Group #14544962 — subgroups

Subgroups — #14544962

Order 2176.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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	#11763810	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#595	C₂	1	1	yes	maximal, characteristic
3	#11763682	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×D₁₇	—	2	1	no	—
6	C₂×C₄⋊#595	C₂²	1	2	yes	—
7	C₂×C₄×#595	—	2	1	no	—
8	C₂×#63242	C₂²	1	2	yes	—
9	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
10	#10587630	—	2	1	no	—
11	C₄⋊C₄×D₁₇	—	2	1	no	—
12	C₄⋊C₄×D₁₇	C₂²	1	2	yes	—
13	#10587574	—	2	1	no	—
14	#10587574	C₄	1	2	yes	complemented
15	C₄²×D₁₇	C₂²	1	1	yes	characteristic
16	#10587555	C₄	1	1	yes	characteristic, complemented
17	(C₂×#595)⋊C₄	—	2	1	no	—
18	#10587544	—	2	1	no	—
19	C₄⋊#3624	—	2	1	no	—
20	C₂²×#595	—	2	2	no	—
21	C₂×Q₈×C₁₇	—	2	1	no	—
22	Q₈×D₁₇	—	4	4	no	—
23	C₂×C₄×D₁₇	—	2	2	no	—
24	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₃₄	—	2	1	no	—
28	C₂×Dic₃₄	—	2	2	no	—
29	C₂×#3624	—	2	2	no	—
30	C₄⋊#595	—	2	2	no	—
31	C₄⋊#595	Q₈	1	4	yes	complemented
32	C₄×#595	—	4	2	no	—
33	#63242	—	2	4	no	—
34	C₄⋊C₄×C₁₇	—	2	1	no	—
35	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
36	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₄⋊Dic₁₇	—	2	1	no	—
38	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
39	#63225	—	2	2	no	—
40	#63225	—	2	2	no	—
41	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	D₄	1	1	yes	characteristic
43	C₄×Dic₁₇	D₄	1	1	yes	characteristic
44	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×#595	—	2	4	no	—
46	C₂×#595	—	4	2	no	—
47	C₂×#595	—	4	2	no	—
48	Q₈×C₁₇	—	4	2	no	—
49	C₂×C₄×C₁₇	—	2	2	no	—
50	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₁₇	—	2	2	no	—
54	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
57	C₄×D₁₇	—	2	2	no	—
58	C₄×D₁₇	—	4	4	no	—
59	C₄×D₁₇	SD₁₆	1	2	yes	—
60	C₄×D₁₇	Q₁₆	1	2	yes	—
61	C₄×D₁₇	C₂×Q₈	1	2	yes	—
62	C₄×D₁₇	#41	1	2	yes	—
63	Dic₃₄	—	4	4	no	—
64	Dic₃₄	—	4	2	no	—
65	#3624	—	4	2	no	—
66	#1987	—	17	1	no	maximal, Sylow
67	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×D₁₇	C₂²⋊Q₈	1	2	yes	—
69	C₂×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×D₁₇	AGL(1,ℤ/8)	1	2	yes	—
72	#595	—	4	4	no	—
73	C₄×C₁₇	—	2	1	no	—
74	C₄×C₁₇	—	4	2	no	—
75	C₄×C₁₇	#103	1	2	yes	—
76	C₄×C₁₇	C₄×Q₈	1	2	yes	—
77	Dic₁₇	—	2	1	no	—
78	Dic₁₇	—	4	2	no	—
79	Dic₁₇	#103	1	2	yes	—
80	Dic₁₇	#124	1	1	yes	characteristic
81	Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄⋊Q₈	—	17	1	no	—
83	C₂×C₄⋊C₈	—	17	1	no	—
84	C₄×C₄⋊C₄	—	17	1	no	—
85	#340	—	17	4	no	—
86	C₂×C₁₇	#391	1	1	yes	characteristic
87	C₂×C₁₇	C₂×#103	1	1	yes	characteristic
88	C₂×C₁₇	#418	1	1	yes	characteristic
89	D₁₇	#476	1	2	yes	—
90	D₁₇	#477	1	2	yes	—
91	C₂²×Q₈	—	34	1	no	—
92	C₂²×C₈	—	17	2	no	—
93	C₄⋊Q₈	—	17	2	no	—
94	C₄⋊Q₈	—	17	1	no	—
95	C₄⋊Q₈	—	34	1	no	—
96	C₄⋊Q₈	—	34	1	no	—
97	C₄⋊Q₈	#595	1	1	yes	characteristic, complemented
98	C₂×C₄⋊C₄	—	17	2	no	—
99	C₂×C₄⋊C₄	—	17	2	no	—
100	C₂×C₄⋊C₄	—	34	1	no	—
101	C₂×C₄²	—	17	1	no	—
102	C₂×C₄²	—	34	1	no	—
103	C₄⋊C₈	—	34	1	no	—
104	C₄⋊C₈	—	34	1	no	—
105	#95	—	34	1	no	—
106	C₁₇	#1987	1	1	yes	characteristic, complemented, Sylow
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	34	2	no	—
109	C₂×Q₈	—	34	1	no	—
110	C₂×Q₈	—	68	4	no	—
111	C₂²×C₄	—	17	1	no	—
112	C₂²×C₄	—	17	1	no	—
113	C₂²×C₄	—	17	1	no	—
114	C₂²×C₄	—	34	2	no	—
115	C₂²×C₄	—	34	2	no	—
116	C₂×C₈	—	34	4	no	—
117	C₂×C₈	—	34	2	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	17	4	no	—
120	C₄⋊C₄	—	17	2	no	—
121	C₄⋊C₄	—	34	2	no	—
122	C₄⋊C₄	—	34	2	no	—
123	C₄⋊C₄	—	34	2	no	—
124	C₄⋊C₄	—	34	1	no	—
125	C₄⋊C₄	C₂×#595	1	2	yes	—
126	C₄²	—	17	1	no	—
127	C₄²	—	17	1	no	—
128	C₄²	—	17	1	no	—
129	C₄²	—	68	2	no	—
130	C₄²	C₂×#595	1	1	yes	characteristic
131	C₂³	—	17	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	68	2	no	—
134	Q₈	—	68	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	17	1	no	—
137	C₂×C₄	—	17	1	no	—
138	C₂×C₄	—	17	2	no	—
139	C₂×C₄	—	17	2	no	—
140	C₂×C₄	—	17	2	no	—
141	C₂×C₄	—	17	2	no	—
142	C₂×C₄	—	17	1	no	—
143	C₂×C₄	—	34	2	no	—
144	C₂×C₄	—	34	4	no	—
145	C₂×C₄	—	34	2	no	—
146	C₂×C₄	—	68	4	no	—
147	C₂×C₄	—	68	2	no	—
148	C₂×C₄	—	68	2	no	—
149	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
150	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
151	C₂×C₄	C₂²×#595	1	1	yes	characteristic
152	C₈	—	68	2	no	—
153	C₂²	—	17	2	no	—
154	C₂²	—	17	1	no	—
155	C₂²	—	17	1	no	—
156	C₂²	—	17	2	no	—
157	C₂²	C₂×C₂²⋊#595	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	—	17	1	no	—
161	C₄	—	17	1	no	—
162	C₄	—	17	2	no	—
163	C₄	—	34	1	no	—
164	C₄	—	68	4	no	—
165	C₄	—	68	2	no	—
166	C₄	#10587549	1	2	yes	—
167	C₄	Q₈×#595	1	2	yes	—
168	C₂	—	17	2	no	—
169	C₂	—	17	2	no	—
170	C₂	(C₂×#595)⋊Q₈	1	1	yes	characteristic
171	C₂	C₂×#10587549	1	1	yes	characteristic
172	C₂	#11763801	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763810

C₂

yes

maximal, characteristic

C₄×C₄⋊#595

C₂

yes

maximal, characteristic

#11763682

C₂

yes

maximal

C₄⋊Q₈×D₁₇

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₇

—

C₂×C₄⋊#595

C₂²

yes

—

C₂×C₄×#595

—

C₂×#63242

C₂²

yes

—

C₁₇×C₄⋊Q₈

C₄

yes

characteristic, complemented

#10587630

—

C₄⋊C₄×D₁₇

—

C₄⋊C₄×D₁₇

C₂²

yes

—

#10587574

—

#10587574

C₄

yes

complemented

C₄²×D₁₇

C₂²

yes

characteristic

#10587555

C₄

yes

characteristic, complemented

(C₂×#595)⋊C₄

—

#10587544

—

C₄⋊#3624

—

C₂²×#595

—

C₂×Q₈×C₁₇

—

Q₈×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×#3624

—

C₄⋊#595

—

C₄⋊#595

Q₈

yes

complemented

C₄×#595

—

#63242

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

#63225

—

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₂²×D₁₇

C₂×D₄

yes

—

—

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

SD₁₆

yes

—

C₄×D₁₇

Q₁₆

yes

—

C₄×D₁₇

C₂×Q₈

yes

—

C₄×D₁₇

#41

yes

—

Dic₃₄

—

Dic₃₄

—

#3624

—

#1987

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇

C₂²⋊Q₈

yes

—

C₂×D₁₇

C₂×SD₁₆

yes

characteristic

C₂×D₁₇

C₂×Q₁₆

yes

characteristic

C₂×D₁₇

AGL(1,ℤ/8)

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#103

yes

—

C₄×C₁₇

C₄×Q₈

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#103

yes

—

Dic₁₇

#124

yes

characteristic

Dic₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

yes

—

D₁₇

#477

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#595

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄²

—

102

C₂×C₄²

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#95

—

106

C₁₇

#1987

yes

characteristic, complemented, Sylow

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

C₂×#595

yes

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂×#595

yes

characteristic

131

C₂³

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

C₂²⋊#595

yes

characteristic

150

C₂×C₄

C₂²⋊#595

yes

characteristic

151

C₂×C₄

C₂²×#595

yes

characteristic

152

C₈

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂²⋊#595

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#10587549

yes

—

167

C₄

Q₈×#595

yes

—

168

C₂

—

169

C₂

—

170

C₂

(C₂×#595)⋊Q₈

yes

characteristic

171

C₂

C₂×#10587549

yes

characteristic

172

C₂

#11763801

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×C₄⋊Q₈	C₄
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₇	C₂×D₄
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	C₄⋊Q₈	#595
hypocenter	C₁₇	#1987
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×C₄⋊Q₈

C₄

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₇

C₂×D₄

cosocle

C₂×C₄×D₁₇

C₂³

C₄⋊Q₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
9. C₁₇×C₄⋊Q₈	C₄	semidirect
14. #10587574	C₄	semidirect
16. #10587555	C₄	semidirect
31. C₄⋊#595	Q₈	semidirect
97. C₄⋊Q₈	#595	semidirect
106. C₁₇	#1987	semidirect

H ≅ G/N

type

9. C₁₇×C₄⋊Q₈

C₄

semidirect

14. #10587574

C₄

semidirect

16. #10587555

C₄

semidirect

31. C₄⋊#595

Q₈

semidirect

97. C₄⋊Q₈

#595

semidirect

106. C₁₇

#1987

semidirect

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

subgroup	index
#11763810	2
C₄×C₄⋊#595	2
#11763682	2
C₄⋊Q₈×D₁₇	2
#1987	17

subgroup

index

#11763810

C₄×C₄⋊#595

#11763682

C₄⋊Q₈×D₁₇

#1987

Subgroups — #14544962

Conjugacy classes of subgroups

Complements

Maximal subgroups