Group #14544586 — subgroups

Subgroups — #14544586

Order 2176.

232 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 90 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	D₁₇⋊#108	C₂	1	1	yes	maximal, characteristic
2	#11763819	C₂	1	1	yes	maximal, characteristic
3	#11763772	C₂	1	2	yes	maximal
4	#11763680	C₂	1	2	yes	maximal
5	C₄×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
6	#10587677	C₂²	1	2	yes	—
7	D₁₇⋊M₁₆	C₂²	1	2	yes	—
8	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
10	C₄²×D₁₇	C₂²	1	1	yes	characteristic
11	#10587536	C₄	1	2	yes	complemented
12	#10587535	C₄	1	2	yes	complemented
13	#10587534	C₄	1	2	yes	complemented
14	#10587533	C₄	1	2	yes	complemented
15	#10587532	—	2	2	no	—
16	C₈×#595	—	2	2	no	—
17	C₄×C₈×C₁₇	C₄	1	1	yes	characteristic, complemented
18	C₈×Dic₁₇	C₄	1	1	yes	characteristic, complemented
19	C₈×Dic₁₇	C₄	1	1	yes	characteristic, complemented
20	C₄×#3622	C₄	1	1	yes	characteristic, complemented
21	C₂×C₄×D₁₇	—	2	1	no	—
22	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
23	C₂²⋊#595	—	2	2	no	—
24	#63247	—	2	2	no	—
25	C₄⋊#595	C₂×C₄	1	4	yes	—
26	C₄×#595	—	2	2	no	—
27	#63243	C₂×C₄	1	4	yes	—
28	#63242	—	2	2	no	—
29	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₇	—	2	1	no	—
33	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×#3622	C₂×C₄	1	1	yes	characteristic
35	C₂×#3622	C₂×C₄	1	1	yes	characteristic
36	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
37	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
39	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
40	C₈×D₁₇	D₄	1	1	yes	characteristic
41	C₈×D₁₇	D₄	1	1	yes	characteristic
42	C₈×D₁₇	Q₈	1	1	yes	characteristic
43	C₈×D₁₇	Q₈	1	1	yes	characteristic
44	C₂²×D₁₇	#41	1	1	yes	characteristic
45	C₂×#595	—	2	4	no	—
46	C₂×C₄×C₁₇	—	2	1	no	—
47	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₇	—	2	1	no	—
49	C₂×Dic₁₇	#41	1	1	yes	characteristic
50	C₄×D₁₇	—	2	4	no	—
51	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
54	C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
55	#3624	—	2	4	no	—
56	C₈×C₁₇	C₄²	1	1	yes	characteristic
57	C₈×C₁₇	C₄²	1	1	yes	characteristic
58	C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
59	C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
60	#3622	C₄²	1	1	yes	characteristic
61	#3622	C₄²	1	1	yes	characteristic
62	#3622	C₄⋊C₄	1	1	yes	characteristic
63	#3622	C₄⋊C₄	1	1	yes	characteristic
64	#1772	—	17	1	no	maximal, Sylow
65	C₂²×C₁₇	#117	1	1	yes	characteristic
66	C₂×D₁₇	—	2	2	no	—
67	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
68	C₂×D₁₇	C₄×Q₈	1	1	yes	characteristic
69	#595	—	4	4	no	—
70	C₄×C₁₇	—	2	2	no	—
71	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	Dic₁₇	—	2	2	no	—
74	Dic₁₇	C₄×D₄	1	1	yes	characteristic
75	Dic₁₇	C₄×Q₈	1	1	yes	characteristic
76	C₂×#108	—	17	1	no	—
77	#427	—	17	1	no	—
78	#405	—	17	2	no	—
79	C₂×C₄×C₈	—	17	1	no	—
80	#339	—	17	2	no	—
81	C₂×C₁₇	—	2	1	no	—
82	C₂×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
83	D₁₇	—	2	1	no	—
84	D₁₇	#443	1	2	yes	—
85	C₂×M₁₆	—	17	2	no	—
86	C₂²×C₈	—	17	1	no	—
87	C₂²×C₈	—	17	1	no	—
88	#117	—	17	2	no	—
89	C₂×C₄²	—	17	1	no	—
90	#108	—	17	2	no	—
91	#108	—	17	2	no	—
92	#107	—	17	2	no	—
93	#106	—	17	2	no	—
94	#97	—	34	2	no	—
95	C₄×C₈	—	17	1	no	—
96	C₄×C₈	—	17	1	no	—
97	C₄×C₈	—	17	1	no	—
98	C₄×C₈	—	34	2	no	—
99	C₄×C₈	#595	1	1	yes	characteristic, complemented
100	C₁₇	#1772	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₄	—	17	1	no	—
102	C₂²×C₄	—	34	1	no	—
103	M₁₆	—	17	4	no	—
104	M₁₆	—	34	2	no	—
105	C₂×C₈	—	17	1	no	—
106	C₂×C₈	—	17	1	no	—
107	C₂×C₈	—	17	1	no	—
108	C₂×C₈	—	17	1	no	—
109	C₂×C₈	—	17	1	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	—	17	1	no	—
114	C₂×C₈	—	17	1	no	—
115	C₂×C₈	—	34	2	no	—
116	C₂×C₈	C₂×#595	1	1	yes	characteristic
117	C₂×C₈	C₂×#595	1	1	yes	characteristic
118	C₄⋊C₄	—	17	4	no	—
119	C₂²⋊C₄	—	34	2	no	—
120	C₄²	—	17	1	no	—
121	C₄²	—	34	1	no	—
122	C₄²	—	34	2	no	—
123	C₄²	C₂×#595	1	1	yes	characteristic
124	C₂³	—	17	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	17	1	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	—	34	4	no	—
132	C₂×C₄	—	34	1	no	—
133	C₂×C₄	—	34	4	no	—
134	C₂×C₄	C₂²×#595	1	1	yes	characteristic
135	C₈	—	17	1	no	—
136	C₈	—	17	1	no	—
137	C₈	—	17	1	no	—
138	C₈	—	17	1	no	—
139	C₈	—	34	4	no	—
140	C₈	C₄×#595	1	1	yes	characteristic
141	C₈	C₄×#595	1	1	yes	characteristic
142	C₈	C₄⋊#595	1	1	yes	characteristic
143	C₈	C₄⋊#595	1	1	yes	characteristic
144	C₂²	—	17	1	no	—
145	C₂²	—	17	1	no	—
146	C₂²	—	34	2	no	—
147	C₂²	#10587677	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	17	1	no	—
150	C₄	—	17	1	no	—
151	C₄	—	34	2	no	—
152	C₄	—	68	4	no	—
153	C₄	C₂×C₄×#595	1	1	yes	characteristic
154	C₄	C₂×C₄⋊#595	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	17	2	no	—
157	C₂	—	34	1	no	—
158	C₂	C₄×C₄⋊#595	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₇⋊#108

C₂

yes

maximal, characteristic

#11763819

C₂

yes

maximal, characteristic

#11763772

C₂

yes

maximal

#11763680

C₂

yes

maximal

C₄×C₈×D₁₇

C₂

yes

maximal, characteristic

#10587677

C₂²

yes

—

D₁₇⋊M₁₆

C₂²

yes

—

C₂×C₈×D₁₇

C₂²

yes

characteristic

C₂×C₈×D₁₇

C₂²

yes

characteristic

C₄²×D₁₇

C₂²

yes

characteristic

#10587536

C₄

yes

complemented

#10587535

C₄

yes

complemented

#10587534

C₄

yes

complemented

#10587533

C₄

yes

complemented

#10587532

—

C₈×#595

—

C₄×C₈×C₁₇

C₄

yes

characteristic, complemented

C₈×Dic₁₇

C₄

yes

characteristic, complemented

C₈×Dic₁₇

C₄

yes

characteristic, complemented

C₄×#3622

C₄

yes

characteristic, complemented

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂²⋊#595

—

#63247

—

C₄⋊#595

C₂×C₄

yes

—

C₄×#595

—

#63243

C₂×C₄

yes

—

#63242

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

C₈×D₁₇

C₂×C₄

yes

characteristic

C₈×D₁₇

C₂×C₄

yes

characteristic

C₈×D₁₇

C₂×C₄

yes

characteristic

C₈×D₁₇

C₂×C₄

yes

characteristic

C₈×D₁₇

D₄

yes

characteristic

C₈×D₁₇

D₄

yes

characteristic

C₈×D₁₇

Q₈

yes

characteristic

C₈×D₁₇

Q₈

yes

C₂²×D₁₇

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#41

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₂²×C₄

yes

characteristic

C₄×D₁₇

C₂²×C₄

yes

characteristic

C₄×D₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

C₂×Q₈

yes

characteristic

#3624

—

C₈×C₁₇

C₄²

yes

characteristic

C₈×C₁₇

C₄²

yes

characteristic

C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₈×C₁₇

C₄⋊C₄

yes

characteristic

#3622

C₄²

yes

characteristic

#3622

C₄²

yes

characteristic

#3622

C₄⋊C₄

yes

characteristic

#3622

C₄⋊C₄

yes

—

C₂²×C₁₇

yes

C₂×D₁₇

—

C₂×D₁₇

C₄×D₄

yes

characteristic

C₂×D₁₇

C₄×Q₈

yes

characteristic

#595

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄²

yes

characteristic

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

Dic₁₇

—

Dic₁₇

C₄×D₄

yes

characteristic

Dic₁₇

C₄×Q₈

yes

—

—

—

C₂×C₄×C₈

—

#339

—

C₂×C₁₇

—

C₂×C₁₇

C₄×C₄⋊C₄

yes

characteristic

D₁₇

—

D₁₇

#443

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

C₂×C₄²

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#595

yes

characteristic, complemented

100

C₁₇

#1772

yes

characteristic, complemented, Sylow

101

C₂²×C₄

—

102

C₂²×C₄

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂×#595

yes

characteristic

117

C₂×C₈

C₂×#595

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₂×#595

yes

characteristic

124

C₂³

—

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

—

134

C₂×C₄

C₂²×#595

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

C₄×#595

yes

characteristic

141

C₈

C₄×#595

yes

characteristic

142

C₈

C₄⋊#595

yes

characteristic

143

C₈

C₄⋊#595

yes

characteristic

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#10587677

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₂×C₄×#595

yes

characteristic

154

C₄

C₂×C₄⋊#595

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

C₄×C₄⋊#595

yes

characteristic

	subgroup	G/H
center	C₈	C₄⋊#595
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#595
Fitting	C₄×C₈×C₁₇	C₄
socle	C₂×C₁₇	C₄×C₄⋊C₄
prime radical	C₂²×D₁₇	#41
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	C₄×C₈	#595
hypocenter	C₁₇	#1772
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₈

C₄⋊#595

commutator

C₄×C₁₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#595

Fitting

C₄×C₈×C₁₇

C₄

socle

C₂×C₁₇

C₄×C₄⋊C₄

prime radical

C₂²×D₁₇

#41

cosocle

C₂×C₄×D₁₇

C₂³

C₄×C₈

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
11. #10587536	C₄	semidirect
12. #10587535	C₄	semidirect
13. #10587534	C₄	semidirect
14. #10587533	C₄	semidirect
17. C₄×C₈×C₁₇	C₄	semidirect
18. C₈×Dic₁₇	C₄	semidirect
19. C₈×Dic₁₇	C₄	semidirect
20. C₄×#3622	C₄	semidirect
99. C₄×C₈	#595	semidirect
100. C₁₇	#1772	semidirect

H ≅ G/N

type

11. #10587536

C₄

semidirect

12. #10587535

C₄

semidirect

13. #10587534

C₄

semidirect

14. #10587533

C₄

semidirect

17. C₄×C₈×C₁₇

C₄

semidirect

18. C₈×Dic₁₇

C₄

semidirect

19. C₈×Dic₁₇

C₄

semidirect

20. C₄×#3622

C₄

semidirect

99. C₄×C₈

#595

semidirect

100. C₁₇

#1772

semidirect

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

subgroup	index
D₁₇⋊#108	2
#11763819	2
#11763772	2
#11763680	2
C₄×C₈×D₁₇	2
#1772	17

subgroup

index

D₁₇⋊#108

C₄×C₈×D₁₇

Subgroups — #14544586

Conjugacy classes of subgroups

Complements

Maximal subgroups