Finite Group Databasebeta

Subgroups — #65924

Order 320.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×D₅	C₂	1	2	yes	maximal, complemented
3	C₅×#120	C₂	1	1	yes	maximal, characteristic, complemented
4	#4086	C₂	1	1	yes	maximal, characteristic, complemented
5	#4085	C₂	1	2	yes	maximal, complemented
6	#4083	C₂	1	1	yes	maximal, characteristic, complemented
7	#4030	C₂	1	2	yes	maximal, complemented
8	#4028	C₂	1	2	yes	maximal, complemented
9	#4027	C₂	1	2	yes	maximal, complemented
10	C₂⁴×C₅	C₂²	1	1	yes	characteristic, complemented
11	C₂³×D₅	C₂²	1	1	yes	characteristic, complemented
12	C₂³×D₅	C₂²	1	2	yes	complemented
13	C₂×C₅×D₄	C₂²	1	2	yes	complemented
14	C₂×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₂²⋊D₅	—	2	8	no	—
16	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂×C₂²⋊D₅	C₂²	1	2	yes	complemented
18	C₂×C₂²⋊D₅	C₂²	1	2	yes	complemented
19	C₂×C₂²⋊D₅	C₂²	1	2	yes	complemented
20	C₂×C₂²⋊D₅	C₂²	1	2	yes	complemented
21	C₂²×Dic₅	C₂²	1	2	yes	complemented
22	D₄×D₅	—	2	8	no	—
23	C₂×D₂₀	C₂²	1	2	yes	complemented
24	C₂×D₂₀	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₄×D₅	C₂²	1	2	yes	complemented
26	C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
27	C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
28	C₂²⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
29	#685	C₂²	1	2	yes	complemented
30	#685	C₂²	1	2	yes	complemented
31	#683	C₂²	1	2	yes	complemented
32	C₄×Dic₅	C₂²	1	1	yes	characteristic, complemented
33	D₄²	—	5	1	no	maximal, Sylow
34	C₂³×C₅	—	2	4	no	—
35	C₂³×C₅	—	4	1	no	—
36	C₂³×C₅	C₂³	1	1	yes	characteristic
37	C₂³×C₅	C₂³	1	1	yes	characteristic
38	C₂³×C₅	C₂³	1	2	yes	—
39	C₂²×D₅	—	2	4	no	—
40	C₂²×D₅	—	2	4	no	—
41	C₂²×D₅	—	2	4	no	—
42	C₂²×D₅	—	4	1	no	—
43	C₂²×D₅	—	4	2	no	—
44	C₂²×D₅	C₂³	1	2	yes	—
45	C₂²×D₅	C₂³	1	2	yes	—
46	C₅×D₄	—	2	4	no	—
47	C₅×D₄	—	4	1	no	—
48	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₅	C₂³	1	2	yes	—
50	C₂²⋊D₅	—	2	4	no	—
51	C₂²⋊D₅	—	2	4	no	—
52	C₂²⋊D₅	—	2	4	no	—
53	C₂²⋊D₅	—	2	4	no	—
54	C₂²⋊D₅	D₄	1	8	yes	complemented
55	C₂×Dic₅	—	2	4	no	—
56	C₂×Dic₅	C₂³	1	2	yes	—
57	C₂×Dic₅	C₂³	1	2	yes	—
58	D₂₀	—	2	4	no	—
59	D₂₀	—	4	1	no	—
60	C₄×D₅	—	2	4	no	—
61	C₂²×D₄	—	5	2	no	—
62	C₂²×D₄	—	5	2	no	—
63	#127	—	5	1	no	—
64	#121	—	5	2	no	—
65	#121	—	5	2	no	—
66	#120	—	5	1	no	—
67	#120	—	5	2	no	—
68	#120	D₅	1	1	yes	characteristic, complemented
69	C₄×D₄	—	5	2	no	—
70	C₂²×C₅	—	2	2	no	—
71	C₂²×C₅	—	2	2	no	—
72	C₂²×C₅	—	2	2	no	—
73	C₂²×C₅	—	2	2	no	—
74	C₂²×C₅	—	2	1	no	—
75	C₂²×C₅	—	2	1	no	—
76	C₂²×C₅	—	4	4	no	—
77	C₂²×C₅	C₂×D₄	1	4	yes	complemented
78	C₂²×C₅	C₂⁴	1	1	yes	characteristic
79	C₂×D₅	—	2	2	no	—
80	C₂×D₅	—	2	2	no	—
81	C₂×D₅	—	2	2	no	—
82	C₂×D₅	—	2	2	no	—
83	C₂×D₅	—	2	2	no	—
84	C₂×D₅	—	4	4	no	—
85	C₂×D₅	—	4	8	no	—
86	C₂×D₅	C₂×D₄	1	4	yes	complemented
87	C₄×C₅	—	2	2	no	—
88	C₄×C₅	—	2	1	no	—
89	Dic₅	—	2	2	no	—
90	Dic₅	C₂×D₄	1	4	yes	complemented
91	C₂⁴	—	5	1	no	—
92	C₂⁴	—	5	2	no	—
93	C₂⁴	C₂×D₅	1	1	yes	characteristic, complemented
94	C₂×D₄	—	5	2	no	—
95	C₂×D₄	—	5	2	no	—
96	C₂×D₄	—	5	2	no	—
97	C₂×D₄	—	5	2	no	—
98	C₂×D₄	—	5	2	no	—
99	C₂×D₄	—	5	2	no	—
100	C₂×D₄	—	5	1	no	—
101	C₂×D₄	—	10	8	no	—
102	C₂×D₄	—	10	8	no	—
103	C₂×D₄	C₂×D₅	1	2	yes	complemented
104	C₂×D₄	C₂×D₅	1	1	yes	characteristic, complemented
105	C₂²×C₄	—	5	2	no	—
106	C₂²×C₄	—	5	2	no	—
107	C₄⋊C₄	—	5	2	no	—
108	C₂²⋊C₄	—	5	1	no	—
109	C₂²⋊C₄	—	5	2	no	—
110	C₂²⋊C₄	—	5	2	no	—
111	C₂²⋊C₄	C₂×D₅	1	2	yes	complemented
112	C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
113	C₄²	—	5	1	no	—
114	C₂×C₅	—	2	4	no	—
115	C₂×C₅	—	4	1	no	—
116	C₂×C₅	—	4	1	no	—
117	C₂×C₅	C₂²×D₄	1	2	yes	—
118	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
119	D₅	—	2	4	no	—
120	D₅	—	4	2	no	—
121	C₂³	—	2	4	no	—
122	C₂³	—	4	1	no	—
123	C₂³	—	5	2	no	—
124	C₂³	—	5	2	no	—
125	C₂³	—	10	4	no	—
126	C₂³	—	10	4	no	—
127	C₂³	—	10	4	no	—
128	C₂³	—	20	2	no	—
129	C₂³	—	20	1	no	—
130	C₂³	C₂²×D₅	1	1	yes	characteristic
131	C₂³	C₂²×D₅	1	1	yes	characteristic
132	C₂³	C₂²×D₅	1	2	yes	—
133	D₄	—	2	4	no	—
134	D₄	—	4	1	no	—
135	D₄	—	5	8	no	—
136	D₄	—	10	4	no	—
137	D₄	—	10	4	no	—
138	D₄	—	10	4	no	—
139	D₄	—	10	4	no	—
140	D₄	—	10	4	no	—
141	D₄	—	20	1	no	—
142	C₂×C₄	—	5	2	no	—
143	C₂×C₄	—	5	2	no	—
144	C₂×C₄	—	10	4	no	—
145	C₂×C₄	—	10	4	no	—
146	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₂×C₄	C₂²×D₅	1	2	yes	—
148	C₅	D₄²	1	1	yes	characteristic, complemented, Sylow
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	5	4	no	—
157	C₂²	—	10	2	no	—
158	C₂²	—	10	2	no	—
159	C₂²	—	10	2	no	—
160	C₂²	—	10	2	no	—
161	C₂²	—	10	2	no	—
162	C₂²	—	20	8	no	—
163	C₂²	—	20	4	no	—
164	C₂²	D₄×D₅	1	4	yes	complemented
165	C₂²	C₂³×D₅	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	1	no	—
168	C₄	—	5	4	no	—
169	C₄	—	10	2	no	—
170	C₂	—	2	4	no	—
171	C₂	—	4	1	no	—
172	C₂	—	4	1	no	—
173	C₂	—	10	4	no	—
174	C₂	—	20	2	no	—
175	C₂	C₂×D₄×D₅	1	2	yes	—
176	C₂	#4144	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#120	C₂
hypercenter	#120	D₅
hypocenter	C₅	D₄²
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #120

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₅	C₂	semidirect
2. C₂×D₄×D₅	C₂	semidirect
3. C₅×#120	C₂	semidirect
4. #4086	C₂	semidirect
5. #4085	C₂	semidirect
6. #4083	C₂	semidirect
7. #4030	C₂	semidirect
8. #4028	C₂	semidirect
9. #4027	C₂	semidirect
10. C₂⁴×C₅	C₂²	semidirect
11. C₂³×D₅	C₂²	semidirect
12. C₂³×D₅	C₂²	semidirect
13. C₂×C₅×D₄	C₂²	semidirect
14. C₂×C₅×D₄	C₂²	semidirect
17. C₂×C₂²⋊D₅	C₂²	semidirect
18. C₂×C₂²⋊D₅	C₂²	semidirect
19. C₂×C₂²⋊D₅	C₂²	semidirect
20. C₂×C₂²⋊D₅	C₂²	semidirect
21. C₂²×Dic₅	C₂²	semidirect
23. C₂×D₂₀	C₂²	semidirect
24. C₂×D₂₀	C₂²	semidirect
25. C₂×C₄×D₅	C₂²	semidirect
26. C₅×C₂²⋊C₄	C₂²	semidirect
27. C₅×C₂²⋊C₄	C₂²	semidirect
28. C₂²⋊Dic₅	C₂²	semidirect
29. #685	C₂²	semidirect
30. #685	C₂²	semidirect
31. #683	C₂²	semidirect
32. C₄×Dic₅	C₂²	semidirect
54. C₂²⋊D₅	D₄	semidirect
68. #120	D₅	semidirect
77. C₂²×C₅	C₂×D₄	semidirect
86. C₂×D₅	C₂×D₄	semidirect
90. Dic₅	C₂×D₄	semidirect
93. C₂⁴	C₂×D₅	semidirect
103. C₂×D₄	C₂×D₅	semidirect
104. C₂×D₄	C₂×D₅	semidirect
111. C₂²⋊C₄	C₂×D₅	semidirect
112. C₂²⋊C₄	C₂×D₅	semidirect
148. C₅	D₄²	semidirect
164. C₂²	D₄×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊D₅	2
C₂×D₄×D₅	2
C₅×#120	2
#4086	2
#4085	2
#4083	2
#4030	2
#4028	2
#4027	2
D₄²	5