Subgroups — #11998675

Order 1280.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic
2	#10602631	C₂	1	4	yes	maximal
3	#10600559	C₂	1	1	yes	maximal, characteristic
4	#10600517	C₂	1	1	yes	maximal, characteristic
5	C₅×#533	C₂²	1	2	yes	—
6	C₅×#500	C₄	1	2	yes	complemented
7	C₅×#499	—	2	1	no	—
8	C₅×#499	—	2	1	no	—
9	C₅×#498	C₄	1	2	yes	complemented
10	C₅×#427	—	2	1	no	—
11	C₅×#427	—	2	1	no	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×#3950	—	2	1	no	—
14	C₂×#3949	—	2	1	no	—
15	#65225	C₂²	1	2	yes	—
16	#65214	C₂²	1	2	yes	—
17	#64758	—	2	8	no	—
18	#64675	—	2	1	no	—
19	C₈×#168	—	2	1	no	—
20	#9755	—	5	1	no	maximal, Sylow
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₅×#125	—	2	2	no	—
25	C₅×C₂²⋊Q₈	—	2	4	no	—
26	C₄×C₅×Q₈	—	2	4	no	—
27	C₅×#117	—	2	2	no	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₅	—	2	2	no	—
30	C₅×#107	—	2	2	no	—
31	C₅×#107	—	2	2	no	—
32	C₅×#106	—	2	2	no	—
33	C₅×#106	—	2	2	no	—
34	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₅×C₈	D₄	1	1	yes	characteristic
37	C₄×C₅×C₈	D₄	1	1	yes	characteristic
38	#3963	—	2	4	no	—
39	C₂²⋊#168	D₄	1	4	yes	—
40	#3950	—	4	2	no	—
41	#3949	—	4	2	no	—
42	C₄⋊#168	—	2	2	no	—
43	C₄×#168	—	2	2	no	—
44	#3160	D₅	1	1	yes	characteristic
45	#1846	—	5	1	no	—
46	#1518	—	5	4	no	—
47	#1450	—	5	1	no	—
48	C₂×C₅×Q₈	—	2	4	no	—
49	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅	—	4	2	no	—
52	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	SD₁₆	1	2	yes	—
57	C₂×C₅×C₈	SD₁₆	1	2	yes	—
58	C₂×C₅×C₈	Q₁₆	1	2	yes	—
59	C₂×C₅×C₈	Q₁₆	1	2	yes	—
60	C₅×C₄⋊C₄	—	2	4	no	—
61	C₅×C₄⋊C₄	—	2	4	no	—
62	C₅×C₄⋊C₄	—	4	4	no	—
63	C₅×C₂²⋊C₄	—	4	2	no	—
64	C₄²×C₅	—	4	2	no	—
65	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅	#41	1	2	yes	—
68	C₄⋊Dic₅	—	4	2	no	—
69	C₄⋊Dic₅	—	4	2	no	—
70	C₂×#168	—	2	4	no	—
71	#533	C₂×D₅	1	2	yes	—
72	#500	Dic₅	1	2	yes	complemented
73	#499	—	2	1	no	—
74	#499	—	2	1	no	—
75	#498	Dic₅	1	2	yes	complemented
76	#431	—	5	2	no	—
77	#427	—	2	1	no	—
78	#427	—	2	1	no	—
79	C₂×#107	—	10	1	no	—
80	C₂×#106	—	10	1	no	—
81	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
82	#382	—	5	2	no	—
83	#328	—	10	8	no	—
84	#322	—	10	1	no	—
85	C₈²	—	10	1	no	—
86	C₂³×C₅	#127	1	1	yes	characteristic
87	C₅×Q₈	—	8	4	no	—
88	C₂×C₄×C₅	—	4	2	no	—
89	C₂×C₄×C₅	—	4	4	no	—
90	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₅	#121	1	2	yes	—
93	C₂×C₄×C₅	#124	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×Dic₅	—	8	4	no	—
99	C₂×Dic₅	—	8	2	no	—
100	C₅×C₈	#103	1	2	yes	—
101	C₅×C₈	#103	1	2	yes	—
102	C₅×C₈	#103	1	2	yes	—
103	C₅×C₈	#103	1	2	yes	—
104	#168	—	4	4	no	—
105	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
107	C₄⋊Q₈	C₂×Dic₅	1	4	yes	—
108	#125	—	2	2	no	—
109	C₂²⋊Q₈	—	2	4	no	—
110	C₄×Q₈	—	2	4	no	—
111	#117	—	2	2	no	—
112	C₂×C₄⋊C₄	—	10	2	no	—
113	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
114	#107	—	2	2	no	—
115	#107	—	2	2	no	—
116	#107	—	20	2	no	—
117	#106	—	2	2	no	—
118	#106	—	2	2	no	—
119	#106	—	20	2	no	—
120	C₄⋊C₈	—	10	2	no	—
121	C₂²⋊C₈	—	5	4	no	—
122	C₄×C₈	—	10	2	no	—
123	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
124	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
126	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
127	#95	—	10	4	no	—
128	C₂²×C₅	—	2	2	no	—
129	C₂²×C₅	#390	1	1	yes	characteristic
130	C₂²×C₅	#492	1	1	yes	characteristic
131	C₂²×C₅	#494	1	1	yes	characteristic
132	C₄×C₅	—	2	2	no	—
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	—	8	4	no	—
135	C₄×C₅	C₂×#103	1	1	yes	characteristic
136	C₄×C₅	C₂×#103	1	1	yes	characteristic
137	C₄×C₅	#487	1	1	yes	characteristic
138	C₄×C₅	#487	1	1	yes	characteristic
139	Dic₅	—	16	2	no	—
140	C₂×Q₈	—	2	4	no	—
141	C₂²×C₄	—	20	2	no	—
142	C₂²×C₄	D₄×D₅	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
144	C₂×C₈	—	10	4	no	—
145	C₂×C₈	#688	1	2	yes	—
146	C₂×C₈	#688	1	2	yes	—
147	C₂×C₈	#689	1	2	yes	—
148	C₂×C₈	#689	1	2	yes	—
149	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
150	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
152	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	20	2	no	—
157	C₄⋊C₄	—	20	2	no	—
158	C₂²⋊C₄	—	4	2	no	—
159	C₄²	—	4	2	no	—
160	C₄²	#711	1	2	yes	—
161	C₄²	C₂²×Dic₅	1	1	yes	characteristic
162	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
163	C₂×C₅	—	2	1	no	—
164	C₂×C₅	—	2	1	no	—
165	C₂×C₅	#1956	1	1	yes	characteristic
166	C₂×C₅	#1976	1	2	yes	—
167	C₂³	#4086	1	1	yes	characteristic
168	Q₈	—	8	4	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	40	4	no	—
172	C₂×C₄	—	40	2	no	—
173	C₂×C₄	D₄×Dic₅	1	2	yes	—
174	C₂×C₄	#4082	1	1	yes	characteristic
175	C₂×C₄	#4084	1	2	yes	—
176	C₂×C₄	C₂×#688	1	1	yes	characteristic
177	C₂×C₄	C₂×#688	1	1	yes	characteristic
178	C₂×C₄	C₂×#689	1	1	yes	characteristic
179	C₂×C₄	C₂×#689	1	1	yes	characteristic
180	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
181	C₈	—	20	4	no	—
182	C₈	#3967	1	2	yes	—
183	C₈	#3967	1	2	yes	—
184	C₈	#3967	1	2	yes	—
185	C₈	#3967	1	2	yes	—
186	C₅	#9755	1	1	yes	characteristic, complemented, Sylow
187	C₂²	—	2	2	no	—
188	C₂²	#65374	1	1	yes	characteristic
189	C₂²	#65381	1	1	yes	characteristic
190	C₂²	#65510	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	8	4	no	—
194	C₄	—	80	2	no	—
195	C₄	#65370	1	1	yes	characteristic
196	C₄	#65370	1	1	yes	characteristic
197	C₄	C₂×#3967	1	1	yes	characteristic
198	C₄	C₂×#3967	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#10602578	1	1	yes	characteristic
202	C₂	#10602879	1	2	yes	—

	subgroup	G/H
center	C₂²	#65510
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#3160	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#3160	D₅
hypocenter	C₅	#9755
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
6. C₅×#500	C₄	semidirect
9. C₅×#498	C₄	semidirect
72. #500	Dic₅	semidirect
75. #498	Dic₅	semidirect
186. C₅	#9755	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3160	2
#10602631	2
#10600559	2
#10600517	2
#9755	5