Group #14530338 — subgroups

Subgroups — #14530338

Order 2176.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#11762858	C₂	1	1	yes	maximal, characteristic, complemented
2	#11762836	C₂	1	2	yes	maximal, complemented
3	#11762827	C₂	1	2	yes	maximal
4	#11762777	C₂	1	1	yes	maximal, characteristic
5	C₁₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#63252	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#117	C₂²	1	2	yes	—
9	C₂×#63223	C₂²	1	2	yes	—
10	C₂²×#3622	C₂²	1	1	yes	characteristic
11	#10587491	—	2	2	no	—
12	#10587490	—	2	2	no	—
13	#10587476	—	2	4	no	—
14	C₄⋊#3622	—	2	4	no	—
15	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
16	C₂×C₂²⋊D₁₇	—	2	1	no	—
17	#63252	—	2	2	no	—
18	#63252	—	2	2	no	—
19	C₂×D₆₈	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₁₇	—	2	1	no	—
21	C₂×Dic₃₄	Q₈	1	1	yes	characteristic
22	M₁₆×C₁₇	—	2	2	no	—
23	C₂×C₈×C₁₇	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₂²⋊C₄×C₁₇	—	2	2	no	—
26	C₄²×C₁₇	D₄	1	4	yes	—
27	#63223	—	2	2	no	—
28	#63223	—	2	2	no	—
29	C₂×#3622	—	2	2	no	—
30	C₂×#3622	—	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	—	2	4	no	—
35	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₇	#41	1	2	yes	—
39	C₂²⋊D₁₇	—	4	2	no	—
40	C₂×Dic₁₇	—	2	1	no	—
41	D₆₈	—	2	2	no	—
42	C₄×D₁₇	—	4	1	no	—
43	C₄×D₁₇	—	4	1	no	—
44	Dic₃₄	—	2	2	no	—
45	C₈×C₁₇	—	2	2	no	—
46	#3622	—	4	1	no	—
47	#3622	—	4	2	no	—
48	#2098	—	17	1	no	maximal, Sylow
49	C₂²×C₁₇	—	4	1	no	—
50	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₇	#123	1	2	yes	—
52	C₂×D₁₇	—	4	2	no	—
53	C₂×D₁₇	—	4	1	no	—
54	C₄×C₁₇	—	4	4	no	—
55	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
56	C₄×C₁₇	#123	1	1	yes	characteristic
57	C₄×C₁₇	#127	1	1	yes	characteristic
58	Dic₁₇	—	4	1	no	—
59	#430	—	17	1	no	—
60	#424	—	17	2	no	—
61	#421	—	17	2	no	—
62	#408	—	17	1	no	—
63	H(3,ℤ/4)	D₁₇	1	1	yes	characteristic, complemented
64	C₂×C₁₇	—	2	2	no	—
65	C₂×C₁₇	—	2	1	no	—
66	C₂×C₁₇	#399	1	1	yes	characteristic
67	D₁₇	—	8	1	no	—
68	C₂×#41	—	17	1	no	—
69	C₂×M₁₆	—	17	2	no	—
70	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
71	C₂²×C₈	—	17	1	no	—
72	#117	C₂×D₁₇	1	2	yes	—
73	#108	—	34	2	no	—
74	C₄⋊C₈	—	34	4	no	—
75	#104	—	34	4	no	—
76	C₂²⋊C₈	—	34	2	no	—
77	C₁₇	#2098	1	1	yes	characteristic, complemented, Sylow
78	#41	—	34	2	no	—
79	#41	—	34	2	no	—
80	C₂×Q₈	—	17	1	no	—
81	C₂×D₄	—	17	1	no	—
82	C₂×D₄	—	34	1	no	—
83	C₂²×C₄	—	34	1	no	—
84	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	34	2	no	—
87	M₁₆	—	34	2	no	—
88	C₂×C₈	—	34	1	no	—
89	C₂×C₈	—	34	2	no	—
90	C₂×C₈	—	34	2	no	—
91	C₂×C₈	D₆₈	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₆₈	1	4	yes	—
95	C₂³	—	34	1	no	—
96	C₂³	#63254	1	1	yes	characteristic
97	Q₈	—	34	2	no	—
98	D₄	—	34	2	no	—
99	D₄	—	68	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	34	1	no	—
102	C₂×C₄	—	68	1	no	—
103	C₂×C₄	—	68	1	no	—
104	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
105	C₂×C₄	C₂×D₆₈	1	2	yes	—
106	C₂×C₄	#63254	1	2	yes	—
107	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	68	1	no	—
110	C₈	—	68	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	68	2	no	—
113	C₂²	—	68	1	no	—
114	C₂²	#10587572	1	2	yes	—
115	C₂²	#10587583	1	1	yes	characteristic
116	C₄	—	4	4	no	—
117	C₄	—	68	1	no	—
118	C₄	#10587560	1	1	yes	characteristic
119	C₄	#10587572	1	1	yes	characteristic
120	C₄	#10587583	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	136	1	no	—
124	C₂	#11762746	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762858

C₂

yes

maximal, characteristic, complemented

#11762836

C₂

yes

maximal, complemented

#11762827

C₂

yes

maximal

#11762777

C₂

yes

maximal, characteristic

C₁₇×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#63252

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

—

C₂×#63223

C₂²

yes

—

C₂²×#3622

C₂²

yes

—

—

—

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₂²⋊D₁₇

—

#63252

—

#63252

—

C₂×D₆₈

Q₈

yes

characteristic

C₂×C₄×D₁₇

—

C₂×Dic₃₄

Q₈

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

D₄

yes

—

—

—

—

—

—

C₂³×C₁₇

#41

yes

characteristic

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇

#123

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#399

yes

characteristic

D₁₇

—

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

—

#117

C₂×D₁₇

yes

—

#108

—

C₄⋊C₈

—

#104

—

C₂²⋊C₈

—

C₁₇

#2098

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₆₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₆₈

yes

—

C₂³

—

C₂³

#63254

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₆₈

yes

characteristic

105

C₂×C₄

C₂×D₆₈

yes

—

106

C₂×C₄

#63254

yes

—

107

C₂×C₄

Q₈×D₁₇

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#10587572

yes

—

115

C₂²

#10587583

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

#10587560

yes

characteristic

119

C₄

#10587572

yes

characteristic

120

C₄

#10587583

yes

—

121

C₂

—

122

C₂

—

123

C₂

—

136

—

124

C₂

#11762746

yes

characteristic

	subgroup	G/H
center	C₄	#10587560
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×H(3,ℤ/4)	C₂
socle	C₂×C₁₇	#399
prime radical	C₂×#63252	C₂²
hypercenter	H(3,ℤ/4)	D₁₇
hypocenter	C₁₇	#2098
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#10587560

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×H(3,ℤ/4)

C₂

C₂×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #11762858	C₂	semidirect
2. #11762836	C₂	semidirect
5. C₁₇×H(3,ℤ/4)	C₂	semidirect
63. H(3,ℤ/4)	D₁₇	semidirect
77. C₁₇	#2098	semidirect

H ≅ G/N

type

1. #11762858

C₂

semidirect

2. #11762836

C₂

semidirect

5. C₁₇×H(3,ℤ/4)

C₂

semidirect

63. H(3,ℤ/4)

D₁₇

semidirect

77. C₁₇

#2098

semidirect

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

subgroup	index
#11762858	2
#11762836	2
#11762827	2
#11762777	2
C₁₇×H(3,ℤ/4)	2
#2098	17

subgroup

index

C₁₇×H(3,ℤ/4)

Subgroups — #14530338

Conjugacy classes of subgroups

Complements

Maximal subgroups