Subgroups — #11992169

Order 1280.

368 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#433	C₂	1	1	yes	maximal, characteristic
2	#10602364	C₂	1	2	yes	maximal
3	#10600522	C₂	1	2	yes	maximal, complemented
4	#10598151	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#117	C₂²	1	1	yes	characteristic
6	C₅×#433	—	2	2	no	—
7	C₅×#433	—	2	2	no	—
8	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₅×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#65214	C₂²	1	4	yes	—
15	#64748	—	2	2	no	—
16	#64745	—	2	2	no	—
17	#64678	—	2	2	no	—
18	#64677	—	2	2	no	—
19	#8603	—	5	1	no	maximal, Sylow
20	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₅×#117	—	2	2	no	—
26	C₅×#117	—	2	2	no	—
27	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
28	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
29	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
30	C₂×C₅×C₂²⋊C₄	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₅×C₄⋊C₈	—	2	1	no	—
34	C₅×C₄⋊C₈	—	2	2	no	—
35	C₅×C₄⋊C₈	—	2	1	no	—
36	C₅×C₂²⋊C₈	—	2	1	no	—
37	C₅×C₂²⋊C₈	—	2	2	no	—
38	C₅×C₂²⋊C₈	—	2	1	no	—
39	C₅×#97	—	2	2	no	—
40	C₄×C₅×C₈	D₄	1	2	yes	—
41	C₄×C₅×C₈	D₄	1	2	yes	—
42	C₂²⋊#168	—	2	4	no	—
43	C₄⋊#168	—	2	4	no	—
44	C₄×#168	—	2	4	no	—
45	C₂×#433	D₅	1	1	yes	characteristic
46	#1564	—	5	2	no	—
47	#1458	—	5	2	no	—
48	#1275	—	5	2	no	—
49	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	—	2	1	no	—
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	—	2	2	no	—
54	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₅×C₄⋊C₄	—	2	2	no	—
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₂²⋊C₄	—	2	2	no	—
72	C₅×C₂²⋊C₄	—	2	2	no	—
73	C₄²×C₅	—	2	1	no	—
74	C₄²×C₅	—	2	1	no	—
75	C₄²×C₅	C₂×D₄	1	2	yes	—
76	C₄²×C₅	#41	1	2	yes	—
77	C₂×#168	—	4	4	no	—
78	C₂×#117	C₂×D₅	1	1	yes	characteristic
79	#433	—	2	2	no	—
80	#433	—	2	2	no	—
81	#431	—	5	4	no	—
82	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
83	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
84	C₂×C₂²⋊C₈	Dic₅	1	1	yes	characteristic
85	C₂×C₂²⋊C₈	Dic₅	1	1	yes	characteristic
86	C₂×#97	C₂×D₅	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
88	#335	—	10	2	no	—
89	#334	—	10	2	no	—
90	#324	—	10	2	no	—
91	#323	—	10	2	no	—
92	C₂³×C₅	—	2	1	no	—
93	C₂³×C₅	—	2	2	no	—
94	C₂³×C₅	—	4	1	no	—
95	C₂³×C₅	—	4	1	no	—
96	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	4	1	no	—
102	C₂×C₄×C₅	—	4	1	no	—
103	C₂×C₄×C₅	—	4	1	no	—
104	C₂×C₄×C₅	—	4	1	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	—	4	1	no	—
107	C₂×C₄×C₅	—	4	1	no	—
108	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	#117	1	1	yes	characteristic
110	C₂×C₄×C₅	#121	1	2	yes	—
111	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₅	#131	1	2	yes	—
113	C₂×C₄×C₅	#131	1	2	yes	—
114	C₅×C₈	—	2	1	no	—
115	C₅×C₈	—	2	2	no	—
116	C₅×C₈	—	2	1	no	—
117	C₅×C₈	—	4	1	no	—
118	C₅×C₈	—	4	1	no	—
119	#168	—	8	4	no	—
120	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
122	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
125	#117	—	2	2	no	—
126	#117	—	2	2	no	—
127	C₂×C₄⋊C₄	Dic₁₀	1	1	yes	characteristic
128	C₂×C₄⋊C₄	D₂₀	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	Dic₁₀	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	D₂₀	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	1	no	—
136	C₄⋊C₈	—	10	4	no	—
137	C₂²⋊C₈	—	2	1	no	—
138	C₂²⋊C₈	—	2	2	no	—
139	C₂²⋊C₈	—	2	1	no	—
140	C₂²⋊C₈	—	10	4	no	—
141	#97	—	2	2	no	—
142	C₄×C₈	—	10	4	no	—
143	C₄×C₈	C₂²⋊D₅	1	2	yes	—
144	C₄×C₈	C₂²⋊D₅	1	2	yes	—
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	—	2	1	no	—
147	C₂²×C₅	—	4	1	no	—
148	C₂²×C₅	—	4	1	no	—
149	C₂²×C₅	—	4	1	no	—
150	C₂²×C₅	—	8	2	no	—
151	C₂²×C₅	#380	1	1	yes	characteristic
152	C₂²×C₅	#424	1	2	yes	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	4	1	no	—
156	C₄×C₅	—	4	1	no	—
157	C₄×C₅	—	4	1	no	—
158	C₄×C₅	#408	1	1	yes	characteristic
159	C₄×C₅	#408	1	1	yes	characteristic
160	C₄×C₅	#433	1	2	yes	—
161	C₂⁴	C₄⋊Dic₅	1	1	yes	characteristic
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
167	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
168	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
169	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
170	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	20	4	no	—
179	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
180	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
181	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	#709	1	2	yes	—
189	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
190	C₂×C₅	—	2	2	no	—
191	C₂×C₅	—	8	1	no	—
192	C₂×C₅	#1798	1	1	yes	characteristic
193	C₂×C₅	#1812	1	1	yes	characteristic
194	C₂×C₅	#1813	1	1	yes	characteristic
195	C₂³	—	2	1	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	4	1	no	—
198	C₂³	—	4	1	no	—
199	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	#4070	1	2	yes	—
212	C₂×C₄	#4072	1	1	yes	characteristic
213	C₂×C₄	#4076	1	2	yes	—
214	C₂×C₄	#4094	1	2	yes	—
215	C₂×C₄	#4094	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	2	1	no	—
220	C₈	—	4	1	no	—
221	C₈	—	4	1	no	—
222	C₈	—	40	4	no	—
223	C₅	#8603	1	1	yes	characteristic, complemented, Sylow
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	8	2	no	—
230	C₂²	#65288	1	2	yes	—
231	C₂²	#65501	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	#65289	1	2	yes	—
238	C₄	#65523	1	1	yes	characteristic
239	C₄	#65523	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	—	8	1	no	—
242	C₂	#10602376	1	1	yes	characteristic
243	C₂	#10602805	1	1	yes	characteristic
244	C₂	#10602807	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#4072
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#433	C₂
socle	C₂²×C₅	#380
prime radical	C₂⁴×C₅	C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#433	D₅
hypocenter	C₅	#8603
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
3. #10600522	C₂	semidirect
4. #10598151	C₂	semidirect
223. C₅	#8603	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#433	2
#10602364	2
#10600522	2
#10598151	2
#8603	5