Group #16811210 — subgroups

Subgroups — #16811210

Order 3392.

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

normal subgroups only

This group has 250 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	#13133212	C₂	1	2	yes	maximal, complemented
2	#13133203	C₂	1	2	yes	maximal, complemented
3	#13133201	C₂	1	2	yes	maximal, complemented
4	#13133196	C₂	1	1	yes	maximal, characteristic, complemented
5	#13133189	C₂	1	2	yes	maximal, complemented
6	#13133185	C₂	1	2	yes	maximal, complemented
7	D₄×Dic₅₃	C₂	1	2	yes	maximal, complemented
8	C₂×#11719921	C₂	1	1	yes	maximal, characteristic, complemented
9	#121×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₅₃	C₂²	1	2	yes	—
12	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₅₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₅₃	C₂²	1	2	yes	—
16	C₂²×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₅₃	C₂²	1	2	yes	complemented
18	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
19	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
20	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
21	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
22	C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
23	#11719921	C₂²	1	2	yes	—
24	#11719921	C₂²	1	2	yes	—
25	#11719921	C₂²	1	2	yes	—
26	#11719921	C₂²	1	2	yes	—
27	C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
28	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
29	#11719919	C₂²	1	2	yes	complemented
30	C₄×Dic₅₃	C₂²	1	2	yes	complemented
31	C₂³×C₅₃	C₂³	1	2	yes	—
32	C₂³×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×D₅₃	—	2	2	no	—
34	C₂²×D₅₃	—	4	1	no	—
35	C₂²×D₅₃	—	4	1	no	—
36	C₂²×D₅₃	C₂³	1	2	yes	—
37	D₄×C₅₃	—	2	4	no	—
38	D₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂²⋊D₅₃	—	4	2	no	—
44	C₂²⋊D₅₃	—	4	2	no	—
45	C₂×Dic₅₃	—	4	1	no	—
46	C₂×Dic₅₃	—	4	2	no	—
47	C₂×Dic₅₃	C₂³	1	2	yes	—
48	C₂×Dic₅₃	C₂³	1	2	yes	—
49	C₂×Dic₅₃	C₂³	1	2	yes	—
50	Dic₁₀₆	—	4	1	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	—	2	2	no	—
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	#41	1	2	yes	—
57	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂×D₅₃	—	2	2	no	—
59	C₂×D₅₃	—	2	2	no	—
60	C₂×D₅₃	—	2	2	no	—
61	C₂×D₅₃	—	4	4	no	—
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	1	no	—
65	Dic₅₃	—	2	2	no	—
66	Dic₅₃	—	2	2	no	—
67	Dic₅₃	—	2	2	no	—
68	C₂×C₅₃	—	2	2	no	—
69	C₂×C₅₃	—	4	2	no	—
70	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
71	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
73	D₅₃	—	4	2	no	—
74	#538	—	53	1	no	maximal, Sylow
75	C₅₃	#538	1	1	yes	characteristic, complemented, Sylow
76	#126	—	53	2	no	—
77	#124	—	53	2	no	—
78	#123	—	53	2	no	—
79	C₂²⋊Q₈	—	53	1	no	—
80	#121	—	53	2	no	—
81	#121	D₅₃	1	1	yes	characteristic, complemented
82	#120	—	53	2	no	—
83	C₄×D₄	—	53	2	no	—
84	C₂×C₂²⋊C₄	—	53	1	no	—
85	C₂⁴	—	53	1	no	—
86	C₂×Q₈	—	53	1	no	—
87	C₂×D₄	—	53	2	no	—
88	C₂×D₄	—	53	2	no	—
89	C₂×D₄	C₂×D₅₃	1	2	yes	—
90	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
91	C₂²×C₄	—	53	1	no	—
92	C₂²×C₄	—	53	2	no	—
93	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
94	C₄⋊C₄	—	53	2	no	—
95	C₄⋊C₄	—	53	2	no	—
96	C₄⋊C₄	—	53	1	no	—
97	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
98	C₂²⋊C₄	—	53	2	no	—
99	C₂²⋊C₄	—	53	2	no	—
100	C₂²⋊C₄	—	53	2	no	—
101	C₂²⋊C₄	—	53	2	no	—
102	C₂²⋊C₄	—	53	2	no	—
103	C₂²⋊C₄	—	53	2	no	—
104	C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
105	C₄²	—	53	2	no	—
106	C₂³	—	53	2	no	—
107	C₂³	—	106	2	no	—
108	C₂³	—	212	1	no	—
109	C₂³	—	212	1	no	—
110	C₂³	C₂²×D₅₃	1	2	yes	—
111	C₂³	C₂²×D₅₃	1	1	yes	characteristic
112	Q₈	—	212	1	no	—
113	D₄	—	2	4	no	—
114	D₄	—	4	1	no	—
115	D₄	—	212	2	no	—
116	D₄	—	212	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	53	2	no	—
119	C₂×C₄	—	53	2	no	—
120	C₂×C₄	—	53	2	no	—
121	C₂×C₄	—	212	1	no	—
122	C₂×C₄	—	212	2	no	—
123	C₂×C₄	C₂²×D₅₃	1	2	yes	—
124	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	106	2	no	—
132	C₂²	—	106	2	no	—
133	C₂²	—	106	2	no	—
134	C₂²	—	212	4	no	—
135	C₂²	#11719947	1	2	yes	—
136	C₂²	C₂³×D₅₃	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	106	2	no	—
141	C₄	—	106	2	no	—
142	C₄	—	106	2	no	—
143	C₂	—	2	2	no	—
144	C₂	—	4	2	no	—
145	C₂	—	212	2	no	—
146	C₂	C₂×#11719947	1	1	yes	characteristic
147	C₂	#13133262	1	1	yes	characteristic
148	C₂	#13133263	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133212

C₂

yes

maximal, complemented

#13133203

C₂

yes

maximal, complemented

#13133201

C₂

yes

maximal, complemented

#13133196

C₂

yes

maximal, characteristic, complemented

#13133189

C₂

yes

maximal, complemented

#13133185

C₂

yes

maximal, complemented

D₄×Dic₅₃

C₂

yes

maximal, complemented

C₂×#11719921

C₂

yes

maximal, characteristic, complemented

#121×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

C₂²

yes

—

C₂×D₄×C₅₃

C₂²

yes

characteristic, complemented

C₂²×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊D₅₃

C₂²

yes

—

C₂×C₂²⋊D₅₃

C₂²

yes

—

C₂²×Dic₅₃

C₂²

yes

characteristic

C₂²×Dic₅₃

C₂²

yes

complemented

C₂×Dic₁₀₆

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₅₃

C₂²

yes

characteristic, complemented

C₂²⋊C₄×C₅₃

C₂²

yes

complemented

C₂²⋊Dic₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

complemented

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

complemented

C₄⋊Dic₅₃

C₂²

yes

characteristic, complemented

#11719919

C₂²

yes

complemented

C₄×Dic₅₃

C₂²

yes

complemented

C₂³×C₅₃

C₂³

yes

—

C₂³×C₅₃

C₂³

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂³

yes

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

—

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#41

yes

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#41

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#538

—

maximal, Sylow

C₅₃

#538

yes

characteristic, complemented, Sylow

#126

—

#124

—

#123

—

C₂²⋊Q₈

—

#121

—

#121

D₅₃

yes

characteristic, complemented

#120

—

C₄×D₄

—

C₂×C₂²⋊C₄

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₅₃

yes

—

C₂×D₄

C₂×D₅₃

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅₃

yes

characteristic, complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

C₂×D₅₃

yes

complemented

105

C₄²

—

106

C₂³

—

107

C₂³

—

106

—

108

C₂³

—

212

—

109

C₂³

—

212

—

110

C₂³

C₂²×D₅₃

yes

—

111

C₂³

C₂²×D₅₃

yes

characteristic

112

Q₈

—

212

—

113

D₄

—

114

D₄

—

115

D₄

—

212

—

116

D₄

—

212

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

212

—

122

C₂×C₄

—

212

—

123

C₂×C₄

C₂²×D₅₃

yes

—

124

C₂×C₄

C₂²×D₅₃

yes

characteristic

125

C₂×C₄

C₂²×D₅₃

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

106

—

132

C₂²

—

106

—

133

C₂²

—

106

—

134

C₂²

—

212

—

135

C₂²

#11719947

yes

—

136

C₂²

C₂³×D₅₃

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

106

—

141

C₄

—

106

—

142

C₄

—

106

—

143

C₂

—

144

C₂

—

145

C₂

—

212

—

146

C₂

C₂×#11719947

yes

characteristic

147

C₂

#13133262

yes

characteristic

148

C₂

#13133263

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	#121×C₅₃	C₂
hypercenter	#121	D₅₃
hypocenter	C₅₃	#538
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting

#121×C₅₃

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. #13133212	C₂	semidirect
2. #13133203	C₂	semidirect
3. #13133201	C₂	semidirect
4. #13133196	C₂	semidirect
5. #13133189	C₂	semidirect
6. #13133185	C₂	semidirect
7. D₄×Dic₅₃	C₂	semidirect
8. C₂×#11719921	C₂	semidirect
9. #121×C₅₃	C₂	semidirect
12. C₂×D₄×C₅₃	C₂²	semidirect
17. C₂²×Dic₅₃	C₂²	semidirect
18. C₂×Dic₁₀₆	C₂²	semidirect
19. C₄⋊C₄×C₅₃	C₂²	semidirect
20. C₂²⋊C₄×C₅₃	C₂²	semidirect
22. C₂²⋊Dic₅₃	C₂²	semidirect
27. C₄⋊Dic₅₃	C₂²	semidirect
28. C₄⋊Dic₅₃	C₂²	semidirect
29. #11719919	C₂²	semidirect
30. C₄×Dic₅₃	C₂²	semidirect
75. C₅₃	#538	semidirect
81. #121	D₅₃	semidirect
90. C₂×D₄	C₂×D₅₃	semidirect
97. C₄⋊C₄	C₂×D₅₃	semidirect
104. C₂²⋊C₄	C₂×D₅₃	semidirect

H ≅ G/N

type

1. #13133212

C₂

semidirect

2. #13133203

C₂

semidirect

3. #13133201

C₂

semidirect

4. #13133196

C₂

semidirect

5. #13133189

C₂

semidirect

6. #13133185

C₂

semidirect

7. D₄×Dic₅₃

C₂

semidirect

8. C₂×#11719921

C₂

semidirect

9. #121×C₅₃

C₂

semidirect

12. C₂×D₄×C₅₃

C₂²

semidirect

17. C₂²×Dic₅₃

C₂²

semidirect

18. C₂×Dic₁₀₆

C₂²

semidirect

19. C₄⋊C₄×C₅₃

C₂²

semidirect

20. C₂²⋊C₄×C₅₃

C₂²

semidirect

22. C₂²⋊Dic₅₃

C₂²

semidirect

27. C₄⋊Dic₅₃

C₂²

semidirect

28. C₄⋊Dic₅₃

C₂²

semidirect

29. #11719919

C₂²

semidirect

30. C₄×Dic₅₃

C₂²

semidirect

75. C₅₃

#538

semidirect

81. #121

D₅₃

semidirect

90. C₂×D₄

C₂×D₅₃

semidirect

97. C₄⋊C₄

C₂×D₅₃

semidirect

104. C₂²⋊C₄

C₂×D₅₃

semidirect

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

subgroup	index
#13133212	2
#13133203	2
#13133201	2
#13133196	2
#13133189	2
#13133185	2
D₄×Dic₅₃	2
C₂×#11719921	2
#121×C₅₃	2
#538	53

subgroup

index

D₄×Dic₅₃

#121×C₅₃

Subgroups — #16811210

Conjugacy classes of subgroups

Complements

Maximal subgroups