Group #14529520 — subgroups

Subgroups — #14529520

Order 2176.

240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 105 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
2	C₂×#10587486	C₂	1	1	yes	maximal, characteristic
3	#11762705	C₂	1	2	yes	maximal
4	#11762571	C₂	1	2	yes	maximal
5	C₂×C₁₇×#106	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
9	C₂×#63225	—	2	2	no	—
10	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂²×#3622	C₂²	1	1	yes	characteristic
12	C₁₇×#106	—	2	2	no	—
13	#10587502	—	2	2	no	—
14	#10587489	C₄	1	4	yes	complemented
15	#10587486	—	2	2	no	—
16	C₂²×C₄×C₁₇	—	2	2	no	—
17	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₇	—	2	2	no	—
19	C₂²×Dic₁₇	D₄	1	2	yes	—
20	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₁₇	D₄	1	2	yes	—
22	C₂×C₈×C₁₇	Q₈	1	2	yes	—
23	C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₄⋊Dic₁₇	—	2	2	no	—
26	C₄⋊Dic₁₇	C₂×C₄	1	4	yes	—
27	#63225	—	4	4	no	—
28	C₄×Dic₁₇	—	2	1	no	—
29	C₄×Dic₁₇	—	2	1	no	—
30	C₂×#3622	—	2	1	no	—
31	C₂×#3622	—	2	1	no	—
32	C₂×#3622	Q₈	1	2	yes	—
33	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇	—	4	2	no	—
35	C₂×C₄×C₁₇	—	4	2	no	—
36	C₂×C₄×C₁₇	—	4	2	no	—
37	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂×Dic₁₇	—	2	4	no	—
44	C₂×Dic₁₇	—	2	2	no	—
45	C₂×Dic₁₇	—	2	2	no	—
46	C₂×Dic₁₇	—	2	2	no	—
47	C₂×Dic₁₇	—	4	2	no	—
48	C₂×Dic₁₇	—	4	2	no	—
49	C₈×C₁₇	C₄⋊C₄	1	4	yes	complemented
50	#3622	—	2	2	no	—
51	#1943	—	17	1	no	maximal, Sylow
52	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₁₇	#121	1	1	yes	characteristic
54	C₂²×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
55	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
56	C₂²×C₁₇	#137	1	2	yes	—
57	C₄×C₁₇	—	8	2	no	—
58	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₄×C₁₇	#125	1	1	yes	characteristic
62	Dic₁₇	—	4	4	no	—
63	Dic₁₇	—	4	2	no	—
64	C₂×#107	—	17	1	no	—
65	C₂×#106	D₁₇	1	1	yes	characteristic
66	C₂×#97	—	17	1	no	—
67	#389	—	17	2	no	—
68	#340	—	17	2	no	—
69	C₂×C₁₇	#389	1	1	yes	characteristic
70	C₂×C₁₇	#441	1	1	yes	characteristic
71	C₂×C₁₇	#442	1	1	yes	characteristic
72	C₂×C₁₇	#468	1	2	yes	—
73	C₂×C₁₇	#501	1	2	yes	—
74	C₂²×C₈	—	17	1	no	—
75	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	17	2	no	—
77	C₂×C₄⋊C₄	—	34	2	no	—
78	C₂×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
79	C₂×C₄²	—	17	1	no	—
80	#107	—	17	4	no	—
81	#106	—	2	2	no	—
82	#97	—	34	2	no	—
83	#95	—	34	2	no	—
84	C₁₇	#1943	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	17	2	no	—
87	C₂²×C₄	—	34	2	no	—
88	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
89	C₂×C₈	—	17	2	no	—
90	C₂×C₈	—	34	1	no	—
91	C₂×C₈	—	34	1	no	—
92	C₂×C₈	Dic₃₄	1	2	yes	—
93	C₂×C₈	C₄×D₁₇	1	2	yes	—
94	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	17	4	no	—
98	C₄⋊C₄	—	34	2	no	—
99	C₄⋊C₄	—	68	4	no	—
100	C₄²	—	34	1	no	—
101	C₄²	—	34	1	no	—
102	C₂³	D₄×D₁₇	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	34	4	no	—
107	C₂×C₄	—	34	2	no	—
108	C₂×C₄	—	34	2	no	—
109	C₂×C₄	—	34	2	no	—
110	C₂×C₄	—	68	2	no	—
111	C₂×C₄	—	68	2	no	—
112	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
114	C₂×C₄	#63254	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
116	C₂×C₄	#63256	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
118	C₈	—	34	2	no	—
119	C₈	#63225	1	4	yes	complemented
120	C₂²	#10587574	1	1	yes	characteristic
121	C₂²	#10587579	1	1	yes	characteristic
122	C₂²	#10587600	1	2	yes	—
123	C₂²	#10587601	1	2	yes	—
124	C₂²	#10587624	1	1	yes	characteristic
125	C₄	—	8	2	no	—
126	C₄	—	68	2	no	—
127	C₄	—	68	4	no	—
128	C₄	#10587573	1	1	yes	characteristic
129	C₄	#10587576	1	1	yes	characteristic
130	C₄	C₂×#63225	1	1	yes	characteristic
131	C₄	#10587632	1	1	yes	characteristic
132	C₂	#11762705	1	1	yes	characteristic
133	C₂	#11762915	1	1	yes	characteristic
134	C₂	#11762920	1	1	yes	characteristic
135	C₂	#11763052	1	2	yes	—
136	C₂	#11763216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587489

C₂

yes

maximal, characteristic

C₂×#10587486

C₂

yes

maximal, characteristic

#11762705

C₂

yes

maximal

#11762571

C₂

yes

maximal

C₂×C₁₇×#106

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63225

—

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

characteristic

C₁₇×#106

—

#10587502

—

#10587489

C₄

yes

complemented

#10587486

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

D₄

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

Q₈

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

Q₈

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

C₄⋊C₄

yes

—

—

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₂²×C₁₇

#121

yes

characteristic

C₂²×C₁₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₇

#137

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₄×Q₈

yes

characteristic

C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₇

#125

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂×#107

—

C₂×#106

D₁₇

yes

characteristic

C₂×#97

—

#389

—

#340

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₂×C₁₇

#501

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄²

—

—

—

—

—

C₁₇

#1943

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₃₄

yes

—

C₂×C₈

C₄×D₁₇

yes

—

C₂×C₈

C₂²⋊D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₂³

D₄×D₁₇

yes

characteristic

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

C₂×Dic₃₄

yes

characteristic

113

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

114

C₂×C₄

#63254

yes

characteristic

115

C₂×C₄

Q₈×D₁₇

yes

characteristic

116

C₂×C₄

#63256

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

118

C₈

—

119

C₈

#63225

yes

complemented

120

C₂²

#10587574

yes

characteristic

121

C₂²

#10587579

yes

characteristic

122

C₂²

#10587600

yes

—

123

C₂²

#10587601

yes

—

124

C₂²

#10587624

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#10587573

yes

characteristic

129

C₄

#10587576

yes

characteristic

130

C₄

C₂×#63225

yes

characteristic

131

C₄

#10587632

yes

characteristic

132

C₂

#11762705

yes

characteristic

133

C₂

#11762915

yes

characteristic

134

C₂

#11762920

yes

characteristic

135

C₂

#11763052

yes

—

136

C₂

#11763216

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#106	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#106	D₁₇
hypocenter	C₁₇	#1943
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×#106

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
14. #10587489	C₄	semidirect
49. C₈×C₁₇	C₄⋊C₄	semidirect
84. C₁₇	#1943	semidirect
119. C₈	#63225	semidirect

H ≅ G/N

type

14. #10587489

C₄

semidirect

49. C₈×C₁₇

C₄⋊C₄

semidirect

84. C₁₇

#1943

semidirect

119. C₈

#63225

semidirect

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

subgroup	index
C₂×#10587489	2
C₂×#10587486	2
#11762705	2
#11762571	2
C₂×C₁₇×#106	2
#1943	17

subgroup

index

C₂×C₁₇×#106

Subgroups — #14529520

Conjugacy classes of subgroups

Complements

Maximal subgroups