Group #14530939 — subgroups

Subgroups — #14530939

Order 2176.

136 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 49 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	#11763210	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#11762636	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762635	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762627	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762610	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#357	C₂	1	1	yes	maximal, characteristic
8	C₂×D₈×C₁₇	C₂²	1	1	yes	characteristic
9	D₄×Dic₁₇	—	2	1	no	—
10	#10587576	—	2	1	no	—
11	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×#107	C₂²	1	1	yes	characteristic
13	#10587503	—	2	1	no	—
14	#10587489	C₂²	1	1	yes	characteristic
15	#10587489	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63214	C₂²	1	1	yes	characteristic
18	#10587479	—	2	1	no	—
19	C₂×D₄×C₁₇	—	2	1	no	—
20	C₂²×Dic₁₇	—	4	1	no	—
21	D₈×C₁₇	—	2	1	no	—
22	D₈×C₁₇	Q₈	1	2	yes	—
23	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₇	—	2	1	no	—
25	C₂²⋊Dic₁₇	—	4	1	no	—
26	C₄⋊Dic₁₇	—	2	1	no	—
27	C₄⋊Dic₁₇	—	2	1	no	—
28	C₄⋊Dic₁₇	—	4	1	no	—
29	#63225	—	4	1	no	—
30	C₄×Dic₁₇	D₄	1	1	yes	characteristic
31	C₂×#3622	D₄	1	1	yes	characteristic
32	C₁₆×C₁₇	—	2	1	no	—
33	#63214	—	2	1	no	—
34	C₂³×C₁₇	—	4	1	no	—
35	D₄×C₁₇	—	2	2	no	—
36	D₄×C₁₇	—	4	1	no	—
37	C₂×C₄×C₁₇	—	4	1	no	—
38	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₁₇	—	4	1	no	—
40	C₂×Dic₁₇	—	4	1	no	—
41	C₂×Dic₁₇	—	8	2	no	—
42	C₂×Dic₁₇	D₈	1	1	yes	characteristic
43	C₂×Dic₁₇	D₈	1	1	yes	characteristic
44	C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₁₇	#41	1	1	yes	characteristic
46	#3622	—	2	1	no	—
47	#2228	—	17	1	no	maximal, Sylow
48	C₂²×C₁₇	—	4	2	no	—
49	C₂²×C₁₇	—	8	1	no	—
50	C₂²×C₁₇	—	8	1	no	—
51	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
52	C₄×C₁₇	—	8	1	no	—
53	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₁₇	#137	1	1	yes	characteristic
55	Dic₁₇	—	2	1	no	—
56	Dic₁₇	—	2	1	no	—
57	Dic₁₇	—	8	1	no	—
58	Dic₁₇	—	8	1	no	—
59	#499	—	17	1	no	—
60	C₄×D₈	—	17	1	no	—
61	#367	—	17	1	no	—
62	#366	—	17	1	no	—
63	C₄⋊C₁₆	—	17	1	no	—
64	#357	—	17	1	no	—
65	#357	D₁₇	1	1	yes	characteristic
66	C₂×C₁₇	—	8	2	no	—
67	C₂×C₁₇	#474	1	1	yes	characteristic
68	C₂×C₁₇	#508	1	1	yes	characteristic
69	C₂×C₁₇	#509	1	1	yes	characteristic
70	C₂×D₈	C₂×D₁₇	1	1	yes	characteristic
71	#125	—	34	1	no	—
72	C₄×D₄	—	34	1	no	—
73	C₂×C₁₆	—	17	1	no	—
74	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
75	#107	—	17	1	no	—
76	#107	—	17	1	no	—
77	#107	C₂×D₁₇	1	1	yes	characteristic
78	#106	—	34	1	no	—
79	#102	—	34	1	no	—
80	C₄×C₈	—	17	1	no	—
81	C₁₇	#2228	1	1	yes	characteristic, complemented, Sylow
82	C₂×D₄	—	2	1	no	—
83	C₂²×C₄	—	68	1	no	—
84	D₈	—	2	1	no	—
85	D₈	Dic₃₄	1	2	yes	—
86	C₂×C₈	—	17	1	no	—
87	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	34	1	no	—
90	C₄⋊C₄	—	34	1	no	—
91	C₄⋊C₄	—	68	1	no	—
92	C₄⋊C₄	—	68	1	no	—
93	C₂²⋊C₄	—	68	1	no	—
94	C₄²	—	17	1	no	—
95	C₁₆	—	2	1	no	—
96	C₁₆	—	34	1	no	—
97	C₂³	—	4	1	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	17	1	no	—
102	C₂×C₄	—	17	1	no	—
103	C₂×C₄	—	68	1	no	—
104	C₂×C₄	—	68	1	no	—
105	C₂×C₄	—	136	2	no	—
106	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
107	C₈	—	34	1	no	—
108	C₈	C₂×Dic₃₄	1	1	yes	characteristic
109	C₈	#63254	1	1	yes	characteristic
110	C₂²	—	4	2	no	—
111	C₂²	—	8	1	no	—
112	C₂²	—	8	1	no	—
113	C₂²	D₈×D₁₇	1	1	yes	characteristic
114	C₄	—	8	1	no	—
115	C₄	—	34	1	no	—
116	C₄	—	34	1	no	—
117	C₄	—	136	1	no	—
118	C₄	—	136	1	no	—
119	C₄	#10587564	1	1	yes	characteristic
120	C₄	#10587601	1	1	yes	characteristic
121	C₂	—	8	2	no	—
122	C₂	#11763087	1	1	yes	characteristic
123	C₂	#11763238	1	1	yes	characteristic
124	C₂	#11763246	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763210

C₂

yes

maximal, characteristic, complemented

D₈×Dic₁₇

C₂

yes

maximal, characteristic

#11762636

C₂

yes

maximal, characteristic, complemented

#11762635

C₂

yes

maximal, characteristic, complemented

#11762627

C₂

yes

maximal, characteristic, complemented

#11762610

C₂

yes

maximal, characteristic

C₁₇×#357

C₂

yes

maximal, characteristic

C₂×D₈×C₁₇

C₂²

yes

characteristic

D₄×Dic₁₇

—

#10587576

—

C₂×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#107

C₂²

yes

characteristic

#10587503

—

#10587489

C₂²

yes

characteristic

#10587489

C₂²

yes

characteristic

C₈×Dic₁₇

C₂²

yes

characteristic

C₂×#63214

C₂²

yes

characteristic

#10587479

—

C₂×D₄×C₁₇

—

C₂²×Dic₁₇

—

D₈×C₁₇

—

D₈×C₁₇

Q₈

yes

—

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₇

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

C₁₆×C₁₇

—

#63214

—

C₂³×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

D₈

yes

characteristic

C₂×Dic₁₇

D₈

yes

characteristic

C₈×C₁₇

C₂×Q₈

yes

C₈×C₁₇

yes

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₇

#137

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

#499

—

C₄×D₈

—

#367

—

#366

—

C₄⋊C₁₆

—

#357

—

#357

D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×D₈

C₂×D₁₇

yes

characteristic

#125

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₁₇

yes

—

—

C₂×D₁₇

yes

characteristic

#106

—

#102

—

C₄×C₈

—

C₁₇

#2228

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

Dic₃₄

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

136

—

106

C₂×C₄

D₄×D₁₇

yes

characteristic

107

C₈

—

108

C₈

C₂×Dic₃₄

yes

characteristic

109

C₈

#63254

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

D₈×D₁₇

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

136

—

118

C₄

—

136

—

119

C₄

#10587564

yes

characteristic

120

C₄

#10587601

yes

characteristic

121

C₂

—

122

C₂

#11763087

yes

characteristic

123

C₂

#11763238

yes

characteristic

124

C₂

#11763246

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#357	C₂
socle	C₂²×C₁₇	C₂×D₈
prime radical	C₂×D₈×C₁₇	C₂²
hypercenter	#357	D₁₇
hypocenter	C₁₇	#2228
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₁₇

commutator, cosocle

C₂×C₈×C₁₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₇

Fitting

C₁₇×#357

C₂

socle

C₂²×C₁₇

C₂×D₈

prime radical

C₂×D₈×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. #11763210	C₂	semidirect
3. #11762636	C₂	semidirect
4. #11762635	C₂	semidirect
5. #11762627	C₂	semidirect
81. C₁₇	#2228	semidirect

H ≅ G/N

type

1. #11763210

C₂

semidirect

3. #11762636

C₂

semidirect

4. #11762635

C₂

semidirect

5. #11762627

C₂

semidirect

81. C₁₇

#2228

semidirect

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

subgroup	index
#11763210	2
D₈×Dic₁₇	2
#11762636	2
#11762635	2
#11762627	2
#11762610	2
C₁₇×#357	2
#2228	17

subgroup

index

D₈×Dic₁₇

C₁₇×#357

Subgroups — #14530939

Conjugacy classes of subgroups

Complements

Maximal subgroups