Group #14583523 — subgroups

Subgroups — #14583523

Order 2752.

216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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	#13063320	C₂	1	2	yes	maximal, complemented
2	#13063314	C₂	1	2	yes	maximal, complemented
3	#13063313	C₂	1	1	yes	maximal, characteristic, complemented
4	#13063310	C₂	1	2	yes	maximal, complemented
5	#13063305	C₂	1	1	yes	maximal, characteristic, complemented
6	#13063301	C₂	1	2	yes	maximal, complemented
7	#13063300	C₂	1	1	yes	maximal, characteristic
8	#13063290	C₂	1	1	yes	maximal, characteristic, complemented
9	#13063287	C₂	1	1	yes	maximal, characteristic, complemented
10	#13063282	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₄₃	C₂²	1	2	yes	—
13	C₂×D₁₇₂	C₂²	1	1	yes	characteristic
14	C₂×D₁₇₂	C₂²	1	2	yes	—
15	C₂×C₄×D₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄₃	C₂²	1	2	yes	—
17	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
23	(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
24	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
28	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
29	C₄⋊Dic₄₃	C₂²	1	2	yes	—
30	Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
31	Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
32	Dic₄₃⋊C₄	C₂²	1	2	yes	—
33	C₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₄₃	C₂²	1	2	yes	—
35	C₂²×D₄₃	C₂³	1	1	yes	characteristic
36	C₂²×D₄₃	C₂³	1	2	yes	—
37	Q₈×C₄₃	—	4	2	no	—
38	C₂×C₄×C₄₃	C₂³	1	2	yes	—
39	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₄₃	C₂³	1	2	yes	—
42	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
44	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₄₃	C₂³	1	2	yes	—
46	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
47	D₁₇₂	—	2	2	no	—
48	D₁₇₂	—	4	2	no	—
49	C₄×D₄₃	—	2	2	no	—
50	C₄×D₄₃	—	4	2	no	—
51	Dic₈₆	—	2	2	no	—
52	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
53	C₂×D₄₃	—	2	1	no	—
54	C₂×D₄₃	—	2	2	no	—
55	C₂×D₄₃	—	2	2	no	—
56	C₂×D₄₃	—	2	1	no	—
57	C₂×D₄₃	—	2	1	no	—
58	C₂×D₄₃	—	2	2	no	—
59	C₄×C₄₃	—	2	2	no	—
60	C₄×C₄₃	—	2	2	no	—
61	C₄×C₄₃	—	2	1	no	—
62	C₄×C₄₃	—	2	1	no	—
63	C₄×C₄₃	#41	1	2	yes	—
64	Dic₄₃	—	2	2	no	—
65	Dic₄₃	—	2	1	no	—
66	Dic₄₃	—	2	1	no	—
67	Dic₄₃	—	2	1	no	—
68	C₂×C₄₃	C₂×#41	1	1	yes	characteristic
69	C₂×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
70	C₂×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
71	D₄₃	—	4	1	no	—
72	D₄₃	—	4	2	no	—
73	#542	—	43	1	no	maximal, Sylow
74	C₄₃	#542	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
76	#126	—	43	2	no	—
77	#124	—	43	2	no	—
78	#124	—	43	1	no	—
79	#123	—	43	2	no	—
80	C₂²⋊Q₈	—	43	2	no	—
81	C₂²⋊Q₈	—	43	1	no	—
82	#121	—	43	1	no	—
83	C₄×Q₈	—	43	1	no	—
84	C₄×D₄	—	43	1	no	—
85	#117	—	43	1	no	—
86	C₂×Q₈	—	43	1	no	—
87	C₂×Q₈	C₂×D₄₃	1	2	yes	—
88	C₂×D₄	—	43	1	no	—
89	C₂×D₄	—	43	2	no	—
90	C₂²×C₄	—	43	1	no	—
91	C₂²×C₄	—	43	2	no	—
92	C₄⋊C₄	—	43	1	no	—
93	C₄⋊C₄	—	43	1	no	—
94	C₄⋊C₄	—	43	2	no	—
95	C₄⋊C₄	—	43	2	no	—
96	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₄₃	1	2	yes	—
99	C₂²⋊C₄	—	43	1	no	—
100	C₂²⋊C₄	—	43	1	no	—
101	C₂²⋊C₄	—	43	2	no	—
102	C₂²⋊C₄	—	43	2	no	—
103	C₂²⋊C₄	—	43	2	no	—
104	C₂²⋊C₄	—	43	2	no	—
105	C₂²⋊C₄	—	43	2	no	—
106	C₄²	—	43	1	no	—
107	C₄²	—	43	2	no	—
108	C₄²	C₂×D₄₃	1	1	yes	characteristic
109	C₂³	—	43	1	no	—
110	C₂³	—	43	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	86	2	no	—
113	D₄	—	86	2	no	—
114	D₄	—	172	2	no	—
115	C₂×C₄	—	43	1	no	—
116	C₂×C₄	—	43	2	no	—
117	C₂×C₄	—	43	1	no	—
118	C₂×C₄	—	43	1	no	—
119	C₂×C₄	—	86	2	no	—
120	C₂×C₄	—	172	2	no	—
121	C₂×C₄	C₂²×D₄₃	1	2	yes	—
122	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₄₃	1	2	yes	—
125	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
126	C₂²	—	86	1	no	—
127	C₂²	—	86	2	no	—
128	C₂²	—	86	2	no	—
129	C₂²	—	86	1	no	—
130	C₂²	—	86	1	no	—
131	C₂²	—	86	2	no	—
132	C₂²	C₂³×D₄₃	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	2	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	86	1	no	—
138	C₄	—	86	2	no	—
139	C₄	—	86	1	no	—
140	C₄	—	86	1	no	—
141	C₄	#10621902	1	2	yes	—
142	C₂	—	172	1	no	—
143	C₂	—	172	2	no	—
144	C₂	C₂×#10621902	1	1	yes	characteristic
145	C₂	#13063370	1	1	yes	characteristic
146	C₂	#13063373	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13063320

C₂

yes

maximal, complemented

#13063314

C₂

yes

maximal, complemented

#13063313

C₂

yes

maximal, characteristic, complemented

#13063310

C₂

yes

maximal, complemented

#13063305

C₂

yes

maximal, characteristic, complemented

#13063301

C₂

yes

maximal, complemented

#13063300

C₂

yes

maximal, characteristic

#13063290

C₂

yes

maximal, characteristic, complemented

#13063287

C₂

yes

maximal, characteristic, complemented

#13063282

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₃

C₂²

yes

—

C₂×D₁₇₂

C₂²

yes

characteristic

C₂×D₁₇₂

C₂²

yes

—

C₂×C₄×D₄₃

C₂²

yes

characteristic

C₂×C₄×D₄₃

C₂²

yes

—

C₂×Dic₈₆

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

—

C₄²×C₄₃

C₂²

yes

characteristic

(C₂×D₄₃)⋊C₄

C₂²

yes

characteristic

(C₂×D₄₃)⋊C₄

C₂²

yes

characteristic

(C₂×D₄₃)⋊C₄

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

characteristic

Dic₄₃⋊C₄

C₂²

yes

characteristic

Dic₄₃⋊C₄

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

characteristic

C₄×Dic₄₃

C₂²

yes

—

C₂²×D₄₃

C₂³

yes

characteristic

C₂²×D₄₃

C₂³

yes

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

characteristic

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₂²×C₄₃

C₂⁴

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#41

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

C₂×C₄₃

C₂×#41

yes

characteristic

C₂×C₄₃

H(5,𝔽₂)

yes

characteristic

C₂×C₄₃

2₋¹⁺⁴

yes

characteristic

D₄₃

—

D₄₃

—

#542

—

maximal, Sylow

C₄₃

#542

yes

characteristic, complemented, Sylow

C₄⋊Q₈

D₄₃

yes

characteristic, complemented

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₄⋊C₄

C₂×D₄₃

yes

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₂×D₄₃

yes

characteristic

109

C₂³

—

110

C₂³

—

111

Q₈

—

112

Q₈

—

113

D₄

—

114

D₄

—

172

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

172

—

121

C₂×C₄

C₂²×D₄₃

yes

—

122

C₂×C₄

C₂²×D₄₃

yes

characteristic

123

C₂×C₄

C₂²×D₄₃

yes

characteristic

124

C₂×C₄

C₂²×D₄₃

yes

—

125

C₂×C₄

C₂²×D₄₃

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×D₄₃

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#10621902

yes

—

142

C₂

—

172

—

143

C₂

—

172

—

144

C₂

C₂×#10621902

yes

characteristic

145

C₂

#13063370

yes

characteristic

146

C₂

#13063373

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	C₄⋊Q₈×C₄₃	C₂
prime radical	#13063300	C₂
hypercenter	C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#542
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃

commutator, socle, cosocle

C₂²×C₄₃

C₂⁴

Fitting

C₄⋊Q₈×C₄₃

C₂

prime radical

#13063300

C₂

hypercenter

C₄⋊Q₈

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #13063320	C₂	semidirect
2. #13063314	C₂	semidirect
3. #13063313	C₂	semidirect
4. #13063310	C₂	semidirect
5. #13063305	C₂	semidirect
6. #13063301	C₂	semidirect
8. #13063290	C₂	semidirect
9. #13063287	C₂	semidirect
10. #13063282	C₂	semidirect
11. C₄⋊Q₈×C₄₃	C₂	semidirect
74. C₄₃	#542	semidirect
75. C₄⋊Q₈	D₄₃	semidirect

H ≅ G/N

type

1. #13063320

C₂

semidirect

2. #13063314

C₂

semidirect

3. #13063313

C₂

semidirect

4. #13063310

C₂

semidirect

5. #13063305

C₂

semidirect

6. #13063301

C₂

semidirect

8. #13063290

C₂

semidirect

9. #13063287

C₂

semidirect

10. #13063282

C₂

semidirect

11. C₄⋊Q₈×C₄₃

C₂

semidirect

74. C₄₃

#542

semidirect

75. C₄⋊Q₈

D₄₃

semidirect

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

subgroup	index
#13063320	2
#13063314	2
#13063313	2
#13063310	2
#13063305	2
#13063301	2
#13063300	2
#13063290	2
#13063287	2
#13063282	2
C₄⋊Q₈×C₄₃	2
#542	43

subgroup

index

C₄⋊Q₈×C₄₃

Subgroups — #14583523

Conjugacy classes of subgroups

Complements

Maximal subgroups