Group #14544901 — subgroups

Subgroups — #14544901

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×#10587533	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	2	no	—
12	#10587533	—	2	1	no	—
13	#10587533	—	2	1	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	2	yes	—
27	C₄×#595	—	2	1	no	—
28	#63243	—	2	1	no	—
29	#63242	—	2	1	no	—
30	M₁₆×C₁₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	#63223	C₂×C₄	1	2	yes	—
33	C₂×#3622	C₂×C₄	1	1	yes	characteristic
34	#63218	—	2	2	no	—
35	C₈×D₁₇	—	2	2	no	—
36	C₈×D₁₇	D₄	1	1	yes	characteristic
37	C₈×D₁₇	D₄	1	1	yes	characteristic
38	C₈×D₁₇	Q₈	1	1	yes	characteristic
39	C₈×D₁₇	Q₈	1	1	yes	characteristic
40	C₂²×D₁₇	#41	1	1	yes	characteristic
41	C₂×#595	—	2	2	no	—
42	C₂×#595	—	4	2	no	—
43	C₂×#595	—	8	2	no	—
44	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
47	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
48	C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₄×D₁₇	#41	1	1	yes	characteristic
50	#3624	—	2	2	no	—
51	C₈×C₁₇	—	2	2	no	—
52	C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
53	C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
54	#3622	—	2	2	no	—
55	#3622	C₄⋊C₄	1	1	yes	characteristic
56	#3622	C₄⋊C₄	1	1	yes	characteristic
57	#1951	—	17	1	no	maximal, Sylow
58	C₂²×C₁₇	C₄×Q₈	1	1	yes	characteristic
59	C₂×D₁₇	—	2	2	no	—
60	C₂×D₁₇	#121	1	1	yes	characteristic
61	C₂×D₁₇	#125	1	1	yes	characteristic
62	#595	—	4	2	no	—
63	#595	—	8	2	no	—
64	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
66	Dic₁₇	C₂²⋊Q₈	1	1	yes	characteristic
67	Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×#108	—	17	1	no	—
69	#428	—	17	2	no	—
70	C₂×#106	—	17	1	no	—
71	#405	—	17	1	no	—
72	#343	—	17	2	no	—
73	C₂×C₁₇	—	2	1	no	—
74	C₂×C₁₇	#389	1	1	yes	characteristic
75	D₁₇	—	2	1	no	—
76	D₁₇	#471	1	2	yes	—
77	C₂×M₁₆	—	17	1	no	—
78	C₂×M₁₆	—	17	2	no	—
79	C₂²×C₈	—	17	1	no	—
80	#117	—	17	1	no	—
81	C₂×C₄⋊C₄	—	17	2	no	—
82	#108	—	17	2	no	—
83	#108	—	17	1	no	—
84	#108	#595	1	1	yes	characteristic, complemented
85	#107	—	34	2	no	—
86	#106	—	34	2	no	—
87	#106	—	34	1	no	—
88	#106	—	34	1	no	—
89	#97	—	34	1	no	—
90	C₄×C₈	—	34	1	no	—
91	C₁₇	#1951	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₄	—	17	1	no	—
93	C₂²×C₄	—	34	2	no	—
94	M₁₆	—	17	2	no	—
95	M₁₆	—	17	2	no	—
96	M₁₆	—	34	2	no	—
97	M₁₆	—	34	1	no	—
98	M₁₆	C₂×#595	1	2	yes	—
99	C₂×C₈	—	17	1	no	—
100	C₂×C₈	—	17	1	no	—
101	C₂×C₈	—	17	1	no	—
102	C₂×C₈	—	17	1	no	—
103	C₂×C₈	—	17	1	no	—
104	C₂×C₈	—	34	2	no	—
105	C₂×C₈	—	34	1	no	—
106	C₂×C₈	C₂×#595	1	1	yes	characteristic
107	C₄⋊C₄	—	17	2	no	—
108	C₄⋊C₄	—	34	2	no	—
109	C₄⋊C₄	—	34	2	no	—
110	C₂²⋊C₄	—	34	1	no	—
111	C₄²	—	34	1	no	—
112	C₂³	—	17	1	no	—
113	C₂×C₄	—	17	1	no	—
114	C₂×C₄	—	17	1	no	—
115	C₂×C₄	—	17	1	no	—
116	C₂×C₄	—	17	1	no	—
117	C₂×C₄	—	17	1	no	—
118	C₂×C₄	—	34	2	no	—
119	C₂×C₄	—	68	2	no	—
120	C₂×C₄	—	136	2	no	—
121	C₂×C₄	C₂²×#595	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	17	1	no	—
124	C₈	—	17	1	no	—
125	C₈	—	34	2	no	—
126	C₈	—	34	2	no	—
127	C₈	C₄⋊#595	1	1	yes	characteristic
128	C₈	C₄⋊#595	1	1	yes	characteristic
129	C₂²	—	17	1	no	—
130	C₂²	—	17	1	no	—
131	C₂²	—	34	2	no	—
132	C₂²	Q₈×#595	1	1	yes	characteristic
133	C₄	—	17	1	no	—
134	C₄	—	17	1	no	—
135	C₄	—	68	2	no	—
136	C₄	—	136	2	no	—
137	C₄	C₂×C₄⋊#595	1	1	yes	characteristic
138	C₄	D₄×#595	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	17	2	no	—
141	C₂	—	34	1	no	—
142	C₂	#11763760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763820

C₂

yes

maximal

C₂×#10587533

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

—

C₄×#595

—

#63243

—

#63242

—

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

#1951

—

maximal, Sylow

C₂²×C₁₇

C₄×Q₈

yes

characteristic

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

#471

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

—

—

yes

characteristic, complemented

—

—

—

—

—

C₄×C₈

—

C₁₇

#1951

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×#595

yes

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×#595

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₂³

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

136

—

121

C₂×C₄

C₂²×#595

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

C₄⋊#595

yes

characteristic

128

C₈

C₄⋊#595

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

Q₈×#595

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

136

—

137

C₄

C₂×C₄⋊#595

yes

characteristic

138

C₄

D₄×#595

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

142

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	#1951
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
84. #108	#595	semidirect
91. C₁₇	#1951	semidirect

H ≅ G/N

type

17. C₁₇×#108

C₄

semidirect

18. #10587493

C₄

semidirect

19. #10587490

C₄

semidirect

84. #108

#595

semidirect

91. C₁₇

#1951

semidirect

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

subgroup	index
#11763820	2
C₂×#10587533	2
#11763772	2
#11763688	2
#108×D₁₇	2
#1951	17

subgroup

index

#108×D₁₇

Subgroups — #14544901

Conjugacy classes of subgroups

Complements

Maximal subgroups