Group #14544506 — subgroups

Subgroups — #14544506

Order 2176.

200 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 78 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	#11763772	C₂	1	2	yes	maximal
2	#11763678	C₂	1	1	yes	maximal, characteristic
3	#11763677	C₂	1	1	yes	maximal, characteristic
4	#11763676	C₂	1	2	yes	maximal
5	#11762766	C₂	1	1	yes	maximal, characteristic
6	#10587677	C₂²	1	2	yes	—
7	D₁₇⋊M₁₆	C₂²	1	2	yes	—
8	C₂×#63218	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
10	#10587558	C₂²	1	1	yes	characteristic
11	#10587532	C₄	1	4	yes	complemented
12	C₈×#595	C₄	1	4	yes	complemented
13	C₁₇×#97	C₄	1	1	yes	characteristic, complemented
14	#10587486	C₄	1	1	yes	characteristic, complemented
15	C₈×Dic₁₇	C₄	1	1	yes	characteristic, complemented
16	C₄×#3622	C₄	1	1	yes	characteristic, complemented
17	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
18	C₂²⋊#595	—	2	2	no	—
19	#63247	—	2	2	no	—
20	C₄⋊#595	—	2	2	no	—
21	C₄×#595	C₂×C₄	1	4	yes	—
22	#63243	—	2	2	no	—
23	#63242	C₂×C₄	1	4	yes	—
24	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
27	#63227	—	2	1	no	—
28	#63225	—	2	1	no	—
29	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×#3622	C₂×C₄	1	1	yes	characteristic
31	C₂×#3622	C₂×C₄	1	1	yes	characteristic
32	#63218	—	2	2	no	—
33	C₈×D₁₇	C₂×C₄	1	2	yes	—
34	C₈×D₁₇	C₂×C₄	1	2	yes	—
35	C₂²×D₁₇	#41	1	1	yes	characteristic
36	C₂×#595	—	2	4	no	—
37	C₂×C₄×C₁₇	—	2	1	no	—
38	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₁₇	—	2	1	no	—
40	C₂×Dic₁₇	#41	1	1	yes	characteristic
41	C₄×D₁₇	C₂²×C₄	1	2	yes	—
42	C₄×D₁₇	#41	1	2	yes	—
43	#3624	—	2	4	no	—
44	C₈×C₁₇	—	2	1	no	—
45	C₈×C₁₇	C₄²	1	2	yes	—
46	#3622	—	2	1	no	—
47	#3622	C₄²	1	1	yes	characteristic
48	#3622	C₄²	1	1	yes	characteristic
49	#1747	—	17	1	no	maximal, Sylow
50	C₂²×C₁₇	#117	1	1	yes	characteristic
51	C₂×D₁₇	—	4	1	no	—
52	C₂×D₁₇	#117	1	2	yes	—
53	#595	—	4	4	no	—
54	C₄×C₁₇	—	4	1	no	—
55	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₁₇	#117	1	1	yes	characteristic
57	Dic₁₇	—	4	1	no	—
58	Dic₁₇	#117	1	2	yes	—
59	#405	—	17	2	no	—
60	#405	—	17	1	no	—
61	#338	—	17	1	no	—
62	H(3,ℤ/4)	—	17	2	no	—
63	H(3,ℤ/4)	—	17	1	no	—
64	C₂×C₁₇	—	2	1	no	—
65	C₂×C₁₇	#376	1	1	yes	characteristic
66	D₁₇	—	2	2	no	—
67	C₂×M₁₆	—	17	2	no	—
68	C₂×M₁₆	—	17	1	no	—
69	C₂²×C₈	—	17	1	no	—
70	#117	—	17	2	no	—
71	#117	—	17	1	no	—
72	#97	—	17	4	no	—
73	#97	—	17	1	no	—
74	#97	#595	1	1	yes	characteristic, complemented
75	C₄×C₈	—	17	4	no	—
76	C₄×C₈	—	17	1	no	—
77	C₄×C₈	—	17	1	no	—
78	C₁₇	#1747	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	—	17	1	no	—
80	M₁₆	—	34	2	no	—
81	M₁₆	—	34	2	no	—
82	M₁₆	—	34	2	no	—
83	C₂×C₈	—	17	1	no	—
84	C₂×C₈	—	17	2	no	—
85	C₂×C₈	—	17	2	no	—
86	C₂×C₈	—	17	4	no	—
87	C₂×C₈	—	17	1	no	—
88	C₂×C₈	C₂×#595	1	1	yes	characteristic
89	C₂×C₈	C₂×#595	1	1	yes	characteristic
90	C₄⋊C₄	—	34	1	no	—
91	C₄⋊C₄	—	34	2	no	—
92	C₂²⋊C₄	—	34	1	no	—
93	C₂²⋊C₄	—	34	2	no	—
94	C₄²	—	17	4	no	—
95	C₄²	—	17	1	no	—
96	C₄²	C₂×#595	1	1	yes	characteristic
97	C₂³	—	17	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	17	1	no	—
100	C₂×C₄	—	17	2	no	—
101	C₂×C₄	—	17	2	no	—
102	C₂×C₄	—	34	4	no	—
103	C₂×C₄	—	34	1	no	—
104	C₂×C₄	C₂²×#595	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	17	1	no	—
107	C₈	—	17	1	no	—
108	C₈	—	34	4	no	—
109	C₈	—	34	1	no	—
110	C₈	C₄×#595	1	2	yes	—
111	C₂²	—	17	2	no	—
112	C₂²	—	68	1	no	—
113	C₂²	#10587677	1	1	yes	characteristic
114	C₄	—	4	1	no	—
115	C₄	—	17	2	no	—
116	C₄	—	68	1	no	—
117	C₄	—	68	4	no	—
118	C₄	C₂×C₄×#595	1	1	yes	characteristic
119	C₄	#10587677	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	34	2	no	—
122	C₂	#11763745	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763772

C₂

yes

maximal

#11763678

C₂

yes

maximal, characteristic

#11763677

C₂

yes

maximal, characteristic

#11763676

C₂

yes

maximal

#11762766

C₂

yes

maximal, characteristic

#10587677

C₂²

yes

—

D₁₇⋊M₁₆

C₂²

yes

—

C₂×#63218

C₂²

yes

characteristic

C₂×C₈×D₁₇

C₂²

yes

characteristic

#10587558

C₂²

yes

characteristic

#10587532

C₄

yes

complemented

C₈×#595

C₄

yes

complemented

C₁₇×#97

C₄

yes

characteristic, complemented

#10587486

C₄

yes

characteristic, complemented

C₈×Dic₁₇

C₄

yes

characteristic, complemented

C₄×#3622

C₄

yes

characteristic, complemented

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂²⋊#595

—

#63247

—

C₄⋊#595

—

C₄×#595

C₂×C₄

yes

—

#63243

—

#63242

C₂×C₄

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

#63227

—

#63225

—

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

#63218

—

C₈×D₁₇

C₂×C₄

yes

—

C₈×D₁₇

C₂×C₄

yes

—

C₂²×D₁₇

#41

yes

characteristic

C₂×#595

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#41

yes

characteristic

C₄×D₁₇

C₂²×C₄

yes

—

C₄×D₁₇

#41

yes

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

C₄²

yes

—

#3622

—

#3622

C₄²

yes

characteristic

#3622

C₄²

yes

—

C₂²×C₁₇

yes

C₂×D₁₇

—

C₂×D₁₇

#117

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄²

yes

characteristic

C₄×C₁₇

#117

yes

characteristic

Dic₁₇

—

Dic₁₇

#117

yes

—

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#376

yes

characteristic

D₁₇

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

yes

characteristic, complemented

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#1747

yes

characteristic, complemented, Sylow

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#595

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂²×#595

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

C₄×#595

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

#10587677

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₂×C₄×#595

yes

characteristic

119

C₄

#10587677

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

#11763745

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₄×#595
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×#97	C₄
socle	C₂×C₁₇	#376
prime radical	C₂²×D₁₇	#41
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	#97	#595
hypocenter	C₁₇	#1747
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #97

subgroup

G/H

center

C₄

C₂×C₄×#595

commutator

C₄×C₁₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×#97

C₄

C₂×C₁₇

C₂²×D₁₇

C₂×C₄×D₁₇

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₁₇

C₁ ◁ C₄ ◁ #97

N	H ≅ G/N	type
11. #10587532	C₄	semidirect
12. C₈×#595	C₄	semidirect
13. C₁₇×#97	C₄	semidirect
14. #10587486	C₄	semidirect
15. C₈×Dic₁₇	C₄	semidirect
16. C₄×#3622	C₄	semidirect
74. #97	#595	semidirect
78. C₁₇	#1747	semidirect

H ≅ G/N

type

11. #10587532

C₄

semidirect

12. C₈×#595

C₄

semidirect

13. C₁₇×#97

C₄

semidirect

14. #10587486

C₄

semidirect

15. C₈×Dic₁₇

C₄

semidirect

16. C₄×#3622

C₄

semidirect

74. #97

#595

semidirect

78. C₁₇

#1747

semidirect

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

subgroup	index
#11763772	2
#11763678	2
#11763677	2
#11763676	2
#11762766	2
#1747	17

subgroup

index

Subgroups — #14544506

Conjugacy classes of subgroups

Complements

Maximal subgroups