Group #17951717 — subgroups

Subgroups — #17951717

Order 4352.

294 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 96 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	#14544502	C₂	1	1	yes	maximal, characteristic
2	#14543998	C₂	1	1	yes	maximal, characteristic
3	#14543609	C₂	1	2	yes	maximal
4	#14543607	C₂	1	2	yes	maximal
5	#14527069	C₂	1	1	yes	maximal, characteristic
6	#11763827	C₂²	1	2	yes	—
7	#11763768	C₂²	1	1	yes	characteristic
8	#11763745	C₂²	1	1	yes	characteristic
9	#11763674	C₄	1	4	yes	—
10	#11763651	C₄	1	4	yes	complemented
11	#11762860	C₂²	1	2	yes	—
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	#11762530	C₄	1	1	yes	characteristic, complemented
14	C₈×#3622	C₄	1	2	yes	complemented
15	C₁₇×#322	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×#595	C₂×C₄	1	2	yes	—
17	C₂×#63242	C₂×C₄	1	2	yes	—
18	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
19	C₄²×D₁₇	C₂³	1	1	yes	characteristic
20	(C₂×#595)⋊C₄	—	2	2	no	—
21	#10587544	—	2	2	no	—
22	#10587543	—	2	2	no	—
23	#10587542	—	2	2	no	—
24	C₄×#3624	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	#10587491	—	2	2	no	—
28	#10587485	—	2	2	no	—
29	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
30	C₄×#3622	C₂×C₄	1	2	yes	—
31	C₄×#3622	C₂×C₄	1	1	yes	characteristic
32	C₂²×#595	—	2	2	no	—
33	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₇	#41	1	2	yes	—
35	C₂×#3624	—	2	2	no	—
36	C₂×#3624	—	2	4	no	—
37	C₄×#595	—	2	4	no	—
38	#63242	—	2	4	no	—
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₂×C₈×C₁₇	C₄²	1	2	yes	—
41	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	#41	1	2	yes	—
44	C₂×#3622	—	2	2	no	—
45	C₂×#3622	C₄²	1	2	yes	—
46	C₈×D₁₇	—	2	4	no	—
47	#8142	—	17	1	no	maximal, Sylow
48	C₂²×D₁₇	#117	1	1	yes	characteristic
49	C₂×#595	—	4	4	no	—
50	C₂×#595	—	4	2	no	—
51	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#117	1	2	yes	—
53	C₂×Dic₁₇	#117	1	2	yes	—
54	C₂×Dic₁₇	#117	1	1	yes	characteristic
55	C₄×D₁₇	—	4	2	no	—
56	C₄×D₁₇	#131	1	4	yes	—
57	#3624	—	4	4	no	—
58	#3624	—	4	2	no	—
59	C₈×C₁₇	—	2	2	no	—
60	C₈×C₁₇	—	4	2	no	—
61	#3622	—	2	2	no	—
62	#3622	—	4	2	no	—
63	#1746	—	17	1	no	—
64	#1452	—	17	1	no	—
65	#1450	—	17	1	no	—
66	#1275	—	17	2	no	—
67	#1274	—	17	2	no	—
68	C₂²×C₁₇	#376	1	1	yes	characteristic
69	C₂×D₁₇	—	2	2	no	—
70	C₂×D₁₇	#433	1	2	yes	—
71	#595	—	8	2	no	—
72	C₄×C₁₇	—	2	2	no	—
73	C₄×C₁₇	#405	1	2	yes	—
74	Dic₁₇	—	2	2	no	—
75	Dic₁₇	#433	1	2	yes	—
76	#431	—	17	2	no	—
77	#431	—	17	2	no	—
78	C₂×#97	—	17	1	no	—
79	C₂×C₄×C₈	—	17	1	no	—
80	#376	—	17	1	no	—
81	#336	—	17	4	no	—
82	#322	—	17	4	no	—
83	#322	—	17	1	no	—
84	#322	#595	1	1	yes	characteristic, complemented
85	C₈²	—	17	2	no	—
86	C₂×C₁₇	#1746	1	1	yes	characteristic
87	C₂×C₁₇	#1747	1	2	yes	—
88	D₁₇	—	2	2	no	—
89	C₂²×C₈	—	17	2	no	—
90	C₂²×C₈	—	17	2	no	—
91	C₂×C₄²	—	17	1	no	—
92	C₂×C₄²	—	17	2	no	—
93	C₄⋊C₈	—	34	2	no	—
94	C₄⋊C₈	—	34	2	no	—
95	C₂²⋊C₈	—	34	2	no	—
96	C₂²⋊C₈	—	34	2	no	—
97	#97	—	34	2	no	—
98	C₄×C₈	—	17	2	no	—
99	C₄×C₈	—	17	4	no	—
100	C₄×C₈	—	17	1	no	—
101	C₄×C₈	—	17	2	no	—
102	C₄×C₈	C₂×#595	1	2	yes	—
103	C₄×C₈	C₂×#595	1	1	yes	characteristic
104	#95	—	34	2	no	—
105	C₁₇	#8142	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₄	—	17	1	no	—
107	C₂²×C₄	—	17	2	no	—
108	C₂²×C₄	—	34	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	17	2	no	—
111	C₂×C₈	—	34	4	no	—
112	C₂×C₈	—	34	4	no	—
113	C₂×C₈	—	34	4	no	—
114	C₂×C₈	—	34	2	no	—
115	C₂×C₈	—	34	2	no	—
116	C₂×C₈	C₄×#595	1	2	yes	—
117	C₄²	—	17	1	no	—
118	C₄²	—	17	2	no	—
119	C₄²	—	34	4	no	—
120	C₄²	C₂²×#595	1	1	yes	characteristic
121	C₂³	—	17	1	no	—
122	C₂×C₄	—	17	1	no	—
123	C₂×C₄	—	17	4	no	—
124	C₂×C₄	—	17	2	no	—
125	C₂×C₄	—	68	2	no	—
126	C₂×C₄	—	68	2	no	—
127	C₂×C₄	—	68	4	no	—
128	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
129	C₂×C₄	#10587677	1	2	yes	—
130	C₈	—	2	2	no	—
131	C₈	—	4	2	no	—
132	C₈	—	34	2	no	—
133	C₈	—	68	2	no	—
134	C₈	—	68	2	no	—
135	C₈	—	68	4	no	—
136	C₂²	—	17	2	no	—
137	C₂²	—	34	2	no	—
138	C₂²	#11763745	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	17	2	no	—
141	C₄	—	34	2	no	—
142	C₄	—	136	2	no	—
143	C₄	#11763772	1	2	yes	—
144	C₂	—	34	2	no	—
145	C₂	#14544502	1	1	yes	characteristic
146	C₂	#14544506	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544502

C₂

yes

maximal, characteristic

#14543998

C₂

yes

maximal, characteristic

#14543609

C₂

yes

maximal

#14543607

C₂

yes

maximal

#14527069

C₂

yes

maximal, characteristic

#11763827

C₂²

yes

—

#11763768

C₂²

yes

characteristic

#11763745

C₂²

yes

characteristic

#11763674

C₄

yes

—

#11763651

C₄

yes

complemented

#11762860

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762530

C₄

yes

characteristic, complemented

C₈×#3622

C₄

yes

complemented

C₁₇×#322

C₄

yes

characteristic, complemented

C₂×C₄×#595

C₂×C₄

yes

—

C₂×#63242

C₂×C₄

yes

—

C₂×C₈×D₁₇

C₂×C₄

yes

—

C₄²×D₁₇

C₂³

yes

characteristic

(C₂×#595)⋊C₄

—

—

—

—

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587491

—

#10587485

—

C₈×Dic₁₇

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×D₁₇

#41

yes

—

—

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

#41

yes

—

C₂×#3622

—

C₂×#3622

C₄²

yes

—

C₈×D₁₇

—

—

C₂²×D₁₇

yes

—

—

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×Dic₁₇

#117

yes

—

C₂×Dic₁₇

#117

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

#131

yes

—

#3624

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

—

—

—

C₂²×C₁₇

#376

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

#433

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

#405

yes

—

Dic₁₇

—

Dic₁₇

#433

yes

—

#431

—

#431

—

C₂×#97

—

C₂×C₄×C₈

—

—

—

—

—

yes

characteristic, complemented

C₈²

—

C₂×C₁₇

#1746

yes

characteristic

C₂×C₁₇

#1747

yes

—

D₁₇

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

C₂×#595

yes

—

103

C₄×C₈

C₂×#595

yes

characteristic

104

#95

—

105

C₁₇

#8142

yes

characteristic, complemented, Sylow

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄×#595

yes

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂²×#595

yes

characteristic

121

C₂³

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₄×#595

yes

characteristic

129

C₂×C₄

#10587677

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₂²

—

137

C₂²

—

138

C₂²

#11763745

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

136

—

143

C₄

#11763772

yes

—

144

C₂

—

145

C₂

#14544502

yes

characteristic

146

C₂

#14544506

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#322	C₄
socle	C₂²×C₁₇	#376
prime radical	C₂²×D₁₇	#117
cosocle	C₄²×D₁₇	C₂³
hypercenter	#322	#595
hypocenter	C₁₇	#8142
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×C₄×#595

commutator

C₂×C₄×C₁₇

C₂×C₄²

Frattini

C₄²

C₂²×#595

Fitting

C₁₇×#322

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄²×D₁₇

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ #322

N	H ≅ G/N	type
10. #11763651	C₄	semidirect
13. #11762530	C₄	semidirect
14. C₈×#3622	C₄	semidirect
15. C₁₇×#322	C₄	semidirect
84. #322	#595	semidirect
105. C₁₇	#8142	semidirect

H ≅ G/N

type

10. #11763651

C₄

semidirect

13. #11762530

C₄

semidirect

14. C₈×#3622

C₄

semidirect

15. C₁₇×#322

C₄

semidirect

84. #322

#595

semidirect

105. C₁₇

#8142

semidirect

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

subgroup	index
#14544502	2
#14543998	2
#14543609	2
#14543607	2
#14527069	2
#8142	17

subgroup

index

Subgroups — #17951717

Conjugacy classes of subgroups

Complements

Maximal subgroups