Group #16928013 — subgroups

Subgroups — #16928013

Order 4352.

262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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	#14528922	C₂	1	1	yes	maximal, characteristic
2	#14528915	C₂	1	1	yes	maximal, characteristic
3	#14526643	C₂	1	4	yes	maximal
4	C₁₇×#1450	C₂	1	1	yes	maximal, characteristic
5	#11762833	C₂²	1	2	yes	—
6	#11762675	C₂²	1	2	yes	—
7	#11762574	—	2	2	no	—
8	#11762572	—	2	2	no	—
9	C₁₇×#431	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₁₇×#322	C₄	1	2	yes	complemented
12	C₈²×C₁₇	C₄	1	2	yes	complemented
13	C₂²×C₈×C₁₇	D₄	1	2	yes	—
14	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₇	—	2	2	no	—
16	C₂×#63223	—	2	2	no	—
17	C₁₇×C₄⋊C₈	—	2	2	no	—
18	C₁₇×C₂²⋊C₈	—	2	2	no	—
19	C₄×C₈×C₁₇	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₇	Q₈	1	2	yes	—
23	#10587502	—	2	4	no	—
24	C₄⋊#3622	—	2	4	no	—
25	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	#41	1	2	yes	—
27	C₂²×Dic₁₇	—	4	2	no	—
28	C₂×C₈×C₁₇	—	2	4	no	—
29	C₂×C₈×C₁₇	—	2	4	no	—
30	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
32	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₁₇	#41	1	2	yes	—
35	C₄⋊Dic₁₇	—	4	4	no	—
36	#63223	—	4	4	no	—
37	C₂×#3622	—	4	2	no	—
38	#8792	—	17	1	no	maximal, Sylow
39	C₂³×C₁₇	#124	1	1	yes	characteristic
40	C₂×C₄×C₁₇	—	4	2	no	—
41	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇	#117	1	2	yes	—
43	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
44	C₂×C₄×C₁₇	#125	1	2	yes	—
45	C₂×C₄×C₁₇	#135	1	4	yes	—
46	C₂×Dic₁₇	—	8	4	no	—
47	C₂×Dic₁₇	—	8	2	no	—
48	C₈×C₁₇	—	2	2	no	—
49	C₈×C₁₇	—	2	2	no	—
50	C₈×C₁₇	—	4	4	no	—
51	#3622	—	8	2	no	—
52	#1837	—	17	1	no	—
53	#1835	—	17	1	no	—
54	#1450	D₁₇	1	1	yes	characteristic
55	#1286	—	17	4	no	—
56	C₂²×C₁₇	—	2	2	no	—
57	C₂²×C₁₇	#382	1	1	yes	characteristic
58	C₂²×C₁₇	#488	1	2	yes	—
59	C₄×C₁₇	—	2	2	no	—
60	C₄×C₁₇	—	2	2	no	—
61	C₄×C₁₇	#427	1	2	yes	—
62	C₄×C₁₇	#488	1	2	yes	—
63	Dic₁₇	—	16	2	no	—
64	#431	C₂×D₁₇	1	2	yes	—
65	#423	—	17	2	no	—
66	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
67	#382	—	17	2	no	—
68	#341	—	34	2	no	—
69	#340	—	34	2	no	—
70	#322	Dic₁₇	1	2	yes	complemented
71	C₈²	Dic₁₇	1	2	yes	complemented
72	C₂×C₁₇	—	2	2	no	—
73	C₂×C₁₇	#1833	1	2	yes	—
74	C₂×C₁₇	#1835	1	1	yes	characteristic
75	C₂×M₁₆	—	34	2	no	—
76	C₂²×C₈	D₆₈	1	2	yes	—
77	C₂×C₄⋊C₄	—	34	2	no	—
78	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
79	C₄⋊C₈	—	2	2	no	—
80	C₄⋊C₈	—	34	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₄×C₈	Dic₃₄	1	2	yes	—
83	C₄×C₈	C₂×Dic₁₇	1	4	yes	—
84	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
86	#95	—	34	4	no	—
87	C₁₇	#8792	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄	—	68	2	no	—
89	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
90	C₂²×C₄	#63252	1	2	yes	—
91	M₁₆	—	68	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	68	2	no	—
95	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
97	C₄⋊C₄	—	68	4	no	—
98	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
99	C₄²	#63252	1	2	yes	—
100	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
101	C₂³	#10587561	1	1	yes	characteristic
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	136	4	no	—
104	C₂×C₄	—	136	2	no	—
105	C₂×C₄	#10587556	1	2	yes	—
106	C₂×C₄	#10587561	1	1	yes	characteristic
107	C₂×C₄	#10587590	1	4	yes	—
108	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
109	C₂×C₄	#10587612	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	4	4	no	—
113	C₈	—	136	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#11762675	1	1	yes	characteristic
116	C₂²	#11763164	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	272	2	no	—
120	C₄	#11762847	1	2	yes	—
121	C₄	#11763164	1	2	yes	—
122	C₂	—	2	2	no	—
123	C₂	#14528908	1	2	yes	—
124	C₂	#14528915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528922

C₂

yes

maximal, characteristic

#14528915

C₂

yes

maximal, characteristic

#14526643

C₂

yes

maximal

C₁₇×#1450

C₂

yes

maximal, characteristic

#11762833

C₂²

yes

—

#11762675

C₂²

yes

—

#11762574

—

#11762572

—

C₁₇×#431

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#322

C₄

yes

complemented

C₈²×C₁₇

C₄

yes

complemented

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×#63223

—

C₁₇×C₄⋊C₈

—

C₁₇×C₂²⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

Q₈

yes

—

#10587502

—

C₄⋊#3622

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

characteristic

C₄²×C₁₇

#41

yes

—

C₄⋊Dic₁₇

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

#125

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

D₁₇

yes

characteristic

#1286

—

C₂²×C₁₇

—

C₂²×C₁₇

#382

yes

characteristic

C₂²×C₁₇

#488

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#427

yes

—

C₄×C₁₇

#488

yes

—

Dic₁₇

—

#431

C₂×D₁₇

yes

—

#423

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

—

Dic₁₇

yes

complemented

C₈²

Dic₁₇

yes

complemented

C₂×C₁₇

—

C₂×C₁₇

#1833

yes

—

C₂×C₁₇

#1835

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

D₆₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

Dic₃₄

yes

—

C₄×C₈

C₂×Dic₁₇

yes

—

C₄×C₈

C₂×Dic₁₇

yes

characteristic

C₄×C₈

C₂×Dic₁₇

yes

characteristic

#95

—

C₁₇

#8792

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂×D₆₈

yes

characteristic

C₂²×C₄

#63252

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₁₇

yes

—

C₂×C₈

C₄⋊Dic₁₇

yes

—

C₄⋊C₄

—

C₄²

C₂×Dic₃₄

yes

characteristic

C₄²

#63252

yes

—

100

C₄²

C₂²×Dic₁₇

yes

characteristic

101

C₂³

#10587561

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

136

—

104

C₂×C₄

—

136

—

105

C₂×C₄

#10587556

yes

—

106

C₂×C₄

#10587561

yes

characteristic

107

C₂×C₄

#10587590

yes

—

108

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

109

C₂×C₄

#10587612

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

136

—

114

C₂²

—

115

C₂²

#11762675

yes

characteristic

116

C₂²

#11763164

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

272

—

120

C₄

#11762847

yes

—

121

C₄

#11763164

yes

—

122

C₂

—

123

C₂

#14528908

yes

—

124

C₂

#14528915

yes

characteristic

	subgroup	G/H
center	C₂²	#11762675
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1450	C₂
socle	C₂²×C₁₇	#382
prime radical	C₂³×C₁₇	#124
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1450	D₁₇
hypocenter	C₁₇	#8792
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#11762675

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1450

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
11. C₁₇×#322	C₄	semidirect
12. C₈²×C₁₇	C₄	semidirect
70. #322	Dic₁₇	semidirect
71. C₈²	Dic₁₇	semidirect
87. C₁₇	#8792	semidirect

H ≅ G/N

type

11. C₁₇×#322

C₄

semidirect

12. C₈²×C₁₇

C₄

semidirect

70. #322

Dic₁₇

semidirect

71. C₈²

Dic₁₇

semidirect

87. C₁₇

#8792

semidirect

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

subgroup	index
#14528922	2
#14528915	2
#14526643	2
C₁₇×#1450	2
#8792	17

subgroup

index

C₁₇×#1450

Subgroups — #16928013

Conjugacy classes of subgroups

Complements

Maximal subgroups