Group #16954806 — subgroups

Subgroups — #16954806

Order 4352.

218 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#14526979	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₇×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
7	#11762643	—	2	2	no	—
8	C₂×#10587465	C₂²	1	2	yes	—
9	C₁₆×Dic₁₇	C₄	1	2	yes	—
10	C₁₆×Dic₁₇	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×M₆₄	—	2	2	no	—
13	C₂×C₁₇×C₃₂	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
16	C₂²×#3622	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	C₈×Dic₁₇	C₈	1	2	yes	—
22	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
23	C₂×#63214	—	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	C₂²×Dic₁₇	C₁₆	1	2	yes	—
29	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
32	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×Dic₁₇	C₁₆	1	1	yes	characteristic
34	C₄×Dic₁₇	C₁₆	1	1	yes	characteristic
35	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
36	C₂×#3622	—	2	2	no	—
37	C₂×#3622	C₂×C₈	1	2	yes	—
38	C₁₆×C₁₇	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	#63214	—	2	2	no	—
42	#13627	—	17	1	no	maximal, Sylow
43	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
45	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
47	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×Dic₁₇	—	2	4	no	—
49	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
50	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
51	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
52	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
54	#3622	—	2	2	no	—
55	C₂×M₆₄	—	17	1	no	—
56	C₂×M₆₄	D₁₇	1	1	yes	characteristic
57	C₂×C₄×C₁₆	—	17	1	no	—
58	C₄⋊C₃₂	—	17	4	no	—
59	C₂²×C₁₇	—	2	2	no	—
60	C₂²×C₁₇	C₄⋊C₁₆	1	2	yes	—
61	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
62	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
63	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
65	Dic₁₇	—	2	2	no	—
66	Dic₁₇	M₆₄	1	4	yes	complemented
67	C₂²×C₁₆	—	17	1	no	—
68	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
69	C₂×C₄×C₈	—	17	1	no	—
70	M₆₄	—	2	2	no	—
71	M₆₄	—	34	2	no	—
72	C₂×C₃₂	—	17	2	no	—
73	C₂×C₃₂	C₂×D₁₇	1	2	yes	—
74	C₄×C₁₆	—	17	2	no	—
75	C₄×C₁₆	—	17	2	no	—
76	C₂×C₁₇	—	2	2	no	—
77	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
78	C₂×C₁₇	C₂×M₆₄	1	2	yes	—
79	C₂²×C₈	—	17	1	no	—
80	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
81	C₂×C₄²	—	17	1	no	—
82	C₂×C₁₆	—	17	2	no	—
83	C₂×C₁₆	—	34	2	no	—
84	C₂×C₁₆	Dic₃₄	1	2	yes	—
85	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
86	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
87	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
88	C₄×C₈	—	17	2	no	—
89	C₄×C₈	—	17	2	no	—
90	C₃₂	—	2	2	no	—
91	C₃₂	—	34	2	no	—
92	C₁₇	#13627	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	17	2	no	—
94	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
95	C₂×C₈	—	17	2	no	—
96	C₂×C₈	—	34	2	no	—
97	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
98	C₂×C₈	#63218	1	2	yes	—
99	C₂×C₈	#63225	1	2	yes	—
100	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
101	C₄²	—	17	1	no	—
102	C₄²	—	17	1	no	—
103	C₄²	—	17	2	no	—
104	C₁₆	—	34	2	no	—
105	C₁₆	#63225	1	2	yes	—
106	C₁₆	C₂×Dic₃₄	1	1	yes	characteristic
107	C₁₆	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
108	C₂³	C₁₆×D₁₇	1	1	yes	characteristic
109	C₂×C₄	—	17	2	no	—
110	C₂×C₄	—	17	2	no	—
111	C₂×C₄	—	34	4	no	—
112	C₂×C₄	C₁₆×D₁₇	1	1	yes	characteristic
113	C₂×C₄	#10587469	1	2	yes	—
114	C₂×C₄	#10587485	1	2	yes	—
115	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
116	C₈	—	34	2	no	—
117	C₈	#10587485	1	2	yes	—
118	C₈	C₂×#63218	1	1	yes	characteristic
119	C₈	C₂×#63225	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#11762627	1	2	yes	—
122	C₂²	C₂×C₁₆×D₁₇	1	1	yes	characteristic
123	C₄	—	17	4	no	—
124	C₄	—	34	2	no	—
125	C₄	#11762627	1	2	yes	—
126	C₄	C₂×#10587485	1	1	yes	characteristic
127	C₄	C₂×#10587469	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	C₂×#11762627	1	1	yes	characteristic
130	C₂	D₁₇×M₆₄	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762643

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₁₇

C₂

yes

maximal, characteristic

#14526979

C₂

yes

maximal, complemented

C₂×C₁₇×M₆₄

C₂

yes

maximal, characteristic

C₂²×#63214

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₄

yes

characteristic

#11762643

—

C₂×#10587465

C₂²

yes

—

C₁₆×Dic₁₇

C₄

yes

—

C₁₆×Dic₁₇

C₂²

yes

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×M₆₄

—

C₂×C₁₇×C₃₂

C₂²

yes

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₈

yes

characteristic

C₂²×#3622

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

C₈×Dic₁₇

C₈

yes

—

C₈×Dic₁₇

C₂×C₄

yes

—

C₂×#63214

—

C₂×#63214

C₂×C₄

yes

—

C₁₇×C₃₂

—

#10587465

—

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂²×Dic₁₇

C₁₆

yes

—

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

C₄×Dic₁₇

C₁₆

yes

characteristic

C₄×Dic₁₇

C₁₆

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

—

C₂×#3622

—

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×Dic₁₇

C₂×C₁₆

yes

—

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

—

C₂×M₆₄

D₁₇

yes

characteristic

C₂×C₄×C₁₆

—

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

—

Dic₁₇

M₆₄

yes

complemented

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₁₇

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₁₇

C₂×M₆₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₄×D₁₇

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Dic₃₄

yes

—

C₂×C₁₆

C₄×D₁₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₇

yes

—

C₂×C₁₆

C₂²×D₁₇

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₁₇

#13627

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₈×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₁₇

yes

characteristic

C₂×C₈

#63218

yes

—

C₂×C₈

#63225

yes

—

100

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₁₆

—

105

C₁₆

#63225

yes

—

106

C₁₆

C₂×Dic₃₄

yes

characteristic

107

C₁₆

C₂×C₂²⋊D₁₇

yes

characteristic

108

C₂³

C₁₆×D₁₇

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₁₆×D₁₇

yes

characteristic

113

C₂×C₄

#10587469

yes

—

114

C₂×C₄

#10587485

yes

—

115

C₂×C₄

C₂×C₈×D₁₇

yes

characteristic

116

C₈

—

117

C₈

#10587485

yes

—

118

C₈

C₂×#63218

yes

characteristic

119

C₈

C₂×#63225

yes

characteristic

120

C₂²

—

121

C₂²

#11762627

yes

—

122

C₂²

C₂×C₁₆×D₁₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

#11762627

yes

—

126

C₄

C₂×#10587485

yes

characteristic

127

C₄

C₂×#10587469

yes

characteristic

128

C₂

—

129

C₂

C₂×#11762627

yes

characteristic

130

C₂

D₁₇×M₆₄

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₇
commutator, socle	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂×C₁₇×M₆₄	C₂
prime radical	C₂³×C₁₇	C₂×C₁₆
cosocle	C₂×C₁₆×C₁₇	C₂³
hypercenter	C₂×M₆₄	D₁₇
hypocenter	C₁₇	#13627
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₁₇

commutator, socle

C₂²×C₁₇

C₂²×C₁₆

Fitting

C₂×C₁₇×M₆₄

C₂

prime radical

C₂³×C₁₇

C₂×C₁₆

cosocle

C₂×C₁₆×C₁₇

C₂³

hypercenter

C₂×M₆₄

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. #14526979	C₂	semidirect
66. Dic₁₇	M₆₄	semidirect
92. C₁₇	#13627	semidirect

H ≅ G/N

type

3. #14526979

C₂

semidirect

66. Dic₁₇

M₆₄

semidirect

92. C₁₇

#13627

semidirect

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

subgroup	index
C₂×#11762643	2
C₂×C₁₆×Dic₁₇	2
#14526979	2
C₂×C₁₇×M₆₄	2
#13627	17

subgroup

index

C₂×#11762643

C₂×C₁₆×Dic₁₇

#14526979

C₂×C₁₇×M₆₄

#13627

Subgroups — #16954806

Conjugacy classes of subgroups

Complements

Maximal subgroups