Group #17960067 — subgroups

Subgroups — #17960067

Order 4352.

234 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 112 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	#14544331	C₂	1	2	yes	maximal
2	#14543810	C₂	1	4	yes	maximal, complemented
3	C₂×#11762584	C₂	1	1	yes	maximal, characteristic
4	#11763825	—	2	2	no	—
5	#11763672	—	2	2	no	—
6	#11763670	C₂²	1	4	yes	—
7	C₂²×#63214	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
9	#11762584	C₄	1	2	yes	—
10	#11762584	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
14	C₂²×#3622	Q₈	1	1	yes	characteristic
15	C₂²⋊#3624	—	4	2	no	—
16	C₄⋊#3624	—	2	4	no	—
17	C₄×#3624	—	4	2	no	—
18	C₂×C₁₆×C₁₇	—	2	2	no	—
19	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
20	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₈×Dic₁₇	D₄	1	1	yes	characteristic
22	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
23	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
24	C₂×#63214	C₈	1	4	yes	complemented
25	C₂×#63214	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×Dic₁₇	SD₁₆	1	2	yes	—
28	C₂×#3624	—	4	4	no	—
29	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
32	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₄×Dic₁₇	SD₁₆	1	2	yes	—
35	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×#3622	—	2	2	no	—
37	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
38	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
39	C₁₆×C₁₇	—	2	2	no	—
40	#63214	C₂×C₈	1	4	yes	—
41	#11999	—	17	1	no	maximal, Sylow
42	C₂³×C₁₇	#106	1	1	yes	characteristic
43	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#108	1	2	yes	—
46	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×Dic₁₇	—	4	2	no	—
48	C₂×Dic₁₇	#106	1	2	yes	—
49	C₂×Dic₁₇	C₂×SD₁₆	1	2	yes	—
50	#3624	—	8	4	no	—
51	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
52	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
53	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
54	#3622	—	2	1	no	—
55	#3622	—	2	1	no	—
56	C₂×#346	—	17	1	no	—
57	#1564	—	17	2	no	—
58	#1368	—	17	4	no	—
59	C₂²×C₁₇	—	2	2	no	—
60	C₂²×C₁₇	#334	1	2	yes	—
61	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
62	C₄×C₁₇	#334	1	2	yes	—
63	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
65	Dic₁₇	—	2	2	no	—
66	Dic₁₇	—	2	2	no	—
67	C₂²×C₁₆	—	17	1	no	—
68	C₂²×C₁₆	#595	1	1	yes	characteristic
69	#431	—	34	2	no	—
70	C₂×C₄×C₈	—	17	1	no	—
71	#346	—	17	2	no	—
72	#346	—	17	2	no	—
73	#335	—	34	2	no	—
74	#334	—	17	4	no	—
75	C₂×C₁₇	—	2	2	no	—
76	C₂×C₁₇	C₂×#334	1	1	yes	characteristic
77	C₂×C₁₇	#2155	1	2	yes	—
78	C₂²×C₈	—	17	1	no	—
79	C₂²×C₈	C₂×#595	1	1	yes	characteristic
80	C₂×C₄²	—	17	1	no	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	—	17	4	no	—
83	C₂×C₁₆	—	17	2	no	—
84	C₂×C₁₆	C₂×#595	1	2	yes	—
85	C₄⋊C₈	—	34	4	no	—
86	C₂²⋊C₈	—	68	2	no	—
87	C₄×C₈	—	17	1	no	—
88	C₄×C₈	—	17	1	no	—
89	C₄×C₈	—	17	1	no	—
90	C₄×C₈	—	17	1	no	—
91	C₄×C₈	—	68	2	no	—
92	C₁₇	#11999	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	17	2	no	—
94	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
95	C₂×C₈	—	17	1	no	—
96	C₂×C₈	—	17	1	no	—
97	C₂×C₈	—	34	2	no	—
98	C₂×C₈	—	68	4	no	—
99	C₂×C₈	#63242	1	2	yes	—
100	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
101	C₂×C₈	#63247	1	2	yes	—
102	C₂×C₈	C₂²×#595	1	1	yes	characteristic
103	C₄²	—	17	1	no	—
104	C₄²	—	17	1	no	—
105	C₄²	—	17	2	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	—	17	4	no	—
108	C₂³	#10587533	1	1	yes	characteristic
109	C₂×C₄	—	17	2	no	—
110	C₂×C₄	—	17	2	no	—
111	C₂×C₄	—	68	2	no	—
112	C₂×C₄	#10587533	1	1	yes	characteristic
113	C₂×C₄	#10587536	1	2	yes	—
114	C₂×C₄	#10587544	1	2	yes	—
115	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
116	C₈	—	34	1	no	—
117	C₈	—	34	1	no	—
118	C₈	—	136	4	no	—
119	C₈	#10587544	1	2	yes	—
120	C₈	C₂×#63242	1	1	yes	characteristic
121	C₈	C₂×#63247	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	#11763671	1	2	yes	—
124	C₂²	C₂×#10587533	1	1	yes	characteristic
125	C₄	—	34	2	no	—
126	C₄	—	34	2	no	—
127	C₄	#11763671	1	2	yes	—
128	C₄	C₂×#10587544	1	1	yes	characteristic
129	C₄	C₂×#10587536	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	C₂×#11763671	1	1	yes	characteristic
132	C₂	#14545131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544331

C₂

yes

maximal

#14543810

C₂

yes

maximal, complemented

C₂×#11762584

C₂

yes

maximal, characteristic

#11763825

—

#11763672

—

#11763670

C₂²

yes

—

C₂²×#63214

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762584

C₄

yes

—

#11762584

C₂²

yes

—

C₂²×C₁₆×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

D₄

yes

characteristic

C₂²×#3622

Q₈

yes

characteristic

C₂²⋊#3624

—

C₄⋊#3624

—

C₄×#3624

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₈

yes

complemented

C₂×#63214

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₇

SD₁₆

yes

—

C₂×#3624

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

—

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₁₆×C₁₇

—

#63214

C₂×C₈

yes

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

#106

yes

characteristic

C₂×C₄×C₁₇

#108

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#106

yes

—

C₂×Dic₁₇

C₂×SD₁₆

yes

—

#3624

—

C₈×C₁₇

C₄⋊C₈

yes

—

C₈×C₁₇

C₂²×C₈

yes

characteristic

C₈×C₁₇

C₂×M₁₆

yes

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#334

yes

—

C₂²×C₁₇

C₂×#106

yes

characteristic

C₄×C₁₇

#334

yes

—

C₄×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₇

C₂×#108

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂²×C₁₆

—

C₂²×C₁₆

#595

yes

characteristic

#431

—

C₂×C₄×C₈

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#334

yes

characteristic

C₂×C₁₇

#2155

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×#595

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#595

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#11999

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₄⋊#595

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#63242

yes

—

100

C₂×C₈

C₄⋊#595

yes

characteristic

101

C₂×C₈

#63247

yes

—

102

C₂×C₈

C₂²×#595

yes

characteristic

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₁₆

—

107

C₁₆

—

108

C₂³

#10587533

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#10587533

yes

characteristic

113

C₂×C₄

#10587536

yes

—

114

C₂×C₄

#10587544

yes

—

115

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

136

—

119

C₈

#10587544

yes

—

120

C₈

C₂×#63242

yes

characteristic

121

C₈

C₂×#63247

yes

characteristic

122

C₂²

—

123

C₂²

#11763671

yes

—

124

C₂²

C₂×#10587533

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

#11763671

yes

—

128

C₄

C₂×#10587544

yes

characteristic

129

C₄

C₂×#10587536

yes

characteristic

130

C₂

—

131

C₂

C₂×#11763671

yes

characteristic

132

C₂

#14545131

yes

—

	subgroup	G/H
center	C₂×C₄	#10587533
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂³×C₁₇	#106
cosocle	C₈×Dic₁₇	C₂³
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	#11999
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₄

#10587533

commutator

C₈×C₁₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×#595

Fitting

C₂²×C₁₆×C₁₇

C₄

C₂²×C₁₇

C₂³×C₁₇

C₈×Dic₁₇

C₂³

C₂²×C₁₆

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. #14543810	C₂	semidirect
24. C₂×#63214	C₈	semidirect
92. C₁₇	#11999	semidirect

H ≅ G/N

type

2. #14543810

C₂

semidirect

24. C₂×#63214

C₈

semidirect

92. C₁₇

#11999

semidirect

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

subgroup	index
#14544331	2
#14543810	2
C₂×#11762584	2
#11999	17

subgroup

index

Subgroups — #17960067

Conjugacy classes of subgroups

Complements

Maximal subgroups