Group #13093098 — subgroups

Subgroups — #13093098

Order 1472.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂₃×#120	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₂₃	C₂	1	2	yes	maximal, complemented
3	#10625107	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625106	C₂	1	1	yes	maximal, characteristic, complemented
5	#10625106	C₂	1	2	yes	maximal, complemented
6	D₄×Dic₂₃	C₂	1	2	yes	maximal, complemented
7	#10625050	C₂	1	2	yes	maximal, complemented
8	#10625049	C₂	1	2	yes	maximal, complemented
9	#10625048	C₂	1	2	yes	maximal, complemented
10	C₂⁴×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₃	C₂²	1	2	yes	—
14	C₂²×Dic₂₃	C₂²	1	2	yes	—
15	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
17	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
20	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
21	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
22	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
23	C₂²⋊Dic₂₃	C₂²	1	2	yes	complemented
24	C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic, complemented
25	C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
26	Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
27	Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
28	Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
29	C₄×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
30	C₄×Dic₂₃	C₂²	1	2	yes	complemented
31	C₂³×C₂₃	—	2	4	no	—
32	C₂³×C₂₃	—	4	1	no	—
33	C₂³×C₂₃	C₂³	1	1	yes	characteristic
34	C₂³×C₂₃	C₂³	1	1	yes	characteristic
35	C₂³×C₂₃	C₂³	1	2	yes	—
36	D₄×C₂₃	—	2	4	no	—
37	D₄×C₂₃	—	4	1	no	—
38	C₂×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×Dic₂₃	—	2	4	no	—
41	C₂×Dic₂₃	—	4	2	no	—
42	C₂×Dic₂₃	—	4	1	no	—
43	C₂×Dic₂₃	C₂³	1	2	yes	—
44	C₂×Dic₂₃	C₂³	1	2	yes	—
45	C₂×Dic₂₃	C₂³	1	2	yes	—
46	C₂×Dic₂₃	C₂³	1	2	yes	—
47	Dic₄₆	—	4	1	no	—
48	C₂²×C₂₃	—	2	2	no	—
49	C₂²×C₂₃	—	2	2	no	—
50	C₂²×C₂₃	—	2	2	no	—
51	C₂²×C₂₃	—	2	2	no	—
52	C₂²×C₂₃	—	2	1	no	—
53	C₂²×C₂₃	—	2	1	no	—
54	C₂²×C₂₃	—	4	4	no	—
55	C₂²×C₂₃	#41	1	4	yes	—
56	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
57	C₄×C₂₃	—	2	2	no	—
58	C₄×C₂₃	—	2	1	no	—
59	Dic₂₃	—	2	2	no	—
60	Dic₂₃	—	2	2	no	—
61	Dic₂₃	—	2	2	no	—
62	Dic₂₃	—	2	2	no	—
63	#551	—	23	1	no	maximal, Sylow
64	C₂×C₂₃	—	2	4	no	—
65	C₂×C₂₃	—	4	1	no	—
66	C₂×C₂₃	—	4	1	no	—
67	C₂×C₂₃	C₂×#41	1	2	yes	—
68	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
69	#126	—	23	2	no	—
70	#124	—	23	1	no	—
71	#123	—	23	2	no	—
72	#123	—	23	1	no	—
73	C₂²⋊Q₈	—	23	2	no	—
74	#120	D₂₃	1	1	yes	characteristic
75	C₄×D₄	—	23	2	no	—
76	#117	—	23	2	no	—
77	C₂×C₂²⋊C₄	—	23	2	no	—
78	C₂₃	#551	1	1	yes	characteristic, complemented, Sylow
79	C₂⁴	C₂×D₂₃	1	1	yes	characteristic
80	C₂×Q₈	—	23	1	no	—
81	C₂×D₄	C₂×D₂₃	1	2	yes	—
82	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
83	C₂²×C₄	—	23	2	no	—
84	C₂²×C₄	—	23	2	no	—
85	C₂²×C₄	—	23	1	no	—
86	C₄⋊C₄	—	23	2	no	—
87	C₄⋊C₄	—	23	2	no	—
88	C₄⋊C₄	—	23	2	no	—
89	C₄⋊C₄	—	23	2	no	—
90	C₂²⋊C₄	—	23	2	no	—
91	C₂²⋊C₄	—	23	2	no	—
92	C₂²⋊C₄	—	23	2	no	—
93	C₂²⋊C₄	—	23	2	no	—
94	C₂²⋊C₄	—	23	2	no	—
95	C₂²⋊C₄	—	23	1	no	—
96	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
97	C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
98	C₄²	—	23	1	no	—
99	C₄²	—	23	2	no	—
100	C₂³	—	2	4	no	—
101	C₂³	—	4	1	no	—
102	C₂³	C₂²×D₂₃	1	1	yes	characteristic
103	C₂³	C₂²×D₂₃	1	1	yes	characteristic
104	C₂³	C₂²×D₂₃	1	2	yes	—
105	Q₈	—	92	1	no	—
106	D₄	—	2	4	no	—
107	D₄	—	4	1	no	—
108	C₂×C₄	—	23	2	no	—
109	C₂×C₄	—	23	2	no	—
110	C₂×C₄	—	23	2	no	—
111	C₂×C₄	—	23	2	no	—
112	C₂×C₄	—	46	4	no	—
113	C₂×C₄	—	92	1	no	—
114	C₂×C₄	—	92	2	no	—
115	C₂×C₄	C₂²×D₂₃	1	2	yes	—
116	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	4	4	no	—
124	C₂²	#67290	1	4	yes	—
125	C₂²	C₂³×D₂₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	46	2	no	—
129	C₄	—	46	2	no	—
130	C₄	—	46	2	no	—
131	C₄	—	46	2	no	—
132	C₂	—	2	4	no	—
133	C₂	—	4	1	no	—
134	C₂	—	4	1	no	—
135	C₂	C₂×#67290	1	2	yes	—
136	C₂	#10625155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#120

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₂₃

C₂

yes

maximal, complemented

#10625107

C₂

yes

maximal, characteristic, complemented

#10625106

C₂

yes

maximal, characteristic, complemented

#10625106

C₂

yes

maximal, complemented

D₄×Dic₂₃

C₂

yes

maximal, complemented

#10625050

C₂

yes

maximal, complemented

#10625049

C₂

yes

maximal, complemented

#10625048

C₂

yes

maximal, complemented

C₂⁴×C₂₃

C₂²

yes

characteristic

C₂×D₄×C₂₃

C₂²

yes

—

C₂×D₄×C₂₃

C₂²

yes

characteristic

C₂²×Dic₂₃

C₂²

yes

—

C₂²×Dic₂₃

C₂²

yes

—

C₂²×Dic₂₃

C₂²

yes

characteristic, complemented

C₂×Dic₄₆

C₂²

yes

characteristic, complemented

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

complemented

C₂²⋊Dic₂₃

C₂²

yes

characteristic, complemented

C₄⋊Dic₂₃

C₂²

yes

complemented

Dic₂₃⋊C₄

C₂²

yes

complemented

Dic₂₃⋊C₄

C₂²

yes

complemented

Dic₂₃⋊C₄

C₂²

yes

complemented

C₄×Dic₂₃

C₂²

yes

characteristic, complemented

C₄×Dic₂₃

C₂²

yes

complemented

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

C₂³

yes

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#41

yes

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

#551

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

—

C₂×C₂₃

yes

—

—

—

—

C₂²⋊Q₈

—

#120

D₂₃

yes

characteristic

C₄×D₄

—

#117

—

C₂×C₂²⋊C₄

—

C₂₃

#551

yes

characteristic, complemented, Sylow

C₂⁴

C₂×D₂₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

C₂×D₂₃

yes

—

C₂×D₄

C₂×D₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

—

C₂²⋊C₄

C₂×D₂₃

yes

characteristic

C₄²

—

C₄²

—

100

C₂³

—

101

C₂³

—

102

C₂³

C₂²×D₂₃

yes

characteristic

103

C₂³

C₂²×D₂₃

yes

characteristic

104

C₂³

C₂²×D₂₃

yes

—

105

Q₈

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂²×D₂₃

yes

—

116

C₂×C₄

C₂²×D₂₃

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#67290

yes

—

125

C₂²

C₂³×D₂₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

C₂×#67290

yes

—

136

C₂

#10625155

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting, prime radical	C₂₃×#120	C₂
hypercenter	#120	D₂₃
hypocenter	C₂₃	#551
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #120

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting, prime radical

C₂₃×#120

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #120

N	H ≅ G/N	type
2. C₂×C₂²⋊Dic₂₃	C₂	semidirect
3. #10625107	C₂	semidirect
4. #10625106	C₂	semidirect
5. #10625106	C₂	semidirect
6. D₄×Dic₂₃	C₂	semidirect
7. #10625050	C₂	semidirect
8. #10625049	C₂	semidirect
9. #10625048	C₂	semidirect
15. C₂²×Dic₂₃	C₂²	semidirect
16. C₂×Dic₄₆	C₂²	semidirect
23. C₂²⋊Dic₂₃	C₂²	semidirect
24. C₂²⋊Dic₂₃	C₂²	semidirect
25. C₄⋊Dic₂₃	C₂²	semidirect
26. Dic₂₃⋊C₄	C₂²	semidirect
27. Dic₂₃⋊C₄	C₂²	semidirect
28. Dic₂₃⋊C₄	C₂²	semidirect
29. C₄×Dic₂₃	C₂²	semidirect
30. C₄×Dic₂₃	C₂²	semidirect
78. C₂₃	#551	semidirect

H ≅ G/N

type

2. C₂×C₂²⋊Dic₂₃

C₂

semidirect

3. #10625107

C₂

semidirect

4. #10625106

C₂

semidirect

5. #10625106

C₂

semidirect

6. D₄×Dic₂₃

C₂

semidirect

7. #10625050

C₂

semidirect

8. #10625049

C₂

semidirect

9. #10625048

C₂

semidirect

15. C₂²×Dic₂₃

C₂²

semidirect

16. C₂×Dic₄₆

C₂²

semidirect

23. C₂²⋊Dic₂₃

C₂²

semidirect

24. C₂²⋊Dic₂₃

C₂²

semidirect

25. C₄⋊Dic₂₃

C₂²

semidirect

26. Dic₂₃⋊C₄

C₂²

semidirect

27. Dic₂₃⋊C₄

C₂²

semidirect

28. Dic₂₃⋊C₄

C₂²

semidirect

29. C₄×Dic₂₃

C₂²

semidirect

30. C₄×Dic₂₃

C₂²

semidirect

78. C₂₃

#551

semidirect

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

subgroup	index
C₂₃×#120	2
C₂×C₂²⋊Dic₂₃	2
#10625107	2
#10625106	2
D₄×Dic₂₃	2
#10625050	2
#10625049	2
#10625048	2
#551	23

subgroup

index

C₂₃×#120

C₂×C₂²⋊Dic₂₃

D₄×Dic₂₃

Subgroups — #13093098

Conjugacy classes of subgroups

Complements

Maximal subgroups