Finite Group Databasebeta

Subgroups — #10610249

Order 640.

392 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₂×#4018	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#65415	C₂	1	2	yes	maximal
4	#65407	C₂	1	2	yes	maximal, complemented
5	C₄×#681	C₂	1	1	yes	maximal, characteristic, complemented
6	#64998	C₂	1	4	yes	maximal, complemented
7	#64979	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×#685	C₄	1	2	yes	—
12	C₂×#683	C₄	1	2	yes	—
13	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×#681	C₂²	1	2	yes	—
15	#4018	C₄	1	4	yes	—
16	#4018	C₂²	1	2	yes	—
17	#4018	C₂²	1	2	yes	—
18	C₅×#97	C₂²	1	2	yes	complemented
19	C₄×C₅×C₈	C₂²	1	2	yes	complemented
20	#3952	C₂²	1	8	yes	—
21	#3946	C₂²	1	8	yes	complemented
22	#3935	C₂²	1	2	yes	complemented
23	C₄×#168	C₂²	1	2	yes	complemented
24	#2921	—	5	1	no	maximal, Sylow
25	C₂³×D₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₅	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₅	—	2	4	no	—
31	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₅	C₂³	1	2	yes	—
33	C₅×M₁₆	—	2	4	no	—
34	C₂×C₅×C₈	C₂³	1	4	yes	—
35	C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₄²×C₅	C₂³	1	1	yes	characteristic
38	#685	C₂×C₄	1	4	yes	—
39	#685	C₂×C₄	1	4	yes	—
40	#683	C₂×C₄	1	4	yes	—
41	#683	C₂×C₄	1	4	yes	—
42	C₄×Dic₅	C₂×C₄	1	2	yes	—
43	C₄×Dic₅	C₂³	1	2	yes	—
44	#681	—	2	4	no	—
45	C₂×#168	C₂³	1	4	yes	—
46	C₂×#117	—	5	1	no	—
47	#433	—	5	4	no	—
48	#433	—	5	4	no	—
49	#423	—	5	2	no	—
50	#407	—	5	2	no	—
51	C₄×M₁₆	—	5	1	no	—
52	C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
53	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₅	—	2	2	no	—
55	C₂²×D₅	—	4	2	no	—
56	C₂²×D₅	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₅	—	4	2	no	—
58	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅	#41	1	4	yes	—
62	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₅	—	2	2	no	—
64	C₂×Dic₅	—	4	2	no	—
65	C₂×Dic₅	C₂²×C₄	1	4	yes	—
66	C₂×Dic₅	C₂²×C₄	1	2	yes	—
67	C₄×D₅	—	2	4	no	—
68	C₄×D₅	—	2	4	no	—
69	C₅×C₈	—	2	4	no	—
70	#168	—	2	4	no	—
71	C₂³×C₄	—	5	1	no	—
72	C₂×M₁₆	—	5	2	no	—
73	C₂×M₁₆	C₂×D₅	1	2	yes	—
74	#117	—	5	4	no	—
75	#117	—	5	2	no	—
76	#117	—	5	2	no	—
77	C₂×C₄⋊C₄	—	5	2	no	—
78	C₂×C₂²⋊C₄	—	5	2	no	—
79	C₂×C₄²	—	5	1	no	—
80	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
81	C₄⋊C₈	—	5	8	no	—
82	C₂²⋊C₈	—	5	8	no	—
83	#97	—	5	2	no	—
84	#97	C₂×D₅	1	2	yes	complemented
85	C₄×C₈	—	5	2	no	—
86	C₄×C₈	C₂×D₅	1	2	yes	complemented
87	C₂²×C₅	—	2	2	no	—
88	C₂²×C₅	#117	1	2	yes	—
89	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₅	—	2	2	no	—
91	C₂×D₅	—	2	4	no	—
92	C₂×D₅	—	4	4	no	—
93	C₄×C₅	—	2	2	no	—
94	C₄×C₅	—	2	2	no	—
95	C₄×C₅	#117	1	2	yes	—
96	C₄×C₅	C₂×#41	1	2	yes	—
97	Dic₅	—	2	2	no	—
98	Dic₅	—	2	4	no	—
99	C₂⁴	—	5	1	no	—
100	C₂²×C₄	—	5	1	no	—
101	C₂²×C₄	—	5	2	no	—
102	C₂²×C₄	—	5	2	no	—
103	C₂²×C₄	—	5	2	no	—
104	C₂²×C₄	—	10	4	no	—
105	C₂²×C₄	C₄×D₅	1	2	yes	—
106	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
107	M₁₆	—	2	4	no	—
108	M₁₆	—	10	4	no	—
109	C₂×C₈	—	5	4	no	—
110	C₂×C₈	C₂²×D₅	1	4	yes	—
111	C₄⋊C₄	—	5	4	no	—
112	C₄⋊C₄	—	5	4	no	—
113	C₂²⋊C₄	—	5	4	no	—
114	C₂²⋊C₄	—	5	4	no	—
115	C₄²	—	5	2	no	—
116	C₄²	—	5	2	no	—
117	C₄²	C₄×D₅	1	2	yes	—
118	C₄²	C₂²×D₅	1	1	yes	characteristic
119	C₄²	C₂²×D₅	1	1	yes	characteristic
120	C₂×C₅	—	2	2	no	—
121	C₂×C₅	C₂×#117	1	1	yes	characteristic
122	C₂×C₅	#568	1	2	yes	—
123	D₅	—	4	2	no	—
124	C₂³	—	5	2	no	—
125	C₂³	—	10	2	no	—
126	C₂³	—	20	2	no	—
127	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	5	4	no	—
130	C₂×C₄	—	5	2	no	—
131	C₂×C₄	—	10	2	no	—
132	C₂×C₄	—	10	4	no	—
133	C₂×C₄	—	10	4	no	—
134	C₂×C₄	—	20	2	no	—
135	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
138	C₂×C₄	#709	1	4	yes	—
139	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
140	C₈	—	2	4	no	—
141	C₈	—	10	4	no	—
142	C₅	#2921	1	1	yes	characteristic, complemented, Sylow
143	C₂²	—	2	2	no	—
144	C₂²	—	10	2	no	—
145	C₂²	—	10	4	no	—
146	C₂²	—	20	4	no	—
147	C₂²	#4018	1	2	yes	—
148	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	10	2	no	—
152	C₄	—	10	4	no	—
153	C₄	#4018	1	2	yes	—
154	C₄	C₂×#709	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	—	20	2	no	—
157	C₂	C₂×#4018	1	1	yes	characteristic
158	C₂	#66079	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₄×C₅×M₁₆	C₂
prime radical	C₂³×D₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	C₄×M₁₆	D₅
hypocenter	C₅	#2921
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×M₁₆	C₂	semidirect
4. #65407	C₂	semidirect
5. C₄×#681	C₂	semidirect
6. #64998	C₂	semidirect
7. #64979	C₂	semidirect
18. C₅×#97	C₂²	semidirect
19. C₄×C₅×C₈	C₂²	semidirect
21. #3946	C₂²	semidirect
22. #3935	C₂²	semidirect
23. C₄×#168	C₂²	semidirect
52. C₄×M₁₆	D₅	semidirect
84. #97	C₂×D₅	semidirect
86. C₄×C₈	C₂×D₅	semidirect
142. C₅	#2921	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4018	2
C₄×C₅×M₁₆	2
#65415	2
#65407	2
C₄×#681	2
#64998	2
#64979	2
#2921	5