Group #14577176 — subgroups

Subgroups — C₂×#11927260 #14577176

Order 2432.

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	#11927260	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10597393	C₂	1	4	yes	maximal, complemented
3	C₂×#10597341	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₇₆	C₂	1	2	yes	maximal, complemented
5	C₂×#10597338	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
8	C₂²×D₇₆	C₂²	1	2	yes	—
9	C₂²×D₇₆	C₂²	1	4	yes	—
10	C₂²×C₄×D₁₉	C₂²	1	4	yes	—
11	C₁₉×C₄⋊Q₈	C₂²	1	8	yes	complemented
12	C₂×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
13	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	#10597393	C₂²	1	32	yes	complemented
15	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
16	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	8	yes	—
17	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₉	C₂²	1	4	yes	—
19	#10597341	C₂²	1	32	yes	complemented
20	#10597338	C₂²	1	32	yes	complemented
21	C₄×D₇₆	C₂²	1	16	yes	complemented
22	C₂³×D₁₉	C₂³	1	4	yes	—
23	C₂×Q₈×C₁₉	—	4	4	no	—
24	C₂×Q₈×C₁₉	C₂³	1	8	yes	—
25	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
27	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
28	C₂²×Dic₁₉	C₂³	1	4	yes	—
29	C₂×D₇₆	—	2	8	no	—
30	C₂×D₇₆	—	4	8	no	—
31	C₂×D₇₆	C₂³	1	8	yes	—
32	C₂×D₇₆	C₂³	1	16	yes	—
33	C₂×C₄×D₁₉	—	2	16	no	—
34	C₂×C₄×D₁₉	C₂³	1	16	yes	—
35	C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
36	C₄²×C₁₉	C₂³	1	4	yes	—
37	(C₂×D₁₉)⋊C₄	C₂³	1	16	yes	—
38	(C₂×D₁₉)⋊C₄	C₂³	1	32	yes	—
39	C₄⋊Dic₁₉	C₂³	1	8	yes	—
40	C₄×Dic₁₉	C₂³	1	16	yes	—
41	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
42	C₂²×D₁₉	—	2	8	no	—
43	C₂²×D₁₉	—	2	8	no	—
44	C₂²×D₁₉	—	2	8	no	—
45	C₂²×D₁₉	C₂⁴	1	8	yes	—
46	Q₈×C₁₉	—	4	8	no	—
47	C₂×C₄×C₁₉	—	2	8	no	—
48	C₂×C₄×C₁₉	—	2	2	no	—
49	C₂×C₄×C₁₉	#41	1	8	yes	—
50	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
51	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
52	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
53	C₂×Dic₁₉	—	2	8	no	—
54	C₂×Dic₁₉	C₂⁴	1	8	yes	—
55	D₇₆	—	2	16	no	—
56	D₇₆	—	4	16	no	—
57	C₄×D₁₉	—	2	32	no	—
58	C₂×#555	—	19	1	no	maximal, Sylow
59	C₂²×C₁₉	C₂×#41	1	4	yes	—
60	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
61	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
62	C₂×D₁₉	—	2	8	no	—
63	C₂×D₁₉	—	2	8	no	—
64	C₂×D₁₉	—	2	8	no	—
65	C₂×D₁₉	—	4	16	no	—
66	C₄×C₁₉	—	2	8	no	—
67	C₄×C₁₉	—	2	2	no	—
68	C₄×C₁₉	C₂×#41	1	8	yes	—
69	Dic₁₉	—	2	8	no	—
70	#555	—	19	16	no	—
71	C₂×C₄⋊Q₈	D₁₉	1	1	yes	characteristic, complemented
72	C₂×#124	—	19	4	no	—
73	C₂×#121	—	19	4	no	—
74	C₂×C₄×D₄	—	19	2	no	—
75	C₂×#117	—	19	4	no	—
76	C₂×C₁₉	#555	1	4	yes	complemented
77	C₂×C₁₉	C₂²×#41	1	2	yes	—
78	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
79	D₁₉	—	4	8	no	—
80	C₂²×Q₈	C₂×D₁₉	1	2	yes	—
81	C₂²×D₄	—	19	2	no	—
82	C₂²×D₄	—	19	4	no	—
83	C₂³×C₄	—	19	4	no	—
84	C₄⋊Q₈	C₂×D₁₉	1	8	yes	complemented
85	#124	—	19	32	no	—
86	#121	—	19	32	no	—
87	C₄×D₄	—	19	16	no	—
88	#117	—	19	32	no	—
89	C₂×C₄⋊C₄	—	19	2	no	—
90	C₂×C₄⋊C₄	C₂×D₁₉	1	4	yes	—
91	C₂×C₂²⋊C₄	—	19	8	no	—
92	C₂×C₂²⋊C₄	—	19	4	no	—
93	C₂×C₄²	—	19	4	no	—
94	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
95	C₁₉	C₂×#555	1	1	yes	characteristic, complemented, Sylow
96	C₂⁴	—	19	4	no	—
97	C₂×Q₈	—	4	4	no	—
98	C₂×Q₈	C₂²×D₁₉	1	8	yes	—
99	C₂×D₄	—	19	8	no	—
100	C₂×D₄	—	19	16	no	—
101	C₂×D₄	—	38	8	no	—
102	C₂×D₄	—	76	8	no	—
103	C₂²×C₄	—	19	4	no	—
104	C₂²×C₄	—	19	16	no	—
105	C₂²×C₄	—	38	16	no	—
106	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
107	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
109	C₄⋊C₄	—	19	8	no	—
110	C₄⋊C₄	C₂²×D₁₉	1	16	yes	—
111	C₂²⋊C₄	—	19	32	no	—
112	C₂²⋊C₄	—	19	16	no	—
113	C₄²	—	19	16	no	—
114	C₄²	C₂²×D₁₉	1	4	yes	—
115	C₂³	—	19	8	no	—
116	C₂³	—	38	8	no	—
117	C₂³	—	38	8	no	—
118	C₂³	—	38	8	no	—
119	C₂³	C₂³×D₁₉	1	1	yes	characteristic
120	Q₈	—	4	8	no	—
121	D₄	—	38	16	no	—
122	D₄	—	76	16	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	19	8	no	—
126	C₂×C₄	—	38	8	no	—
127	C₂×C₄	—	38	32	no	—
128	C₂×C₄	#64545	1	8	yes	—
129	C₂×C₄	C₂³×D₁₉	1	8	yes	—
130	C₂×C₄	C₂³×D₁₉	1	2	yes	—
131	C₂×C₄	C₂³×D₁₉	1	4	yes	—
132	C₂²	—	38	8	no	—
133	C₂²	—	38	8	no	—
134	C₂²	—	38	8	no	—
135	C₂²	—	76	16	no	—
136	C₂²	#10597430	1	2	yes	—
137	C₂²	C₂×#64545	1	4	yes	—
138	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
139	C₄	—	2	8	no	—
140	C₄	—	2	2	no	—
141	C₄	—	38	8	no	—
142	C₄	C₂×#64545	1	8	yes	—
143	C₂	—	76	8	no	—
144	C₂	#11927260	1	4	yes	direct product factor
145	C₂	C₂²×#64545	1	2	yes	—
146	C₂	C₂×#10597430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11927260

C₂

yes

maximal, direct product factor

C₂×#10597393

C₂

yes

maximal, complemented

C₂×#10597341

C₂

yes

maximal, complemented

C₂×C₄×D₇₆

C₂

yes

maximal, complemented

C₂×#10597338

C₂

yes

maximal, complemented

C₂×C₁₉×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₉

C₂²

yes

—

C₂²×D₇₆

C₂²

yes

—

C₂²×D₇₆

C₂²

yes

—

C₂²×C₄×D₁₉

C₂²

yes

—

C₁₉×C₄⋊Q₈

C₂²

yes

complemented

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

#10597393

C₂²

yes

complemented

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×C₄×Dic₁₉

C₂²

yes

—

#10597341

C₂²

yes

complemented

#10597338

C₂²

yes

complemented

C₄×D₇₆

C₂²

yes

complemented

C₂³×D₁₉

C₂³

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

C₂³

yes

—

C₂×D₇₆

C₂³

yes

—

C₂×C₄×D₁₉

—

C₂×C₄×D₁₉

C₂³

yes

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₄²×C₁₉

C₂³

yes

—

(C₂×D₁₉)⋊C₄

C₂³

yes

—

(C₂×D₁₉)⋊C₄

C₂³

yes

—

C₄⋊Dic₁₉

C₂³

yes

—

C₄×Dic₁₉

C₂³

yes

—

C₂³×C₁₉

C₂⁴

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

C₂⁴

yes

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#41

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

C₂⁴

yes

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

C₂×#555

—

maximal, Sylow

C₂²×C₁₉

C₂×#41

yes

—

C₂²×C₁₉

H(5,𝔽₂)

yes

—

C₂²×C₁₉

C₂⁵

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×#41

yes

—

Dic₁₉

—

#555

—

C₂×C₄⋊Q₈

D₁₉

yes

characteristic, complemented

C₂×#124

—

C₂×#121

—

C₂×C₄×D₄

—

C₂×#117

—

C₂×C₁₉

#555

yes

complemented

C₂×C₁₉

C₂²×#41

yes

—

C₂×C₁₉

C₂×H(5,𝔽₂)

yes

characteristic

D₁₉

—

C₂²×Q₈

C₂×D₁₉

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

—

C₄⋊Q₈

C₂×D₁₉

yes

complemented

#124

—

#121

—

C₄×D₄

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₉

yes

characteristic

C₁₉

C₂×#555

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₉

yes

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₁₉

yes

—

107

C₂²×C₄

C₂²×D₁₉

yes

characteristic

108

C₂²×C₄

C₂²×D₁₉

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂²×D₁₉

yes

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²×D₁₉

yes

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

C₂³×D₁₉

yes

characteristic

120

Q₈

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#64545

yes

—

129

C₂×C₄

C₂³×D₁₉

yes

—

130

C₂×C₄

C₂³×D₁₉

yes

—

131

C₂×C₄

C₂³×D₁₉

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#10597430

yes

—

137

C₂²

C₂×#64545

yes

—

138

C₂²

C₂⁴×D₁₉

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#64545

yes

—

143

C₂

—

144

C₂

#11927260

yes

direct product factor

145

C₂

C₂²×#64545

yes

—

146

C₂

C₂×#10597430

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₉
commutator, cosocle	C₂²×C₁₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₁₉×C₄⋊Q₈	C₂
socle	C₂³×C₁₉	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	C₂×#555
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₁₉

commutator, cosocle

C₂²×C₁₉

C₂⁵

Frattini

C₂²

C₂⁴×D₁₉

Fitting

C₂×C₁₉×C₄⋊Q₈

C₂

socle

C₂³×C₁₉

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. #11927260	C₂	direct (N × H)
2. C₂×#10597393	C₂	semidirect
3. C₂×#10597341	C₂	semidirect
4. C₂×C₄×D₇₆	C₂	semidirect
5. C₂×#10597338	C₂	semidirect
6. C₂×C₁₉×C₄⋊Q₈	C₂	semidirect
11. C₁₉×C₄⋊Q₈	C₂²	semidirect
14. #10597393	C₂²	semidirect
19. #10597341	C₂²	semidirect
20. #10597338	C₂²	semidirect
21. C₄×D₇₆	C₂²	semidirect
71. C₂×C₄⋊Q₈	D₁₉	semidirect
76. C₂×C₁₉	#555	semidirect
84. C₄⋊Q₈	C₂×D₁₉	semidirect
95. C₁₉	C₂×#555	semidirect
144. C₂	#11927260	direct (N × H)

H ≅ G/N

type

1. #11927260

C₂

direct (N × H)

2. C₂×#10597393

C₂

semidirect

3. C₂×#10597341

C₂

semidirect

4. C₂×C₄×D₇₆

C₂

semidirect

5. C₂×#10597338

C₂

semidirect

6. C₂×C₁₉×C₄⋊Q₈

C₂

semidirect

11. C₁₉×C₄⋊Q₈

C₂²

semidirect

14. #10597393

C₂²

semidirect

19. #10597341

C₂²

semidirect

20. #10597338

C₂²

semidirect

21. C₄×D₇₆

C₂²

semidirect

71. C₂×C₄⋊Q₈

D₁₉

semidirect

76. C₂×C₁₉

#555

semidirect

84. C₄⋊Q₈

C₂×D₁₉

semidirect

95. C₁₉

C₂×#555

semidirect

144. C₂

#11927260

direct (N × H)

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

subgroup	index
#11927260	2
C₂×#10597393	2
C₂×#10597341	2
C₂×C₄×D₇₆	2
C₂×#10597338	2
C₂×C₁₉×C₄⋊Q₈	2
C₂×#555	19

subgroup

index

C₂×C₄×D₇₆

C₂×C₁₉×C₄⋊Q₈

Subgroups — C₂×#11927260 #14577176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#11927260 #14577176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11927260 #14577176