Subgroups — #10617444

Order 640.

452 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66264	C₂	1	2	yes	maximal, complemented
2	D₅×#135	C₂	1	1	yes	maximal, characteristic
3	#65795	C₂	1	2	yes	maximal, complemented
4	#65741	C₂	1	2	yes	maximal, complemented
5	#65736	C₂	1	4	yes	maximal, complemented
6	#65733	C₂	1	2	yes	maximal, complemented
7	C₂×#3995	C₂	1	1	yes	maximal, characteristic, complemented
8	#65718	C₂	1	1	yes	maximal, characteristic, complemented
9	D₅×#41	C₂²	1	2	yes	—
10	#4133	C₂²	1	4	yes	complemented
11	#4131	—	2	4	no	—
12	#4131	C₂²	1	4	yes	complemented
13	#4130	C₂²	1	1	yes	characteristic
14	D₅⋊M₁₆	—	2	2	no	—
15	D₅⋊M₁₆	C₂²	1	1	yes	characteristic
16	C₂×#700	—	2	2	no	—
17	C₂×#700	C₂²	1	2	yes	—
18	C₅×#135	C₄	1	1	yes	characteristic, complemented
19	#4096	C₄	1	2	yes	complemented
20	#4065	C₂²	1	1	yes	characteristic
21	D₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
22	#4062	C₂²	1	2	yes	—
23	D₅×SD₁₆	C₄	1	2	yes	complemented
24	#4058	C₂²	1	1	yes	characteristic
25	D₅×D₈	C₄	1	1	yes	characteristic, complemented
26	#4050	C₄	1	1	yes	characteristic, complemented
27	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
28	#4010	C₂²	1	4	yes	complemented
29	#4008	C₂²	1	4	yes	complemented
30	#3995	C₂²	1	2	yes	complemented
31	#3995	C₂²	1	2	yes	complemented
32	#3992	C₂²	1	2	yes	complemented
33	C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	complemented
34	#2957	—	5	1	no	maximal, Sylow
35	C₅×#41	C₂×C₄	1	2	yes	—
36	#713	C₂³	1	2	yes	—
37	Q₈×D₅	C₂×C₄	1	2	yes	—
38	#711	—	2	2	no	—
39	#711	C₂³	1	2	yes	—
40	D₄×D₅	—	2	2	no	—
41	D₄×D₅	C₂×C₄	1	2	yes	—
42	#709	—	2	2	no	—
43	#709	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₅	—	2	2	no	—
45	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
46	C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
47	#704	—	2	4	no	—
48	#704	D₄	1	2	yes	—
49	C₂×#170	—	2	2	no	—
50	C₂×#170	—	2	4	no	—
51	C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
52	C₄×AGL(1,𝔽₅)	C₂³	1	2	yes	—
53	#700	—	2	2	no	—
54	#700	—	2	4	no	—
55	#700	D₄	1	2	yes	—
56	#700	C₂³	1	4	yes	—
57	D₅⋊C₈	—	2	4	no	—
58	D₅⋊C₈	C₂³	1	2	yes	—
59	C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
60	C₅×SD₁₆	C₂×C₄	1	2	yes	—
61	C₅×D₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
63	#689	C₂×C₄	1	2	yes	—
64	#688	C₂×C₄	1	2	yes	—
65	#687	C₂×C₄	1	2	yes	—
66	#686	C₂×C₄	1	2	yes	—
67	C₂×#168	C₂×C₄	1	1	yes	characteristic
68	Dic₂₀	C₂×C₄	1	1	yes	characteristic
69	D₄₀	C₂×C₄	1	1	yes	characteristic
70	#677	C₂×C₄	1	2	yes	—
71	C₈×D₅	C₂×C₄	1	1	yes	characteristic
72	C₈×D₅	C₂×C₄	1	1	yes	characteristic
73	C₈×D₅	C₂³	1	1	yes	characteristic
74	C₈×D₅	C₂³	1	1	yes	characteristic
75	C₂×#135	—	5	1	no	—
76	#568	—	5	2	no	—
77	#444	—	5	2	no	—
78	#444	—	5	2	no	—
79	#443	—	5	4	no	—
80	C₂×#108	—	5	1	no	—
81	#421	—	5	2	no	—
82	#405	—	5	1	no	—
83	C₂²×D₅	—	2	2	no	—
84	C₂²×D₅	#41	1	1	yes	characteristic
85	C₂×AGL(1,𝔽₅)	—	2	2	no	—
86	C₅×Q₈	C₂²×C₄	1	2	yes	—
87	C₅×D₄	—	2	2	no	—
88	C₅×D₄	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂²⋊D₅	—	2	2	no	—
92	C₂²⋊D₅	—	2	2	no	—
93	C₂²⋊D₅	—	2	2	no	—
94	C₂×Dic₅	—	2	2	no	—
95	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
96	D₂₀	—	2	2	no	—
97	D₂₀	C₂²×C₄	1	2	yes	—
98	C₄×D₅	—	2	2	no	—
99	C₄×D₅	—	2	2	no	—
100	C₄×D₅	—	2	2	no	—
101	C₄×D₅	C₂²×C₄	1	1	yes	characteristic
102	C₄×D₅	C₂×D₄	1	1	yes	characteristic
103	C₄×D₅	#41	1	1	yes	characteristic
104	C₄×D₅	C₂⁴	1	1	yes	characteristic
105	Dic₁₀	—	2	2	no	—
106	Dic₁₀	C₂²×C₄	1	2	yes	—
107	#170	—	2	4	no	—
108	#170	C₂×D₄	1	4	yes	—
109	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
111	#168	C₂²×C₄	1	1	yes	characteristic
112	#168	C₂²×C₄	1	1	yes	characteristic
113	C₂×#41	—	5	2	no	—
114	#135	—	5	2	no	—
115	#135	—	5	1	no	—
116	#135	—	5	2	no	—
117	#135	—	5	1	no	—
118	#135	—	5	1	no	—
119	#135	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
120	C₂×Q₁₆	—	5	1	no	—
121	C₂×SD₁₆	—	5	2	no	—
122	C₂×D₈	—	5	1	no	—
123	#131	—	5	4	no	—
124	#131	—	5	4	no	—
125	#131	—	10	4	no	—
126	C₂×M₁₆	—	5	1	no	—
127	C₂×M₁₆	—	5	2	no	—
128	C₂×M₁₆	—	10	2	no	—
129	C₂×M₁₆	—	10	2	no	—
130	C₂²×C₈	—	5	1	no	—
131	#117	—	5	1	no	—
132	#108	—	5	2	no	—
133	#108	—	5	2	no	—
134	#104	—	5	4	no	—
135	#104	—	5	4	no	—
136	#97	—	5	2	no	—
137	C₄×C₈	—	5	2	no	—
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
140	C₂×D₅	—	2	2	no	—
141	C₂×D₅	—	4	1	no	—
142	C₂×D₅	—	8	2	no	—
143	C₂×D₅	C₄×D₄	1	1	yes	characteristic
144	C₂×D₅	C₂×#41	1	1	yes	characteristic
145	AGL(1,𝔽₅)	—	4	2	no	—
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	C₄×D₄	1	1	yes	characteristic
148	C₄×C₅	C₂³×C₄	1	1	yes	characteristic
149	Dic₅	—	2	2	no	—
150	Dic₅	C₄×D₄	1	1	yes	characteristic
151	Dic₅	C₂²×D₄	1	1	yes	characteristic
152	#41	—	5	2	no	—
153	#41	—	5	2	no	—
154	#41	—	5	2	no	—
155	#41	—	10	2	no	—
156	#41	—	10	2	no	—
157	#41	C₂×AGL(1,𝔽₅)	1	2	yes	—
158	C₂×Q₈	—	5	2	no	—
159	C₂×D₄	—	5	2	no	—
160	C₂×D₄	—	10	2	no	—
161	C₂²×C₄	—	5	1	no	—
162	C₂²×C₄	—	10	2	no	—
163	Q₁₆	—	5	2	no	—
164	Q₁₆	—	5	1	no	—
165	Q₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
166	SD₁₆	—	5	2	no	—
167	SD₁₆	—	5	2	no	—
168	SD₁₆	—	5	2	no	—
169	SD₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
170	D₈	—	5	2	no	—
171	D₈	—	5	1	no	—
172	D₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
173	M₁₆	—	5	2	no	—
174	M₁₆	—	5	2	no	—
175	M₁₆	—	5	4	no	—
176	M₁₆	—	10	4	no	—
177	M₁₆	—	10	4	no	—
178	M₁₆	—	10	2	no	—
179	C₂×C₈	—	5	2	no	—
180	C₂×C₈	—	5	1	no	—
181	C₂×C₈	—	5	1	no	—
182	C₂×C₈	—	5	1	no	—
183	C₂×C₈	—	5	1	no	—
184	C₂×C₈	—	5	1	no	—
185	C₂×C₈	—	10	4	no	—
186	C₂×C₈	—	10	4	no	—
187	C₂×C₈	—	10	2	no	—
188	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
189	C₄⋊C₄	—	10	1	no	—
190	C₂²⋊C₄	—	10	1	no	—
191	C₄²	—	5	2	no	—
192	C₂×C₅	—	2	1	no	—
193	C₂×C₅	—	4	2	no	—
194	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
195	D₅	—	2	1	no	—
196	D₅	—	2	1	no	—
197	D₅	—	4	2	no	—
198	C₂³	—	5	1	no	—
199	C₂³	—	10	2	no	—
200	Q₈	—	5	2	no	—
201	Q₈	—	10	2	no	—
202	Q₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
203	D₄	—	2	2	no	—
204	D₄	—	5	2	no	—
205	D₄	—	10	2	no	—
206	D₄	—	10	2	no	—
207	D₄	—	10	2	no	—
208	D₄	—	10	2	no	—
209	D₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	5	1	no	—
212	C₂×C₄	—	5	1	no	—
213	C₂×C₄	—	5	1	no	—
214	C₂×C₄	—	5	1	no	—
215	C₂×C₄	—	5	1	no	—
216	C₂×C₄	—	10	2	no	—
217	C₂×C₄	—	10	2	no	—
218	C₂×C₄	—	10	2	no	—
219	C₂×C₄	—	10	2	no	—
220	C₂×C₄	—	10	2	no	—
221	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₈	—	5	4	no	—
223	C₈	—	5	1	no	—
224	C₈	—	5	1	no	—
225	C₈	—	10	4	no	—
226	C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
228	C₅	#2957	1	1	yes	characteristic, complemented, Sylow
229	C₂²	—	2	2	no	—
230	C₂²	—	5	1	no	—
231	C₂²	—	5	1	no	—
232	C₂²	—	10	2	no	—
233	C₂²	—	20	1	no	—
234	C₂²	—	40	2	no	—
235	C₂²	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	5	1	no	—
238	C₄	—	5	1	no	—
239	C₄	—	10	2	no	—
240	C₄	—	20	2	no	—
241	C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
242	C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₂	—	2	1	no	—
244	C₂	—	4	2	no	—
245	C₂	—	10	1	no	—
246	C₂	—	10	1	no	—
247	C₂	—	20	2	no	—
248	C₂	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂³×C₄
Frattini	C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#135	C₄
socle	C₂×C₅	C₂×C₄×D₄
prime radical	D₅×#135	C₂
cosocle	C₄×D₅	C₂⁴
hypercenter	#135	AGL(1,𝔽₅)
hypocenter	C₅	#2957
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #66264	C₂	semidirect
3. #65795	C₂	semidirect
4. #65741	C₂	semidirect
5. #65736	C₂	semidirect
6. #65733	C₂	semidirect
7. C₂×#3995	C₂	semidirect
8. #65718	C₂	semidirect
10. #4133	C₂²	semidirect
12. #4131	C₂²	semidirect
18. C₅×#135	C₄	semidirect
19. #4096	C₄	semidirect
21. D₅×Q₁₆	C₄	semidirect
23. D₅×SD₁₆	C₄	semidirect
25. D₅×D₈	C₄	semidirect
26. #4050	C₄	semidirect
28. #4010	C₂²	semidirect
29. #4008	C₂²	semidirect
30. #3995	C₂²	semidirect
31. #3995	C₂²	semidirect
32. #3992	C₂²	semidirect
33. C₈×AGL(1,𝔽₅)	C₂²	semidirect
119. #135	AGL(1,𝔽₅)	semidirect
228. C₅	#2957	semidirect

Maximal subgroups

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

subgroup	index
#66264	2
D₅×#135	2
#65795	2
#65741	2
#65736	2
#65733	2
C₂×#3995	2
#65718	2
#2957	5