Group #17960240 — subgroups

Subgroups — #17960240

Order 4352.

218 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 88 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	#14545198	C₂	1	2	yes	maximal
2	#14543813	C₂	1	1	yes	maximal, characteristic
3	#14543751	C₂	1	2	yes	maximal
4	C₈×#63216	C₂	1	1	yes	maximal, characteristic
5	#14527590	C₂	1	1	yes	maximal, characteristic
6	#11763730	—	2	2	no	—
7	C₄⋊#63216	C₂²	1	2	yes	—
8	#11763695	—	2	2	no	—
9	C₄×#63216	C₂²	1	1	yes	characteristic
10	C₄×#63216	C₂²	1	1	yes	characteristic
11	#11762933	C₂²	1	2	yes	—
12	C₄×Dic₆₈	C₄	1	1	yes	characteristic
13	#11762539	C₄	1	2	yes	—
14	C₈×#3622	C₂²	1	1	yes	characteristic
15	C₁₇×#335	C₄	1	1	yes	characteristic
16	C₂×Dic₆₈	C₈	1	1	yes	characteristic
17	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
18	C₂×#63216	—	2	1	no	—
19	C₂×#63216	—	2	2	no	—
20	C₂×#63216	D₄	1	1	yes	characteristic
21	C₂×#63216	D₄	1	1	yes	characteristic
22	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	#10587489	C₈	1	1	yes	characteristic
25	#10587487	C₈	1	2	yes	—
26	#10587485	—	2	2	no	—
27	C₈×Dic₁₇	—	2	2	no	—
28	C₄×#3622	C₂×C₄	1	1	yes	characteristic
29	C₄×#3622	C₂³	1	1	yes	characteristic
30	C₂×Dic₃₄	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₁₇	—	2	2	no	—
32	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
36	#63225	—	2	2	no	—
37	C₄×Dic₁₇	—	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	Dic₆₈	C₁₆	1	4	yes	complemented
42	#63216	—	2	2	no	—
43	#63216	—	2	1	no	—
44	#63216	—	4	2	no	—
45	#63216	Q₁₆	1	2	yes	complemented
46	C₁₆×Q₁₆	—	17	1	no	maximal, Sylow
47	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
50	C₂×Dic₁₇	—	2	2	no	—
51	C₂×Dic₁₇	—	2	2	no	—
52	Dic₃₄	—	2	2	no	—
53	Dic₃₄	C₂×C₁₆	1	4	yes	—
54	C₈×C₁₇	—	2	1	no	—
55	C₈×C₁₇	—	4	2	no	—
56	C₈×C₁₇	C₂×C₁₆	1	2	yes	—
57	#3622	—	2	2	no	—
58	#3622	—	2	1	no	—
59	#3622	C₂×Q₁₆	1	1	yes	characteristic
60	#3622	#135	1	1	yes	characteristic
61	Q₈×C₁₆	—	17	2	no	—
62	C₈×Q₁₆	—	17	1	no	—
63	#1370	—	17	1	no	—
64	Q₈⋊C₁₆	—	17	2	no	—
65	C₈×C₁₆	—	17	1	no	—
66	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
67	C₄×C₁₇	—	2	1	no	—
68	C₄×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
69	C₄×C₁₇	#443	1	1	yes	characteristic
70	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
71	C₄×C₁₇	#504	1	1	yes	characteristic
72	Dic₁₇	—	2	4	no	—
73	Dic₁₇	—	4	2	no	—
74	C₈×Q₈	—	17	2	no	—
75	C₄×Q₁₆	—	17	1	no	—
76	C₄⋊C₁₆	—	17	2	no	—
77	C₄⋊C₁₆	—	34	2	no	—
78	C₄×C₁₆	—	17	1	no	—
79	C₄×C₁₆	—	17	1	no	—
80	C₄×C₁₆	—	34	2	no	—
81	#335	#595	1	1	yes	characteristic
82	Q₈⋊C₈	—	17	2	no	—
83	C₈²	—	17	1	no	—
84	C₂×C₁₇	C₈×Q₁₆	1	1	yes	characteristic
85	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
86	C₂×C₁₇	#2170	1	1	yes	characteristic
87	C₂×Q₁₆	—	17	1	no	—
88	C₄×Q₈	—	17	2	no	—
89	C₂×C₁₆	—	17	1	no	—
90	C₂×C₁₆	—	17	1	no	—
91	C₂×C₁₆	—	34	1	no	—
92	C₂×C₁₆	—	34	2	no	—
93	#107	—	17	1	no	—
94	C₄⋊C₈	—	34	2	no	—
95	C₄⋊C₈	C₂×#595	1	2	yes	—
96	#103	—	17	2	no	—
97	C₄×C₈	—	17	1	no	—
98	C₄×C₈	—	17	1	no	—
99	C₄×C₈	—	34	2	no	—
100	C₄×C₈	C₂×#595	1	1	yes	characteristic
101	C₁₇	C₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	17	2	no	—
103	Q₁₆	—	17	4	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	17	1	no	—
106	C₂×C₈	—	17	1	no	—
107	C₂×C₈	—	34	1	no	—
108	C₂×C₈	#63242	1	1	yes	characteristic
109	C₂×C₈	#63242	1	1	yes	characteristic
110	C₄⋊C₄	—	17	2	no	—
111	C₄⋊C₄	—	34	2	no	—
112	C₄²	—	34	2	no	—
113	C₄²	C₂²×#595	1	1	yes	characteristic
114	C₁₆	—	17	2	no	—
115	C₁₆	—	34	1	no	—
116	C₁₆	—	34	2	no	—
117	C₁₆	—	68	2	no	—
118	Q₈	—	17	4	no	—
119	Q₈	—	34	2	no	—
120	C₂×C₄	—	34	2	no	—
121	C₂×C₄	—	34	2	no	—
122	C₂×C₄	C₂×#63242	1	1	yes	characteristic
123	C₂×C₄	D₄×#595	1	1	yes	characteristic
124	C₂×C₄	#10587680	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	4	2	no	—
127	C₈	—	17	1	no	—
128	C₈	—	17	1	no	—
129	C₈	—	34	2	no	—
130	C₈	—	34	1	no	—
131	C₈	#10587529	1	2	yes	—
132	C₂²	#11763837	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	34	4	no	—
135	C₄	—	68	2	no	—
136	C₄	Q₁₆×#595	1	1	yes	characteristic
137	C₄	#11763851	1	1	yes	characteristic
138	C₄	C₂×#10587529	1	1	yes	characteristic
139	C₄	#11763868	1	1	yes	characteristic
140	C₂	#14544372	1	1	yes	characteristic
141	C₂	#14545163	1	1	yes	characteristic
142	C₂	#14545173	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545198

C₂

yes

maximal

#14543813

C₂

yes

maximal, characteristic

#14543751

C₂

yes

maximal

C₈×#63216

C₂

yes

maximal, characteristic

#14527590

C₂

yes

maximal, characteristic

#11763730

—

C₄⋊#63216

C₂²

yes

—

#11763695

—

C₄×#63216

C₂²

yes

characteristic

C₄×#63216

C₂²

yes

characteristic

#11762933

C₂²

yes

—

C₄×Dic₆₈

C₄

yes

characteristic

#11762539

C₄

yes

—

C₈×#3622

C₂²

yes

characteristic

C₁₇×#335

C₄

yes

characteristic

C₂×Dic₆₈

C₈

yes

characteristic

C₄×Dic₃₄

C₂×C₄

yes

—

C₂×#63216

—

C₂×#63216

—

C₂×#63216

D₄

yes

characteristic

C₂×#63216

D₄

yes

characteristic

C₁₇×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587489

C₈

yes

characteristic

#10587487

C₈

yes

—

#10587485

—

C₈×Dic₁₇

—

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂³

yes

characteristic

C₂×Dic₃₄

C₂×C₈

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊Dic₁₇

C₂×C₈

yes

—

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

C₂×D₄

yes

yes

Dic₆₈

C₁₆

yes

—

—

—

Q₁₆

yes

complemented

C₁₆×Q₁₆

—

maximal, Sylow

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

#131

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

C₂×C₁₆

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₁₆

yes

—

#3622

—

#3622

—

#3622

C₂×Q₁₆

yes

yes

Q₈×C₁₆

—

C₈×Q₁₆

—

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₂²×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₄×Q₁₆

yes

characteristic

C₄×C₁₇

#443

yes

characteristic

C₄×C₁₇

C₂²×C₁₆

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

#595

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂×C₁₇

C₈×Q₁₆

yes

characteristic

C₂×C₁₇

D₄×C₁₆

yes

characteristic

C₂×C₁₇

#2170

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×#595

yes

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×#595

yes

characteristic

101

C₁₇

C₁₆×Q₁₆

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

#63242

yes

characteristic

109

C₂×C₈

#63242

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂²×#595

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×#63242

yes

characteristic

123

C₂×C₄

D₄×#595

yes

characteristic

124

C₂×C₄

#10587680

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

#10587529

yes

—

132

C₂²

#11763837

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

Q₁₆×#595

yes

characteristic

137

C₄

#11763851

yes

characteristic

138

C₄

C₂×#10587529

yes

characteristic

139

C₄

#11763868

yes

characteristic

140

C₂

#14544372

yes

characteristic

141

C₂

#14545163

yes

characteristic

142

C₂

#14545173

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#335	C₄
socle, prime radical	C₂²×C₁₇	C₈×D₄
cosocle	C₄×#3622	C₂³
hypercenter	#335	#595
hypocenter	C₁₇	C₁₆×Q₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#595

commutator

C₄×C₁₇

C₂²×C₁₆

Frattini

C₄²

C₂²×#595

Fitting

C₁₇×#335

C₄

socle, prime radical

C₂²×C₁₇

C₈×D₄

C₂³

C₁₇

C₁₆×Q₁₆

LCS

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
41. Dic₆₈	C₁₆	semidirect
45. #63216	Q₁₆	semidirect
101. C₁₇	C₁₆×Q₁₆	semidirect

H ≅ G/N

type

41. Dic₆₈

C₁₆

semidirect

45. #63216

Q₁₆

semidirect

101. C₁₇

C₁₆×Q₁₆

semidirect

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

subgroup	index
#14545198	2
#14543813	2
#14543751	2
C₈×#63216	2
#14527590	2
C₁₆×Q₁₆	17

subgroup

index

C₁₆×Q₁₆

Subgroups — #17960240

Conjugacy classes of subgroups

Complements

Maximal subgroups