Group #16949060 — subgroups

Subgroups — #16949060

Order 4352.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14531059	C₂	1	1	yes	maximal, characteristic
2	#14531057	C₂	1	2	yes	maximal
3	#14528401	C₂	1	2	yes	maximal
4	C₁₆×#3622	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#2252	C₂	1	1	yes	maximal, characteristic
6	#11763214	C₂²	1	2	yes	—
7	#11763212	—	2	2	no	—
8	#11763188	—	2	2	no	—
9	#11762635	—	2	2	no	—
10	#11762634	—	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×#63231	—	4	4	no	—
18	#10587555	—	2	2	no	—
19	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
20	C₁₇×#107	—	2	2	no	—
21	C₁₇×#107	—	2	2	no	—
22	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
23	#10587489	—	2	2	no	—
24	#10587489	—	2	2	no	—
25	C₂×#63214	D₄	1	2	yes	—
26	#10587478	—	4	4	no	—
27	C₄×#3622	D₄	1	1	yes	characteristic
28	C₄×#3622	D₄	1	1	yes	characteristic
29	C₂×Q₈×C₁₇	—	4	2	no	—
30	C₂×Dic₃₄	—	4	2	no	—
31	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₁₇	—	4	2	no	—
34	C₄⋊C₄×C₁₇	—	4	2	no	—
35	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
36	#63231	—	8	8	no	—
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	C₁₆×C₁₇	Q₁₆	1	4	yes	complemented
42	#63214	Q₁₆	1	4	yes	complemented
43	#12521	—	17	1	no	maximal, Sylow
44	Q₈×C₁₇	—	8	4	no	—
45	C₂×C₄×C₁₇	—	8	2	no	—
46	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
49	C₂×Dic₁₇	—	8	2	no	—
50	Dic₃₄	—	8	4	no	—
51	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
52	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
53	#3622	D₁₆	1	4	yes	—
54	#3622	Q₃₂	1	4	yes	—
55	#2252	—	17	2	no	—
56	#2252	—	17	1	no	—
57	#2252	D₁₇	1	1	yes	characteristic
58	#1713	—	17	2	no	—
59	C₈×C₁₆	—	17	1	no	—
60	C₂²×C₁₇	#493	1	1	yes	characteristic
61	C₄×C₁₇	—	16	2	no	—
62	C₄×C₁₇	#494	1	1	yes	characteristic
63	C₄×C₁₇	#494	1	1	yes	characteristic
64	C₄×C₁₇	C₂×D₁₆	1	2	yes	—
65	C₄×C₁₇	C₂×Q₃₂	1	2	yes	—
66	Dic₁₇	—	16	2	no	—
67	#500	—	17	2	no	—
68	#500	—	34	2	no	—
69	#500	C₂×D₁₇	1	2	yes	—
70	#494	—	34	2	no	—
71	#366	—	2	2	no	—
72	#366	—	34	4	no	—
73	#366	—	34	2	no	—
74	C₄×C₁₆	—	17	1	no	—
75	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
76	C₈²	—	17	1	no	—
77	C₂×C₁₇	#1713	1	1	yes	characteristic
78	C₂×C₁₇	#2246	1	1	yes	characteristic
79	C₂×C₁₇	#2247	1	1	yes	characteristic
80	C₂×Q₁₆	—	68	4	no	—
81	C₄⋊Q₈	—	2	2	no	—
82	C₄⋊Q₈	—	34	2	no	—
83	C₂×C₁₆	—	17	2	no	—
84	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
85	#107	—	2	2	no	—
86	#107	—	2	2	no	—
87	#107	—	34	2	no	—
88	#107	—	34	2	no	—
89	#107	—	68	4	no	—
90	C₄×C₈	—	17	1	no	—
91	C₄×C₈	—	17	1	no	—
92	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
93	C₁₇	#12521	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	68	2	no	—
96	Q₁₆	—	136	8	no	—
97	C₂×C₈	—	17	2	no	—
98	C₂×C₈	—	17	2	no	—
99	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	68	2	no	—
104	C₄⋊C₄	—	68	2	no	—
105	C₄²	D₄×D₁₇	1	1	yes	characteristic
106	C₁₆	—	17	4	no	—
107	C₁₆	#63231	1	4	yes	complemented
108	Q₈	—	8	4	no	—
109	Q₈	—	136	4	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	136	2	no	—
112	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
113	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
114	C₂×C₄	#10587626	1	1	yes	characteristic
115	C₈	—	17	4	no	—
116	C₈	—	17	4	no	—
117	C₈	Q₁₆×D₁₇	1	2	yes	—
118	C₈	C₂×#63231	1	2	yes	—
119	C₂²	#11763184	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	272	2	no	—
122	C₄	#11763187	1	1	yes	characteristic
123	C₄	#11763188	1	1	yes	characteristic
124	C₄	D₁₆×D₁₇	1	2	yes	—
125	C₄	Q₃₂×D₁₇	1	2	yes	—
126	C₂	#14528401	1	1	yes	characteristic
127	C₂	#14531032	1	1	yes	characteristic
128	C₂	#14531035	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531059

C₂

yes

maximal, characteristic

#14531057

C₂

yes

maximal

#14528401

C₂

yes

maximal

C₁₆×#3622

C₂

yes

maximal, characteristic

C₁₇×#2252

C₂

yes

maximal, characteristic

#11763214

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#3622

C₂²

yes

characteristic

C₁₇×#500

C₂²

yes

—

C₁₇×#366

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊Q₈

—

C₂×#63231

—

#10587555

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₁₇×#107

—

C₁₇×#107

—

C₄×C₈×C₁₇

C₂³

yes

—

—

D₄

yes

—

#10587478

—

C₄×#3622

D₄

yes

characteristic

C₄×#3622

D₄

yes

characteristic

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

#63231

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

D₈

yes

—

C₂×#3622

D₈

yes

—

C₁₆×C₁₇

Q₁₆

yes

complemented

#63214

Q₁₆

yes

complemented

#12521

—

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

—

C₈×C₁₇

C₂×Q₁₆

yes

—

C₈×C₁₇

C₂×Q₁₆

yes

—

#3622

D₁₆

yes

—

#3622

Q₃₂

yes

—

#2252

—

#2252

—

#2252

D₁₇

yes

characteristic

#1713

—

C₈×C₁₆

—

C₂²×C₁₇

#493

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×D₁₆

yes

—

C₄×C₁₇

C₂×Q₃₂

yes

—

Dic₁₇

—

#500

—

#500

—

#500

C₂×D₁₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₈²

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₇

yes

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12521

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

136

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₁₇

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

D₄×D₁₇

yes

characteristic

106

C₁₆

—

107

C₁₆

#63231

yes

complemented

108

Q₈

—

109

Q₈

—

136

—

110

C₂×C₄

—

111

C₂×C₄

—

136

—

112

C₂×C₄

D₈×D₁₇

yes

characteristic

113

C₂×C₄

D₈×D₁₇

yes

characteristic

114

C₂×C₄

#10587626

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

Q₁₆×D₁₇

yes

—

118

C₈

C₂×#63231

yes

—

119

C₂²

#11763184

yes

characteristic

120

C₄

—

121

C₄

—

272

—

122

C₄

#11763187

yes

characteristic

123

C₄

#11763188

yes

characteristic

124

C₄

D₁₆×D₁₇

yes

—

125

C₄

Q₃₂×D₁₇

yes

—

126

C₂

#14528401

yes

characteristic

127

C₂

#14531032

yes

characteristic

128

C₂

#14531035

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	#12521
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
41. C₁₆×C₁₇	Q₁₆	semidirect
42. #63214	Q₁₆	semidirect
93. C₁₇	#12521	semidirect
107. C₁₆	#63231	semidirect

H ≅ G/N

type

41. C₁₆×C₁₇

Q₁₆

semidirect

42. #63214

Q₁₆

semidirect

93. C₁₇

#12521

semidirect

107. C₁₆

#63231

semidirect

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

subgroup	index
#14531059	2
#14531057	2
#14528401	2
C₁₆×#3622	2
C₁₇×#2252	2
#12521	17

subgroup

index

#14531059

#14531057

#14528401

C₁₆×#3622

C₁₇×#2252

#12521

Subgroups — #16949060

Conjugacy classes of subgroups

Complements

Maximal subgroups