Group #19134289 — subgroups

Subgroups — #19134289

Order 5504.

480 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 221 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	C₂²×#10621877	C₂	1	2	yes	maximal
2	#14582654	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#10621877	—	2	4	no	—
5	C₂×#10621877	—	2	4	no	—
6	C₂×#10621877	C₄	1	8	yes	complemented
7	C₂²×C₄₃⋊C₈	C₂²	1	4	yes	—
8	C₂³×C₄×C₄₃	C₄	1	2	yes	—
9	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₄₃	—	2	4	no	—
11	C₂×C₄²×C₄₃	C₄	1	2	yes	—
12	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
13	C₂⁴×C₄₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₄₃	—	2	8	no	—
15	C₂²×C₄×C₄₃	—	2	4	no	—
16	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	D₄	1	4	yes	—
20	C₂²×C₄×C₄₃	D₄	1	2	yes	—
21	C₂²×C₄×C₄₃	Q₈	1	2	yes	—
22	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	C₄²×C₄₃	—	2	4	no	—
24	C₄²×C₄₃	—	2	4	no	—
25	#10621877	—	2	16	no	—
26	C₂×C₄₃⋊C₈	—	2	4	no	—
27	C₂×C₄₃⋊C₈	—	2	4	no	—
28	C₂×C₄₃⋊C₈	C₂×C₄	1	8	yes	complemented
29	C₂³×C₄₃	—	2	4	no	—
30	C₂³×C₄₃	C₄²	1	2	yes	—
31	C₂³×C₄₃	C₂²⋊C₄	1	2	yes	—
32	C₂³×C₄₃	C₄⋊C₄	1	2	yes	—
33	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₃	—	2	16	no	—
35	C₂×C₄×C₄₃	—	2	4	no	—
36	C₂×C₄×C₄₃	—	2	4	no	—
37	C₂×C₄×C₄₃	C₄²	1	2	yes	—
38	C₂×C₄×C₄₃	C₂²⋊C₄	1	8	yes	—
39	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
41	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₄₃	M₁₆	1	8	yes	complemented
43	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
47	C₄₃⋊C₈	—	2	8	no	—
48	C₂²×C₄₃	—	2	4	no	—
49	C₂²×C₄₃	—	2	4	no	—
50	C₂²×C₄₃	—	2	4	no	—
51	C₂²×C₄₃	#95	1	4	yes	—
52	C₂²×C₄₃	C₂×C₄²	1	1	yes	characteristic
53	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄₃	C₂×M₁₆	1	4	yes	—
56	C₄×C₄₃	—	2	8	no	—
57	C₄×C₄₃	#95	1	4	yes	—
58	C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
59	C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
60	#1728	—	43	1	no	maximal, Sylow
61	C₂×C₄₃	—	2	4	no	—
62	C₂×C₄₃	C₂×#95	1	1	yes	characteristic
63	C₂×C₄₃	C₄×M₁₆	1	2	yes	—
64	C₂×C₄₃	#407	1	2	yes	—
65	C₂×C₄₃	#423	1	2	yes	—
66	C₂²×M₁₆	—	43	2	no	—
67	C₂²×C₄²	D₄₃	1	1	yes	characteristic
68	#336	—	43	4	no	—
69	C₄₃	#1728	1	1	yes	characteristic, complemented, Sylow
70	C₂³×C₄	Dic₄₃	1	2	yes	—
71	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
72	C₂×M₁₆	—	43	8	no	—
73	C₂×M₁₆	—	86	4	no	—
74	C₂×M₁₆	—	86	4	no	—
75	C₂²×C₈	—	43	4	no	—
76	C₂×C₄²	—	2	4	no	—
77	C₂×C₄²	Dic₄₃	1	2	yes	—
78	C₂×C₄²	C₂×D₄₃	1	2	yes	—
79	C₂⁴	C₂×Dic₄₃	1	1	yes	characteristic
80	C₂²×C₄	—	2	4	no	—
81	C₂²×C₄	—	2	8	no	—
82	C₂²×C₄	Dic₈₆	1	2	yes	—
83	C₂²×C₄	C₄×D₄₃	1	4	yes	—
84	C₂²×C₄	D₁₇₂	1	2	yes	—
85	C₂²×C₄	C₂×Dic₄₃	1	4	yes	—
86	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₄₃	1	4	yes	—
88	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
89	M₁₆	—	86	16	no	—
90	C₂×C₈	—	43	8	no	—
91	C₂×C₈	—	86	4	no	—
92	C₂×C₈	—	86	4	no	—
93	C₄²	—	2	4	no	—
94	C₄²	—	2	4	no	—
95	C₂³	—	2	4	no	—
96	C₂³	C₄×Dic₄₃	1	2	yes	—
97	C₂³	C₄⋊Dic₄₃	1	2	yes	—
98	C₂³	C₂²⋊Dic₄₃	1	2	yes	—
99	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
100	C₂×C₄	—	2	16	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	#10621877	1	8	yes	complemented
104	C₂×C₄	C₄×Dic₄₃	1	2	yes	—
105	C₂×C₄	Dic₄₃⋊C₄	1	8	yes	—
106	C₂×C₄	C₄⋊Dic₄₃	1	2	yes	—
107	C₂×C₄	(C₂×D₄₃)⋊C₄	1	8	yes	—
108	C₂×C₄	C₂²⋊Dic₄₃	1	2	yes	—
109	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
111	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
113	C₈	—	86	8	no	—
114	C₂²	—	2	4	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	4	no	—
117	C₂²	C₄₃⋊#95	1	4	yes	—
118	C₂²	C₂×C₄×Dic₄₃	1	1	yes	characteristic
119	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
120	C₂²	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
121	C₂²	C₂×#10621877	1	4	yes	—
122	C₄	—	2	8	no	—
123	C₄	C₄₃⋊#95	1	4	yes	—
124	C₄	C₂×(C₂×D₄₃)⋊C₄	1	2	yes	—
125	C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
126	C₂	—	2	4	no	—
127	C₂	C₂×C₄₃⋊#95	1	1	yes	characteristic
128	C₂	C₄×#10621877	1	2	yes	—
129	C₂	#14582863	1	2	yes	—
130	C₂	#14582923	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10621877

C₂

yes

maximal

#14582654

C₂

yes

maximal, complemented

C₂²×C₄²×C₄₃

C₂

yes

maximal, characteristic

C₂×#10621877

—

C₂×#10621877

—

C₂×#10621877

C₄

yes

complemented

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂³×C₄×C₄₃

C₄

yes

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₄

yes

—

C₂×C₄²×C₄₃

C₂²

yes

—

C₂⁴×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

complemented

C₂³×C₄₃

—

C₂³×C₄₃

C₄²

yes

—

C₂³×C₄₃

C₂²⋊C₄

yes

—

C₂³×C₄₃

C₄⋊C₄

yes

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄²

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

M₁₆

yes

complemented

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#95

yes

—

C₂²×C₄₃

C₂×C₄²

yes

characteristic

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₃

C₂×M₁₆

yes

—

C₄×C₄₃

—

C₄×C₄₃

#95

yes

—

C₄×C₄₃

C₂×C₂²⋊C₄

yes

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

—

#1728

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂×#95

yes

characteristic

C₂×C₄₃

C₄×M₁₆

yes

—

C₂×C₄₃

#407

yes

—

C₂×C₄₃

#423

yes

—

C₂²×M₁₆

—

C₂²×C₄²

D₄₃

yes

characteristic

#336

—

C₄₃

#1728

yes

characteristic, complemented, Sylow

C₂³×C₄

Dic₄₃

yes

—

C₂³×C₄

C₂×D₄₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

Dic₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

—

C₂⁴

C₂×Dic₄₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₈₆

yes

—

C₂²×C₄

C₄×D₄₃

yes

—

C₂²×C₄

D₁₇₂

yes

—

C₂²×C₄

C₂×Dic₄₃

yes

—

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

—

C₂²×C₄

C₂²×D₄₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₂³

—

C₂³

C₄×Dic₄₃

yes

—

C₂³

C₄⋊Dic₄₃

yes

—

C₂³

C₂²⋊Dic₄₃

yes

—

C₂³

C₂²×Dic₄₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#10621877

yes

complemented

104

C₂×C₄

C₄×Dic₄₃

yes

—

105

C₂×C₄

Dic₄₃⋊C₄

yes

—

106

C₂×C₄

C₄⋊Dic₄₃

yes

—

107

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

108

C₂×C₄

C₂²⋊Dic₄₃

yes

—

109

C₂×C₄

C₂×Dic₈₆

yes

characteristic

110

C₂×C₄

C₂×C₄×D₄₃

yes

—

111

C₂×C₄

C₂×D₁₇₂

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₄₃⋊#95

yes

—

118

C₂²

C₂×C₄×Dic₄₃

yes

characteristic

119

C₂²

C₂×C₂²⋊Dic₄₃

yes

characteristic

120

C₂²

C₂×C₄⋊Dic₄₃

yes

characteristic

121

C₂²

C₂×#10621877

yes

—

122

C₄

—

123

C₄

C₄₃⋊#95

yes

—

124

C₄

C₂×(C₂×D₄₃)⋊C₄

yes

—

125

C₄

C₂×Dic₄₃⋊C₄

yes

—

126

C₂

—

127

C₂

C₂×C₄₃⋊#95

yes

characteristic

128

C₂

C₄×#10621877

yes

—

129

C₂

#14582863

yes

—

130

C₂

#14582923

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂×C₄²
Fitting	C₂²×C₄²×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×C₄
hypercenter	C₂²×C₄²	D₄₃
hypocenter	C₄₃	#1728
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₃

commutator

C₂²×C₄₃

C₂×C₄²

Fitting

C₂²×C₄²×C₄₃

C₂

socle

C₂³×C₄₃

C₂²×C₄

hypercenter

C₂²×C₄²

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. #14582654	C₂	semidirect
6. C₂×#10621877	C₄	semidirect
28. C₂×C₄₃⋊C₈	C₂×C₄	semidirect
42. C₂×C₄×C₄₃	M₁₆	semidirect
69. C₄₃	#1728	semidirect
103. C₂×C₄	#10621877	semidirect

H ≅ G/N

type

2. #14582654

C₂

semidirect

6. C₂×#10621877

C₄

semidirect

28. C₂×C₄₃⋊C₈

C₂×C₄

semidirect

42. C₂×C₄×C₄₃

M₁₆

semidirect

69. C₄₃

#1728

semidirect

103. C₂×C₄

#10621877

semidirect

Subgroups — #19134289

Conjugacy classes of subgroups

Complements