Group #19148136 — subgroups

Subgroups — C₂×#14583615 #19148136

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	#14583615	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13063314	C₂	1	4	yes	maximal, complemented
3	C₂×#13063300	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₁₇₂	C₂	1	2	yes	maximal, complemented
5	C₂×#13063283	C₂	1	4	yes	maximal, complemented
6	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₇₂	C₂²	1	2	yes	—
8	C₂²×D₁₇₂	C₂²	1	4	yes	—
9	C₂²×C₄×D₄₃	C₂²	1	4	yes	—
10	#13063314	C₂²	1	32	yes	complemented
11	#13063300	C₂²	1	32	yes	complemented
12	C₄×D₁₇₂	C₂²	1	16	yes	complemented
13	#13063283	C₂²	1	32	yes	complemented
14	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
15	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	8	yes	—
16	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄×Dic₄₃	C₂²	1	4	yes	—
18	C₂²×Q₈×C₄₃	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₄₃	C₂²	1	8	yes	complemented
20	C₂×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
21	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	C₂³×D₄₃	C₂³	1	4	yes	—
23	C₂×Q₈×C₄₃	—	4	4	no	—
24	C₂×Q₈×C₄₃	C₂³	1	8	yes	—
25	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
28	C₂²×Dic₄₃	C₂³	1	4	yes	—
29	C₂×D₁₇₂	—	2	8	no	—
30	C₂×D₁₇₂	—	4	8	no	—
31	C₂×D₁₇₂	C₂³	1	8	yes	—
32	C₂×D₁₇₂	C₂³	1	16	yes	—
33	C₂×C₄×D₄₃	—	2	16	no	—
34	C₂×C₄×D₄₃	C₂³	1	16	yes	—
35	C₄⋊C₄×C₄₃	C₂³	1	16	yes	—
36	C₄²×C₄₃	C₂³	1	4	yes	—
37	(C₂×D₄₃)⋊C₄	C₂³	1	32	yes	—
38	(C₂×D₄₃)⋊C₄	C₂³	1	16	yes	—
39	C₄⋊Dic₄₃	C₂³	1	8	yes	—
40	C₄×Dic₄₃	C₂³	1	16	yes	—
41	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
42	C₂²×D₄₃	—	2	8	no	—
43	C₂²×D₄₃	—	2	8	no	—
44	C₂²×D₄₃	—	2	8	no	—
45	C₂²×D₄₃	C₂⁴	1	8	yes	—
46	Q₈×C₄₃	—	4	8	no	—
47	C₂×C₄×C₄₃	—	2	8	no	—
48	C₂×C₄×C₄₃	—	2	2	no	—
49	C₂×C₄×C₄₃	#41	1	8	yes	—
50	C₂×C₄×C₄₃	C₂⁴	1	8	yes	—
51	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
52	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
53	C₂×Dic₄₃	—	2	8	no	—
54	C₂×Dic₄₃	C₂⁴	1	8	yes	—
55	D₁₇₂	—	2	16	no	—
56	D₁₇₂	—	4	16	no	—
57	C₄×D₄₃	—	2	32	no	—
58	C₂²×C₄₃	C₂×#41	1	4	yes	—
59	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
60	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
61	C₂×D₄₃	—	2	8	no	—
62	C₂×D₄₃	—	2	8	no	—
63	C₂×D₄₃	—	2	8	no	—
64	C₂×D₄₃	—	4	16	no	—
65	C₄×C₄₃	—	2	8	no	—
66	C₄×C₄₃	—	2	2	no	—
67	C₄×C₄₃	C₂×#41	1	8	yes	—
68	Dic₄₃	—	2	8	no	—
69	C₂×#555	—	43	1	no	maximal, Sylow
70	C₂×C₄₃	#555	1	4	yes	complemented
71	C₂×C₄₃	C₂²×#41	1	2	yes	—
72	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
73	D₄₃	—	4	8	no	—
74	#555	—	43	16	no	—
75	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
76	C₂×#124	—	43	4	no	—
77	C₂×#121	—	43	4	no	—
78	C₂×C₄×D₄	—	43	2	no	—
79	C₂×#117	—	43	4	no	—
80	C₄₃	C₂×#555	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	C₂×D₄₃	1	2	yes	—
82	C₂²×D₄	—	43	2	no	—
83	C₂²×D₄	—	43	4	no	—
84	C₂³×C₄	—	43	4	no	—
85	C₄⋊Q₈	C₂×D₄₃	1	8	yes	complemented
86	#124	—	43	32	no	—
87	#121	—	43	32	no	—
88	C₄×D₄	—	43	16	no	—
89	#117	—	43	32	no	—
90	C₂×C₄⋊C₄	—	43	2	no	—
91	C₂×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
92	C₂×C₂²⋊C₄	—	43	8	no	—
93	C₂×C₂²⋊C₄	—	43	4	no	—
94	C₂×C₄²	—	43	4	no	—
95	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
96	C₂⁴	—	43	4	no	—
97	C₂×Q₈	—	4	4	no	—
98	C₂×Q₈	C₂²×D₄₃	1	8	yes	—
99	C₂×D₄	—	43	8	no	—
100	C₂×D₄	—	43	16	no	—
101	C₂×D₄	—	86	8	no	—
102	C₂×D₄	—	172	8	no	—
103	C₂²×C₄	—	43	4	no	—
104	C₂²×C₄	—	43	16	no	—
105	C₂²×C₄	—	86	16	no	—
106	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
107	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
109	C₄⋊C₄	—	43	8	no	—
110	C₄⋊C₄	C₂²×D₄₃	1	16	yes	—
111	C₂²⋊C₄	—	43	32	no	—
112	C₂²⋊C₄	—	43	16	no	—
113	C₄²	—	43	16	no	—
114	C₄²	C₂²×D₄₃	1	4	yes	—
115	C₂³	—	43	8	no	—
116	C₂³	—	86	8	no	—
117	C₂³	—	86	8	no	—
118	C₂³	—	86	8	no	—
119	C₂³	C₂³×D₄₃	1	1	yes	characteristic
120	Q₈	—	4	8	no	—
121	D₄	—	86	16	no	—
122	D₄	—	172	16	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	43	8	no	—
126	C₂×C₄	—	86	8	no	—
127	C₂×C₄	—	86	32	no	—
128	C₂×C₄	#10621902	1	8	yes	—
129	C₂×C₄	C₂³×D₄₃	1	8	yes	—
130	C₂×C₄	C₂³×D₄₃	1	2	yes	—
131	C₂×C₄	C₂³×D₄₃	1	4	yes	—
132	C₂²	—	86	8	no	—
133	C₂²	—	86	8	no	—
134	C₂²	—	86	8	no	—
135	C₂²	—	172	16	no	—
136	C₂²	C₂×#10621902	1	4	yes	—
137	C₂²	#13063370	1	2	yes	—
138	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
139	C₄	—	2	8	no	—
140	C₄	—	2	2	no	—
141	C₄	—	86	8	no	—
142	C₄	C₂×#10621902	1	8	yes	—
143	C₂	—	172	8	no	—
144	C₂	#14583615	1	4	yes	direct product factor
145	C₂	C₂²×#10621902	1	2	yes	—
146	C₂	C₂×#13063370	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583615

C₂

yes

maximal, direct product factor

C₂×#13063314

C₂

yes

maximal, complemented

C₂×#13063300

C₂

yes

maximal, complemented

C₂×C₄×D₁₇₂

C₂

yes

maximal, complemented

C₂×#13063283

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

—

C₂²×D₁₇₂

C₂²

yes

—

C₂²×C₄×D₄₃

C₂²

yes

—

#13063314

C₂²

yes

complemented

#13063300

C₂²

yes

complemented

C₄×D₁₇₂

C₂²

yes

complemented

#13063283

C₂²

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

—

C₄⋊Q₈×C₄₃

C₂²

yes

complemented

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂³×D₄₃

C₂³

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂³

yes

—

C₂×D₁₇₂

C₂³

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₄²×C₄₃

C₂³

yes

—

(C₂×D₄₃)⋊C₄

C₂³

yes

—

(C₂×D₄₃)⋊C₄

C₂³

yes

—

C₄⋊Dic₄₃

C₂³

yes

—

C₄×Dic₄₃

C₂³

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂⁴

yes

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂⁴

yes

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₂²×C₄₃

C₂×#41

yes

—

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂²×C₄₃

C₂⁵

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#41

yes

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

C₂²×#41

yes

—

C₂×C₄₃

C₂×H(5,𝔽₂)

yes

characteristic

D₄₃

—

#555

—

C₂×C₄⋊Q₈

D₄₃

yes

characteristic, complemented

C₂×#124

—

C₂×#121

—

C₂×C₄×D₄

—

C₂×#117

—

C₄₃

C₂×#555

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₃

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

—

C₄⋊Q₈

C₂×D₄₃

yes

complemented

#124

—

#121

—

C₄×D₄

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₄₃

yes

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

172

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₄₃

yes

—

107

C₂²×C₄

C₂²×D₄₃

yes

characteristic

108

C₂²×C₄

C₂²×D₄₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂²×D₄₃

yes

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²×D₄₃

yes

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

C₂³×D₄₃

yes

characteristic

120

Q₈

—

121

D₄

—

122

D₄

—

172

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#10621902

yes

—

129

C₂×C₄

C₂³×D₄₃

yes

—

130

C₂×C₄

C₂³×D₄₃

yes

—

131

C₂×C₄

C₂³×D₄₃

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

172

—

136

C₂²

C₂×#10621902

yes

—

137

C₂²

#13063370

yes

—

138

C₂²

C₂⁴×D₄₃

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#10621902

yes

—

143

C₂

—

172

—

144

C₂

#14583615

yes

direct product factor

145

C₂

C₂²×#10621902

yes

—

146

C₂

C₂×#13063370

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	C₂×#555
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂⁵

Frattini

C₂²

C₂⁴×D₄₃

Fitting

C₂×C₄⋊Q₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. #14583615	C₂	direct (N × H)
2. C₂×#13063314	C₂	semidirect
3. C₂×#13063300	C₂	semidirect
4. C₂×C₄×D₁₇₂	C₂	semidirect
5. C₂×#13063283	C₂	semidirect
6. C₂×C₄⋊Q₈×C₄₃	C₂	semidirect
10. #13063314	C₂²	semidirect
11. #13063300	C₂²	semidirect
12. C₄×D₁₇₂	C₂²	semidirect
13. #13063283	C₂²	semidirect
19. C₄⋊Q₈×C₄₃	C₂²	semidirect
70. C₂×C₄₃	#555	semidirect
75. C₂×C₄⋊Q₈	D₄₃	semidirect
80. C₄₃	C₂×#555	semidirect
85. C₄⋊Q₈	C₂×D₄₃	semidirect
144. C₂	#14583615	direct (N × H)

H ≅ G/N

type

1. #14583615

C₂

direct (N × H)

2. C₂×#13063314

C₂

semidirect

3. C₂×#13063300

C₂

semidirect

4. C₂×C₄×D₁₇₂

C₂

semidirect

5. C₂×#13063283

C₂

semidirect

6. C₂×C₄⋊Q₈×C₄₃

C₂

semidirect

10. #13063314

C₂²

semidirect

11. #13063300

C₂²

semidirect

12. C₄×D₁₇₂

C₂²

semidirect

13. #13063283

C₂²

semidirect

19. C₄⋊Q₈×C₄₃

C₂²

semidirect

70. C₂×C₄₃

#555

semidirect

75. C₂×C₄⋊Q₈

D₄₃

semidirect

80. C₄₃

C₂×#555

semidirect

85. C₄⋊Q₈

C₂×D₄₃

semidirect

144. C₂

#14583615

direct (N × H)

Subgroups — C₂×#14583615 #19148136

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14583615 #19148136

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583615 #19148136