Group #17961660 — subgroups

Subgroups — #17961660

Order 4352.

382 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 152 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×#11763732	C₂	1	4	yes	maximal
2	#14545116	C₂	1	2	yes	maximal
3	C₄²×#3622	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587538	C₂²	1	4	yes	—
5	#11763732	—	2	8	no	—
6	C₄⋊#63216	—	2	4	no	—
7	C₂×C₄×#3622	C₄	1	2	yes	—
8	C₂×C₄×#3622	C₄	1	1	yes	characteristic
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	C₂×C₄×#3622	C₂²	1	2	yes	—
11	C₄³×C₁₇	C₄	1	1	yes	characteristic
12	C₂×C₄²×C₁₇	—	2	2	no	—
13	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
16	C₂²×#3622	C₂×C₄	1	2	yes	—
17	C₂²×#3622	C₂³	1	1	yes	characteristic
18	#10587538	—	2	8	no	—
19	C₂×#63216	—	2	4	no	—
20	C₄×#3622	—	2	4	no	—
21	C₄×#3622	C₈	1	2	yes	—
22	C₄×#3622	C₈	1	2	yes	—
23	C₄×#3622	D₄	1	2	yes	—
24	C₄×#3622	D₄	1	4	yes	—
25	C₄×#3622	Q₈	1	4	yes	—
26	C₄×#3622	Q₈	1	2	yes	—
27	C₂²×C₄×C₁₇	—	2	2	no	—
28	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
29	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₁₇	—	2	8	no	—
31	C₄²×C₁₇	—	2	2	no	—
32	C₄²×C₁₇	C₄⋊C₄	1	4	yes	—
33	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
34	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
36	C₂×#3622	—	2	2	no	—
37	C₂×#3622	C₄⋊C₄	1	2	yes	—
38	C₂×#3622	C₄⋊C₄	1	4	yes	—
39	C₂×#3622	C₂×C₈	1	4	yes	—
40	C₂×#3622	M₁₆	1	4	yes	—
41	C₂×#3622	C₂×D₄	1	1	yes	characteristic
42	C₂×#3622	C₂×D₄	1	2	yes	—
43	C₂×#3622	C₂×Q₈	1	2	yes	—
44	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
45	#63216	—	4	4	no	—
46	#12900	—	17	1	no	maximal, Sylow
47	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₇	—	2	8	no	—
49	C₂×C₄×C₁₇	—	2	4	no	—
50	C₂×C₄×C₁₇	—	2	4	no	—
51	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
53	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
57	#3622	—	2	2	no	—
58	#3622	C₄⋊C₈	1	8	yes	—
59	#3622	#127	1	1	yes	characteristic
60	#3622	C₄⋊Q₈	1	1	yes	characteristic
61	#3622	C₄⋊Q₈	1	2	yes	—
62	C₂×#364	—	17	4	no	—
63	#2150	—	17	2	no	—
64	C₄²×C₈	—	17	1	no	—
65	C₂²×C₁₇	—	2	2	no	—
66	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₂²×C₁₇	#423	1	2	yes	—
68	C₄×C₁₇	—	2	8	no	—
69	C₄×C₁₇	—	2	2	no	—
70	C₄×C₁₇	#364	1	4	yes	complemented
71	C₄×C₁₇	#384	1	1	yes	characteristic
72	C₄×C₁₇	C₂×C₄⋊C₈	1	2	yes	—
73	C₄×C₁₇	#423	1	1	yes	characteristic
74	C₂×M₃₂	—	17	4	no	—
75	C₂×C₄×C₈	—	17	2	no	—
76	C₂×C₄×C₈	—	17	1	no	—
77	C₂×C₄×C₈	—	17	1	no	—
78	C₂×C₄×C₈	—	17	2	no	—
79	C₄³	#595	1	1	yes	characteristic
80	#364	—	34	8	no	—
81	C₄⋊C₁₆	—	34	4	no	—
82	C₂×C₁₇	—	2	2	no	—
83	C₂×C₁₇	#1842	1	1	yes	characteristic
84	C₂×C₁₇	C₂×#364	1	2	yes	—
85	C₂²×C₈	—	17	2	no	—
86	C₂²×C₈	—	17	1	no	—
87	C₂²×C₈	—	17	1	no	—
88	C₂×C₄²	—	2	2	no	—
89	C₂×C₄²	C₂×#595	1	2	yes	—
90	C₂×C₄²	C₂×#595	1	1	yes	characteristic
91	M₃₂	—	34	8	no	—
92	C₂×C₁₆	—	34	4	no	—
93	C₄×C₈	—	17	2	no	—
94	C₄×C₈	—	17	2	no	—
95	C₄×C₈	—	17	2	no	—
96	C₄×C₈	—	17	4	no	—
97	C₄×C₈	—	17	4	no	—
98	C₄×C₈	—	17	2	no	—
99	C₄×C₈	—	34	4	no	—
100	C₁₇	#12900	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	#63247	1	2	yes	—
103	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
104	C₂×C₈	—	17	4	no	—
105	C₂×C₈	—	17	4	no	—
106	C₂×C₈	—	17	2	no	—
107	C₂×C₈	—	17	4	no	—
108	C₂×C₈	—	17	2	no	—
109	C₂×C₈	—	17	1	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	2	no	—
112	C₂×C₈	—	34	2	no	—
113	C₄²	—	2	8	no	—
114	C₄²	—	2	2	no	—
115	C₄²	#63242	1	1	yes	characteristic
116	C₄²	#63242	1	1	yes	characteristic
117	C₄²	C₄⋊#595	1	4	yes	—
118	C₄²	C₄⋊#595	1	2	yes	—
119	C₁₆	—	68	4	no	—
120	C₂³	C₂×#63247	1	1	yes	characteristic
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	#10587544	1	2	yes	—
125	C₂×C₄	#10587544	1	2	yes	—
126	C₂×C₄	C₂×#63242	1	1	yes	characteristic
127	C₂×C₄	D₁₇⋊M₁₆	1	2	yes	—
128	C₂×C₄	C₂×C₄⋊#595	1	2	yes	—
129	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
130	C₈	—	17	1	no	—
131	C₈	—	17	2	no	—
132	C₈	—	17	1	no	—
133	C₈	—	17	8	no	—
134	C₈	—	34	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×#10587544	1	1	yes	characteristic
137	C₂²	#11763812	1	2	yes	—
138	C₄	—	2	8	no	—
139	C₄	—	2	2	no	—
140	C₄	#11763732	1	4	yes	complemented
141	C₄	#11763755	1	1	yes	characteristic
142	C₄	#11763810	1	2	yes	—
143	C₄	#11763814	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	#14544717	1	1	yes	characteristic
146	C₂	C₂×#11763732	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763732

C₂

yes

maximal

#14545116

C₂

yes

maximal

C₄²×#3622

C₂

yes

maximal, characteristic

C₂×#10587538

C₂²

yes

—

#11763732

—

C₄⋊#63216

—

C₂×C₄×#3622

C₄

yes

—

C₂×C₄×#3622

C₄

yes

characteristic

C₂×C₄×#3622

C₂²

yes

characteristic

C₂×C₄×#3622

C₂²

yes

—

C₄³×C₁₇

C₄

yes

characteristic

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂²×#3622

C₂×C₄

yes

characteristic

C₂²×#3622

C₂×C₄

yes

—

C₂²×#3622

C₂³

yes

—

—

—

C₈

yes

—

C₄×#3622

C₈

yes

—

C₄×#3622

D₄

yes

—

C₄×#3622

D₄

yes

—

C₄×#3622

Q₈

yes

—

C₄×#3622

Q₈

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

M₁₆

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₂×C₈

yes

—

C₂×#3622

M₁₆

yes

—

C₂×#3622

C₂×D₄

yes

characteristic

C₂×#3622

C₂×D₄

yes

—

C₂×#3622

C₂×Q₈

yes

—

C₂×#3622

C₂×Q₈

yes

—

—

C₂³×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

#3622

—

#3622

C₄⋊C₈

yes

—

yes

C₄⋊Q₈

yes

characteristic

#3622

C₄⋊Q₈

yes

—

C₂×#364

—

#2150

—

C₄²×C₈

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₇

#423

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×C₄⋊C₈

yes

—

C₄×C₁₇

#423

yes

characteristic

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

#595

yes

characteristic

#364

—

C₄⋊C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

#1842

yes

characteristic

C₂×C₁₇

C₂×#364

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₂×#595

yes

—

C₂×C₄²

C₂×#595

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₁₇

#12900

yes

characteristic, complemented, Sylow

101

C₂²×C₄

—

102

C₂²×C₄

#63247

yes

—

103

C₂²×C₄

C₂²×#595

yes

characteristic

104

C₂×C₈

—

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

C₄²

—

114

C₄²

—

115

C₄²

#63242

yes

characteristic

116

C₄²

#63242

yes

characteristic

117

C₄²

C₄⋊#595

yes

—

118

C₄²

C₄⋊#595

yes

—

119

C₁₆

—

120

C₂³

C₂×#63247

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#10587544

yes

—

125

C₂×C₄

#10587544

yes

—

126

C₂×C₄

C₂×#63242

yes

characteristic

127

C₂×C₄

D₁₇⋊M₁₆

yes

—

128

C₂×C₄

C₂×C₄⋊#595

yes

—

129

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

C₂×#10587544

yes

characteristic

137

C₂²

#11763812

yes

—

138

C₄

—

139

C₄

—

140

C₄

#11763732

yes

complemented

141

C₄

#11763755

yes

characteristic

142

C₄

#11763810

yes

—

143

C₄

#11763814

yes

characteristic

144

C₂

—

145

C₂

#14544717

yes

characteristic

146

C₂

C₂×#11763732

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₄³×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×#3622	C₂³
hypercenter	C₄³	#595
hypocenter	C₁₇	#12900
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#595

Fitting

C₄³×C₁₇

C₄

socle

C₂²×C₁₇

C₂×C₄⋊C₈

prime radical

C₂³×C₁₇

C₂×M₁₆

cosocle

C₂²×#3622

C₂³

C₄³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
70. C₄×C₁₇	#364	semidirect
100. C₁₇	#12900	semidirect
140. C₄	#11763732	semidirect

H ≅ G/N

type

70. C₄×C₁₇

#364

semidirect

100. C₁₇

#12900

semidirect

140. C₄

#11763732

semidirect

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

subgroup	index
C₂×#11763732	2
#14545116	2
C₄²×#3622	2
#12900	17

subgroup

index

C₂×#11763732

#14545116

C₄²×#3622

#12900

Subgroups — #17961660

Conjugacy classes of subgroups

Complements

Maximal subgroups