Group #14529932 — subgroups

Subgroups — #14529932

Order 2176.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#10587481	C₂	1	2	yes	maximal
2	#11762715	C₂	1	1	yes	maximal, characteristic
3	#11762571	C₂	1	2	yes	maximal
4	#11762564	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#125	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄	C₂²	1	1	yes	characteristic
7	C₁₇×#125	—	2	4	no	—
8	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
12	C₂²×#3622	C₂²	1	2	yes	—
13	#10587502	—	2	2	no	—
14	#10587481	—	2	4	no	—
15	C₂²×C₄×C₁₇	—	2	2	no	—
16	C₂²×C₄×C₁₇	D₄	1	2	yes	—
17	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₇	—	2	1	no	—
19	C₂²×Dic₁₇	—	2	1	no	—
20	C₂×Dic₃₄	—	2	2	no	—
21	C₂×Dic₃₄	—	2	2	no	—
22	C₂×Dic₃₄	—	2	1	no	—
23	C₂×Dic₃₄	Q₈	1	2	yes	—
24	C₄⋊C₄×C₁₇	—	2	8	no	—
25	C₄⋊C₄×C₁₇	—	2	2	no	—
26	C₄⋊C₄×C₁₇	D₄	1	4	yes	—
27	C₄²×C₁₇	—	2	2	no	—
28	C₄⋊Dic₁₇	—	2	2	no	—
29	C₂×#3622	—	2	2	no	—
30	C₂×#3622	—	2	2	no	—
31	C₂×#3622	—	2	2	no	—
32	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇	—	2	2	no	—
34	C₂×C₄×C₁₇	—	2	4	no	—
35	C₂×C₄×C₁₇	—	2	2	no	—
36	C₂×C₄×C₁₇	—	2	2	no	—
37	C₂×C₄×C₁₇	—	4	2	no	—
38	C₂×C₄×C₁₇	—	4	2	no	—
39	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₇	#41	1	2	yes	—
42	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂×Dic₁₇	—	2	2	no	—
44	C₂×Dic₁₇	—	4	1	no	—
45	C₂×Dic₁₇	—	4	1	no	—
46	C₂×Dic₁₇	—	4	1	no	—
47	C₂×Dic₁₇	—	4	2	no	—
48	Dic₃₄	—	2	4	no	—
49	Dic₃₄	—	4	2	no	—
50	#3622	—	4	2	no	—
51	#2026	—	17	1	no	maximal, Sylow
52	C₂²×C₁₇	#120	1	1	yes	characteristic
53	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₇	#124	1	1	yes	characteristic
55	C₂²×C₁₇	#135	1	2	yes	—
56	C₂²×C₁₇	#137	1	2	yes	—
57	C₄×C₁₇	—	4	2	no	—
58	C₄×C₁₇	—	4	4	no	—
59	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₁₇	#124	1	2	yes	—
61	Dic₁₇	—	4	2	no	—
62	Dic₁₇	—	8	1	no	—
63	C₂×#125	D₁₇	1	1	yes	characteristic
64	C₂×#103	—	17	2	no	—
65	#391	—	17	1	no	—
66	#340	—	17	2	no	—
67	#336	—	17	1	no	—
68	C₂×C₁₇	#393	1	1	yes	characteristic
69	C₂×C₁₇	#452	1	2	yes	—
70	C₂×C₁₇	#479	1	2	yes	—
71	C₂×C₁₇	#488	1	1	yes	characteristic
72	C₂×C₁₇	#491	1	1	yes	characteristic
73	C₂²×Q₈	—	17	1	no	—
74	C₂²×C₈	—	17	2	no	—
75	#125	—	2	4	no	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	17	1	no	—
78	C₂×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
79	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
80	#103	—	34	4	no	—
81	#95	—	34	2	no	—
82	C₁₇	#2026	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	17	2	no	—
84	C₂×Q₈	—	34	1	no	—
85	C₂×Q₈	—	34	2	no	—
86	C₂×Q₈	—	34	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	34	1	no	—
89	C₂²×C₄	—	34	1	no	—
90	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
92	C₂×C₈	—	34	2	no	—
93	C₂×C₈	—	34	2	no	—
94	C₂×C₈	—	34	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	34	2	no	—
98	C₄⋊C₄	C₂²⋊D₁₇	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
101	Q₈	—	34	4	no	—
102	Q₈	—	68	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	34	2	no	—
110	C₂×C₄	—	68	1	no	—
111	C₂×C₄	—	68	1	no	—
112	C₂×C₄	—	68	1	no	—
113	C₂×C₄	—	68	2	no	—
114	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
115	C₂×C₄	#63256	1	2	yes	—
116	C₂×C₄	#63256	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
118	C₈	—	68	2	no	—
119	C₂²	#10587632	1	1	yes	characteristic
120	C₂²	#10587633	1	1	yes	characteristic
121	C₂²	#10587636	1	2	yes	—
122	C₂²	#10587637	1	2	yes	—
123	C₂²	#10587639	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	68	2	no	—
127	C₄	—	136	1	no	—
128	C₄	#10587632	1	2	yes	—
129	C₄	#10587633	1	2	yes	—
130	C₂	#11762720	1	1	yes	characteristic
131	C₂	#11762971	1	2	yes	—
132	C₂	#11763113	1	2	yes	—
133	C₂	#11763166	1	1	yes	characteristic
134	C₂	#11763179	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587481

C₂

yes

maximal

#11762715

C₂

yes

maximal, characteristic

#11762571

C₂

yes

maximal

#11762564

C₂

yes

maximal, characteristic

C₂×C₁₇×#125

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₂²

yes

characteristic

C₁₇×#125

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

—

#10587502

—

#10587481

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

Q₈

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

C₂²⋊Q₈

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#137

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

#124

yes

—

Dic₁₇

—

Dic₁₇

—

C₂×#125

D₁₇

yes

—

—

—

—

C₂×C₁₇

#393

yes

characteristic

C₂×C₁₇

#452

yes

—

C₂×C₁₇

#479

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#103

—

#95

—

C₁₇

#2026

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₇

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

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

Q₈×D₁₇

yes

characteristic

115

C₂×C₄

#63256

yes

—

116

C₂×C₄

#63256

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

118

C₈

—

119

C₂²

#10587632

yes

characteristic

120

C₂²

#10587633

yes

characteristic

121

C₂²

#10587636

yes

—

122

C₂²

#10587637

yes

—

123

C₂²

#10587639

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

136

—

128

C₄

#10587632

yes

—

129

C₄

#10587633

yes

—

130

C₂

#11762720

yes

characteristic

131

C₂

#11762971

yes

—

132

C₂

#11763113

yes

—

133

C₂

#11763166

yes

characteristic

134

C₂

#11763179

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#125	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	C₂×#125	D₁₇
hypocenter	C₁₇	#2026
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×#125

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
82. C₁₇	#2026	semidirect

H ≅ G/N

type

82. C₁₇

#2026

semidirect

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

subgroup	index
C₂×#10587481	2
#11762715	2
#11762571	2
#11762564	2
C₂×C₁₇×#125	2
#2026	17

subgroup

index

C₂×C₁₇×#125

Subgroups — #14529932

Conjugacy classes of subgroups

Complements

Maximal subgroups