Group #16948924 — subgroups

Subgroups — #16948924

Order 4352.

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

normal subgroups only

This group has 224 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	#14530834	C₂	1	2	yes	maximal
2	#14528402	C₂	1	1	yes	maximal, characteristic
3	#14526848	C₂	1	1	yes	maximal, characteristic
4	#14526832	C₂	1	2	yes	maximal
5	C₁₇×#1591	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587500	—	2	2	no	—
7	#11763214	—	2	1	no	—
8	#11763214	—	2	1	no	—
9	#11763189	C₂²	1	2	yes	—
10	C₄⋊#63214	C₂²	1	2	yes	—
11	#11762612	—	2	2	no	—
12	C₁₇×#446	—	2	1	no	—
13	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#335	C₂²	1	1	yes	characteristic
15	C₁₇×Q₈⋊C₈	—	2	1	no	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₇	D₄	1	1	yes	characteristic
18	C₄×Q₈×C₁₇	—	2	1	no	—
19	C₂×Dic₆₈	—	2	2	no	—
20	C₂×Dic₆₈	—	2	2	no	—
21	#10587555	—	2	2	no	—
22	C₁₇×#107	D₄	1	1	yes	characteristic
23	C₁₇×C₄⋊C₈	—	2	1	no	—
24	C₁₇×C₄⋊C₈	—	4	1	no	—
25	C₁₇×#103	—	2	1	no	—
26	C₁₇×#97	—	4	1	no	—
27	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	#10587500	—	4	4	no	—
31	#10587489	—	4	2	no	—
32	#10587489	—	4	1	no	—
33	#10587489	—	4	1	no	—
34	C₂×#63214	—	2	2	no	—
35	C₂×Q₈×C₁₇	—	2	1	no	—
36	C₂×Dic₃₄	—	4	2	no	—
37	Q₁₆×C₁₇	—	2	2	no	—
38	C₂×C₈×C₁₇	—	2	1	no	—
39	C₂×C₈×C₁₇	—	4	1	no	—
40	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
41	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₁₇	—	2	1	no	—
45	C₄⋊C₄×C₁₇	—	4	1	no	—
46	C₄²×C₁₇	—	4	1	no	—
47	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₁₇	—	4	2	no	—
49	C₄⋊Dic₁₇	—	4	2	no	—
50	Dic₆₈	—	4	2	no	—
51	Dic₆₈	—	4	2	no	—
52	Dic₆₈	—	8	2	no	—
53	#63214	—	4	2	no	—
54	#12494	—	17	1	no	maximal, Sylow
55	Q₈×C₁₇	—	4	1	no	—
56	Q₈×C₁₇	—	4	1	no	—
57	C₂×C₄×C₁₇	—	4	1	no	—
58	C₂×C₄×C₁₇	—	4	1	no	—
59	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×Dic₁₇	—	8	2	no	—
63	Dic₃₄	—	8	2	no	—
64	Dic₃₄	—	8	2	no	—
65	C₈×C₁₇	—	2	2	no	—
66	C₈×C₁₇	—	2	1	no	—
67	C₈×C₁₇	—	2	1	no	—
68	C₈×C₁₇	—	8	1	no	—
69	C₈×C₁₇	Q₃₂	1	2	yes	complemented
70	C₈×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
71	#2209	—	17	2	no	—
72	#1713	—	17	1	no	—
73	#1591	D₁₇	1	1	yes	characteristic
74	#1367	—	17	1	no	—
75	#1362	—	17	2	no	—
76	C₂²×C₁₇	#459	1	1	yes	characteristic
77	C₄×C₁₇	—	2	1	no	—
78	C₄×C₁₇	—	8	1	no	—
79	C₄×C₁₇	—	8	1	no	—
80	C₄×C₁₇	#469	1	1	yes	characteristic
81	C₄×C₁₇	#472	1	1	yes	characteristic
82	C₄×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
83	C₄×C₁₇	#510	1	1	yes	characteristic
84	Dic₁₇	—	16	2	no	—
85	C₂×Q₃₂	—	34	2	no	—
86	#500	—	34	1	no	—
87	#500	—	34	1	no	—
88	#494	—	17	2	no	—
89	#446	—	2	1	no	—
90	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
91	C₄⋊C₁₆	—	17	2	no	—
92	#358	—	34	2	no	—
93	#335	C₂×D₁₇	1	1	yes	characteristic
94	Q₈⋊C₈	—	2	1	no	—
95	C₈²	C₂×D₁₇	1	1	yes	characteristic
96	C₂×C₁₇	#1687	1	1	yes	characteristic
97	C₂×C₁₇	#2209	1	1	yes	characteristic
98	C₂×C₁₇	#2214	1	1	yes	characteristic
99	C₂×Q₁₆	—	34	2	no	—
100	C₂×Q₁₆	—	34	2	no	—
101	C₂×Q₁₆	D₆₈	1	1	yes	characteristic
102	C₄⋊Q₈	—	34	2	no	—
103	C₄×Q₈	—	2	1	no	—
104	Q₃₂	—	68	4	no	—
105	C₂×C₁₆	—	34	2	no	—
106	#107	—	68	2	no	—
107	#107	—	68	1	no	—
108	#107	—	68	1	no	—
109	#107	D₆₈	1	1	yes	characteristic
110	C₄⋊C₈	—	2	1	no	—
111	C₄⋊C₈	—	4	1	no	—
112	#103	—	2	1	no	—
113	#97	—	4	1	no	—
114	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
115	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
116	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
117	C₁₇	#12494	1	1	yes	characteristic, complemented, Sylow
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	—	68	2	no	—
120	Q₁₆	—	2	2	no	—
121	Q₁₆	—	68	2	no	—
122	Q₁₆	—	68	2	no	—
123	Q₁₆	—	136	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	#63228	1	1	yes	characteristic
127	C₂×C₈	#63228	1	1	yes	characteristic
128	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
129	C₂×C₈	#63252	1	1	yes	characteristic
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	68	2	no	—
133	C₄⋊C₄	—	68	2	no	—
134	C₄²	—	4	1	no	—
135	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
136	C₁₆	—	68	2	no	—
137	Q₈	—	4	1	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	136	2	no	—
140	Q₈	—	136	2	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	136	2	no	—
144	C₂×C₄	#10587616	1	1	yes	characteristic
145	C₂×C₄	C₂×#63228	1	1	yes	characteristic
146	C₂×C₄	#10587635	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	8	1	no	—
151	C₈	#10587500	1	2	yes	complemented
152	C₈	#10587594	1	2	yes	—
153	C₂²	#11763004	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	8	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	272	2	no	—
158	C₄	#11763062	1	1	yes	characteristic
159	C₄	#11763079	1	1	yes	characteristic
160	C₄	C₂×#10587500	1	1	yes	characteristic
161	C₄	#11763249	1	1	yes	characteristic
162	C₂	#14528263	1	1	yes	characteristic
163	C₂	#14530834	1	1	yes	characteristic
164	C₂	#14530865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530834

C₂

yes

maximal

#14528402

C₂

yes

maximal, characteristic

#14526848

C₂

yes

maximal, characteristic

#14526832

C₂

yes

maximal

C₁₇×#1591

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄⋊#63214

C₂²

yes

—

#11762612

—

C₁₇×#446

—

C₄×Q₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#335

C₂²

yes

characteristic

C₁₇×Q₈⋊C₈

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

D₄

yes

characteristic

C₄×Q₈×C₁₇

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

#10587555

—

C₁₇×#107

D₄

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×#103

—

C₁₇×#97

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

—

—

—

—

—

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

Dic₆₈

—

#63214

—

#12494

—

maximal, Sylow

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

Q₃₂

yes

complemented

C₈×C₁₇

AGL(1,ℤ/8)

yes

—

#2209

—

#1713

—

#1591

D₁₇

yes

characteristic

#1367

—

#1362

—

C₂²×C₁₇

#459

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×Q₃₂

yes

characteristic

C₄×C₁₇

#510

yes

characteristic

Dic₁₇

—

C₂×Q₃₂

—

—

—

—

—

C₄×Q₁₆

C₂×D₁₇

yes

characteristic

C₄⋊C₁₆

—

#358

—

#335

C₂×D₁₇

yes

characteristic

Q₈⋊C₈

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

C₂×Q₁₆

D₆₈

yes

characteristic

102

C₄⋊Q₈

—

103

C₄×Q₈

—

104

Q₃₂

—

105

C₂×C₁₆

—

106

#107

—

107

#107

—

108

#107

—

109

#107

D₆₈

yes

characteristic

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

#103

—

113

#97

—

114

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

115

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

116

C₄×C₈

C₂²×D₁₇

yes

characteristic

117

C₁₇

#12494

yes

characteristic, complemented, Sylow

118

C₂×Q₈

—

119

C₂×Q₈

—

120

Q₁₆

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

136

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

#63228

yes

characteristic

127

C₂×C₈

#63228

yes

characteristic

128

C₂×C₈

C₂×D₆₈

yes

characteristic

129

C₂×C₈

#63252

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

—

135

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

136

C₁₆

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

136

—

140

Q₈

—

136

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

136

—

144

C₂×C₄

#10587616

yes

characteristic

145

C₂×C₄

C₂×#63228

yes

characteristic

146

C₂×C₄

#10587635

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

#10587500

yes

complemented

152

C₈

#10587594

yes

—

153

C₂²

#11763004

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

272

—

158

C₄

#11763062

yes

characteristic

159

C₄

#11763079

yes

characteristic

160

C₄

C₂×#10587500

yes

characteristic

161

C₄

#11763249

yes

characteristic

162

C₂

#14528263

yes

characteristic

163

C₂

#14530834

yes

characteristic

164

C₂

#14530865

yes

characteristic

	subgroup	G/H
center	C₂²	#11763004
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1591	C₂
socle, prime radical	C₂²×C₁₇	#459
hypercenter	#1591	D₁₇
hypocenter	C₁₇	#12494
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1591

subgroup

G/H

center

C₂²

#11763004

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1591

C₂

C₂²×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1591

N	H ≅ G/N	type
69. C₈×C₁₇	Q₃₂	semidirect
117. C₁₇	#12494	semidirect
151. C₈	#10587500	semidirect

H ≅ G/N

type

69. C₈×C₁₇

Q₃₂

semidirect

117. C₁₇

#12494

semidirect

151. C₈

#10587500

semidirect

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

subgroup	index
#14530834	2
#14528402	2
#14526848	2
#14526832	2
C₁₇×#1591	2
#12494	17

subgroup

index

C₁₇×#1591

Subgroups — #16948924

Conjugacy classes of subgroups

Complements

Maximal subgroups