Group #16948983 — subgroups

Subgroups — #16948983

Order 4352.

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

normal subgroups only

This group has 262 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	#14531037	C₂	1	1	yes	maximal, characteristic
2	#14531013	C₂	1	2	yes	maximal
3	#14528382	C₂	1	2	yes	maximal
4	C₁₆×#3622	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#2252	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₃₆	—	2	2	no	—
7	#11763205	—	2	2	no	—
8	#11763189	C₂²	1	2	yes	—
9	#11763159	—	2	2	no	—
10	#11762612	—	2	4	no	—
11	C₄×#63214	C₂²	1	1	yes	characteristic
12	C₈×#3622	C₂²	1	1	yes	characteristic
13	C₁₇×#500	C₂²	1	2	yes	—
14	C₁₇×#366	—	2	2	no	—
15	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×C₄⋊Q₈	—	2	2	no	—
17	C₂×Dic₆₈	—	2	2	no	—
18	C₂×Dic₆₈	—	2	2	no	—
19	#10587555	—	2	2	no	—
20	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
21	C₁₇×#107	—	2	2	no	—
22	C₁₇×#107	—	2	2	no	—
23	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	#10587506	—	4	4	no	—
25	C₂×#63214	D₄	1	2	yes	—
26	#10587479	—	4	4	no	—
27	C₄×#3622	D₄	1	1	yes	characteristic
28	C₄×#3622	D₄	1	1	yes	characteristic
29	Dic₁₃₆	—	4	4	no	—
30	C₂×Q₈×C₁₇	—	4	2	no	—
31	C₂×Dic₃₄	—	4	2	no	—
32	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₁₇	—	4	2	no	—
35	C₄⋊C₄×C₁₇	—	4	2	no	—
36	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₁₇	—	4	2	no	—
38	C₄⋊Dic₁₇	—	4	2	no	—
39	C₂×#3622	D₈	1	2	yes	—
40	C₂×#3622	D₈	1	2	yes	—
41	Dic₆₈	—	4	4	no	—
42	Dic₆₈	—	4	4	no	—
43	C₁₆×C₁₇	SD₁₆	1	4	yes	—
44	#63214	—	2	2	no	—
45	#12504	—	17	1	no	maximal, Sylow
46	Q₈×C₁₇	—	8	4	no	—
47	C₂×C₄×C₁₇	—	8	2	no	—
48	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₁₇	—	8	2	no	—
52	Dic₃₄	—	8	2	no	—
53	Dic₃₄	—	8	2	no	—
54	C₈×C₁₇	C₂×SD₁₆	1	2	yes	—
55	C₈×C₁₇	#135	1	2	yes	—
56	#3622	—	2	2	no	—
57	#3622	Q₃₂	1	4	yes	complemented
58	#2252	D₁₇	1	1	yes	characteristic
59	#2247	—	17	1	no	—
60	#2241	—	17	2	no	—
61	#1707	—	17	2	no	—
62	C₈×C₁₆	—	17	1	no	—
63	C₂²×C₁₇	#493	1	1	yes	characteristic
64	C₄×C₁₇	—	16	2	no	—
65	C₄×C₁₇	#492	1	1	yes	characteristic
66	C₄×C₁₇	#495	1	1	yes	characteristic
67	C₄×C₁₇	C₂×Q₃₂	1	2	yes	—
68	C₄×C₁₇	#508	1	2	yes	—
69	Dic₁₇	—	16	2	no	—
70	C₂×Q₃₂	—	34	2	no	—
71	#500	C₂×D₁₇	1	2	yes	—
72	#498	—	34	2	no	—
73	#494	—	17	2	no	—
74	#487	—	34	2	no	—
75	#366	—	2	2	no	—
76	#358	—	34	4	no	—
77	C₄×C₁₆	—	17	1	no	—
78	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
79	C₈²	—	17	1	no	—
80	C₂×C₁₇	#1706	1	1	yes	characteristic
81	C₂×C₁₇	#2247	1	1	yes	characteristic
82	C₂×C₁₇	#2249	1	1	yes	characteristic
83	C₂×Q₁₆	—	34	2	no	—
84	C₂×Q₁₆	—	34	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	C₄⋊Q₈	—	34	2	no	—
87	Q₃₂	—	68	4	no	—
88	C₂×C₁₆	—	17	2	no	—
89	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
90	#107	—	2	2	no	—
91	#107	—	2	2	no	—
92	#106	—	68	4	no	—
93	#103	—	68	4	no	—
94	C₄×C₈	—	17	1	no	—
95	C₄×C₈	—	17	1	no	—
96	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
97	C₁₇	#12504	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	68	2	no	—
100	Q₁₆	—	68	4	no	—
101	Q₁₆	—	68	4	no	—
102	C₂×C₈	—	17	2	no	—
103	C₂×C₈	—	17	2	no	—
104	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	68	2	no	—
109	C₄⋊C₄	—	68	2	no	—
110	C₄²	D₄×D₁₇	1	1	yes	characteristic
111	C₁₆	—	34	2	no	—
112	C₁₆	#63230	1	4	yes	—
113	Q₈	—	8	4	no	—
114	Q₈	—	136	2	no	—
115	Q₈	—	136	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	136	2	no	—
118	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
119	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
120	C₂×C₄	#10587626	1	1	yes	characteristic
121	C₈	—	17	4	no	—
122	C₈	—	34	2	no	—
123	C₈	#10587605	1	2	yes	—
124	C₈	C₂×#63230	1	2	yes	—
125	C₂²	#11763184	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	272	2	no	—
128	C₄	#11763183	1	1	yes	characteristic
129	C₄	#11763194	1	1	yes	characteristic
130	C₄	Q₃₂×D₁₇	1	2	yes	—
131	C₄	#11763238	1	2	yes	—
132	C₂	#14528377	1	1	yes	characteristic
133	C₂	#14531035	1	1	yes	characteristic
134	C₂	#14531042	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531037

C₂

yes

maximal, characteristic

#14531013

C₂

yes

maximal

#14528382

C₂

yes

maximal

C₁₆×#3622

C₂

yes

maximal, characteristic

C₁₇×#2252

C₂

yes

maximal, characteristic

C₂×Dic₁₃₆

—

#11763205

—

#11763189

C₂²

yes

—

#11763159

—

#11762612

—

C₄×#63214

C₂²

yes

characteristic

C₈×#3622

C₂²

yes

characteristic

C₁₇×#500

C₂²

yes

—

C₁₇×#366

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊Q₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

#10587555

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₁₇×#107

—

C₁₇×#107

—

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587506

—

C₂×#63214

D₄

yes

—

#10587479

—

C₄×#3622

D₄

yes

characteristic

C₄×#3622

D₄

yes

characteristic

Dic₁₃₆

—

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

D₈

yes

—

C₂×#3622

D₈

yes

—

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

SD₁₆

yes

—

#63214

—

#12504

—

maximal, Sylow

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

C₂×SD₁₆

yes

—

C₈×C₁₇

#135

yes

—

#3622

—

#3622

Q₃₂

yes

complemented

#2252

D₁₇

yes

—

—

—

C₈×C₁₆

—

C₂²×C₁₇

#493

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×Q₃₂

yes

—

C₄×C₁₇

#508

yes

—

Dic₁₇

—

C₂×Q₃₂

—

#500

C₂×D₁₇

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₈²

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₇

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12504

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₁₇

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

D₄×D₁₇

yes

characteristic

111

C₁₆

—

112

C₁₆

#63230

yes

—

113

Q₈

—

114

Q₈

—

136

—

115

Q₈

—

136

—

116

C₂×C₄

—

117

C₂×C₄

—

136

—

118

C₂×C₄

D₈×D₁₇

yes

characteristic

119

C₂×C₄

D₈×D₁₇

yes

characteristic

120

C₂×C₄

#10587626

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

#10587605

yes

—

124

C₈

C₂×#63230

yes

—

125

C₂²

#11763184

yes

characteristic

126

C₄

—

127

C₄

—

272

—

128

C₄

#11763183

yes

characteristic

129

C₄

#11763194

yes

characteristic

130

C₄

Q₃₂×D₁₇

yes

—

131

C₄

#11763238

yes

—

132

C₂

#14528377

yes

characteristic

133

C₂

#14531035

yes

characteristic

134

C₂

#14531042

yes

characteristic

	subgroup	G/H
center	C₂²	#11763184
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#2252	C₂
socle, prime radical	C₂²×C₁₇	#493
hypercenter	#2252	D₁₇
hypocenter	C₁₇	#12504
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#11763184

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#2252

C₂

C₂²×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
57. #3622	Q₃₂	semidirect
97. C₁₇	#12504	semidirect

H ≅ G/N

type

57. #3622

Q₃₂

semidirect

97. C₁₇

#12504

semidirect

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

subgroup	index
#14531037	2
#14531013	2
#14528382	2
C₁₆×#3622	2
C₁₇×#2252	2
#12504	17

subgroup

index

#14531037

#14531013

#14528382

C₁₆×#3622

C₁₇×#2252

#12504

Subgroups — #16948983

Conjugacy classes of subgroups

Complements

Maximal subgroups