Finite Group Databasebeta

Subgroups — #10603962

Order 640.

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

normal subgroups only

This group has 224 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	#65776	C₂	1	1	yes	maximal, characteristic
2	C₂×#4008	C₂	1	1	yes	maximal, characteristic
3	C₂×#3995	C₂	1	1	yes	maximal, characteristic, complemented
4	#65701	C₂	1	1	yes	maximal, characteristic, complemented
5	#65061	C₂	1	1	yes	maximal, characteristic
6	(C₂×AGL(1,𝔽₅))⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#64875	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#711	C₂²	1	1	yes	characteristic
9	C₂×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
10	C₂×#700	C₂²	1	1	yes	characteristic
11	C₂×#700	C₂²	1	1	yes	characteristic
12	C₂×D₅⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
14	#4038	C₂²	1	1	yes	characteristic
15	C₂²⋊#170	—	2	1	no	—
16	C₂²⋊#170	—	2	1	no	—
17	#4008	—	2	2	no	—
18	#4002	—	2	1	no	—
19	C₄×#170	—	2	1	no	—
20	#3995	—	2	2	no	—
21	C₅×#102	C₄	1	1	yes	characteristic, complemented
22	#3964	C₄	1	1	yes	characteristic, complemented
23	#3948	C₄	1	1	yes	characteristic, complemented
24	#3942	C₄	1	1	yes	characteristic, complemented
25	#1929	—	5	1	no	maximal, Sylow
26	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
27	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₂²⋊D₅	—	2	1	no	—
29	C₂²×Dic₅	—	2	1	no	—
30	#711	—	2	1	no	—
31	#711	—	2	1	no	—
32	#711	—	2	1	no	—
33	#711	—	2	1	no	—
34	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
35	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
36	C₂×#170	—	2	1	no	—
37	C₂×#170	—	2	1	no	—
38	C₂×#170	D₄	1	1	yes	characteristic
39	C₂×#170	D₄	1	1	yes	characteristic
40	C₄×AGL(1,𝔽₅)	—	2	1	no	—
41	C₄×AGL(1,𝔽₅)	—	2	1	no	—
42	#700	—	2	1	no	—
43	#700	—	2	1	no	—
44	#700	—	2	1	no	—
45	#700	D₄	1	2	yes	—
46	D₅⋊C₈	—	2	1	no	—
47	D₅⋊C₈	—	2	1	no	—
48	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	#685	—	2	1	no	—
51	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
52	C₄×Dic₅	—	2	1	no	—
53	C₂×#168	C₂×C₄	1	1	yes	characteristic
54	C₈×D₅	—	2	1	no	—
55	C₈×D₅	—	2	1	no	—
56	C₂×#108	—	5	1	no	—
57	C₂×#104	—	5	1	no	—
58	#416	—	5	1	no	—
59	#408	—	5	1	no	—
60	#405	—	5	1	no	—
61	#339	—	5	1	no	—
62	#336	—	5	1	no	—
63	C₂³×C₅	—	2	1	no	—
64	C₂²×D₅	#41	1	1	yes	characteristic
65	C₂×AGL(1,𝔽₅)	—	2	2	no	—
66	C₂×AGL(1,𝔽₅)	—	4	1	no	—
67	C₂×AGL(1,𝔽₅)	—	4	1	no	—
68	C₅×D₄	—	2	1	no	—
69	C₅×D₄	—	2	1	no	—
70	C₂×C₄×C₅	—	2	1	no	—
71	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²⋊D₅	—	4	1	no	—
73	C₂²⋊D₅	—	4	1	no	—
74	C₂×Dic₅	—	2	1	no	—
75	C₂×Dic₅	—	2	1	no	—
76	C₂×Dic₅	—	4	1	no	—
77	C₂×Dic₅	—	4	1	no	—
78	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
79	C₄×D₅	C₂×D₄	1	1	yes	characteristic
80	C₄×D₅	#41	1	1	yes	characteristic
81	C₄×D₅	#41	1	1	yes	characteristic
82	C₄×D₅	#41	1	1	yes	characteristic
83	Dic₁₀	—	2	1	no	—
84	Dic₁₀	—	2	1	no	—
85	#170	—	2	2	no	—
86	#170	—	2	1	no	—
87	#170	—	2	1	no	—
88	#170	—	4	1	no	—
89	C₅×C₈	—	2	1	no	—
90	#168	—	2	1	no	—
91	C₂×#41	—	5	1	no	—
92	C₂×M₁₆	—	5	1	no	—
93	C₂×M₁₆	—	5	1	no	—
94	C₂²×C₈	—	5	1	no	—
95	C₂²×C₈	—	5	1	no	—
96	#117	—	5	1	no	—
97	C₂×C₄²	—	5	1	no	—
98	#108	—	10	2	no	—
99	#104	—	10	2	no	—
100	#103	—	5	1	no	—
101	#103	—	5	1	no	—
102	#102	—	5	1	no	—
103	#102	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
104	C₂²⋊C₈	—	10	1	no	—
105	C₂²⋊C₈	—	10	1	no	—
106	#97	—	10	1	no	—
107	C₄×C₈	—	10	1	no	—
108	C₂²×C₅	—	4	1	no	—
109	C₂²×C₅	—	4	1	no	—
110	C₂²×C₅	—	4	1	no	—
111	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
112	C₂×D₅	—	2	1	no	—
113	C₂×D₅	—	2	1	no	—
114	C₂×D₅	#124	1	1	yes	characteristic
115	C₂×D₅	#126	1	1	yes	characteristic
116	AGL(1,𝔽₅)	—	4	2	no	—
117	C₄×C₅	—	4	1	no	—
118	C₄×C₅	#117	1	1	yes	characteristic
119	C₄×C₅	C₄×D₄	1	1	yes	characteristic
120	Dic₅	—	4	1	no	—
121	Dic₅	—	4	1	no	—
122	Dic₅	#121	1	1	yes	characteristic
123	Dic₅	#123	1	1	yes	characteristic
124	#41	—	10	1	no	—
125	#41	—	10	1	no	—
126	#41	—	10	1	no	—
127	#41	—	10	1	no	—
128	C₂×Q₈	—	5	1	no	—
129	C₂×D₄	—	10	1	no	—
130	C₂×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₂²×C₄	—	5	1	no	—
132	C₂²×C₄	—	10	1	no	—
133	C₂²×C₄	—	10	1	no	—
134	M₁₆	—	5	2	no	—
135	M₁₆	—	10	1	no	—
136	M₁₆	—	10	1	no	—
137	M₁₆	—	10	1	no	—
138	C₂×C₈	—	5	1	no	—
139	C₂×C₈	—	5	1	no	—
140	C₂×C₈	—	5	1	no	—
141	C₂×C₈	—	10	1	no	—
142	C₂×C₈	—	10	1	no	—
143	C₂×C₈	—	10	1	no	—
144	C₂×C₈	—	10	1	no	—
145	C₂×C₈	—	10	1	no	—
146	C₂×C₈	—	10	1	no	—
147	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₄⋊C₄	—	5	1	no	—
149	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂²⋊C₄	—	10	1	no	—
151	C₄²	—	10	1	no	—
152	C₄²	—	10	1	no	—
153	C₄²	—	10	1	no	—
154	C₂×C₅	—	8	1	no	—
155	C₂×C₅	#388	1	1	yes	characteristic
156	C₂×C₅	#443	1	1	yes	characteristic
157	C₂×C₅	#444	1	1	yes	characteristic
158	D₅	—	2	1	no	—
159	D₅	—	2	1	no	—
160	C₂³	—	2	1	no	—
161	C₂³	—	5	1	no	—
162	Q₈	—	10	1	no	—
163	Q₈	—	10	1	no	—
164	D₄	—	2	1	no	—
165	D₄	—	2	1	no	—
166	D₄	—	20	1	no	—
167	D₄	—	20	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	5	1	no	—
170	C₂×C₄	—	5	1	no	—
171	C₂×C₄	—	5	1	no	—
172	C₂×C₄	—	5	1	no	—
173	C₂×C₄	—	5	1	no	—
174	C₂×C₄	—	10	2	no	—
175	C₂×C₄	—	10	1	no	—
176	C₂×C₄	—	10	1	no	—
177	C₂×C₄	—	20	1	no	—
178	C₂×C₄	—	20	1	no	—
179	C₂×C₄	—	20	1	no	—
180	C₂×C₄	—	20	1	no	—
181	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₈	—	2	1	no	—
183	C₈	—	10	1	no	—
184	C₈	—	10	1	no	—
185	C₈	—	10	2	no	—
186	C₈	—	10	1	no	—
187	C₈	—	20	1	no	—
188	C₅	#1929	1	1	yes	characteristic, complemented, Sylow
189	C₂²	—	4	1	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	1	no	—
192	C₂²	—	5	1	no	—
193	C₂²	—	5	1	no	—
194	C₂²	—	10	1	no	—
195	C₂²	—	10	1	no	—
196	C₂²	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₄	—	4	1	no	—
198	C₄	—	5	1	no	—
199	C₄	—	5	1	no	—
200	C₄	—	20	2	no	—
201	C₄	—	20	1	no	—
202	C₄	—	20	1	no	—
203	C₄	#4130	1	1	yes	characteristic
204	C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₂	—	8	1	no	—
206	C₂	—	10	1	no	—
207	C₂	—	10	1	no	—
208	C₂	#65700	1	1	yes	characteristic
209	C₂	#65733	1	1	yes	characteristic
210	C₂	#65736	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×#102	C₄
socle	C₂²×C₅	C₄×D₄
prime radical	C₂×#711	C₂²
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#102	AGL(1,𝔽₅)
hypocenter	C₅	#1929
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
3. C₂×#3995	C₂	semidirect
4. #65701	C₂	semidirect
6. (C₂×AGL(1,𝔽₅))⋊C₈	C₂	semidirect
7. #64875	C₂	semidirect
21. C₅×#102	C₄	semidirect
22. #3964	C₄	semidirect
23. #3948	C₄	semidirect
24. #3942	C₄	semidirect
103. #102	AGL(1,𝔽₅)	semidirect
188. C₅	#1929	semidirect

Maximal subgroups

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

subgroup	index
#65776	2
C₂×#4008	2
C₂×#3995	2
#65701	2
#65061	2
(C₂×AGL(1,𝔽₅))⋊C₈	2
#64875	2
#1929	5