Group #16939430 — subgroups

Subgroups — #16939430

Order 4352.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#14528941	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×#10587488	—	2	6	no	—
6	C₂×#10587492	—	2	6	no	—
7	#11762679	C₂²	1	3	yes	—
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	6	no	—
14	#10587560	—	2	3	no	—
15	#10587560	Q₈	1	2	yes	—
16	C₄×C₈×C₁₇	—	2	6	no	—
17	#10587492	—	4	12	no	—
18	#10587488	—	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	SD₁₆	1	12	yes	—
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	12	no	—
33	#10683	—	17	1	no	maximal, Sylow
34	C₂³×C₁₇	#127	1	1	yes	characteristic
35	C₂²×D₁₇	—	4	2	no	—
36	C₂²×D₁₇	—	8	3	no	—
37	C₂²×D₁₇	—	8	3	no	—
38	C₂×C₄×C₁₇	C₂²⋊Q₈	1	3	yes	—
39	C₂×C₄×C₁₇	#123	1	3	yes	—
40	C₂×C₄×C₁₇	C₂×SD₁₆	1	6	yes	—
41	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	6	yes	—
42	C₂×Dic₁₇	—	8	3	no	—
43	C₂×Dic₁₇	—	8	6	no	—
44	D₆₈	—	4	12	no	—
45	D₆₈	—	8	6	no	—
46	C₈×C₁₇	—	2	12	no	—
47	#1957	—	17	3	no	—
48	#1844	—	17	3	no	—
49	#1273	D₁₇	1	1	yes	characteristic, complemented
50	C₂²×C₁₇	#399	1	1	yes	characteristic
51	C₂²×C₁₇	#492	1	3	yes	—
52	C₂²×C₁₇	#496	1	3	yes	—
53	C₂×D₁₇	—	8	6	no	—
54	C₂×D₁₇	—	16	1	no	—
55	C₂×D₁₇	—	16	3	no	—
56	C₄×C₁₇	#476	1	6	yes	—
57	C₄×C₁₇	#481	1	6	yes	—
58	Dic₁₇	—	16	3	no	—
59	C₂×#127	—	17	1	no	—
60	C₂×#106	—	34	6	no	—
61	C₂×#102	—	34	6	no	—
62	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
63	#384	—	17	3	no	—
64	C₂×C₁₇	#1705	1	1	yes	characteristic
65	C₂×C₁₇	#1715	1	3	yes	—
66	C₂×C₁₇	#2094	1	3	yes	—
67	D₁₇	—	16	2	no	—
68	C₂²×D₄	—	34	3	no	—
69	C₂²×C₈	D₆₈	1	6	yes	—
70	#127	—	17	2	no	—
71	#127	—	34	3	no	—
72	C₂×C₄⋊C₄	—	34	3	no	—
73	C₂×C₄⋊C₄	—	34	6	no	—
74	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
75	#106	—	68	12	no	—
76	#102	—	68	12	no	—
77	C₄×C₈	—	2	6	no	—
78	C₁₇	#10683	1	1	yes	characteristic, complemented, Sylow
79	C₂⁴	—	68	1	no	—
80	C₂×D₄	—	34	6	no	—
81	C₂×D₄	—	68	3	no	—
82	C₂×D₄	—	68	6	no	—
83	C₂×D₄	—	68	6	no	—
84	C₂²×C₄	—	68	3	no	—
85	C₂²×C₄	C₂×D₆₈	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	#63219	1	12	yes	—
89	C₄⋊C₄	—	68	6	no	—
90	C₄⋊C₄	—	68	12	no	—
91	C₄²	#63254	1	3	yes	—
92	C₄²	Q₈×D₁₇	1	1	yes	characteristic
93	C₂³	—	68	2	no	—
94	C₂³	—	136	3	no	—
95	C₂³	—	136	3	no	—
96	C₂³	#10587560	1	1	yes	characteristic
97	D₄	—	68	12	no	—
98	D₄	—	136	6	no	—
99	C₂×C₄	—	136	3	no	—
100	C₂×C₄	—	136	6	no	—
101	C₂×C₄	#10587572	1	3	yes	—
102	C₂×C₄	#10587583	1	3	yes	—
103	C₂×C₄	C₂×#63219	1	6	yes	—
104	C₂×C₄	#10587594	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	272	1	no	—
107	C₂²	—	272	3	no	—
108	C₂²	—	136	6	no	—
109	C₂²	#11762746	1	1	yes	characteristic
110	C₂²	#11763182	1	3	yes	—
111	C₂²	#11763198	1	3	yes	—
112	C₄	—	272	3	no	—
113	C₄	#11763100	1	6	yes	—
114	C₄	#11763128	1	6	yes	—
115	C₂	—	272	2	no	—
116	C₂	#14528371	1	1	yes	characteristic
117	C₂	#14528412	1	3	yes	—
118	C₂	#14530316	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529590

C₂

yes

maximal

#14528941

C₂

yes

maximal, complemented

C₁₇×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#10587560

C₂²

yes

—

—

C₂²

yes

—

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

—

#10587560

—

#10587560

Q₈

yes

—

C₄×C₈×C₁₇

—

#10587492

—

#10587488

—

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

SD₁₆

yes

—

C₄²×C₁₇

C₂×Q₈

yes

characteristic

C₄²×C₁₇

#41

yes

—

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

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

C₈×C₁₇

—

#1957

—

#1844

—

#1273

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

#399

yes

characteristic

C₂²×C₁₇

#492

yes

—

C₂²×C₁₇

#496

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

#476

yes

—

C₄×C₁₇

#481

yes

—

Dic₁₇

—

C₂×#127

—

C₂×#106

—

C₂×#102

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

#384

—

C₂×C₁₇

#1705

yes

characteristic

C₂×C₁₇

#1715

yes

—

C₂×C₁₇

#2094

yes

—

D₁₇

—

C₂²×D₄

—

C₂²×C₈

D₆₈

yes

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

#106

—

#102

—

C₄×C₈

—

C₁₇

#10683

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

#63219

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

#63254

yes

—

C₄²

Q₈×D₁₇

yes

characteristic

C₂³

—

C₂³

—

136

—

C₂³

—

136

—

C₂³

#10587560

yes

characteristic

D₄

—

D₄

—

136

—

C₂×C₄

—

136

—

100

C₂×C₄

—

136

—

101

C₂×C₄

#10587572

yes

—

102

C₂×C₄

#10587583

yes

—

103

C₂×C₄

C₂×#63219

yes

—

104

C₂×C₄

#10587594

yes

—

105

C₈

—

106

C₂²

—

272

—

107

C₂²

—

272

—

108

C₂²

—

136

—

109

C₂²

#11762746

yes

characteristic

110

C₂²

#11763182

yes

—

111

C₂²

#11763198

yes

—

112

C₄

—

272

—

113

C₄

#11763100

yes

—

114

C₄

#11763128

yes

—

115

C₂

—

272

—

116

C₂

#14528371

yes

characteristic

117

C₂

#14528412

yes

—

118

C₂

#14530316

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	#10683
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. #14528941	C₂	semidirect
3. C₁₇×#1273	C₂	semidirect
49. #1273	D₁₇	semidirect
78. C₁₇	#10683	semidirect

H ≅ G/N

type

2. #14528941

C₂

semidirect

3. C₁₇×#1273

C₂

semidirect

49. #1273

D₁₇

semidirect

78. C₁₇

#10683

semidirect

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

subgroup	index
#14529590	2
#14528941	2
C₁₇×#1273	2
#10683	17

subgroup

index

#14529590

#14528941

C₁₇×#1273

#10683

Subgroups — #16939430

Conjugacy classes of subgroups

Complements

Maximal subgroups