Group #14544970 — subgroups

Subgroups — #14544970

Order 2176.

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

normal subgroups only

This group has 284 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	#11763814	C₂	1	1	yes	maximal, characteristic
2	C₄²×#595	C₂	1	1	yes	maximal, characteristic
3	#11763679	C₂	1	4	yes	maximal
4	#11763365	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#595	—	2	2	no	—
6	C₂×C₄×#595	C₂²	1	2	yes	—
7	C₂×#63243	C₂²	1	2	yes	—
8	C₁₇×#125	C₄	1	1	yes	characteristic, complemented
9	#10587582	—	2	2	no	—
10	#10587578	C₂²	1	2	yes	—
11	#10587574	C₄	1	2	yes	complemented
12	#10587573	—	2	2	no	—
13	C₄²×D₁₇	C₂²	1	1	yes	characteristic
14	#10587556	C₄	1	1	yes	characteristic, complemented
15	#10587544	—	2	2	no	—
16	C₂²×#595	—	2	2	no	—
17	C₂×C₄×D₁₇	D₄	1	2	yes	—
18	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₃₄	—	2	2	no	—
20	C₂×#3624	—	2	2	no	—
21	C₄×#595	—	2	8	no	—
22	C₄×#595	—	2	2	no	—
23	C₄×#595	Q₈	1	4	yes	complemented
24	#63243	—	2	4	no	—
25	C₄⋊C₄×C₁₇	—	2	2	no	—
26	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
27	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	#63227	—	2	2	no	—
29	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
30	#63225	—	2	2	no	—
31	#63225	—	2	2	no	—
32	C₄×Dic₁₇	—	2	2	no	—
33	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₇	D₄	1	2	yes	—
35	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×#595	—	2	8	no	—
37	C₂×#595	—	2	4	no	—
38	C₂×C₄×C₁₇	—	2	2	no	—
39	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₇	—	2	2	no	—
42	C₂×Dic₁₇	C₂²⋊C₄	1	2	yes	—
43	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
44	C₄×D₁₇	—	2	4	no	—
45	C₄×D₁₇	C₂×Q₈	1	2	yes	—
46	C₄×D₁₇	#41	1	2	yes	—
47	Dic₃₄	—	4	4	no	—
48	#3624	—	4	2	no	—
49	#1994	—	17	1	no	maximal, Sylow
50	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×D₁₇	—	2	2	no	—
52	C₂×D₁₇	C₂²⋊Q₈	1	2	yes	—
53	#595	—	2	8	no	—
54	C₄×C₁₇	—	2	2	no	—
55	C₄×C₁₇	—	4	2	no	—
56	C₄×C₁₇	C₄×Q₈	1	2	yes	—
57	Dic₁₇	—	4	2	no	—
58	Dic₁₇	#104	1	4	yes	—
59	Dic₁₇	#124	1	1	yes	characteristic
60	Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
61	#533	—	17	1	no	—
62	#423	—	17	1	no	—
63	C₄³	—	17	1	no	—
64	#339	—	17	4	no	—
65	C₂×C₁₇	#391	1	1	yes	characteristic
66	C₂×C₁₇	C₂×#104	1	2	yes	—
67	D₁₇	—	2	2	no	—
68	C₂×M₁₆	—	17	2	no	—
69	C₄⋊Q₈	—	17	2	no	—
70	#125	—	17	1	no	—
71	#125	#595	1	1	yes	characteristic, complemented
72	C₂²⋊Q₈	—	34	2	no	—
73	C₄×Q₈	—	34	2	no	—
74	#117	—	17	2	no	—
75	C₂×C₄²	—	17	1	no	—
76	C₂×C₄²	—	17	2	no	—
77	C₂×C₄²	—	34	2	no	—
78	C₄⋊C₈	—	34	2	no	—
79	C₁₇	#1994	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₈	—	34	2	no	—
81	C₂²×C₄	—	17	1	no	—
82	C₂²×C₄	—	17	2	no	—
83	C₂²×C₄	—	34	2	no	—
84	M₁₆	—	34	4	no	—
85	C₂×C₈	—	34	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	17	2	no	—
88	C₄⋊C₄	—	34	2	no	—
89	C₄⋊C₄	—	34	2	no	—
90	C₄⋊C₄	C₂×#595	1	2	yes	—
91	C₂²⋊C₄	—	34	2	no	—
92	C₄²	—	17	2	no	—
93	C₄²	—	17	1	no	—
94	C₄²	—	17	4	no	—
95	C₄²	—	34	2	no	—
96	C₄²	—	34	2	no	—
97	C₄²	—	34	8	no	—
98	C₄²	C₂×#595	1	1	yes	characteristic
99	C₂³	—	17	1	no	—
100	Q₈	—	68	4	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	17	1	no	—
103	C₂×C₄	—	17	2	no	—
104	C₂×C₄	—	17	2	no	—
105	C₂×C₄	—	17	2	no	—
106	C₂×C₄	—	34	4	no	—
107	C₂×C₄	—	34	4	no	—
108	C₂×C₄	—	34	8	no	—
109	C₂×C₄	—	34	2	no	—
110	C₂×C₄	C₂²⋊#595	1	2	yes	—
111	C₂×C₄	C₂²×#595	1	1	yes	characteristic
112	C₈	—	68	2	no	—
113	C₂²	—	17	2	no	—
114	C₂²	—	34	2	no	—
115	C₂²	C₂×C₂²⋊#595	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	17	1	no	—
119	C₄	—	17	1	no	—
120	C₄	—	17	4	no	—
121	C₄	—	34	8	no	—
122	C₄	—	68	2	no	—
123	C₄	Q₈×#595	1	2	yes	—
124	C₂	—	34	2	no	—
125	C₂	(C₂×#595)⋊Q₈	1	1	yes	characteristic
126	C₂	D₁₇⋊#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763814

C₂

yes

maximal, characteristic

C₄²×#595

C₂

yes

maximal, characteristic

#11763679

C₂

yes

maximal

#11763365

C₂

yes

maximal, characteristic

C₂×C₄×#595

—

C₂×C₄×#595

C₂²

yes

—

C₂×#63243

C₂²

yes

—

C₁₇×#125

C₄

yes

characteristic, complemented

#10587582

—

#10587578

C₂²

yes

—

#10587574

C₄

yes

complemented

#10587573

—

C₄²×D₁₇

C₂²

yes

characteristic

#10587556

C₄

yes

characteristic, complemented

#10587544

—

C₂²×#595

—

C₂×C₄×D₁₇

D₄

yes

—

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

—

—

—

Q₈

yes

complemented

#63243

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

#63227

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

—

C₂²×D₁₇

C₂×D₄

yes

characteristic

C₂×#595

—

C₂×#595

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂²⋊C₄

yes

—

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₂×Q₈

yes

—

C₄×D₁₇

#41

yes

—

Dic₃₄

—

#3624

—

#1994

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

C₂²⋊Q₈

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄×Q₈

yes

—

Dic₁₇

—

Dic₁₇

#104

yes

—

Dic₁₇

#124

yes

characteristic

Dic₁₇

C₄⋊Q₈

yes

characteristic

#533

—

#423

—

C₄³

—

#339

—

C₂×C₁₇

#391

yes

characteristic

C₂×C₁₇

C₂×#104

yes

—

D₁₇

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

#595

yes

characteristic, complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₁₇

#1994

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#595

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#595

yes

characteristic

C₂³

—

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²⋊#595

yes

—

111

C₂×C₄

C₂²×#595

yes

characteristic

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₂²⋊#595

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

Q₈×#595

yes

—

124

C₂

—

125

C₂

(C₂×#595)⋊Q₈

yes

characteristic

126

C₂

D₁₇⋊#104

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×#125	C₄
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₇	C₂×D₄
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	#125	#595
hypocenter	C₁₇	#1994
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×#125

C₄

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₇

C₂×D₄

cosocle

C₂×C₄×D₁₇

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
8. C₁₇×#125	C₄	semidirect
11. #10587574	C₄	semidirect
14. #10587556	C₄	semidirect
23. C₄×#595	Q₈	semidirect
71. #125	#595	semidirect
79. C₁₇	#1994	semidirect

H ≅ G/N

type

8. C₁₇×#125

C₄

semidirect

11. #10587574

C₄

semidirect

14. #10587556

C₄

semidirect

23. C₄×#595

Q₈

semidirect

71. #125

#595

semidirect

79. C₁₇

#1994

semidirect

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

subgroup	index
#11763814	2
C₄²×#595	2
#11763679	2
#11763365	2
#1994	17

subgroup

index

C₄²×#595

Subgroups — #14544970

Conjugacy classes of subgroups

Complements

Maximal subgroups