Group #16939399 — subgroups

Subgroups — #16939399

Order 4352.

478 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 141 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	#14529590	C₂	1	3	yes	maximal
2	#14528912	C₂	1	3	yes	maximal, complemented
3	C₁₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10587560	C₂²	1	1	yes	characteristic
5	C₂×#10587492	—	2	6	no	—
6	#11762679	C₂²	1	3	yes	—
7	#11762572	—	2	6	no	—
8	C₂×C₄×C₈×C₁₇	C₂²	1	3	yes	—
9	C₂²×D₆₈	—	2	3	no	—
10	C₂²×C₈×C₁₇	D₄	1	6	yes	—
11	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₇	—	2	3	no	—
13	C₂×C₄⋊Dic₁₇	—	2	3	no	—
14	C₂×C₄⋊Dic₁₇	—	2	3	no	—
15	#10587560	—	2	3	no	—
16	#10587560	Q₈	1	2	yes	—
17	C₄×C₈×C₁₇	—	2	6	no	—
18	#10587492	—	4	12	no	—
19	C₂³×D₁₇	—	4	1	no	—
20	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
21	C₂²×Dic₁₇	—	4	3	no	—
22	C₂×D₆₈	—	2	6	no	—
23	C₂×D₆₈	—	4	6	no	—
24	C₂×D₆₈	—	4	6	no	—
25	C₂×D₆₈	—	4	3	no	—
26	C₂×C₈×C₁₇	—	2	6	no	—
27	C₂×C₈×C₁₇	—	2	6	no	—
28	C₂×C₈×C₁₇	—	2	6	no	—
29	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₁₇	#41	1	3	yes	—
31	C₄⋊Dic₁₇	—	4	6	no	—
32	C₄⋊Dic₁₇	—	4	6	no	—
33	C₄⋊Dic₁₇	—	4	6	no	—
34	#10678	—	17	1	no	maximal, Sylow
35	C₂³×C₁₇	#127	1	1	yes	characteristic
36	C₂²×D₁₇	—	4	2	no	—
37	C₂²×D₁₇	—	8	3	no	—
38	C₂²×D₁₇	—	8	3	no	—
39	C₂×C₄×C₁₇	C₂²⋊Q₈	1	3	yes	—
40	C₂×C₄×C₁₇	#123	1	3	yes	—
41	C₂×C₄×C₁₇	#135	1	6	yes	—
42	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	6	yes	—
43	C₂×Dic₁₇	—	8	3	no	—
44	C₂×Dic₁₇	—	8	3	no	—
45	C₂×Dic₁₇	—	8	3	no	—
46	D₆₈	—	4	12	no	—
47	D₆₈	—	8	6	no	—
48	C₈×C₁₇	—	4	6	no	—
49	#1957	—	17	3	no	—
50	#1834	—	17	3	no	—
51	#1273	D₁₇	1	1	yes	characteristic, complemented
52	C₂²×C₁₇	#399	1	1	yes	characteristic
53	C₂²×C₁₇	#495	1	3	yes	—
54	C₂²×C₁₇	#496	1	3	yes	—
55	C₂×D₁₇	—	8	6	no	—
56	C₂×D₁₇	—	16	1	no	—
57	C₂×D₁₇	—	16	3	no	—
58	C₄×C₁₇	#478	1	6	yes	—
59	C₄×C₁₇	#482	1	6	yes	—
60	Dic₁₇	—	16	3	no	—
61	C₂×#127	—	17	1	no	—
62	C₂×#102	—	34	6	no	—
63	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
64	#384	—	17	3	no	—
65	#340	—	34	6	no	—
66	C₂×C₁₇	#1711	1	1	yes	characteristic
67	C₂×C₁₇	#1717	1	3	yes	—
68	C₂×C₁₇	#2092	1	3	yes	—
69	D₁₇	—	16	2	no	—
70	C₂²×D₄	—	34	3	no	—
71	C₂²×C₈	D₆₈	1	6	yes	—
72	#127	—	17	2	no	—
73	#127	—	34	3	no	—
74	C₂×C₄⋊C₄	—	34	3	no	—
75	C₂×C₄⋊C₄	—	34	3	no	—
76	C₂×C₄⋊C₄	—	34	3	no	—
77	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
78	#102	—	68	12	no	—
79	C₄×C₈	—	2	6	no	—
80	C₁₇	#10678	1	1	yes	characteristic, complemented, Sylow
81	C₂⁴	—	68	1	no	—
82	C₂×D₄	—	34	6	no	—
83	C₂×D₄	—	68	3	no	—
84	C₂×D₄	—	68	6	no	—
85	C₂×D₄	—	68	6	no	—
86	C₂²×C₄	—	68	3	no	—
87	C₂²×C₄	C₂×D₆₈	1	3	yes	—
88	C₂×C₈	—	2	6	no	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	2	6	no	—
91	C₄⋊C₄	—	68	6	no	—
92	C₄⋊C₄	—	68	6	no	—
93	C₄⋊C₄	—	68	6	no	—
94	C₄²	#63254	1	3	yes	—
95	C₄²	Q₈×D₁₇	1	1	yes	characteristic
96	C₂³	—	68	2	no	—
97	C₂³	—	136	3	no	—
98	C₂³	—	136	3	no	—
99	C₂³	#10587560	1	1	yes	characteristic
100	D₄	—	68	12	no	—
101	D₄	—	136	6	no	—
102	C₂×C₄	—	136	3	no	—
103	C₂×C₄	—	136	3	no	—
104	C₂×C₄	—	136	3	no	—
105	C₂×C₄	#10587572	1	3	yes	—
106	C₂×C₄	#10587583	1	3	yes	—
107	C₂×C₄	#10587590	1	6	yes	—
108	C₂×C₄	#10587594	1	6	yes	—
109	C₈	—	4	6	no	—
110	C₂²	—	272	1	no	—
111	C₂²	—	272	3	no	—
112	C₂²	—	136	6	no	—
113	C₂²	#11762746	1	1	yes	characteristic
114	C₂²	#11763193	1	3	yes	—
115	C₂²	#11763198	1	3	yes	—
116	C₄	—	272	3	no	—
117	C₄	#11763110	1	6	yes	—
118	C₄	#11763134	1	6	yes	—
119	C₂	—	272	2	no	—
120	C₂	#14528396	1	1	yes	characteristic
121	C₂	#14528423	1	3	yes	—
122	C₂	#14530304	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529590

C₂

yes

maximal

#14528912

C₂

yes

maximal, complemented

C₁₇×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#10587560

C₂²

yes

characteristic

C₂×#10587492

—

#11762679

C₂²

yes

—

#11762572

—

C₂×C₄×C₈×C₁₇

C₂²

yes

—

C₂²×D₆₈

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

#10587560

—

#10587560

Q₈

yes

—

C₄×C₈×C₁₇

—

#10587492

—

C₂³×D₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×Dic₁₇

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×Q₈

yes

characteristic

C₄²×C₁₇

#41

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

C₂³×C₁₇

yes

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

C₈×C₁₇

—

#1957

—

#1834

—

#1273

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

#399

yes

characteristic

C₂²×C₁₇

#495

yes

—

C₂²×C₁₇

#496

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

#478

yes

—

C₄×C₁₇

#482

yes

—

Dic₁₇

—

C₂×#127

—

C₂×#102

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

#384

—

#340

—

C₂×C₁₇

#1711

yes

characteristic

C₂×C₁₇

#1717

yes

—

C₂×C₁₇

#2092

yes

—

D₁₇

—

C₂²×D₄

—

C₂²×C₈

D₆₈

yes

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

#102

—

C₄×C₈

—

C₁₇

#10678

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₆₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

#63254

yes

—

C₄²

Q₈×D₁₇

yes

characteristic

C₂³

—

C₂³

—

136

—

C₂³

—

136

—

C₂³

#10587560

yes

characteristic

100

D₄

—

101

D₄

—

136

—

102

C₂×C₄

—

136

—

103

C₂×C₄

—

136

—

104

C₂×C₄

—

136

—

105

C₂×C₄

#10587572

yes

—

106

C₂×C₄

#10587583

yes

—

107

C₂×C₄

#10587590

yes

—

108

C₂×C₄

#10587594

yes

—

109

C₈

—

110

C₂²

—

272

—

111

C₂²

—

272

—

112

C₂²

—

136

—

113

C₂²

#11762746

yes

characteristic

114

C₂²

#11763193

yes

—

115

C₂²

#11763198

yes

—

116

C₄

—

272

—

117

C₄

#11763110

yes

—

118

C₄

#11763134

yes

—

119

C₂

—

272

—

120

C₂

#14528396

yes

characteristic

121

C₂

#14528423

yes

—

122

C₂

#14530304

yes

—

	subgroup	G/H
center	C₂³	#10587560
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1273	C₂
socle	C₂³×C₁₇	#127
prime radical	C₂×#10587560	C₂²
hypercenter	#1273	D₁₇
hypocenter	C₁₇	#10678
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#10587560

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1273

C₂

C₂³×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #14528912	C₂	semidirect
3. C₁₇×#1273	C₂	semidirect
51. #1273	D₁₇	semidirect
80. C₁₇	#10678	semidirect

H ≅ G/N

type

2. #14528912

C₂

semidirect

3. C₁₇×#1273

C₂

semidirect

51. #1273

D₁₇

semidirect

80. C₁₇

#10678

semidirect

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

subgroup	index
#14529590	2
#14528912	2
C₁₇×#1273	2
#10678	17

subgroup

index

#14529590

#14528912

C₁₇×#1273

#10678

Subgroups — #16939399

Conjugacy classes of subgroups

Complements

Maximal subgroups