Group #17981472 — subgroups

Subgroups — #17981472

Order 4352.

182 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 71 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×#11763918	C₂	1	2	yes	maximal
2	C₂×#11763696	C₂	1	1	yes	maximal, characteristic
3	#11763918	—	2	2	no	—
4	#11763918	—	2	2	no	—
5	C₂×#10587528	C₄	1	4	yes	complemented
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	—	2	1	no	—
25	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
26	C₂×#3624	—	2	1	no	—
27	C₄×#595	—	2	1	no	—
28	C₄×#595	C₁₆	1	2	yes	—
29	#63242	—	2	1	no	—
30	#63242	C₁₆	1	2	yes	—
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	#7466	—	17	1	no	maximal, Sylow
42	C₂²×D₁₇	M₃₂	1	1	yes	characteristic
43	C₂×#595	—	2	2	no	—
44	C₂×#595	—	4	1	no	—
45	C₂×#595	—	4	1	no	—
46	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
47	C₂×Dic₁₇	M₃₂	1	1	yes	characteristic
48	C₄×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
49	C₄×D₁₇	C₄⋊C₈	1	1	yes	characteristic
50	C₄×D₁₇	C₂×C₁₆	1	1	yes	characteristic
51	C₄×D₁₇	C₂×C₁₆	1	1	yes	characteristic
52	#3624	—	2	2	no	—
53	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
54	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
55	#3622	#95	1	1	yes	characteristic
56	#3622	#97	1	1	yes	characteristic
57	C₂×M₆₄	—	17	2	no	—
58	C₂×#346	—	17	1	no	—
59	C₂²×C₁₇	#346	1	1	yes	characteristic
60	C₂×D₁₇	—	2	1	no	—
61	C₂×D₁₇	—	2	1	no	—
62	C₂×D₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
63	C₂×D₁₇	C₄⋊C₁₆	1	1	yes	characteristic
64	#595	—	4	2	no	—
65	C₄×C₁₇	#336	1	1	yes	characteristic
66	C₄×C₁₇	C₄×C₁₆	1	1	yes	characteristic
67	Dic₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
68	Dic₁₇	C₄⋊C₁₆	1	1	yes	characteristic
69	C₂²×C₁₆	—	17	1	no	—
70	C₂²×C₁₆	—	17	1	no	—
71	C₂×C₄×C₈	—	17	1	no	—
72	M₆₄	—	34	2	no	—
73	M₆₄	—	34	2	no	—
74	C₂×C₃₂	—	17	4	no	—
75	#346	—	34	2	no	—
76	C₂×C₁₇	#1374	1	1	yes	characteristic
77	C₂×C₁₇	#1398	1	1	yes	characteristic
78	C₂×C₁₇	#1398	1	1	yes	characteristic
79	D₁₇	—	2	1	no	—
80	D₁₇	—	2	1	no	—
81	C₂²×C₈	—	17	1	no	—
82	C₂²×C₈	—	17	1	no	—
83	C₂×C₄²	—	17	1	no	—
84	C₂×C₁₆	—	17	2	no	—
85	C₂×C₁₆	—	17	2	no	—
86	C₂×C₁₆	—	17	1	no	—
87	C₂×C₁₆	—	17	1	no	—
88	C₂×C₁₆	—	17	1	no	—
89	C₂×C₁₆	—	34	1	no	—
90	C₂×C₁₆	—	34	1	no	—
91	C₂×C₁₆	#3633	1	1	yes	characteristic
92	C₄×C₈	—	34	1	no	—
93	C₄×C₈	—	34	1	no	—
94	C₃₂	—	34	4	no	—
95	C₁₇	#7466	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₄	—	17	1	no	—
97	C₂²×C₄	—	34	1	no	—
98	C₂×C₈	—	17	1	no	—
99	C₂×C₈	—	17	1	no	—
100	C₂×C₈	—	17	1	no	—
101	C₂×C₈	—	17	1	no	—
102	C₂×C₈	—	17	1	no	—
103	C₂×C₈	—	17	2	no	—
104	C₂×C₈	—	34	1	no	—
105	C₂×C₈	—	34	1	no	—
106	C₂×C₈	C₂×#3633	1	1	yes	characteristic
107	C₄²	—	17	2	no	—
108	C₄²	—	34	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	17	2	no	—
111	C₁₆	—	17	2	no	—
112	C₁₆	—	34	1	no	—
113	C₂³	—	17	1	no	—
114	C₂×C₄	—	17	1	no	—
115	C₂×C₄	—	17	1	no	—
116	C₂×C₄	—	17	1	no	—
117	C₂×C₄	—	17	1	no	—
118	C₂×C₄	—	17	1	no	—
119	C₂×C₄	—	34	2	no	—
120	C₂×C₄	—	68	1	no	—
121	C₂×C₄	—	68	1	no	—
122	C₂×C₄	C₄×#3633	1	1	yes	characteristic
123	C₈	—	17	1	no	—
124	C₈	—	17	1	no	—
125	C₈	—	34	2	no	—
126	C₈	C₄⋊#3633	1	1	yes	characteristic
127	C₈	C₂²⋊#3633	1	1	yes	characteristic
128	C₂²	—	17	1	no	—
129	C₂²	—	17	1	no	—
130	C₂²	—	34	1	no	—
131	C₂²	—	34	1	no	—
132	C₂²	#11763904	1	1	yes	characteristic
133	C₄	—	17	1	no	—
134	C₄	—	17	1	no	—
135	C₄	—	68	2	no	—
136	C₄	(C₂×#3633)⋊C₄	1	1	yes	characteristic
137	C₄	#11763902	1	1	yes	characteristic
138	C₂	—	34	1	no	—
139	C₂	—	34	1	no	—
140	C₂	#14545855	1	1	yes	characteristic
141	C₂	#14545874	1	1	yes	characteristic
142	C₂	#14545875	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763918

C₂

yes

maximal

C₂×#11763696

C₂

yes

maximal, characteristic

#11763918

—

#11763918

—

C₂×#10587528

C₄

yes

complemented

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

—

C₂×C₄×D₁₇

C₂×C₈

yes

—

—

C₁₆

yes

—

#63242

—

#63242

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

—

#7466

—

maximal, Sylow

C₂²×D₁₇

M₃₂

yes

—

—

—

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×Dic₁₇

M₃₂

yes

characteristic

C₄×D₁₇

C₂²⋊C₈

yes

characteristic

C₄×D₁₇

C₄⋊C₈

yes

characteristic

C₄×D₁₇

C₂×C₁₆

yes

characteristic

C₄×D₁₇

C₂×C₁₆

yes

characteristic

#3624

—

C₈×C₁₇

C₂²⋊C₈

yes

characteristic

C₈×C₁₇

C₄⋊C₈

yes

yes

yes

C₂×M₆₄

—

C₂×#346

—

C₂²×C₁₇

#346

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

C₂²⋊C₁₆

yes

characteristic

C₂×D₁₇

C₄⋊C₁₆

yes

characteristic

#595

—

C₄×C₁₇

#336

yes

characteristic

C₄×C₁₇

C₄×C₁₆

yes

characteristic

Dic₁₇

C₂²⋊C₁₆

yes

characteristic

Dic₁₇

C₄⋊C₁₆

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

#346

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

D₁₇

—

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

#7466

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×#3633

yes

characteristic

107

C₄²

—

108

C₄²

—

109

C₁₆

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₂³

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₄×#3633

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

C₄⋊#3633

yes

characteristic

127

C₈

C₂²⋊#3633

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#11763904

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

(C₂×#3633)⋊C₄

yes

characteristic

137

C₄

#11763902

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#14545855

yes

characteristic

141

C₂

#14545874

yes

characteristic

142

C₂

#14545875

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#3633
commutator	C₄×C₁₇	C₄×C₁₆
Frattini	C₂×C₈	C₂×#3633
Fitting	C₂×C₁₆×C₁₇	C₈
socle	C₂²×C₁₇	#346
prime radical	C₂²×D₁₇	M₃₂
cosocle	C₂×#10587529	C₂²
hypercenter	C₂×C₁₆	#3633
hypocenter	C₁₇	#7466
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

subgroup

G/H

center

C₂×C₄

C₄×#3633

commutator

C₄×C₁₇

C₄×C₁₆

Frattini

C₂×C₈

C₂×#3633

Fitting

C₂×C₁₆×C₁₇

C₈

socle

C₂²×C₁₇

#346

prime radical

C₂²×D₁₇

M₃₂

C₂²

C₂×C₁₆

C₁₇

G ▷ C₄×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

N	H ≅ G/N	type
5. C₂×#10587528	C₄	semidirect
95. C₁₇	#7466	semidirect

H ≅ G/N

type

5. C₂×#10587528

C₄

semidirect

95. C₁₇

#7466

semidirect

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

subgroup	index
C₂×#11763918	2
C₂×#11763696	2
#7466	17

subgroup

index

C₂×#11763918

C₂×#11763696

#7466

Subgroups — #17981472

Conjugacy classes of subgroups

Complements

Maximal subgroups