Group #16949017 — subgroups

Subgroups — #16949017

Order 4352.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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	#14531011	C₂	1	2	yes	maximal, complemented
2	#14531007	C₂	1	2	yes	maximal
3	#14528394	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#63214	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1709	C₂	1	1	yes	maximal, characteristic
6	#11763214	C₂²	1	2	yes	—
7	#11763160	—	2	2	no	—
8	#11762613	—	2	4	no	—
9	#11762610	—	2	4	no	—
10	C₄×#63214	C₂²	1	2	yes	—
11	C₁₇×#494	C₂²	1	1	yes	characteristic
12	C₁₇×#493	C₂²	1	1	yes	characteristic
13	C₁₇×#486	—	2	2	no	—
14	C₈²×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₁₇	—	2	2	no	—
16	C₂×D₈×C₁₇	—	2	2	no	—
17	C₁₇×C₄⋊Q₈	—	2	1	no	—
18	C₁₇×#127	—	2	1	no	—
19	#10587555	—	2	2	no	—
20	C₁₇×#102	—	4	4	no	—
21	C₄×C₈×C₁₇	D₄	1	2	yes	—
22	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
23	#10587489	—	2	2	no	—
24	#10587489	—	2	2	no	—
25	#10587487	—	4	4	no	—
26	C₂×#63214	D₄	1	4	yes	—
27	C₂×Q₈×C₁₇	—	4	1	no	—
28	C₂×D₄×C₁₇	—	4	1	no	—
29	C₂×D₄×C₁₇	—	4	2	no	—
30	C₂×Dic₃₄	—	4	2	no	—
31	Q₁₆×C₁₇	—	4	4	no	—
32	D₈×C₁₇	—	4	4	no	—
33	C₂×C₈×C₁₇	D₈	1	4	yes	—
34	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
35	C₄⋊C₄×C₁₇	—	4	2	no	—
36	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₁₇	—	4	2	no	—
38	C₄⋊Dic₁₇	—	4	2	no	—
39	#63214	—	2	4	no	—
40	#12511	—	17	1	no	maximal, Sylow
41	C₂³×C₁₇	—	8	1	no	—
42	Q₈×C₁₇	—	8	2	no	—
43	D₄×C₁₇	—	8	2	no	—
44	D₄×C₁₇	—	8	4	no	—
45	C₂×C₄×C₁₇	—	8	1	no	—
46	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
48	C₂×Dic₁₇	—	8	2	no	—
49	Dic₃₄	—	8	4	no	—
50	C₈×C₁₇	—	2	4	no	—
51	C₈×C₁₇	#135	1	4	yes	—
52	#2241	—	17	2	no	—
53	#2240	—	17	2	no	—
54	#1710	—	17	1	no	—
55	#1709	D₁₇	1	1	yes	characteristic
56	C₈×C₁₆	—	17	1	no	—
57	C₂²×C₁₇	—	16	1	no	—
58	C₂²×C₁₇	—	16	2	no	—
59	C₂²×C₁₇	#493	1	1	yes	characteristic
60	C₄×C₁₇	—	16	1	no	—
61	C₄×C₁₇	#495	1	2	yes	—
62	C₄×C₁₇	#508	1	4	yes	—
63	Dic₁₇	—	16	2	no	—
64	#500	—	17	2	no	—
65	#494	C₂×D₁₇	1	1	yes	characteristic
66	#493	C₂×D₁₇	1	1	yes	characteristic
67	#487	—	34	2	no	—
68	#486	—	2	2	no	—
69	#358	—	34	4	no	—
70	#357	—	34	4	no	—
71	C₄×C₁₆	—	17	2	no	—
72	C₈²	C₂×D₁₇	1	1	yes	characteristic
73	C₂×C₁₇	—	32	1	no	—
74	C₂×C₁₇	#1709	1	1	yes	characteristic
75	C₂×C₁₇	#2249	1	2	yes	—
76	C₂×Q₁₆	—	2	2	no	—
77	C₂×D₈	—	2	2	no	—
78	C₄⋊Q₈	—	2	1	no	—
79	C₄⋊Q₈	—	34	2	no	—
80	#127	—	2	1	no	—
81	C₂×C₁₆	—	17	4	no	—
82	#107	—	34	2	no	—
83	#107	—	34	2	no	—
84	#103	—	68	4	no	—
85	#102	—	4	4	no	—
86	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
87	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
88	C₁₇	#12511	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	68	2	no	—
91	C₂×D₄	—	4	2	no	—
92	C₂×D₄	—	4	1	no	—
93	Q₁₆	—	4	4	no	—
94	D₈	—	4	4	no	—
95	C₂×C₈	#63228	1	4	yes	—
96	C₂×C₈	D₄×D₁₇	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	68	2	no	—
99	C₄⋊C₄	—	68	2	no	—
100	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
101	C₁₆	—	34	4	no	—
102	C₂³	—	8	1	no	—
103	Q₈	—	8	2	no	—
104	Q₈	—	136	4	no	—
105	D₄	—	8	4	no	—
106	D₄	—	8	2	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	136	2	no	—
109	C₂×C₄	C₂×#63228	1	2	yes	—
110	C₂×C₄	#10587626	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	#10587598	1	4	yes	—
113	C₂²	—	16	1	no	—
114	C₂²	—	16	2	no	—
115	C₂²	#11763185	1	1	yes	characteristic
116	C₄	—	16	1	no	—
117	C₄	—	272	2	no	—
118	C₄	#11763192	1	2	yes	—
119	C₄	#11763239	1	4	yes	—
120	C₂	—	32	1	no	—
121	C₂	#14528389	1	1	yes	characteristic
122	C₂	#14531043	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14531011

C₂

yes

maximal, complemented

#14531007

C₂

yes

maximal

#14528394

C₂

yes

maximal, characteristic, complemented

C₈×#63214

C₂

yes

maximal, characteristic, complemented

C₁₇×#1709

C₂

yes

maximal, characteristic

#11763214

C₂²

yes

—

—

—

—

C₂²

yes

—

C₁₇×#494

C₂²

yes

characteristic

C₁₇×#493

C₂²

yes

characteristic

C₁₇×#486

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

—

C₂×D₈×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₁₇×#127

—

#10587555

—

C₁₇×#102

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

—

—

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×Dic₃₄

—

Q₁₆×C₁₇

—

D₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63214

—

#12511

—

maximal, Sylow

C₂³×C₁₇

—

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

#135

yes

—

—

—

—

D₁₇

yes

characteristic

C₈×C₁₆

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#493

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

#495

yes

—

C₄×C₁₇

#508

yes

—

Dic₁₇

—

#500

—

#494

C₂×D₁₇

yes

characteristic

#493

C₂×D₁₇

yes

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#1709

yes

characteristic

C₂×C₁₇

#2249

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

C₂²⋊D₁₇

yes

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12511

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

#63228

yes

—

C₂×C₈

D₄×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

101

C₁₆

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

136

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

136

—

109

C₂×C₄

C₂×#63228

yes

—

110

C₂×C₄

#10587626

yes

characteristic

111

C₈

—

112

C₈

#10587598

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

#11763185

yes

characteristic

116

C₄

—

117

C₄

—

272

—

118

C₄

#11763192

yes

—

119

C₄

#11763239

yes

—

120

C₂

—

121

C₂

#14528389

yes

characteristic

122

C₂

#14531043

yes

—

	subgroup	G/H
center	C₂²	#11763185
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1709	C₂
socle	C₂²×C₁₇	#493
prime radical	C₁₇×#493	C₂²
hypercenter	#1709	D₁₇
hypocenter	C₁₇	#12511
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#11763185

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1709

C₂

socle

C₂²×C₁₇

#493

prime radical

C₁₇×#493

C₂²

D₁₇

C₁₇

G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #14531011	C₂	semidirect
3. #14528394	C₂	semidirect
4. C₈×#63214	C₂	semidirect
88. C₁₇	#12511	semidirect

H ≅ G/N

type

1. #14531011

C₂

semidirect

3. #14528394

C₂

semidirect

4. C₈×#63214

C₂

semidirect

88. C₁₇

#12511

semidirect

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

subgroup	index
#14531011	2
#14531007	2
#14528394	2
C₈×#63214	2
C₁₇×#1709	2
#12511	17

subgroup

index

C₁₇×#1709

Subgroups — #16949017

Conjugacy classes of subgroups

Complements

Maximal subgroups