Group #14530226 — subgroups

Subgroups — #14530226

Order 2176.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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×#10587489	C₂	1	1	yes	maximal, characteristic, complemented
2	#11762811	C₂	1	2	yes	maximal
3	#11762775	C₂	1	1	yes	maximal, characteristic
4	#11762577	C₂	1	2	yes	maximal, complemented
5	C₁₇×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#63252	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
10	C₂×#63223	C₂²	1	1	yes	characteristic
11	#10587492	—	2	2	no	—
12	#10587491	—	2	2	no	—
13	#10587489	—	2	2	no	—
14	#10587487	—	2	2	no	—
15	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
16	C₂×C₂²⋊D₁₇	—	2	1	no	—
17	C₂²×Dic₁₇	—	2	2	no	—
18	#63252	—	2	2	no	—
19	#63252	—	2	2	no	—
20	C₂×D₆₈	D₄	1	1	yes	characteristic
21	C₂×C₄×D₁₇	—	2	1	no	—
22	C₂×Dic₃₄	D₄	1	1	yes	characteristic
23	M₁₆×C₁₇	—	2	2	no	—
24	M₁₆×C₁₇	—	2	2	no	—
25	C₂×C₈×C₁₇	—	2	2	no	—
26	C₂×C₈×C₁₇	—	2	2	no	—
27	C₂×C₈×C₁₇	D₄	1	2	yes	—
28	C₄⋊Dic₁₇	—	2	2	no	—
29	C₄⋊Dic₁₇	—	2	2	no	—
30	#63223	—	2	2	no	—
31	C₂×#3622	—	2	1	no	—
32	C₂³×C₁₇	#41	1	1	yes	characteristic
33	C₂²×D₁₇	—	2	1	no	—
34	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	#41	1	2	yes	—
37	C₂×C₄×C₁₇	#41	1	2	yes	—
38	C₂²⋊D₁₇	—	4	2	no	—
39	C₂×Dic₁₇	—	2	1	no	—
40	C₂×Dic₁₇	—	4	2	no	—
41	C₂×Dic₁₇	—	8	2	no	—
42	D₆₈	—	2	2	no	—
43	C₄×D₁₇	—	4	1	no	—
44	C₄×D₁₇	—	4	1	no	—
45	Dic₃₄	—	2	2	no	—
46	C₈×C₁₇	—	2	2	no	—
47	C₈×C₁₇	—	4	2	no	—
48	#3622	—	4	1	no	—
49	#2078	—	17	1	no	maximal, Sylow
50	C₂²×C₁₇	—	2	1	no	—
51	C₂²×C₁₇	—	2	1	no	—
52	C₂²×C₁₇	#121	1	1	yes	characteristic
53	C₂²×C₁₇	#126	1	2	yes	—
54	C₂×D₁₇	—	4	1	no	—
55	C₂×D₁₇	—	4	1	no	—
56	C₂×D₁₇	—	4	1	no	—
57	C₄×C₁₇	#123	1	2	yes	—
58	C₄×C₁₇	#123	1	1	yes	characteristic
59	C₄×C₁₇	#124	1	1	yes	characteristic
60	Dic₁₇	—	4	1	no	—
61	Dic₁₇	—	8	2	no	—
62	C₂×#107	—	17	1	no	—
63	#417	—	17	2	no	—
64	#408	—	17	1	no	—
65	#343	—	17	2	no	—
66	#341	D₁₇	1	1	yes	characteristic, complemented
67	C₂×C₁₇	—	2	2	no	—
68	C₂×C₁₇	#397	1	1	yes	characteristic
69	C₂×C₁₇	#472	1	1	yes	characteristic
70	C₂×C₁₇	#473	1	1	yes	characteristic
71	D₁₇	—	8	1	no	—
72	C₂×#41	—	17	1	no	—
73	C₂×M₁₆	—	17	1	no	—
74	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
75	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	17	2	no	—
77	#107	—	34	2	no	—
78	#103	—	34	2	no	—
79	#102	—	34	2	no	—
80	C₂²⋊C₈	—	34	2	no	—
81	C₁₇	#2078	1	1	yes	characteristic, complemented, Sylow
82	#41	—	34	2	no	—
83	#41	—	34	2	no	—
84	C₂×Q₈	—	17	1	no	—
85	C₂×D₄	—	17	1	no	—
86	C₂×D₄	—	34	1	no	—
87	C₂²×C₄	—	34	1	no	—
88	C₂²×C₄	—	34	2	no	—
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	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	34	1	no	—
96	C₂×C₈	D₆₈	1	2	yes	—
97	C₄⋊C₄	—	34	2	no	—
98	C₄⋊C₄	—	34	2	no	—
99	C₂³	—	34	1	no	—
100	C₂³	#63256	1	1	yes	characteristic
101	Q₈	—	34	2	no	—
102	D₄	—	34	2	no	—
103	D₄	—	68	2	no	—
104	C₂×C₄	—	34	1	no	—
105	C₂×C₄	—	68	1	no	—
106	C₂×C₄	—	68	1	no	—
107	C₂×C₄	—	68	2	no	—
108	C₂×C₄	—	136	2	no	—
109	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
110	C₂×C₄	#63252	1	2	yes	—
111	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
112	C₂×C₄	#63254	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	4	2	no	—
115	C₈	—	68	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	68	1	no	—
119	C₂²	—	68	1	no	—
120	C₂²	—	68	1	no	—
121	C₂²	#10587581	1	1	yes	characteristic
122	C₂²	#10587584	1	2	yes	—
123	C₄	—	68	1	no	—
124	C₄	—	136	2	no	—
125	C₄	#10587561	1	1	yes	characteristic
126	C₄	#10587569	1	2	yes	—
127	C₄	#10587572	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	—	136	1	no	—
130	C₂	#11762736	1	1	yes	characteristic
131	C₂	#11763080	1	1	yes	characteristic
132	C₂	#11763085	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587489

C₂

yes

maximal, characteristic, complemented

#11762811

C₂

yes

maximal

#11762775

C₂

yes

maximal, characteristic

#11762577

C₂

yes

maximal, complemented

C₁₇×#341

C₂

yes

maximal, characteristic, complemented

C₂×#63252

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63223

C₂²

yes

—

—

—

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₂²⋊D₁₇

—

C₂²×Dic₁₇

—

#63252

—

#63252

—

C₂×D₆₈

D₄

yes

characteristic

C₂×C₄×D₁₇

—

C₂×Dic₃₄

D₄

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63223

—

C₂×#3622

—

C₂³×C₁₇

#41

yes

characteristic

C₂²×D₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#2078

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#121

yes

characteristic

C₂²×C₁₇

#126

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

#123

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

#103

—

#102

—

C₂²⋊C₈

—

C₁₇

#2078

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₆₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

100

C₂³

#63256

yes

characteristic

101

Q₈

—

102

D₄

—

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

136

—

109

C₂×C₄

C₂×D₆₈

yes

characteristic

110

C₂×C₄

#63252

yes

—

111

C₂×C₄

D₄×D₁₇

yes

characteristic

112

C₂×C₄

#63254

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#10587581

yes

characteristic

122

C₂²

#10587584

yes

—

123

C₄

—

124

C₄

—

136

—

125

C₄

#10587561

yes

characteristic

126

C₄

#10587569

yes

—

127

C₄

#10587572

yes

characteristic

128

C₂

—

129

C₂

—

136

—

130

C₂

#11762736

yes

characteristic

131

C₂

#11763080

yes

characteristic

132

C₂

#11763085

yes

characteristic

	subgroup	G/H
center	C₂²	#10587581
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#341	C₂
socle	C₂²×C₁₇	#121
prime radical	C₂×#63252	C₂²
hypercenter	#341	D₁₇
hypocenter	C₁₇	#2078
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#10587581

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#341

C₂

C₂²×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
1. C₂×#10587489	C₂	semidirect
4. #11762577	C₂	semidirect
5. C₁₇×#341	C₂	semidirect
66. #341	D₁₇	semidirect
81. C₁₇	#2078	semidirect

H ≅ G/N

type

1. C₂×#10587489

C₂

semidirect

4. #11762577

C₂

semidirect

5. C₁₇×#341

C₂

semidirect

66. #341

D₁₇

semidirect

81. C₁₇

#2078

semidirect

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

subgroup	index
C₂×#10587489	2
#11762811	2
#11762775	2
#11762577	2
C₁₇×#341	2
#2078	17

subgroup

index

C₁₇×#341

Subgroups — #14530226

Conjugacy classes of subgroups

Complements

Maximal subgroups