Group #14544898 — subgroups

Subgroups — #14544898

Order 2176.

200 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 74 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×#10587536	C₂	1	2	yes	maximal
2	#11763813	C₂	1	1	yes	maximal, characteristic
3	C₄×#63243	C₂	1	1	yes	maximal, characteristic
4	#11763685	C₂	1	2	yes	maximal
5	C₄⋊C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#63243	C₂²	1	2	yes	—
7	C₂×#63243	C₂²	1	1	yes	characteristic
8	C₂×#63243	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₁₇	C₂²	1	2	yes	—
10	C₄²×D₁₇	C₂²	1	1	yes	characteristic
11	#10587542	—	2	1	no	—
12	C₄⋊#3624	—	2	1	no	—
13	C₄⋊#3624	—	2	1	no	—
14	C₄×#3624	—	2	1	no	—
15	#10587536	—	2	4	no	—
16	C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
17	#10587485	C₄	1	2	yes	—
18	C₄⋊#3622	C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₇	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₁₇	Q₈	1	1	yes	characteristic
21	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
22	C₂×#3624	—	2	1	no	—
23	C₂×#3624	—	2	1	no	—
24	C₂×#3624	—	2	2	no	—
25	#63243	—	2	1	no	—
26	#63243	—	2	1	no	—
27	#63243	—	2	2	no	—
28	#63243	—	2	2	no	—
29	#63243	D₄	1	2	yes	—
30	#63243	Q₈	1	2	yes	—
31	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₇	D₄	1	1	yes	characteristic
35	C₄×Dic₁₇	D₄	1	1	yes	characteristic
36	C₂×#3622	C₂×C₄	1	2	yes	—
37	C₈×D₁₇	—	2	2	no	—
38	C₈×D₁₇	—	2	2	no	—
39	C₂²×D₁₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
46	C₄×D₁₇	—	2	2	no	—
47	C₄×D₁₇	—	4	1	no	—
48	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
49	C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₄×D₁₇	#41	1	1	yes	characteristic
51	C₄×D₁₇	#41	1	1	yes	characteristic
52	#3624	—	2	2	no	—
53	#3624	—	2	2	no	—
54	#3624	—	4	2	no	—
55	C₈×C₁₇	—	2	2	no	—
56	#3622	—	2	2	no	—
57	#1950	—	17	1	no	maximal, Sylow
58	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×D₁₇	—	2	1	no	—
60	C₂×D₁₇	—	2	1	no	—
61	C₂×D₁₇	#125	1	1	yes	characteristic
62	C₂×D₁₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₄×C₁₇	—	2	1	no	—
64	C₄×C₁₇	#108	1	2	yes	complemented
65	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
66	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
67	Dic₁₇	—	2	1	no	—
68	Dic₁₇	—	2	1	no	—
69	Dic₁₇	#121	1	1	yes	characteristic
70	Dic₁₇	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×#108	—	17	2	no	—
72	#423	—	17	1	no	—
73	C₂×C₄⋊C₈	—	17	1	no	—
74	C₄×M₁₆	—	17	1	no	—
75	#341	—	17	2	no	—
76	C₂×C₁₇	#389	1	1	yes	characteristic
77	C₂×C₁₇	C₂×#108	1	1	yes	characteristic
78	C₂×C₁₇	#430	1	1	yes	characteristic
79	D₁₇	—	2	1	no	—
80	D₁₇	—	2	1	no	—
81	C₂×M₁₆	—	17	1	no	—
82	C₂×M₁₆	—	17	1	no	—
83	C₂×M₁₆	—	17	2	no	—
84	C₂²×C₈	—	17	2	no	—
85	C₂×C₄²	—	17	1	no	—
86	#108	—	34	4	no	—
87	C₄⋊C₈	—	17	2	no	—
88	C₄⋊C₈	—	17	1	no	—
89	C₄⋊C₈	—	34	1	no	—
90	C₄⋊C₈	—	34	1	no	—
91	C₄⋊C₈	#595	1	1	yes	characteristic
92	#97	—	34	1	no	—
93	C₄×C₈	—	34	1	no	—
94	C₁₇	#1950	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	M₁₆	—	17	2	no	—
99	M₁₆	—	17	2	no	—
100	M₁₆	—	34	1	no	—
101	M₁₆	—	34	1	no	—
102	M₁₆	—	34	2	no	—
103	M₁₆	—	34	2	no	—
104	C₂×C₈	—	17	2	no	—
105	C₂×C₈	—	34	2	no	—
106	C₂×C₈	—	34	2	no	—
107	C₂×C₈	—	34	1	no	—
108	C₂×C₈	—	34	1	no	—
109	C₂×C₈	—	34	2	no	—
110	C₂×C₈	C₂×#595	1	2	yes	—
111	C₄²	—	17	1	no	—
112	C₄²	—	17	1	no	—
113	C₄²	—	17	1	no	—
114	C₄²	C₂×#595	1	1	yes	characteristic
115	C₂³	—	17	1	no	—
116	C₂×C₄	—	17	1	no	—
117	C₂×C₄	—	17	1	no	—
118	C₂×C₄	—	17	1	no	—
119	C₂×C₄	—	17	1	no	—
120	C₂×C₄	—	17	1	no	—
121	C₂×C₄	—	17	1	no	—
122	C₂×C₄	—	17	1	no	—
123	C₂×C₄	—	34	2	no	—
124	C₂×C₄	—	68	1	no	—
125	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
126	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
127	C₂×C₄	C₂²×#595	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	34	2	no	—
130	C₈	—	34	2	no	—
131	C₈	—	34	2	no	—
132	C₈	—	68	2	no	—
133	C₂²	—	17	1	no	—
134	C₂²	—	17	1	no	—
135	C₂²	—	34	1	no	—
136	C₂²	—	34	1	no	—
137	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	17	1	no	—
140	C₄	—	17	1	no	—
141	C₄	—	34	1	no	—
142	C₄	—	34	1	no	—
143	C₄	#10587536	1	2	yes	complemented
144	C₄	D₄×#595	1	1	yes	characteristic
145	C₄	Q₈×#595	1	1	yes	characteristic
146	C₂	—	34	1	no	—
147	C₂	—	34	1	no	—
148	C₂	#11763760	1	1	yes	characteristic
149	C₂	C₂×#10587536	1	1	yes	characteristic
150	C₂	#11763824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587536

C₂

yes

maximal

#11763813

C₂

yes

maximal, characteristic

C₄×#63243

C₂

yes

maximal, characteristic

#11763685

C₂

yes

maximal

C₄⋊C₈×D₁₇

C₂

yes

maximal, characteristic

C₂×#63243

C₂²

yes

—

C₂×#63243

C₂²

yes

characteristic

C₂×#63243

C₂²

yes

characteristic

C₂×C₈×D₁₇

C₂²

yes

—

C₄²×D₁₇

C₂²

yes

—

—

—

—

—

C₁₇×C₄⋊C₈

C₄

yes

characteristic

#10587485

C₄

yes

—

C₄⋊#3622

C₄

yes

characteristic

C₂×C₄×D₁₇

Q₈

yes

characteristic

C₂×C₄×D₁₇

Q₈

yes

characteristic

C₂×C₄×D₁₇

C₂³

yes

—

—

—

—

—

—

—

D₄

yes

—

#63243

Q₈

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

—

C₈×D₁₇

—

C₈×D₁₇

—

C₂²×D₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

C₂×Q₈

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

—

—

—

C₈×C₁₇

—

#3622

—

#1950

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#125

yes

characteristic

C₂×D₁₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

#108

yes

complemented

C₄×C₁₇

C₄×D₄

yes

characteristic

C₄×C₁₇

C₄×Q₈

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#121

yes

characteristic

Dic₁₇

C₂²⋊Q₈

yes

characteristic

C₂×#108

—

#423

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#341

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

D₁₇

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

#595

yes

characteristic

#97

—

C₄×C₈

—

C₁₇

#1950

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂×#595

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×#595

yes

characteristic

115

C₂³

—

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

C₄⋊#595

yes

characteristic

126

C₂×C₄

C₄⋊#595

yes

characteristic

127

C₂×C₄

C₂²×#595

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×C₄⋊#595

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#10587536

yes

complemented

144

C₄

D₄×#595

yes

characteristic

145

C₄

Q₈×#595

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

#11763760

yes

characteristic

149

C₂

C₂×#10587536

yes

characteristic

150

C₂

#11763824

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×C₄⋊C₈	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₇	C₂×Q₈
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	C₄⋊C₈	#595
hypocenter	C₁₇	#1950
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×C₄⋊C₈

C₄

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₇

C₂×Q₈

cosocle

C₂×C₄×D₁₇

C₂³

C₄⋊C₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
64. C₄×C₁₇	#108	semidirect
94. C₁₇	#1950	semidirect
143. C₄	#10587536	semidirect

H ≅ G/N

type

64. C₄×C₁₇

#108

semidirect

94. C₁₇

#1950

semidirect

143. C₄

#10587536

semidirect

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

subgroup	index
C₂×#10587536	2
#11763813	2
C₄×#63243	2
#11763685	2
C₄⋊C₈×D₁₇	2
#1950	17

subgroup

index

C₄⋊C₈×D₁₇

Subgroups — #14544898

Conjugacy classes of subgroups

Complements

Maximal subgroups