Subgroups — #11992539

Order 1280.

350 conjugacy classes of subgroups.

The subgroup lattice has 244 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	#10602631	C₂	1	2	yes	maximal
3	#10600556	C₂	1	1	yes	maximal, characteristic
4	#10600522	C₂	1	1	yes	maximal, characteristic
5	#10598204	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	complemented
9	C₅×C₈×Q₈	—	2	1	no	—
10	C₅×C₈×Q₈	C₄	1	2	yes	complemented
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	#65225	C₂²	1	2	yes	—
16	#65214	C₂²	1	2	yes	—
17	#64771	—	2	1	no	—
18	#64771	—	2	1	no	—
19	#64758	—	2	4	no	—
20	#64678	—	2	1	no	—
21	#64677	—	2	1	no	—
22	#8681	—	5	1	no	maximal, Sylow
23	C₂×C₅×M₁₆	—	2	1	no	—
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₅×C₄⋊Q₈	Q₈	1	1	yes	characteristic
27	C₅×C₄⋊Q₈	Q₈	1	1	yes	characteristic
28	C₅×#125	—	2	1	no	—
29	C₅×C₂²⋊Q₈	—	2	1	no	—
30	C₅×C₂²⋊Q₈	D₄	1	2	yes	—
31	C₄×C₅×Q₈	—	2	1	no	—
32	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
33	C₅×#117	—	2	1	no	—
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊Dic₅	—	2	2	no	—
36	C₅×C₄⋊C₈	—	2	2	no	—
37	C₅×C₄⋊C₈	—	2	2	no	—
38	C₅×C₄⋊C₈	—	2	1	no	—
39	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₅×#97	—	2	2	no	—
41	C₄×C₅×C₈	—	2	2	no	—
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	#3963	—	2	2	no	—
47	#3963	—	2	2	no	—
48	C₂²⋊#168	—	2	2	no	—
49	C₄⋊#168	—	2	2	no	—
50	C₄×#168	—	2	2	no	—
51	#2960	D₅	1	1	yes	characteristic
52	#1835	—	5	1	no	—
53	#1564	—	5	1	no	—
54	#1518	—	5	2	no	—
55	#1286	—	5	2	no	—
56	C₂×C₅×Q₈	—	2	1	no	—
57	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	#41	1	1	yes	characteristic
61	C₂²×Dic₅	—	4	2	no	—
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	2	no	—
72	C₅×C₄⋊C₄	—	2	1	no	—
73	C₅×C₄⋊C₄	—	2	1	no	—
74	C₅×C₄⋊C₄	—	2	1	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×Q₈	1	1	yes	characteristic
81	C₄²×C₅	#41	1	1	yes	characteristic
82	C₄⋊Dic₅	—	4	2	no	—
83	C₄⋊Dic₅	—	4	2	no	—
84	C₂×#168	—	4	2	no	—
85	#533	C₂×D₅	1	1	yes	characteristic
86	#446	—	2	1	no	—
87	#446	Dic₅	1	2	yes	complemented
88	C₈×Q₈	—	2	1	no	—
89	C₈×Q₈	Dic₅	1	2	yes	complemented
90	#431	—	5	2	no	—
91	#424	—	2	1	no	—
92	#423	C₂×D₅	1	1	yes	characteristic
93	#405	—	2	1	no	—
94	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
95	#382	—	5	2	no	—
96	#340	—	10	1	no	—
97	#340	—	10	1	no	—
98	#335	—	10	1	no	—
99	#334	—	10	1	no	—
100	#328	—	10	4	no	—
101	C₂³×C₅	#121	1	1	yes	characteristic
102	C₅×Q₈	—	4	2	no	—
103	C₅×Q₈	—	4	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	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	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	#117	1	1	yes	characteristic
111	C₂×C₄×C₅	#121	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅	#135	1	2	yes	—
115	C₂×C₄×C₅	#137	1	2	yes	—
116	C₂×Dic₅	—	8	2	no	—
117	C₂×Dic₅	—	8	2	no	—
118	C₂×Dic₅	—	8	2	no	—
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	—	2	1	no	—
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	—	4	2	no	—
124	#168	—	8	2	no	—
125	C₂×M₁₆	—	2	1	no	—
126	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
127	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
128	C₄⋊Q₈	Dic₁₀	1	1	yes	characteristic
129	C₄⋊Q₈	Dic₁₀	1	1	yes	characteristic
130	#125	—	2	1	no	—
131	C₂²⋊Q₈	—	2	1	no	—
132	C₂²⋊Q₈	D₂₀	1	2	yes	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×Q₈	C₂×Dic₅	1	2	yes	—
135	#117	—	2	1	no	—
136	C₂×C₄⋊C₄	—	10	2	no	—
137	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	—	10	2	no	—
142	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
143	C₂²⋊C₈	—	10	2	no	—
144	#97	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	10	2	no	—
147	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
148	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
151	#95	—	10	2	no	—
152	#95	—	10	2	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	#380	1	1	yes	characteristic
156	C₂²×C₅	#464	1	2	yes	—
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	4	2	no	—
162	C₄×C₅	—	4	2	no	—
163	C₄×C₅	#416	1	1	yes	characteristic
164	C₄×C₅	#419	1	1	yes	characteristic
165	C₄×C₅	#479	1	2	yes	—
166	Dic₅	—	16	2	no	—
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	C₄⋊Dic₅	1	2	yes	—
169	C₂²×C₄	—	20	2	no	—
170	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
171	C₂²×C₄	#709	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
173	M₁₆	—	4	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	20	2	no	—
178	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
180	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	20	2	no	—
188	C₄⋊C₄	—	20	2	no	—
189	C₄⋊C₄	C₄⋊Dic₅	1	2	yes	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
193	C₄²	#709	1	1	yes	characteristic
194	C₄²	C₂²×Dic₅	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
196	C₂×C₅	—	2	2	no	—
197	C₂×C₅	#1805	1	1	yes	characteristic
198	C₂×C₅	#1812	1	1	yes	characteristic
199	C₂×C₅	#1814	1	1	yes	characteristic
200	C₂³	#4076	1	1	yes	characteristic
201	Q₈	—	4	2	no	—
202	Q₈	—	4	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	40	2	no	—
208	C₂×C₄	—	40	2	no	—
209	C₂×C₄	—	40	2	no	—
210	C₂×C₄	#4070	1	1	yes	characteristic
211	C₂×C₄	#4070	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
213	C₂×C₄	#4072	1	1	yes	characteristic
214	C₂×C₄	#4076	1	1	yes	characteristic
215	C₂×C₄	#4088	1	2	yes	—
216	C₂×C₄	#4096	1	2	yes	—
217	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	4	2	no	—
223	C₈	—	40	2	no	—
224	C₅	#8681	1	1	yes	characteristic, complemented, Sylow
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#65317	1	2	yes	—
228	C₂²	#65501	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	80	2	no	—
236	C₄	#65311	1	2	yes	—
237	C₄	#65514	1	1	yes	characteristic
238	C₄	#65522	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	#10602420	1	1	yes	characteristic
241	C₂	#10602805	1	1	yes	characteristic
242	C₂	#10602806	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	#121
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2960	D₅
hypocenter	C₅	#8681
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
8. C₅×#446	C₄	semidirect
10. C₅×C₈×Q₈	C₄	semidirect
87. #446	Dic₅	semidirect
89. C₈×Q₈	Dic₅	semidirect
224. C₅	#8681	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
#10602631	2
#10600556	2
#10600522	2
#10598204	2
#8681	5