Finite Group Databasebeta

Subgroups — #10615548

Order 640.

738 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 249 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	#66283	C₂	1	1	yes	maximal, characteristic
2	C₅×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	#41×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#65428	C₂	1	6	yes	maximal, complemented
5	C₂×#3957	C₂	1	6	yes	maximal, complemented
6	C₂×C₅×#41	C₂²	1	1	yes	characteristic
7	#4150	C₄	1	4	yes	complemented
8	#4149	C₄	1	4	yes	complemented
9	D₅×#41	—	2	4	no	—
10	#4147	—	2	1	no	—
11	#4144	—	2	3	no	—
12	C₂×#709	C₂²	1	6	yes	—
13	C₅×#131	—	2	4	no	—
14	C₂×C₅×M₁₆	C₂²	1	6	yes	complemented
15	Q₈×Dic₅	—	2	1	no	—
16	D₄×Dic₅	—	2	3	no	—
17	#4072	C₂²	1	3	yes	complemented
18	C₂×C₄×Dic₅	C₂²	1	3	yes	complemented
19	#3957	C₂²	1	4	yes	complemented
20	#3957	C₂²	1	12	yes	complemented
21	#2901	—	5	1	no	maximal, Sylow
22	C₅×#41	C₂×C₄	1	4	yes	—
23	C₅×#41	D₄	1	4	yes	—
24	C₂×C₅×Q₈	D₄	1	1	yes	characteristic
25	C₂×C₅×D₄	D₄	1	3	yes	—
26	C₂²×C₄×C₅	C₂³	1	3	yes	—
27	C₂×C₂²⋊D₅	—	2	6	no	—
28	C₂²×Dic₅	—	2	3	no	—
29	#713	—	2	4	no	—
30	Q₈×D₅	—	2	4	no	—
31	#711	—	2	12	no	—
32	D₄×D₅	—	2	12	no	—
33	#709	—	2	6	no	—
34	#709	—	2	6	no	—
35	#709	C₂×C₄	1	12	yes	complemented
36	#709	C₂³	1	4	yes	—
37	C₂×D₂₀	C₂×C₄	1	6	yes	complemented
38	C₂×C₄×D₅	—	2	6	no	—
39	C₂×Dic₁₀	C₂×C₄	1	6	yes	complemented
40	C₅×M₁₆	—	2	6	no	—
41	C₅×M₁₆	C₂³	1	4	yes	—
42	C₂×C₅×C₈	—	2	6	no	—
43	C₂²⋊Dic₅	—	2	3	no	—
44	C₄⋊Dic₅	—	2	3	no	—
45	C₄×Dic₅	—	2	3	no	—
46	C₄×Dic₅	C₂³	1	1	yes	characteristic
47	C₄×Dic₅	C₂³	1	3	yes	—
48	#585	—	5	1	no	—
49	#568	D₅	1	1	yes	characteristic, complemented
50	C₄×#41	—	5	1	no	—
51	#421	—	5	6	no	—
52	C₂×#104	—	5	6	no	—
53	C₂³×C₅	C₂×D₄	1	3	yes	—
54	C₂²×D₅	—	2	6	no	—
55	C₅×Q₈	C₂²⋊C₄	1	4	yes	—
56	C₅×D₄	C₂²⋊C₄	1	12	yes	—
57	C₂×C₄×C₅	C₂²×C₄	1	6	yes	—
58	C₂×C₄×C₅	C₂×D₄	1	3	yes	—
59	C₂×C₄×C₅	C₂×D₄	1	6	yes	—
60	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₅	—	2	12	no	—
62	C₂²⋊D₅	—	2	4	no	—
63	C₂×Dic₅	—	2	6	no	—
64	C₂×Dic₅	—	2	1	no	—
65	C₂×Dic₅	—	2	3	no	—
66	C₂×Dic₅	—	4	6	no	—
67	D₂₀	—	2	6	no	—
68	D₂₀	C₂²×C₄	1	4	yes	—
69	C₄×D₅	—	2	12	no	—
70	C₄×D₅	—	2	4	no	—
71	Dic₁₀	—	2	6	no	—
72	Dic₁₀	C₂²×C₄	1	4	yes	—
73	C₅×C₈	—	2	4	no	—
74	2₋¹⁺⁴	—	5	4	no	—
75	2₋¹⁺⁴	—	10	1	no	—
76	H(5,𝔽₂)	—	5	4	no	—
77	H(5,𝔽₂)	—	10	3	no	—
78	C₂×#41	—	5	6	no	—
79	C₂×#41	—	10	4	no	—
80	C₂×#41	C₂×D₅	1	1	yes	characteristic
81	#131	—	2	4	no	—
82	C₂×M₁₆	C₂×D₅	1	6	yes	complemented
83	C₄×Q₈	—	10	1	no	—
84	C₄×D₄	—	10	3	no	—
85	#117	—	5	3	no	—
86	C₂×C₄²	—	5	3	no	—
87	#104	—	5	4	no	—
88	#104	—	5	12	no	—
89	C₂²×C₅	—	4	3	no	—
90	C₂²×C₅	C₂×C₂²⋊C₄	1	6	yes	—
91	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
92	C₂×D₅	—	2	4	no	—
93	C₂×D₅	—	8	6	no	—
94	C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
95	C₄×C₅	C₂³×C₄	1	1	yes	characteristic
96	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
97	Dic₅	—	2	2	no	—
98	Dic₅	—	2	4	no	—
99	Dic₅	—	4	3	no	—
100	#41	—	5	12	no	—
101	#41	—	5	4	no	—
102	#41	—	10	4	no	—
103	#41	—	10	12	no	—
104	#41	—	10	6	no	—
105	#41	—	10	6	no	—
106	#41	C₄×D₅	1	4	yes	—
107	#41	C₂²⋊D₅	1	4	yes	—
108	C₂×Q₈	—	5	6	no	—
109	C₂×Q₈	—	10	4	no	—
110	C₂×Q₈	D₂₀	1	1	yes	characteristic
111	C₂×D₄	—	5	6	no	—
112	C₂×D₄	—	10	6	no	—
113	C₂×D₄	—	10	12	no	—
114	C₂×D₄	D₂₀	1	3	yes	—
115	C₂²×C₄	—	10	6	no	—
116	C₂²×C₄	—	10	3	no	—
117	C₂²×C₄	C₂²×D₅	1	3	yes	—
118	M₁₆	—	2	6	no	—
119	M₁₆	C₂²×D₅	1	4	yes	—
120	C₂×C₈	—	2	6	no	—
121	C₄⋊C₄	—	10	3	no	—
122	C₂²⋊C₄	—	10	3	no	—
123	C₄²	—	5	1	no	—
124	C₄²	—	5	3	no	—
125	C₄²	—	10	3	no	—
126	C₂×C₅	—	2	1	no	—
127	C₂×C₅	—	2	6	no	—
128	C₂×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	D₅	—	4	4	no	—
130	C₂³	—	10	6	no	—
131	C₂³	C₂×D₂₀	1	3	yes	—
132	Q₈	—	5	4	no	—
133	Q₈	—	10	6	no	—
134	Q₈	#685	1	4	yes	—
135	D₄	—	5	4	no	—
136	D₄	—	10	4	no	—
137	D₄	—	10	12	no	—
138	D₄	—	10	6	no	—
139	D₄	#685	1	12	yes	—
140	C₂×C₄	—	10	1	no	—
141	C₂×C₄	—	10	6	no	—
142	C₂×C₄	—	10	4	no	—
143	C₂×C₄	—	10	12	no	—
144	C₂×C₄	—	10	3	no	—
145	C₂×C₄	—	20	6	no	—
146	C₂×C₄	C₂×C₄×D₅	1	6	yes	—
147	C₂×C₄	C₂×D₂₀	1	3	yes	—
148	C₂×C₄	C₂×C₂²⋊D₅	1	6	yes	—
149	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
150	C₈	—	2	4	no	—
151	C₅	#2901	1	1	yes	characteristic, complemented, Sylow
152	C₂²	—	4	3	no	—
153	C₂²	—	10	4	no	—
154	C₂²	—	40	6	no	—
155	C₂²	C₂×#685	1	6	yes	—
156	C₂²	C₂²×D₂₀	1	1	yes	characteristic
157	C₄	—	10	2	no	—
158	C₄	—	10	4	no	—
159	C₄	—	20	3	no	—
160	C₄	C₂×#685	1	6	yes	—
161	C₄	C₂²×C₄×D₅	1	1	yes	characteristic
162	C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	6	no	—
165	C₂	—	20	4	no	—
166	C₂	C₂²×#685	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₅
commutator	C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅
Fitting	C₅×#568	C₂
socle	C₂×C₅	C₂²×C₂²⋊C₄
prime radical	#66283	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#568	D₅
hypocenter	C₅	#2901
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

Complements

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

N	H ≅ G/N	type
2. C₅×#568	C₂	semidirect
3. #41×Dic₅	C₂	semidirect
4. #65428	C₂	semidirect
5. C₂×#3957	C₂	semidirect
7. #4150	C₄	semidirect
8. #4149	C₄	semidirect
14. C₂×C₅×M₁₆	C₂²	semidirect
17. #4072	C₂²	semidirect
18. C₂×C₄×Dic₅	C₂²	semidirect
19. #3957	C₂²	semidirect
20. #3957	C₂²	semidirect
35. #709	C₂×C₄	semidirect
37. C₂×D₂₀	C₂×C₄	semidirect
39. C₂×Dic₁₀	C₂×C₄	semidirect
49. #568	D₅	semidirect
82. C₂×M₁₆	C₂×D₅	semidirect
151. C₅	#2901	semidirect

Maximal subgroups

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

subgroup	index
#66283	2
C₅×#568	2
#41×Dic₅	2
#65428	2
C₂×#3957	2
#2901	5