Finite Group Databasebeta

Subgroups — #10604677

Order 640.

312 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 66 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	#66260	C₂	1	1	yes	maximal, characteristic
2	#66058	C₂	1	1	yes	maximal, characteristic
3	#65785	C₂	1	1	yes	maximal, characteristic
4	#65782	C₂	1	2	yes	maximal, complemented
5	#64863	C₂	1	2	yes	maximal, complemented
6	C₂×#713	C₂²	1	1	yes	characteristic
7	C₂×Q₈×D₅	C₂²	1	1	yes	characteristic
8	#4133	—	2	4	no	—
9	#4130	C₂²	1	2	yes	—
10	D₅⋊M₁₆	—	2	2	no	—
11	D₅⋊M₁₆	C₂²	1	2	yes	—
12	C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
13	#4093	C₄	1	1	yes	characteristic, complemented
14	#4090	C₄	1	1	yes	characteristic, complemented
15	#4042	—	2	2	no	—
16	#4040	—	2	2	no	—
17	#4021	C₄	1	1	yes	characteristic, complemented
18	#4018	C₂²	1	1	yes	characteristic
19	#4009	—	2	4	no	—
20	#3999	—	2	2	no	—
21	#1909	—	5	1	no	maximal, Sylow
22	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
24	#713	—	2	2	no	—
25	#713	—	2	2	no	—
26	Q₈×D₅	—	2	2	no	—
27	Q₈×D₅	—	2	2	no	—
28	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
29	C₂×D₂₀	D₄	1	2	yes	—
30	C₂×C₄×D₅	—	2	1	no	—
31	C₂×C₄×D₅	D₄	1	2	yes	—
32	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₀	—	2	1	no	—
34	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
35	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
36	#704	—	2	2	no	—
37	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
38	C₄×AGL(1,𝔽₅)	—	2	2	no	—
39	#700	—	2	4	no	—
40	#700	—	2	2	no	—
41	D₅⋊C₈	—	2	4	no	—
42	D₅⋊C₈	D₄	1	4	yes	—
43	C₅×C₄⋊C₄	—	2	2	no	—
44	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	#685	—	2	2	no	—
46	#685	—	2	2	no	—
47	#685	—	2	1	no	—
48	#683	—	2	2	no	—
49	#683	—	2	1	no	—
50	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
51	#568	—	5	1	no	—
52	#536	—	5	1	no	—
53	#419	—	5	2	no	—
54	#410	—	5	1	no	—
55	H(3,ℤ/4)	—	5	2	no	—
56	C₂²×D₅	C₂²⋊C₄	1	2	yes	—
57	C₂²×D₅	C₂×D₄	1	1	yes	characteristic
58	C₂×AGL(1,𝔽₅)	—	4	2	no	—
59	C₅×Q₈	—	2	2	no	—
60	C₅×Q₈	—	2	2	no	—
61	C₂×C₄×C₅	—	2	1	no	—
62	C₂×C₄×C₅	—	2	1	no	—
63	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×Dic₅	—	2	1	no	—
66	C₂×Dic₅	—	2	1	no	—
67	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
68	D₂₀	—	2	4	no	—
69	D₂₀	—	2	2	no	—
70	C₄×D₅	—	2	2	no	—
71	C₄×D₅	—	2	2	no	—
72	C₄×D₅	—	4	2	no	—
73	C₄×D₅	C₂×D₄	1	2	yes	—
74	C₄×D₅	#41	1	2	yes	—
75	Dic₁₀	—	2	2	no	—
76	Dic₁₀	—	4	1	no	—
77	Dic₁₀	—	4	1	no	—
78	#170	—	2	4	no	—
79	C₂×#41	—	5	1	no	—
80	C₂²×Q₈	—	5	1	no	—
81	#131	—	10	4	no	—
82	C₂×M₁₆	—	5	2	no	—
83	C₂×M₁₆	—	10	2	no	—
84	C₄⋊Q₈	—	5	1	no	—
85	C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
86	#124	—	5	1	no	—
87	#124	—	5	1	no	—
88	#123	—	10	2	no	—
89	C₂²⋊Q₈	—	10	2	no	—
90	#117	—	5	1	no	—
91	#117	—	5	2	no	—
92	#103	—	10	4	no	—
93	#99	—	10	2	no	—
94	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×D₅	—	2	2	no	—
96	C₂×D₅	—	4	1	no	—
97	C₂×D₅	—	4	2	no	—
98	C₂×D₅	#121	1	2	yes	—
99	AGL(1,𝔽₅)	—	8	2	no	—
100	C₄×C₅	—	2	2	no	—
101	C₄×C₅	—	4	1	no	—
102	C₄×C₅	—	4	1	no	—
103	C₄×C₅	C₄×D₄	1	2	yes	—
104	Dic₅	—	4	1	no	—
105	Dic₅	—	4	1	no	—
106	Dic₅	#120	1	1	yes	characteristic
107	Dic₅	#123	1	1	yes	characteristic
108	#41	—	10	2	no	—
109	#41	—	10	2	no	—
110	C₂×Q₈	—	5	1	no	—
111	C₂×Q₈	—	10	1	no	—
112	C₂×Q₈	—	10	2	no	—
113	C₂×Q₈	—	10	2	no	—
114	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
116	C₂×D₄	—	5	2	no	—
117	C₂×D₄	—	5	1	no	—
118	C₂²×C₄	—	5	1	no	—
119	C₂²×C₄	—	5	2	no	—
120	C₂²×C₄	—	10	1	no	—
121	M₁₆	—	10	2	no	—
122	M₁₆	—	10	2	no	—
123	M₁₆	—	10	4	no	—
124	C₂×C₈	—	5	4	no	—
125	C₂×C₈	—	10	4	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	10	1	no	—
128	C₄⋊C₄	—	10	2	no	—
129	C₄⋊C₄	—	10	2	no	—
130	C₂²⋊C₄	—	10	1	no	—
131	C₂²⋊C₄	—	10	2	no	—
132	C₂²⋊C₄	—	10	2	no	—
133	C₂²⋊C₄	—	10	2	no	—
134	C₄²	—	5	1	no	—
135	C₄²	—	10	2	no	—
136	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₂×C₅	—	2	1	no	—
138	C₂×C₅	#386	1	1	yes	characteristic
139	D₅	—	2	2	no	—
140	D₅	—	4	2	no	—
141	C₂³	—	5	1	no	—
142	C₂³	—	5	2	no	—
143	Q₈	—	2	2	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	10	2	no	—
146	Q₈	—	20	1	no	—
147	Q₈	—	20	1	no	—
148	D₄	—	10	4	no	—
149	D₄	—	10	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	5	1	no	—
153	C₂×C₄	—	5	2	no	—
154	C₂×C₄	—	5	2	no	—
155	C₂×C₄	—	10	2	no	—
156	C₂×C₄	—	10	2	no	—
157	C₂×C₄	—	10	1	no	—
158	C₂×C₄	—	10	1	no	—
159	C₂×C₄	—	20	2	no	—
160	C₂×C₄	—	20	2	no	—
161	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
162	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₈	—	10	4	no	—
164	C₅	#1909	1	1	yes	characteristic, complemented, Sylow
165	C₂²	—	5	2	no	—
166	C₂²	—	10	2	no	—
167	C₂²	—	20	1	no	—
168	C₂²	—	20	2	no	—
169	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	5	1	no	—
174	C₄	—	5	1	no	—
175	C₄	—	20	1	no	—
176	C₄	—	20	1	no	—
177	C₄	—	40	2	no	—
178	C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
179	C₂	—	2	1	no	—
180	C₂	—	10	2	no	—
181	C₂	—	20	2	no	—
182	C₂	#65779	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#65779
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₄⋊Q₈	C₄
socle	C₂×C₅	#386
prime radical	C₂×#713	C₂²
cosocle	C₂×C₄×D₅	C₂³
hypercenter	C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#1909
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. #65782	C₂	semidirect
5. #64863	C₂	semidirect
12. C₅×C₄⋊Q₈	C₄	semidirect
13. #4093	C₄	semidirect
14. #4090	C₄	semidirect
17. #4021	C₄	semidirect
85. C₄⋊Q₈	AGL(1,𝔽₅)	semidirect
164. C₅	#1909	semidirect

Maximal subgroups

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

subgroup	index
#66260	2
#66058	2
#65785	2
#65782	2
#64863	2
#1909	5