Group #16954506 — subgroups

Subgroups — #16954506

Order 4352.

238 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂²×#10587465	C₂	1	1	yes	maximal, characteristic
2	#14530380	C₂	1	1	yes	maximal, characteristic
3	#14526917	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₁₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₇×M₆₄	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₄	1	1	yes	characteristic
7	#11762764	C₄	1	1	yes	characteristic
8	C₂×#10587465	C₄	1	4	yes	complemented
9	C₂×#10587465	C₂²	1	1	yes	characteristic
10	C₂×#10587465	C₂²	1	1	yes	characteristic
11	#11762584	C₄	1	2	yes	—
12	C₁₆×Dic₁₇	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×M₆₄	C₄	1	4	yes	complemented
15	C₂×C₁₇×C₃₂	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	#10587612	C₈	1	1	yes	characteristic
18	C₂×#63223	C₈	1	1	yes	characteristic
19	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
23	#10587486	C₈	1	2	yes	—
24	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
25	#10587483	C₈	1	4	yes	—
26	C₂×#63214	C₂×C₄	1	2	yes	—
27	C₁₇×C₃₂	C₂×C₄	1	4	yes	—
28	#10587465	C₂×C₄	1	2	yes	—
29	#10587465	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₁₇	C₄²	1	2	yes	—
32	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²⋊Dic₁₇	C₁₆	1	2	yes	—
36	C₄⋊Dic₁₇	C₁₆	1	2	yes	—
37	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
38	#63223	C₁₆	1	4	yes	—
39	C₂×#3622	C₂×C₈	1	2	yes	—
40	C₁₆×C₁₇	C₄²	1	2	yes	—
41	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	#63214	C₂×C₈	1	4	yes	—
44	#13549	—	17	1	no	maximal, Sylow
45	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
49	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
50	C₂×Dic₁₇	C₂×C₁₆	1	4	yes	—
51	C₈×C₁₇	C₄×C₈	1	2	yes	—
52	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
53	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
54	#3622	C₂×C₁₆	1	4	yes	—
55	C₂×M₆₄	D₁₇	1	1	yes	characteristic
56	C₂²×C₃₂	—	17	1	no	—
57	#2108	—	17	1	no	—
58	#1397	—	17	2	no	—
59	C₄×C₃₂	—	17	2	no	—
60	C₂²×C₁₇	—	2	2	no	—
61	C₂²×C₁₇	C₄×C₁₆	1	2	yes	—
62	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
63	C₄×C₁₇	C₄×C₁₆	1	2	yes	—
64	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
66	Dic₁₇	—	2	4	no	—
67	C₂×M₃₂	—	17	1	no	—
68	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
69	#405	—	17	1	no	—
70	M₆₄	Dic₁₇	1	4	yes	complemented
71	C₂×C₃₂	—	17	4	no	—
72	C₂×C₃₂	—	17	1	no	—
73	C₂×C₃₂	—	17	1	no	—
74	C₂×C₃₂	C₂×D₁₇	1	2	yes	—
75	#346	—	17	2	no	—
76	C₄×C₁₆	—	17	2	no	—
77	C₂×C₁₇	—	2	2	no	—
78	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
79	C₂×C₁₇	#2258	1	2	yes	—
80	C₂×M₁₆	—	17	1	no	—
81	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
82	#117	—	17	1	no	—
83	M₃₂	—	17	4	no	—
84	C₂×C₁₆	—	17	2	no	—
85	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
86	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
87	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
88	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
89	#97	—	17	2	no	—
90	C₄×C₈	—	17	2	no	—
91	C₃₂	—	17	2	no	—
92	C₃₂	—	17	2	no	—
93	C₃₂	C₂×Dic₁₇	1	4	yes	—
94	C₁₇	#13549	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
96	M₁₆	—	17	4	no	—
97	C₂×C₈	—	17	2	no	—
98	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
99	C₂×C₈	C₈×D₁₇	1	2	yes	—
100	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
101	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
102	C₄⋊C₄	—	17	2	no	—
103	C₂²⋊C₄	—	17	2	no	—
104	C₄²	—	17	2	no	—
105	C₁₆	—	17	4	no	—
106	C₁₆	C₄×Dic₁₇	1	2	yes	—
107	C₁₆	C₂×C₄×D₁₇	1	1	yes	characteristic
108	C₁₆	C₂²×Dic₁₇	1	1	yes	characteristic
109	C₂³	C₁₆×D₁₇	1	1	yes	characteristic
110	C₂×C₄	—	17	4	no	—
111	C₂×C₄	C₁₆×D₁₇	1	2	yes	—
112	C₂×C₄	C₁₆×D₁₇	1	1	yes	characteristic
113	C₂×C₄	C₈×Dic₁₇	1	2	yes	—
114	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
115	C₈	—	17	4	no	—
116	C₈	C₈×Dic₁₇	1	2	yes	—
117	C₈	C₂×C₈×D₁₇	1	1	yes	characteristic
118	C₈	C₂×C₄×Dic₁₇	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₁₆×Dic₁₇	1	2	yes	—
121	C₂²	C₂×C₁₆×D₁₇	1	1	yes	characteristic
122	C₄	—	34	4	no	—
123	C₄	C₁₆×Dic₁₇	1	2	yes	—
124	C₄	C₂×C₈×Dic₁₇	1	1	yes	characteristic
125	C₄	C₂×C₁₆×D₁₇	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₁₆×Dic₁₇	1	1	yes	characteristic
128	C₂	#14531079	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10587465

C₂

yes

maximal, characteristic

#14530380

C₂

yes

maximal, characteristic

#14526917

C₂

yes

maximal, complemented

C₃₂×Dic₁₇

C₂

yes

maximal, complemented

C₂×C₁₇×M₆₄

C₂

yes

maximal, characteristic

C₂×#10587483

C₄

yes

characteristic

#11762764

C₄

yes

characteristic

C₂×#10587465

C₄

yes

complemented

C₂×#10587465

C₂²

yes

characteristic

C₂×#10587465

C₂²

yes

characteristic

#11762584

C₄

yes

—

C₁₆×Dic₁₇

C₂²

yes

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×M₆₄

C₄

yes

complemented

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

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

#10587486

C₈

yes

—

C₈×Dic₁₇

C₂×C₄

yes

—

#10587483

C₈

yes

—

C₂×#63214

C₂×C₄

yes

—

C₁₇×C₃₂

C₂×C₄

yes

—

#10587465

C₂×C₄

yes

—

#10587465

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₄²

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂²⋊Dic₁₇

C₁₆

yes

—

C₄⋊Dic₁₇

C₁₆

yes

—

C₄×Dic₁₇

C₂×C₈

yes

—

#63223

C₁₆

yes

—

C₂×#3622

C₂×C₈

yes

—

C₁₆×C₁₇

C₄²

yes

—

C₁₆×C₁₇

C₂²×C₄

yes

characteristic

C₁₆×C₁₇

C₂²×C₄

yes

characteristic

#63214

C₂×C₈

yes

—

#13549

—

maximal, Sylow

C₂³×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

—

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×C₁₆

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×Dic₁₇

C₂×C₁₆

yes

—

C₈×C₁₇

C₄×C₈

yes

—

C₈×C₁₇

C₂×C₄²

yes

characteristic

C₈×C₁₇

C₂²×C₈

yes

characteristic

#3622

C₂×C₁₆

yes

—

C₂×M₆₄

D₁₇

yes

characteristic

C₂²×C₃₂

—

#2108

—

#1397

—

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

C₂²×C₁₆

yes

characteristic

Dic₁₇

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

#405

—

M₆₄

Dic₁₇

yes

complemented

C₂×C₃₂

—

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

C₄×D₁₇

yes

—

C₂×C₁₆

C₄×D₁₇

yes

characteristic

C₂×C₁₆

C₂×Dic₁₇

yes

—

C₂×C₁₆

C₂²×D₁₇

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

C₂×Dic₁₇

yes

—

C₁₇

#13549

yes

characteristic, complemented, Sylow

C₂²×C₄

C₈×D₁₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₈×D₁₇

yes

characteristic

C₂×C₈

C₈×D₁₇

yes

—

100

C₂×C₈

C₄×Dic₁₇

yes

—

101

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

C₄×Dic₁₇

yes

—

107

C₁₆

C₂×C₄×D₁₇

yes

characteristic

108

C₁₆

C₂²×Dic₁₇

yes

characteristic

109

C₂³

C₁₆×D₁₇

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

C₁₆×D₁₇

yes

—

112

C₂×C₄

C₁₆×D₁₇

yes

characteristic

113

C₂×C₄

C₈×Dic₁₇

yes

—

114

C₂×C₄

C₂×C₈×D₁₇

yes

characteristic

115

C₈

—

116

C₈

C₈×Dic₁₇

yes

—

117

C₈

C₂×C₈×D₁₇

yes

characteristic

118

C₈

C₂×C₄×Dic₁₇

yes

characteristic

119

C₂²

—

120

C₂²

C₁₆×Dic₁₇

yes

—

121

C₂²

C₂×C₁₆×D₁₇

yes

characteristic

122

C₄

—

123

C₄

C₁₆×Dic₁₇

yes

—

124

C₄

C₂×C₈×Dic₁₇

yes

characteristic

125

C₄

C₂×C₁₆×D₁₇

yes

characteristic

126

C₂

—

127

C₂

C₂×C₁₆×Dic₁₇

yes

characteristic

128

C₂

#14531079

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₇
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Fitting	C₂×C₁₇×M₆₄	C₂
socle	C₂²×C₁₇	C₂²×C₁₆
prime radical	C₂³×C₁₇	C₂×C₁₆
cosocle	C₂×C₁₆×C₁₇	C₂³
hypercenter	C₂×M₆₄	D₁₇
hypocenter	C₁₇	#13549
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₁₇

commutator

C₂×C₁₇

C₂×C₄×C₁₆

Fitting

C₂×C₁₇×M₆₄

C₂

socle

C₂²×C₁₇

C₂²×C₁₆

prime radical

C₂³×C₁₇

C₂×C₁₆

cosocle

C₂×C₁₆×C₁₇

C₂³

hypercenter

C₂×M₆₄

D₁₇

C₁₇

G ▷ C₂×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. #14526917	C₂	semidirect
4. C₃₂×Dic₁₇	C₂	semidirect
8. C₂×#10587465	C₄	semidirect
14. C₁₇×M₆₄	C₄	semidirect
70. M₆₄	Dic₁₇	semidirect
94. C₁₇	#13549	semidirect

H ≅ G/N

type

3. #14526917

C₂

semidirect

4. C₃₂×Dic₁₇

C₂

semidirect

8. C₂×#10587465

C₄

semidirect

14. C₁₇×M₆₄

C₄

semidirect

70. M₆₄

Dic₁₇

semidirect

94. C₁₇

#13549

semidirect

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

subgroup	index
C₂²×#10587465	2
#14530380	2
#14526917	2
C₃₂×Dic₁₇	2
C₂×C₁₇×M₆₄	2
#13549	17

subgroup

index

C₂²×#10587465

#14530380

#14526917

C₃₂×Dic₁₇

C₂×C₁₇×M₆₄

#13549

Subgroups — #16954506

Conjugacy classes of subgroups

Complements

Maximal subgroups