Finite Group Databasebeta

Subgroups — #10604079

Order 640.

186 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 60 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	#65708	C₂	1	1	yes	maximal, characteristic
2	#65701	C₂	1	1	yes	maximal, characteristic
3	#65169	C₂	1	1	yes	maximal, characteristic
4	#64896	C₂	1	1	yes	maximal, characteristic
5	(C₂×AGL(1,𝔽₅))⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#64875	C₂	1	1	yes	maximal, characteristic
7	#64875	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
9	C₂×#700	C₂²	1	1	yes	characteristic
10	C₂×#700	C₂²	1	1	yes	characteristic
11	C₂×D₅⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
13	#4038	C₂²	1	1	yes	characteristic
14	#4038	C₂²	1	1	yes	characteristic
15	#4004	—	2	1	no	—
16	#4002	—	2	1	no	—
17	C₄⋊#170	—	2	1	no	—
18	C₄×#170	—	2	1	no	—
19	C₅×#107	C₄	1	1	yes	characteristic, complemented
20	#3950	C₄	1	1	yes	characteristic, complemented
21	#3940	C₄	1	1	yes	characteristic, complemented
22	#3940	C₄	1	1	yes	characteristic, complemented
23	#1922	—	5	1	no	maximal, Sylow
24	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
25	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
26	C₂×#170	—	2	1	no	—
27	C₂×#170	—	2	1	no	—
28	C₂×#170	D₄	1	1	yes	characteristic
29	C₂×#170	D₄	1	1	yes	characteristic
30	C₄×AGL(1,𝔽₅)	—	2	1	no	—
31	C₄×AGL(1,𝔽₅)	—	2	1	no	—
32	#700	—	2	1	no	—
33	#700	—	2	1	no	—
34	#700	—	2	1	no	—
35	#700	Q₈	1	2	yes	—
36	D₅⋊C₈	—	2	1	no	—
37	D₅⋊C₈	—	2	1	no	—
38	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	#685	—	2	1	no	—
42	#685	—	2	1	no	—
43	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₅	—	2	1	no	—
46	C₄×Dic₅	—	2	1	no	—
47	C₂×#168	C₂×C₄	1	1	yes	characteristic
48	C₈×D₅	—	2	1	no	—
49	C₈×D₅	—	2	1	no	—
50	#427	—	5	1	no	—
51	#424	—	5	1	no	—
52	#405	—	5	1	no	—
53	#341	—	5	1	no	—
54	#339	—	5	1	no	—
55	#339	—	5	1	no	—
56	#336	—	5	1	no	—
57	C₂²×D₅	#41	1	1	yes	characteristic
58	C₂×AGL(1,𝔽₅)	—	2	2	no	—
59	C₂×AGL(1,𝔽₅)	—	4	1	no	—
60	C₂×AGL(1,𝔽₅)	—	4	1	no	—
61	C₂×C₄×C₅	—	2	1	no	—
62	C₂×C₄×C₅	—	2	1	no	—
63	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×Dic₅	—	2	1	no	—
65	C₂×Dic₅	—	2	1	no	—
66	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
67	C₄×D₅	C₂×Q₈	1	1	yes	characteristic
68	C₄×D₅	#41	1	1	yes	characteristic
69	C₄×D₅	#41	1	1	yes	characteristic
70	C₄×D₅	#41	1	1	yes	characteristic
71	#170	—	2	2	no	—
72	#170	—	2	1	no	—
73	#170	—	2	1	no	—
74	#170	—	4	1	no	—
75	C₅×C₈	—	2	1	no	—
76	#168	—	2	1	no	—
77	C₂×M₁₆	—	5	1	no	—
78	C₂×M₁₆	—	5	1	no	—
79	C₂²×C₈	—	5	1	no	—
80	C₂²×C₈	—	5	1	no	—
81	#117	—	5	1	no	—
82	#117	—	5	1	no	—
83	C₂×C₄²	—	5	1	no	—
84	#107	—	5	1	no	—
85	#107	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
86	#106	—	5	1	no	—
87	#106	—	5	1	no	—
88	C₄⋊C₈	—	10	1	no	—
89	C₄⋊C₈	—	10	1	no	—
90	#97	—	10	1	no	—
91	C₄×C₈	—	10	1	no	—
92	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
93	C₂×D₅	—	2	1	no	—
94	C₂×D₅	—	2	1	no	—
95	C₂×D₅	#125	1	1	yes	characteristic
96	C₂×D₅	#126	1	1	yes	characteristic
97	AGL(1,𝔽₅)	—	4	2	no	—
98	C₄×C₅	—	4	1	no	—
99	C₄×C₅	—	4	1	no	—
100	C₄×C₅	#117	1	1	yes	characteristic
101	C₄×C₅	C₄×Q₈	1	1	yes	characteristic
102	Dic₅	—	4	1	no	—
103	Dic₅	—	4	1	no	—
104	Dic₅	C₂²⋊Q₈	1	1	yes	characteristic
105	Dic₅	#123	1	1	yes	characteristic
106	C₂²×C₄	—	5	1	no	—
107	C₂²×C₄	—	10	1	no	—
108	M₁₆	—	5	2	no	—
109	M₁₆	—	10	1	no	—
110	M₁₆	—	10	1	no	—
111	M₁₆	—	10	1	no	—
112	C₂×C₈	—	5	1	no	—
113	C₂×C₈	—	5	1	no	—
114	C₂×C₈	—	5	1	no	—
115	C₂×C₈	—	10	1	no	—
116	C₂×C₈	—	10	1	no	—
117	C₂×C₈	—	10	1	no	—
118	C₂×C₈	—	10	1	no	—
119	C₂×C₈	—	10	1	no	—
120	C₂×C₈	—	10	1	no	—
121	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₄⋊C₄	—	5	1	no	—
123	C₄⋊C₄	—	5	1	no	—
124	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₂²⋊C₄	—	10	1	no	—
127	C₂²⋊C₄	—	10	1	no	—
128	C₄²	—	10	1	no	—
129	C₄²	—	10	1	no	—
130	C₄²	—	10	1	no	—
131	C₄²	—	10	1	no	—
132	C₂×C₅	#387	1	1	yes	characteristic
133	C₂×C₅	#443	1	1	yes	characteristic
134	C₂×C₅	#444	1	1	yes	characteristic
135	D₅	—	2	1	no	—
136	D₅	—	2	1	no	—
137	C₂³	—	5	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	5	1	no	—
141	C₂×C₄	—	5	1	no	—
142	C₂×C₄	—	5	1	no	—
143	C₂×C₄	—	5	1	no	—
144	C₂×C₄	—	5	1	no	—
145	C₂×C₄	—	10	2	no	—
146	C₂×C₄	—	10	1	no	—
147	C₂×C₄	—	10	1	no	—
148	C₂×C₄	—	20	1	no	—
149	C₂×C₄	—	20	1	no	—
150	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	10	1	no	—
153	C₈	—	10	1	no	—
154	C₈	—	10	2	no	—
155	C₈	—	10	1	no	—
156	C₈	—	20	1	no	—
157	C₅	#1922	1	1	yes	characteristic, complemented, Sylow
158	C₂²	—	5	1	no	—
159	C₂²	—	5	1	no	—
160	C₂²	—	10	1	no	—
161	C₂²	—	10	1	no	—
162	C₂²	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	5	1	no	—
166	C₄	—	5	1	no	—
167	C₄	—	20	2	no	—
168	C₄	—	20	1	no	—
169	C₄	—	20	1	no	—
170	C₄	#4130	1	1	yes	characteristic
171	C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₂	—	10	1	no	—
173	C₂	—	10	1	no	—
174	C₂	#65711	1	1	yes	characteristic
175	C₂	#65732	1	1	yes	characteristic
176	C₂	#65741	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#107	C₄
socle	C₂²×C₅	C₄×D₄
prime radical	C₂²×D₅	#41
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#107	AGL(1,𝔽₅)
hypocenter	C₅	#1922
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
19. C₅×#107	C₄	semidirect
20. #3950	C₄	semidirect
21. #3940	C₄	semidirect
22. #3940	C₄	semidirect
85. #107	AGL(1,𝔽₅)	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
#65708	2
#65701	2
#65169	2
#64896	2
(C₂×AGL(1,𝔽₅))⋊C₈	2
#64875	2
#1922	5