Group #14544902 — subgroups

Subgroups — #14544902

Order 2176.

196 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 68 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	#11763820	C₂	1	2	yes	maximal
2	C₂×#10587534	C₂	1	1	yes	maximal, characteristic
3	#11763772	C₂	1	1	yes	maximal, characteristic
4	#11763688	C₂	1	2	yes	maximal
5	#108×D₁₇	C₂	1	1	yes	maximal, characteristic
6	#10587677	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#595	C₂²	1	2	yes	—
8	D₁₇⋊M₁₆	C₂²	1	1	yes	characteristic
9	M₁₆×D₁₇	C₂²	1	2	yes	—
10	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
11	#10587534	—	2	1	no	—
12	#10587534	—	2	1	no	—
13	#10587534	—	2	2	no	—
14	#10587533	—	2	2	no	—
15	#10587532	—	2	1	no	—
16	C₈×#595	—	2	1	no	—
17	C₁₇×#108	C₄	1	1	yes	characteristic, complemented
18	#10587493	C₄	1	2	yes	complemented
19	#10587490	C₄	1	1	yes	characteristic, complemented
20	C₂²×#595	—	2	2	no	—
21	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
22	C₂²⋊#595	—	2	1	no	—
23	#63247	Q₈	1	2	yes	—
24	C₄⋊#595	—	2	2	no	—
25	C₄⋊#595	—	2	2	no	—
26	C₄⋊#595	D₄	1	1	yes	characteristic
27	C₄⋊#595	D₄	1	1	yes	characteristic
28	C₄×#595	—	2	1	no	—
29	#63243	—	2	1	no	—
30	#63242	—	2	1	no	—
31	M₁₆×C₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	#63223	C₂×C₄	1	2	yes	—
34	C₂×#3622	C₂×C₄	1	1	yes	characteristic
35	#63218	—	2	2	no	—
36	C₈×D₁₇	—	2	2	no	—
37	C₈×D₁₇	D₄	1	1	yes	characteristic
38	C₈×D₁₇	D₄	1	1	yes	characteristic
39	C₈×D₁₇	Q₈	1	1	yes	characteristic
40	C₈×D₁₇	Q₈	1	1	yes	characteristic
41	C₂²×D₁₇	#41	1	1	yes	characteristic
42	C₂×#595	—	2	1	no	—
43	C₂×#595	—	2	1	no	—
44	C₂×#595	—	4	2	no	—
45	C₂×#595	—	8	2	no	—
46	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
49	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
50	C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₄×D₁₇	#41	1	1	yes	characteristic
52	#3624	—	2	2	no	—
53	C₈×C₁₇	—	2	2	no	—
54	C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
55	C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
56	#3622	—	2	2	no	—
57	#3622	C₄⋊C₄	1	1	yes	characteristic
58	#3622	C₄⋊C₄	1	1	yes	characteristic
59	#1952	—	17	1	no	maximal, Sylow
60	C₂²×C₁₇	C₄×Q₈	1	1	yes	characteristic
61	C₂×D₁₇	—	2	1	no	—
62	C₂×D₁₇	—	2	1	no	—
63	C₂×D₁₇	#121	1	1	yes	characteristic
64	C₂×D₁₇	#125	1	1	yes	characteristic
65	#595	—	4	1	no	—
66	#595	—	4	1	no	—
67	#595	—	8	2	no	—
68	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
70	Dic₁₇	C₂²⋊Q₈	1	1	yes	characteristic
71	Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×#108	—	17	1	no	—
73	#428	—	17	2	no	—
74	C₂×#107	—	17	1	no	—
75	#405	—	17	1	no	—
76	#343	—	17	2	no	—
77	C₂×C₁₇	—	2	1	no	—
78	C₂×C₁₇	#389	1	1	yes	characteristic
79	D₁₇	—	2	1	no	—
80	D₁₇	#472	1	1	yes	characteristic
81	D₁₇	#473	1	1	yes	characteristic
82	C₂×M₁₆	—	17	1	no	—
83	C₂×M₁₆	—	17	2	no	—
84	C₂²×C₈	—	17	1	no	—
85	#117	—	17	1	no	—
86	C₂×C₄⋊C₄	—	17	2	no	—
87	#108	—	17	2	no	—
88	#108	—	17	1	no	—
89	#108	#595	1	1	yes	characteristic, complemented
90	#107	—	34	2	no	—
91	#107	—	34	1	no	—
92	#107	—	34	1	no	—
93	#106	—	34	2	no	—
94	#97	—	34	1	no	—
95	C₄×C₈	—	34	1	no	—
96	C₁₇	#1952	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₄	—	17	1	no	—
98	C₂²×C₄	—	34	2	no	—
99	M₁₆	—	17	2	no	—
100	M₁₆	—	17	2	no	—
101	M₁₆	—	34	2	no	—
102	M₁₆	—	34	1	no	—
103	M₁₆	C₂×#595	1	2	yes	—
104	C₂×C₈	—	17	1	no	—
105	C₂×C₈	—	17	1	no	—
106	C₂×C₈	—	17	1	no	—
107	C₂×C₈	—	17	1	no	—
108	C₂×C₈	—	17	1	no	—
109	C₂×C₈	—	34	2	no	—
110	C₂×C₈	—	34	1	no	—
111	C₂×C₈	C₂×#595	1	1	yes	characteristic
112	C₄⋊C₄	—	17	1	no	—
113	C₄⋊C₄	—	17	1	no	—
114	C₄⋊C₄	—	34	2	no	—
115	C₄⋊C₄	—	34	2	no	—
116	C₂²⋊C₄	—	34	1	no	—
117	C₄²	—	34	1	no	—
118	C₂³	—	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	—	17	1	no	—
124	C₂×C₄	—	34	1	no	—
125	C₂×C₄	—	34	1	no	—
126	C₂×C₄	—	68	2	no	—
127	C₂×C₄	—	136	2	no	—
128	C₂×C₄	C₂²×#595	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	17	1	no	—
131	C₈	—	17	1	no	—
132	C₈	—	34	2	no	—
133	C₈	—	34	2	no	—
134	C₈	C₄⋊#595	1	1	yes	characteristic
135	C₈	C₄⋊#595	1	1	yes	characteristic
136	C₂²	—	17	1	no	—
137	C₂²	—	17	1	no	—
138	C₂²	—	34	1	no	—
139	C₂²	—	34	1	no	—
140	C₂²	Q₈×#595	1	1	yes	characteristic
141	C₄	—	17	1	no	—
142	C₄	—	17	1	no	—
143	C₄	—	68	1	no	—
144	C₄	—	68	1	no	—
145	C₄	—	136	2	no	—
146	C₄	C₂×C₄⋊#595	1	1	yes	characteristic
147	C₄	D₄×#595	1	1	yes	characteristic
148	C₂	—	2	1	no	—
149	C₂	—	17	1	no	—
150	C₂	—	17	1	no	—
151	C₂	—	34	1	no	—
152	C₂	#11763760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763820

C₂

yes

maximal

C₂×#10587534

C₂

yes

maximal, characteristic

#11763772

C₂

yes

maximal, characteristic

#11763688

C₂

yes

maximal

#108×D₁₇

C₂

yes

maximal, characteristic

#10587677

C₂²

yes

characteristic

C₂×C₄⋊#595

C₂²

yes

—

D₁₇⋊M₁₆

C₂²

yes

characteristic

M₁₆×D₁₇

C₂²

yes

—

C₂×C₈×D₁₇

C₂²

yes

—

—

—

—

—

—

C₁₇×#108

C₄

yes

characteristic, complemented

#10587493

C₄

yes

complemented

#10587490

C₄

yes

characteristic, complemented

C₂²×#595

—

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂²⋊#595

—

#63247

Q₈

yes

—

C₄⋊#595

—

C₄⋊#595

—

C₄⋊#595

D₄

yes

characteristic

C₄⋊#595

D₄

yes

—

—

—

M₁₆×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

#63223

C₂×C₄

yes

—

C₂×#3622

C₂×C₄

yes

characteristic

#63218

—

C₈×D₁₇

—

C₈×D₁₇

D₄

yes

characteristic

C₈×D₁₇

D₄

yes

characteristic

C₈×D₁₇

Q₈

yes

characteristic

C₈×D₁₇

Q₈

yes

C₂²×D₁₇

yes

—

—

—

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

C₂×Q₈

yes

characteristic

C₄×D₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

C₂×Q₈

yes

C₄×D₁₇

yes

—

C₈×C₁₇

—

C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₈×C₁₇

C₄⋊C₄

yes

characteristic

#3622

—

#3622

C₄⋊C₄

yes

characteristic

#3622

C₄⋊C₄

yes

characteristic

#1952

—

maximal, Sylow

C₂²×C₁₇

C₄×Q₈

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

—

—

—

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₄×D₄

yes

characteristic

Dic₁₇

C₂²⋊Q₈

yes

characteristic

Dic₁₇

C₄⋊Q₈

yes

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#389

yes

characteristic

D₁₇

—

D₁₇

yes

D₁₇

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

—

—

yes

characteristic, complemented

—

—

—

—

—

C₄×C₈

—

C₁₇

#1952

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

C₂×#595

yes

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂×#595

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₂³

—

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

—

136

—

128

C₂×C₄

C₂²×#595

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

C₄⋊#595

yes

characteristic

135

C₈

C₄⋊#595

yes

characteristic

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

Q₈×#595

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

136

—

146

C₄

C₂×C₄⋊#595

yes

characteristic

147

C₄

D₄×#595

yes

characteristic

148

C₂

—

149

C₂

—

150

C₂

—

151

C₂

—

152

C₂

#11763760

yes

characteristic

	subgroup	G/H
center	C₂	#11763760
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×#108	C₄
socle	C₂×C₁₇	#389
prime radical	C₂²×D₁₇	#41
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	#108	#595
hypocenter	C₁₇	#1952
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#11763760

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×#108

C₄

C₂×C₁₇

C₂²×D₁₇

C₂×C₄×D₁₇

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
17. C₁₇×#108	C₄	semidirect
18. #10587493	C₄	semidirect
19. #10587490	C₄	semidirect
89. #108	#595	semidirect
96. C₁₇	#1952	semidirect

H ≅ G/N

type

17. C₁₇×#108

C₄

semidirect

18. #10587493

C₄

semidirect

19. #10587490

C₄

semidirect

89. #108

#595

semidirect

96. C₁₇

#1952

semidirect

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

subgroup	index
#11763820	2
C₂×#10587534	2
#11763772	2
#11763688	2
#108×D₁₇	2
#1952	17

subgroup

index

#108×D₁₇

Subgroups — #14544902

Conjugacy classes of subgroups

Complements

Maximal subgroups