Subgroups — #11992579

Order 1280.

346 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2960	C₂	1	1	yes	maximal, characteristic
2	#10602623	C₂	1	2	yes	maximal
3	#10600584	C₂	1	1	yes	maximal, characteristic
4	C₂×#64678	C₂	1	1	yes	maximal, characteristic
5	#10598202	C₂	1	2	yes	maximal
6	C₅×#533	C₂²	1	1	yes	characteristic
7	C₅×#446	—	2	1	no	—
8	C₅×#446	C₄	1	2	yes	—
9	C₅×C₈×Q₈	—	2	1	no	—
10	C₅×C₈×Q₈	C₄	1	2	yes	—
11	C₅×#424	—	2	1	no	—
12	C₅×#423	C₂²	1	1	yes	characteristic
13	C₅×#405	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#65213	C₂²	1	2	yes	—
16	C₂×C₄⋊#168	C₂²	1	2	yes	—
17	#64772	—	2	1	no	—
18	#64772	—	2	1	no	—
19	#64767	—	2	2	no	—
20	#64766	—	2	2	no	—
21	#64678	—	2	1	no	—
22	#64678	—	2	1	no	—
23	#8689	—	5	1	no	maximal, Sylow
24	C₂×C₅×M₁₆	—	2	1	no	—
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
28	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
29	C₅×#125	—	2	1	no	—
30	C₅×C₂²⋊Q₈	—	2	1	no	—
31	C₅×C₂²⋊Q₈	Q₈	1	2	yes	—
32	C₄×C₅×Q₈	—	2	1	no	—
33	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
34	C₅×#117	—	2	1	no	—
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₂×#681	—	2	2	no	—
37	C₂²×#168	—	2	2	no	—
38	C₅×C₄⋊C₈	—	2	2	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	1	no	—
41	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₅×#97	—	2	2	no	—
43	C₄×C₅×C₈	—	2	2	no	—
44	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	D₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₄⋊#168	—	2	2	no	—
49	C₄⋊#168	—	2	2	no	—
50	C₄⋊#168	—	2	2	no	—
51	C₄⋊#168	—	2	2	no	—
52	#2960	D₅	1	1	yes	characteristic
53	#1837	—	5	1	no	—
54	C₂×#335	—	5	1	no	—
55	#1550	—	5	2	no	—
56	#1292	—	5	2	no	—
57	C₂×C₅×Q₈	—	2	1	no	—
58	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	#41	1	1	yes	characteristic
62	C₅×M₁₆	—	4	2	no	—
63	C₂×C₅×C₈	—	2	2	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	1	yes	characteristic
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	—	2	1	no	—
72	C₅×C₄⋊C₄	—	2	1	no	—
73	C₅×C₄⋊C₄	—	2	1	no	—
74	C₅×C₄⋊C₄	—	2	2	no	—
75	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
76	C₅×C₂²⋊C₄	—	2	2	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅	#41	1	1	yes	characteristic
82	#681	—	4	2	no	—
83	#681	—	4	2	no	—
84	C₂×#168	—	4	2	no	—
85	C₂×#168	—	4	2	no	—
86	C₂×#168	—	8	2	no	—
87	#533	C₂×D₅	1	1	yes	characteristic
88	#446	—	2	1	no	—
89	#446	Dic₅	1	2	yes	—
90	C₈×Q₈	—	2	1	no	—
91	C₈×Q₈	Dic₅	1	2	yes	—
92	#424	—	2	1	no	—
93	#423	—	5	2	no	—
94	#423	C₂×D₅	1	1	yes	characteristic
95	C₂×C₄⋊C₈	—	5	2	no	—
96	#405	—	2	1	no	—
97	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
98	#341	—	10	1	no	—
99	#341	—	10	1	no	—
100	#335	—	10	1	no	—
101	#335	—	10	1	no	—
102	#333	—	10	2	no	—
103	#332	—	10	2	no	—
104	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
105	C₅×Q₈	—	4	2	no	—
106	C₅×Q₈	—	4	1	no	—
107	C₅×Q₈	—	4	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	4	1	no	—
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	#117	1	1	yes	characteristic
116	C₂×C₄×C₅	#121	1	1	yes	characteristic
117	C₂×C₄×C₅	#121	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅	#135	1	2	yes	—
120	C₂×C₄×C₅	#137	1	2	yes	—
121	C₅×C₈	—	2	1	no	—
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	—	2	1	no	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	4	2	no	—
126	#168	—	8	2	no	—
127	#168	—	8	2	no	—
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	10	2	no	—
130	C₂²×C₈	—	10	2	no	—
131	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
132	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
133	C₄⋊Q₈	D₂₀	1	1	yes	characteristic
134	C₄⋊Q₈	D₂₀	1	1	yes	characteristic
135	#125	—	2	1	no	—
136	C₂²⋊Q₈	—	2	1	no	—
137	C₂²⋊Q₈	Dic₁₀	1	2	yes	—
138	C₄×Q₈	—	2	1	no	—
139	C₄×Q₈	C₂×Dic₅	1	2	yes	—
140	#117	—	2	1	no	—
141	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	1	no	—
145	C₄⋊C₈	—	10	2	no	—
146	C₄⋊C₈	—	10	2	no	—
147	C₄⋊C₈	—	10	2	no	—
148	C₄⋊C₈	—	10	2	no	—
149	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
150	#97	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
153	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
156	C₂²×C₅	—	2	1	no	—
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	#380	1	1	yes	characteristic
159	C₂²×C₅	#479	1	2	yes	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	4	2	no	—
164	C₄×C₅	—	4	2	no	—
165	C₄×C₅	#416	1	1	yes	characteristic
166	C₄×C₅	#419	1	1	yes	characteristic
167	C₄×C₅	#464	1	2	yes	—
168	C₄×C₅	#472	1	1	yes	characteristic
169	C₄×C₅	#473	1	1	yes	characteristic
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	C₄⋊Dic₅	1	2	yes	—
172	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
173	C₂²×C₄	#709	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
175	M₁₆	—	4	2	no	—
176	M₁₆	—	20	2	no	—
177	M₁₆	—	20	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	20	2	no	—
182	C₂×C₈	—	20	2	no	—
183	C₂×C₈	—	40	2	no	—
184	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	C₄⋊Dic₅	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	C₂×D₂₀	1	1	yes	characteristic
197	C₄²	#709	1	1	yes	characteristic
198	C₄²	C₂²×Dic₅	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
200	C₂×C₅	—	2	2	no	—
201	C₂×C₅	#1805	1	1	yes	characteristic
202	C₂×C₅	#1813	1	1	yes	characteristic
203	C₂×C₅	#1816	1	1	yes	characteristic
204	C₂³	#4070	1	1	yes	characteristic
205	Q₈	—	4	2	no	—
206	Q₈	—	4	1	no	—
207	Q₈	—	4	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	#4070	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
215	C₂×C₄	#4072	1	1	yes	characteristic
216	C₂×C₄	#4076	1	1	yes	characteristic
217	C₂×C₄	#4076	1	1	yes	characteristic
218	C₂×C₄	#4088	1	2	yes	—
219	C₂×C₄	#4096	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	4	2	no	—
226	C₈	—	40	2	no	—
227	C₈	—	40	2	no	—
228	C₅	#8689	1	1	yes	characteristic, complemented, Sylow
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#65311	1	2	yes	—
232	C₂²	#65501	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	#65317	1	2	yes	—
239	C₄	#65431	1	1	yes	characteristic
240	C₄	#65432	1	1	yes	characteristic
241	C₄	#65514	1	1	yes	characteristic
242	C₄	#65522	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#10602420	1	1	yes	characteristic
245	C₂	#10602807	1	1	yes	characteristic
246	C₂	#10602809	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65501
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2960	C₂
socle	C₂²×C₅	#380
prime radical	C₂³×C₅	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2960	D₅
hypocenter	C₅	#8689
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
228. C₅	#8689	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2960	2
#10602623	2
#10600584	2
C₂×#64678	2
#10598202	2
#8689	5