Group #14530135 — subgroups

Subgroups — #14530135

Order 2176.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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	#11763440	C₂	1	1	yes	maximal, characteristic
2	#11762851	C₂	1	2	yes	maximal
3	#11762674	C₂	1	1	yes	maximal, characteristic
4	#11762570	C₂	1	2	yes	maximal
5	C₁₇×#343	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
8	#10587612	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
10	#10587610	—	2	2	no	—
11	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
12	#10587576	—	2	2	no	—
13	#10587574	—	2	2	no	—
14	#10587502	—	2	2	no	—
15	#10587489	—	2	2	no	—
16	#10587488	—	2	2	no	—
17	C₂²×C₄×C₁₇	D₄	1	2	yes	—
18	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₇	—	2	1	no	—
20	C₂²×Dic₁₇	—	2	1	no	—
21	C₂×Dic₃₄	—	2	2	no	—
22	M₁₆×C₁₇	—	2	2	no	—
23	M₁₆×C₁₇	Q₈	1	4	yes	complemented
24	C₂×C₈×C₁₇	—	2	2	no	—
25	C₄⋊C₄×C₁₇	—	2	2	no	—
26	C₂²⋊Dic₁₇	—	2	2	no	—
27	C₄⋊Dic₁₇	—	2	2	no	—
28	C₄⋊Dic₁₇	—	2	2	no	—
29	C₄⋊Dic₁₇	D₄	1	2	yes	—
30	C₄⋊Dic₁₇	D₄	1	2	yes	—
31	#63225	—	2	4	no	—
32	C₄×Dic₁₇	—	2	2	no	—
33	C₄×Dic₁₇	—	2	1	no	—
34	C₄×Dic₁₇	Q₈	1	2	yes	complemented
35	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	—	2	2	no	—
37	C₂×C₄×C₁₇	—	4	2	no	—
38	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
42	C₂×Dic₁₇	—	2	2	no	—
43	C₂×Dic₁₇	—	2	2	no	—
44	C₂×Dic₁₇	—	2	2	no	—
45	C₂×Dic₁₇	—	4	1	no	—
46	C₂×Dic₁₇	—	4	1	no	—
47	C₂×Dic₁₇	—	4	2	no	—
48	C₂×Dic₁₇	—	4	1	no	—
49	Dic₃₄	—	8	2	no	—
50	C₈×C₁₇	—	2	4	no	—
51	#2061	—	17	1	no	maximal, Sylow
52	C₂²×C₁₇	—	2	1	no	—
53	C₂²×C₁₇	—	2	1	no	—
54	C₂²×C₁₇	#120	1	1	yes	characteristic
55	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×C₁₇	—	4	2	no	—
58	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
60	Dic₁₇	—	4	2	no	—
61	Dic₁₇	—	4	2	no	—
62	Dic₁₇	—	4	2	no	—
63	Dic₁₇	—	8	1	no	—
64	#544	—	17	1	no	—
65	#428	—	17	2	no	—
66	#382	—	17	1	no	—
67	#343	D₁₇	1	1	yes	characteristic
68	#339	—	17	2	no	—
69	C₂×C₁₇	—	2	1	no	—
70	C₂×C₁₇	—	2	1	no	—
71	C₂×C₁₇	#395	1	1	yes	characteristic
72	C₂×C₁₇	#454	1	1	yes	characteristic
73	C₂×C₁₇	#455	1	1	yes	characteristic
74	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
75	C₄⋊Q₈	—	34	2	no	—
76	#125	—	34	2	no	—
77	C₂²⋊Q₈	—	34	2	no	—
78	#117	—	17	2	no	—
79	C₂×C₄⋊C₄	—	17	1	no	—
80	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
81	C₂×C₄²	—	17	1	no	—
82	#107	—	34	2	no	—
83	#106	—	34	2	no	—
84	#95	—	34	2	no	—
85	C₁₇	#2061	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₈	—	34	2	no	—
87	C₂²×C₄	—	34	1	no	—
88	C₂²×C₄	—	34	1	no	—
89	C₂²×C₄	D₆₈	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
91	M₁₆	—	2	2	no	—
92	M₁₆	Dic₃₄	1	4	yes	complemented
93	C₂×C₈	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	17	2	no	—
96	C₄⋊C₄	—	17	2	no	—
97	C₄⋊C₄	—	34	2	no	—
98	C₄⋊C₄	—	34	4	no	—
99	C₄⋊C₄	—	34	2	no	—
100	C₂²⋊C₄	—	34	2	no	—
101	C₄²	—	17	2	no	—
102	C₄²	—	34	2	no	—
103	C₄²	—	34	1	no	—
104	C₂³	C₂×D₆₈	1	1	yes	characteristic
105	Q₈	—	136	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	34	2	no	—
109	C₂×C₄	—	34	2	no	—
110	C₂×C₄	—	34	2	no	—
111	C₂×C₄	—	68	1	no	—
112	C₂×C₄	—	68	1	no	—
113	C₂×C₄	—	68	2	no	—
114	C₂×C₄	—	68	1	no	—
115	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
116	C₂×C₄	D₄×D₁₇	1	2	yes	—
117	C₂×C₄	#63254	1	1	yes	characteristic
118	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#10587555	1	1	yes	characteristic
123	C₂²	#10587568	1	1	yes	characteristic
124	C₂²	#10587583	1	1	yes	characteristic
125	C₄	—	4	2	no	—
126	C₄	—	68	2	no	—
127	C₄	—	68	2	no	—
128	C₄	—	68	2	no	—
129	C₄	—	136	1	no	—
130	C₄	#10587564	1	2	yes	—
131	C₄	#10587574	1	2	yes	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#11762730	1	1	yes	characteristic
135	C₂	#11762984	1	1	yes	characteristic
136	C₂	#11762989	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763440

C₂

yes

maximal, characteristic

#11762851

C₂

yes

maximal

#11762674

C₂

yes

maximal, characteristic

#11762570

C₂

yes

maximal

C₁₇×#343

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

#10587612

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

#10587610

—

C₂×C₄×Dic₁₇

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

M₁₆×C₁₇

Q₈

yes

complemented

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

—

C₄⋊Dic₁₇

D₄

yes

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

Q₈

yes

complemented

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

#2061

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#120

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

C₄⋊Q₈

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

—

—

—

D₁₇

yes

characteristic

#339

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

C₂×D₁₇

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₂×C₄²

—

#107

—

#106

—

#95

—

C₁₇

#2061

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₆₈

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

Dic₃₄

yes

complemented

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₂³

C₂×D₆₈

yes

characteristic

105

Q₈

—

136

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂×Dic₃₄

yes

—

116

C₂×C₄

D₄×D₁₇

yes

—

117

C₂×C₄

#63254

yes

characteristic

118

C₂×C₄

Q₈×D₁₇

yes

characteristic

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

#10587555

yes

characteristic

123

C₂²

#10587568

yes

characteristic

124

C₂²

#10587583

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

136

—

130

C₄

#10587564

yes

—

131

C₄

#10587574

yes

—

132

C₂

—

133

C₂

—

134

C₂

#11762730

yes

characteristic

135

C₂

#11762984

yes

characteristic

136

C₂

#11762989

yes

characteristic

	subgroup	G/H
center	C₂²	#10587568
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#343	C₂
socle	C₂²×C₁₇	#120
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#343	D₁₇
hypocenter	C₁₇	#2061
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#10587568

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#343

C₂

socle

C₂²×C₁₇

#120

prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
23. M₁₆×C₁₇	Q₈	semidirect
34. C₄×Dic₁₇	Q₈	semidirect
85. C₁₇	#2061	semidirect
92. M₁₆	Dic₃₄	semidirect

H ≅ G/N

type

23. M₁₆×C₁₇

Q₈

semidirect

34. C₄×Dic₁₇

Q₈

semidirect

85. C₁₇

#2061

semidirect

92. M₁₆

Dic₃₄

semidirect

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

subgroup	index
#11763440	2
#11762851	2
#11762674	2
#11762570	2
C₁₇×#343	2
#2061	17

subgroup

index

C₁₇×#343

Subgroups — #14530135

Conjugacy classes of subgroups

Complements

Maximal subgroups