Group #19137511 — subgroups

Subgroups — Q₈×C₄⋊Dic₄₃ #19137511

Order 5504.

568 conjugacy classes of subgroups; only the 333 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 333 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×Q₈×Dic₄₃	C₂	1	2	yes	maximal
2	#14582798	C₂	1	6	yes	maximal
3	#14582769	C₂	1	3	yes	maximal
4	C₄×C₄⋊Dic₄₃	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₄₃	—	2	8	no	—
7	C₂×C₄⋊Dic₄₃	C₂²	1	6	yes	—
8	C₂×C₄⋊Dic₄₃	C₂²	1	3	yes	—
9	C₂×C₄⋊Dic₄₃	C₂²	1	6	yes	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₄₃	C₂²	1	6	yes	—
12	C₄₃⋊#95	C₂²	1	6	yes	—
13	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₄₃	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₄₃	C₂²	1	3	yes	—
16	C₂×C₄²×C₄₃	C₂²	1	3	yes	—
17	C₂×Q₈×C₄₃	C₂×C₄	1	4	yes	—
18	C₂×Q₈×C₄₃	D₄	1	4	yes	—
19	C₂×Q₈×C₄₃	Q₈	1	4	yes	—
20	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
21	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
22	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₄₃	C₂³	1	6	yes	—
24	C₂²×Dic₄₃	C₂³	1	2	yes	—
25	C₄⋊C₄×C₄₃	C₂×C₄	1	12	yes	—
26	C₄²×C₄₃	C₂×C₄	1	12	yes	—
27	C₄⋊Dic₄₃	—	2	12	no	—
28	C₄⋊Dic₄₃	—	2	6	no	—
29	C₄⋊Dic₄₃	—	4	6	no	—
30	C₄⋊Dic₄₃	Q₈	1	4	yes	direct product factor
31	C₄×Dic₄₃	—	2	12	no	—
32	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
33	Q₈×C₄₃	C₄⋊C₄	1	16	yes	complemented
34	C₂×C₄×C₄₃	—	2	3	no	—
35	C₂×C₄×C₄₃	—	2	3	no	—
36	C₂×C₄×C₄₃	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₄₃	C₂²×C₄	1	6	yes	—
38	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₄₃	C₂×D₄	1	6	yes	—
40	C₂×C₄×C₄₃	C₂×Q₈	1	6	yes	—
41	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₄₃	#41	1	2	yes	—
43	C₂×Dic₄₃	—	2	2	no	—
44	C₂×Dic₄₃	—	2	2	no	—
45	C₂×Dic₄₃	—	2	6	no	—
46	C₂×Dic₄₃	—	2	6	no	—
47	C₂×Dic₄₃	—	2	6	no	—
48	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
49	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
50	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
54	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
56	C₄×C₄₃	—	2	6	no	—
57	C₄×C₄₃	C₂×C₄⋊C₄	1	12	yes	—
58	C₄×C₄₃	C₄×Q₈	1	4	yes	complemented
59	Dic₄₃	—	2	4	no	—
60	Dic₄₃	—	4	6	no	—
61	Q₈×C₄⋊C₄	—	43	1	no	maximal, Sylow
62	C₂×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
64	C₂×C₄₃	#520	1	1	yes	characteristic
65	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
66	C₂×C₄₃	#550	1	1	yes	characteristic
67	C₂×C₄₃	#557	1	1	yes	characteristic
68	C₂×C₄₃	Q₈²	1	1	yes	characteristic
69	C₂×C₄×Q₈	—	43	2	no	—
70	C₂×C₄×Q₈	D₄₃	1	1	yes	characteristic
71	#389	—	43	6	no	—
72	#384	—	43	3	no	—
73	C₄×C₄⋊C₄	—	43	3	no	—
74	C₄₃	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
76	C₄×Q₈	—	86	8	no	—
77	C₄×Q₈	Dic₄₃	1	8	yes	complemented
78	C₂×C₄⋊C₄	—	43	1	no	—
79	C₂×C₄⋊C₄	—	43	6	no	—
80	C₂×C₄⋊C₄	—	43	3	no	—
81	C₂×C₄⋊C₄	—	43	6	no	—
82	C₂×C₄⋊C₄	C₂×D₄₃	1	3	yes	—
83	C₂×C₄²	—	43	6	no	—
84	C₂×C₄²	C₂×D₄₃	1	3	yes	—
85	#95	—	43	6	no	—
86	C₂×Q₈	Dic₈₆	1	4	yes	—
87	C₂×Q₈	D₁₇₂	1	4	yes	—
88	C₂×Q₈	C₂×Dic₄₃	1	4	yes	—
89	C₂²×C₄	—	43	2	no	—
90	C₂²×C₄	—	43	6	no	—
91	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
92	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
93	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
94	C₄⋊C₄	—	43	4	no	—
95	C₄⋊C₄	—	86	12	no	—
96	C₄⋊C₄	—	86	6	no	—
97	C₄⋊C₄	—	172	6	no	—
98	C₄⋊C₄	C₂×Dic₄₃	1	12	yes	—
99	C₄²	—	86	12	no	—
100	C₄²	C₂×Dic₄₃	1	12	yes	—
101	C₂³	C₂³×D₄₃	1	1	yes	characteristic
102	Q₈	C₄⋊Dic₄₃	1	16	yes	direct product factor
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	43	4	no	—
106	C₂×C₄	—	86	2	no	—
107	C₂×C₄	—	86	2	no	—
108	C₂×C₄	—	86	6	no	—
109	C₂×C₄	—	86	6	no	—
110	C₂×C₄	—	86	6	no	—
111	C₂×C₄	C₂×Dic₈₆	1	6	yes	—
112	C₂×C₄	C₂×D₁₇₂	1	6	yes	—
113	C₂×C₄	Q₈×D₄₃	1	2	yes	—
114	C₂×C₄	#10621902	1	2	yes	—
115	C₂×C₄	C₂²×Dic₄₃	1	6	yes	—
116	C₂×C₄	C₂²×Dic₄₃	1	6	yes	—
117	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
118	C₂²	C₂³×Dic₄₃	1	1	yes	characteristic
119	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
120	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
121	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
122	C₂²	C₂×#10621902	1	1	yes	characteristic
123	C₂²	#13063371	1	1	yes	characteristic
124	C₂²	#13063372	1	1	yes	characteristic
125	C₄	—	2	6	no	—
126	C₄	—	86	4	no	—
127	C₄	—	172	6	no	—
128	C₄	C₂×C₄⋊Dic₄₃	1	12	yes	—
129	C₄	Q₈×Dic₄₃	1	4	yes	complemented
130	C₂	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
131	C₂	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
132	C₂	#14583382	1	1	yes	characteristic
133	C₂	Q₈×D₁₇₂	1	1	yes	characteristic
134	C₂	#14583571	1	1	yes	characteristic
135	C₂	#14583628	1	1	yes	characteristic
136	C₂	Q₈×Dic₈₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₄₃

C₂

yes

maximal

#14582798

C₂

yes

maximal

#14582769

C₂

yes

maximal

C₄×C₄⋊Dic₄₃

C₂

yes

maximal

C₂×C₄×Q₈×C₄₃

C₂

yes

maximal, characteristic

Q₈×Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

characteristic

C₄×Q₈×C₄₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

—

C₂×Q₈×C₄₃

C₂×C₄

yes

—

C₂×Q₈×C₄₃

D₄

yes

—

C₂×Q₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

Q₈

yes

direct product factor

C₄×Dic₄₃

—

C₂³×C₄₃

C₂⁴

yes

characteristic

Q₈×C₄₃

C₄⋊C₄

yes

complemented

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×Q₈

yes

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₃

2₋¹⁺⁴

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

—

C₄×C₄₃

C₄×Q₈

yes

complemented

Dic₄₃

—

Dic₄₃

—

Q₈×C₄⋊C₄

—

maximal, Sylow

C₂×C₄₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₄₃

#520

yes

characteristic

C₂×C₄₃

D₄×Q₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

Q₈²

yes

characteristic

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₄₃

yes

characteristic

#389

—

#384

—

C₄×C₄⋊C₄

—

C₄₃

Q₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

Dic₄₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

—

#95

—

C₂×Q₈

Dic₈₆

yes

—

C₂×Q₈

D₁₇₂

yes

—

C₂×Q₈

C₂×Dic₄₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₃

yes

—

C₂²×C₄

C₂²×D₄₃

yes

—

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

172

—

C₄⋊C₄

C₂×Dic₄₃

yes

—

C₄²

—

100

C₄²

C₂×Dic₄₃

yes

—

101

C₂³

C₂³×D₄₃

yes

characteristic

102

Q₈

C₄⋊Dic₄₃

yes

direct product factor

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×Dic₈₆

yes

—

112

C₂×C₄

C₂×D₁₇₂

yes

—

113

C₂×C₄

Q₈×D₄₃

yes

—

114

C₂×C₄

#10621902

yes

—

115

C₂×C₄

C₂²×Dic₄₃

yes

—

116

C₂×C₄

C₂²×Dic₄₃

yes

—

117

C₂×C₄

C₂²×Dic₄₃

yes

—

118

C₂²

C₂³×Dic₄₃

yes

characteristic

119

C₂²

C₂²×D₁₇₂

yes

characteristic

120

C₂²

C₂×Q₈×D₄₃

yes

characteristic

121

C₂²

C₂²×Dic₈₆

yes

characteristic

122

C₂²

C₂×#10621902

yes

characteristic

123

C₂²

#13063371

yes

characteristic

124

C₂²

#13063372

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

172

—

128

C₄

C₂×C₄⋊Dic₄₃

yes

—

129

C₄

Q₈×Dic₄₃

yes

complemented

130

C₂

C₂²×C₄⋊Dic₄₃

yes

characteristic

131

C₂

C₂×Q₈×Dic₄₃

yes

characteristic

132

C₂

#14583382

yes

characteristic

133

C₂

Q₈×D₁₇₂

yes

characteristic

134

C₂

#14583571

yes

characteristic

135

C₂

#14583628

yes

characteristic

136

C₂

Q₈×Dic₈₆

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₄₃
hypocenter	C₄₃	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂⁴

hypercenter

C₂×C₄×Q₈

D₄₃

hypocenter

C₄₃

Q₈×C₄⋊C₄

LCS

G ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
14. C₄×Q₈×C₄₃	C₄	semidirect
30. C₄⋊Dic₄₃	Q₈	direct (N × H)
33. Q₈×C₄₃	C₄⋊C₄	semidirect
58. C₄×C₄₃	C₄×Q₈	semidirect
74. C₄₃	Q₈×C₄⋊C₄	semidirect
77. C₄×Q₈	Dic₄₃	semidirect
102. Q₈	C₄⋊Dic₄₃	direct (N × H)
129. C₄	Q₈×Dic₄₃	semidirect

H ≅ G/N

type

14. C₄×Q₈×C₄₃

C₄

semidirect

30. C₄⋊Dic₄₃

Q₈

direct (N × H)

33. Q₈×C₄₃

C₄⋊C₄

semidirect

58. C₄×C₄₃

C₄×Q₈

semidirect

74. C₄₃

Q₈×C₄⋊C₄

semidirect

77. C₄×Q₈

Dic₄₃

semidirect

102. Q₈

C₄⋊Dic₄₃

direct (N × H)

129. C₄

Q₈×Dic₄₃

semidirect

Subgroups — Q₈×C₄⋊Dic₄₃ #19137511

Conjugacy classes of subgroups

Complements

Subgroups — Q8×C4⋊Dic43 #19137511

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×C₄⋊Dic₄₃ #19137511