Group #19139624 — subgroups

Subgroups — #19139624

Order 5504.

556 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 197 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×#13063324	C₂	1	4	yes	maximal
2	#14582819	C₂	1	4	yes	maximal
3	#14582795	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₄₃	C₂	1	2	yes	maximal
5	C₄₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₈₆	C₂²	1	4	yes	—
7	#13063324	—	2	16	no	—
8	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	C₂×Dic₄₃⋊C₄	C₂²	1	8	yes	—
11	C₂×C₄×Dic₄₃	C₂²	1	4	yes	—
12	C₄₃⋊#95	C₂²	1	4	yes	—
13	C₄₃⋊#95	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
20	C₂²×Dic₄₃	C₂³	1	4	yes	—
21	C₂²×Dic₄₃	C₂³	1	4	yes	—
22	C₂×Dic₈₆	—	2	16	no	—
23	C₂×Dic₈₆	—	4	4	no	—
24	C₄⋊C₄×C₄₃	—	2	8	no	—
25	C₄⋊C₄×C₄₃	—	4	2	no	—
26	C₄²×C₄₃	Q₈	1	4	yes	complemented
27	C₄⋊Dic₄₃	D₄	1	8	yes	—
28	C₄⋊Dic₄₃	Q₈	1	8	yes	complemented
29	Dic₄₃⋊C₄	—	2	16	no	—
30	C₄×Dic₄₃	—	2	8	no	—
31	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₄₃	—	2	1	no	—
33	C₂×C₄×C₄₃	—	2	1	no	—
34	C₂×C₄×C₄₃	—	2	8	no	—
35	C₂×C₄×C₄₃	—	2	4	no	—
36	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
40	C₂×Dic₄₃	—	2	4	no	—
41	C₂×Dic₄₃	—	2	4	no	—
42	C₂×Dic₄₃	—	2	4	no	—
43	C₂×Dic₄₃	—	2	4	no	—
44	C₂×Dic₄₃	C₂×D₄	1	8	yes	—
45	C₂×Dic₄₃	C₂×Q₈	1	8	yes	—
46	Dic₈₆	—	4	16	no	—
47	C₂²×C₄₃	C₂²×D₄	1	2	yes	—
48	C₂²×C₄₃	C₂²×Q₈	1	2	yes	—
49	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
50	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
52	C₄×C₄₃	—	2	2	no	—
53	C₄×C₄₃	—	4	4	no	—
54	C₄×C₄₃	C₄⋊Q₈	1	8	yes	complemented
55	Dic₄₃	—	2	8	no	—
56	Dic₄₃	—	2	8	no	—
57	#2551	—	43	1	no	maximal, Sylow
58	C₂×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
59	C₂×C₄₃	#544	1	1	yes	characteristic
60	C₂×C₄₃	D₄²	1	1	yes	characteristic
61	C₂×C₄₃	D₄×Q₈	1	2	yes	—
62	C₂×C₄₃	Q₈²	1	1	yes	characteristic
63	C₂×C₄⋊Q₈	—	43	4	no	—
64	#395	—	43	4	no	—
65	#389	—	43	4	no	—
66	#384	D₄₃	1	1	yes	characteristic
67	C₄×C₄⋊C₄	—	43	2	no	—
68	C₄₃	#2551	1	1	yes	characteristic, complemented, Sylow
69	C₂²×Q₈	—	43	4	no	—
70	C₄⋊Q₈	—	86	16	no	—
71	C₂×C₄⋊C₄	—	43	2	no	—
72	C₂×C₄⋊C₄	—	43	2	no	—
73	C₂×C₄⋊C₄	—	43	8	no	—
74	C₂×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
75	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
76	C₂×C₄²	—	43	4	no	—
77	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
78	#95	—	43	4	no	—
79	#95	—	43	4	no	—
80	C₂×Q₈	—	86	16	no	—
81	C₂×Q₈	—	172	4	no	—
82	C₂²×C₄	—	43	4	no	—
83	C₂²×C₄	—	43	4	no	—
84	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
85	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	43	8	no	—
90	C₄⋊C₄	—	43	8	no	—
91	C₄⋊C₄	—	86	16	no	—
92	C₄²	—	86	8	no	—
93	C₄²	Dic₈₆	1	4	yes	complemented
94	C₂³	C₂³×D₄₃	1	1	yes	characteristic
95	Q₈	—	172	16	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	43	8	no	—
101	C₂×C₄	—	43	8	no	—
102	C₂×C₄	—	86	4	no	—
103	C₂×C₄	—	86	4	no	—
104	C₂×C₄	—	86	4	no	—
105	C₂×C₄	—	86	4	no	—
106	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
107	C₂×C₄	C₂×Dic₈₆	1	4	yes	—
108	C₂×C₄	D₄×D₄₃	1	4	yes	—
109	C₂×C₄	Q₈×D₄₃	1	4	yes	—
110	C₂²	C₂×D₄×D₄₃	1	2	yes	—
111	C₂²	C₂×Q₈×D₄₃	1	2	yes	—
112	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
113	C₂²	#13063370	1	1	yes	characteristic
114	C₂²	#13063373	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	4	4	no	—
117	C₄	—	86	8	no	—
118	C₄	—	86	8	no	—
119	C₄	#13063324	1	8	yes	complemented
120	C₂	C₂×#13063324	1	2	yes	—
121	C₂	#14583530	1	1	yes	characteristic
122	C₂	#14583534	1	1	yes	characteristic
123	C₂	#14583564	1	2	yes	—
124	C₂	#14583629	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063324

C₂

yes

maximal

#14582819

C₂

yes

maximal

#14582795

C₂

yes

maximal

C₄×C₄⋊Dic₄₃

C₂

yes

maximal

C₄₃×#384

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

—

#13063324

—

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

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

Q₈

yes

complemented

C₄⋊Dic₄₃

D₄

yes

—

C₄⋊Dic₄₃

Q₈

yes

complemented

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×Q₈

yes

—

Dic₈₆

—

C₂²×C₄₃

C₂²×D₄

yes

—

C₂²×C₄₃

C₂²×Q₈

yes

—

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₃

2₋¹⁺⁴

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄⋊Q₈

yes

complemented

Dic₄₃

—

Dic₄₃

—

#2551

—

maximal, Sylow

C₂×C₄₃

C₂×C₄⋊Q₈

yes

—

C₂×C₄₃

#544

yes

characteristic

C₂×C₄₃

D₄²

yes

characteristic

C₂×C₄₃

D₄×Q₈

yes

—

C₂×C₄₃

Q₈²

yes

characteristic

C₂×C₄⋊Q₈

—

#395

—

#389

—

#384

D₄₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄₃

#2551

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

172

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₃

yes

—

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

Dic₈₆

yes

complemented

C₂³

C₂³×D₄₃

yes

characteristic

Q₈

—

172

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₈₆

yes

—

107

C₂×C₄

C₂×Dic₈₆

yes

—

108

C₂×C₄

D₄×D₄₃

yes

—

109

C₂×C₄

Q₈×D₄₃

yes

—

110

C₂²

C₂×D₄×D₄₃

yes

—

111

C₂²

C₂×Q₈×D₄₃

yes

—

112

C₂²

C₂²×Dic₈₆

yes

characteristic

113

C₂²

#13063370

yes

characteristic

114

C₂²

#13063373

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#13063324

yes

complemented

120

C₂

C₂×#13063324

yes

—

121

C₂

#14583530

yes

characteristic

122

C₂

#14583534

yes

characteristic

123

C₂

#14583564

yes

—

124

C₂

#14583629

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄₃×#384	C₂
hypercenter	#384	D₄₃
hypocenter	C₄₃	#2551
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₄₃×#384

C₂

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₄₃	Q₈	semidirect
28. C₄⋊Dic₄₃	Q₈	semidirect
54. C₄×C₄₃	C₄⋊Q₈	semidirect
68. C₄₃	#2551	semidirect
93. C₄²	Dic₈₆	semidirect
119. C₄	#13063324	semidirect

H ≅ G/N

type

26. C₄²×C₄₃

Q₈

semidirect

28. C₄⋊Dic₄₃

Q₈

semidirect

54. C₄×C₄₃

C₄⋊Q₈

semidirect

68. C₄₃

#2551

semidirect

93. C₄²

Dic₈₆

semidirect

119. C₄

#13063324

semidirect

Subgroups — #19139624

Conjugacy classes of subgroups

Complements