Finite Group Databasebeta

Subgroups — #10599819

Order 640.

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 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	#65855	C₂	1	1	yes	maximal, characteristic
2	C₂×#3955	C₂	1	1	yes	maximal, characteristic
3	C₂×#3955	C₂	1	1	yes	maximal, characteristic
4	C₅×#344	C₂	1	1	yes	maximal, characteristic, complemented
5	#64779	C₂	1	1	yes	maximal, characteristic, complemented
6	#64776	C₂	1	1	yes	maximal, characteristic, complemented
7	#64753	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₀	C₂²	1	1	yes	characteristic
9	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₅×#117	C₂²	1	1	yes	characteristic
12	#4072	C₂²	1	1	yes	characteristic
13	C₂×#681	C₂²	1	1	yes	characteristic
14	C₂×#681	C₂²	1	1	yes	characteristic
15	#4039	—	2	2	no	—
16	D₅×C₂²⋊C₄	C₄	1	4	yes	—
17	C₄×D₂₀	—	2	2	no	—
18	#3955	—	2	2	no	—
19	#3955	—	2	2	no	—
20	#1829	—	5	1	no	maximal, Sylow
21	C₂³×D₅	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×D₂₀	—	2	1	no	—
24	C₂×D₂₀	—	2	1	no	—
25	C₂×D₂₀	—	2	1	no	—
26	C₂×D₂₀	—	2	1	no	—
27	C₂×D₂₀	D₄	1	2	yes	—
28	C₂×D₂₀	D₄	1	2	yes	—
29	C₂×C₄×D₅	—	2	4	no	—
30	C₅×M₁₆	—	2	1	no	—
31	C₅×M₁₆	—	2	1	no	—
32	C₅×M₁₆	—	2	1	no	—
33	C₅×M₁₆	—	2	1	no	—
34	C₂×C₅×C₈	—	2	1	no	—
35	C₂×C₅×C₈	—	2	1	no	—
36	C₅×C₄⋊C₄	—	2	1	no	—
37	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
38	C₄²×C₅	—	2	1	no	—
39	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
40	#685	—	2	4	no	—
41	C₄⋊Dic₅	—	2	1	no	—
42	C₄×Dic₅	—	2	1	no	—
43	#681	—	2	1	no	—
44	#681	—	2	1	no	—
45	#681	—	2	1	no	—
46	#681	—	2	1	no	—
47	C₂×#168	—	2	1	no	—
48	C₂×#168	—	2	1	no	—
49	#518	—	5	1	no	—
50	C₂×#100	—	5	1	no	—
51	C₂×#100	—	5	1	no	—
52	#344	—	5	1	no	—
53	#344	—	5	1	no	—
54	#344	—	5	1	no	—
55	#344	D₅	1	1	yes	characteristic, complemented
56	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂²×D₅	—	2	2	no	—
58	C₂²×D₅	—	2	2	no	—
59	C₂²×D₅	—	8	2	no	—
60	C₂²×D₅	C₂²⋊C₄	1	4	yes	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₄×C₅	#41	1	1	yes	characteristic
66	C₂×C₄×C₅	#41	1	1	yes	characteristic
67	C₂×C₄×C₅	#41	1	1	yes	characteristic
68	C₂×Dic₅	—	2	2	no	—
69	D₂₀	—	2	2	no	—
70	D₂₀	—	2	2	no	—
71	D₂₀	—	2	2	no	—
72	D₂₀	—	2	2	no	—
73	C₄×D₅	—	8	4	no	—
74	C₅×C₈	—	4	1	no	—
75	C₅×C₈	—	4	1	no	—
76	#168	—	4	1	no	—
77	#168	—	4	1	no	—
78	C₂²×D₄	—	5	1	no	—
79	C₂×M₁₆	—	5	1	no	—
80	C₂×M₁₆	—	5	1	no	—
81	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
82	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
83	C₄×D₄	—	10	2	no	—
84	C₄×D₄	—	10	2	no	—
85	#117	—	5	1	no	—
86	#117	C₂×D₅	1	1	yes	characteristic
87	C₂×C₂²⋊C₄	—	5	4	no	—
88	#100	—	10	2	no	—
89	#100	—	10	2	no	—
90	C₂²×C₅	—	4	1	no	—
91	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₅	#117	1	1	yes	characteristic
93	C₂²×C₅	#117	1	1	yes	characteristic
94	C₂×D₅	—	2	4	no	—
95	C₂×D₅	—	4	4	no	—
96	C₂×D₅	—	8	2	no	—
97	C₂×D₅	—	8	2	no	—
98	C₄×C₅	—	4	2	no	—
99	C₄×C₅	#123	1	1	yes	characteristic
100	C₄×C₅	#123	1	1	yes	characteristic
101	C₄×C₅	#123	1	1	yes	characteristic
102	C₄×C₅	#123	1	1	yes	characteristic
103	Dic₅	—	4	2	no	—
104	C₂⁴	—	5	2	no	—
105	C₂×D₄	—	5	2	no	—
106	C₂×D₄	—	5	2	no	—
107	C₂×D₄	—	10	1	no	—
108	C₂×D₄	—	10	1	no	—
109	C₂×D₄	—	10	1	no	—
110	C₂×D₄	—	10	1	no	—
111	C₂²×C₄	—	10	4	no	—
112	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	10	1	no	—
118	M₁₆	—	10	1	no	—
119	M₁₆	—	10	1	no	—
120	M₁₆	—	10	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	10	1	no	—
124	C₂×C₈	—	10	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	10	1	no	—
127	C₂²⋊C₄	—	5	2	no	—
128	C₂²⋊C₄	—	10	4	no	—
129	C₂²⋊C₄	C₄×D₅	1	2	yes	—
130	C₄²	—	2	1	no	—
131	C₄²	—	10	1	no	—
132	C₂×C₅	—	2	1	no	—
133	C₂×C₅	—	2	1	no	—
134	C₂×C₅	—	2	1	no	—
135	C₂×C₅	#381	1	1	yes	characteristic
136	D₅	—	4	4	no	—
137	C₂³	—	5	4	no	—
138	C₂³	—	10	2	no	—
139	C₂³	—	10	2	no	—
140	C₂³	—	40	2	no	—
141	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
142	D₄	—	10	2	no	—
143	D₄	—	10	2	no	—
144	D₄	—	10	2	no	—
145	D₄	—	10	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	10	2	no	—
148	C₂×C₄	—	40	4	no	—
149	C₂×C₄	#709	1	1	yes	characteristic
150	C₂×C₄	#709	1	1	yes	characteristic
151	C₂×C₄	D₄×D₅	1	1	yes	characteristic
152	C₂×C₄	D₄×D₅	1	1	yes	characteristic
153	C₂×C₄	#711	1	1	yes	characteristic
154	C₂×C₄	#713	1	1	yes	characteristic
155	C₈	—	4	1	no	—
156	C₈	—	4	1	no	—
157	C₈	—	20	1	no	—
158	C₈	—	20	1	no	—
159	C₅	#1829	1	1	yes	characteristic, complemented, Sylow
160	C₂²	—	4	1	no	—
161	C₂²	—	10	4	no	—
162	C₂²	—	20	4	no	—
163	C₂²	—	40	2	no	—
164	C₂²	—	40	2	no	—
165	C₂²	#4018	1	1	yes	characteristic
166	C₂²	D₅×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²	#4038	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	20	2	no	—
170	C₄	#4029	1	1	yes	characteristic
171	C₄	#4029	1	1	yes	characteristic
172	C₄	#4040	1	1	yes	characteristic
173	C₄	#4040	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	20	4	no	—
178	C₂	#64953	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#64953
commutator	C₂³×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₅×#344	C₂
socle	C₂×C₅	#381
prime radical	C₂²×D₂₀	C₂²
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	#344	D₅
hypocenter	C₅	#1829
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
4. C₅×#344	C₂	semidirect
5. #64779	C₂	semidirect
6. #64776	C₂	semidirect
7. #64753	C₂	semidirect
55. #344	D₅	semidirect
159. C₅	#1829	semidirect

Maximal subgroups

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

subgroup	index
#65855	2
C₂×#3955	2
C₅×#344	2
#64779	2
#64776	2
#64753	2
#1829	5