Group #16939134 — subgroups

Subgroups — #16939134

Order 4352.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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	#14528925	C₂	1	1	yes	maximal, characteristic
2	#14527456	C₂	1	2	yes	maximal
3	#14526642	C₂	1	2	yes	maximal
4	#14526629	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1960	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	2	yes	—
7	#11762833	C₂²	1	2	yes	—
8	#11762574	—	2	2	no	—
9	#11762555	—	2	4	no	—
10	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₇×#103	—	2	2	no	—
12	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
13	C₁₇×#382	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₁₇	—	2	1	no	—
15	C₂²×C₈×C₁₇	D₄	1	2	yes	—
16	C₁₇×C₄⋊Q₈	—	2	2	no	—
17	C₁₇×C₄⋊Q₈	—	2	1	no	—
18	C₁₇×C₄⋊Q₈	D₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
21	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×#63223	—	2	2	no	—
23	C₁₇×#103	—	4	4	no	—
24	C₁₇×#97	—	2	2	no	—
25	C₁₇×#95	—	2	2	no	—
26	C₂²⋊#3622	—	2	2	no	—
27	C₄⋊#3622	—	2	2	no	—
28	C₄⋊#3622	—	2	2	no	—
29	C₄⋊#3622	—	2	2	no	—
30	C₄×#3622	—	2	2	no	—
31	C₂×Q₈×C₁₇	—	2	2	no	—
32	C₂×Q₈×C₁₇	—	4	1	no	—
33	C₂×Q₈×C₁₇	—	4	2	no	—
34	C₂×Q₈×C₁₇	—	4	2	no	—
35	C₂²×C₄×C₁₇	—	4	1	no	—
36	C₂²×C₄×C₁₇	—	4	1	no	—
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	#41	1	2	yes	—
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₂×C₈×C₁₇	—	2	2	no	—
41	C₂×C₈×C₁₇	—	2	2	no	—
42	C₄⋊C₄×C₁₇	—	2	4	no	—
43	C₄⋊C₄×C₁₇	—	4	2	no	—
44	C₄⋊C₄×C₁₇	—	4	2	no	—
45	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₁₇	#41	1	1	yes	characteristic
47	C₄²×C₁₇	#41	1	2	yes	—
48	#63223	—	4	2	no	—
49	#63223	—	4	2	no	—
50	C₂×#3622	—	4	2	no	—
51	C₂×#3622	—	4	2	no	—
52	#10635	—	17	1	no	maximal, Sylow
53	C₂³×C₁₇	#124	1	1	yes	characteristic
54	Q₈×C₁₇	—	4	4	no	—
55	Q₈×C₁₇	—	8	2	no	—
56	C₂×C₄×C₁₇	—	4	2	no	—
57	C₂×C₄×C₁₇	—	4	2	no	—
58	C₂×C₄×C₁₇	—	8	1	no	—
59	C₂×C₄×C₁₇	—	8	1	no	—
60	C₂×C₄×C₁₇	—	8	2	no	—
61	C₂×C₄×C₁₇	—	8	1	no	—
62	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₁₇	#123	1	2	yes	—
64	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#126	1	2	yes	—
66	C₂×C₄×C₁₇	#135	1	2	yes	—
67	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
68	C₈×C₁₇	—	4	2	no	—
69	#3622	—	8	2	no	—
70	#3622	—	8	2	no	—
71	#1960	D₁₇	1	1	yes	characteristic
72	#1838	—	17	1	no	—
73	#1546	—	17	2	no	—
74	#1286	—	17	2	no	—
75	#1281	—	17	1	no	—
76	C₂²×C₁₇	—	2	2	no	—
77	C₂²×C₁₇	#397	1	1	yes	characteristic
78	C₂²×C₁₇	#488	1	1	yes	characteristic
79	C₂²×C₁₇	#490	1	1	yes	characteristic
80	C₄×C₁₇	—	2	2	no	—
81	C₄×C₁₇	—	2	2	no	—
82	C₄×C₁₇	—	8	2	no	—
83	C₄×C₁₇	—	16	1	no	—
84	C₄×C₁₇	#463	1	2	yes	—
85	C₄×C₁₇	#482	1	2	yes	—
86	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
87	#431	—	17	2	no	—
88	#423	—	17	2	no	—
89	C₂×#103	—	2	2	no	—
90	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
91	#382	C₂×D₁₇	1	1	yes	characteristic
92	#341	—	34	2	no	—
93	#332	—	34	4	no	—
94	C₂×C₁₇	—	2	1	no	—
95	C₂×C₁₇	—	2	1	no	—
96	C₂×C₁₇	#2073	1	1	yes	characteristic
97	C₂×C₁₇	#2082	1	2	yes	—
98	C₂²×Q₈	—	2	1	no	—
99	C₂×M₁₆	—	34	2	no	—
100	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
101	C₄⋊Q₈	—	2	1	no	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	D₆₈	1	2	yes	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
107	C₄⋊C₈	—	34	2	no	—
108	C₄⋊C₈	—	34	2	no	—
109	C₄⋊C₈	—	34	2	no	—
110	#103	—	4	4	no	—
111	C₂²⋊C₈	—	34	2	no	—
112	#97	—	2	2	no	—
113	C₄×C₈	—	34	2	no	—
114	#95	—	2	2	no	—
115	C₁₇	#10635	1	1	yes	characteristic, complemented, Sylow
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	4	1	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	C₂²×C₄	#63254	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
124	M₁₆	—	68	2	no	—
125	M₁₆	—	68	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	68	2	no	—
130	C₂×C₈	—	68	2	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄²	C₂×D₆₈	1	1	yes	characteristic
135	C₄²	#63252	1	1	yes	characteristic
136	C₄²	#63254	1	2	yes	—
137	C₂³	#10587622	1	1	yes	characteristic
138	Q₈	—	4	4	no	—
139	Q₈	—	8	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	#10587565	1	2	yes	—
147	C₂×C₄	#10587572	1	2	yes	—
148	C₂×C₄	#10587616	1	1	yes	characteristic
149	C₂×C₄	#10587618	1	2	yes	—
150	C₂×C₄	#10587621	1	1	yes	characteristic
151	C₂×C₄	#10587636	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	136	2	no	—
154	C₈	—	136	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#11762734	1	1	yes	characteristic
157	C₂²	#11763163	1	1	yes	characteristic
158	C₂²	#11763173	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	1	no	—
163	C₄	#11763024	1	2	yes	—
164	C₄	#11763130	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#14530197	1	1	yes	characteristic
168	C₂	#14530248	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528925

C₂

yes

maximal, characteristic

#14527456

C₂

yes

maximal

#14526642

C₂

yes

maximal

#14526629

C₂

yes

maximal, characteristic

C₁₇×#1960

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

—

#11762833

C₂²

yes

—

#11762574

—

#11762555

—

C₂×C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₁₇×#103

—

C₂×C₁₇×#97

C₂²

yes

characteristic

C₁₇×#382

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

D₄

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₁₇×#103

—

C₁₇×#97

—

C₁₇×#95

—

C₂²⋊#3622

—

—

—

—

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

—

—

—

—

—

—

C₂³×C₁₇

yes

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#123

yes

characteristic

C₂×C₄×C₁₇

#126

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₇

—

#3622

—

#3622

—

#1960

D₁₇

yes

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#463

yes

—

C₄×C₁₇

#482

yes

—

C₂×C₄⋊Q₈

C₂×D₁₇

yes

—

—

—

C₂×#97

C₂×D₁₇

yes

characteristic

#382

C₂×D₁₇

yes

characteristic

#341

—

#332

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#2073

yes

characteristic

C₂×C₁₇

#2082

yes

—

C₂²×Q₈

—

C₂×M₁₆

—

100

C₂²×C₈

C₂²⋊D₁₇

yes

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

D₆₈

yes

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₂²×D₁₇

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

#103

—

111

C₂²⋊C₈

—

112

#97

—

113

C₄×C₈

—

114

#95

—

115

C₁₇

#10635

yes

characteristic, complemented, Sylow

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

#63254

yes

—

123

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

124

M₁₆

—

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

C₂×D₆₈

yes

characteristic

135

C₄²

#63252

yes

characteristic

136

C₄²

#63254

yes

—

137

C₂³

#10587622

yes

characteristic

138

Q₈

—

139

Q₈

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#10587565

yes

—

147

C₂×C₄

#10587572

yes

—

148

C₂×C₄

#10587616

yes

characteristic

149

C₂×C₄

#10587618

yes

—

150

C₂×C₄

#10587621

yes

characteristic

151

C₂×C₄

#10587636

yes

—

152

C₈

—

153

C₈

—

136

—

154

C₈

—

136

—

155

C₂²

—

156

C₂²

#11762734

yes

characteristic

157

C₂²

#11763163

yes

characteristic

158

C₂²

#11763173

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#11763024

yes

—

164

C₄

#11763130

yes

—

165

C₂

—

166

C₂

—

167

C₂

#14530197

yes

characteristic

168

C₂

#14530248

yes

—

	subgroup	G/H
center	C₂²	#11762734
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1960	C₂
socle	C₂²×C₁₇	#397
prime radical	C₂³×C₁₇	#124
hypercenter	#1960	D₁₇
hypocenter	C₁₇	#10635
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

subgroup

G/H

center

C₂²

#11762734

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1960

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

N	H ≅ G/N	type
115. C₁₇	#10635	semidirect

H ≅ G/N

type

115. C₁₇

#10635

semidirect

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

subgroup	index
#14528925	2
#14527456	2
#14526642	2
#14526629	2
C₁₇×#1960	2
#10635	17

subgroup

index

C₁₇×#1960

Subgroups — #16939134

Conjugacy classes of subgroups

Complements

Maximal subgroups