Group #16948683 — subgroups

Subgroups — #16948683

Order 4352.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 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	#14528178	C₂	1	2	yes	maximal
3	#14526856	C₂	1	1	yes	maximal, characteristic
4	#14526838	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
6	#11763214	C₂²	1	2	yes	—
7	#11763019	—	2	2	no	—
8	#11762635	—	2	1	no	—
9	#11762635	—	2	1	no	—
10	#11762562	C₂²	1	2	yes	—
11	#11762554	—	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	C₄×C₁₆×C₁₇	—	2	1	no	—
16	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂×#63231	—	4	2	no	—
19	#10587555	—	2	2	no	—
20	C₂×C₁₆×C₁₇	—	2	1	no	—
21	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
23	C₁₇×C₄⋊C₈	—	2	1	no	—
24	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	—	2	1	no	—
26	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	#10587489	—	2	2	no	—
28	#10587489	—	2	2	no	—
29	#10587481	—	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	#63231	—	8	4	no	—
41	C₄⋊Dic₁₇	—	4	2	no	—
42	C₄⋊Dic₁₇	—	4	2	no	—
43	C₂×#3622	—	4	2	no	—
44	C₁₆×C₁₇	—	2	1	no	—
45	C₁₆×C₁₇	—	4	1	no	—
46	C₁₆×C₁₇	Q₁₆	1	2	yes	complemented
47	#12457	—	17	1	no	maximal, Sylow
48	Q₈×C₁₇	—	2	1	no	—
49	Q₈×C₁₇	D₁₆	1	2	yes	—
50	C₂×C₄×C₁₇	—	2	1	no	—
51	C₂×C₄×C₁₇	—	2	1	no	—
52	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
55	C₂×Dic₁₇	—	8	2	no	—
56	Dic₃₄	—	8	2	no	—
57	Dic₃₄	—	8	2	no	—
58	C₈×C₁₇	—	2	1	no	—
59	C₈×C₁₇	—	4	1	no	—
60	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₈×C₁₇	#137	1	1	yes	characteristic
62	#3622	—	8	2	no	—
63	#2252	—	17	1	no	—
64	Q₈×C₁₆	D₁₇	1	1	yes	characteristic
65	#1674	—	17	2	no	—
66	#1371	—	17	1	no	—
67	#1364	—	17	2	no	—
68	C₂²×C₁₇	#466	1	1	yes	characteristic
69	C₄×C₁₇	—	2	2	no	—
70	C₄×C₁₇	—	2	1	no	—
71	C₄×C₁₇	—	4	1	no	—
72	C₄×C₁₇	#462	1	1	yes	characteristic
73	C₄×C₁₇	#473	1	1	yes	characteristic
74	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
75	C₄×C₁₇	#508	1	1	yes	characteristic
76	Dic₁₇	—	16	2	no	—
77	#500	—	17	2	no	—
78	#462	—	34	2	no	—
79	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
80	#366	—	34	1	no	—
81	#366	—	34	1	no	—
82	C₄⋊C₁₆	—	2	1	no	—
83	C₄⋊C₁₆	C₂×D₁₇	1	1	yes	characteristic
84	C₄×C₁₆	—	2	1	no	—
85	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
86	#335	—	17	2	no	—
87	#332	—	34	2	no	—
88	C₂×C₁₇	#1674	1	1	yes	characteristic
89	C₂×C₁₇	#2215	1	1	yes	characteristic
90	C₂×C₁₇	#2222	1	1	yes	characteristic
91	C₂×Q₁₆	—	68	2	no	—
92	C₄⋊Q₈	—	34	2	no	—
93	C₄×Q₈	D₆₈	1	1	yes	characteristic
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
96	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
97	#107	—	34	2	no	—
98	#107	—	34	2	no	—
99	C₄⋊C₈	—	2	1	no	—
100	C₄⋊C₈	—	34	2	no	—
101	C₄⋊C₈	D₆₈	1	1	yes	characteristic
102	#103	—	68	2	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
105	C₁₇	#12457	1	1	yes	characteristic, complemented, Sylow
106	C₂×Q₈	—	68	2	no	—
107	C₂×Q₈	D₁₃₆	1	1	yes	characteristic
108	Q₁₆	—	136	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	68	2	no	—
111	C₂×C₈	#63252	1	1	yes	characteristic
112	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	68	2	no	—
115	C₄⋊C₄	—	68	2	no	—
116	C₄⋊C₄	D₁₃₆	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₆₈	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#63231	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	136	2	no	—
124	Q₈	—	136	2	no	—
125	Q₈	D₂₇₂	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	136	2	no	—
129	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
130	C₂×C₄	#10587590	1	1	yes	characteristic
131	C₂×C₄	#10587616	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	136	2	no	—
135	C₈	C₂×#63231	1	1	yes	characteristic
136	C₈	#10587637	1	1	yes	characteristic
137	C₂²	#11763045	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	272	2	no	—
142	C₄	#11763019	1	1	yes	characteristic
143	C₄	#11763084	1	1	yes	characteristic
144	C₄	C₂×D₂₇₂	1	1	yes	characteristic
145	C₄	#11763241	1	1	yes	characteristic
146	C₂	#14528178	1	1	yes	characteristic
147	C₂	#14530870	1	1	yes	characteristic
148	C₂	#14530909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531059

C₂

yes

maximal, characteristic

#14528178

C₂

yes

maximal

#14526856

C₂

yes

maximal, characteristic

#14526838

C₂

yes

maximal

Q₈×C₁₆×C₁₇

C₂

yes

maximal, characteristic

#11763214

C₂²

yes

—

—

—

—

C₂²

yes

—

#11762554

—

C₈×Q₈×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊C₁₆

—

C₁₇×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₆×C₁₇

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₄×Q₈×C₁₇

D₄

yes

characteristic

C₂×#63231

—

#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

#63231

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

C₁₆×C₁₇

Q₁₆

yes

complemented

#12457

—

maximal, Sylow

Q₈×C₁₇

—

Q₈×C₁₇

D₁₆

yes

—

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

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×Q₁₆

yes

C₈×C₁₇

yes

—

—

Q₈×C₁₆

D₁₇

yes

—

—

—

C₂²×C₁₇

#466

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×D₁₆

yes

characteristic

C₄×C₁₇

#508

yes

characteristic

Dic₁₇

—

#500

—

#462

—

C₈×Q₈

C₂×D₁₇

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₁₇

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

#335

—

#332

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

D₆₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₇

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

D₆₈

yes

characteristic

102

#103

—

103

C₄×C₈

—

104

C₄×C₈

C₂²×D₁₇

yes

characteristic

105

C₁₇

#12457

yes

characteristic, complemented, Sylow

106

C₂×Q₈

—

107

C₂×Q₈

D₁₃₆

yes

characteristic

108

Q₁₆

—

136

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#63252

yes

characteristic

112

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

D₁₃₆

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₆₈

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#63231

yes

complemented

122

Q₈

—

123

Q₈

—

136

—

124

Q₈

—

136

—

125

Q₈

D₂₇₂

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

136

—

129

C₂×C₄

C₂×D₁₃₆

yes

characteristic

130

C₂×C₄

#10587590

yes

characteristic

131

C₂×C₄

#10587616

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

136

—

135

C₈

C₂×#63231

yes

characteristic

136

C₈

#10587637

yes

characteristic

137

C₂²

#11763045

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

272

—

142

C₄

#11763019

yes

characteristic

143

C₄

#11763084

yes

characteristic

144

C₄

C₂×D₂₇₂

yes

characteristic

145

C₄

#11763241

yes

characteristic

146

C₂

#14528178

yes

characteristic

147

C₂

#14530870

yes

characteristic

148

C₂

#14530909

yes

characteristic

	subgroup	G/H
center	C₂²	#11763045
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	Q₈×C₁₆×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	#466
hypercenter	Q₈×C₁₆	D₁₇
hypocenter	C₁₇	#12457
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#11763045

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

Q₈×C₁₆×C₁₇

C₂

socle, prime radical

C₂²×C₁₇

#466

hypercenter

Q₈×C₁₆

D₁₇

hypocenter

C₁₇

#12457

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₁₇	Q₁₆	semidirect
105. C₁₇	#12457	semidirect
121. C₁₆	#63231	semidirect

H ≅ G/N

type

46. C₁₆×C₁₇

Q₁₆

semidirect

105. C₁₇

#12457

semidirect

121. C₁₆

#63231

semidirect

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

subgroup	index
#14531059	2
#14528178	2
#14526856	2
#14526838	2
Q₈×C₁₆×C₁₇	2
#12457	17

subgroup

index

Q₈×C₁₆×C₁₇

Subgroups — #16948683

Conjugacy classes of subgroups

Complements

Maximal subgroups