Group #14537144 — subgroups

Subgroups — #14537144

Order 2176.

472 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 181 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×#10587556	C₂	1	1	yes	maximal, characteristic
2	#11762704	C₂	1	8	yes	maximal
3	#11762678	C₂	1	2	yes	maximal
4	#11762674	C₂	1	2	yes	maximal
5	C₄²×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₇×#125	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#125	—	2	4	no	—
8	C₂×C₄⋊C₄×C₁₇	C₂²	1	4	yes	—
9	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
12	C₂×#63225	C₂²	1	4	yes	—
13	C₂×#63225	C₂²	1	8	yes	—
14	C₂×C₄×Dic₁₇	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
17	#10587556	—	2	4	no	—
18	#10587502	C₂²	1	8	yes	—
19	C₂²×C₄×C₁₇	C₂³	1	4	yes	—
20	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₇	C₂³	1	4	yes	—
23	C₂²×Dic₁₇	C₂³	1	2	yes	—
24	C₂²×Dic₁₇	C₂³	1	2	yes	—
25	C₄⋊C₄×C₁₇	—	2	8	no	—
26	C₄⋊C₄×C₁₇	—	2	4	no	—
27	C₄²×C₁₇	D₄	1	4	yes	—
28	C₄⋊Dic₁₇	—	2	4	no	—
29	#63225	—	2	8	no	—
30	#63225	—	2	16	no	—
31	C₄×Dic₁₇	—	2	8	no	—
32	C₄×Dic₁₇	Q₈	1	4	yes	—
33	C₄×Dic₁₇	Q₈	1	4	yes	—
34	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₁₇	—	2	4	no	—
36	C₂×C₄×C₁₇	—	2	4	no	—
37	C₂×C₄×C₁₇	—	2	8	no	—
38	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
39	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
42	C₂×Dic₁₇	—	2	4	no	—
43	C₂×Dic₁₇	—	2	4	no	—
44	C₂×Dic₁₇	—	2	8	no	—
45	C₂×Dic₁₇	C₂×Q₈	1	4	yes	—
46	C₂×Dic₁₇	C₂×Q₈	1	4	yes	—
47	C₂×Dic₁₇	#41	1	8	yes	—
48	#2859	—	17	1	no	maximal, Sylow
49	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₁₇	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₁₇	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₁₇	C₂×#41	1	4	yes	—
53	C₄×C₁₇	—	2	4	no	—
54	C₄×C₁₇	—	4	4	no	—
55	C₄×C₁₇	C₄⋊Q₈	1	4	yes	—
56	Dic₁₇	—	2	4	no	—
57	Dic₁₇	—	4	4	no	—
58	Dic₁₇	#125	1	8	yes	complemented
59	C₂×#125	—	17	1	no	—
60	C₂×#125	D₁₇	1	1	yes	characteristic
61	#389	—	17	8	no	—
62	#384	—	17	2	no	—
63	#382	—	17	2	no	—
64	C₄³	—	17	1	no	—
65	C₂×C₁₇	C₂×#125	1	2	yes	—
66	C₂×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₁₇	#532	1	2	yes	—
68	C₂×C₁₇	#533	1	2	yes	—
69	#125	—	2	4	no	—
70	#125	—	34	4	no	—
71	C₂×C₄⋊C₄	—	17	2	no	—
72	C₂×C₄⋊C₄	—	17	4	no	—
73	C₂×C₄⋊C₄	—	17	8	no	—
74	C₂×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
75	C₂×C₄⋊C₄	C₂×D₁₇	1	4	yes	—
76	C₂×C₄²	—	17	1	no	—
77	C₂×C₄²	—	17	1	no	—
78	C₂×C₄²	—	17	4	no	—
79	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
80	#95	—	17	8	no	—
81	C₁₇	#2859	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	17	2	no	—
83	C₂²×C₄	—	17	2	no	—
84	C₂²×C₄	—	17	4	no	—
85	C₂²×C₄	C₂²×D₁₇	1	4	yes	—
86	C₂²×C₄	C₂²×D₁₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
88	C₄⋊C₄	—	2	8	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	34	16	no	—
91	C₄⋊C₄	—	34	8	no	—
92	C₄⋊C₄	—	34	4	no	—
93	C₄²	—	17	4	no	—
94	C₄²	—	17	4	no	—
95	C₄²	—	34	8	no	—
96	C₄²	C₂²⋊D₁₇	1	4	yes	—
97	C₂³	C₂³×D₁₇	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	17	4	no	—
102	C₂×C₄	—	17	8	no	—
103	C₂×C₄	—	17	4	no	—
104	C₂×C₄	—	34	4	no	—
105	C₂×C₄	—	34	8	no	—
106	C₂×C₄	—	34	4	no	—
107	C₂×C₄	Q₈×D₁₇	1	2	yes	—
108	C₂×C₄	Q₈×D₁₇	1	2	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₇	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
111	C₂²	C₂×Q₈×D₁₇	1	1	yes	characteristic
112	C₂²	C₂×Q₈×D₁₇	1	1	yes	characteristic
113	C₂²	#41×D₁₇	1	4	yes	—
114	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
115	C₄	—	2	4	no	—
116	C₄	—	4	4	no	—
117	C₄	—	17	8	no	—
118	C₄	—	34	4	no	—
119	C₄	—	68	4	no	—
120	C₄	#10587630	1	4	yes	—
121	C₂	#125×D₁₇	1	2	yes	—
122	C₂	C₂×#10587630	1	1	yes	characteristic
123	C₂	#11763355	1	2	yes	—
124	C₂	#11763365	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587556

C₂

yes

maximal, characteristic

#11762704

C₂

yes

maximal

#11762678

C₂

yes

maximal

#11762674

C₂

yes

maximal

C₄²×Dic₁₇

C₂

yes

maximal, characteristic

C₂×C₁₇×#125

C₂

yes

maximal, characteristic

C₁₇×#125

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63225

C₂²

yes

—

C₂×#63225

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

#10587556

—

#10587502

C₂²

yes

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

C₂³

yes

—

C₂²×Dic₁₇

C₂³

yes

—

C₂²×Dic₁₇

C₂³

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

D₄

yes

—

C₄⋊Dic₁₇

—

#63225

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

Q₈

yes

—

C₄×Dic₁₇

Q₈

yes

—

C₂³×C₁₇

C₂⁴

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×Q₈

yes

—

C₂×Dic₁₇

C₂×Q₈

yes

—

C₂×Dic₁₇

#41

yes

—

#2859

—

maximal, Sylow

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₂²×C₁₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₇

C₂×#41

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄⋊Q₈

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

yes

—

D₁₇

yes

—

—

—

C₄³

—

C₂×C₁₇

C₂×#125

yes

—

C₂×C₁₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₇

#532

yes

—

C₂×C₁₇

#533

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#95

—

C₁₇

#2859

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₁₇

yes

—

C₂³

C₂³×D₁₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

Q₈×D₁₇

yes

—

108

C₂×C₄

Q₈×D₁₇

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

110

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

111

C₂²

C₂×Q₈×D₁₇

yes

characteristic

112

C₂²

C₂×Q₈×D₁₇

yes

characteristic

113

C₂²

#41×D₁₇

yes

—

114

C₂²

C₂²×C₂²⋊D₁₇

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#10587630

yes

—

121

C₂

#125×D₁₇

yes

—

122

C₂

C₂×#10587630

yes

characteristic

123

C₂

#11763355

yes

—

124

C₂

#11763365

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, prime radical, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₂×C₁₇×#125	C₂
hypercenter	C₂×#125	D₁₇
hypocenter	C₁₇	#2859
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇

commutator, socle, prime radical, cosocle

C₂³×C₁₇

C₂⁴

Fitting

C₂×C₁₇×#125

C₂

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
58. Dic₁₇	#125	semidirect
81. C₁₇	#2859	semidirect

H ≅ G/N

type

58. Dic₁₇

#125

semidirect

81. C₁₇

#2859

semidirect

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

subgroup	index
C₂×#10587556	2
#11762704	2
#11762678	2
#11762674	2
C₄²×Dic₁₇	2
C₂×C₁₇×#125	2
#2859	17

subgroup

index

C₄²×Dic₁₇

C₂×C₁₇×#125

#2859

Subgroups — #14537144

Conjugacy classes of subgroups

Complements

Maximal subgroups