Group #16954809 — subgroups

Subgroups — #16954809

Order 4352.

206 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₂×#11762643	C₂	1	1	yes	maximal, characteristic
2	#14530380	C₂	1	1	yes	maximal, characteristic
3	#14526979	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₇×C₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	C₄	1	1	yes	characteristic
6	#11762764	C₄	1	1	yes	characteristic
7	#11762643	—	2	2	no	—
8	C₂×#10587465	C₂²	1	2	yes	—
9	#11762584	C₄	1	2	yes	—
10	C₁₆×Dic₁₇	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×C₃₂	—	2	2	no	—
13	C₂×C₁₇×C₃₂	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	#10587612	C₈	1	1	yes	characteristic
16	C₂×#63223	C₈	1	1	yes	characteristic
17	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
19	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
20	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
21	#10587486	C₈	1	2	yes	—
22	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
23	#10587483	—	2	2	no	—
24	C₂×#63214	C₂×C₄	1	2	yes	—
25	C₁₇×C₃₂	—	2	2	no	—
26	#10587465	—	2	2	no	—
27	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
31	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²⋊Dic₁₇	C₁₆	1	2	yes	—
33	C₄⋊Dic₁₇	C₁₆	1	2	yes	—
34	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
35	#63223	—	2	2	no	—
36	C₂×#3622	C₂×C₈	1	2	yes	—
37	C₁₆×C₁₇	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₁₇	C₂×Q₈	1	1	yes	characteristic
40	#63214	—	2	2	no	—
41	#13628	—	17	1	no	maximal, Sylow
42	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
46	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
47	C₂×Dic₁₇	C₂×C₁₆	1	4	yes	—
48	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
49	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
51	#3622	—	2	2	no	—
52	C₂×M₆₄	—	17	1	no	—
53	C₂²×C₃₂	D₁₇	1	1	yes	characteristic
54	#2108	—	17	1	no	—
55	C₄⋊C₃₂	—	17	4	no	—
56	C₂²×C₁₇	—	2	2	no	—
57	C₂²×C₁₇	C₄⋊C₁₆	1	2	yes	—
58	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
59	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
60	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
61	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
62	Dic₁₇	—	2	4	no	—
63	C₂×M₃₂	—	17	1	no	—
64	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
65	#405	—	17	1	no	—
66	M₆₄	—	34	2	no	—
67	C₂×C₃₂	—	2	2	no	—
68	C₂×C₃₂	—	17	2	no	—
69	C₂×C₃₂	C₂×D₁₇	1	2	yes	—
70	#346	—	17	2	no	—
71	C₄×C₁₆	—	17	2	no	—
72	C₂×C₁₇	—	2	2	no	—
73	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
74	C₂×C₁₇	#2258	1	2	yes	—
75	C₂×M₁₆	—	17	1	no	—
76	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
77	#117	—	17	1	no	—
78	M₃₂	—	34	2	no	—
79	C₂×C₁₆	—	17	2	no	—
80	C₂×C₁₆	Dic₃₄	1	2	yes	—
81	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
82	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
83	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
84	#97	—	17	2	no	—
85	C₄×C₈	—	17	2	no	—
86	C₃₂	—	2	2	no	—
87	C₃₂	—	34	2	no	—
88	C₁₇	#13628	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
90	M₁₆	—	34	2	no	—
91	C₂×C₈	—	17	2	no	—
92	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
93	C₂×C₈	#63218	1	2	yes	—
94	C₂×C₈	#63225	1	2	yes	—
95	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
96	C₄⋊C₄	—	17	2	no	—
97	C₂²⋊C₄	—	17	2	no	—
98	C₄²	—	17	2	no	—
99	C₁₆	—	34	2	no	—
100	C₁₆	#63225	1	2	yes	—
101	C₁₆	C₂×Dic₃₄	1	1	yes	characteristic
102	C₁₆	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
103	C₂³	C₁₆×D₁₇	1	1	yes	characteristic
104	C₂×C₄	—	17	4	no	—
105	C₂×C₄	C₁₆×D₁₇	1	1	yes	characteristic
106	C₂×C₄	#10587469	1	2	yes	—
107	C₂×C₄	#10587485	1	2	yes	—
108	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
109	C₈	—	34	2	no	—
110	C₈	#10587485	1	2	yes	—
111	C₈	C₂×#63218	1	1	yes	characteristic
112	C₈	C₂×#63225	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	#11762627	1	2	yes	—
115	C₂²	C₂×C₁₆×D₁₇	1	1	yes	characteristic
116	C₄	—	34	4	no	—
117	C₄	#11762627	1	2	yes	—
118	C₄	C₂×#10587485	1	1	yes	characteristic
119	C₄	C₂×#10587469	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×#11762627	1	1	yes	characteristic
122	C₂	#14531077	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762643

C₂

yes

maximal, characteristic

#14530380

C₂

yes

maximal, characteristic

#14526979

C₂

yes

maximal, complemented

C₂²×C₁₇×C₃₂

C₂

yes

maximal, characteristic

C₂×#10587483

C₄

yes

characteristic

#11762764

C₄

yes

characteristic

#11762643

—

C₂×#10587465

C₂²

yes

—

#11762584

C₄

yes

—

C₁₆×Dic₁₇

C₂²

yes

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×C₃₂

—

C₂×C₁₇×C₃₂

C₂²

yes

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587612

C₈

yes

characteristic

C₂×#63223

C₈

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

D₄

yes

—

C₂×C₁₆×C₁₇

Q₈

yes

—

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

#10587486

C₈

yes

—

C₈×Dic₁₇

C₂×C₄

yes

—

#10587483

—

C₂×#63214

C₂×C₄

yes

—

C₁₇×C₃₂

—

#10587465

—

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂²⋊Dic₁₇

C₁₆

yes

—

C₄⋊Dic₁₇

C₁₆

yes

—

C₄×Dic₁₇

C₂×C₈

yes

—

#63223

—

C₂×#3622

C₂×C₈

yes

—

C₁₆×C₁₇

C₄⋊C₄

yes

—

C₁₆×C₁₇

C₂×D₄

yes

characteristic

C₁₆×C₁₇

C₂×Q₈

yes

—

—

C₂³×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

M₃₂

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×Dic₁₇

C₂×C₁₆

yes

—

C₈×C₁₇

C₄⋊C₈

yes

—

C₈×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₇

C₂×M₁₆

yes

characteristic

#3622

—

C₂×M₆₄

—

C₂²×C₃₂

D₁₇

yes

characteristic

#2108

—

C₄⋊C₃₂

—

C₂²×C₁₇

—

C₂²×C₁₇

C₄⋊C₁₆

yes

—

C₂²×C₁₇

C₂²×C₁₆

yes

characteristic

C₄×C₁₇

C₄⋊C₁₆

yes

—

C₄×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₇

C₂×M₃₂

yes

characteristic

Dic₁₇

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₁₇

yes

—

#346

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₁₇

#2258

yes

—

C₂×M₁₆

—

C₂²×C₈

C₄×D₁₇

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

Dic₃₄

yes

—

C₂×C₁₆

C₄×D₁₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₇

yes

—

C₂×C₁₆

C₂²×D₁₇

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₁₇

#13628

yes

characteristic, complemented, Sylow

C₂²×C₄

C₈×D₁₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₈×D₁₇

yes

characteristic

C₂×C₈

#63218

yes

—

C₂×C₈

#63225

yes

—

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

100

C₁₆

#63225

yes

—

101

C₁₆

C₂×Dic₃₄

yes

characteristic

102

C₁₆

C₂×C₂²⋊D₁₇

yes

characteristic

103

C₂³

C₁₆×D₁₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

C₁₆×D₁₇

yes

characteristic

106

C₂×C₄

#10587469

yes

—

107

C₂×C₄

#10587485

yes

—

108

C₂×C₄

C₂×C₈×D₁₇

yes

characteristic

109

C₈

—

110

C₈

#10587485

yes

—

111

C₈

C₂×#63218

yes

characteristic

112

C₈

C₂×#63225

yes

characteristic

113

C₂²

—

114

C₂²

#11762627

yes

—

115

C₂²

C₂×C₁₆×D₁₇

yes

characteristic

116

C₄

—

117

C₄

#11762627

yes

—

118

C₄

C₂×#10587485

yes

characteristic

119

C₄

C₂×#10587469

yes

characteristic

120

C₂

—

121

C₂

C₂×#11762627

yes

characteristic

122

C₂

#14531077

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₇
commutator, socle	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂²×C₁₇×C₃₂	C₂
prime radical	C₂³×C₁₇	C₂×C₁₆
cosocle	C₂×C₁₆×C₁₇	C₂³
hypercenter	C₂²×C₃₂	D₁₇
hypocenter	C₁₇	#13628
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₁₇

commutator, socle

C₂²×C₁₇

C₂²×C₁₆

Fitting

C₂²×C₁₇×C₃₂

C₂

prime radical

C₂³×C₁₇

C₂×C₁₆

cosocle

C₂×C₁₆×C₁₇

C₂³

hypercenter

C₂²×C₃₂

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂

N	H ≅ G/N	type
3. #14526979	C₂	semidirect
88. C₁₇	#13628	semidirect

H ≅ G/N

type

3. #14526979

C₂

semidirect

88. C₁₇

#13628

semidirect

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

subgroup	index
C₂×#11762643	2
#14530380	2
#14526979	2
C₂²×C₁₇×C₃₂	2
#13628	17

subgroup

index

C₂²×C₁₇×C₃₂

Subgroups — #16954809

Conjugacy classes of subgroups

Complements

Maximal subgroups