Group #14544971 — subgroups

Subgroups — #14544971

Order 2176.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#11763813	C₂	1	1	yes	maximal, characteristic
2	#11763745	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	C₂²	1	2	yes	—
6	C₂×#63243	C₂²	1	2	yes	—
7	C₁₇×#125	C₄	1	1	yes	characteristic, complemented
8	#10587582	—	2	2	no	—
9	#10587578	C₂²	1	2	yes	—
10	#10587574	C₄	1	2	yes	complemented
11	#10587573	—	2	2	no	—
12	C₄²×D₁₇	C₂²	1	1	yes	characteristic
13	#10587556	C₄	1	1	yes	characteristic, complemented
14	(C₂×#595)⋊C₄	—	2	2	no	—
15	C₄⋊#3624	—	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	2	no	—
22	C₄×#595	Q₈	1	4	yes	complemented
23	#63243	—	2	4	no	—
24	C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
26	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
27	#63227	—	2	2	no	—
28	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
29	#63225	—	2	2	no	—
30	#63225	—	2	2	no	—
31	C₄×Dic₁₇	—	2	2	no	—
32	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₇	D₄	1	2	yes	—
34	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂×#595	—	2	4	no	—
36	C₂×#595	—	4	4	no	—
37	C₂×C₄×C₁₇	—	2	2	no	—
38	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₁₇	—	2	2	no	—
41	C₂×Dic₁₇	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
43	C₄×D₁₇	—	4	2	no	—
44	C₄×D₁₇	C₂×Q₈	1	2	yes	—
45	C₄×D₁₇	#41	1	2	yes	—
46	Dic₃₄	—	4	4	no	—
47	#3624	—	4	2	no	—
48	#1995	—	17	1	no	maximal, Sylow
49	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂×D₁₇	—	2	2	no	—
51	C₂×D₁₇	C₂²⋊Q₈	1	2	yes	—
52	#595	—	4	4	no	—
53	C₄×C₁₇	—	2	2	no	—
54	C₄×C₁₇	—	4	2	no	—
55	C₄×C₁₇	C₄×Q₈	1	2	yes	—
56	Dic₁₇	—	2	2	no	—
57	Dic₁₇	—	4	2	no	—
58	Dic₁₇	#124	1	1	yes	characteristic
59	Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
60	#533	—	17	1	no	—
61	#423	—	17	1	no	—
62	#376	—	17	1	no	—
63	#339	—	17	4	no	—
64	C₂×C₁₇	#391	1	1	yes	characteristic
65	C₂×C₁₇	#421	1	2	yes	—
66	D₁₇	—	2	2	no	—
67	C₂×M₁₆	—	17	2	no	—
68	C₄⋊Q₈	—	17	2	no	—
69	#125	—	17	1	no	—
70	#125	#595	1	1	yes	characteristic, complemented
71	C₂²⋊Q₈	—	34	2	no	—
72	C₄×Q₈	—	34	2	no	—
73	#117	—	17	2	no	—
74	C₂×C₄²	—	17	1	no	—
75	C₂×C₄²	—	17	2	no	—
76	C₄⋊C₈	—	34	2	no	—
77	#95	—	34	2	no	—
78	C₁₇	#1995	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₈	—	34	2	no	—
80	C₂²×C₄	—	17	1	no	—
81	C₂²×C₄	—	17	2	no	—
82	C₂²×C₄	—	34	2	no	—
83	M₁₆	—	34	4	no	—
84	C₂×C₈	—	34	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	17	2	no	—
87	C₄⋊C₄	—	34	2	no	—
88	C₄⋊C₄	—	34	2	no	—
89	C₄⋊C₄	C₂×#595	1	2	yes	—
90	C₂²⋊C₄	—	34	2	no	—
91	C₄²	—	17	2	no	—
92	C₄²	—	17	1	no	—
93	C₄²	—	17	4	no	—
94	C₄²	—	34	2	no	—
95	C₄²	—	34	2	no	—
96	C₄²	C₂×#595	1	1	yes	characteristic
97	C₂³	—	17	1	no	—
98	Q₈	—	68	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	17	1	no	—
101	C₂×C₄	—	17	2	no	—
102	C₂×C₄	—	17	2	no	—
103	C₂×C₄	—	17	2	no	—
104	C₂×C₄	—	34	4	no	—
105	C₂×C₄	—	34	2	no	—
106	C₂×C₄	—	68	2	no	—
107	C₂×C₄	—	68	4	no	—
108	C₂×C₄	C₂²⋊#595	1	2	yes	—
109	C₂×C₄	C₂²×#595	1	1	yes	characteristic
110	C₈	—	68	2	no	—
111	C₂²	—	17	2	no	—
112	C₂²	—	34	2	no	—
113	C₂²	C₂×C₂²⋊#595	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	17	1	no	—
117	C₄	—	17	1	no	—
118	C₄	—	34	2	no	—
119	C₄	—	68	4	no	—
120	C₄	—	68	2	no	—
121	C₄	Q₈×#595	1	2	yes	—
122	C₂	—	34	2	no	—
123	C₂	(C₂×#595)⋊Q₈	1	1	yes	characteristic
124	C₂	#11763806	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763813

C₂

yes

maximal, characteristic

#11763745

C₂

yes

maximal, characteristic

#11763679

C₂

yes

maximal

#11763365

C₂

yes

maximal, characteristic

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

(C₂×#595)⋊C₄

—

C₄⋊#3624

—

C₂²×#595

—

C₂×C₄×D₁₇

D₄

yes

—

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

C₂×#3624

—

C₄×#595

—

C₄×#595

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

—

#1995

—

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

—

Dic₁₇

#124

yes

characteristic

Dic₁₇

C₄⋊Q₈

yes

—

—

—

—

C₂×C₁₇

#391

yes

characteristic

C₂×C₁₇

#421

yes

—

D₁₇

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

#595

yes

characteristic, complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

#95

—

C₁₇

#1995

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

C₂×#595

yes

characteristic

C₂³

—

Q₈

—

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

—

108

C₂×C₄

C₂²⋊#595

yes

—

109

C₂×C₄

C₂²×#595

yes

characteristic

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₂²⋊#595

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Q₈×#595

yes

—

122

C₂

—

123

C₂

(C₂×#595)⋊Q₈

yes

characteristic

124

C₂

#11763806

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	#1995
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
7. C₁₇×#125	C₄	semidirect
10. #10587574	C₄	semidirect
13. #10587556	C₄	semidirect
22. C₄×#595	Q₈	semidirect
70. #125	#595	semidirect
78. C₁₇	#1995	semidirect

H ≅ G/N

type

7. C₁₇×#125

C₄

semidirect

10. #10587574

C₄

semidirect

13. #10587556

C₄

semidirect

22. C₄×#595

Q₈

semidirect

70. #125

#595

semidirect

78. C₁₇

#1995

semidirect

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

subgroup	index
#11763813	2
#11763745	2
#11763679	2
#11763365	2
#1995	17

subgroup

index

Subgroups — #14544971

Conjugacy classes of subgroups

Complements

Maximal subgroups