Group #14539529 — subgroups

Subgroups — #14539529

Order 2176.

358 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 133 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	#11763366	C₂	1	2	yes	maximal
2	#11763207	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763189	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763160	C₂	1	2	yes	maximal, complemented
5	#11763067	C₂	1	4	yes	maximal
6	#11762917	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
10	#10587612	—	2	2	no	—
11	#10587610	—	2	2	no	—
12	#10587610	C₂²	1	4	yes	—
13	C₂×Dic₆₈	C₂²	1	4	yes	—
14	#10587556	—	2	2	no	—
15	#10587555	C₂²	1	2	yes	—
16	#10587555	C₂²	1	2	yes	—
17	C₄×Dic₃₄	—	2	2	no	—
18	C₄×Dic₃₄	C₂²	1	4	yes	—
19	C₁₇×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	#10587488	C₂²	1	4	yes	—
23	#10587487	C₂²	1	8	yes	—
24	C₂²×C₄×C₁₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₃₄	—	2	2	no	—
27	C₂×Dic₃₄	C₂³	1	4	yes	—
28	M₁₆×C₁₇	—	2	4	no	—
29	C₂×C₈×C₁₇	C₂³	1	2	yes	—
30	C₂×C₈×C₁₇	C₂³	1	2	yes	—
31	C₄²×C₁₇	D₄	1	2	yes	—
32	C₄²×C₁₇	C₂³	1	1	yes	characteristic
33	C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₁₇	—	2	4	no	—
35	C₄⋊Dic₁₇	—	2	2	no	—
36	C₄⋊Dic₁₇	—	2	2	no	—
37	C₄⋊Dic₁₇	—	2	2	no	—
38	C₄⋊Dic₁₇	C₂³	1	4	yes	—
39	#63225	—	2	4	no	—
40	#63225	—	2	4	no	—
41	C₄×Dic₁₇	—	2	4	no	—
42	Dic₆₈	—	2	8	no	—
43	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	—	2	4	no	—
45	C₂×C₄×C₁₇	—	2	2	no	—
46	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₇	—	2	4	no	—
51	C₂×Dic₁₇	—	2	4	no	—
52	Dic₃₄	—	2	8	no	—
53	Dic₃₄	—	4	4	no	—
54	C₈×C₁₇	—	2	2	no	—
55	C₈×C₁₇	#41	1	4	yes	—
56	#3185	—	17	1	no	maximal, Sylow
57	C₂²×C₁₇	—	2	1	no	—
58	C₂²×C₁₇	—	2	2	no	—
59	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₁₇	—	2	1	no	—
61	C₄×C₁₇	—	2	2	no	—
62	C₄×C₁₇	#137	1	4	yes	—
63	C₄×C₁₇	C₂×#41	1	2	yes	—
64	Dic₁₇	—	4	4	no	—
65	Dic₁₇	—	4	4	no	—
66	#533	—	17	2	no	—
67	#498	—	17	1	no	—
68	#494	—	17	1	no	—
69	#487	—	17	2	no	—
70	#470	—	17	4	no	—
71	#441	—	17	4	no	—
72	C₄×M₁₆	D₁₇	1	1	yes	characteristic
73	C₂×C₁₇	—	4	1	no	—
74	C₂×C₁₇	#532	1	1	yes	characteristic
75	C₂×C₁₇	C₂×#137	1	2	yes	—
76	C₂×Q₁₆	—	17	4	no	—
77	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
78	C₄⋊Q₈	—	17	2	no	—
79	C₄⋊Q₈	—	17	2	no	—
80	#125	—	34	2	no	—
81	C₂²⋊Q₈	—	17	4	no	—
82	C₂²⋊Q₈	—	34	2	no	—
83	C₄×Q₈	—	17	4	no	—
84	C₄×Q₈	—	34	2	no	—
85	#117	—	34	2	no	—
86	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
87	#106	—	17	4	no	—
88	#103	—	17	8	no	—
89	#97	C₂×D₁₇	1	2	yes	—
90	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
91	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
92	C₁₇	#3185	1	1	yes	characteristic, complemented, Sylow
93	C₂×Q₈	—	17	4	no	—
94	C₂×Q₈	—	34	2	no	—
95	C₂²×C₄	D₆₈	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
97	Q₁₆	—	34	8	no	—
98	M₁₆	—	2	4	no	—
99	C₂×C₈	C₂²×D₁₇	1	2	yes	—
100	C₂×C₈	C₂²×D₁₇	1	2	yes	—
101	C₄⋊C₄	—	17	4	no	—
102	C₄⋊C₄	—	34	2	no	—
103	C₄⋊C₄	—	34	2	no	—
104	C₄⋊C₄	—	34	4	no	—
105	C₄⋊C₄	—	34	4	no	—
106	C₄⋊C₄	—	34	2	no	—
107	C₂²⋊C₄	—	34	4	no	—
108	C₄²	—	34	4	no	—
109	C₄²	D₆₈	1	2	yes	—
110	C₄²	C₂²×D₁₇	1	1	yes	characteristic
111	C₄²	C₂²×D₁₇	1	1	yes	characteristic
112	C₂³	C₂×D₆₈	1	1	yes	characteristic
113	Q₈	—	34	8	no	—
114	Q₈	—	68	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	34	4	no	—
118	C₂×C₄	—	34	4	no	—
119	C₂×C₄	C₂×D₆₈	1	2	yes	—
120	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
121	C₂×C₄	C₂×D₆₈	1	2	yes	—
122	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	#63252	1	4	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	C₂²×D₆₈	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	68	4	no	—
131	C₄	—	68	4	no	—
132	C₄	#10587595	1	4	yes	—
133	C₄	C₂×#63252	1	2	yes	—
134	C₂	—	4	1	no	—
135	C₂	#11763359	1	1	yes	characteristic
136	C₂	C₂×#10587595	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763366

C₂

yes

maximal

#11763207

C₂

yes

maximal, characteristic, complemented

#11763189

C₂

yes

maximal, characteristic, complemented

#11763160

C₂

yes

maximal, complemented

#11763067

C₂

yes

maximal

#11762917

C₂

yes

maximal, complemented

C₄×M₁₆×C₁₇

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

—

—

C₂²

yes

—

C₂×Dic₆₈

C₂²

yes

—

#10587556

—

#10587555

C₂²

yes

—

#10587555

C₂²

yes

—

C₄×Dic₃₄

—

C₄×Dic₃₄

C₂²

yes

—

C₁₇×#97

C₂²

yes

—

C₄×C₈×C₁₇

C₂²

yes

characteristic

C₄×C₈×C₁₇

C₂²

yes

characteristic

#10587488

C₂²

yes

—

#10587487

C₂²

yes

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

C₂³

yes

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂³

yes

—

C₂×C₈×C₁₇

C₂³

yes

—

C₄²×C₁₇

D₄

yes

—

C₄²×C₁₇

C₂³

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂³

yes

—

#63225

—

#63225

—

C₄×Dic₁₇

—

Dic₆₈

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

#41

yes

—

#3185

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#137

yes

—

C₄×C₁₇

C₂×#41

yes

—

Dic₁₇

—

Dic₁₇

—

—

—

—

—

—

—

C₄×M₁₆

D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#532

yes

characteristic

C₂×C₁₇

C₂×#137

yes

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₁₇

yes

—

—

C₂×D₁₇

yes

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#3185

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

D₆₈

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

C₂²×D₁₇

yes

—

100

C₂×C₈

C₂²×D₁₇

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

D₆₈

yes

—

110

C₄²

C₂²×D₁₇

yes

characteristic

111

C₄²

C₂²×D₁₇

yes

characteristic

112

C₂³

C₂×D₆₈

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×D₆₈

yes

—

120

C₂×C₄

C₂×D₆₈

yes

characteristic

121

C₂×C₄

C₂×D₆₈

yes

—

122

C₂×C₄

C₂³×D₁₇

yes

characteristic

123

C₈

—

124

C₈

#63252

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₆₈

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#10587595

yes

—

133

C₄

C₂×#63252

yes

—

134

C₂

—

135

C₂

#11763359

yes

characteristic

136

C₂

C₂×#10587595

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₄×M₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	C₄×M₁₆	D₁₇
hypocenter	C₁₇	#3185
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₆₈

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₄×M₁₆×C₁₇

C₂

socle

C₂²×C₁₇

C₂²×D₄

prime radical

C₂³×C₁₇

C₂×D₄

hypercenter

C₄×M₁₆

D₁₇

hypocenter

C₁₇

#3185

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #11763207	C₂	semidirect
3. #11763189	C₂	semidirect
4. #11763160	C₂	semidirect
6. #11762917	C₂	semidirect
92. C₁₇	#3185	semidirect

H ≅ G/N

type

2. #11763207

C₂

semidirect

3. #11763189

C₂

semidirect

4. #11763160

C₂

semidirect

6. #11762917

C₂

semidirect

92. C₁₇

#3185

semidirect

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

subgroup	index
#11763366	2
#11763207	2
#11763189	2
#11763160	2
#11763067	2
#11762917	2
C₄×M₁₆×C₁₇	2
#3185	17

subgroup

index

C₄×M₁₆×C₁₇

Subgroups — #14539529

Conjugacy classes of subgroups

Complements

Maximal subgroups