Group #14528833 — subgroups

Subgroups — #14528833

Order 2176.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#10587593	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587493	C₂	1	2	yes	maximal, complemented
3	#11762791	C₂	1	2	yes	maximal
4	#11762567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#338	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#63252	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
8	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×#63223	C₂²	1	2	yes	—
10	C₂²×#3622	C₂²	1	1	yes	characteristic
11	#10587593	—	2	4	no	—
12	C₂×#63218	C₄	1	2	yes	—
13	C₂×C₈×D₁₇	C₄	1	2	yes	—
14	#10587495	—	2	2	no	—
15	#10587494	—	2	2	no	—
16	#10587493	—	2	4	no	—
17	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₂²⋊D₁₇	Q₈	1	2	yes	—
19	#63252	—	2	2	no	—
20	#63252	—	2	2	no	—
21	C₂×D₆₈	D₄	1	1	yes	characteristic
22	C₂×C₄×D₁₇	C₂×C₄	1	2	yes	—
23	C₂×Dic₃₄	D₄	1	1	yes	characteristic
24	M₁₆×C₁₇	—	2	2	no	—
25	M₁₆×C₁₇	—	2	2	no	—
26	M₁₆×C₁₇	—	2	2	no	—
27	C₂×C₈×C₁₇	—	2	2	no	—
28	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	#63223	—	2	2	no	—
30	#63223	—	2	2	no	—
31	C₂×#3622	—	2	2	no	—
32	C₂×#3622	C₂×C₄	1	1	yes	characteristic
33	C₂×#3622	C₂×C₄	1	1	yes	characteristic
34	C₂×#3622	D₄	1	4	yes	—
35	#63218	—	2	4	no	—
36	C₈×D₁₇	—	2	4	no	—
37	C₂³×C₁₇	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₁₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇	#41	1	2	yes	—
43	C₂²⋊D₁₇	—	2	4	no	—
44	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
45	D₆₈	—	2	2	no	—
46	C₄×D₁₇	—	2	2	no	—
47	C₄×D₁₇	—	2	2	no	—
48	Dic₃₄	—	2	2	no	—
49	C₈×C₁₇	—	2	2	no	—
50	C₈×C₁₇	—	4	2	no	—
51	#3622	—	4	2	no	—
52	#3622	C₂²⋊C₄	1	2	yes	—
53	#3622	C₂²⋊C₄	1	2	yes	—
54	#1813	—	17	1	no	maximal, Sylow
55	C₂²×C₁₇	—	4	1	no	—
56	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₇	C₂²⋊Q₈	1	2	yes	—
58	C₂×D₁₇	—	2	2	no	—
59	C₂×D₁₇	—	4	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	Dic₁₇	—	2	2	no	—
64	C₂×#131	—	17	1	no	—
65	C₂×#108	—	17	2	no	—
66	#413	—	17	2	no	—
67	#338	—	17	1	no	—
68	#338	D₁₇	1	1	yes	characteristic, complemented
69	C₂×C₁₇	—	2	1	no	—
70	C₂×C₁₇	—	2	2	no	—
71	C₂×C₁₇	#380	1	1	yes	characteristic
72	D₁₇	—	4	2	no	—
73	C₂×#41	—	17	1	no	—
74	#131	—	34	4	no	—
75	C₂×M₁₆	—	17	2	no	—
76	C₂×M₁₆	—	17	2	no	—
77	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
78	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
79	C₂²×C₈	—	17	1	no	—
80	C₂²×C₈	—	17	2	no	—
81	#108	—	34	4	no	—
82	#101	—	34	2	no	—
83	#100	—	34	2	no	—
84	C₁₇	#1813	1	1	yes	characteristic, complemented, Sylow
85	#41	—	34	2	no	—
86	#41	—	34	2	no	—
87	C₂×Q₈	—	17	1	no	—
88	C₂×D₄	—	17	2	no	—
89	C₂×D₄	—	17	1	no	—
90	C₂²×C₄	—	17	2	no	—
91	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	34	4	no	—
96	M₁₆	—	34	2	no	—
97	M₁₆	—	34	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	17	1	no	—
100	C₂×C₈	—	17	1	no	—
101	C₂×C₈	—	17	4	no	—
102	C₂×C₈	—	34	4	no	—
103	C₂×C₈	—	34	2	no	—
104	C₂×C₈	C₄×D₁₇	1	2	yes	—
105	C₂³	—	17	2	no	—
106	C₂³	Q₈×D₁₇	1	1	yes	characteristic
107	Q₈	—	34	2	no	—
108	D₄	—	34	4	no	—
109	D₄	—	34	2	no	—
110	C₂×C₄	—	17	2	no	—
111	C₂×C₄	—	34	2	no	—
112	C₂×C₄	—	34	2	no	—
113	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
114	C₂×C₄	#63252	1	2	yes	—
115	C₂×C₄	D₄×D₁₇	1	2	yes	—
116	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	17	2	no	—
120	C₈	—	17	2	no	—
121	C₈	—	68	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	34	2	no	—
124	C₂²	—	68	2	no	—
125	C₂²	C₄⋊C₄×D₁₇	1	1	yes	characteristic
126	C₂²	#10587582	1	2	yes	—
127	C₄	—	34	2	no	—
128	C₄	#10587558	1	1	yes	characteristic
129	C₄	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
130	C₄	#10587570	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	68	2	no	—
134	C₂	#11762668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587593

C₂

yes

maximal, characteristic

C₂×#10587493

C₂

yes

maximal, complemented

#11762791

C₂

yes

maximal

#11762567

C₂

yes

maximal, characteristic, complemented

C₁₇×#338

C₂

yes

maximal, characteristic, complemented

C₂×#63252

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

—

C₂²×#3622

C₂²

yes

characteristic

#10587593

—

C₂×#63218

C₄

yes

—

C₂×C₈×D₁₇

C₄

yes

—

#10587495

—

#10587494

—

#10587493

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₂²⋊D₁₇

Q₈

yes

—

#63252

—

#63252

—

C₂×D₆₈

D₄

yes

characteristic

C₂×C₄×D₁₇

C₂×C₄

yes

—

C₂×Dic₃₄

D₄

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

C₂×#3622

D₄

yes

—

#63218

—

C₈×D₁₇

—

C₂³×C₁₇

C₂×Q₈

yes

characteristic

C₂²×D₁₇

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

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

C₄⋊C₄

yes

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3622

C₂²⋊C₄

yes

—

#3622

C₂²⋊C₄

yes

—

#1813

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

C₂×C₂²⋊C₄

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#380

yes

characteristic

D₁₇

—

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#108

—

#101

—

#100

—

C₁₇

#1813

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₄×D₁₇

yes

—

105

C₂³

—

106

C₂³

Q₈×D₁₇

yes

characteristic

107

Q₈

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

114

C₂×C₄

#63252

yes

—

115

C₂×C₄

D₄×D₁₇

yes

—

116

C₂×C₄

D₄×D₁₇

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₄⋊C₄×D₁₇

yes

characteristic

126

C₂²

#10587582

yes

—

127

C₄

—

128

C₄

#10587558

yes

characteristic

129

C₄

C₂²⋊C₄×D₁₇

yes

characteristic

130

C₄

#10587570

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

134

C₂

#11762668

yes

characteristic

	subgroup	G/H
center	C₄	#10587558
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#338	C₂
socle	C₂×C₁₇	#380
prime radical	C₂×#63252	C₂²
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#338	D₁₇
hypocenter	C₁₇	#1813
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#10587558

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#338

C₂

C₂×C₁₇

C₂²

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#10587493	C₂	semidirect
4. #11762567	C₂	semidirect
5. C₁₇×#338	C₂	semidirect
68. #338	D₁₇	semidirect
84. C₁₇	#1813	semidirect

H ≅ G/N

type

2. C₂×#10587493

C₂

semidirect

4. #11762567

C₂

semidirect

5. C₁₇×#338

C₂

semidirect

68. #338

D₁₇

semidirect

84. C₁₇

#1813

semidirect

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

subgroup	index
C₂×#10587593	2
C₂×#10587493	2
#11762791	2
#11762567	2
C₁₇×#338	2
#1813	17

subgroup

index

C₁₇×#338

Subgroups — #14528833

Conjugacy classes of subgroups

Complements

Maximal subgroups