Group #17982810 — subgroups

Subgroups — #17982810

Order 4352.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#14546017	C₂	1	2	yes	maximal, complemented
2	#14545884	C₂	1	1	yes	maximal, characteristic, complemented
3	#14545880	C₂	1	2	yes	maximal, complemented
4	#14545870	C₂	1	1	yes	maximal, characteristic, complemented
5	#14545171	C₂	1	1	yes	maximal, characteristic
6	#11763917	—	2	2	no	—
7	#11763917	C₂²	1	2	yes	—
8	C₂×#10587528	—	2	2	no	—
9	C₂×#10587528	C₂²	1	1	yes	characteristic
10	C₂×#10587528	C₂²	1	1	yes	characteristic
11	#11763867	C₂²	1	2	yes	—
12	#11763731	C₄	1	1	yes	characteristic
13	#11763707	C₄	1	2	yes	—
14	#11763695	C₂²	1	1	yes	characteristic
15	#11762926	C₄	1	1	yes	characteristic
16	C₁₇×#135	C₈	1	1	yes	characteristic
17	#10587634	C₂×C₄	1	2	yes	—
18	#10587538	—	2	2	no	—
19	#10587538	C₂×C₄	1	2	yes	—
20	C₂×#63216	—	2	2	no	—
21	C₂×#63216	C₂×C₄	1	1	yes	characteristic
22	C₂×#63216	C₂³	1	1	yes	characteristic
23	#10587528	—	2	2	no	—
24	#10587528	—	2	1	no	—
25	#10587528	D₄	1	2	yes	—
26	#10587508	C₈	1	2	yes	—
27	#10587490	C₈	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	#41×C₁₇	C₂×C₈	1	2	yes	—
30	Q₁₆×C₁₇	C₁₆	1	1	yes	characteristic
31	SD₁₆×C₁₇	C₁₆	1	2	yes	—
32	D₈×C₁₇	C₁₆	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
35	#63223	—	2	2	no	—
36	#63223	C₂×C₈	1	2	yes	—
37	C₂×#3622	—	2	2	no	—
38	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
39	#63216	—	2	2	no	—
40	#63216	—	2	1	no	—
41	#63216	C₂×D₄	1	1	yes	characteristic
42	#63216	#41	1	1	yes	characteristic
43	#13648	—	17	1	no	maximal, Sylow
44	Q₈×C₁₇	C₂×C₁₆	1	2	yes	—
45	D₄×C₁₇	—	2	2	no	—
46	D₄×C₁₇	C₂×C₁₆	1	2	yes	—
47	C₂×C₄×C₁₇	—	2	2	no	—
48	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₁₇	—	2	1	no	—
50	C₈×C₁₇	C₂×C₁₆	1	1	yes	characteristic
51	C₈×C₁₇	C₂×C₁₆	1	1	yes	characteristic
52	#3622	—	2	2	no	—
53	#3622	C₄×D₄	1	1	yes	characteristic
54	#3622	#131	1	1	yes	characteristic
55	#2258	—	17	2	no	—
56	#2170	—	17	1	no	—
57	#1422	—	17	1	no	—
58	#1401	—	17	2	no	—
59	#1397	—	17	1	no	—
60	C₂²×C₁₇	—	2	2	no	—
61	C₂²×C₁₇	#504	1	1	yes	characteristic
62	C₄×C₁₇	—	2	2	no	—
63	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
64	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
65	Dic₁₇	—	4	1	no	—
66	#504	—	17	2	no	—
67	#443	—	17	1	no	—
68	M₆₄	—	17	2	no	—
69	M₆₄	—	34	2	no	—
70	C₂×C₃₂	—	17	1	no	—
71	C₂×C₃₂	—	17	1	no	—
72	C₂×C₃₂	—	34	2	no	—
73	#364	—	17	1	no	—
74	#350	—	17	2	no	—
75	C₄×C₁₆	—	17	1	no	—
76	C₂×C₁₇	—	2	1	no	—
77	C₂×C₁₇	—	4	2	no	—
78	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
79	#135	#3633	1	1	yes	characteristic
80	#131	—	17	2	no	—
81	M₃₂	—	17	2	no	—
82	M₃₂	—	34	2	no	—
83	C₂×C₁₆	—	17	1	no	—
84	C₂×C₁₆	—	17	1	no	—
85	C₂×C₁₆	—	34	2	no	—
86	#108	—	17	1	no	—
87	#104	—	17	2	no	—
88	C₄×C₈	—	17	1	no	—
89	C₃₂	—	17	2	no	—
90	C₃₂	—	34	1	no	—
91	C₃₂	—	34	2	no	—
92	C₁₇	#13648	1	1	yes	characteristic, complemented, Sylow
93	#41	C₂×#3633	1	2	yes	—
94	Q₁₆	#63241	1	1	yes	characteristic
95	SD₁₆	#63241	1	2	yes	—
96	D₈	#63241	1	1	yes	characteristic
97	M₁₆	—	17	2	no	—
98	M₁₆	—	34	2	no	—
99	C₂×C₈	—	17	1	no	—
100	C₂×C₈	—	34	2	no	—
101	C₂×C₈	C₂×#3633	1	1	yes	characteristic
102	C₄²	—	17	1	no	—
103	C₁₆	—	17	1	no	—
104	C₁₆	—	17	1	no	—
105	C₁₆	—	34	1	no	—
106	C₁₆	—	34	2	no	—
107	Q₈	C₂×#63241	1	2	yes	—
108	D₄	—	2	2	no	—
109	D₄	C₂×#63241	1	2	yes	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	34	1	no	—
112	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
113	C₈	—	17	1	no	—
114	C₈	—	17	1	no	—
115	C₈	—	34	2	no	—
116	C₈	C₂×#63241	1	1	yes	characteristic
117	C₈	C₂×#63241	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	#11763930	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	68	1	no	—
122	C₄	D₄×#3633	1	1	yes	characteristic
123	C₄	C₂²×#63241	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	4	2	no	—
126	C₂	D₄×#63241	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14546017

C₂

yes

maximal, complemented

#14545884

C₂

yes

maximal, characteristic, complemented

#14545880

C₂

yes

maximal, complemented

#14545870

C₂

yes

maximal, characteristic, complemented

#14545171

C₂

yes

maximal, characteristic

#11763917

—

#11763917

C₂²

yes

—

C₂×#10587528

—

C₂×#10587528

C₂²

yes

characteristic

C₂×#10587528

C₂²

yes

characteristic

#11763867

C₂²

yes

—

#11763731

C₄

yes

characteristic

#11763707

C₄

yes

—

#11763695

C₂²

yes

characteristic

#11762926

C₄

yes

characteristic

C₁₇×#135

C₈

yes

characteristic

#10587634

C₂×C₄

yes

—

#10587538

—

#10587538

C₂×C₄

yes

—

C₂×#63216

—

C₂×#63216

C₂×C₄

yes

characteristic

C₂×#63216

C₂³

yes

—

—

D₄

yes

—

#10587508

C₈

yes

—

#10587490

C₈

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

#41×C₁₇

C₂×C₈

yes

—

Q₁₆×C₁₇

C₁₆

yes

characteristic

SD₁₆×C₁₇

C₁₆

yes

—

D₈×C₁₇

C₁₆

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

characteristic

#63223

—

#63223

C₂×C₈

yes

—

C₂×#3622

—

C₂×#3622

C₂²×C₄

yes

—

—

C₂×D₄

yes

yes

—

Q₈×C₁₇

C₂×C₁₆

yes

—

D₄×C₁₇

—

D₄×C₁₇

C₂×C₁₆

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×Dic₁₇

—

C₈×C₁₇

C₂×C₁₆

yes

characteristic

C₈×C₁₇

C₂×C₁₆

yes

characteristic

#3622

—

#3622

C₄×D₄

yes

yes

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#504

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

C₂²×C₁₆

yes

characteristic

Dic₁₇

—

#504

—

#443

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#364

—

#350

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

D₄×C₁₆

yes

yes

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#104

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

—

C₁₇

#13648

yes

characteristic, complemented, Sylow

#41

C₂×#3633

yes

—

Q₁₆

#63241

yes

characteristic

SD₁₆

#63241

yes

—

D₈

#63241

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×#3633

yes

characteristic

102

C₄²

—

103

C₁₆

—

104

C₁₆

—

105

C₁₆

—

106

C₁₆

—

107

Q₈

C₂×#63241

yes

—

108

D₄

—

109

D₄

C₂×#63241

yes

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×#3633

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

C₂×#63241

yes

characteristic

117

C₈

C₂×#63241

yes

characteristic

118

C₂²

—

119

C₂²

#11763930

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

D₄×#3633

yes

characteristic

123

C₄

C₂²×#63241

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

D₄×#63241

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#3633
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#3633
Fitting, prime radical	C₁₇×#135	C₈
socle	C₂×C₁₇	D₄×C₁₆
cosocle	C₂×#63216	C₂³
hypercenter	#135	#3633
hypocenter	C₁₇	#13648
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₄

D₄×#3633

commutator

C₄×C₁₇

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#3633

Fitting, prime radical

C₁₇×#135

C₈

socle

C₂×C₁₇

D₄×C₁₆

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. #14546017	C₂	semidirect
2. #14545884	C₂	semidirect
3. #14545880	C₂	semidirect
4. #14545870	C₂	semidirect
92. C₁₇	#13648	semidirect

H ≅ G/N

type

1. #14546017

C₂

semidirect

2. #14545884

C₂

semidirect

3. #14545880

C₂

semidirect

4. #14545870

C₂

semidirect

92. C₁₇

#13648

semidirect

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

subgroup	index
#14546017	2
#14545884	2
#14545880	2
#14545870	2
#14545171	2
#13648	17

subgroup

index

Subgroups — #17982810

Conjugacy classes of subgroups

Complements

Maximal subgroups