Group #14577749 — subgroups

Subgroups — C₂×#11927281 #14577749

Order 2432.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 469 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	#11927281	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10597322	C₂	1	2	yes	maximal, complemented
3	C₂×#10597381	C₂	1	3	yes	maximal, complemented
4	C₂×#10597385	C₂	1	6	yes	maximal, complemented
5	C₂×#10597383	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₁₉	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₁₉	C₂²	1	3	yes	—
9	C₂²×D₄×C₁₉	C₂²	1	3	yes	complemented
10	C₂²×C₂²⋊D₁₉	C₂²	1	6	yes	—
11	C₂³×Dic₁₉	C₂²	1	3	yes	complemented
12	C₁₉×#127	C₂²	1	8	yes	complemented
13	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊Dic₁₉	C₂²	1	3	yes	—
15	C₂×C₂²⋊Dic₁₉	C₂²	1	3	yes	complemented
16	#10597385	C₂²	1	48	yes	complemented
17	#10597383	C₂²	1	24	yes	complemented
18	#10597381	C₂²	1	24	yes	complemented
19	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	6	yes	—
20	C₂×Dic₁₉⋊C₄	C₂²	1	6	yes	complemented
21	#10597322	C₂²	1	16	yes	complemented
22	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
23	C₂⁴×C₁₉	C₂³	1	3	yes	—
24	C₂³×D₁₉	—	4	6	no	—
25	C₂³×D₁₉	C₂³	1	2	yes	—
26	C₂³×D₁₉	C₂³	1	4	yes	—
27	C₂×D₄×C₁₉	—	4	6	no	—
28	C₂×D₄×C₁₉	—	4	6	no	—
29	C₂×D₄×C₁₉	C₂³	1	12	yes	—
30	C₂×D₄×C₁₉	C₂³	1	12	yes	complemented
31	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
32	C₂×C₂²⋊D₁₉	—	4	12	no	—
33	C₂×C₂²⋊D₁₉	C₂³	1	24	yes	—
34	C₂²×Dic₁₉	—	4	6	no	—
35	C₂²×Dic₁₉	C₂³	1	6	yes	—
36	C₂²×Dic₁₉	C₂³	1	12	yes	complemented
37	C₄²×C₁₉	C₂³	1	4	yes	complemented
38	C₂²⋊Dic₁₉	C₂³	1	12	yes	—
39	C₂²⋊Dic₁₉	C₂³	1	12	yes	complemented
40	(C₂×D₁₉)⋊C₄	C₂³	1	24	yes	—
41	Dic₁₉⋊C₄	C₂³	1	24	yes	complemented
42	C₂³×C₁₉	—	2	6	no	—
43	C₂³×C₁₉	—	2	12	no	—
44	C₂³×C₁₉	—	2	6	no	—
45	C₂³×C₁₉	C₂⁴	1	2	yes	—
46	C₂³×C₁₉	C₂⁴	1	6	yes	—
47	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
48	C₂²×D₁₉	—	2	12	no	—
49	C₂²×D₁₉	—	4	12	no	—
50	C₂²×D₁₉	—	4	16	no	—
51	C₂²×D₁₉	C₂⁴	1	4	yes	—
52	D₄×C₁₉	—	4	12	no	—
53	D₄×C₁₉	—	4	12	no	—
54	C₂×C₄×C₁₉	—	2	6	no	—
55	C₂×C₄×C₁₉	C₂⁴	1	6	yes	—
56	C₂²⋊D₁₉	—	4	24	no	—
57	C₂×Dic₁₉	—	2	12	no	—
58	C₂×Dic₁₉	—	4	12	no	—
59	C₂×Dic₁₉	C₂⁴	1	12	yes	—
60	C₂×#560	—	19	1	no	maximal, Sylow
61	C₂²×C₁₉	—	2	6	no	—
62	C₂²×C₁₉	—	2	12	no	—
63	C₂²×C₁₉	—	2	6	no	—
64	C₂²×C₁₉	—	4	4	no	—
65	C₂²×C₁₉	—	4	12	no	—
66	C₂²×C₁₉	H(5,𝔽₂)	1	6	yes	—
67	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
68	C₂×D₁₉	—	2	12	no	—
69	C₂×D₁₉	—	4	16	no	—
70	C₂×D₁₉	—	4	8	no	—
71	C₄×C₁₉	—	2	6	no	—
72	Dic₁₉	—	2	12	no	—
73	#560	—	19	16	no	—
74	C₂×#127	D₁₉	1	1	yes	characteristic, complemented
75	C₂×#126	—	19	2	no	—
76	C₂×#123	—	19	3	no	—
77	C₂×#121	—	19	6	no	—
78	C₂×#120	—	19	3	no	—
79	C₂×C₁₉	—	4	2	no	—
80	C₂×C₁₉	—	4	6	no	—
81	C₂×C₁₉	#560	1	4	yes	complemented
82	C₂×C₁₉	C₂×H(5,𝔽₂)	1	3	yes	—
83	D₁₉	—	4	4	no	—
84	C₂⁵	—	19	1	no	—
85	C₂²×D₄	—	19	6	no	—
86	C₂²×D₄	C₂×D₁₉	1	3	yes	—
87	C₂²×D₄	C₂×D₁₉	1	3	yes	complemented
88	C₂³×C₄	—	19	3	no	—
89	#127	C₂×D₁₉	1	8	yes	complemented
90	#126	—	19	16	no	—
91	#123	—	19	24	no	—
92	#121	—	19	48	no	—
93	#120	—	19	24	no	—
94	C₂×C₄⋊C₄	—	19	6	no	—
95	C₂×C₂²⋊C₄	—	19	3	no	—
96	C₂×C₂²⋊C₄	—	19	3	no	—
97	C₂×C₂²⋊C₄	—	19	6	no	—
98	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic, complemented
99	C₁₉	C₂×#560	1	1	yes	characteristic, complemented, Sylow
100	C₂⁴	—	19	2	no	—
101	C₂⁴	—	19	4	no	—
102	C₂⁴	—	76	6	no	—
103	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
104	C₂⁴	C₂²×D₁₉	1	3	yes	—
105	C₂×D₄	—	4	6	no	—
106	C₂×D₄	—	4	6	no	—
107	C₂×D₄	—	19	24	no	—
108	C₂×D₄	—	76	12	no	—
109	C₂×D₄	C₂²×D₁₉	1	12	yes	—
110	C₂×D₄	C₂²×D₁₉	1	12	yes	complemented
111	C₂²×C₄	—	19	6	no	—
112	C₂²×C₄	—	19	12	no	—
113	C₂²×C₄	—	76	6	no	—
114	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
115	C₄⋊C₄	—	19	24	no	—
116	C₂²⋊C₄	—	19	12	no	—
117	C₂²⋊C₄	—	19	12	no	—
118	C₂²⋊C₄	—	19	24	no	—
119	C₄²	C₂²×D₁₉	1	4	yes	complemented
120	C₂³	—	2	6	no	—
121	C₂³	—	2	12	no	—
122	C₂³	—	2	6	no	—
123	C₂³	—	19	4	no	—
124	C₂³	—	38	12	no	—
125	C₂³	—	76	12	no	—
126	C₂³	—	76	16	no	—
127	C₂³	C₂³×D₁₉	1	2	yes	—
128	C₂³	C₂³×D₁₉	1	6	yes	—
129	C₂³	C₂³×D₁₉	1	1	yes	characteristic
130	D₄	—	4	12	no	—
131	D₄	—	4	12	no	—
132	D₄	—	76	24	no	—
133	C₂×C₄	—	2	6	no	—
134	C₂×C₄	—	19	12	no	—
135	C₂×C₄	—	38	12	no	—
136	C₂×C₄	—	76	12	no	—
137	C₂×C₄	C₂³×D₁₉	1	6	yes	—
138	C₂²	—	2	6	no	—
139	C₂²	—	2	12	no	—
140	C₂²	—	2	6	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	12	no	—
143	C₂²	—	38	12	no	—
144	C₂²	—	76	16	no	—
145	C₂²	—	76	8	no	—
146	C₂²	#10597430	1	6	yes	—
147	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	—	38	12	no	—
150	C₂	—	4	2	no	—
151	C₂	—	4	6	no	—
152	C₂	—	76	4	no	—
153	C₂	#11927281	1	4	yes	direct product factor
154	C₂	C₂×#10597430	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11927281

C₂

yes

maximal, direct product factor

C₂×#10597322

C₂

yes

maximal, complemented

C₂×#10597381

C₂

yes

maximal, complemented

C₂×#10597385

C₂

yes

maximal, complemented

C₂×#10597383

C₂

yes

maximal, complemented

C₂×C₁₉×#127

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₁₉

C₂²

yes

characteristic

C₂²×D₄×C₁₉

C₂²

yes

—

C₂²×D₄×C₁₉

C₂²

yes

complemented

C₂²×C₂²⋊D₁₉

C₂²

yes

—

C₂³×Dic₁₉

C₂²

yes

complemented

C₁₉×#127

C₂²

yes

complemented

C₂×C₄²×C₁₉

C₂²

yes

characteristic, complemented

C₂×C₂²⋊Dic₁₉

C₂²

yes

—

C₂×C₂²⋊Dic₁₉

C₂²

yes

complemented

#10597385

C₂²

yes

complemented

#10597383

C₂²

yes

complemented

#10597381

C₂²

yes

complemented

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×Dic₁₉⋊C₄

C₂²

yes

complemented

#10597322

C₂²

yes

complemented

C₂⁴×C₁₉

C₂³

yes

characteristic

C₂⁴×C₁₉

C₂³

yes

—

C₂³×D₁₉

—

C₂³×D₁₉

C₂³

yes

—

C₂³×D₁₉

C₂³

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₂³

yes

—

C₂×D₄×C₁₉

C₂³

yes

complemented

C₂²×C₄×C₁₉

C₂³

yes

—

C₂×C₂²⋊D₁₉

—

C₂×C₂²⋊D₁₉

C₂³

yes

—

C₂²×Dic₁₉

—

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

complemented

C₄²×C₁₉

C₂³

yes

complemented

C₂²⋊Dic₁₉

C₂³

yes

—

C₂²⋊Dic₁₉

C₂³

yes

complemented

(C₂×D₁₉)⋊C₄

C₂³

yes

—

Dic₁₉⋊C₄

C₂³

yes

complemented

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂⁴

yes

—

C₂³×C₁₉

C₂⁴

yes

—

C₂³×C₁₉

C₂⁴

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

C₂⁴

yes

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂⁴

yes

—

C₂²⋊D₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

C₂⁴

yes

—

C₂×#560

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

H(5,𝔽₂)

yes

—

C₂²×C₁₉

C₂⁵

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

Dic₁₉

—

#560

—

C₂×#127

D₁₉

yes

characteristic, complemented

—

—

—

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#560

yes

complemented

C₂×C₁₉

C₂×H(5,𝔽₂)

yes

—

D₁₉

—

C₂⁵

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₉

yes

—

C₂²×D₄

C₂×D₁₉

yes

complemented

C₂³×C₄

—

#127

C₂×D₁₉

yes

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₉

yes

characteristic, complemented

C₁₉

C₂×#560

yes

characteristic, complemented, Sylow

100

C₂⁴

—

101

C₂⁴

—

102

C₂⁴

—

103

C₂⁴

C₂²×D₁₉

yes

characteristic

104

C₂⁴

C₂²×D₁₉

yes

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₁₉

yes

—

110

C₂×D₄

C₂²×D₁₉

yes

complemented

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²×D₁₉

yes

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

C₂²×D₁₉

yes

complemented

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂³×D₁₉

yes

—

128

C₂³

C₂³×D₁₉

yes

—

129

C₂³

C₂³×D₁₉

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂³×D₁₉

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#10597430

yes

—

147

C₂²

C₂⁴×D₁₉

yes

characteristic

148

C₄

—

149

C₄

—

150

C₂

—

151

C₂

—

152

C₂

—

153

C₂

#11927281

yes

direct product factor

154

C₂

C₂×#10597430

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₁₉
commutator, cosocle	C₂²×C₁₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₁₉×#127	C₂
socle	C₂³×C₁₉	C₂⁴
hypercenter	C₂×#127	D₁₉
hypocenter	C₁₉	C₂×#560
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂³×D₁₉

commutator, cosocle

C₂²×C₁₉

C₂⁵

Frattini

C₂²

C₂⁴×D₁₉

Fitting

C₂×C₁₉×#127

C₂

socle

C₂³×C₁₉

C₂⁴

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. #11927281	C₂	direct (N × H)
2. C₂×#10597322	C₂	semidirect
3. C₂×#10597381	C₂	semidirect
4. C₂×#10597385	C₂	semidirect
5. C₂×#10597383	C₂	semidirect
6. C₂×C₁₉×#127	C₂	semidirect
9. C₂²×D₄×C₁₉	C₂²	semidirect
11. C₂³×Dic₁₉	C₂²	semidirect
12. C₁₉×#127	C₂²	semidirect
13. C₂×C₄²×C₁₉	C₂²	semidirect
15. C₂×C₂²⋊Dic₁₉	C₂²	semidirect
16. #10597385	C₂²	semidirect
17. #10597383	C₂²	semidirect
18. #10597381	C₂²	semidirect
20. C₂×Dic₁₉⋊C₄	C₂²	semidirect
21. #10597322	C₂²	semidirect
30. C₂×D₄×C₁₉	C₂³	semidirect
36. C₂²×Dic₁₉	C₂³	semidirect
37. C₄²×C₁₉	C₂³	semidirect
39. C₂²⋊Dic₁₉	C₂³	semidirect
41. Dic₁₉⋊C₄	C₂³	semidirect
74. C₂×#127	D₁₉	semidirect
81. C₂×C₁₉	#560	semidirect
87. C₂²×D₄	C₂×D₁₉	semidirect
89. #127	C₂×D₁₉	semidirect
98. C₂×C₄²	C₂×D₁₉	semidirect
99. C₁₉	C₂×#560	semidirect
110. C₂×D₄	C₂²×D₁₉	semidirect
119. C₄²	C₂²×D₁₉	semidirect
153. C₂	#11927281	direct (N × H)

H ≅ G/N

type

1. #11927281

C₂

direct (N × H)

2. C₂×#10597322

C₂

semidirect

3. C₂×#10597381

C₂

semidirect

4. C₂×#10597385

C₂

semidirect

5. C₂×#10597383

C₂

semidirect

6. C₂×C₁₉×#127

C₂

semidirect

9. C₂²×D₄×C₁₉

C₂²

semidirect

11. C₂³×Dic₁₉

C₂²

semidirect

12. C₁₉×#127

C₂²

semidirect

13. C₂×C₄²×C₁₉

C₂²

semidirect

15. C₂×C₂²⋊Dic₁₉

C₂²

semidirect

16. #10597385

C₂²

semidirect

17. #10597383

C₂²

semidirect

18. #10597381

C₂²

semidirect

20. C₂×Dic₁₉⋊C₄

C₂²

semidirect

21. #10597322

C₂²

semidirect

30. C₂×D₄×C₁₉

C₂³

semidirect

36. C₂²×Dic₁₉

C₂³

semidirect

37. C₄²×C₁₉

C₂³

semidirect

39. C₂²⋊Dic₁₉

C₂³

semidirect

41. Dic₁₉⋊C₄

C₂³

semidirect

74. C₂×#127

D₁₉

semidirect

81. C₂×C₁₉

#560

semidirect

87. C₂²×D₄

C₂×D₁₉

semidirect

89. #127

C₂×D₁₉

semidirect

98. C₂×C₄²

C₂×D₁₉

semidirect

99. C₁₉

C₂×#560

semidirect

110. C₂×D₄

C₂²×D₁₉

semidirect

119. C₄²

C₂²×D₁₉

semidirect

153. C₂

#11927281

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11927281	2
C₂×#10597322	2
C₂×#10597381	2
C₂×#10597385	2
C₂×#10597383	2
C₂×C₁₉×#127	2
C₂×#560	19

subgroup

index

C₂×C₁₉×#127

Subgroups — C₂×#11927281 #14577749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#11927281 #14577749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11927281 #14577749