Group #14528820 — subgroups

Subgroups — #14528820

Order 2176.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×#3622	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587493	C₂	1	2	yes	maximal, complemented
3	#11762773	C₂	1	1	yes	maximal, characteristic
4	#11762573	C₂	1	2	yes	maximal, complemented
5	C₁₇×#407	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
8	C₂×#63223	C₂²	1	2	yes	—
9	C₂²×#3622	—	2	4	no	—
10	C₂²×#3622	C₄	1	4	yes	—
11	C₂²×#3622	C₂²	1	1	yes	characteristic
12	C₂²×#3622	C₂²	1	1	yes	characteristic
13	C₁₇×C₂²⋊C₈	—	2	2	no	—
14	C₂²⋊#3622	—	2	2	no	—
15	#10587493	—	2	4	no	—
16	C₂⁴×C₁₇	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	—	2	2	no	—
18	C₂²×C₄×C₁₇	—	2	2	no	—
19	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₇	—	2	2	no	—
24	M₁₆×C₁₇	—	2	2	no	—
25	C₂×C₈×C₁₇	—	2	2	no	—
26	#63223	—	2	2	no	—
27	#63223	—	2	2	no	—
28	C₂×#3622	—	2	2	no	—
29	C₂×#3622	—	2	8	no	—
30	C₂×#3622	—	2	2	no	—
31	C₂×#3622	C₂×C₄	1	2	yes	—
32	C₂×#3622	C₂×C₄	1	2	yes	—
33	C₂×#3622	D₄	1	4	yes	—
34	C₂³×C₁₇	—	2	4	no	—
35	C₂³×C₁₇	—	2	2	no	—
36	C₂³×C₁₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₇	—	2	2	no	—
39	C₂×C₄×C₁₇	—	2	2	no	—
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	—	2	4	no	—
42	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#41	1	2	yes	—
47	C₈×C₁₇	—	4	2	no	—
48	#3622	—	2	2	no	—
49	#3622	—	4	2	no	—
50	#3622	C₂²⋊C₄	1	4	yes	—
51	#1810	—	17	1	no	maximal, Sylow
52	C₂²×C₁₇	—	2	8	no	—
53	C₂²×C₁₇	—	2	2	no	—
54	C₂²×C₁₇	—	2	2	no	—
55	C₂²×C₁₇	—	2	2	no	—
56	C₂²×C₁₇	#108	1	4	yes	complemented
57	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₇	—	2	2	no	—
60	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₁₇	#117	1	1	yes	characteristic
62	C₄×C₁₇	#121	1	2	yes	—
63	C₂³×C₈	—	17	1	no	—
64	C₂×#108	—	17	2	no	—
65	#407	—	17	1	no	—
66	#407	D₁₇	1	1	yes	characteristic
67	#341	—	17	2	no	—
68	C₂×C₁₇	—	2	4	no	—
69	C₂×C₁₇	—	2	2	no	—
70	C₂×C₁₇	#380	1	1	yes	characteristic
71	C₂×C₁₇	C₂×#108	1	2	yes	—
72	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
73	C₂×M₁₆	—	17	2	no	—
74	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
75	C₂²×C₈	—	17	1	no	—
76	C₂²×C₈	—	17	4	no	—
77	C₂²×C₈	—	17	1	no	—
78	C₂²×C₈	—	34	4	no	—
79	#108	—	34	4	no	—
80	C₂²⋊C₈	—	2	2	no	—
81	C₂²⋊C₈	—	34	2	no	—
82	C₁₇	#1810	1	1	yes	characteristic, complemented, Sylow
83	C₂⁴	Dic₃₄	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	Dic₃₄	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₁₇	1	2	yes	—
88	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	34	2	no	—
93	M₁₆	—	34	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	17	2	no	—
96	C₂×C₈	—	17	4	no	—
97	C₂×C₈	—	17	2	no	—
98	C₂×C₈	—	34	8	no	—
99	C₂×C₈	—	34	2	no	—
100	C₂×C₈	—	34	2	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	2	2	no	—
103	C₂³	#63225	1	2	yes	—
104	C₂³	C₂×Dic₃₄	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	#63225	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₇	1	2	yes	—
112	C₂×C₄	#63254	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	17	4	no	—
116	C₈	—	34	2	no	—
117	C₈	—	68	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#10587493	1	4	yes	complemented
123	C₂²	#10587564	1	2	yes	—
124	C₂²	C₂×#63225	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	#10587563	1	1	yes	characteristic
127	C₄	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
128	C₄	#10587625	1	2	yes	—
129	C₂	—	2	4	no	—
130	C₂	—	2	2	no	—
131	C₂	#11762667	1	1	yes	characteristic
132	C₂	C₂×#10587493	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#3622

C₂

yes

maximal, characteristic

C₂×#10587493

C₂

yes

maximal, complemented

#11762773

C₂

yes

maximal, characteristic

#11762573

C₂

yes

maximal, complemented

C₁₇×#407

C₂

yes

maximal, characteristic

C₂³×C₄×C₁₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂×#63223

C₂²

yes

—

C₂²×#3622

—

C₂²×#3622

C₄

yes

—

C₂²×#3622

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

characteristic

C₁₇×C₂²⋊C₈

—

C₂²⋊#3622

—

#10587493

—

C₂⁴×C₁₇

Q₈

yes

characteristic

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

Q₈

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#3622

C₂×C₄

yes

—

C₂×#3622

D₄

yes

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

C₄⋊C₄

yes

—

C₂³×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₈×C₁₇

—

#3622

—

#3622

—

#3622

C₂²⋊C₄

yes

—

#1810

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#108

yes

complemented

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₂²⋊C₄

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₂³×C₈

—

C₂×#108

—

#407

—

#407

D₁₇

yes

characteristic

#341

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#380

yes

characteristic

C₂×C₁₇

C₂×#108

yes

—

C₂³×C₄

C₂×D₁₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₁₇

#1810

yes

characteristic, complemented, Sylow

C₂⁴

Dic₃₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₃₄

yes

characteristic

C₂²×C₄

C₄×D₁₇

yes

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂³

—

102

C₂³

—

103

C₂³

#63225

yes

—

104

C₂³

C₂×Dic₃₄

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#63225

yes

—

110

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

111

C₂×C₄

D₄×D₁₇

yes

—

112

C₂×C₄

#63254

yes

—

113

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#10587493

yes

complemented

123

C₂²

#10587564

yes

—

124

C₂²

C₂×#63225

yes

characteristic

125

C₄

—

126

C₄

#10587563

yes

characteristic

127

C₄

C₂²⋊C₄×D₁₇

yes

characteristic

128

C₄

#10587625

yes

—

129

C₂

—

130

C₂

—

131

C₂

#11762667

yes

characteristic

132

C₂

C₂×#10587493

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#407	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₇	Q₈
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#407	D₁₇
hypocenter	C₁₇	#1810
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#407

C₂

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂⁴×C₁₇

Q₈

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#10587493	C₂	semidirect
4. #11762573	C₂	semidirect
56. C₂²×C₁₇	#108	semidirect
82. C₁₇	#1810	semidirect
122. C₂²	#10587493	semidirect

H ≅ G/N

type

2. C₂×#10587493

C₂

semidirect

4. #11762573

C₂

semidirect

56. C₂²×C₁₇

#108

semidirect

82. C₁₇

#1810

semidirect

122. C₂²

#10587493

semidirect

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

subgroup	index
C₂³×#3622	2
C₂×#10587493	2
#11762773	2
#11762573	2
C₁₇×#407	2
#1810	17

subgroup

index

C₂³×#3622

C₁₇×#407

Subgroups — #14528820

Conjugacy classes of subgroups

Complements

Maximal subgroups