Group #5881 — subgroups

Subgroups — C₃×#551 #5881

Order 192.

248 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 150 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	C₃×#126	C₂	1	2	yes	maximal, complemented
2	C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#123	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#123	C₂	1	2	yes	maximal, complemented
5	C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₃×#120	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
8	C₃×#117	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
10	#551	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂⁴×C₃	C₂²	1	1	yes	characteristic
12	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃×D₄	C₂²	1	2	yes	—
14	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄	C₂²	1	2	yes	—
16	C₂²×C₃×C₄	C₂²	1	2	yes	—
17	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
18	C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
19	C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
22	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
29	C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
30	C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
31	C₃×C₄²	C₂²	1	2	yes	complemented
32	#126	C₂×C₃	1	2	yes	complemented
33	#124	C₂×C₃	1	1	yes	characteristic, complemented
34	#123	C₂×C₃	1	1	yes	characteristic, complemented
35	#123	C₂×C₃	1	2	yes	complemented
36	C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
37	#120	C₂×C₃	1	1	yes	characteristic
38	C₄×D₄	C₂×C₃	1	2	yes	complemented
39	#117	C₂×C₃	1	2	yes	complemented
40	C₂×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
41	C₂³×C₃	—	2	4	no	—
42	C₂³×C₃	—	4	1	no	—
43	C₂³×C₃	C₂³	1	1	yes	characteristic
44	C₂³×C₃	C₂³	1	1	yes	characteristic
45	C₂³×C₃	C₂³	1	2	yes	—
46	C₃×Q₈	—	4	1	no	—
47	C₃×D₄	—	2	4	no	—
48	C₃×D₄	—	4	1	no	—
49	C₂×C₃×C₄	—	2	4	no	—
50	C₂×C₃×C₄	—	4	2	no	—
51	C₂×C₃×C₄	—	4	1	no	—
52	C₂×C₃×C₄	C₂³	1	2	yes	—
53	C₂×C₃×C₄	C₂³	1	2	yes	—
54	C₂×C₃×C₄	C₂³	1	2	yes	—
55	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄	C₂³	1	2	yes	—
57	C₂×C₃×C₄	C₂³	1	2	yes	—
58	C₂⁴	C₂²×C₃	1	1	yes	characteristic
59	C₂×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
60	C₂×D₄	C₂²×C₃	1	2	yes	—
61	C₂×D₄	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₄	C₂²×C₃	1	2	yes	—
63	C₂²×C₄	C₂²×C₃	1	2	yes	—
64	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic, complemented
65	C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
66	C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
67	C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
68	C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
69	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
70	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
75	C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
76	C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
77	C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
78	C₄²	C₂²×C₃	1	2	yes	complemented
79	C₂²×C₃	—	2	2	no	—
80	C₂²×C₃	—	2	2	no	—
81	C₂²×C₃	—	2	2	no	—
82	C₂²×C₃	—	2	2	no	—
83	C₂²×C₃	—	2	1	no	—
84	C₂²×C₃	—	2	1	no	—
85	C₂²×C₃	—	4	4	no	—
86	C₂²×C₃	#41	1	4	yes	—
87	C₂²×C₃	C₂⁴	1	1	yes	characteristic
88	C₃×C₄	—	2	2	no	—
89	C₃×C₄	—	2	2	no	—
90	C₃×C₄	—	2	2	no	—
91	C₃×C₄	—	2	2	no	—
92	C₃×C₄	—	2	2	no	—
93	C₃×C₄	—	2	1	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	4	1	no	—
96	C₂³	C₂³×C₃	1	1	yes	characteristic
97	C₂³	C₂³×C₃	1	1	yes	characteristic
98	C₂³	C₂³×C₃	1	2	yes	—
99	Q₈	—	4	1	no	—
100	D₄	—	2	4	no	—
101	D₄	—	4	1	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	C₂³×C₃	1	2	yes	—
106	C₂×C₄	C₂³×C₃	1	2	yes	—
107	C₂×C₄	C₂³×C₃	1	2	yes	—
108	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₄	C₂³×C₃	1	2	yes	—
110	C₂×C₄	C₂³×C₃	1	2	yes	—
111	C₂×C₃	—	2	4	no	—
112	C₂×C₃	—	4	1	no	—
113	C₂×C₃	—	4	1	no	—
114	C₂×C₃	C₂×#41	1	2	yes	—
115	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	4	4	no	—
123	C₂²	C₃×#41	1	4	yes	—
124	C₂²	C₂⁴×C₃	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₃	#551	1	1	yes	characteristic, direct product factor, Sylow
132	C₂	—	2	4	no	—
133	C₂	—	4	1	no	—
134	C₂	—	4	1	no	—
135	C₂	C₂×C₃×#41	1	2	yes	—
136	C₂	C₃×H(5,𝔽₂)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#126

C₂

yes

maximal, complemented

C₃×#124

C₂

yes

maximal, characteristic, complemented

C₃×#123

C₂

yes

maximal, characteristic, complemented

C₃×#123

C₂

yes

maximal, complemented

C₃×C₂²⋊Q₈

C₂

yes

maximal, complemented

C₃×#120

C₂

yes

maximal, characteristic

C₃×C₄×D₄

C₂

yes

maximal, complemented

C₃×#117

C₂

yes

maximal, complemented

C₂×C₃×C₂²⋊C₄

C₂

yes

maximal, complemented

#551

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂⁴×C₃

C₂²

yes

characteristic

C₂×C₃×Q₈

C₂²

yes

characteristic, complemented

C₂×C₃×D₄

C₂²

yes

—

C₂×C₃×D₄

C₂²

yes

characteristic

C₂²×C₃×C₄

C₂²

yes

—

C₂²×C₃×C₄

C₂²

yes

—

C₂²×C₃×C₄

C₂²

yes

characteristic, complemented

C₃×C₄⋊C₄

C₂²

yes

complemented

C₃×C₄⋊C₄

C₂²

yes

complemented

C₃×C₄⋊C₄

C₂²

yes

complemented

C₃×C₄⋊C₄

C₂²

yes

complemented

C₃×C₂²⋊C₄

C₂²

yes

—

C₃×C₂²⋊C₄

C₂²

yes

—

C₃×C₂²⋊C₄

C₂²

yes

—

C₃×C₂²⋊C₄

C₂²

yes

—

C₃×C₂²⋊C₄

C₂²

yes

—

C₃×C₂²⋊C₄

C₂²

yes

characteristic

C₃×C₂²⋊C₄

C₂²

yes

complemented

C₃×C₂²⋊C₄

C₂²

yes

characteristic, complemented

C₃×C₄²

C₂²

yes

characteristic, complemented

C₃×C₄²

C₂²

yes

complemented

#126

C₂×C₃

yes

complemented

#124

C₂×C₃

yes

characteristic, complemented

#123

C₂×C₃

yes

characteristic, complemented

#123

C₂×C₃

yes

complemented

C₂²⋊Q₈

C₂×C₃

yes

complemented

#120

C₂×C₃

yes

characteristic

C₄×D₄

C₂×C₃

yes

complemented

#117

C₂×C₃

yes

complemented

C₂×C₂²⋊C₄

C₂×C₃

yes

complemented

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂³

yes

characteristic

C₂³×C₃

C₂³

yes

characteristic

C₂³×C₃

C₂³

yes

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂³

yes

—

C₂×C₃×C₄

C₂³

yes

—

C₂×C₃×C₄

C₂³

yes

—

C₂×C₃×C₄

C₂³

yes

characteristic

C₂×C₃×C₄

C₂³

yes

—

C₂×C₃×C₄

C₂³

yes

—

C₂⁴

C₂²×C₃

yes

characteristic

C₂×Q₈

C₂²×C₃

yes

characteristic, complemented

C₂×D₄

C₂²×C₃

yes

—

C₂×D₄

C₂²×C₃

yes

characteristic

C₂²×C₄

C₂²×C₃

yes

—

C₂²×C₄

C₂²×C₃

yes

—

C₂²×C₄

C₂²×C₃

yes

characteristic, complemented

C₄⋊C₄

C₂²×C₃

yes

complemented

C₄⋊C₄

C₂²×C₃

yes

complemented

C₄⋊C₄

C₂²×C₃

yes

complemented

C₄⋊C₄

C₂²×C₃

yes

complemented

C₂²⋊C₄

C₂²×C₃

yes

—

C₂²⋊C₄

C₂²×C₃

yes

—

C₂²⋊C₄

C₂²×C₃

yes

—

C₂²⋊C₄

C₂²×C₃

yes

—

C₂²⋊C₄

C₂²×C₃

yes

—

C₂²⋊C₄

C₂²×C₃

yes

characteristic

C₂²⋊C₄

C₂²×C₃

yes

complemented

C₂²⋊C₄

C₂²×C₃

yes

characteristic, complemented

C₄²

C₂²×C₃

yes

characteristic, complemented

C₄²

C₂²×C₃

yes

complemented

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#41

yes

—

C₂²×C₃

C₂⁴

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₂³

—

C₂³

—

C₂³

C₂³×C₃

yes

characteristic

C₂³

C₂³×C₃

yes

characteristic

C₂³

C₂³×C₃

yes

—

Q₈

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂³×C₃

yes

—

106

C₂×C₄

C₂³×C₃

yes

—

107

C₂×C₄

C₂³×C₃

yes

—

108

C₂×C₄

C₂³×C₃

yes

characteristic

109

C₂×C₄

C₂³×C₃

yes

—

110

C₂×C₄

C₂³×C₃

yes

—

111

C₂×C₃

—

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂×C₃

C₂×#41

yes

—

115

C₂×C₃

H(5,𝔽₂)

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

C₃×#41

yes

—

124

C₂²

C₂⁴×C₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₃

#551

yes

characteristic, direct product factor, Sylow

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

C₂×C₃×#41

yes

—

136

C₂

C₃×H(5,𝔽₂)

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₃×#120	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₃

C₂⁴

commutator, Frattini, cosocle

C₂²

C₂⁴×C₃

C₂

G ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₃ ◁ G

N	H ≅ G/N	type
1. C₃×#126	C₂	semidirect
2. C₃×#124	C₂	semidirect
3. C₃×#123	C₂	semidirect
4. C₃×#123	C₂	semidirect
5. C₃×C₂²⋊Q₈	C₂	semidirect
7. C₃×C₄×D₄	C₂	semidirect
8. C₃×#117	C₂	semidirect
9. C₂×C₃×C₂²⋊C₄	C₂	semidirect
10. #551	C₃	direct (N × H)
12. C₂×C₃×Q₈	C₂²	semidirect
17. C₂²×C₃×C₄	C₂²	semidirect
18. C₃×C₄⋊C₄	C₂²	semidirect
19. C₃×C₄⋊C₄	C₂²	semidirect
20. C₃×C₄⋊C₄	C₂²	semidirect
21. C₃×C₄⋊C₄	C₂²	semidirect
28. C₃×C₂²⋊C₄	C₂²	semidirect
29. C₃×C₂²⋊C₄	C₂²	semidirect
30. C₃×C₄²	C₂²	semidirect
31. C₃×C₄²	C₂²	semidirect
32. #126	C₂×C₃	semidirect
33. #124	C₂×C₃	semidirect
34. #123	C₂×C₃	semidirect
35. #123	C₂×C₃	semidirect
36. C₂²⋊Q₈	C₂×C₃	semidirect
38. C₄×D₄	C₂×C₃	semidirect
39. #117	C₂×C₃	semidirect
40. C₂×C₂²⋊C₄	C₂×C₃	semidirect
59. C₂×Q₈	C₂²×C₃	semidirect
64. C₂²×C₄	C₂²×C₃	semidirect
65. C₄⋊C₄	C₂²×C₃	semidirect
66. C₄⋊C₄	C₂²×C₃	semidirect
67. C₄⋊C₄	C₂²×C₃	semidirect
68. C₄⋊C₄	C₂²×C₃	semidirect
75. C₂²⋊C₄	C₂²×C₃	semidirect
76. C₂²⋊C₄	C₂²×C₃	semidirect
77. C₄²	C₂²×C₃	semidirect
78. C₄²	C₂²×C₃	semidirect
131. C₃	#551	direct (N × H)

H ≅ G/N

type

1. C₃×#126

C₂

semidirect

2. C₃×#124

C₂

semidirect

3. C₃×#123

C₂

semidirect

4. C₃×#123

C₂

semidirect

5. C₃×C₂²⋊Q₈

C₂

semidirect

7. C₃×C₄×D₄

C₂

semidirect

8. C₃×#117

C₂

semidirect

9. C₂×C₃×C₂²⋊C₄

C₂

semidirect

10. #551

C₃

direct (N × H)

12. C₂×C₃×Q₈

C₂²

semidirect

17. C₂²×C₃×C₄

C₂²

semidirect

18. C₃×C₄⋊C₄

C₂²

semidirect

19. C₃×C₄⋊C₄

C₂²

semidirect

20. C₃×C₄⋊C₄

C₂²

semidirect

21. C₃×C₄⋊C₄

C₂²

semidirect

28. C₃×C₂²⋊C₄

C₂²

semidirect

29. C₃×C₂²⋊C₄

C₂²

semidirect

30. C₃×C₄²

C₂²

semidirect

31. C₃×C₄²

C₂²

semidirect

32. #126

C₂×C₃

semidirect

33. #124

C₂×C₃

semidirect

34. #123

C₂×C₃

semidirect

35. #123

C₂×C₃

semidirect

36. C₂²⋊Q₈

C₂×C₃

semidirect

38. C₄×D₄

C₂×C₃

semidirect

39. #117

C₂×C₃

semidirect

40. C₂×C₂²⋊C₄

C₂×C₃

semidirect

59. C₂×Q₈

C₂²×C₃

semidirect

64. C₂²×C₄

C₂²×C₃

semidirect

65. C₄⋊C₄

C₂²×C₃

semidirect

66. C₄⋊C₄

C₂²×C₃

semidirect

67. C₄⋊C₄

C₂²×C₃

semidirect

68. C₄⋊C₄

C₂²×C₃

semidirect

75. C₂²⋊C₄

C₂²×C₃

semidirect

76. C₂²⋊C₄

C₂²×C₃

semidirect

77. C₄²

C₂²×C₃

semidirect

78. C₄²

C₂²×C₃

semidirect

131. C₃

#551

direct (N × H)

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

subgroup	index
C₃×#126	2
C₃×#124	2
C₃×#123	2
C₃×C₂²⋊Q₈	2
C₃×#120	2
C₃×C₄×D₄	2
C₃×#117	2
C₂×C₃×C₂²⋊C₄	2
#551	3

subgroup

index

C₃×C₂²⋊Q₈

C₃×C₄×D₄

C₂×C₃×C₂²⋊C₄

Subgroups — C₃×#551 #5881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#551 #5881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#551 #5881