Group #16950776 — subgroups

Subgroups — #16950776

Order 4352.

306 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 179 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×#11762639	C₂	1	4	yes	maximal
2	#14528956	C₂	1	2	yes	maximal
3	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587490	—	2	2	no	—
5	C₂×#10587490	C₂²	1	4	yes	—
6	#11762833	—	2	2	no	—
7	#11762639	—	2	8	no	—
8	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×#346	C₄	1	4	yes	—
11	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
13	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
14	C₂×#63223	—	2	4	no	—
15	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
16	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
17	C₂×C₁₆×C₁₇	Q₈	1	4	yes	—
18	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₇	Q₈	1	2	yes	—
20	#10587490	—	2	8	no	—
21	#10587490	—	4	4	no	—
22	C₄⋊#3622	—	4	4	no	—
23	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
24	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
25	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₁₇	D₈	1	2	yes	—
27	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
28	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
30	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
31	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
33	C₄²×C₁₇	D₈	1	2	yes	—
34	#63223	—	4	4	no	—
35	#63223	—	4	4	no	—
36	C₂×#3622	—	4	4	no	—
37	C₁₆×C₁₇	C₄⋊C₄	1	8	yes	—
38	#12910	—	17	1	no	maximal, Sylow
39	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
40	C₂×C₄×C₁₇	—	4	2	no	—
41	C₂×C₄×C₁₇	#107	1	4	yes	—
42	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
48	C₈×C₁₇	#107	1	4	yes	—
49	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
50	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
51	#3622	—	8	4	no	—
52	C₂×#368	—	17	4	no	—
53	C₂×#346	D₁₇	1	1	yes	characteristic
54	#1848	—	17	2	no	—
55	C₂²×C₁₇	—	2	2	no	—
56	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
57	C₂²×C₁₇	#494	1	1	yes	characteristic
58	C₂²×C₁₇	#500	1	1	yes	characteristic
59	C₄×C₁₇	—	2	4	no	—
60	C₄×C₁₇	#384	1	1	yes	characteristic
61	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
62	C₄×C₁₇	#493	1	1	yes	characteristic
63	C₄×C₁₇	#500	1	1	yes	characteristic
64	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
65	C₂×#108	—	17	4	no	—
66	C₂×#108	—	34	2	no	—
67	#423	—	34	2	no	—
68	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
69	#368	—	34	8	no	—
70	#346	Dic₁₇	1	4	yes	—
71	C₂×C₁₇	—	2	1	no	—
72	C₂×C₁₇	—	2	1	no	—
73	C₂×C₁₇	#1845	1	1	yes	characteristic
74	C₂×C₁₇	#2161	1	2	yes	—
75	C₂×M₁₆	—	34	4	no	—
76	C₂²×C₈	D₆₈	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
78	C₂×C₄²	D₆₈	1	1	yes	characteristic
79	C₂×C₁₆	Dic₃₄	1	4	yes	—
80	C₂×C₁₆	D₆₈	1	4	yes	—
81	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
82	#108	—	34	8	no	—
83	#108	—	68	4	no	—
84	C₄⋊C₈	—	68	4	no	—
85	C₄×C₈	Dic₃₄	1	2	yes	—
86	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
87	C₁₇	#12910	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄	Dic₆₈	1	2	yes	—
89	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
90	M₁₆	—	68	4	no	—
91	M₁₆	—	68	4	no	—
92	C₂×C₈	—	68	4	no	—
93	C₂×C₈	D₁₃₆	1	2	yes	—
94	C₂×C₈	Dic₆₈	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
96	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
97	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
98	C₂×C₈	C₂×D₆₈	1	2	yes	—
99	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
100	C₄²	D₁₃₆	1	2	yes	—
101	C₄²	C₄⋊Dic₁₇	1	2	yes	—
102	C₁₆	C₄⋊Dic₁₇	1	8	yes	—
103	C₂³	C₂×Dic₆₈	1	1	yes	characteristic
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	#10587489	1	4	yes	—
106	C₂×C₄	#10587555	1	1	yes	characteristic
107	C₂×C₄	#10587560	1	1	yes	characteristic
108	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
109	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
110	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
112	C₈	—	136	4	no	—
113	C₈	#10587489	1	4	yes	—
114	C₈	#10587555	1	2	yes	—
115	C₈	C₂×C₄⋊Dic₁₇	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#10587489	1	1	yes	characteristic
118	C₂²	#11763189	1	1	yes	characteristic
119	C₂²	#11763214	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	#11762679	1	1	yes	characteristic
122	C₄	C₂×#10587489	1	1	yes	characteristic
123	C₄	#11763186	1	1	yes	characteristic
124	C₄	#11763214	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	#14528944	1	1	yes	characteristic
128	C₂	#14530579	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762639

C₂

yes

maximal

#14528956

C₂

yes

maximal

C₂×C₁₇×#346

C₂

yes

maximal, characteristic

C₂×#10587490

—

C₂×#10587490

C₂²

yes

—

#11762833

—

#11762639

—

C₂²×C₁₆×C₁₇

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#346

C₄

yes

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×#63223

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₂×C₁₆×C₁₇

Q₈

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

#10587490

—

#10587490

—

C₄⋊#3622

—

C₂²×C₄×C₁₇

Q₁₆

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

D₈

yes

—

#63223

—

#63223

—

C₂×#3622

—

C₁₆×C₁₇

C₄⋊C₄

yes

—

#12910

—

maximal, Sylow

C₂³×C₁₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#107

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₁₆

yes

characteristic

C₈×C₁₇

#107

yes

—

C₈×C₁₇

C₂×C₄⋊C₄

yes

—

C₈×C₁₇

C₄⋊Q₈

yes

—

#3622

—

C₂×#368

—

C₂×#346

D₁₇

yes

characteristic

#1848

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₂²×C₁₆

C₂×D₁₇

yes

—

C₂×#108

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

#368

—

#346

Dic₁₇

yes

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1845

yes

characteristic

C₂×C₁₇

#2161

yes

—

C₂×M₁₆

—

C₂²×C₈

D₆₈

yes

characteristic

C₂²×C₈

C₂²×D₁₇

yes

characteristic

C₂×C₄²

D₆₈

yes

characteristic

C₂×C₁₆

Dic₃₄

yes

—

C₂×C₁₆

D₆₈

yes

—

C₂×C₁₆

C₂×Dic₁₇

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

Dic₃₄

yes

—

C₄×C₈

C₂×Dic₁₇

yes

—

C₁₇

#12910

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₆₈

yes

—

C₂²×C₄

C₂×D₆₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

D₁₃₆

yes

—

C₂×C₈

Dic₆₈

yes

—

C₂×C₈

C₄⋊Dic₁₇

yes

—

C₂×C₈

C₂×Dic₃₄

yes

characteristic

C₂×C₈

C₂×Dic₃₄

yes

—

C₂×C₈

C₂×D₆₈

yes

—

C₂×C₈

C₂²×Dic₁₇

yes

characteristic

100

C₄²

D₁₃₆

yes

—

101

C₄²

C₄⋊Dic₁₇

yes

—

102

C₁₆

C₄⋊Dic₁₇

yes

—

103

C₂³

C₂×Dic₆₈

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#10587489

yes

—

106

C₂×C₄

#10587555

yes

characteristic

107

C₂×C₄

#10587560

yes

characteristic

108

C₂×C₄

C₂×D₁₃₆

yes

characteristic

109

C₂×C₄

C₂×D₁₃₆

yes

characteristic

110

C₂×C₄

C₂×Dic₆₈

yes

characteristic

111

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

112

C₈

—

136

—

113

C₈

#10587489

yes

—

114

C₈

#10587555

yes

—

115

C₈

C₂×C₄⋊Dic₁₇

yes

—

116

C₂²

—

117

C₂²

C₂×#10587489

yes

characteristic

118

C₂²

#11763189

yes

characteristic

119

C₂²

#11763214

yes

characteristic

120

C₄

—

121

C₄

#11762679

yes

characteristic

122

C₄

C₂×#10587489

yes

characteristic

123

C₄

#11763186

yes

characteristic

124

C₄

#11763214

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

#14528944

yes

characteristic

128

C₂

#14530579

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#346	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×C₁₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	#12910
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₁₃₆

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₂×C₁₇×#346

C₂

socle

C₂²×C₁₇

C₂×#107

prime radical

C₂³×C₁₇

C₂×Q₁₆

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
87. C₁₇	#12910	semidirect

H ≅ G/N

type

87. C₁₇

#12910

semidirect

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

subgroup	index
C₂×#11762639	2
#14528956	2
C₂×C₁₇×#346	2
#12910	17

subgroup

index

C₂×#11762639

#14528956

C₂×C₁₇×#346

#12910

Subgroups — #16950776

Conjugacy classes of subgroups

Complements

Maximal subgroups