Group #14540238 — subgroups

Subgroups — #14540238

Order 2176.

322 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 117 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	#11763421	C₂	1	2	yes	maximal
2	#11763117	C₂	1	4	yes	maximal, complemented
3	#11763067	C₂	1	2	yes	maximal
4	#11763050	C₂	1	2	yes	maximal
5	#11763020	C₂	1	4	yes	maximal, complemented
6	C₁₇×#424	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#117	C₂²	1	1	yes	characteristic
10	#10587610	C₂²	1	4	yes	—
11	C₂×Dic₆₈	C₂²	1	2	yes	—
12	C₂×Dic₆₈	C₂²	1	2	yes	—
13	#10587575	—	2	4	no	—
14	#10587565	—	2	4	no	—
15	#10587556	C₂²	1	2	yes	—
16	#10587555	C₂²	1	2	yes	—
17	C₄×Dic₃₄	C₂²	1	4	yes	—
18	C₁₇×C₄⋊C₈	C₂²	1	4	yes	—
19	#10587489	C₂²	1	2	yes	—
20	#10587488	C₂²	1	2	yes	—
21	#10587487	C₂²	1	4	yes	—
22	#10587487	C₂²	1	4	yes	—
23	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₃₄	C₂³	1	4	yes	—
25	M₁₆×C₁₇	—	4	1	no	—
26	C₂×C₈×C₁₇	—	4	1	no	—
27	C₂×C₈×C₁₇	C₂³	1	2	yes	—
28	C₂×C₈×C₁₇	C₂³	1	2	yes	—
29	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
30	C₂²⋊C₄×C₁₇	D₄	1	2	yes	—
31	C₄²×C₁₇	C₂³	1	2	yes	—
32	C₂²⋊Dic₁₇	—	2	4	no	—
33	C₄⋊Dic₁₇	—	2	4	no	—
34	C₄⋊Dic₁₇	C₂³	1	4	yes	—
35	#63225	—	2	4	no	—
36	#63225	—	2	4	no	—
37	#63225	—	2	4	no	—
38	C₄×Dic₁₇	—	2	4	no	—
39	Dic₆₈	—	4	2	no	—
40	Dic₆₈	—	4	2	no	—
41	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇	—	2	1	no	—
43	C₂×C₄×C₁₇	—	2	1	no	—
44	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₁₇	—	2	4	no	—
48	C₂×Dic₁₇	—	2	4	no	—
49	Dic₃₄	—	2	4	no	—
50	Dic₃₄	—	2	4	no	—
51	C₈×C₁₇	—	2	2	no	—
52	C₈×C₁₇	—	2	2	no	—
53	#3261	—	17	1	no	maximal, Sylow
54	C₂²×C₁₇	—	2	1	no	—
55	C₂²×C₁₇	—	2	2	no	—
56	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₁₇	—	2	1	no	—
58	C₄×C₁₇	—	2	4	no	—
59	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
60	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
61	Dic₁₇	—	4	4	no	—
62	Dic₁₇	—	4	4	no	—
63	#541	—	17	2	no	—
64	#479	—	17	4	no	—
65	#470	—	17	2	no	—
66	#467	—	17	2	no	—
67	#462	—	17	4	no	—
68	#424	D₁₇	1	1	yes	characteristic
69	C₂×C₁₇	—	4	1	no	—
70	C₂×C₁₇	#537	1	1	yes	characteristic
71	C₂×C₁₇	#578	1	2	yes	—
72	C₂×Q₁₆	—	17	2	no	—
73	C₂×Q₁₆	—	17	2	no	—
74	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
75	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
76	C₄⋊Q₈	—	17	2	no	—
77	#126	—	34	4	no	—
78	#125	—	17	2	no	—
79	#125	—	34	4	no	—
80	C₂²⋊Q₈	—	17	4	no	—
81	C₄×Q₈	—	17	4	no	—
82	#117	C₂×D₁₇	1	1	yes	characteristic
83	#107	—	17	2	no	—
84	#106	—	17	2	no	—
85	C₄⋊C₈	C₂×D₁₇	1	4	yes	—
86	#103	—	17	4	no	—
87	#103	—	17	4	no	—
88	C₁₇	#3261	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	17	4	no	—
90	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
91	Q₁₆	—	68	2	no	—
92	Q₁₆	—	68	2	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂²×D₁₇	1	2	yes	—
96	C₂×C₈	C₂²×D₁₇	1	2	yes	—
97	C₄⋊C₄	—	17	4	no	—
98	C₄⋊C₄	—	34	4	no	—
99	C₄⋊C₄	—	34	4	no	—
100	C₄⋊C₄	—	34	4	no	—
101	C₄⋊C₄	—	34	4	no	—
102	C₄⋊C₄	D₆₈	1	2	yes	—
103	C₂²⋊C₄	—	34	4	no	—
104	C₂²⋊C₄	D₆₈	1	2	yes	—
105	C₄²	—	34	4	no	—
106	C₄²	C₂²×D₁₇	1	2	yes	—
107	C₂³	C₂×D₆₈	1	1	yes	characteristic
108	Q₈	—	34	4	no	—
109	Q₈	—	34	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	34	4	no	—
113	C₂×C₄	—	34	4	no	—
114	C₂×C₄	C₂×D₆₈	1	4	yes	—
115	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂²×D₆₈	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	68	4	no	—
125	C₄	—	68	4	no	—
126	C₄	#10587691	1	1	yes	characteristic
127	C₄	#10587694	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#11763390	1	1	yes	characteristic
130	C₂	#11763626	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763421

C₂

yes

maximal

#11763117

C₂

yes

maximal, complemented

#11763067

C₂

yes

maximal

#11763050

C₂

yes

maximal

#11763020

C₂

yes

maximal, complemented

C₁₇×#424

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

characteristic

#10587610

C₂²

yes

—

C₂×Dic₆₈

C₂²

yes

—

C₂×Dic₆₈

C₂²

yes

—

#10587575

—

#10587565

—

#10587556

C₂²

yes

—

#10587555

C₂²

yes

—

C₄×Dic₃₄

C₂²

yes

—

C₁₇×C₄⋊C₈

C₂²

yes

—

#10587489

C₂²

yes

—

#10587488

C₂²

yes

—

#10587487

C₂²

yes

—

#10587487

C₂²

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

C₂³

yes

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂³

yes

—

C₂×C₈×C₁₇

C₂³

yes

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₂²⋊C₄×C₁₇

D₄

yes

—

C₄²×C₁₇

C₂³

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂³

yes

—

#63225

—

#63225

—

#63225

—

C₄×Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

#3261

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

H(5,𝔽₂)

yes

characteristic

C₄×C₁₇

2₋¹⁺⁴

yes

characteristic

Dic₁₇

—

Dic₁₇

—

—

—

—

—

—

D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#537

yes

characteristic

C₂×C₁₇

#578

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₄⋊Q₈

—

#126

—

#125

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₇

yes

characteristic

#107

—

#106

—

C₄⋊C₈

C₂×D₁₇

yes

—

#103

—

#103

—

C₁₇

#3261

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₇

yes

—

C₂×C₈

C₂²×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

D₆₈

yes

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

D₆₈

yes

—

105

C₄²

—

106

C₄²

C₂²×D₁₇

yes

—

107

C₂³

C₂×D₆₈

yes

characteristic

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×D₆₈

yes

—

115

C₂×C₄

C₂×D₆₈

yes

characteristic

116

C₂×C₄

C₂³×D₁₇

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×D₆₈

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#10587691

yes

characteristic

127

C₄

#10587694

yes

characteristic

128

C₂

—

129

C₂

#11763390

yes

characteristic

130

C₂

#11763626

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#424	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#424	D₁₇
hypocenter	C₁₇	#3261
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₆₈

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₁₇×#424

C₂

socle

C₂²×C₁₇

C₂²×D₄

prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
2. #11763117	C₂	semidirect
5. #11763020	C₂	semidirect
88. C₁₇	#3261	semidirect

H ≅ G/N

type

2. #11763117

C₂

semidirect

5. #11763020

C₂

semidirect

88. C₁₇

#3261

semidirect

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

subgroup	index
#11763421	2
#11763117	2
#11763067	2
#11763050	2
#11763020	2
C₁₇×#424	2
#3261	17

subgroup

index

C₁₇×#424

Subgroups — #14540238

Conjugacy classes of subgroups

Complements

Maximal subgroups