Group #16949052 — subgroups

Subgroups — #16949052

Order 4352.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14531040	C₂	1	2	yes	maximal, complemented
2	#14531034	C₂	1	2	yes	maximal, complemented
3	#14528399	C₂	1	1	yes	maximal, characteristic, complemented
4	#14528390	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10587471	—	2	4	no	—
7	C₂×D₂₇₂	—	2	4	no	—
8	#11763189	C₂²	1	1	yes	characteristic
9	#11763186	—	2	2	no	—
10	#11763186	C₂²	1	1	yes	characteristic
11	#11763186	C₂²	1	2	yes	—
12	#11763156	—	2	2	no	—
13	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
14	C₈²×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₆₈	—	2	2	no	—
16	C₂×D₁₃₆	—	2	2	no	—
17	C₂×D₁₃₆	—	2	2	no	—
18	C₂×D₁₃₆	—	2	2	no	—
19	C₂×D₁₃₆	—	4	4	no	—
20	#10587560	—	2	1	no	—
21	#10587560	—	2	2	no	—
22	#10587555	—	2	1	no	—
23	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
24	C₄×C₈×C₁₇	D₄	1	2	yes	—
25	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	#10587492	—	4	4	no	—
27	#10587471	—	4	8	no	—
28	D₂₇₂	—	4	8	no	—
29	C₂×D₆₈	—	4	1	no	—
30	C₂×D₆₈	—	4	2	no	—
31	C₂×D₆₈	—	4	2	no	—
32	C₂×D₆₈	—	4	4	no	—
33	C₂×Dic₃₄	—	4	1	no	—
34	C₂×C₈×C₁₇	D₈	1	4	yes	—
35	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
36	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₁₇	—	4	2	no	—
38	Dic₆₈	—	4	4	no	—
39	D₁₃₆	—	4	4	no	—
40	D₁₃₆	—	4	4	no	—
41	D₁₃₆	—	4	4	no	—
42	D₁₃₆	—	8	8	no	—
43	C₁₆×C₁₇	D₈	1	8	yes	complemented
44	#12519	—	17	1	no	maximal, Sylow
45	C₂²×D₁₇	—	8	1	no	—
46	C₂²×D₁₇	—	8	2	no	—
47	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
49	C₂×Dic₁₇	—	8	1	no	—
50	D₆₈	—	8	2	no	—
51	D₆₈	—	8	2	no	—
52	D₆₈	—	8	2	no	—
53	D₆₈	—	8	4	no	—
54	D₆₈	—	8	8	no	—
55	Dic₃₄	—	8	2	no	—
56	C₈×C₁₇	—	2	4	no	—
57	C₈×C₁₇	C₂×D₈	1	4	yes	—
58	#2248	—	17	2	no	—
59	#2246	—	17	2	no	—
60	#1712	—	17	1	no	—
61	#1709	—	17	1	no	—
62	#1311	D₁₇	1	1	yes	characteristic, complemented
63	C₂²×C₁₇	#493	1	1	yes	characteristic
64	C₂×D₁₇	—	16	1	no	—
65	C₂×D₁₇	—	16	2	no	—
66	C₂×D₁₇	—	16	2	no	—
67	C₂×D₁₇	—	16	4	no	—
68	C₄×C₁₇	#493	1	2	yes	—
69	C₄×C₁₇	#509	1	4	yes	—
70	Dic₁₇	—	16	1	no	—
71	C₂×SD₃₂	—	34	4	no	—
72	C₂×D₁₆	—	34	4	no	—
73	#494	—	17	1	no	—
74	#493	—	17	1	no	—
75	#493	—	17	2	no	—
76	#493	—	34	2	no	—
77	#486	—	34	2	no	—
78	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
79	C₈²	C₂×D₁₇	1	1	yes	characteristic
80	C₂×C₁₇	#1712	1	1	yes	characteristic
81	C₂×C₁₇	#2250	1	2	yes	—
82	D₁₇	—	32	1	no	—
83	D₁₇	—	32	2	no	—
84	C₂×Q₁₆	—	34	2	no	—
85	C₂×D₈	—	34	2	no	—
86	C₂×D₈	—	34	2	no	—
87	C₂×D₈	—	34	2	no	—
88	C₂×D₈	—	68	4	no	—
89	C₄⋊Q₈	—	34	1	no	—
90	#127	—	34	1	no	—
91	#127	—	34	2	no	—
92	SD₃₂	—	68	8	no	—
93	D₁₆	—	68	8	no	—
94	C₂×C₁₆	D₆₈	1	4	yes	—
95	#102	—	68	4	no	—
96	C₄×C₈	D₆₈	1	2	yes	—
97	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
98	C₁₇	#12519	1	1	yes	characteristic, complemented, Sylow
99	C₂×Q₈	—	68	1	no	—
100	C₂×D₄	—	68	1	no	—
101	C₂×D₄	—	68	2	no	—
102	C₂×D₄	—	68	2	no	—
103	C₂×D₄	—	68	4	no	—
104	Q₁₆	—	68	4	no	—
105	D₈	—	68	4	no	—
106	D₈	—	68	4	no	—
107	D₈	—	68	4	no	—
108	D₈	—	136	8	no	—
109	C₂×C₈	D₁₃₆	1	4	yes	—
110	C₂×C₈	C₂×D₆₈	1	2	yes	—
111	C₄⋊C₄	—	68	2	no	—
112	C₄²	C₂×D₆₈	1	1	yes	characteristic
113	C₁₆	D₁₃₆	1	8	yes	complemented
114	C₂³	—	136	1	no	—
115	C₂³	—	136	2	no	—
116	Q₈	—	136	2	no	—
117	D₄	—	136	8	no	—
118	D₄	—	136	4	no	—
119	D₄	—	136	2	no	—
120	D₄	—	136	2	no	—
121	D₄	—	136	2	no	—
122	C₂×C₄	—	136	1	no	—
123	C₂×C₄	#10587560	1	1	yes	characteristic
124	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	C₂×D₁₃₆	1	4	yes	—
127	C₂²	—	272	1	no	—
128	C₂²	—	272	2	no	—
129	C₂²	—	272	4	no	—
130	C₂²	—	272	2	no	—
131	C₂²	#11763186	1	1	yes	characteristic
132	C₄	—	272	1	no	—
133	C₄	#11763186	1	2	yes	—
134	C₄	#11763247	1	4	yes	—
135	C₂	—	544	2	no	—
136	C₂	—	544	1	no	—
137	C₂	#14528399	1	1	yes	characteristic
138	C₂	#14531050	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14531040

C₂

yes

maximal, complemented

#14531034

C₂

yes

maximal, complemented

#14528399

C₂

yes

maximal, characteristic, complemented

#14528390

C₂

yes

maximal, characteristic, complemented

C₁₇×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#10587471

—

C₂×D₂₇₂

—

#11763189

C₂²

yes

characteristic

#11763186

—

#11763186

C₂²

yes

characteristic

#11763186

C₂²

yes

—

#11763156

—

C₄×C₁₆×C₁₇

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

#10587560

—

#10587560

—

#10587555

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587492

—

#10587471

—

D₂₇₂

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

Dic₆₈

—

D₁₃₆

—

D₁₃₆

—

D₁₃₆

—

D₁₃₆

—

C₁₆×C₁₇

D₈

yes

complemented

#12519

—

maximal, Sylow

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

C₂×D₈

yes

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

#493

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

#493

yes

—

C₄×C₁₇

#509

yes

—

Dic₁₇

—

C₂×SD₃₂

—

C₂×D₁₆

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₇

yes

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

D₁₇

—

D₁₇

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

D₆₈

yes

—

#102

—

C₄×C₈

D₆₈

yes

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12519

yes

characteristic, complemented, Sylow

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

Q₁₆

—

105

D₈

—

106

D₈

—

107

D₈

—

108

D₈

—

136

—

109

C₂×C₈

D₁₃₆

yes

—

110

C₂×C₈

C₂×D₆₈

yes

—

111

C₄⋊C₄

—

112

C₄²

C₂×D₆₈

yes

characteristic

113

C₁₆

D₁₃₆

yes

complemented

114

C₂³

—

136

—

115

C₂³

—

136

—

116

Q₈

—

136

—

117

D₄

—

136

—

118

D₄

—

136

—

119

D₄

—

136

—

120

D₄

—

136

—

121

D₄

—

136

—

122

C₂×C₄

—

136

—

123

C₂×C₄

#10587560

yes

characteristic

124

C₂×C₄

C₂×D₁₃₆

yes

—

125

C₈

—

126

C₈

C₂×D₁₃₆

yes

—

127

C₂²

—

272

—

128

C₂²

—

272

—

129

C₂²

—

272

—

130

C₂²

—

272

—

131

C₂²

#11763186

yes

characteristic

132

C₄

—

272

—

133

C₄

#11763186

yes

—

134

C₄

#11763247

yes

—

135

C₂

—

544

—

136

C₂

—

544

—

137

C₂

#14528399

yes

characteristic

138

C₂

#14531050

yes

—

	subgroup	G/H
center	C₂²	#11763186
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1311	C₂
socle	C₂²×C₁₇	#493
hypercenter	#1311	D₁₇
hypocenter	C₁₇	#12519
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#11763186

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1311

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #14531040	C₂	semidirect
2. #14531034	C₂	semidirect
3. #14528399	C₂	semidirect
4. #14528390	C₂	semidirect
5. C₁₇×#1311	C₂	semidirect
43. C₁₆×C₁₇	D₈	semidirect
62. #1311	D₁₇	semidirect
98. C₁₇	#12519	semidirect
113. C₁₆	D₁₃₆	semidirect

H ≅ G/N

type

1. #14531040

C₂

semidirect

2. #14531034

C₂

semidirect

3. #14528399

C₂

semidirect

4. #14528390

C₂

semidirect

5. C₁₇×#1311

C₂

semidirect

43. C₁₆×C₁₇

D₈

semidirect

62. #1311

D₁₇

semidirect

98. C₁₇

#12519

semidirect

113. C₁₆

D₁₃₆

semidirect

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

subgroup	index
#14531040	2
#14531034	2
#14528399	2
#14528390	2
C₁₇×#1311	2
#12519	17

subgroup

index

C₁₇×#1311

Subgroups — #16949052

Conjugacy classes of subgroups

Complements

Maximal subgroups