Group #19148184 — subgroups

Subgroups — C₂×#14583635 #19148184

Order 5504.

1240 conjugacy classes of subgroups; only the 501 normal subgroups are shown.

normal subgroups only

This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 501 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	#14583635	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13063312	C₂	1	4	yes	maximal, complemented
3	C₂×#13063306	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Dic₈₆	C₂	1	2	yes	maximal, complemented
5	C₂×D₄×Dic₄₃	C₂	1	4	yes	maximal, complemented
6	C₂×#127×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₈₆	C₂²	1	2	yes	—
8	C₂³×Dic₄₃	C₂²	1	4	yes	—
9	#13063312	C₂²	1	32	yes	complemented
10	#13063306	C₂²	1	32	yes	complemented
11	C₄×Dic₈₆	C₂²	1	16	yes	complemented
12	D₄×Dic₄₃	C₂²	1	32	yes	complemented
13	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
14	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₄₃	C₂²	1	8	yes	—
16	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
17	C₂×C₄×Dic₄₃	C₂²	1	4	yes	—
18	C₂²×D₄×C₄₃	C₂²	1	4	yes	—
19	C₂²×D₄×C₄₃	C₂²	1	2	yes	—
20	#127×C₄₃	C₂²	1	8	yes	—
21	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄₃	C₂³	1	4	yes	—
23	C₂×D₄×C₄₃	—	2	16	no	—
24	C₂×D₄×C₄₃	—	4	4	no	—
25	C₂×D₄×C₄₃	C₂³	1	16	yes	—
26	C₂×D₄×C₄₃	C₂³	1	8	yes	—
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
29	C₂²×Dic₄₃	—	4	8	no	—
30	C₂²×Dic₄₃	C₂³	1	4	yes	—
31	C₂²×Dic₄₃	C₂³	1	4	yes	—
32	C₂²×Dic₄₃	C₂³	1	16	yes	—
33	C₂×Dic₈₆	—	2	8	no	—
34	C₂×Dic₈₆	C₂³	1	8	yes	—
35	C₄²×C₄₃	C₂³	1	4	yes	—
36	C₂²⋊Dic₄₃	C₂³	1	32	yes	—
37	C₂²⋊Dic₄₃	C₂³	1	16	yes	—
38	C₄⋊Dic₄₃	C₂³	1	8	yes	—
39	Dic₄₃⋊C₄	C₂³	1	16	yes	—
40	C₄×Dic₄₃	C₂³	1	16	yes	—
41	C₂³×C₄₃	—	2	16	no	—
42	C₂³×C₄₃	—	2	8	no	—
43	C₂³×C₄₃	C₂⁴	1	8	yes	—
44	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
45	D₄×C₄₃	—	2	32	no	—
46	D₄×C₄₃	—	4	8	no	—
47	C₂×C₄×C₄₃	—	2	2	no	—
48	C₂×C₄×C₄₃	#41	1	8	yes	—
49	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
50	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
51	C₂×Dic₄₃	—	2	8	no	—
52	C₂×Dic₄₃	—	2	8	no	—
53	C₂×Dic₄₃	—	4	16	no	—
54	C₂×Dic₄₃	C₂⁴	1	8	yes	—
55	C₂×Dic₄₃	C₂⁴	1	8	yes	—
56	Dic₈₆	—	2	16	no	—
57	C₂²×C₄₃	—	2	16	no	—
58	C₂²×C₄₃	—	2	8	no	—
59	C₂²×C₄₃	—	4	16	no	—
60	C₂²×C₄₃	C₂×#41	1	4	yes	—
61	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
62	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
63	C₄×C₄₃	—	2	2	no	—
64	C₄×C₄₃	C₂×#41	1	8	yes	—
65	Dic₄₃	—	2	8	no	—
66	Dic₄₃	—	2	8	no	—
67	C₂×#559	—	43	1	no	maximal, Sylow
68	C₂×C₄₃	—	4	8	no	—
69	C₂×C₄₃	#559	1	4	yes	complemented
70	C₂×C₄₃	C₂²×#41	1	2	yes	—
71	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
72	#559	—	43	16	no	—
73	C₂×#127	D₄₃	1	1	yes	characteristic
74	C₂×#124	—	43	4	no	—
75	C₂×#123	—	43	4	no	—
76	C₂×C₄×Q₈	—	43	2	no	—
77	C₂×C₄×D₄	—	43	4	no	—
78	C₄₃	C₂×#559	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₈	—	43	2	no	—
80	C₂²×D₄	C₂×D₄₃	1	4	yes	—
81	C₂²×D₄	C₂×D₄₃	1	2	yes	—
82	C₂³×C₄	—	43	4	no	—
83	#127	C₂×D₄₃	1	8	yes	—
84	#124	—	43	32	no	—
85	#123	—	43	32	no	—
86	C₄×Q₈	—	43	16	no	—
87	C₄×D₄	—	43	32	no	—
88	C₂×C₄⋊C₄	—	43	2	no	—
89	C₂×C₄⋊C₄	—	43	4	no	—
90	C₂×C₂²⋊C₄	—	43	8	no	—
91	C₂×C₂²⋊C₄	—	43	4	no	—
92	C₂×C₄²	—	43	4	no	—
93	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
94	C₂⁴	C₂²×D₄₃	1	4	yes	—
95	C₂×Q₈	—	43	8	no	—
96	C₂×Q₈	—	86	8	no	—
97	C₂×D₄	—	2	16	no	—
98	C₂×D₄	—	4	4	no	—
99	C₂×D₄	C₂²×D₄₃	1	16	yes	—
100	C₂×D₄	C₂²×D₄₃	1	8	yes	—
101	C₂²×C₄	—	43	4	no	—
102	C₂²×C₄	—	43	4	no	—
103	C₂²×C₄	—	43	16	no	—
104	C₂²×C₄	—	172	8	no	—
105	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
107	C₄⋊C₄	—	43	8	no	—
108	C₄⋊C₄	—	43	16	no	—
109	C₂²⋊C₄	—	43	32	no	—
110	C₂²⋊C₄	—	43	16	no	—
111	C₄²	—	43	16	no	—
112	C₄²	C₂²×D₄₃	1	4	yes	—
113	C₂³	—	2	16	no	—
114	C₂³	—	2	8	no	—
115	C₂³	C₂³×D₄₃	1	8	yes	—
116	C₂³	C₂³×D₄₃	1	1	yes	characteristic
117	Q₈	—	86	16	no	—
118	D₄	—	2	32	no	—
119	D₄	—	4	8	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	43	8	no	—
122	C₂×C₄	—	43	8	no	—
123	C₂×C₄	—	86	8	no	—
124	C₂×C₄	—	86	8	no	—
125	C₂×C₄	—	172	16	no	—
126	C₂×C₄	#10621900	1	8	yes	—
127	C₂×C₄	C₂³×D₄₃	1	2	yes	—
128	C₂×C₄	C₂³×D₄₃	1	4	yes	—
129	C₂²	—	2	16	no	—
130	C₂²	—	2	8	no	—
131	C₂²	—	4	16	no	—
132	C₂²	C₂×#10621900	1	4	yes	—
133	C₂²	#13063370	1	2	yes	—
134	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	86	8	no	—
137	C₄	—	86	8	no	—
138	C₄	C₂×#10621900	1	8	yes	—
139	C₂	—	4	8	no	—
140	C₂	#14583635	1	4	yes	direct product factor
141	C₂	C₂²×#10621900	1	2	yes	—
142	C₂	C₂×#13063370	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583635

C₂

yes

maximal, direct product factor

C₂×#13063312

C₂

yes

maximal, complemented

C₂×#13063306

C₂

yes

maximal, complemented

C₂×C₄×Dic₈₆

C₂

yes

maximal, complemented

C₂×D₄×Dic₄₃

C₂

yes

maximal, complemented

C₂×#127×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

—

C₂³×Dic₄₃

C₂²

yes

—

#13063312

C₂²

yes

complemented

#13063306

C₂²

yes

complemented

C₄×Dic₈₆

C₂²

yes

complemented

D₄×Dic₄₃

C₂²

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₂²×D₄×C₄₃

C₂²

yes

—

C₂²×D₄×C₄₃

C₂²

yes

—

#127×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂⁴×C₄₃

C₂³

yes

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂³

yes

—

C₂×D₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂³

yes

—

C₄²×C₄₃

C₂³

yes

—

C₂²⋊Dic₄₃

C₂³

yes

—

C₂²⋊Dic₄₃

C₂³

yes

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

C₂³

yes

—

C₄×Dic₄₃

C₂³

yes

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂⁴

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂⁴

yes

—

C₂×Dic₄₃

C₂⁴

yes

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×#41

yes

—

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂²×C₄₃

C₂⁵

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

C₂×#41

yes

—

Dic₄₃

—

Dic₄₃

—

C₂×#559

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

#559

yes

complemented

C₂×C₄₃

C₂²×#41

yes

—

C₂×C₄₃

C₂×H(5,𝔽₂)

yes

characteristic

#559

—

C₂×#127

D₄₃

yes

characteristic

C₂×#124

—

C₂×#123

—

C₂×C₄×Q₈

—

C₂×C₄×D₄

—

C₄₃

C₂×#559

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

C₂×D₄₃

yes

—

C₂²×D₄

C₂×D₄₃

yes

—

C₂³×C₄

—

#127

C₂×D₄₃

yes

—

#124

—

#123

—

C₄×Q₈

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₂⁴

C₂²×D₄₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₄₃

yes

—

100

C₂×D₄

C₂²×D₄₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

172

—

105

C₂²×C₄

C₂²×D₄₃

yes

characteristic

106

C₂²×C₄

C₂²×D₄₃

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

C₂²×D₄₃

yes

—

113

C₂³

—

114

C₂³

—

115

C₂³

C₂³×D₄₃

yes

—

116

C₂³

C₂³×D₄₃

yes

characteristic

117

Q₈

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

172

—

126

C₂×C₄

#10621900

yes

—

127

C₂×C₄

C₂³×D₄₃

yes

—

128

C₂×C₄

C₂³×D₄₃

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×#10621900

yes

—

133

C₂²

#13063370

yes

—

134

C₂²

C₂⁴×D₄₃

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

C₂×#10621900

yes

—

139

C₂

—

140

C₂

#14583635

yes

direct product factor

141

C₂

C₂²×#10621900

yes

—

142

C₂

C₂×#13063370

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₂×#127×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×#127	D₄₃
hypocenter	C₄₃	C₂×#559
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂⁵

Frattini

C₂²

C₂⁴×D₄₃

Fitting

C₂×#127×C₄₃

C₂

socle

C₂³×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. #14583635	C₂	direct (N × H)
2. C₂×#13063312	C₂	semidirect
3. C₂×#13063306	C₂	semidirect
4. C₂×C₄×Dic₈₆	C₂	semidirect
5. C₂×D₄×Dic₄₃	C₂	semidirect
9. #13063312	C₂²	semidirect
10. #13063306	C₂²	semidirect
11. C₄×Dic₈₆	C₂²	semidirect
12. D₄×Dic₄₃	C₂²	semidirect
69. C₂×C₄₃	#559	semidirect
78. C₄₃	C₂×#559	semidirect
140. C₂	#14583635	direct (N × H)

H ≅ G/N

type

1. #14583635

C₂

direct (N × H)

2. C₂×#13063312

C₂

semidirect

3. C₂×#13063306

C₂

semidirect

4. C₂×C₄×Dic₈₆

C₂

semidirect

5. C₂×D₄×Dic₄₃

C₂

semidirect

9. #13063312

C₂²

semidirect

10. #13063306

C₂²

semidirect

11. C₄×Dic₈₆

C₂²

semidirect

12. D₄×Dic₄₃

C₂²

semidirect

69. C₂×C₄₃

#559

semidirect

78. C₄₃

C₂×#559

semidirect

140. C₂

#14583635

direct (N × H)

Subgroups — C₂×#14583635 #19148184

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14583635 #19148184

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583635 #19148184