₂²	1	2	yes	—
8	C₂×C₄×#595	—	2	2	no	—
9	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
11	C₄²×D₁₇	C₂²	1	1	yes	characteristic
12	(C₂×#595)⋊C₄	—	2	2	no	—
13	#10587534	C₄	1	4	yes	complemented
14	#10587533	C₄	1	4	yes	complemented
15	C₁₇×#97	C₄	1	1	yes	characteristic, complemented
16	#10587486	C₄	1	1	yes	characteristic, complemented
17	#10587486	C₄	1	1	yes	characteristic, complemented
18	#10587474	C₄	1	1	yes	characteristic, complemented
19	C₂²×#595	—	2	2	no	—
20	C₂²×#595	—	2	2	no	—
21	C₂×C₄×D₁₇	D₄	1	2	yes	—
22	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
23	C₄⋊#595	—	2	2	no	—
24	C₄⋊#595	—	2	2	no	—
25	C₄⋊#595	C₂×C₄	1	4	yes	—
26	C₄⋊#595	C₂×C₄	1	4	yes	—
27	C₄×#595	—	4	4	no	—
28	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₇	Q₈	1	2	yes	—
33	C₂×#3622	C₂×C₄	1	1	yes	characteristic
34	C₂×#3622	C₂×C₄	1	1	yes	characteristic
35	C₈×D₁₇	—	2	1	no	—
36	C₈×D₁₇	—	2	1	no	—
37	C₈×D₁₇	C₂×C₄	1	2	yes	—
38	C₈×D₁₇	C₂×C₄	1	2	yes	—
39	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×#595	—	2	4	no	—
41	C₂×#595	—	2	4	no	—
42	C₂×#595	—	4	2	no	—
43	C₂×#595	—	4	2	no	—
44	C₂×#595	—	4	2	no	—
45	C₂×#595	—	4	2	no	—
46	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
49	C₂×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₄×D₁₇	—	2	2	no	—
51	C₄×D₁₇	—	2	2	no	—
52	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₄×D₁₇	#41	1	1	yes	characteristic
55	C₄×D₁₇	#41	1	1	yes	characteristic
56	C₈×C₁₇	—	2	1	no	—
57	C₈×C₁₇	C₄²	1	2	yes	complemented
58	#3622	—	2	1	no	—
59	#3622	C₄²	1	2	yes	complemented
60	#1776	—	17	1	no	maximal, Sylow
61	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
63	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
64	C₂×D₁₇	AGL(1,ℤ/8)	1	2	yes	—
65	C₂×D₁₇	#137	1	2	yes	—
66	#595	—	4	4	no	—
67	#595	—	4	4	no	—
68	C₄×C₁₇	—	2	2	no	—
69	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
70	C₄×C₁₇	#117	1	1	yes	characteristic
71	Dic₁₇	—	2	2	no	—
72	Dic₁₇	C₄×Q₈	1	1	yes	characteristic
73	Dic₁₇	C₄×Q₈	1	1	yes	characteristic
74	C₂×#107	—	17	1	no	—
75	C₂×#106	—	17	1	no	—
76	C₂×#97	—	17	1	no	—
77	C₄×C₄⋊C₄	—	17	2	no	—
78	#340	—	17	2	no	—
79	C₂×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₁₇	#428	1	2	yes	—
81	D₁₇	#440	1	2	yes	—
82	D₁₇	#441	1	1	yes	characteristic
83	D₁₇	#442	1	1	yes	characteristic
84	C₂²×C₈	—	17	1	no	—
85	C₂²×C₈	—	17	1	no	—
86	C₂×C₄⋊C₄	—	17	2	no	—
87	C₂×C₄⋊C₄	—	17	2	no	—
88	C₂×C₄²	—	17	1	no	—
89	C₂×C₄²	—	34	2	no	—
90	#107	—	17	4	no	—
91	#106	—	17	4	no	—
92	#97	—	17	1	no	—
93	#97	—	17	1	no	—
94	#97	—	17	1	no	—
95	#97	#595	1	1	yes	characteristic, complemented
96	#95	—	34	2	no	—
97	C₁₇	#1776	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₄	—	17	1	no	—
99	C₂²×C₄	—	17	2	no	—
100	C₂²×C₄	—	34	2	no	—
101	C₂²×C₄	—	34	2	no	—
102	C₂×C₈	—	17	1	no	—
103	C₂×C₈	—	17	2	no	—
104	C₂×C₈	—	17	2	no	—
105	C₂×C₈	—	17	1	no	—
106	C₂×C₈	—	34	1	no	—
107	C₂×C₈	—	34	1	no	—
108	C₂×C₈	C₂×#595	1	1	yes	characteristic
109	C₂×C₈	C₂×#595	1	1	yes	characteristic
110	C₄⋊C₄	—	17	4	no	—
111	C₄⋊C₄	—	17	4	no	—
112	C₄⋊C₄	—	34	2	no	—
113	C₄⋊C₄	—	34	2	no	—
114	C₄²	—	17	2	no	—
115	C₄²	—	17	1	no	—
116	C₄²	—	68	4	no	—
117	C₄²	C₂×#595	1	1	yes	characteristic
118	C₂³	—	17	1	no	—
119	C₂×C₄	—	17	1	no	—
120	C₂×C₄	—	17	1	no	—
121	C₂×C₄	—	17	1	no	—
122	C₂×C₄	—	17	1	no	—
123	C₂×C₄	—	17	1	no	—
124	C₂×C₄	—	17	2	no	—
125	C₂×C₄	—	34	2	no	—
126	C₂×C₄	—	34	2	no	—
127	C₂×C₄	—	34	4	no	—
128	C₂×C₄	—	34	4	no	—
129	C₂×C₄	—	68	2	no	—
130	C₂×C₄	—	68	2	no	—
131	C₂×C₄	—	68	2	no	—
132	C₂×C₄	—	68	2	no	—
133	C₂×C₄	C₄⋊#595	1	2	yes	—
134	C₂×C₄	C₂²×#595	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	—	17	2	no	—
137	C₈	—	34	1	no	—
138	C₈	C₄×#595	1	2	yes	complemented
139	C₂²	—	17	1	no	—
140	C₂²	—	17	1	no	—
141	C₂²	—	17	2	no	—
142	C₂²	—	17	2	no	—
143	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	17	1	no	—
146	C₄	—	17	1	no	—
147	C₄	—	34	2	no	—
148	C₄	—	68	4	no	—
149	C₄	—	68	4	no	—
150	C₄	C₂×C₄×#595	1	1	yes	characteristic
151	C₄	#10587677	1	1	yes	characteristic
152	C₂	—	17	1	no	—
153	C₂	—	17	2	no	—
154	C₂	—	17	1	no	—
155	C₂	C₄×C₄⋊#595	1	1	yes	characteristic
156	C₂	#11763820	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587534

C₂

yes

maximal, characteristic

C₂×#10587533

C₂

yes

maximal, characteristic

C₄×C₄⋊#595

C₂

yes

maximal

#11763683

C₂

yes

maximal

#97×D₁₇

C₂

yes

maximal, characteristic

C₂×C₄⋊#595

C₂²

yes

—

C₂×C₄⋊#595

C₂²

yes

—

C₂×C₄×#595

—

C₂×C₈×D₁₇

C₂²

yes

characteristic

C₂×C₈×D₁₇

C₂²

yes

characteristic

C₄²×D₁₇

C₂²

yes

characteristic

(C₂×#595)⋊C₄

—

#10587534

C₄

yes

complemented

#10587533

C₄

yes

complemented

C₁₇×#97

C₄

yes

characteristic, complemented

#10587486

C₄

yes

characteristic, complemented

#10587486

C₄

yes

characteristic, complemented

#10587474

C₄

yes

characteristic, complemented

C₂²×#595

—

C₂²×#595

—

C₂×C₄×D₁₇

D₄

yes

—

C₂×C₄×D₁₇

C₂³

yes

—

—

C₂×C₄

yes

—

C₄⋊#595

C₂×C₄

yes

—

C₄×#595

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

Q₈

yes

—

C₂×#3622

C₂×C₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

C₂×C₄

yes

—

C₈×D₁₇

C₂×C₄

yes

—

C₂²×D₁₇

C₂×D₄

yes

—

—

—

—

—

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₂×Dic₁₇

C₂×Q₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂²×C₄

yes

characteristic

C₄×D₁₇

C₂²×C₄

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

C₈×C₁₇

—

C₈×C₁₇

C₄²

yes

complemented

#3622

—

#3622

C₄²

yes

complemented

#1776

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₇

C₄×D₄

yes

characteristic

C₂×D₁₇

C₄×D₄

yes

characteristic

C₂×D₁₇

AGL(1,ℤ/8)

yes

—

C₂×D₁₇

#137

yes

—

#595

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄²

yes

characteristic

C₄×C₁₇

#117

yes

characteristic

Dic₁₇

—

Dic₁₇

C₄×Q₈

yes

characteristic

Dic₁₇

C₄×Q₈

yes

characteristic

C₂×#107

—

C₂×#106

—

C₂×#97

—

C₄×C₄⋊C₄

—

#340

—

C₂×C₁₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₇

#428

yes

—

D₁₇

#440

yes

—

D₁₇

yes

D₁₇

yes

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

—

—

—

—

—

yes

characteristic, complemented

#95

—

C₁₇

#1776

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂×#595

yes

characteristic

109

C₂×C₈

C₂×#595

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₂×#595

yes

characteristic

118

C₂³

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄⋊#595

yes

—

134

C₂×C₄

C₂²×#595

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

C₄×#595

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×C₄⋊#595

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₂×C₄×#595

yes

characteristic

151

C₄

#10587677

yes

characteristic

152

C₂

—

153

C₂

—

154

C₂

—

155

C₂

C₄×C₄⋊#595

yes

characteristic

156

C₂

#11763820

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#595
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×#97	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₇	C₂×D₄
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	#97	#595
hypocenter	C₁₇	#1776
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#595

commutator

C₄×C₁₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×#97

C₄

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₇

C₂×D₄

cosocle

C₂×C₄×D₁₇

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
13. #10587534	C₄	semidirect
14. #10587533	C₄	semidirect
15. C₁₇×#97	C₄	semidirect
16. #10587486	C₄	semidirect
17. #10587486	C₄	semidirect
18. #10587474	C₄	semidirect
57. C₈×C₁₇	C₄²	semidirect
59. #3622	C₄²	semidirect
95. #97	#595	semidirect
97. C₁₇	#1776	semidirect
138. C₈	C₄×#595	semidirect

H ≅ G/N

type

13. #10587534

C₄

semidirect

14. #10587533

C₄

semidirect

15. C₁₇×#97

C₄

semidirect

16. #10587486

C₄

semidirect

17. #10587486

C₄

semidirect

18. #10587474

C₄

semidirect

57. C₈×C₁₇

C₄²

semidirect

59. #3622

C₄²

semidirect

95. #97

#595

semidirect

97. C₁₇

#1776

semidirect

138. C₈

C₄×#595

semidirect

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

subgroup	index
C₂×#10587534	2
C₂×#10587533	2
C₄×C₄⋊#595	2
#11763683	2
#97×D₁₇	2
#1776	17

subgroup

index

C₄×C₄⋊#595

#97×D₁₇

Subgroups — #14544598

Conjugacy classes of subgroups

Complements

Maximal subgroups