Group #65921 — subgroups

Subgroups — #65921

Order 320.

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	C₅×#120	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₅	C₂	1	2	yes	maximal, complemented
3	#4082	C₂	1	1	yes	maximal, characteristic, complemented
4	#4081	C₂	1	1	yes	maximal, characteristic, complemented
5	#4081	C₂	1	2	yes	maximal, complemented
6	D₄×Dic₅	C₂	1	2	yes	maximal, complemented
7	#4025	C₂	1	2	yes	maximal, complemented
8	#4024	C₂	1	2	yes	maximal, complemented
9	#4023	C₂	1	2	yes	maximal, complemented
10	C₂⁴×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×D₄	C₂²	1	2	yes	—
12	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂²×Dic₅	C₂²	1	2	yes	—
14	C₂²×Dic₅	C₂²	1	2	yes	—
15	C₂²×Dic₅	C₂²	1	1	yes	characteristic, complemented
16	C₂×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
17	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
19	C₂²⋊Dic₅	C₂²	1	2	yes	—
20	C₂²⋊Dic₅	C₂²	1	2	yes	—
21	C₂²⋊Dic₅	C₂²	1	2	yes	—
22	C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₂²⋊Dic₅	C₂²	1	2	yes	complemented
24	C₂²⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
25	C₄⋊Dic₅	C₂²	1	2	yes	complemented
26	#683	C₂²	1	2	yes	complemented
27	#683	C₂²	1	2	yes	complemented
28	#683	C₂²	1	2	yes	complemented
29	C₄×Dic₅	C₂²	1	1	yes	characteristic, complemented
30	C₄×Dic₅	C₂²	1	2	yes	complemented
31	#551	—	5	1	no	maximal, Sylow
32	C₂³×C₅	—	2	4	no	—
33	C₂³×C₅	—	4	1	no	—
34	C₂³×C₅	C₂³	1	1	yes	characteristic
35	C₂³×C₅	C₂³	1	1	yes	characteristic
36	C₂³×C₅	C₂³	1	2	yes	—
37	C₅×D₄	—	2	4	no	—
38	C₅×D₄	—	4	1	no	—
39	C₂×C₄×C₅	C₂³	1	2	yes	—
40	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×Dic₅	—	2	4	no	—
42	C₂×Dic₅	—	4	2	no	—
43	C₂×Dic₅	—	4	1	no	—
44	C₂×Dic₅	C₂³	1	2	yes	—
45	C₂×Dic₅	C₂³	1	2	yes	—
46	C₂×Dic₅	C₂³	1	2	yes	—
47	C₂×Dic₅	C₂³	1	2	yes	—
48	Dic₁₀	—	4	1	no	—
49	#126	—	5	2	no	—
50	#124	—	5	1	no	—
51	#123	—	5	2	no	—
52	#123	—	5	1	no	—
53	C₂²⋊Q₈	—	5	2	no	—
54	#120	D₅	1	1	yes	characteristic
55	C₄×D₄	—	5	2	no	—
56	#117	—	5	2	no	—
57	C₂×C₂²⋊C₄	—	5	2	no	—
58	C₂²×C₅	—	2	2	no	—
59	C₂²×C₅	—	2	2	no	—
60	C₂²×C₅	—	2	2	no	—
61	C₂²×C₅	—	2	2	no	—
62	C₂²×C₅	—	2	1	no	—
63	C₂²×C₅	—	2	1	no	—
64	C₂²×C₅	—	4	4	no	—
65	C₂²×C₅	#41	1	4	yes	—
66	C₂²×C₅	C₂⁴	1	1	yes	characteristic
67	C₄×C₅	—	2	2	no	—
68	C₄×C₅	—	2	1	no	—
69	Dic₅	—	2	2	no	—
70	Dic₅	—	2	2	no	—
71	Dic₅	—	2	2	no	—
72	Dic₅	—	2	2	no	—
73	C₂⁴	C₂×D₅	1	1	yes	characteristic
74	C₂×Q₈	—	5	1	no	—
75	C₂×D₄	C₂×D₅	1	2	yes	—
76	C₂×D₄	C₂×D₅	1	1	yes	characteristic
77	C₂²×C₄	—	5	2	no	—
78	C₂²×C₄	—	5	2	no	—
79	C₂²×C₄	—	5	1	no	—
80	C₄⋊C₄	—	5	2	no	—
81	C₄⋊C₄	—	5	2	no	—
82	C₄⋊C₄	—	5	2	no	—
83	C₄⋊C₄	—	5	2	no	—
84	C₂²⋊C₄	—	5	2	no	—
85	C₂²⋊C₄	—	5	2	no	—
86	C₂²⋊C₄	—	5	2	no	—
87	C₂²⋊C₄	—	5	2	no	—
88	C₂²⋊C₄	—	5	2	no	—
89	C₂²⋊C₄	—	5	1	no	—
90	C₂²⋊C₄	C₂×D₅	1	2	yes	—
91	C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
92	C₄²	—	5	1	no	—
93	C₄²	—	5	2	no	—
94	C₂×C₅	—	2	4	no	—
95	C₂×C₅	—	4	1	no	—
96	C₂×C₅	—	4	1	no	—
97	C₂×C₅	C₂×#41	1	2	yes	—
98	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂³	—	2	4	no	—
100	C₂³	—	4	1	no	—
101	C₂³	C₂²×D₅	1	1	yes	characteristic
102	C₂³	C₂²×D₅	1	1	yes	characteristic
103	C₂³	C₂²×D₅	1	2	yes	—
104	Q₈	—	20	1	no	—
105	D₄	—	2	4	no	—
106	D₄	—	4	1	no	—
107	C₂×C₄	—	5	2	no	—
108	C₂×C₄	—	5	2	no	—
109	C₂×C₄	—	5	2	no	—
110	C₂×C₄	—	5	2	no	—
111	C₂×C₄	—	10	4	no	—
112	C₂×C₄	—	20	1	no	—
113	C₂×C₄	—	20	2	no	—
114	C₂×C₄	C₂²×D₅	1	2	yes	—
115	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
116	C₅	#551	1	1	yes	characteristic, complemented, Sylow
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	#711	1	4	yes	—
125	C₂²	C₂³×D₅	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	10	2	no	—
129	C₄	—	10	2	no	—
130	C₄	—	10	2	no	—
131	C₄	—	10	2	no	—
132	C₂	—	2	4	no	—
133	C₂	—	4	1	no	—
134	C₂	—	4	1	no	—
135	C₂	C₂×#711	1	2	yes	—
136	C₂	#4144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#120

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₅

C₂

yes

maximal, complemented

#4082

C₂

yes

maximal, characteristic, complemented

#4081

C₂

yes

maximal, characteristic, complemented

#4081

C₂

yes

maximal, complemented

D₄×Dic₅

C₂

yes

maximal, complemented

#4025

C₂

yes

maximal, complemented

#4024

C₂

yes

maximal, complemented

#4023

C₂

yes

maximal, complemented

C₂⁴×C₅

C₂²

yes

characteristic

C₂×C₅×D₄

C₂²

yes

—

C₂×C₅×D₄

C₂²

yes

characteristic

C₂²×Dic₅

C₂²

yes

—

C₂²×Dic₅

C₂²

yes

—

C₂²×Dic₅

C₂²

yes

characteristic, complemented

C₂×Dic₁₀

C₂²

yes

characteristic, complemented

C₅×C₂²⋊C₄

C₂²

yes

—

C₅×C₂²⋊C₄

C₂²

yes

characteristic

C₂²⋊Dic₅

C₂²

yes

—

C₂²⋊Dic₅

C₂²

yes

—

C₂²⋊Dic₅

C₂²

yes

—

C₂²⋊Dic₅

C₂²

yes

—

C₂²⋊Dic₅

C₂²

yes

complemented

C₂²⋊Dic₅

C₂²

yes

characteristic, complemented

C₄⋊Dic₅

C₂²

yes

complemented

#683

C₂²

yes

complemented

#683

C₂²

yes

complemented

#683

C₂²

yes

complemented

C₄×Dic₅

C₂²

yes

characteristic, complemented

C₄×Dic₅

C₂²

yes

complemented

#551

—

maximal, Sylow

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂³

yes

characteristic

C₂³×C₅

C₂³

yes

characteristic

C₂³×C₅

C₂³

yes

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

Dic₁₀

—

—

—

—

—

C₂²⋊Q₈

—

#120

D₅

yes

characteristic

C₄×D₄

—

#117

—

C₂×C₂²⋊C₄

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#41

yes

—

C₂²×C₅

C₂⁴

yes

characteristic

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂⁴

C₂×D₅

yes

characteristic

C₂×Q₈

—

C₂×D₄

C₂×D₅

yes

—

C₂×D₄

C₂×D₅

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×D₅

yes

—

C₂²⋊C₄

C₂×D₅

yes

characteristic

C₄²

—

C₄²

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂×#41

yes

—

C₂×C₅

H(5,𝔽₂)

yes

characteristic

C₂³

—

100

C₂³

—

101

C₂³

C₂²×D₅

yes

characteristic

102

C₂³

C₂²×D₅

yes

characteristic

103

C₂³

C₂²×D₅

yes

—

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×D₅

yes

—

115

C₂×C₄

C₂²×D₅

yes

characteristic

116

C₅

#551

yes

characteristic, complemented, Sylow

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#711

yes

—

125

C₂²

C₂³×D₅

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

C₂×#711

yes

—

136

C₂

#4144

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting, prime radical	C₅×#120	C₂
hypercenter	#120	D₅
hypocenter	C₅	#551
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #120

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

Fitting, prime radical

C₂

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #120

N	H ≅ G/N	type
2. C₂×C₂²⋊Dic₅	C₂	semidirect
3. #4082	C₂	semidirect
4. #4081	C₂	semidirect
5. #4081	C₂	semidirect
6. D₄×Dic₅	C₂	semidirect
7. #4025	C₂	semidirect
8. #4024	C₂	semidirect
9. #4023	C₂	semidirect
15. C₂²×Dic₅	C₂²	semidirect
16. C₂×Dic₁₀	C₂²	semidirect
23. C₂²⋊Dic₅	C₂²	semidirect
24. C₂²⋊Dic₅	C₂²	semidirect
25. C₄⋊Dic₅	C₂²	semidirect
26. #683	C₂²	semidirect
27. #683	C₂²	semidirect
28. #683	C₂²	semidirect
29. C₄×Dic₅	C₂²	semidirect
30. C₄×Dic₅	C₂²	semidirect
116. C₅	#551	semidirect

H ≅ G/N

type

2. C₂×C₂²⋊Dic₅

C₂

semidirect

3. #4082

C₂

semidirect

4. #4081

C₂

semidirect

5. #4081

C₂

semidirect

6. D₄×Dic₅

C₂

semidirect

7. #4025

C₂

semidirect

8. #4024

C₂

semidirect

9. #4023

C₂

semidirect

15. C₂²×Dic₅

C₂²

semidirect

16. C₂×Dic₁₀

C₂²

semidirect

23. C₂²⋊Dic₅

C₂²

semidirect

24. C₂²⋊Dic₅

C₂²

semidirect

25. C₄⋊Dic₅

C₂²

semidirect

26. #683

C₂²

semidirect

27. #683

C₂²

semidirect

28. #683

C₂²

semidirect

29. C₄×Dic₅

C₂²

semidirect

30. C₄×Dic₅

C₂²

semidirect

116. C₅

#551

semidirect

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

subgroup	index
C₅×#120	2
C₂×C₂²⋊Dic₅	2
#4082	2
#4081	2
D₄×Dic₅	2
#4025	2
#4024	2
#4023	2
#551	5

subgroup

index

C₂×C₂²⋊Dic₅

D₄×Dic₅

Subgroups — #65921

Conjugacy classes of subgroups

Complements

Maximal subgroups