Group #17981478 — subgroups

Subgroups — #17981478

Order 4352.

182 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14546015	C₂	1	2	yes	maximal
2	C₂×#11763696	C₂	1	1	yes	maximal, characteristic
3	#11763918	—	2	2	no	—
4	#11763917	C₄	1	4	yes	complemented
5	#11763916	—	2	2	no	—
6	C₂×#10587529	C₂²	1	1	yes	characteristic
7	C₂×C₈×#595	C₄	1	1	yes	characteristic
8	#11763696	—	2	2	no	—
9	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic
10	C₂×C₄×#595	C₈	1	1	yes	characteristic
11	C₂×#63242	C₈	1	1	yes	characteristic
12	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂×#63216	C₂×C₄	1	2	yes	—
14	C₈×#595	—	2	1	no	—
15	C₈×#595	—	2	1	no	—
16	#10587529	D₄	1	1	yes	characteristic
17	#10587529	D₄	1	2	yes	—
18	#10587529	Q₈	1	1	yes	characteristic
19	#10587528	—	2	4	no	—
20	C₂×C₁₆×C₁₇	C₈	1	1	yes	characteristic
21	C₂×#63214	C₈	1	1	yes	characteristic
22	C₁₆×D₁₇	—	2	1	no	—
23	C₁₆×D₁₇	—	2	1	no	—
24	C₂²×#595	C₁₆	1	2	yes	—
25	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
26	C₂×#3624	C₁₆	1	2	yes	—
27	C₄×#595	—	2	1	no	—
28	C₄×#595	—	2	1	no	—
29	#63242	—	2	1	no	—
30	#63242	—	2	1	no	—
31	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₂×#3622	C₄²	1	1	yes	characteristic
33	C₈×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₈×D₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₈×D₁₇	M₁₆	1	1	yes	characteristic
36	C₈×D₁₇	M₁₆	1	1	yes	characteristic
37	#63216	C₂²⋊C₄	1	2	yes	—
38	#63216	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₁₇	—	2	1	no	—
40	#63214	—	2	1	no	—
41	#7468	—	17	1	no	maximal, Sylow
42	C₂²×D₁₇	C₂×C₁₆	1	1	yes	characteristic
43	C₂×#595	—	2	2	no	—
44	C₂×#595	—	4	2	no	—
45	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
46	C₂×Dic₁₇	C₂×C₁₆	1	1	yes	characteristic
47	C₄×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
48	C₄×D₁₇	C₄⋊C₈	1	1	yes	characteristic
49	C₄×D₁₇	M₃₂	1	1	yes	characteristic
50	C₄×D₁₇	M₃₂	1	1	yes	characteristic
51	#3624	—	2	2	no	—
52	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
53	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
54	#3622	#95	1	1	yes	characteristic
55	#3622	#97	1	1	yes	characteristic
56	C₂×M₆₄	—	17	2	no	—
57	C₂×#346	—	17	1	no	—
58	C₂²×C₁₇	C₄×C₁₆	1	1	yes	characteristic
59	C₂×D₁₇	—	2	2	no	—
60	C₂×D₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
61	C₂×D₁₇	C₄⋊C₁₆	1	1	yes	characteristic
62	#595	—	4	2	no	—
63	C₄×C₁₇	#336	1	1	yes	characteristic
64	C₄×C₁₇	#346	1	1	yes	characteristic
65	Dic₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
66	Dic₁₇	C₄⋊C₁₆	1	1	yes	characteristic
67	C₂²×C₁₆	—	17	1	no	—
68	C₂²×C₁₆	—	17	1	no	—
69	C₂×C₄×C₈	—	17	1	no	—
70	M₆₄	—	17	4	no	—
71	M₆₄	—	34	2	no	—
72	C₂×C₃₂	—	34	2	no	—
73	#346	—	34	2	no	—
74	C₂×C₁₇	—	2	1	no	—
75	C₂×C₁₇	#1374	1	1	yes	characteristic
76	D₁₇	—	2	1	no	—
77	D₁₇	#1400	1	2	yes	—
78	C₂²×C₈	—	17	1	no	—
79	C₂²×C₈	—	17	1	no	—
80	C₂×C₄²	—	17	1	no	—
81	C₂×C₁₆	—	17	2	no	—
82	C₂×C₁₆	—	17	2	no	—
83	C₂×C₁₆	—	17	1	no	—
84	C₂×C₁₆	—	17	1	no	—
85	C₂×C₁₆	—	17	1	no	—
86	C₂×C₁₆	—	34	1	no	—
87	C₂×C₁₆	—	34	1	no	—
88	C₂×C₁₆	#3633	1	1	yes	characteristic
89	C₄×C₈	—	34	1	no	—
90	C₄×C₈	—	34	1	no	—
91	C₃₂	—	34	4	no	—
92	C₁₇	#7468	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	17	1	no	—
94	C₂²×C₄	—	17	2	no	—
95	C₂×C₈	—	17	1	no	—
96	C₂×C₈	—	17	1	no	—
97	C₂×C₈	—	17	1	no	—
98	C₂×C₈	—	17	1	no	—
99	C₂×C₈	—	17	1	no	—
100	C₂×C₈	—	17	2	no	—
101	C₂×C₈	—	34	1	no	—
102	C₂×C₈	—	34	1	no	—
103	C₂×C₈	C₂×#3633	1	1	yes	characteristic
104	C₄²	—	34	1	no	—
105	C₄²	—	34	1	no	—
106	C₁₆	—	2	1	no	—
107	C₁₆	—	17	2	no	—
108	C₁₆	—	17	2	no	—
109	C₁₆	—	34	1	no	—
110	C₂³	—	17	1	no	—
111	C₂×C₄	—	17	1	no	—
112	C₂×C₄	—	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	2	no	—
117	C₂×C₄	—	68	2	no	—
118	C₂×C₄	C₄×#3633	1	1	yes	characteristic
119	C₈	—	17	1	no	—
120	C₈	—	17	1	no	—
121	C₈	—	34	2	no	—
122	C₈	C₄⋊#3633	1	1	yes	characteristic
123	C₈	C₂²⋊#3633	1	1	yes	characteristic
124	C₂²	—	17	1	no	—
125	C₂²	—	17	1	no	—
126	C₂²	—	34	2	no	—
127	C₂²	#11763902	1	1	yes	characteristic
128	C₄	—	17	1	no	—
129	C₄	—	17	1	no	—
130	C₄	—	68	2	no	—
131	C₄	(C₂×#3633)⋊C₄	1	1	yes	characteristic
132	C₄	#11763904	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	17	2	no	—
135	C₂	—	34	1	no	—
136	C₂	#14545855	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14546015

C₂

yes

maximal

C₂×#11763696

C₂

yes

maximal, characteristic

#11763918

—

#11763917

C₄

yes

complemented

#11763916

—

C₂×#10587529

C₂²

yes

characteristic

C₂×C₈×#595

C₄

yes

characteristic

#11763696

—

C₂×C₁₆×D₁₇

C₄

yes

characteristic

C₂×C₄×#595

C₈

yes

characteristic

C₂×#63242

C₈

yes

characteristic

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×#63216

C₂×C₄

yes

—

C₈×#595

—

C₈×#595

—

#10587529

D₄

yes

characteristic

#10587529

D₄

yes

—

#10587529

Q₈

yes

characteristic

#10587528

—

C₂×C₁₆×C₁₇

C₈

yes

characteristic

C₂×#63214

C₈

yes

characteristic

C₁₆×D₁₇

—

C₁₆×D₁₇

—

C₂²×#595

C₁₆

yes

—

C₂×C₄×D₁₇

C₂×C₈

yes

characteristic

C₂×#3624

C₁₆

yes

—

—

—

—

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×#3622

C₄²

yes

characteristic

C₈×D₁₇

C₂²⋊C₄

yes

characteristic

C₈×D₁₇

C₄⋊C₄

yes

characteristic

C₈×D₁₇

M₁₆

yes

characteristic

C₈×D₁₇

M₁₆

yes

characteristic

#63216

C₂²⋊C₄

yes

—

#63216

C₄⋊C₄

yes

—

C₁₆×C₁₇

—

#63214

—

#7468

—

maximal, Sylow

C₂²×D₁₇

C₂×C₁₆

yes

characteristic

C₂×#595

—

C₂×#595

—

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×Dic₁₇

C₂×C₁₆

yes

characteristic

C₄×D₁₇

C₂²⋊C₈

yes

characteristic

C₄×D₁₇

C₄⋊C₈

yes

characteristic

C₄×D₁₇

M₃₂

yes

characteristic

C₄×D₁₇

M₃₂

yes

characteristic

#3624

—

C₈×C₁₇

C₂²⋊C₈

yes

characteristic

C₈×C₁₇

C₄⋊C₈

yes

yes

yes

C₂×M₆₄

—

C₂×#346

—

C₂²×C₁₇

C₄×C₁₆

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

C₂²⋊C₁₆

yes

characteristic

C₂×D₁₇

C₄⋊C₁₆

yes

characteristic

#595

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

C₂²⋊C₁₆

yes

characteristic

Dic₁₇

C₄⋊C₁₆

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

#346

—

C₂×C₁₇

—

C₂×C₁₇

#1374

yes

characteristic

D₁₇

—

D₁₇

#1400

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#3633

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₁₇

#7468

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₂×#3633

yes

characteristic

104

C₄²

—

105

C₄²

—

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

—

110

C₂³

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₄×#3633

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

C₄⋊#3633

yes

characteristic

123

C₈

C₂²⋊#3633

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#11763902

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

(C₂×#3633)⋊C₄

yes

characteristic

132

C₄

#11763904

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

—

136

C₂

#14545855

yes

characteristic

	subgroup	G/H
center	C₈	C₂²⋊#3633
commutator	C₂²×C₁₇	C₄×C₁₆
Frattini	C₂×C₈	C₂×#3633
Fitting	C₂×C₁₆×C₁₇	C₈
socle	C₂×C₁₇	#1374
prime radical	C₂²×D₁₇	C₂×C₁₆
cosocle	C₂×#10587529	C₂²
hypercenter	C₂×C₁₆	#3633
hypocenter	C₁₇	#7468
LCS	G ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₈

C₂²⋊#3633

commutator

C₂²×C₁₇

C₄×C₁₆

Frattini

C₂×C₈

C₂×#3633

Fitting

C₂×C₁₆×C₁₇

C₈

socle

C₂×C₁₇

#1374

prime radical

C₂²×D₁₇

C₂×C₁₆

C₂²

C₂×C₁₆

C₁₇

G ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
4. #11763917	C₄	semidirect
92. C₁₇	#7468	semidirect

H ≅ G/N

type

4. #11763917

C₄

semidirect

92. C₁₇

#7468

semidirect

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

subgroup	index
#14546015	2
C₂×#11763696	2
#7468	17

subgroup

index

#14546015

C₂×#11763696

#7468

Subgroups — #17981478

Conjugacy classes of subgroups

Complements

Maximal subgroups