Group #19140716 — subgroups

Subgroups — #19140716

Order 5504.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13063325	C₂	1	1	yes	maximal, characteristic
2	#14582829	C₂	1	4	yes	maximal
3	#14582819	C₂	1	4	yes	maximal
4	#14582768	C₂	1	2	yes	maximal
5	#14582764	C₂	1	2	yes	maximal
6	#14582745	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₈₆	C₂²	1	2	yes	—
9	#13063325	—	2	4	no	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
12	C₂×Dic₄₃⋊C₄	C₂²	1	8	yes	—
13	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
14	C₄₃⋊#95	C₂²	1	2	yes	—
15	C₄₃⋊#95	C₂²	1	8	yes	—
16	C₄₃⋊#95	C₂²	1	2	yes	—
17	#125×C₄₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
24	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₄₃	C₂³	1	4	yes	—
29	C₂²×Dic₄₃	C₂³	1	4	yes	—
30	C₂×Dic₈₆	—	2	4	no	—
31	C₂×Dic₈₆	—	4	4	no	—
32	C₄⋊C₄×C₄₃	—	2	4	no	—
33	C₄⋊C₄×C₄₃	—	2	4	no	—
34	C₄⋊C₄×C₄₃	—	2	2	no	—
35	C₄⋊C₄×C₄₃	—	2	2	no	—
36	C₄²×C₄₃	D₄	1	4	yes	—
37	C₄⋊Dic₄₃	—	2	4	no	—
38	C₄⋊Dic₄₃	—	2	4	no	—
39	Dic₄₃⋊C₄	—	4	8	no	—
40	C₄×Dic₄₃	Q₈	1	8	yes	complemented
41	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	4	no	—
47	C₂×C₄×C₄₃	—	2	4	no	—
48	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
52	C₂×Dic₄₃	—	2	4	no	—
53	C₂×Dic₄₃	—	2	4	no	—
54	C₂×Dic₄₃	—	2	4	no	—
55	C₂×Dic₄₃	—	2	8	no	—
56	C₂×Dic₄₃	C₂×Q₈	1	8	yes	—
57	Dic₈₆	—	4	8	no	—
58	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
59	C₂²×C₄₃	C₂²×Q₈	1	2	yes	—
60	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
61	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
62	C₄×C₄₃	—	2	2	no	—
63	C₄×C₄₃	—	2	2	no	—
64	C₄×C₄₃	—	4	2	no	—
65	C₄×C₄₃	—	4	2	no	—
66	C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
67	Dic₄₃	—	2	8	no	—
68	Dic₄₃	—	4	4	no	—
69	#2660	—	43	1	no	maximal, Sylow
70	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄₃	#535	1	1	yes	characteristic
72	C₂×C₄₃	#536	1	1	yes	characteristic
73	C₂×C₄₃	#544	1	4	yes	—
74	C₂×C₄⋊Q₈	—	43	1	no	—
75	C₂×#125	D₄₃	1	1	yes	characteristic
76	#398	—	43	4	no	—
77	#395	—	43	4	no	—
78	#384	—	43	2	no	—
79	#382	—	43	2	no	—
80	#376	—	43	1	no	—
81	C₄₃	#2660	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	—	43	2	no	—
83	C₄⋊Q₈	—	86	4	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	43	2	no	—
86	C₂×C₄⋊C₄	—	43	2	no	—
87	C₂×C₄⋊C₄	—	43	8	no	—
88	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
91	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
92	C₂×C₄²	—	43	2	no	—
93	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
94	#95	—	43	2	no	—
95	#95	—	43	2	no	—
96	#95	—	43	8	no	—
97	C₂×Q₈	—	86	4	no	—
98	C₂×Q₈	—	172	4	no	—
99	C₂²×C₄	—	43	4	no	—
100	C₂²×C₄	—	43	4	no	—
101	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
102	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
103	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	86	4	no	—
111	C₄⋊C₄	—	86	4	no	—
112	C₄⋊C₄	—	172	8	no	—
113	C₄²	—	43	8	no	—
114	C₄²	C₂²⋊D₄₃	1	4	yes	—
115	C₂³	C₂³×D₄₃	1	1	yes	characteristic
116	Q₈	—	172	8	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	43	8	no	—
124	C₂×C₄	—	86	4	no	—
125	C₂×C₄	—	86	4	no	—
126	C₂×C₄	—	86	4	no	—
127	C₂×C₄	—	86	8	no	—
128	C₂×C₄	Q₈×D₄₃	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
132	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
133	C₂²	C₂×Q₈×D₄₃	1	2	yes	—
134	C₂²	#13063371	1	2	yes	—
135	C₂²	#13063372	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	86	8	no	—
141	C₄	—	172	4	no	—
142	C₄	#13063323	1	4	yes	—
143	C₂	C₂×#13063323	1	1	yes	characteristic
144	C₂	#14583463	1	1	yes	characteristic
145	C₂	#14583470	1	1	yes	characteristic
146	C₂	#14583529	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063325

C₂

yes

maximal, characteristic

#14582829

C₂

yes

maximal

#14582819

C₂

yes

maximal

#14582768

C₂

yes

maximal

#14582764

C₂

yes

maximal

#14582745

C₂

yes

maximal, characteristic

C₂×#125×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

—

#13063325

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

#125×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

Q₈

yes

complemented

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×Q₈

yes

—

Dic₈₆

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

—

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄⋊Q₈

yes

—

Dic₄₃

—

Dic₄₃

—

#2660

—

maximal, Sylow

C₂×C₄₃

C₂×C₄⋊Q₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄⋊Q₈

—

C₂×#125

D₄₃

yes

—

—

—

—

—

C₄₃

#2660

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

172

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₄₃

yes

—

102

C₂²×C₄

C₂²×D₄₃

yes

—

103

C₂²×C₄

C₂²×D₄₃

yes

characteristic

104

C₂²×C₄

C₂²×D₄₃

yes

characteristic

105

C₂²×C₄

C₂²×D₄₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

172

—

113

C₄²

—

114

C₄²

C₂²⋊D₄₃

yes

—

115

C₂³

C₂³×D₄₃

yes

characteristic

116

Q₈

—

172

—

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

Q₈×D₄₃

yes

—

129

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

131

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

132

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

133

C₂²

C₂×Q₈×D₄₃

yes

—

134

C₂²

#13063371

yes

—

135

C₂²

#13063372

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

172

—

142

C₄

#13063323

yes

—

143

C₂

C₂×#13063323

yes

characteristic

144

C₂

#14583463

yes

characteristic

145

C₂

#14583470

yes

characteristic

146

C₂

#14583529

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₂×#125×C₄₃	C₂
hypercenter	C₂×#125	D₄₃
hypocenter	C₄₃	#2660
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₂×#125×C₄₃

C₂

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
40. C₄×Dic₄₃	Q₈	semidirect
81. C₄₃	#2660	semidirect

H ≅ G/N

type

40. C₄×Dic₄₃

Q₈

semidirect

81. C₄₃

#2660

semidirect

Subgroups — #19140716

Conjugacy classes of subgroups

Complements