Group #16948697 — subgroups

Subgroups — #16948697

Order 4352.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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	#14530692	C₂	1	1	yes	maximal, characteristic
2	#14528175	C₂	1	2	yes	maximal
3	#14526858	C₂	1	1	yes	maximal, characteristic
4	#14526839	C₂	1	2	yes	maximal
5	C₁₇×#2252	C₂	1	1	yes	maximal, characteristic
6	#11763016	—	2	2	no	—
7	C₈×Dic₃₄	C₂²	1	1	yes	characteristic
8	C₄⋊#63214	C₂²	1	1	yes	characteristic
9	#11762627	—	2	1	no	—
10	#11762584	—	2	1	no	—
11	#11762562	C₂²	1	2	yes	—
12	#11762555	—	2	2	no	—
13	C₁₇×#500	C₂²	1	2	yes	—
14	C₁₇×#366	—	2	1	no	—
15	C₁₇×#366	—	2	1	no	—
16	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×C₄⋊Q₈	—	2	2	no	—
18	C₂×#63231	—	4	2	no	—
19	C₄×Dic₃₄	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
22	C₁₇×#107	—	2	2	no	—
23	C₁₇×#107	—	2	2	no	—
24	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	#10587506	—	4	2	no	—
26	#10587485	—	2	1	no	—
27	C₈×Dic₁₇	—	2	1	no	—
28	C₂×#63214	—	2	1	no	—
29	C₄⋊#3622	—	2	2	no	—
30	C₄⋊#3622	D₄	1	1	yes	characteristic
31	C₂×Q₈×C₁₇	—	4	2	no	—
32	C₂×Dic₃₄	D₈	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
35	C₄⋊C₄×C₁₇	—	4	2	no	—
36	C₄⋊C₄×C₁₇	—	4	2	no	—
37	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
38	#63231	—	8	4	no	—
39	C₄⋊Dic₁₇	D₈	1	1	yes	characteristic
40	#63225	—	2	1	no	—
41	C₄×Dic₁₇	—	2	1	no	—
42	C₂×#3622	—	2	1	no	—
43	C₂×#3622	—	4	2	no	—
44	C₁₆×C₁₇	—	2	1	no	—
45	C₁₆×C₁₇	Q₁₆	1	2	yes	complemented
46	#63214	—	4	1	no	—
47	#12460	—	17	1	no	maximal, Sylow
48	Q₈×C₁₇	—	8	2	no	—
49	Q₈×C₁₇	—	8	2	no	—
50	C₂×C₄×C₁₇	—	8	2	no	—
51	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
54	C₂×Dic₁₇	—	2	1	no	—
55	C₂×Dic₁₇	—	2	1	no	—
56	Dic₃₄	—	2	2	no	—
57	C₈×C₁₇	—	2	1	no	—
58	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₁₇	#137	1	1	yes	characteristic
60	#3622	—	4	1	no	—
61	#3622	—	8	2	no	—
62	#2252	D₁₇	1	1	yes	characteristic
63	#2183	—	17	1	no	—
64	#1674	—	17	2	no	—
65	#1372	—	17	1	no	—
66	#1364	—	17	2	no	—
67	C₂²×C₁₇	#466	1	1	yes	characteristic
68	C₄×C₁₇	—	2	1	no	—
69	C₄×C₁₇	—	16	2	no	—
70	C₄×C₁₇	#462	1	1	yes	characteristic
71	C₄×C₁₇	#473	1	1	yes	characteristic
72	C₄×C₁₇	#509	1	2	yes	—
73	Dic₁₇	—	4	1	no	—
74	Dic₁₇	—	4	1	no	—
75	#500	C₂×D₁₇	1	2	yes	—
76	#462	—	34	2	no	—
77	C₈×Q₈	—	17	1	no	—
78	#366	—	2	1	no	—
79	#366	—	2	1	no	—
80	C₄⋊C₁₆	—	17	1	no	—
81	C₄⋊C₁₆	—	34	1	no	—
82	#346	—	34	1	no	—
83	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
84	#335	—	17	2	no	—
85	#332	—	34	2	no	—
86	C₂×C₁₇	#1674	1	1	yes	characteristic
87	C₂×C₁₇	#2218	1	1	yes	characteristic
88	C₂×C₁₇	#2222	1	1	yes	characteristic
89	C₂×Q₁₆	—	68	2	no	—
90	C₄⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	17	1	no	—
92	C₂×C₁₆	—	34	1	no	—
93	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
94	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	C₄⋊C₈	—	17	1	no	—
98	C₄⋊C₈	—	34	1	no	—
99	C₄⋊C₈	—	34	2	no	—
100	#103	—	68	2	no	—
101	C₄×C₈	—	34	1	no	—
102	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
103	C₁₇	#12460	1	1	yes	characteristic, complemented, Sylow
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	17	1	no	—
106	Q₁₆	—	136	4	no	—
107	C₂×C₈	—	34	1	no	—
108	C₂×C₈	—	68	2	no	—
109	C₂×C₈	#63254	1	1	yes	characteristic
110	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	17	1	no	—
114	C₄⋊C₄	—	34	1	no	—
115	C₄²	—	34	1	no	—
116	C₄²	D₄×D₁₇	1	1	yes	characteristic
117	C₁₆	—	2	1	no	—
118	C₁₆	—	68	1	no	—
119	C₁₆	#63231	1	2	yes	complemented
120	Q₈	—	8	2	no	—
121	Q₈	—	8	2	no	—
122	Q₈	—	34	2	no	—
123	C₂×C₄	—	8	2	no	—
124	C₂×C₄	—	34	1	no	—
125	C₂×C₄	—	34	1	no	—
126	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
127	C₂×C₄	#10587598	1	1	yes	characteristic
128	C₂×C₄	#10587624	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	68	1	no	—
131	C₈	—	136	2	no	—
132	C₈	#10587628	1	1	yes	characteristic
133	C₈	C₂×#63231	1	1	yes	characteristic
134	C₂²	#11763042	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	16	2	no	—
137	C₄	—	68	1	no	—
138	C₄	—	68	1	no	—
139	C₄	#11763016	1	1	yes	characteristic
140	C₄	#11763081	1	1	yes	characteristic
141	C₄	#11763243	1	2	yes	—
142	C₂	#14528175	1	1	yes	characteristic
143	C₂	#14530884	1	1	yes	characteristic
144	C₂	#14530906	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530692

C₂

yes

maximal, characteristic

#14528175

C₂

yes

maximal

#14526858

C₂

yes

maximal, characteristic

#14526839

C₂

yes

maximal

C₁₇×#2252

C₂

yes

maximal, characteristic

#11763016

—

C₈×Dic₃₄

C₂²

yes

characteristic

C₄⋊#63214

C₂²

yes

—

—

C₂²

yes

—

#11762555

—

C₁₇×#500

C₂²

yes

—

C₁₇×#366

—

C₁₇×#366

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊Q₈

—

C₂×#63231

—

C₄×Dic₃₄

D₄

yes

characteristic

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₁₇×#107

—

C₁₇×#107

—

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587506

—

#10587485

—

C₈×Dic₁₇

—

C₂×#63214

—

C₄⋊#3622

—

C₄⋊#3622

D₄

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

D₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

#63231

—

C₄⋊Dic₁₇

D₈

yes

characteristic

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₁₆×C₁₇

—

C₁₆×C₁₇

Q₁₆

yes

—

—

Q₈×C₁₇

—

Q₈×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

—

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

C₂×Q₁₆

yes

C₈×C₁₇

yes

—

—

D₁₇

yes

—

—

—

—

C₂²×C₁₇

#466

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

Dic₁₇

—

#500

C₂×D₁₇

yes

—

#462

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

#335

—

#332

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₇

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

100

#103

—

101

C₄×C₈

—

102

C₄×C₈

C₂²×D₁₇

yes

characteristic

103

C₁₇

#12460

yes

characteristic, complemented, Sylow

104

C₂×Q₈

—

105

C₂×Q₈

—

106

Q₁₆

—

136

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

#63254

yes

characteristic

110

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

D₄×D₁₇

yes

characteristic

117

C₁₆

—

118

C₁₆

—

119

C₁₆

#63231

yes

complemented

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₈×D₁₇

yes

characteristic

127

C₂×C₄

#10587598

yes

characteristic

128

C₂×C₄

#10587624

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

136

—

132

C₈

#10587628

yes

characteristic

133

C₈

C₂×#63231

yes

characteristic

134

C₂²

#11763042

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#11763016

yes

characteristic

140

C₄

#11763081

yes

characteristic

141

C₄

#11763243

yes

—

142

C₂

#14528175

yes

characteristic

143

C₂

#14530884

yes

characteristic

144

C₂

#14530906

yes

characteristic

	subgroup	G/H
center	C₂²	#11763042
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#2252	C₂
socle, prime radical	C₂²×C₁₇	#466
hypercenter	#2252	D₁₇
hypocenter	C₁₇	#12460
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#11763042

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
45. C₁₆×C₁₇	Q₁₆	semidirect
103. C₁₇	#12460	semidirect
119. C₁₆	#63231	semidirect

H ≅ G/N

type

45. C₁₆×C₁₇

Q₁₆

semidirect

103. C₁₇

#12460

semidirect

119. C₁₆

#63231

semidirect

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

subgroup	index
#14530692	2
#14528175	2
#14526858	2
#14526839	2
C₁₇×#2252	2
#12460	17

subgroup

index

C₁₇×#2252

Subgroups — #16948697

Conjugacy classes of subgroups

Complements

Maximal subgroups