Subgroups — #13015255

Order 1280.

526 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10603839	C₂	1	1	yes	maximal, characteristic
2	#10603839	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
4	#10603807	C₂	1	1	yes	maximal, characteristic
5	#10603806	C₂	1	1	yes	maximal, characteristic
6	#10603806	C₂	1	1	yes	maximal, characteristic
7	D₅×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×AGL(1,𝔽₅)	C₄	1	4	yes	—
9	C₄²×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
10	C₄×D₅⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×D₅⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×D₅⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
14	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
16	(C₂×AGL(1,𝔽₅))⋊C₈	C₄	1	4	yes	—
17	(C₂×AGL(1,𝔽₅))⋊C₈	C₄	1	4	yes	—
18	#170⋊C₈	C₄	1	4	yes	complemented
19	#170⋊C₈	C₄	1	4	yes	complemented
20	#64879	C₄	1	4	yes	complemented
21	C₅×#322	C₄	1	1	yes	characteristic, complemented
22	#64675	C₄	1	1	yes	characteristic, complemented
23	#64674	C₄	1	1	yes	characteristic, complemented
24	#64674	C₄	1	1	yes	characteristic, complemented
25	#7843	—	5	1	no	maximal, Sylow
26	C₂×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
27	C₂×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
28	C₂×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
29	C₂×D₅⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×D₅⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×D₅⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
35	C₄²×D₅	C₂³	1	1	yes	characteristic
36	C₄×#170	C₂×C₄	1	2	yes	—
37	C₄×#170	C₂×C₄	1	2	yes	—
38	C₄×#170	C₂×C₄	1	2	yes	—
39	C₄×#170	C₂×C₄	1	2	yes	—
40	C₄×#170	C₂×C₄	1	2	yes	—
41	C₄×#170	C₂×C₄	1	2	yes	—
42	C₈×AGL(1,𝔽₅)	—	2	8	no	—
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₈×Dic₅	C₂×C₄	1	2	yes	—
47	C₈×Dic₅	C₂×C₄	1	2	yes	—
48	C₈×Dic₅	C₂×C₄	1	2	yes	—
49	C₄×#168	C₂×C₄	1	1	yes	characteristic
50	C₄×#168	C₂×C₄	1	1	yes	characteristic
51	C₄×#168	C₂×C₄	1	1	yes	characteristic
52	#1744	—	5	1	no	—
53	#1744	—	5	1	no	—
54	C₄²×C₈	—	5	1	no	—
55	C₂×#322	—	5	1	no	—
56	C₂×#322	—	5	1	no	—
57	C₂×#322	—	5	1	no	—
58	C₂×#322	—	5	1	no	—
59	C₂²×AGL(1,𝔽₅)	C₄²	1	4	yes	—
60	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×#170	C₄²	1	4	yes	—
64	C₂×#170	C₄²	1	4	yes	—
65	C₂×#170	C₄²	1	4	yes	—
66	C₄×AGL(1,𝔽₅)	—	2	4	no	—
67	C₄×AGL(1,𝔽₅)	—	2	4	no	—
68	C₄×AGL(1,𝔽₅)	M₁₆	1	8	yes	—
69	D₅⋊C₈	—	2	4	no	—
70	D₅⋊C₈	—	2	4	no	—
71	D₅⋊C₈	—	2	4	no	—
72	C₂×C₅×C₈	C₄²	1	2	yes	—
73	C₂×C₅×C₈	C₄²	1	2	yes	—
74	C₂×C₅×C₈	C₄²	1	2	yes	—
75	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
77	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
78	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×#168	C₄²	1	2	yes	—
80	C₂×#168	C₄²	1	2	yes	—
81	C₂×#168	C₄²	1	2	yes	—
82	C₈×D₅	—	2	4	no	—
83	C₈×D₅	—	2	4	no	—
84	C₈×D₅	—	2	4	no	—
85	C₂×C₄×C₈	—	5	1	no	—
86	C₂×C₄×C₈	—	5	1	no	—
87	C₂×C₄×C₈	—	5	1	no	—
88	C₂×C₄×C₈	—	5	1	no	—
89	C₂×C₄×C₈	—	5	1	no	—
90	C₂×C₄×C₈	—	5	1	no	—
91	C₂×C₄×C₈	—	5	4	no	—
92	C₄³	—	5	1	no	—
93	#336	—	5	4	no	—
94	#336	—	5	4	no	—
95	#322	—	5	4	no	—
96	#322	—	5	4	no	—
97	#322	—	5	4	no	—
98	#322	—	5	1	no	—
99	#322	—	5	1	no	—
100	#322	—	5	1	no	—
101	#322	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
102	C₂²×D₅	C₂×C₄²	1	1	yes	characteristic
103	C₂×AGL(1,𝔽₅)	—	2	4	no	—
104	C₂×AGL(1,𝔽₅)	—	2	4	no	—
105	C₂×AGL(1,𝔽₅)	#97	1	8	yes	—
106	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
109	C₂×Dic₅	C₂×C₄²	1	1	yes	characteristic
110	C₂×Dic₅	C₂×C₄²	1	1	yes	characteristic
111	C₂×Dic₅	C₂×C₄²	1	1	yes	characteristic
112	C₄×D₅	C₂×M₁₆	1	4	yes	—
113	C₄×D₅	#131	1	4	yes	—
114	C₄×D₅	#131	1	4	yes	—
115	#170	—	2	4	no	—
116	#170	—	2	4	no	—
117	#170	—	2	4	no	—
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	—	2	2	no	—
120	C₅×C₈	—	2	2	no	—
121	#168	—	2	2	no	—
122	#168	—	2	2	no	—
123	#168	—	2	2	no	—
124	C₂²×C₈	—	5	2	no	—
125	C₂²×C₈	—	5	2	no	—
126	C₂²×C₈	—	5	2	no	—
127	C₂²×C₈	—	5	2	no	—
128	C₂²×C₈	—	5	2	no	—
129	C₂²×C₈	—	5	2	no	—
130	C₂×C₄²	—	5	2	no	—
131	C₂×C₄²	—	5	1	no	—
132	C₂×C₄²	—	5	2	no	—
133	C₂×C₄²	—	5	2	no	—
134	C₄×C₈	—	5	2	no	—
135	C₄×C₈	—	5	2	no	—
136	C₄×C₈	—	5	2	no	—
137	C₄×C₈	—	5	2	no	—
138	C₄×C₈	—	5	2	no	—
139	C₄×C₈	—	5	2	no	—
140	C₄×C₈	—	5	2	no	—
141	C₄×C₈	—	5	2	no	—
142	C₄×C₈	—	5	2	no	—
143	C₄×C₈	—	5	1	no	—
144	C₄×C₈	—	5	1	no	—
145	C₄×C₈	—	5	1	no	—
146	C₄×C₈	—	10	8	no	—
147	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂²×C₅	C₄³	1	1	yes	characteristic
151	C₂×D₅	C₂×#97	1	2	yes	—
152	C₂×D₅	#405	1	2	yes	—
153	C₂×D₅	#405	1	2	yes	—
154	AGL(1,𝔽₅)	—	2	8	no	—
155	C₄×C₅	C₄×M₁₆	1	2	yes	—
156	C₄×C₅	#405	1	2	yes	—
157	C₄×C₅	#405	1	2	yes	—
158	Dic₅	C₄×M₁₆	1	2	yes	—
159	Dic₅	#405	1	2	yes	—
160	Dic₅	#405	1	2	yes	—
161	C₂²×C₄	—	5	1	no	—
162	C₂²×C₄	—	5	1	no	—
163	C₂²×C₄	—	5	1	no	—
164	C₂²×C₄	—	5	4	no	—
165	C₂×C₈	—	5	4	no	—
166	C₂×C₈	—	5	4	no	—
167	C₂×C₈	—	5	4	no	—
168	C₂×C₈	—	5	2	no	—
169	C₂×C₈	—	5	2	no	—
170	C₂×C₈	—	5	2	no	—
171	C₂×C₈	—	10	4	no	—
172	C₂×C₈	—	10	4	no	—
173	C₂×C₈	—	10	4	no	—
174	C₂×C₈	—	10	4	no	—
175	C₂×C₈	—	10	4	no	—
176	C₂×C₈	—	10	4	no	—
177	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
178	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
179	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
180	C₄²	—	5	8	no	—
181	C₄²	—	5	1	no	—
182	C₄²	—	5	1	no	—
183	C₄²	—	5	1	no	—
184	C₄²	—	10	4	no	—
185	C₄²	—	10	4	no	—
186	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
187	C₂×C₅	C₄×#97	1	1	yes	characteristic
188	C₂×C₅	#1726	1	1	yes	characteristic
189	C₂×C₅	#1726	1	1	yes	characteristic
190	D₅	#1452	1	4	yes	—
191	C₂³	—	5	1	no	—
192	C₂×C₄	—	5	1	no	—
193	C₂×C₄	—	5	1	no	—
194	C₂×C₄	—	5	1	no	—
195	C₂×C₄	—	5	4	no	—
196	C₂×C₄	—	5	4	no	—
197	C₂×C₄	—	5	4	no	—
198	C₂×C₄	—	5	8	no	—
199	C₂×C₄	—	10	4	no	—
200	C₂×C₄	—	10	4	no	—
201	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	10	4	no	—
208	C₈	—	10	4	no	—
209	C₈	—	10	4	no	—
210	C₈	—	10	2	no	—
211	C₈	—	10	2	no	—
212	C₈	—	10	2	no	—
213	C₅	#7843	1	1	yes	characteristic, complemented, Sylow
214	C₂²	—	5	2	no	—
215	C₂²	—	5	2	no	—
216	C₂²	—	5	2	no	—
217	C₂²	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₄	—	5	2	no	—
219	C₄	—	5	2	no	—
220	C₄	—	5	2	no	—
221	C₄	—	10	8	no	—
222	C₄	#65718	1	2	yes	—
223	C₄	M₁₆×AGL(1,𝔽₅)	1	2	yes	—
224	C₄	#65728	1	2	yes	—
225	C₂	—	5	4	no	—
226	C₂	#10603838	1	1	yes	characteristic
227	C₂	AGL(1,𝔽₅)×#97	1	1	yes	characteristic
228	C₂	#10603877	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₄³
Fitting	C₅×#322	C₄
prime radical	C₂²×D₅	C₂×C₄²
cosocle	C₄²×D₅	C₂³
hypercenter	#322	AGL(1,𝔽₅)
hypocenter	C₅	#7843
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
18. #170⋊C₈	C₄	semidirect
19. #170⋊C₈	C₄	semidirect
20. #64879	C₄	semidirect
21. C₅×#322	C₄	semidirect
22. #64675	C₄	semidirect
23. #64674	C₄	semidirect
24. #64674	C₄	semidirect
101. #322	AGL(1,𝔽₅)	semidirect
213. C₅	#7843	semidirect

Maximal subgroups

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

subgroup	index
#10603839	2
C₄×C₈×AGL(1,𝔽₅)	2
#10603807	2
#10603806	2
D₅×#322	2
#7843	5