Group #14529608 — subgroups

Subgroups — #14529608

Order 2176.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#10587560	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587492	C₂	1	4	yes	maximal, complemented
3	#11762675	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₆₈	—	2	2	no	—
6	C₂²×D₆₈	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
10	#10587560	—	2	2	no	—
11	#10587560	C₄	1	4	yes	complemented
12	C₁₇×#97	—	2	2	no	—
13	#10587502	—	2	2	no	—
14	#10587492	—	2	8	no	—
15	C₂³×D₁₇	—	2	2	no	—
16	C₂²×C₄×C₁₇	D₄	1	2	yes	—
17	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₇	—	2	2	no	—
19	C₂×D₆₈	—	2	4	no	—
20	C₂×D₆₈	—	2	4	no	—
21	C₂×D₆₈	—	2	8	no	—
22	C₂×D₆₈	—	2	2	no	—
23	C₂×D₆₈	—	4	8	no	—
24	C₂×D₆₈	C₂×C₄	1	4	yes	—
25	C₂×C₈×C₁₇	—	2	2	no	—
26	C₂×C₈×C₁₇	—	2	2	no	—
27	C₂×C₈×C₁₇	D₄	1	4	yes	complemented
28	C₄²×C₁₇	C₂×C₄	1	2	yes	—
29	C₄²×C₁₇	D₄	1	2	yes	—
30	C₄⋊Dic₁₇	—	2	4	no	—
31	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₁₇	—	2	4	no	—
33	C₂²×D₁₇	—	4	2	no	—
34	C₂²×D₁₇	—	4	2	no	—
35	C₂²×D₁₇	—	4	8	no	—
36	C₂×C₄×C₁₇	—	2	4	no	—
37	C₂×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇	#41	1	2	yes	—
42	C₂×Dic₁₇	—	4	2	no	—
43	C₂×Dic₁₇	—	4	4	no	—
44	D₆₈	—	2	8	no	—
45	D₆₈	—	4	16	no	—
46	D₆₈	—	4	4	no	—
47	C₈×C₁₇	—	2	4	no	—
48	#1961	—	17	1	no	maximal, Sylow
49	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₇	#124	1	1	yes	characteristic
51	C₂²×C₁₇	#127	1	1	yes	characteristic
52	C₂²×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
53	C₂×D₁₇	—	4	4	no	—
54	C₂×D₁₇	—	4	8	no	—
55	C₂×D₁₇	—	8	4	no	—
56	C₂×D₁₇	—	8	4	no	—
57	C₄×C₁₇	—	2	4	no	—
58	C₄×C₁₇	C₄×D₄	1	2	yes	—
59	C₄×C₁₇	#121	1	2	yes	—
60	Dic₁₇	—	8	2	no	—
61	C₂×#127	—	17	1	no	—
62	C₂×#102	—	17	4	no	—
63	C₂×#97	D₁₇	1	1	yes	characteristic, complemented
64	#382	—	17	1	no	—
65	C₂×C₁₇	#390	1	1	yes	characteristic
66	C₂×C₁₇	#418	1	2	yes	—
67	C₂×C₁₇	#490	1	2	yes	—
68	C₂×C₁₇	#496	1	2	yes	—
69	D₁₇	—	8	4	no	—
70	C₂²×D₄	—	17	2	no	—
71	C₂²×D₄	—	34	2	no	—
72	C₂²×C₈	C₂×D₁₇	1	2	yes	—
73	#127	—	17	4	no	—
74	#127	—	34	2	no	—
75	C₂×C₄⋊C₄	—	17	2	no	—
76	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
77	#102	—	34	8	no	—
78	#97	—	2	2	no	—
79	#95	—	34	2	no	—
80	C₁₇	#1961	1	1	yes	characteristic, complemented, Sylow
81	C₂⁴	—	34	2	no	—
82	C₂×D₄	—	17	4	no	—
83	C₂×D₄	—	34	4	no	—
84	C₂×D₄	—	34	4	no	—
85	C₂×D₄	—	34	2	no	—
86	C₂×D₄	—	34	8	no	—
87	C₂×D₄	—	68	8	no	—
88	C₂²×C₄	—	34	2	no	—
89	C₂²×C₄	D₆₈	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₆₈	1	4	yes	complemented
94	C₄⋊C₄	—	34	4	no	—
95	C₄²	C₄×D₁₇	1	2	yes	—
96	C₄²	C₂²⋊D₁₇	1	2	yes	—
97	C₂³	—	34	4	no	—
98	C₂³	—	68	2	no	—
99	C₂³	—	68	2	no	—
100	C₂³	—	68	8	no	—
101	C₂³	C₂×D₆₈	1	1	yes	characteristic
102	D₄	—	34	8	no	—
103	D₄	—	68	16	no	—
104	D₄	—	68	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	68	2	no	—
107	C₂×C₄	—	68	4	no	—
108	C₂×C₄	#63227	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
110	C₂×C₄	C₂×D₆₈	1	2	yes	—
111	C₂×C₄	#63252	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
113	C₈	—	2	4	no	—
114	C₂²	—	68	4	no	—
115	C₂²	—	68	8	no	—
116	C₂²	—	136	4	no	—
117	C₂²	—	136	4	no	—
118	C₂²	#10587560	1	1	yes	characteristic
119	C₂²	#10587561	1	1	yes	characteristic
120	C₂²	#10587594	1	4	yes	—
121	C₂²	C₂×#63227	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	136	2	no	—
124	C₄	C₄×D₆₈	1	2	yes	—
125	C₄	#10587616	1	2	yes	—
126	C₂	—	136	4	no	—
127	C₂	#11762711	1	1	yes	characteristic
128	C₂	#11762815	1	2	yes	—
129	C₂	#11763174	1	2	yes	—
130	C₂	#11763198	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587560

C₂

yes

maximal, characteristic

C₂×#10587492

C₂

yes

maximal, complemented

#11762675

C₂

yes

maximal, characteristic, complemented

C₂×C₁₇×#97

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₈

—

C₂²×D₆₈

C₂²

yes

—

C₂²×C₈×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

#10587560

—

#10587560

C₄

yes

complemented

C₁₇×#97

—

#10587502

—

#10587492

—

C₂³×D₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

C₂×C₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

complemented

C₄²×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

D₄

yes

—

C₄⋊Dic₁₇

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊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

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

D₆₈

—

C₈×C₁₇

—

#1961

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄×D₄

yes

—

C₄×C₁₇

#121

yes

—

Dic₁₇

—

C₂×#127

—

C₂×#102

—

C₂×#97

D₁₇

yes

characteristic, complemented

#382

—

C₂×C₁₇

#390

yes

characteristic

C₂×C₁₇

#418

yes

—

C₂×C₁₇

#490

yes

—

C₂×C₁₇

#496

yes

—

D₁₇

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

C₂×D₁₇

yes

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

—

—

—

C₁₇

#1961

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

D₆₈

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₆₈

yes

complemented

C₄⋊C₄

—

C₄²

C₄×D₁₇

yes

—

C₄²

C₂²⋊D₁₇

yes

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×D₆₈

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#63227

yes

—

109

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

110

C₂×C₄

C₂×D₆₈

yes

—

111

C₂×C₄

#63252

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

136

—

117

C₂²

—

136

—

118

C₂²

#10587560

yes

characteristic

119

C₂²

#10587561

yes

characteristic

120

C₂²

#10587594

yes

—

121

C₂²

C₂×#63227

yes

characteristic

122

C₄

—

123

C₄

—

136

—

124

C₄

C₄×D₆₈

yes

—

125

C₄

#10587616

yes

—

126

C₂

—

136

—

127

C₂

#11762711

yes

characteristic

128

C₂

#11762815

yes

—

129

C₂

#11763174

yes

—

130

C₂

#11763198

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#97	C₂
socle	C₂³×C₁₇	C₂×D₄
prime radical	C₂×#10587560	C₂
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#97	D₁₇
hypocenter	C₁₇	#1961
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₆₈

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×#97

C₂

socle

C₂³×C₁₇

C₂×D₄

prime radical

C₂×#10587560

C₂

cosocle

C₂²×C₄×C₁₇

C₂³

hypercenter

C₂×#97

D₁₇

hypocenter

C₁₇

#1961

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#10587492	C₂	semidirect
3. #11762675	C₂	semidirect
4. C₂×C₁₇×#97	C₂	semidirect
11. #10587560	C₄	semidirect
27. C₂×C₈×C₁₇	D₄	semidirect
63. C₂×#97	D₁₇	semidirect
80. C₁₇	#1961	semidirect
93. C₂×C₈	D₆₈	semidirect

H ≅ G/N

type

2. C₂×#10587492

C₂

semidirect

3. #11762675

C₂

semidirect

4. C₂×C₁₇×#97

C₂

semidirect

11. #10587560

C₄

semidirect

27. C₂×C₈×C₁₇

D₄

semidirect

63. C₂×#97

D₁₇

semidirect

80. C₁₇

#1961

semidirect

93. C₂×C₈

D₆₈

semidirect

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

subgroup	index
C₂×#10587560	2
C₂×#10587492	2
#11762675	2
C₂×C₁₇×#97	2
#1961	17

subgroup

index

C₂×C₁₇×#97

Subgroups — #14529608

Conjugacy classes of subgroups

Complements

Maximal subgroups