Subgroups — #12065050

Order 1280.

712 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2920	C₂	1	1	yes	maximal, characteristic
2	#10613780	C₂	1	2	yes	maximal
3	#10601537	C₂	1	8	yes	maximal, complemented
4	#10600642	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₅×#432	C₄	1	4	yes	—
7	C₅×#431	C₂²	1	4	yes	—
8	C₅×#423	C₄	1	2	yes	—
9	C₅×#407	C₄	1	2	yes	—
10	C₄×C₅×M₁₆	C₂²	1	2	yes	—
11	#65496	C₄	1	4	yes	—
12	#65215	C₄	1	8	yes	—
13	#65214	C₂²	1	8	yes	—
14	#65213	C₄	1	4	yes	—
15	C₄×#681	C₂²	1	4	yes	—
16	#64675	C₂²	1	4	yes	complemented
17	#64674	C₂²	1	8	yes	complemented
18	C₈×#168	C₂²	1	4	yes	complemented
19	#17339	—	5	1	no	maximal, Sylow
20	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂³	1	2	yes	—
29	C₂×#681	C₂×C₄	1	8	yes	—
30	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
31	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
32	C₅×#97	C₂×C₄	1	4	yes	—
33	C₄×C₅×C₈	C₂³	1	4	yes	—
34	C₂²⋊#168	C₂×C₄	1	16	yes	—
35	C₄⋊#168	C₂×C₄	1	16	yes	—
36	#3935	C₂×C₄	1	8	yes	—
37	C₄×#168	C₂³	1	8	yes	—
38	#2920	—	5	2	no	—
39	#2920	D₅	1	1	yes	characteristic
40	#1455	—	5	4	no	—
41	#1454	—	5	8	no	—
42	C₂⁴×C₅	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₅	—	4	4	no	—
44	C₂²×C₄×C₅	—	4	2	no	—
45	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₄²	1	2	yes	—
47	C₂²×C₄×C₅	C₄²	1	2	yes	—
48	C₂²×C₄×C₅	C₄²	1	4	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
53	C₅×M₁₆	—	4	4	no	—
54	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
55	C₄²×C₅	C₄²	1	4	yes	—
56	C₄²×C₅	C₄²	1	2	yes	—
57	C₄²×C₅	C₂²×C₄	1	2	yes	—
58	C₄²×C₅	C₂²×C₄	1	4	yes	—
59	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₅	C₂²×C₄	1	2	yes	—
61	C₄²×C₅	C₂⁴	1	1	yes	characteristic
62	#681	—	4	8	no	—
63	C₂×#168	C₂²×C₄	1	16	yes	—
64	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
65	#432	—	5	8	no	—
66	#432	Dic₅	1	4	yes	—
67	#431	—	5	8	no	—
68	#431	C₂×D₅	1	4	yes	—
69	#423	—	5	4	no	—
70	#423	Dic₅	1	2	yes	—
71	#407	—	5	4	no	—
72	#407	Dic₅	1	2	yes	—
73	C₄×M₁₆	—	5	4	no	—
74	C₄×M₁₆	C₂×D₅	1	2	yes	—
75	#322	—	5	8	no	—
76	#322	—	5	4	no	—
77	C₈²	—	5	4	no	—
78	C₂³×C₅	—	4	1	no	—
79	C₂³×C₅	—	4	2	no	—
80	C₂³×C₅	C₂×C₄²	1	2	yes	—
81	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅	—	2	4	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	8	no	—
85	C₂×C₄×C₅	—	2	4	no	—
86	C₂×C₄×C₅	—	4	2	no	—
87	C₂×C₄×C₅	—	4	1	no	—
88	C₂×C₄×C₅	—	4	4	no	—
89	C₂×C₄×C₅	—	4	2	no	—
90	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
92	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
94	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
96	C₅×C₈	—	2	8	no	—
97	#168	—	2	16	no	—
98	C₂³×C₄	C₄×D₅	1	2	yes	—
99	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
100	C₂×M₁₆	—	5	8	no	—
101	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
102	C₂×C₄²	C₄×D₅	1	2	yes	—
103	C₂×C₄²	C₄×D₅	1	4	yes	—
104	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₅	1	2	yes	—
106	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
107	C₂×C₄²	C₂²×D₅	1	2	yes	—
108	C₄⋊C₈	—	5	16	no	—
109	C₄⋊C₈	C₂×Dic₅	1	8	yes	—
110	C₂²⋊C₈	—	5	16	no	—
111	C₂²⋊C₈	C₂×Dic₅	1	8	yes	—
112	#97	—	5	8	no	—
113	#97	C₂×Dic₅	1	4	yes	—
114	C₄×C₈	—	5	8	no	—
115	C₄×C₈	C₂²×D₅	1	4	yes	—
116	C₂²×C₅	—	2	1	no	—
117	C₂²×C₅	—	2	2	no	—
118	C₂²×C₅	—	2	2	no	—
119	C₂²×C₅	—	2	4	no	—
120	C₂²×C₅	—	4	4	no	—
121	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
122	C₄×C₅	—	2	1	no	—
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	2	4	no	—
126	C₄×C₅	#568	1	4	yes	—
127	C₄×C₅	#568	1	2	yes	—
128	C₂⁴	C₄×Dic₅	1	1	yes	characteristic
129	C₂²×C₄	—	4	4	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
132	C₂²×C₄	C₄×Dic₅	1	2	yes	—
133	C₂²×C₄	C₄×Dic₅	1	2	yes	—
134	C₂²×C₄	C₄×Dic₅	1	4	yes	—
135	C₂²×C₄	C₂×C₄×D₅	1	4	yes	—
136	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
139	M₁₆	—	4	4	no	—
140	M₁₆	—	20	8	no	—
141	C₂×C₈	—	5	16	no	—
142	C₂×C₈	C₂²×Dic₅	1	8	yes	—
143	C₄²	C₄×Dic₅	1	4	yes	—
144	C₄²	C₄×Dic₅	1	2	yes	—
145	C₄²	C₂×C₄×D₅	1	2	yes	—
146	C₄²	C₂×C₄×D₅	1	4	yes	—
147	C₄²	C₂²×Dic₅	1	1	yes	characteristic
148	C₄²	C₂²×Dic₅	1	2	yes	—
149	C₄²	C₂³×D₅	1	1	yes	characteristic
150	C₂×C₅	—	4	1	no	—
151	C₂×C₅	—	4	2	no	—
152	C₂×C₅	#2873	1	1	yes	characteristic
153	C₂×C₅	#2873	1	2	yes	—
154	C₂³	—	4	1	no	—
155	C₂³	—	4	2	no	—
156	C₂³	C₂×C₄×Dic₅	1	2	yes	—
157	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
168	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
169	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
170	C₂×C₄	C₂²×C₄×D₅	1	2	yes	—
171	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
172	C₈	—	2	8	no	—
173	C₈	—	10	16	no	—
174	C₅	#17339	1	1	yes	characteristic, complemented, Sylow
175	C₂²	—	2	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	4	no	—
179	C₂²	—	4	4	no	—
180	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	2	4	no	—
185	C₄	#66079	1	4	yes	—
186	C₄	#66153	1	2	yes	—
187	C₂	—	4	1	no	—
188	C₂	—	4	2	no	—
189	C₂	#10613933	1	1	yes	characteristic
190	C₂	#10615478	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅
commutator, socle	C₂²×C₅	C₂²×C₄²
Fitting	C₅×#2920	C₂
prime radical	C₂⁴×C₅	C₄²
cosocle	C₄²×C₅	C₂⁴
hypercenter	#2920	D₅
hypocenter	C₅	#17339
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #10601537	C₂	semidirect
4. #10600642	C₂	semidirect
16. #64675	C₂²	semidirect
17. #64674	C₂²	semidirect
18. C₈×#168	C₂²	semidirect
174. C₅	#17339	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2920	2
#10613780	2
#10601537	2
#10600642	2
#17339	5