Group #14530112 — subgroups

Subgroups — #14530112

Order 2176.

282 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×#10587555	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587478	C₂	1	2	yes	maximal
3	#11762571	C₂	1	2	yes	maximal
4	#11762564	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#389	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄	—	2	1	no	—
7	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
8	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₇	—	2	1	no	—
12	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
13	C₂²×#3622	C₂²	1	2	yes	—
14	#10587555	—	2	4	no	—
15	C₁₇×#95	—	2	1	no	—
16	#10587478	—	2	4	no	—
17	C₂²×C₄×C₁₇	—	2	1	no	—
18	C₂²×C₄×C₁₇	—	2	1	no	—
19	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₇	—	2	2	no	—
23	C₂×Dic₃₄	—	2	4	no	—
24	C₂×Dic₃₄	—	4	2	no	—
25	C₂×Dic₃₄	—	4	2	no	—
26	C₄⋊C₄×C₁₇	—	2	1	no	—
27	C₄⋊C₄×C₁₇	—	2	2	no	—
28	C₄⋊C₄×C₁₇	—	4	2	no	—
29	C₄⋊C₄×C₁₇	Q₈	1	2	yes	complemented
30	C₄²×C₁₇	—	2	2	no	—
31	C₄⋊Dic₁₇	—	2	4	no	—
32	C₄⋊Dic₁₇	—	2	2	no	—
33	C₄⋊Dic₁₇	D₄	1	4	yes	—
34	C₂×#3622	—	2	2	no	—
35	C₂×#3622	—	2	2	no	—
36	C₂×#3622	Q₈	1	4	yes	complemented
37	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	—	2	1	no	—
39	C₂×C₄×C₁₇	—	2	1	no	—
40	C₂×C₄×C₁₇	—	2	1	no	—
41	C₂×C₄×C₁₇	—	2	1	no	—
42	C₂×C₄×C₁₇	—	2	2	no	—
43	C₂×C₄×C₁₇	—	4	1	no	—
44	C₂×C₄×C₁₇	—	4	1	no	—
45	C₂×C₄×C₁₇	—	4	1	no	—
46	C₂×C₄×C₁₇	—	4	1	no	—
47	C₂×C₄×C₁₇	—	4	1	no	—
48	C₂×C₄×C₁₇	D₈	1	2	yes	—
49	C₂×C₄×C₁₇	Q₁₆	1	2	yes	—
50	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
54	C₂×Dic₁₇	—	2	4	no	—
55	C₂×Dic₁₇	—	4	2	no	—
56	C₂×Dic₁₇	—	4	2	no	—
57	Dic₃₄	—	4	8	no	—
58	#3622	—	2	4	no	—
59	#2057	—	17	1	no	maximal, Sylow
60	C₂²×C₁₇	#120	1	1	yes	characteristic
61	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
64	C₂²×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₁₇	#137	1	1	yes	characteristic
67	C₄×C₁₇	—	2	2	no	—
68	C₄×C₁₇	—	2	2	no	—
69	C₄×C₁₇	—	4	2	no	—
70	C₄×C₁₇	—	8	1	no	—
71	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
73	Dic₁₇	—	4	4	no	—
74	C₂×C₄⋊Q₈	—	17	1	no	—
75	C₂×#107	—	17	2	no	—
76	#389	D₁₇	1	1	yes	characteristic
77	#340	—	17	2	no	—
78	#336	—	17	1	no	—
79	C₂×C₁₇	#395	1	1	yes	characteristic
80	C₂×C₁₇	#447	1	1	yes	characteristic
81	C₂×C₁₇	#451	1	1	yes	characteristic
82	C₂×C₁₇	#474	1	1	yes	characteristic
83	C₂×C₁₇	#477	1	1	yes	characteristic
84	C₂×C₁₇	#500	1	1	yes	characteristic
85	C₂×C₁₇	#501	1	1	yes	characteristic
86	C₂²×Q₈	—	34	1	no	—
87	C₂²×C₈	—	17	2	no	—
88	C₄⋊Q₈	—	34	4	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	17	2	no	—
91	C₂×C₄⋊C₄	—	34	1	no	—
92	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
93	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
94	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
95	#107	—	34	4	no	—
96	#95	—	2	1	no	—
97	C₁₇	#2057	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₈	—	34	4	no	—
99	C₂×Q₈	—	68	2	no	—
100	C₂×Q₈	—	68	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	34	2	no	—
104	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
107	C₂×C₈	—	17	4	no	—
108	C₂×C₈	—	34	2	no	—
109	C₂×C₈	—	34	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	17	4	no	—
114	C₄⋊C₄	—	34	4	no	—
115	C₄⋊C₄	—	34	2	no	—
116	C₄⋊C₄	Dic₃₄	1	2	yes	complemented
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
119	Q₈	—	68	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	34	4	no	—
131	C₂×C₄	—	68	2	no	—
132	C₂×C₄	—	68	2	no	—
133	C₂×C₄	#63228	1	2	yes	—
134	C₂×C₄	#63231	1	2	yes	—
135	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
136	C₂×C₄	#63252	1	1	yes	characteristic
137	C₂×C₄	D₄×D₁₇	1	2	yes	—
138	C₂×C₄	Q₈×D₁₇	1	2	yes	—
139	C₈	—	34	4	no	—
140	C₂²	#10587610	1	1	yes	characteristic
141	C₂²	C₂×#63228	1	1	yes	characteristic
142	C₂²	#10587623	1	1	yes	characteristic
143	C₂²	C₂×#63231	1	1	yes	characteristic
144	C₂²	#10587630	1	1	yes	characteristic
145	C₂²	#10587635	1	1	yes	characteristic
146	C₂²	#10587637	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	8	1	no	—
151	C₄	—	68	4	no	—
152	C₄	#10587574	1	2	yes	—
153	C₄	#10587582	1	2	yes	—
154	C₂	#11762729	1	1	yes	characteristic
155	C₂	#11762942	1	1	yes	characteristic
156	C₂	#11762964	1	1	yes	characteristic
157	C₂	#11763089	1	1	yes	characteristic
158	C₂	#11763104	1	1	yes	characteristic
159	C₂	#11763212	1	1	yes	characteristic
160	C₂	#11763219	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587555

C₂

yes

maximal, characteristic

C₂×#10587478

C₂

yes

maximal

#11762571

C₂

yes

maximal

#11762564

C₂

yes

maximal, characteristic

C₁₇×#389

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂²×#3622

C₂²

yes

—

#10587555

—

C₁₇×#95

—

#10587478

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

Q₈

yes

complemented

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

Q₈

yes

complemented

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

D₈

yes

—

C₂×C₄×C₁₇

Q₁₆

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇

C₂×D₈

yes

characteristic

C₂²×C₁₇

C₂×Q₁₆

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

C₄⋊Q₈

yes

—

Dic₁₇

—

C₂×C₄⋊Q₈

—

C₂×#107

—

#389

D₁₇

yes

characteristic

#340

—

#336

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₂×C₄²

C₂×D₁₇

yes

characteristic

#107

—

#95

—

C₁₇

#2057

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

105

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

106

C₂²×C₄

C₂²×D₁₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

Dic₃₄

yes

complemented

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#63228

yes

—

134

C₂×C₄

#63231

yes

—

135

C₂×C₄

C₂×Dic₃₄

yes

characteristic

136

C₂×C₄

#63252

yes

characteristic

137

C₂×C₄

D₄×D₁₇

yes

—

138

C₂×C₄

Q₈×D₁₇

yes

—

139

C₈

—

140

C₂²

#10587610

yes

characteristic

141

C₂²

C₂×#63228

yes

characteristic

142

C₂²

#10587623

yes

characteristic

143

C₂²

C₂×#63231

yes

characteristic

144

C₂²

#10587630

yes

characteristic

145

C₂²

#10587635

yes

characteristic

146

C₂²

#10587637

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#10587574

yes

—

153

C₄

#10587582

yes

—

154

C₂

#11762729

yes

characteristic

155

C₂

#11762942

yes

characteristic

156

C₂

#11762964

yes

characteristic

157

C₂

#11763089

yes

characteristic

158

C₂

#11763104

yes

characteristic

159

C₂

#11763212

yes

characteristic

160

C₂

#11763219

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#389	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#389	D₁₇
hypocenter	C₁₇	#2057
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#389

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
29. C₄⋊C₄×C₁₇	Q₈	semidirect
36. C₂×#3622	Q₈	semidirect
97. C₁₇	#2057	semidirect
116. C₄⋊C₄	Dic₃₄	semidirect

H ≅ G/N

type

29. C₄⋊C₄×C₁₇

Q₈

semidirect

36. C₂×#3622

Q₈

semidirect

97. C₁₇

#2057

semidirect

116. C₄⋊C₄

Dic₃₄

semidirect

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

subgroup	index
C₂×#10587555	2
C₂×#10587478	2
#11762571	2
#11762564	2
C₁₇×#389	2
#2057	17

subgroup

index

C₁₇×#389

Subgroups — #14530112

Conjugacy classes of subgroups

Complements

Maximal subgroups