Group #14544570 — subgroups

Subgroups — #14544570

Order 2176.

292 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 130 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×C₄⋊#3624	C₂	1	2	yes	maximal
2	C₂×#10587542	C₂	1	2	yes	maximal
3	#11763675	C₂	1	2	yes	maximal
4	C₄²×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#3624	C₂²	1	4	yes	—
6	C₂×C₄²×C₁₇	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₇	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
13	#10587542	—	2	4	no	—
14	C₄⋊#3624	C₄	1	8	yes	complemented
15	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₁₇	C₂³	1	1	yes	characteristic
22	C₂×#3624	—	2	4	no	—
23	C₂×#3624	—	2	4	no	—
24	C₂×#3624	C₂×C₄	1	8	yes	—
25	C₄²×C₁₇	—	2	2	no	—
26	C₄×Dic₁₇	—	2	2	no	—
27	C₄×Dic₁₇	—	2	2	no	—
28	C₄×Dic₁₇	—	2	2	no	—
29	C₄×Dic₁₇	—	2	2	no	—
30	C₄×Dic₁₇	C₂×C₄	1	4	yes	—
31	C₄×Dic₁₇	D₄	1	2	yes	—
32	C₄×Dic₁₇	Q₈	1	2	yes	—
33	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇	—	2	1	no	—
35	C₂×C₄×C₁₇	—	2	1	no	—
36	C₂×C₄×C₁₇	—	2	1	no	—
37	C₂×C₄×C₁₇	—	2	1	no	—
38	C₂×C₄×C₁₇	C₄²	1	2	yes	—
39	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₇	M₁₆	1	2	yes	—
41	C₂×C₄×C₁₇	M₁₆	1	2	yes	—
42	C₂×C₄×C₁₇	M₁₆	1	2	yes	—
43	C₂×Dic₁₇	—	2	1	no	—
44	C₂×Dic₁₇	—	2	1	no	—
45	C₂×Dic₁₇	—	2	1	no	—
46	C₂×Dic₁₇	—	2	1	no	—
47	C₂×Dic₁₇	—	2	1	no	—
48	C₂×Dic₁₇	—	2	1	no	—
49	C₂×Dic₁₇	C₄²	1	2	yes	—
50	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
51	C₂×Dic₁₇	M₁₆	1	2	yes	—
52	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
53	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×Dic₁₇	#41	1	2	yes	—
56	#3624	—	2	8	no	—
57	#1767	—	17	1	no	maximal, Sylow
58	C₂²×C₁₇	C₂×C₄²	1	1	yes	characteristic
59	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₇	#117	1	1	yes	characteristic
61	C₂²×C₁₇	C₂×M₁₆	1	1	yes	characteristic
62	C₂²×C₁₇	C₂×M₁₆	1	1	yes	characteristic
63	C₂²×C₁₇	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₁₇	C₂×M₁₆	1	1	yes	characteristic
65	C₄×C₁₇	—	2	2	no	—
66	C₄×C₁₇	—	2	2	no	—
67	C₄×C₁₇	#97	1	4	yes	complemented
68	Dic₁₇	—	2	2	no	—
69	Dic₁₇	—	2	2	no	—
70	Dic₁₇	—	2	2	no	—
71	Dic₁₇	C₄×D₄	1	2	yes	—
72	Dic₁₇	C₄×Q₈	1	2	yes	—
73	C₂×C₄⋊C₈	—	17	2	no	—
74	C₂×#97	—	17	2	no	—
75	C₄³	—	17	1	no	—
76	#336	—	17	2	no	—
77	C₂×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₁₇	C₂×#97	1	1	yes	characteristic
79	C₂×C₁₇	C₄×M₁₆	1	1	yes	characteristic
80	C₂×C₁₇	#423	1	1	yes	characteristic
81	C₂×C₁₇	#423	1	1	yes	characteristic
82	C₂×C₁₇	#432	1	1	yes	characteristic
83	C₂×C₁₇	#432	1	1	yes	characteristic
84	C₂²×C₈	—	17	4	no	—
85	C₂×C₄²	—	17	1	no	—
86	C₂×C₄²	—	17	1	no	—
87	C₂×C₄²	—	17	1	no	—
88	C₂×C₄²	—	17	1	no	—
89	C₂×C₄²	—	17	1	no	—
90	C₂×C₄²	—	17	1	no	—
91	C₂×C₄²	#595	1	1	yes	characteristic
92	C₄⋊C₈	—	17	8	no	—
93	#97	—	34	4	no	—
94	C₁₇	#1767	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄	—	17	1	no	—
96	C₂²×C₄	—	17	1	no	—
97	C₂²×C₄	—	17	1	no	—
98	C₂²×C₄	—	17	1	no	—
99	C₂²×C₄	C₂×#595	1	1	yes	characteristic
100	C₂²×C₄	C₂×#595	1	1	yes	characteristic
101	C₂²×C₄	C₂×#595	1	1	yes	characteristic
102	C₂×C₈	—	17	8	no	—
103	C₂×C₈	—	34	4	no	—
104	C₂×C₈	—	34	4	no	—
105	C₄²	—	2	2	no	—
106	C₄²	—	17	4	no	—
107	C₄²	—	17	2	no	—
108	C₄²	—	17	2	no	—
109	C₄²	—	34	2	no	—
110	C₄²	—	34	2	no	—
111	C₄²	—	34	2	no	—
112	C₄²	—	34	2	no	—
113	C₂³	C₂²×#595	1	1	yes	characteristic
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	17	2	no	—
119	C₂×C₄	—	17	1	no	—
120	C₂×C₄	—	17	1	no	—
121	C₂×C₄	—	17	2	no	—
122	C₂×C₄	—	17	2	no	—
123	C₂×C₄	—	17	2	no	—
124	C₂×C₄	—	17	2	no	—
125	C₂×C₄	—	34	1	no	—
126	C₂×C₄	—	34	1	no	—
127	C₂×C₄	—	34	1	no	—
128	C₂×C₄	—	34	1	no	—
129	C₂×C₄	—	34	1	no	—
130	C₂×C₄	—	34	1	no	—
131	C₂×C₄	#63243	1	2	yes	—
132	C₂×C₄	#63243	1	2	yes	—
133	C₂×C₄	C₄×#595	1	2	yes	—
134	C₂×C₄	C₄⋊#595	1	2	yes	—
135	C₂×C₄	#63247	1	2	yes	—
136	C₈	—	34	8	no	—
137	C₂²	C₂×#63243	1	1	yes	characteristic
138	C₂²	C₂×#63243	1	1	yes	characteristic
139	C₂²	D₁₇⋊M₁₆	1	1	yes	characteristic
140	C₂²	C₂×C₄×#595	1	1	yes	characteristic
141	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
142	C₂²	#10587677	1	1	yes	characteristic
143	C₂²	C₂×#63247	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	17	2	no	—
147	C₄	—	17	2	no	—
148	C₄	—	34	2	no	—
149	C₄	—	34	2	no	—
150	C₄	—	34	2	no	—
151	C₄	#10587542	1	4	yes	complemented
152	C₂	C₄×C₄⋊#595	1	1	yes	characteristic
153	C₂	C₂×#10587542	1	1	yes	characteristic
154	C₂	C₄×#63243	1	1	yes	characteristic
155	C₂	#11763812	1	1	yes	characteristic
156	C₂	#11763813	1	1	yes	characteristic
157	C₂	#11763830	1	1	yes	characteristic
158	C₂	#11763831	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#3624

C₂

yes

maximal

C₂×#10587542

C₂

yes

maximal

#11763675

C₂

yes

maximal

C₄²×Dic₁₇

C₂

yes

maximal, characteristic

C₂²×#3624

C₂²

yes

—

C₂×C₄²×C₁₇

C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

#10587542

—

C₄⋊#3624

C₄

yes

complemented

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₇

C₂³

yes

—

—

C₂×C₄

yes

—

C₄²×C₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

D₄

yes

—

C₄×Dic₁₇

Q₈

yes

—

C₂³×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄²

yes

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

M₁₆

yes

—

C₂×C₄×C₁₇

M₁₆

yes

—

C₂×C₄×C₁₇

M₁₆

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₄²

yes

—

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₂×Dic₁₇

M₁₆

yes

—

C₂×Dic₁₇

C₂²×C₄

yes

—

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₇

#41

yes

—

#3624

—

#1767

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄²

yes

characteristic

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

#117

yes

characteristic

C₂²×C₁₇

C₂×M₁₆

yes

characteristic

C₂²×C₁₇

C₂×M₁₆

yes

characteristic

C₂²×C₁₇

C₂×M₁₆

yes

characteristic

C₂²×C₁₇

C₂×M₁₆

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#97

yes

complemented

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

C₄×D₄

yes

—

Dic₁₇

C₄×Q₈

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄³

—

#336

—

C₂×C₁₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₇

C₂×#97

yes

characteristic

C₂×C₁₇

C₄×M₁₆

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#595

yes

characteristic

C₄⋊C₈

—

#97

—

C₁₇

#1767

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#595

yes

characteristic

100

C₂²×C₄

C₂×#595

yes

characteristic

101

C₂²×C₄

C₂×#595

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₂³

C₂²×#595

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

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

#63243

yes

—

132

C₂×C₄

#63243

yes

—

133

C₂×C₄

C₄×#595

yes

—

134

C₂×C₄

C₄⋊#595

yes

—

135

C₂×C₄

#63247

yes

—

136

C₈

—

137

C₂²

C₂×#63243

yes

characteristic

138

C₂²

C₂×#63243

yes

characteristic

139

C₂²

D₁₇⋊M₁₆

yes

characteristic

140

C₂²

C₂×C₄×#595

yes

characteristic

141

C₂²

C₂×C₄⋊#595

yes

characteristic

142

C₂²

#10587677

yes

characteristic

143

C₂²

C₂×#63247

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#10587542

yes

complemented

152

C₂

C₄×C₄⋊#595

yes

characteristic

153

C₂

C₂×#10587542

yes

characteristic

154

C₂

C₄×#63243

yes

characteristic

155

C₂

#11763812

yes

characteristic

156

C₂

#11763813

yes

characteristic

157

C₂

#11763830

yes

characteristic

158

C₂

#11763831

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄²
Fitting	C₂×C₄²×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂²×Dic₁₇	C₂³
hypercenter	C₂×C₄²	#595
hypocenter	C₁₇	#1767
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#595

commutator

C₂²×C₁₇

C₂×C₄²

Fitting

C₂×C₄²×C₁₇

C₄

socle, prime radical

C₂³×C₁₇

C₂²×C₄

cosocle

C₂²×Dic₁₇

C₂³

C₂×C₄²

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
14. C₄⋊#3624	C₄	semidirect
67. C₄×C₁₇	#97	semidirect
94. C₁₇	#1767	semidirect
151. C₄	#10587542	semidirect

H ≅ G/N

type

14. C₄⋊#3624

C₄

semidirect

67. C₄×C₁₇

#97

semidirect

94. C₁₇

#1767

semidirect

151. C₄

#10587542

semidirect

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

subgroup	index
C₂×C₄⋊#3624	2
C₂×#10587542	2
#11763675	2
C₄²×Dic₁₇	2
#1767	17

subgroup

index

C₂×C₄⋊#3624

C₂×#10587542

#11763675

C₄²×Dic₁₇

#1767

Subgroups — #14544570

Conjugacy classes of subgroups

Complements

Maximal subgroups