Group #17960271 — subgroups

Subgroups — #17960271

Order 4352.

200 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 86 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	#14545199	C₂	1	2	yes	maximal
2	#14543806	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#63216	C₂	1	2	yes	maximal
4	#14543697	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#3622	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#63216	—	2	2	no	—
7	C₄⋊#63216	C₂²	1	2	yes	—
8	#11763697	—	2	2	no	—
9	C₄×#63216	C₂²	1	1	yes	characteristic
10	C₄×#63216	C₂²	1	1	yes	characteristic
11	Q₈×#3622	C₂²	1	2	yes	—
12	#11762562	C₄	1	1	yes	characteristic
13	Q₈⋊#3622	C₄	1	2	yes	—
14	C₈×#3622	C₂²	1	1	yes	characteristic
15	C₄×Q₁₆×C₁₇	C₄	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₇	C₈	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
18	C₂×#63216	—	2	2	no	—
19	C₂×#63216	—	2	1	no	—
20	C₂×#63216	D₄	1	2	yes	—
21	C₁₇×#107	C₈	1	1	yes	characteristic
22	C₁₇×#103	C₈	1	2	yes	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄⋊#3622	—	2	2	no	—
25	C₄⋊#3622	C₂×C₄	1	2	yes	—
26	C₄×#3622	—	2	2	no	—
27	C₄×#3622	C₂×C₄	1	1	yes	characteristic
28	C₄×#3622	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
30	Q₁₆×C₁₇	—	2	2	no	—
31	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₄⋊C₄×C₁₇	—	2	2	no	—
34	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
35	C₄²×C₁₇	—	2	2	no	—
36	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×#3622	—	2	2	no	—
38	C₂×#3622	—	2	1	no	—
39	C₂×#3622	C₂×D₄	1	1	yes	characteristic
40	C₂×#3622	#41	1	1	yes	characteristic
41	#63216	—	2	2	no	—
42	#63216	—	4	1	no	—
43	#63216	—	4	2	no	—
44	#12061	—	17	1	no	maximal, Sylow
45	Q₈×C₁₇	—	2	2	no	—
46	Q₈×C₁₇	—	2	2	no	—
47	C₂×C₄×C₁₇	—	2	2	no	—
48	C₂×C₄×C₁₇	—	2	2	no	—
49	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
52	C₈×C₁₇	—	2	1	no	—
53	C₈×C₁₇	M₃₂	1	2	yes	—
54	#3622	—	2	1	no	—
55	#3622	—	4	1	no	—
56	#3622	—	4	2	no	—
57	#3622	#137	1	2	yes	—
58	#2183	—	17	2	no	—
59	C₈×Q₁₆	—	17	1	no	—
60	#1367	—	17	1	no	—
61	Q₈⋊C₁₆	—	17	2	no	—
62	#1312	—	17	1	no	—
63	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	—	4	2	no	—
66	C₄×C₁₇	—	4	2	no	—
67	C₄×C₁₇	#441	1	1	yes	characteristic
68	C₄×C₁₇	#444	1	1	yes	characteristic
69	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
70	C₄×C₁₇	#504	1	1	yes	characteristic
71	C₈×Q₈	—	17	2	no	—
72	C₄×Q₁₆	#595	1	1	yes	characteristic
73	C₄⋊C₁₆	—	17	2	no	—
74	C₄⋊C₁₆	—	34	2	no	—
75	#346	—	34	2	no	—
76	C₄×C₁₆	—	17	1	no	—
77	C₄×C₁₆	—	17	1	no	—
78	#335	—	17	1	no	—
79	Q₈⋊C₈	—	17	2	no	—
80	C₈²	—	17	1	no	—
81	C₂×C₁₇	#1579	1	1	yes	characteristic
82	C₂×C₁₇	#2169	1	1	yes	characteristic
83	C₂×C₁₇	#2171	1	1	yes	characteristic
84	C₂×Q₁₆	#3624	1	1	yes	characteristic
85	C₄×Q₈	C₂×#595	1	2	yes	—
86	C₂×C₁₆	—	17	2	no	—
87	C₂×C₁₆	—	34	1	no	—
88	C₂×C₁₆	—	34	2	no	—
89	#107	#3624	1	1	yes	characteristic
90	C₄⋊C₈	—	17	2	no	—
91	C₄⋊C₈	—	34	2	no	—
92	#103	#3624	1	2	yes	—
93	C₄×C₈	—	17	1	no	—
94	C₄×C₈	—	17	1	no	—
95	C₄×C₈	—	34	2	no	—
96	C₄×C₈	C₂×#595	1	1	yes	characteristic
97	C₁₇	#12061	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₈	C₂×#3624	1	2	yes	—
99	Q₁₆	—	2	2	no	—
100	C₂×C₈	—	17	1	no	—
101	C₂×C₈	—	17	1	no	—
102	C₂×C₈	—	34	1	no	—
103	C₂×C₈	—	34	2	no	—
104	C₂×C₈	C₂×#3624	1	1	yes	characteristic
105	C₂×C₈	C₂×#3624	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×#3624	1	2	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂²×#595	1	1	yes	characteristic
110	C₁₆	—	34	2	no	—
111	C₁₆	—	68	2	no	—
112	C₁₆	—	68	1	no	—
113	Q₈	—	2	2	no	—
114	Q₈	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	#10587678	1	1	yes	characteristic
118	C₂×C₄	D₄×#595	1	1	yes	characteristic
119	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	17	2	no	—
122	C₈	—	34	1	no	—
123	C₈	—	68	1	no	—
124	C₈	—	68	2	no	—
125	C₈	#10587538	1	2	yes	—
126	C₂²	D₄×#3624	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	#11763849	1	1	yes	characteristic
131	C₄	#11763855	1	1	yes	characteristic
132	C₄	C₂×#10587538	1	1	yes	characteristic
133	C₄	#11763867	1	1	yes	characteristic
134	C₂	#14544377	1	1	yes	characteristic
135	C₂	#14545168	1	1	yes	characteristic
136	C₂	#14545174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545199

C₂

yes

maximal

#14543806

C₂

yes

maximal, characteristic

Q₈⋊#63216

C₂

yes

maximal

#14543697

C₂

yes

maximal, characteristic

Q₁₆×#3622

C₂

yes

maximal, characteristic

C₄⋊#63216

—

C₄⋊#63216

C₂²

yes

—

#11763697

—

C₄×#63216

C₂²

yes

characteristic

C₄×#63216

C₂²

yes

characteristic

Q₈×#3622

C₂²

yes

—

#11762562

C₄

yes

characteristic

Q₈⋊#3622

C₄

yes

—

C₈×#3622

C₂²

yes

characteristic

C₄×Q₁₆×C₁₇

C₄

yes

characteristic

C₂×Q₁₆×C₁₇

C₈

yes

characteristic

C₄×Q₈×C₁₇

C₂×C₄

yes

—

C₂×#63216

—

C₂×#63216

—

C₂×#63216

D₄

yes

—

C₁₇×#107

C₈

yes

characteristic

C₁₇×#103

C₈

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄⋊#3622

—

C₄⋊#3622

C₂×C₄

yes

—

C₄×#3622

—

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂³

yes

characteristic

C₂×Q₈×C₁₇

C₂×C₈

yes

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₈

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

—

—

C₂×D₄

yes

yes

—

—

—

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

#131

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

M₃₂

yes

—

—

—

—

yes

—

#2183

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₂²×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×M₃₂

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

#595

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

yes

C₄×Q₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

#3624

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#103

#3624

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#595

yes

characteristic

C₁₇

#12061

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×#3624

yes

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂×#3624

yes

characteristic

105

C₂×C₈

C₂×#3624

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×#3624

yes

—

108

C₄²

—

109

C₄²

C₂²×#595

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#10587678

yes

characteristic

118

C₂×C₄

D₄×#595

yes

characteristic

119

C₂×C₄

C₂²×#3624

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

#10587538

yes

—

126

C₂²

D₄×#3624

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#11763849

yes

characteristic

131

C₄

#11763855

yes

characteristic

132

C₄

C₂×#10587538

yes

characteristic

133

C₄

#11763867

yes

characteristic

134

C₂

#14544377

yes

characteristic

135

C₂

#14545168

yes

characteristic

136

C₂

#14545174

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₄×Q₁₆×C₁₇	C₄
socle, prime radical	C₂²×C₁₇	C₈×D₄
cosocle	C₄×#3622	C₂³
hypercenter	C₄×Q₁₆	#595
hypocenter	C₁₇	#12061
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×#595

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₄²

C₂²×#595

Fitting

C₄×Q₁₆×C₁₇

C₄

socle, prime radical

C₂²×C₁₇

C₈×D₄

C₂³

C₄×Q₁₆

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
97. C₁₇	#12061	semidirect

H ≅ G/N

type

97. C₁₇

#12061

semidirect

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

subgroup	index
#14545199	2
#14543806	2
Q₈⋊#63216	2
#14543697	2
Q₁₆×#3622	2
#12061	17

subgroup

index

Q₁₆×#3622

Subgroups — #17960271

Conjugacy classes of subgroups

Complements

Maximal subgroups