Finite Group Databasebeta

Subgroups — #10600692

Order 640.

186 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₅×#405	C₂	1	1	yes	maximal, characteristic
2	#65391	C₂	1	1	yes	maximal, characteristic
3	#65287	C₂	1	1	yes	maximal, characteristic
4	#64778	C₂	1	1	yes	maximal, characteristic
5	#64775	C₂	1	1	yes	maximal, characteristic
6	#64770	C₂	1	1	yes	maximal, characteristic
7	#64752	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₅×#117	C₂²	1	1	yes	characteristic
11	#4072	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×#681	C₂²	1	1	yes	characteristic
14	C₂²×#168	C₂²	1	1	yes	characteristic
15	C₅×#97	—	2	1	no	—
16	C₄×C₅×C₈	—	2	1	no	—
17	#3946	—	2	1	no	—
18	#3946	—	2	1	no	—
19	#3940	C₄	1	1	yes	characteristic, complemented
20	#3940	C₄	1	1	yes	characteristic, complemented
21	#3939	C₄	1	1	yes	characteristic, complemented
22	#3939	C₄	1	1	yes	characteristic, complemented
23	#1922	—	5	1	no	maximal, Sylow
24	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅	—	2	1	no	—
26	C₅×M₁₆	—	2	1	no	—
27	C₅×M₁₆	Q₈	1	2	yes	—
28	C₂×C₅×C₈	—	2	1	no	—
29	C₂×C₅×C₈	—	2	1	no	—
30	C₂×C₅×C₈	—	2	1	no	—
31	C₂×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₅×C₈	D₄	1	1	yes	characteristic
33	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₅×C₂²⋊C₄	—	2	1	no	—
36	C₄²×C₅	—	2	1	no	—
37	C₂²⋊Dic₅	—	2	1	no	—
38	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₅	—	2	1	no	—
41	C₄×Dic₅	—	2	1	no	—
42	C₄×Dic₅	—	2	1	no	—
43	#681	—	2	1	no	—
44	#681	—	2	1	no	—
45	C₂×#168	—	2	1	no	—
46	C₂×#168	—	2	1	no	—
47	C₂×#168	—	2	1	no	—
48	C₂×#168	C₂×C₄	1	1	yes	characteristic
49	C₂×#168	C₂×C₄	1	1	yes	characteristic
50	#427	—	5	1	no	—
51	#424	—	5	1	no	—
52	#405	D₅	1	1	yes	characteristic
53	#341	—	5	1	no	—
54	#339	—	5	1	no	—
55	#339	—	5	1	no	—
56	#336	—	5	1	no	—
57	C₂³×C₅	#41	1	1	yes	characteristic
58	C₂×C₄×C₅	—	2	1	no	—
59	C₂×C₄×C₅	—	2	1	no	—
60	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅	#41	1	1	yes	characteristic
64	C₂×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₄×C₅	#41	1	1	yes	characteristic
66	C₂×Dic₅	—	2	2	no	—
67	C₂×Dic₅	—	2	1	no	—
68	C₂×Dic₅	—	2	1	no	—
69	C₂×Dic₅	—	4	1	no	—
70	C₂×Dic₅	—	4	1	no	—
71	C₅×C₈	—	2	2	no	—
72	C₅×C₈	—	2	1	no	—
73	C₅×C₈	—	2	1	no	—
74	#168	—	2	1	no	—
75	#168	—	2	1	no	—
76	#168	—	4	1	no	—
77	C₂×M₁₆	—	5	1	no	—
78	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
79	C₂²×C₈	—	5	1	no	—
80	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
81	#117	—	5	1	no	—
82	#117	C₂×D₅	1	1	yes	characteristic
83	C₂×C₄²	—	5	1	no	—
84	#107	—	5	1	no	—
85	#107	—	5	1	no	—
86	#106	—	5	1	no	—
87	#106	—	5	1	no	—
88	C₄⋊C₈	—	10	1	no	—
89	C₄⋊C₈	—	10	1	no	—
90	#97	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₂²×C₅	—	2	1	no	—
93	C₂²×C₅	—	2	1	no	—
94	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
95	C₂²×C₅	#125	1	1	yes	characteristic
96	C₂²×C₅	#126	1	1	yes	characteristic
97	C₄×C₅	—	4	1	no	—
98	C₄×C₅	—	4	1	no	—
99	C₄×C₅	#117	1	1	yes	characteristic
100	C₄×C₅	C₄×Q₈	1	1	yes	characteristic
101	C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
102	C₄×C₅	#123	1	1	yes	characteristic
103	Dic₅	—	4	2	no	—
104	Dic₅	—	4	1	no	—
105	Dic₅	—	4	1	no	—
106	C₂²×C₄	—	10	1	no	—
107	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	—	10	1	no	—
110	M₁₆	—	10	1	no	—
111	M₁₆	Dic₁₀	1	2	yes	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	5	1	no	—
116	C₂×C₈	—	5	1	no	—
117	C₂×C₈	—	10	1	no	—
118	C₂×C₈	—	10	1	no	—
119	C₂×C₈	—	10	1	no	—
120	C₂×C₈	C₂²⋊D₅	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₅	1	1	yes	characteristic
122	C₄⋊C₄	—	5	1	no	—
123	C₄⋊C₄	—	5	1	no	—
124	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
125	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
126	C₂²⋊C₄	—	2	1	no	—
127	C₂²⋊C₄	—	10	1	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	10	1	no	—
130	C₄²	—	10	1	no	—
131	C₄²	—	10	1	no	—
132	C₂×C₅	—	2	1	no	—
133	C₂×C₅	—	2	1	no	—
134	C₂×C₅	#387	1	1	yes	characteristic
135	C₂×C₅	#443	1	1	yes	characteristic
136	C₂×C₅	#444	1	1	yes	characteristic
137	C₂³	#709	1	1	yes	characteristic
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	10	2	no	—
141	C₂×C₄	—	10	1	no	—
142	C₂×C₄	—	10	1	no	—
143	C₂×C₄	—	20	1	no	—
144	C₂×C₄	—	20	1	no	—
145	C₂×C₄	C₂×Dic₁₀	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
147	C₂×C₄	#709	1	1	yes	characteristic
148	C₂×C₄	#709	1	1	yes	characteristic
149	C₂×C₄	#709	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	10	1	no	—
155	C₈	—	10	1	no	—
156	C₈	—	20	1	no	—
157	C₅	#1922	1	1	yes	characteristic, complemented, Sylow
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#4016	1	1	yes	characteristic
161	C₂²	#4022	1	1	yes	characteristic
162	C₂²	C₄×C₂²⋊D₅	1	1	yes	characteristic
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	20	2	no	—
166	C₄	—	20	1	no	—
167	C₄	—	20	1	no	—
168	C₄	C₄×Dic₁₀	1	1	yes	characteristic
169	C₄	#4018	1	1	yes	characteristic
170	C₄	#4070	1	1	yes	characteristic
171	C₄	#4075	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#65222	1	1	yes	characteristic
175	C₂	#65433	1	1	yes	characteristic
176	C₂	#65434	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₅×#405	C₂
socle	C₂²×C₅	C₄×D₄
prime radical	C₂³×C₅	#41
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	#405	D₅
hypocenter	C₅	#1922
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
19. #3940	C₄	semidirect
20. #3940	C₄	semidirect
21. #3939	C₄	semidirect
22. #3939	C₄	semidirect
157. C₅	#1922	semidirect

Maximal subgroups

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

subgroup	index
C₅×#405	2
#65391	2
#65287	2
#64778	2
#64775	2
#64770	2
#64752	2
#1922	5