Group #16949048 — subgroups

Subgroups — #16949048

Order 4352.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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	#14531041	C₂	1	2	yes	maximal, complemented
2	#14531034	C₂	1	1	yes	maximal, characteristic
3	#14528391	C₂	1	2	yes	maximal, complemented
4	C₁₆×#3622	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10587499	—	2	4	no	—
7	C₂×D₂₇₂	—	2	2	no	—
8	#11763186	C₂²	1	2	yes	—
9	#11763157	—	2	4	no	—
10	C₄×#63214	C₂²	1	1	yes	characteristic
11	C₈×#3622	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×Q₃₂	—	2	2	no	—
13	C₁₇×#494	C₂²	1	2	yes	—
14	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₁₇	—	2	2	no	—
16	C₂×Q₁₆×C₁₇	—	2	2	no	—
17	C₁₇×C₄⋊Q₈	—	2	2	no	—
18	C₂×D₁₃₆	—	2	2	no	—
19	C₂×D₁₃₆	—	2	2	no	—
20	#10587560	—	2	2	no	—
21	C₁₇×Q₃₂	—	4	4	no	—
22	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
23	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	#10587499	—	4	8	no	—
25	C₂×#63214	D₄	1	2	yes	—
26	#10587480	—	4	8	no	—
27	C₄×#3622	D₄	1	2	yes	—
28	D₂₇₂	—	4	4	no	—
29	C₂×Q₈×C₁₇	—	4	2	no	—
30	C₂×D₆₈	—	4	2	no	—
31	C₂×D₆₈	—	4	4	no	—
32	Q₁₆×C₁₇	—	4	4	no	—
33	Q₁₆×C₁₇	—	4	4	no	—
34	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₇	—	4	4	no	—
37	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×#3622	D₈	1	4	yes	—
39	D₁₃₆	—	4	4	no	—
40	D₁₃₆	—	4	4	no	—
41	C₁₆×C₁₇	D₈	1	4	yes	—
42	#63214	D₈	1	4	yes	—
43	#12518	—	17	1	no	maximal, Sylow
44	C₂²×D₁₇	—	8	2	no	—
45	Q₈×C₁₇	—	8	2	no	—
46	Q₈×C₁₇	—	8	2	no	—
47	C₂×C₄×C₁₇	—	8	2	no	—
48	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
50	D₆₈	—	8	2	no	—
51	D₆₈	—	8	2	no	—
52	D₆₈	—	8	8	no	—
53	C₈×C₁₇	C₂×D₈	1	2	yes	—
54	C₈×C₁₇	C₂×D₈	1	2	yes	—
55	#3622	—	2	4	no	—
56	#2248	—	17	2	no	—
57	#2247	D₁₇	1	1	yes	characteristic, complemented
58	#2246	—	17	1	no	—
59	#1709	—	17	2	no	—
60	C₈×C₁₆	—	17	1	no	—
61	C₂²×C₁₇	#493	1	1	yes	characteristic
62	C₂×D₁₇	—	16	2	no	—
63	C₂×D₁₇	—	16	4	no	—
64	C₄×C₁₇	—	16	2	no	—
65	C₄×C₁₇	#493	1	1	yes	characteristic
66	C₄×C₁₇	#493	1	1	yes	characteristic
67	C₄×C₁₇	#508	1	4	yes	—
68	C₂×Q₃₂	—	2	2	no	—
69	C₂×SD₃₂	—	34	4	no	—
70	C₂×D₁₆	—	34	2	no	—
71	#494	C₂×D₁₇	1	2	yes	—
72	#493	—	17	2	no	—
73	#486	—	34	4	no	—
74	C₄×C₁₆	—	17	1	no	—
75	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
76	C₈²	—	17	1	no	—
77	C₂×C₁₇	#1712	1	1	yes	characteristic
78	C₂×C₁₇	#2249	1	2	yes	—
79	D₁₇	—	32	2	no	—
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×D₈	—	34	2	no	—
83	C₂×D₈	—	34	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	#127	—	34	2	no	—
86	Q₃₂	—	4	4	no	—
87	SD₃₂	—	68	8	no	—
88	D₁₆	—	68	4	no	—
89	C₂×C₁₆	—	17	2	no	—
90	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
91	#102	—	68	8	no	—
92	C₄×C₈	—	17	2	no	—
93	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
94	C₁₇	#12518	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	68	4	no	—
97	C₂×D₄	—	68	2	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	68	4	no	—
101	D₈	—	68	4	no	—
102	C₂×C₈	—	17	4	no	—
103	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
105	C₄⋊C₄	—	4	4	no	—
106	C₄²	D₄×D₁₇	1	1	yes	characteristic
107	C₁₆	—	17	4	no	—
108	C₁₆	#63228	1	4	yes	—
109	C₂³	—	136	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	136	8	no	—
113	D₄	—	136	2	no	—
114	D₄	—	136	2	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	D₈×D₁₇	1	2	yes	—
117	C₂×C₄	#10587626	1	1	yes	characteristic
118	C₈	—	34	4	no	—
119	C₈	D₈×D₁₇	1	2	yes	—
120	C₈	C₂×#63228	1	2	yes	—
121	C₂²	—	272	2	no	—
122	C₂²	—	272	4	no	—
123	C₂²	#11763184	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	#11763184	1	1	yes	characteristic
126	C₄	#11763185	1	1	yes	characteristic
127	C₄	#11763242	1	4	yes	—
128	C₂	—	544	2	no	—
129	C₂	#14528398	1	1	yes	characteristic
130	C₂	#14531046	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14531041

C₂

yes

maximal, complemented

#14531034

C₂

yes

maximal, characteristic

#14528391

C₂

yes

maximal, complemented

C₁₆×#3622

C₂

yes

maximal, characteristic, complemented

C₁₇×#2247

C₂

yes

maximal, characteristic, complemented

C₂×#10587499

—

C₂×D₂₇₂

—

#11763186

C₂²

yes

—

#11763157

—

C₄×#63214

C₂²

yes

characteristic

C₈×#3622

C₂²

yes

characteristic

C₂×C₁₇×Q₃₂

—

C₁₇×#494

C₂²

yes

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

#10587560

—

C₁₇×Q₃₂

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587499

—

C₂×#63214

D₄

yes

—

#10587480

—

C₄×#3622

D₄

yes

—

D₂₇₂

—

C₂×Q₈×C₁₇

—

C₂×D₆₈

—

C₂×D₆₈

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₂×#3622

D₈

yes

—

D₁₃₆

—

D₁₃₆

—

C₁₆×C₁₇

D₈

yes

—

#63214

D₈

yes

—

#12518

—

maximal, Sylow

C₂²×D₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

D₆₈

—

D₆₈

—

D₆₈

—

C₈×C₁₇

C₂×D₈

yes

—

C₈×C₁₇

C₂×D₈

yes

—

#3622

—

#2248

—

#2247

D₁₇

yes

characteristic, complemented

#2246

—

#1709

—

C₈×C₁₆

—

C₂²×C₁₇

#493

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂×D₁₇

yes

—

#493

—

#486

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₈²

—

C₂×C₁₇

#1712

yes

characteristic

C₂×C₁₇

#2249

yes

—

D₁₇

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₇

yes

—

#102

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12518

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

100

D₈

—

101

D₈

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₁₇

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

D₄×D₁₇

yes

characteristic

107

C₁₆

—

108

C₁₆

#63228

yes

—

109

C₂³

—

136

—

110

Q₈

—

111

Q₈

—

112

D₄

—

136

—

113

D₄

—

136

—

114

D₄

—

136

—

115

C₂×C₄

—

116

C₂×C₄

D₈×D₁₇

yes

—

117

C₂×C₄

#10587626

yes

characteristic

118

C₈

—

119

C₈

D₈×D₁₇

yes

—

120

C₈

C₂×#63228

yes

—

121

C₂²

—

272

—

122

C₂²

—

272

—

123

C₂²

#11763184

yes

characteristic

124

C₄

—

125

C₄

#11763184

yes

characteristic

126

C₄

#11763185

yes

characteristic

127

C₄

#11763242

yes

—

128

C₂

—

544

—

129

C₂

#14528398

yes

characteristic

130

C₂

#14531046

yes

—

	subgroup	G/H
center	C₂²	#11763184
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#2247	C₂
socle	C₂²×C₁₇	#493
prime radical	#14531034	C₂
hypercenter	#2247	D₁₇
hypocenter	C₁₇	#12518
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#11763184

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#2247

C₂

C₂²×C₁₇

C₂

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
1. #14531041	C₂	semidirect
3. #14528391	C₂	semidirect
4. C₁₆×#3622	C₂	semidirect
5. C₁₇×#2247	C₂	semidirect
57. #2247	D₁₇	semidirect
94. C₁₇	#12518	semidirect

H ≅ G/N

type

1. #14531041

C₂

semidirect

3. #14528391

C₂

semidirect

4. C₁₆×#3622

C₂

semidirect

5. C₁₇×#2247

C₂

semidirect

57. #2247

D₁₇

semidirect

94. C₁₇

#12518

semidirect

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

subgroup	index
#14531041	2
#14531034	2
#14528391	2
C₁₆×#3622	2
C₁₇×#2247	2
#12518	17

subgroup

index

#14531041

#14531034

#14528391

C₁₆×#3622

C₁₇×#2247

#12518

Subgroups — #16949048

Conjugacy classes of subgroups

Complements

Maximal subgroups