Group #16948705 — subgroups

Subgroups — #16948705

Order 4352.

188 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 71 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	#14528188	C₂	1	2	yes	maximal
3	#14526858	C₂	1	1	yes	maximal, characteristic
4	#14526841	C₂	1	2	yes	maximal
5	C₁₇×#2183	C₂	1	1	yes	maximal, characteristic
6	#11763214	C₂²	1	2	yes	—
7	#11763104	—	2	2	no	—
8	#11762635	—	2	1	no	—
9	#11762635	—	2	1	no	—
10	#11762562	C₂²	1	2	yes	—
11	#11762557	—	2	2	no	—
12	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×C₄⋊C₁₆	—	2	1	no	—
14	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₁₇×#346	—	2	1	no	—
16	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	D₄	1	1	yes	characteristic
18	#10587555	—	2	2	no	—
19	C₂×C₁₆×C₁₇	—	2	1	no	—
20	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
22	C₁₇×C₄⋊C₈	—	2	1	no	—
23	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	—	2	1	no	—
25	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	#10587506	—	4	2	no	—
27	#10587489	—	2	2	no	—
28	#10587489	—	2	2	no	—
29	#10587478	—	4	2	no	—
30	C₄⋊#3622	—	2	2	no	—
31	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
32	C₂×Dic₃₄	—	4	2	no	—
33	C₂×C₈×C₁₇	—	2	1	no	—
34	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
36	C₄⋊C₄×C₁₇	—	2	1	no	—
37	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
38	C₄²×C₁₇	—	2	1	no	—
39	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₁₇	—	4	2	no	—
41	C₄⋊Dic₁₇	—	4	2	no	—
42	C₂×#3622	—	4	2	no	—
43	C₁₆×C₁₇	—	2	1	no	—
44	C₁₆×C₁₇	—	4	1	no	—
45	C₁₆×C₁₇	Q₁₆	1	2	yes	—
46	#12461	—	17	1	no	maximal, Sylow
47	Q₈×C₁₇	—	2	2	no	—
48	C₂×C₄×C₁₇	—	2	1	no	—
49	C₂×C₄×C₁₇	—	2	1	no	—
50	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
53	C₂×Dic₁₇	—	8	2	no	—
54	Dic₃₄	—	16	2	no	—
55	C₈×C₁₇	—	2	1	no	—
56	C₈×C₁₇	—	4	1	no	—
57	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₁₇	#137	1	1	yes	characteristic
59	#3622	—	8	2	no	—
60	#2252	—	17	1	no	—
61	#2183	D₁₇	1	1	yes	characteristic
62	#1676	—	17	2	no	—
63	#1372	—	17	1	no	—
64	#1365	—	17	2	no	—
65	C₂²×C₁₇	#466	1	1	yes	characteristic
66	C₄×C₁₇	—	2	1	no	—
67	C₄×C₁₇	—	4	1	no	—
68	C₄×C₁₇	—	4	1	no	—
69	C₄×C₁₇	#462	1	1	yes	characteristic
70	C₄×C₁₇	#473	1	1	yes	characteristic
71	C₄×C₁₇	#510	1	2	yes	—
72	Dic₁₇	—	16	2	no	—
73	#500	—	17	2	no	—
74	#477	—	34	2	no	—
75	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
76	#366	—	34	1	no	—
77	#366	—	34	1	no	—
78	C₄⋊C₁₆	—	2	1	no	—
79	C₄⋊C₁₆	C₂×D₁₇	1	1	yes	characteristic
80	#346	—	2	1	no	—
81	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
82	#335	—	17	2	no	—
83	#333	—	34	2	no	—
84	C₂×C₁₇	#1674	1	1	yes	characteristic
85	C₂×C₁₇	#2219	1	1	yes	characteristic
86	C₂×C₁₇	#2221	1	1	yes	characteristic
87	C₄⋊Q₈	—	34	2	no	—
88	C₄×Q₈	D₆₈	1	1	yes	characteristic
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
91	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
92	#107	—	34	2	no	—
93	#107	—	34	2	no	—
94	#107	—	68	2	no	—
95	C₄⋊C₈	—	2	1	no	—
96	C₄⋊C₈	—	34	2	no	—
97	C₄⋊C₈	D₆₈	1	1	yes	characteristic
98	#103	—	68	2	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
101	C₁₇	#12461	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	68	2	no	—
103	C₂×Q₈	D₁₃₆	1	1	yes	characteristic
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	68	2	no	—
106	C₂×C₈	#63252	1	1	yes	characteristic
107	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	68	2	no	—
110	C₄⋊C₄	—	68	2	no	—
111	C₄⋊C₄	D₁₃₆	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂×D₆₈	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	1	no	—
116	C₁₆	#63231	1	2	yes	—
117	Q₈	—	2	2	no	—
118	Q₈	—	272	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	136	2	no	—
122	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
123	C₂×C₄	#10587590	1	1	yes	characteristic
124	C₂×C₄	#10587616	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	136	2	no	—
128	C₈	C₂×#63231	1	1	yes	characteristic
129	C₈	#10587637	1	1	yes	characteristic
130	C₂²	#11763045	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	272	2	no	—
135	C₄	#11763019	1	1	yes	characteristic
136	C₄	#11763084	1	1	yes	characteristic
137	C₄	#11763252	1	2	yes	—
138	C₂	#14528178	1	1	yes	characteristic
139	C₂	#14530892	1	1	yes	characteristic
140	C₂	#14530904	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531059

C₂

yes

maximal, characteristic

#14528188

C₂

yes

maximal

#14526858

C₂

yes

maximal, characteristic

#14526841

C₂

yes

maximal

C₁₇×#2183

C₂

yes

maximal, characteristic

#11763214

C₂²

yes

—

—

—

—

C₂²

yes

—

#11762557

—

C₈×Q₈×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊C₁₆

—

C₁₇×C₄⋊C₁₆

C₂²

yes

characteristic

C₁₇×#346

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₄×Q₈×C₁₇

D₄

yes

characteristic

#10587555

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

D₄

yes

characteristic

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂³

yes

—

—

—

—

—

C₂×Q₈×C₁₇

D₈

yes

characteristic

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₈

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

C₁₆×C₁₇

Q₁₆

yes

—

#12461

—

maximal, Sylow

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

#135

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×Q₁₆

yes

C₈×C₁₇

yes

—

—

D₁₇

yes

—

—

—

C₂²×C₁₇

#466

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

#500

—

#477

—

C₈×Q₈

C₂×D₁₇

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₁₇

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

#335

—

#333

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₄⋊Q₈

—

C₄×Q₈

D₆₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₇

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₆₈

yes

characteristic

#103

—

C₄×C₈

—

100

C₄×C₈

C₂²×D₁₇

yes

characteristic

101

C₁₇

#12461

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

D₁₃₆

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

#63252

yes

characteristic

107

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

D₁₃₆

yes

characteristic

112

C₄²

—

113

C₄²

C₂×D₆₈

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

#63231

yes

—

117

Q₈

—

118

Q₈

—

272

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

136

—

122

C₂×C₄

C₂×D₁₃₆

yes

characteristic

123

C₂×C₄

#10587590

yes

characteristic

124

C₂×C₄

#10587616

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

136

—

128

C₈

C₂×#63231

yes

characteristic

129

C₈

#10587637

yes

characteristic

130

C₂²

#11763045

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

272

—

135

C₄

#11763019

yes

characteristic

136

C₄

#11763084

yes

characteristic

137

C₄

#11763252

yes

—

138

C₂

#14528178

yes

characteristic

139

C₂

#14530892

yes

characteristic

140

C₂

#14530904

yes

characteristic

	subgroup	G/H
center	C₂²	#11763045
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#2183	C₂
socle, prime radical	C₂²×C₁₇	#466
hypercenter	#2183	D₁₇
hypocenter	C₁₇	#12461
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

subgroup

G/H

center

C₂²

#11763045

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#2183

C₂

C₂²×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

N	H ≅ G/N	type
101. C₁₇	#12461	semidirect

H ≅ G/N

type

101. C₁₇

#12461

semidirect

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

subgroup	index
#14531059	2
#14528188	2
#14526858	2
#14526841	2
C₁₇×#2183	2
#12461	17

subgroup

index

C₁₇×#2183

Subgroups — #16948705

Conjugacy classes of subgroups

Complements

Maximal subgroups