Group #16937401 — subgroups

Subgroups — #16937401

Order 4352.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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	#14529607	C₂	1	3	yes	maximal
2	#14526665	C₂	1	3	yes	maximal, complemented
3	C₁₇×#2845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10587560	C₂²	1	1	yes	characteristic
5	C₂×#10587480	—	2	6	no	—
6	C₂×#10587474	C₂²	1	3	yes	—
7	C₁₇×#398	—	2	6	no	—
8	C₁₇×#382	C₂²	1	3	yes	—
9	C₂²×D₆₈	—	2	3	no	—
10	C₂×C₄⋊C₄×C₁₇	—	2	6	no	—
11	C₂×C₄⋊C₄×C₁₇	D₄	1	6	yes	—
12	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
13	C₂²×#3622	—	2	3	no	—
14	#10587560	—	2	3	no	—
15	#10587560	Q₈	1	2	yes	—
16	C₁₇×#95	—	2	2	no	—
17	C₁₇×#95	—	2	6	no	—
18	#10587480	—	4	12	no	—
19	#10587474	—	2	6	no	—
20	C₂³×D₁₇	—	4	1	no	—
21	C₂²×C₄×C₁₇	—	2	6	no	—
22	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
23	C₂×D₆₈	—	2	6	no	—
24	C₂×D₆₈	—	4	6	no	—
25	C₂×D₆₈	—	4	6	no	—
26	C₂×D₆₈	—	4	3	no	—
27	C₄⋊C₄×C₁₇	—	4	6	no	—
28	C₄⋊C₄×C₁₇	—	8	6	no	—
29	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₁₇	#41	1	3	yes	—
31	C₂×#3622	—	2	6	no	—
32	C₂×#3622	—	4	3	no	—
33	C₂×#3622	—	4	3	no	—
34	#10360	—	17	1	no	maximal, Sylow
35	C₂³×C₁₇	#120	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	—	2	3	no	—
40	C₂×C₄×C₁₇	—	2	3	no	—
41	C₂×C₄×C₁₇	—	4	6	no	—
42	C₂×C₄×C₁₇	—	4	6	no	—
43	C₂×C₄×C₁₇	—	4	6	no	—
44	C₂×C₄×C₁₇	C₂²⋊Q₈	1	3	yes	—
45	C₂×C₄×C₁₇	#124	1	3	yes	—
46	D₆₈	—	4	12	no	—
47	D₆₈	—	8	6	no	—
48	#3622	—	4	6	no	—
49	#2845	D₁₇	1	1	yes	characteristic, complemented
50	#1961	—	17	3	no	—
51	#1295	—	17	3	no	—
52	C₂²×C₁₇	#393	1	1	yes	characteristic
53	C₂²×C₁₇	#453	1	3	yes	—
54	C₂²×C₁₇	#454	1	3	yes	—
55	C₂×D₁₇	—	8	6	no	—
56	C₂×D₁₇	—	16	3	no	—
57	C₂×D₁₇	—	16	1	no	—
58	C₄×C₁₇	—	2	3	no	—
59	C₄×C₁₇	—	2	3	no	—
60	C₄×C₁₇	—	8	6	no	—
61	C₂×#127	—	17	1	no	—
62	C₂×#102	—	34	6	no	—
63	C₂×#97	—	17	3	no	—
64	#398	—	2	6	no	—
65	#382	C₂×D₁₇	1	3	yes	—
66	C₂×C₁₇	#1656	1	3	yes	—
67	C₂×C₁₇	#1658	1	1	yes	characteristic
68	C₂×C₁₇	#2027	1	3	yes	—
69	D₁₇	—	16	2	no	—
70	C₂²×D₄	—	34	3	no	—
71	C₂²×C₈	—	34	3	no	—
72	#127	—	17	2	no	—
73	#127	—	34	3	no	—
74	C₂×C₄⋊C₄	—	2	6	no	—
75	C₂×C₄⋊C₄	C₂²⋊D₁₇	1	6	yes	—
76	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
77	#102	—	68	12	no	—
78	#97	—	34	6	no	—
79	#95	—	2	2	no	—
80	#95	—	2	6	no	—
81	C₁₇	#10360	1	1	yes	characteristic, complemented, Sylow
82	C₂⁴	—	68	1	no	—
83	C₂×D₄	—	34	6	no	—
84	C₂×D₄	—	68	6	no	—
85	C₂×D₄	—	68	6	no	—
86	C₂×D₄	—	68	3	no	—
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
89	C₂×C₈	—	34	6	no	—
90	C₂×C₈	—	68	3	no	—
91	C₂×C₈	—	68	3	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄⋊C₄	—	8	6	no	—
94	C₄²	Q₈×D₁₇	1	1	yes	characteristic
95	C₄²	#63256	1	3	yes	—
96	C₂³	—	68	2	no	—
97	C₂³	—	136	3	no	—
98	C₂³	—	136	3	no	—
99	C₂³	#10587639	1	1	yes	characteristic
100	D₄	—	68	12	no	—
101	D₄	—	136	6	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	#10587632	1	3	yes	—
108	C₂×C₄	#10587633	1	3	yes	—
109	C₈	—	68	6	no	—
110	C₂²	—	272	3	no	—
111	C₂²	—	272	1	no	—
112	C₂²	—	136	6	no	—
113	C₂²	#11762720	1	1	yes	characteristic
114	C₂²	#11762977	1	3	yes	—
115	C₂²	#11762982	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	272	2	no	—
120	C₂	#14528077	1	3	yes	—
121	C₂	#14528086	1	1	yes	characteristic
122	C₂	#14529937	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529607

C₂

yes

maximal

#14526665

C₂

yes

maximal, complemented

C₁₇×#2845

C₂

yes

maximal, characteristic, complemented

C₂×#10587560

C₂²

yes

characteristic

C₂×#10587480

—

C₂×#10587474

C₂²

yes

—

C₁₇×#398

—

C₁₇×#382

C₂²

yes

—

C₂²×D₆₈

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂²×#3622

—

#10587560

—

#10587560

Q₈

yes

—

C₁₇×#95

—

C₁₇×#95

—

#10587480

—

#10587474

—

C₂³×D₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×Q₈

yes

characteristic

C₄²×C₁₇

#41

yes

—

—

—

—

—

C₂³×C₁₇

yes

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

#124

yes

—

D₆₈

—

D₆₈

—

#3622

—

#2845

D₁₇

yes

characteristic, complemented

#1961

—

#1295

—

C₂²×C₁₇

#393

yes

characteristic

C₂²×C₁₇

#453

yes

—

C₂²×C₁₇

#454

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₂×#127

—

C₂×#102

—

C₂×#97

—

#398

—

#382

C₂×D₁₇

yes

—

C₂×C₁₇

#1656

yes

—

C₂×C₁₇

#1658

yes

characteristic

C₂×C₁₇

#2027

yes

—

D₁₇

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₁₇

yes

—

C₂×C₄²

C₂²×D₁₇

yes

—

—

—

—

C₁₇

#10360

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₁₇

yes

characteristic

C₄²

#63256

yes

—

C₂³

—

C₂³

—

136

—

C₂³

—

136

—

C₂³

#10587639

yes

characteristic

100

D₄

—

101

D₄

—

136

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#10587632

yes

—

108

C₂×C₄

#10587633

yes

—

109

C₈

—

110

C₂²

—

272

—

111

C₂²

—

272

—

112

C₂²

—

136

—

113

C₂²

#11762720

yes

characteristic

114

C₂²

#11762977

yes

—

115

C₂²

#11762982

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

272

—

120

C₂

#14528077

yes

—

121

C₂

#14528086

yes

characteristic

122

C₂

#14529937

yes

—

	subgroup	G/H
center	C₂³	#10587639
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2845	C₂
socle	C₂³×C₁₇	#120
prime radical	C₂×#10587560	C₂²
hypercenter	#2845	D₁₇
hypocenter	C₁₇	#10360
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#10587639

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#2845

C₂

C₂³×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
2. #14526665	C₂	semidirect
3. C₁₇×#2845	C₂	semidirect
49. #2845	D₁₇	semidirect
81. C₁₇	#10360	semidirect

H ≅ G/N

type

2. #14526665

C₂

semidirect

3. C₁₇×#2845

C₂

semidirect

49. #2845

D₁₇

semidirect

81. C₁₇

#10360

semidirect

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

subgroup	index
#14529607	2
#14526665	2
C₁₇×#2845	2
#10360	17

subgroup

index

#14529607

#14526665

C₁₇×#2845

#10360

Subgroups — #16937401

Conjugacy classes of subgroups

Complements

Maximal subgroups