Group #17960135 — subgroups

Subgroups — #17960135

Order 4352.

232 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 96 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	#14544339	C₂	1	2	yes	maximal
2	#14543808	C₂	1	4	yes	maximal, complemented
3	C₄×#10587469	C₂	1	1	yes	maximal, characteristic
4	#11763813	—	2	2	no	—
5	#11763670	—	2	2	no	—
6	#11763670	C₂²	1	4	yes	—
7	C₂×#10587469	C₄	1	2	yes	—
8	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	#11762584	C₂²	1	2	yes	—
10	C₄×#63214	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic
12	C₂×#63243	—	4	2	no	—
13	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
15	C₄²×D₁₇	D₄	1	1	yes	characteristic
16	C₄⋊#3624	—	2	4	no	—
17	C₄⋊#3624	—	4	2	no	—
18	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₁₇	D₄	1	1	yes	characteristic
21	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
22	C₂×#63214	C₂×C₄	1	2	yes	—
23	C₄×#3622	Q₈	1	1	yes	characteristic
24	#10587469	—	2	4	no	—
25	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×D₁₇	SD₁₆	1	2	yes	—
27	C₂×#3624	—	4	4	no	—
28	#63243	—	8	4	no	—
29	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
32	C₄×Dic₁₇	SD₁₆	1	2	yes	—
33	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
35	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
36	C₈×D₁₇	—	2	2	no	—
37	C₈×D₁₇	—	2	2	no	—
38	C₁₆×C₁₇	M₁₆	1	4	yes	complemented
39	#63214	—	2	2	no	—
40	#12020	—	17	1	no	maximal, Sylow
41	C₂²×D₁₇	#106	1	1	yes	characteristic
42	C₂×C₄×C₁₇	#106	1	2	yes	—
43	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₁₇	#106	1	1	yes	characteristic
45	C₂×Dic₁₇	C₂×SD₁₆	1	2	yes	—
46	C₄×D₁₇	—	2	4	no	—
47	C₄×D₁₇	—	2	1	no	—
48	C₄×D₁₇	—	2	1	no	—
49	#3624	—	8	4	no	—
50	C₈×C₁₇	#108	1	2	yes	—
51	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
52	#3622	—	2	1	no	—
53	#3622	—	2	1	no	—
54	C₄×M₃₂	—	17	1	no	—
55	#1566	—	17	2	no	—
56	#1368	—	17	4	no	—
57	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
58	C₂×D₁₇	—	2	1	no	—
59	C₂×D₁₇	—	2	2	no	—
60	C₄×C₁₇	#365	1	4	yes	—
61	C₄×C₁₇	#423	1	1	yes	characteristic
62	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
63	Dic₁₇	—	2	1	no	—
64	Dic₁₇	—	2	2	no	—
65	C₂×M₃₂	—	17	2	no	—
66	#423	—	34	2	no	—
67	C₂×C₄×C₈	—	17	1	no	—
68	#346	—	17	2	no	—
69	C₄×C₁₆	—	17	1	no	—
70	C₄×C₁₆	#595	1	1	yes	characteristic
71	#334	—	17	4	no	—
72	#334	—	34	2	no	—
73	C₂×C₁₇	#1566	1	1	yes	characteristic
74	C₂×C₁₇	C₂×#365	1	2	yes	—
75	D₁₇	—	4	1	no	—
76	C₂×M₁₆	—	68	2	no	—
77	C₂²×C₈	—	17	1	no	—
78	C₂²×C₈	—	17	1	no	—
79	C₂×C₄²	—	17	1	no	—
80	M₃₂	—	34	4	no	—
81	C₂×C₁₆	—	17	2	no	—
82	C₂×C₁₆	C₂×#595	1	2	yes	—
83	C₄⋊C₈	—	34	4	no	—
84	C₄⋊C₈	—	68	2	no	—
85	C₄×C₈	—	17	1	no	—
86	C₄×C₈	—	17	1	no	—
87	C₄×C₈	—	17	1	no	—
88	C₄×C₈	C₂×#595	1	1	yes	characteristic
89	C₁₇	#12020	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₄	—	17	1	no	—
91	C₂²×C₄	—	17	2	no	—
92	M₁₆	—	136	4	no	—
93	C₂×C₈	—	17	1	no	—
94	C₂×C₈	—	17	1	no	—
95	C₂×C₈	—	34	2	no	—
96	C₂×C₈	—	34	2	no	—
97	C₂×C₈	—	68	4	no	—
98	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
99	C₂×C₈	C₂²×#595	1	1	yes	characteristic
100	C₄²	—	17	1	no	—
101	C₄²	—	17	2	no	—
102	C₄²	C₄⋊#595	1	1	yes	characteristic
103	C₁₆	—	34	2	no	—
104	C₁₆	#63243	1	4	yes	complemented
105	C₂³	—	17	1	no	—
106	C₂×C₄	—	17	2	no	—
107	C₂×C₄	—	17	1	no	—
108	C₂×C₄	—	34	4	no	—
109	C₂×C₄	—	34	1	no	—
110	C₂×C₄	—	34	1	no	—
111	C₂×C₄	#10587533	1	2	yes	—
112	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
113	C₈	—	34	1	no	—
114	C₈	—	34	1	no	—
115	C₈	—	136	4	no	—
116	C₈	#10587535	1	2	yes	—
117	C₈	C₂×#63243	1	2	yes	—
118	C₂²	—	34	1	no	—
119	C₂²	—	34	2	no	—
120	C₂²	C₂×#10587533	1	1	yes	characteristic
121	C₄	—	34	1	no	—
122	C₄	—	34	2	no	—
123	C₄	#11763733	1	4	yes	—
124	C₄	#11763813	1	1	yes	characteristic
125	C₄	C₂×#10587535	1	1	yes	characteristic
126	C₂	—	68	1	no	—
127	C₂	#14544339	1	1	yes	characteristic
128	C₂	C₂×#11763733	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544339

C₂

yes

maximal

#14543808

C₂

yes

maximal, complemented

C₄×#10587469

C₂

yes

maximal, characteristic

#11763813

—

#11763670

—

#11763670

C₂²

yes

—

C₂×#10587469

C₄

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762584

C₂²

yes

—

C₄×#63214

C₄

yes

characteristic

C₄×C₁₆×C₁₇

C₄

yes

characteristic

C₂×#63243

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

characteristic

C₄²×D₁₇

D₄

yes

characteristic

C₄⋊#3624

—

C₄⋊#3624

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₂×C₄

yes

—

C₄×#3622

Q₈

yes

characteristic

#10587469

—

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

—

C₂×#3624

—

#63243

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

—

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

M₁₆

yes

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

#106

yes

characteristic

C₂×Dic₁₇

C₂×SD₁₆

yes

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

C₈×C₁₇

#108

yes

—

C₈×C₁₇

C₂×M₁₆

yes

—

#3622

—

#3622

—

C₄×M₃₂

—

#1566

—

#1368

—

C₂²×C₁₇

C₂×#106

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

#365

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂×M₃₂

—

#423

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

yes

—

—

C₂×C₁₇

#1566

yes

characteristic

C₂×C₁₇

C₂×#365

yes

—

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#595

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#595

yes

characteristic

C₁₇

#12020

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

136

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊#595

yes

characteristic

C₂×C₈

C₂²×#595

yes

characteristic

100

C₄²

—

101

C₄²

—

102

C₄²

C₄⋊#595

yes

characteristic

103

C₁₆

—

104

C₁₆

#63243

yes

complemented

105

C₂³

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#10587533

yes

—

112

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

136

—

116

C₈

#10587535

yes

—

117

C₈

C₂×#63243

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#10587533

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

#11763733

yes

—

124

C₄

#11763813

yes

characteristic

125

C₄

C₂×#10587535

yes

characteristic

126

C₂

—

127

C₂

#14544339

yes

characteristic

128

C₂

C₂×#11763733

yes

—

	subgroup	G/H
center	C₂²	C₂×#10587533
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting	C₄×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂²×D₁₇	#106
cosocle	C₈×Dic₁₇	C₂³
hypercenter	C₄×C₁₆	#595
hypocenter	C₁₇	#12020
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#10587533

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×#595

Fitting

C₄×C₁₆×C₁₇

C₄

C₂²×C₁₇

C₂²×D₁₇

C₈×Dic₁₇

C₂³

C₄×C₁₆

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #14543808	C₂	semidirect
38. C₁₆×C₁₇	M₁₆	semidirect
89. C₁₇	#12020	semidirect
104. C₁₆	#63243	semidirect

H ≅ G/N

type

2. #14543808

C₂

semidirect

38. C₁₆×C₁₇

M₁₆

semidirect

89. C₁₇

#12020

semidirect

104. C₁₆

#63243

semidirect

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

subgroup	index
#14544339	2
#14543808	2
C₄×#10587469	2
#12020	17

subgroup

index

Subgroups — #17960135

Conjugacy classes of subgroups

Complements

Maximal subgroups