Subgroups — #12008633

Order 1280.

360 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×#2931	C₂	1	1	yes	maximal, characteristic
2	#10600538	C₂	1	1	yes	maximal, characteristic, complemented
3	#10600537	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598180	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#131	C₄	1	2	yes	—
6	C₅×#532	C₄	1	1	yes	characteristic
7	C₅×#436	—	2	1	no	—
8	C₅×#436	—	2	1	no	—
9	C₅×C₈×D₄	—	2	1	no	—
10	C₅×C₈×D₄	—	2	1	no	—
11	C₅×#423	C₄	1	1	yes	characteristic
12	C₅×#408	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂²⋊#672	C₂²	1	2	yes	—
15	C₂²⋊#672	C₂²	1	2	yes	—
16	C₄⋊#672	—	2	1	no	—
17	C₄⋊#672	—	2	1	no	—
18	#64681	—	2	1	no	—
19	C₄×#672	—	2	1	no	—
20	#11492	—	5	1	no	maximal, Sylow
21	C₂×C₅×#41	C₂×C₄	1	2	yes	—
22	C₅×#131	—	4	4	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
29	C₅×#127	C₈	1	1	yes	characteristic
30	C₅×#124	C₈	1	2	yes	—
31	C₅×#121	—	2	1	no	—
32	C₅×#121	—	2	1	no	—
33	C₄×C₅×D₄	—	2	1	no	—
34	C₄×C₅×D₄	—	2	1	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₅×C₄⋊C₈	—	2	1	no	—
37	C₅×C₄⋊C₈	—	2	1	no	—
38	C₅×C₂²⋊C₈	—	2	2	no	—
39	C₅×C₂²⋊C₈	—	2	2	no	—
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×C₈	D₄	1	1	yes	characteristic
42	C₄×C₅×C₈	D₄	1	1	yes	characteristic
43	C₄×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂×#672	—	2	2	no	—
45	C₂×#672	—	2	2	no	—
46	#2931	D₅	1	1	yes	characteristic
47	#2163	—	5	1	no	—
48	#2162	—	5	1	no	—
49	#1315	—	5	4	no	—
50	C₅×#41	—	4	2	no	—
51	C₅×#41	—	4	2	no	—
52	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
53	C₂×C₅×D₄	—	2	2	no	—
54	C₂×C₅×D₄	—	2	1	no	—
55	C₂×C₅×D₄	—	2	1	no	—
56	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₅×M₁₆	—	4	4	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	—	4	1	no	—
66	C₂×C₅×C₈	—	4	4	no	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₅×C₄⋊C₄	—	2	1	no	—
72	C₅×C₄⋊C₄	—	2	1	no	—
73	C₅×C₂²⋊C₄	—	2	2	no	—
74	C₅×C₂²⋊C₄	—	2	2	no	—
75	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
79	#672	—	4	2	no	—
80	#672	—	4	2	no	—
81	C₂×#131	Dic₅	1	2	yes	—
82	#532	Dic₅	1	1	yes	characteristic
83	#436	—	2	1	no	—
84	#436	—	2	1	no	—
85	C₈×D₄	—	2	1	no	—
86	C₈×D₄	—	2	1	no	—
87	#423	Dic₅	1	1	yes	characteristic
88	#408	C₂×D₅	1	2	yes	—
89	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
90	C₄⋊C₁₆	—	10	1	no	—
91	C₄⋊C₁₆	—	10	1	no	—
92	C₂²⋊C₁₆	—	5	2	no	—
93	C₂²⋊C₁₆	—	5	2	no	—
94	#346	—	10	1	no	—
95	C₄×C₁₆	—	10	1	no	—
96	C₂³×C₅	—	2	2	no	—
97	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₅×Q₈	—	4	2	no	—
99	C₅×D₄	—	4	2	no	—
100	C₅×D₄	—	4	2	no	—
101	C₅×D₄	—	4	4	no	—
102	C₅×D₄	—	4	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	4	1	no	—
108	C₂×C₄×C₅	—	4	2	no	—
109	C₂×C₄×C₅	—	4	2	no	—
110	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₅×C₈	—	2	1	no	—
117	C₅×C₈	—	2	1	no	—
118	C₅×C₈	—	2	1	no	—
119	C₅×C₈	—	4	2	no	—
120	C₅×C₈	#100	1	2	yes	—
121	C₂×#41	C₂×Dic₅	1	2	yes	—
122	#131	—	4	4	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
128	C₄⋊Q₈	#168	1	1	yes	characteristic
129	#127	#168	1	1	yes	characteristic
130	#124	#168	1	2	yes	—
131	#121	—	2	1	no	—
132	#121	—	2	1	no	—
133	C₄×D₄	—	2	1	no	—
134	C₄×D₄	—	2	1	no	—
135	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
136	C₂×C₁₆	—	10	2	no	—
137	C₂×C₁₆	—	10	2	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
144	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
146	C₂²×C₅	—	2	1	no	—
147	C₂²×C₅	—	2	1	no	—
148	C₂²×C₅	—	2	1	no	—
149	C₂²×C₅	—	4	2	no	—
150	C₂²×C₅	—	4	2	no	—
151	C₂²×C₅	—	4	2	no	—
152	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₄×C₅	—	2	1	no	—
154	C₄×C₅	—	2	1	no	—
155	C₄×C₅	—	2	1	no	—
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	—	4	2	no	—
158	C₄×C₅	#350	1	2	yes	—
159	C₄×C₅	#410	1	1	yes	characteristic
160	C₄×C₅	C₂×#100	1	1	yes	characteristic
161	#41	—	4	2	no	—
162	#41	—	4	2	no	—
163	C₂×Q₈	C₂×#168	1	2	yes	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	C₂×#168	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	#681	1	1	yes	characteristic
170	C₂²×C₄	#681	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
172	M₁₆	—	4	4	no	—
173	M₁₆	—	4	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	4	no	—
178	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
181	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₄²	C₂×#168	1	1	yes	characteristic
187	C₄²	C₂×#168	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
190	C₁₆	—	20	2	no	—
191	C₁₆	—	20	2	no	—
192	C₂×C₅	—	4	1	no	—
193	C₂×C₅	—	8	2	no	—
194	C₂×C₅	#1460	1	1	yes	characteristic
195	C₂×C₅	C₂×#350	1	1	yes	characteristic
196	C₂×C₅	#2117	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	C₂×#681	1	1	yes	characteristic
199	Q₈	—	4	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
212	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
213	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
215	C₂×C₄	C₂²×#168	1	1	yes	characteristic
216	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	4	2	no	—
221	C₈	#3965	1	2	yes	—
222	C₅	#11492	1	1	yes	characteristic, complemented, Sylow
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	#64781	1	2	yes	—
236	C₄	C₂×#3965	1	1	yes	characteristic
237	C₄	#65529	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	8	2	no	—
240	C₂	#10602394	1	1	yes	characteristic
241	C₂	C₂×#64781	1	1	yes	characteristic
242	C₂	#10602799	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2931	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₅×#532	C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2931	D₅
hypocenter	C₅	#11492
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. #10600538	C₂	semidirect
3. #10600537	C₂	semidirect
4. #10598180	C₂	semidirect
222. C₅	#11492	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2931	2
#10600538	2
#10600537	2
#10598180	2
#11492	5