Group #14544707 — subgroups

Subgroups — #14544707

Order 2176.

224 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 74 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	#11763755	C₂	1	1	yes	maximal, characteristic
2	#11763753	C₂	1	1	yes	maximal, characteristic
3	#11763683	C₂	1	4	yes	maximal
4	#97×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#595	—	2	2	no	—
6	C₂×C₄⋊#595	C₂²	1	2	yes	—
7	C₂×C₄⋊#595	C₂²	1	2	yes	—
8	C₂×C₈×D₁₇	C₂²	1	2	yes	—
9	C₄²×D₁₇	C₂²	1	1	yes	characteristic
10	(C₂×#595)⋊C₄	—	2	2	no	—
11	C₁₇×#97	C₄	1	1	yes	characteristic, complemented
12	#10587486	C₄	1	2	yes	complemented
13	#10587474	C₄	1	1	yes	characteristic, complemented
14	C₂²×#595	—	2	2	no	—
15	C₂²×#595	—	2	2	no	—
16	C₂×C₄×D₁₇	D₄	1	2	yes	—
17	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
18	C₄⋊#595	—	2	4	no	—
19	C₄⋊#595	—	2	4	no	—
20	C₄⋊#595	—	4	4	no	—
21	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₇	Q₈	1	2	yes	—
25	C₂×#3622	C₂×C₄	1	2	yes	—
26	C₈×D₁₇	—	2	2	no	—
27	C₈×D₁₇	—	2	2	no	—
28	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
29	C₂×#595	—	4	2	no	—
30	C₂×#595	—	4	2	no	—
31	C₂×#595	—	4	4	no	—
32	C₂×#595	—	4	2	no	—
33	C₂×#595	—	4	2	no	—
34	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
37	C₂×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
38	C₄×D₁₇	—	2	2	no	—
39	C₄×D₁₇	—	2	2	no	—
40	C₄×D₁₇	#41	1	2	yes	—
41	C₄×D₁₇	#41	1	2	yes	—
42	C₈×C₁₇	—	2	2	no	—
43	#3622	—	2	2	no	—
44	#1836	—	17	1	no	maximal, Sylow
45	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×D₁₇	#124	1	1	yes	characteristic
47	C₂×D₁₇	#124	1	1	yes	characteristic
48	C₂×D₁₇	AGL(1,ℤ/8)	1	2	yes	—
49	C₂×D₁₇	#137	1	2	yes	—
50	#595	—	8	2	no	—
51	#595	—	8	2	no	—
52	C₄×C₁₇	—	2	2	no	—
53	C₄×C₁₇	#117	1	2	yes	—
54	Dic₁₇	—	2	2	no	—
55	Dic₁₇	#125	1	2	yes	—
56	C₂×#97	—	17	1	no	—
57	#384	—	17	1	no	—
58	#382	—	17	1	no	—
59	#340	—	17	4	no	—
60	C₂×C₁₇	#382	1	1	yes	characteristic
61	C₂×C₁₇	#428	1	2	yes	—
62	D₁₇	#489	1	2	yes	—
63	D₁₇	#490	1	1	yes	characteristic
64	D₁₇	#491	1	1	yes	characteristic
65	C₂²×C₈	—	17	2	no	—
66	C₂×C₄⋊C₄	—	17	2	no	—
67	C₂×C₄⋊C₄	—	17	2	no	—
68	C₂×C₄⋊C₄	—	34	2	no	—
69	C₂×C₄²	—	17	1	no	—
70	#97	—	17	2	no	—
71	#97	—	17	1	no	—
72	#97	#595	1	1	yes	characteristic, complemented
73	#95	—	34	2	no	—
74	C₁₇	#1836	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₄	—	17	1	no	—
76	C₂²×C₄	—	17	2	no	—
77	C₂²×C₄	—	34	2	no	—
78	C₂²×C₄	—	34	2	no	—
79	C₂×C₈	—	17	2	no	—
80	C₂×C₈	—	34	2	no	—
81	C₂×C₈	—	34	2	no	—
82	C₂×C₈	C₂×#595	1	2	yes	—
83	C₄⋊C₄	—	34	4	no	—
84	C₄⋊C₄	—	34	4	no	—
85	C₄⋊C₄	—	68	4	no	—
86	C₄²	—	17	2	no	—
87	C₄²	—	17	1	no	—
88	C₄²	C₂×#595	1	1	yes	characteristic
89	C₂³	—	17	1	no	—
90	C₂×C₄	—	17	1	no	—
91	C₂×C₄	—	17	2	no	—
92	C₂×C₄	—	17	2	no	—
93	C₂×C₄	—	17	2	no	—
94	C₂×C₄	—	34	2	no	—
95	C₂×C₄	—	34	2	no	—
96	C₂×C₄	—	68	2	no	—
97	C₂×C₄	—	68	2	no	—
98	C₂×C₄	—	68	4	no	—
99	C₂×C₄	—	68	2	no	—
100	C₂×C₄	—	68	2	no	—
101	C₂×C₄	C₄⋊#595	1	2	yes	—
102	C₂×C₄	C₂²×#595	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	34	2	no	—
105	C₂²	—	17	1	no	—
106	C₂²	—	17	1	no	—
107	C₂²	—	17	2	no	—
108	C₂²	—	17	2	no	—
109	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	17	2	no	—
112	C₄	—	34	2	no	—
113	C₄	—	136	2	no	—
114	C₄	—	136	2	no	—
115	C₄	#10587677	1	2	yes	—
116	C₂	—	17	1	no	—
117	C₂	—	17	2	no	—
118	C₂	—	17	1	no	—
119	C₂	#11763753	1	1	yes	characteristic
120	C₂	#11763820	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763755

C₂

yes

maximal, characteristic

#11763753

C₂

yes

maximal, characteristic

#11763683

C₂

yes

maximal

#97×D₁₇

C₂

yes

maximal, characteristic

C₂×C₄⋊#595

—

C₂×C₄⋊#595

C₂²

yes

—

C₂×C₄⋊#595

C₂²

yes

—

C₂×C₈×D₁₇

C₂²

yes

—

C₄²×D₁₇

C₂²

yes

characteristic

(C₂×#595)⋊C₄

—

C₁₇×#97

C₄

yes

characteristic, complemented

#10587486

C₄

yes

complemented

#10587474

C₄

yes

characteristic, complemented

C₂²×#595

—

C₂²×#595

—

C₂×C₄×D₁₇

D₄

yes

—

C₂×C₄×D₁₇

C₂³

yes

—

—

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

Q₈

yes

—

C₂×#3622

C₂×C₄

yes

—

C₈×D₁₇

—

C₈×D₁₇

—

C₂²×D₁₇

C₂×D₄

yes

—

—

—

—

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₂×Dic₁₇

C₂×Q₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

#41

yes

—

C₄×D₁₇

#41

yes

—

C₈×C₁₇

—

#3622

—

#1836

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

C₂×D₁₇

yes

C₂×D₁₇

yes

C₂×D₁₇

yes

—

C₂×D₁₇

#137

yes

—

#595

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

#117

yes

—

Dic₁₇

—

Dic₁₇

#125

yes

—

C₂×#97

—

#384

—

#382

—

#340

—

C₂×C₁₇

#382

yes

characteristic

C₂×C₁₇

#428

yes

—

D₁₇

#489

yes

—

D₁₇

yes

D₁₇

yes

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

yes

characteristic, complemented

#95

—

C₁₇

#1836

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#595

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#595

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊#595

yes

—

102

C₂×C₄

C₂²×#595

yes

characteristic

103

C₈

—

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₄⋊#595

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

136

—

114

C₄

—

136

—

115

C₄

#10587677

yes

—

116

C₂

—

117

C₂

—

118

C₂

—

119

C₂

#11763753

yes

characteristic

120

C₂

#11763820

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×#97	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₇	C₂×D₄
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	#97	#595
hypocenter	C₁₇	#1836
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×#97

C₄

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₇

C₂×D₄

cosocle

C₂×C₄×D₁₇

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
11. C₁₇×#97	C₄	semidirect
12. #10587486	C₄	semidirect
13. #10587474	C₄	semidirect
72. #97	#595	semidirect
74. C₁₇	#1836	semidirect

H ≅ G/N

type

11. C₁₇×#97

C₄

semidirect

12. #10587486

C₄

semidirect

13. #10587474

C₄

semidirect

72. #97

#595

semidirect

74. C₁₇

#1836

semidirect

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

subgroup	index
#11763755	2
#11763753	2
#11763683	2
#97×D₁₇	2
#1836	17

subgroup

index

#97×D₁₇

Subgroups — #14544707

Conjugacy classes of subgroups

Complements

Maximal subgroups