Subgroups — C₅×#11368 #11934088

Order 1280.

574 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#430	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#344	C₂	1	2	yes	maximal
3	C₂×C₅×#341	C₂	1	2	yes	maximal
4	C₂×C₅×#338	C₂	1	1	yes	maximal, characteristic
5	C₅×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	1	yes	characteristic
11	C₅×#430	—	2	2	no	—
12	C₅×#430	C₄	1	4	yes	complemented
13	C₂×C₅×#108	—	2	2	no	—
14	C₅×#344	—	2	4	no	—
15	C₅×#341	—	2	4	no	—
16	C₅×#338	—	2	2	no	—
17	C₅×#336	—	2	2	no	—
18	#11368	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₅×M₁₆	D₄	1	4	yes	—
35	C₂×C₅×M₁₆	D₄	1	2	yes	—
36	C₂²×C₅×C₈	—	2	1	no	—
37	C₂²×C₅×C₈	—	2	2	no	—
38	C₂²×C₅×C₈	—	2	2	no	—
39	C₂²×C₅×C₈	—	2	1	no	—
40	C₂²×C₅×C₈	—	2	2	no	—
41	C₂²×C₅×C₈	—	2	2	no	—
42	C₂²×C₅×C₈	D₄	1	2	yes	—
43	C₂²×C₅×C₈	Q₈	1	2	yes	—
44	C₅×#117	—	2	2	no	—
45	C₅×#117	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	—	2	1	no	—
47	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
48	C₂×C₄²×C₅	—	2	1	no	—
49	C₅×#108	—	2	8	no	—
50	C₂×#430	C₂×C₅	1	1	yes	characteristic
51	C₂×#344	C₂×C₅	1	2	yes	—
52	C₂×#341	C₂×C₅	1	2	yes	—
53	C₂×#338	C₂×C₅	1	1	yes	characteristic
54	#1729	C₂×C₅	1	1	yes	characteristic
55	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	—	2	2	no	—
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×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	#41	1	2	yes	—
63	C₂²×C₄×C₅	#41	1	2	yes	—
64	C₂²×C₄×C₅	#41	1	2	yes	—
65	C₂²×C₄×C₅	#41	1	1	yes	characteristic
66	C₂²×C₄×C₅	#41	1	1	yes	characteristic
67	C₅×M₁₆	—	2	4	no	—
68	C₅×M₁₆	—	2	4	no	—
69	C₅×M₁₆	—	2	4	no	—
70	C₅×M₁₆	—	2	4	no	—
71	C₅×M₁₆	—	2	4	no	—
72	C₅×M₁₆	—	2	4	no	—
73	C₅×M₁₆	—	4	2	no	—
74	C₅×M₁₆	—	4	2	no	—
75	C₅×M₁₆	—	4	2	no	—
76	C₅×M₁₆	—	4	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	1	no	—
90	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
91	C₅×C₄⋊C₄	—	4	2	no	—
92	C₅×C₂²⋊C₄	—	2	4	no	—
93	C₄²×C₅	—	4	1	no	—
94	C₄²×C₅	—	4	1	no	—
95	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
96	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
97	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
98	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
99	C₂×#117	C₂²×C₅	1	1	yes	characteristic
100	#430	—	2	2	no	—
101	#430	C₄×C₅	1	4	yes	complemented
102	C₂×#108	—	2	2	no	—
103	#344	—	2	4	no	—
104	#341	—	2	4	no	—
105	#338	—	2	2	no	—
106	#336	—	2	2	no	—
107	C₂³×C₅	—	4	2	no	—
108	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
109	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
111	C₂³×C₅	#125	1	2	yes	—
112	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅	—	4	4	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	#117	1	2	yes	—
117	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅	#121	1	2	yes	—
120	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅	#123	1	2	yes	—
123	C₂×C₄×C₅	#123	1	1	yes	characteristic
124	C₂×C₄×C₅	#123	1	1	yes	characteristic
125	C₂×C₄×C₅	#123	1	2	yes	—
126	C₂×C₄×C₅	#123	1	2	yes	—
127	C₂×C₄×C₅	#124	1	1	yes	characteristic
128	C₂×C₄×C₅	#124	1	1	yes	characteristic
129	C₂×C₄×C₅	#125	1	2	yes	—
130	C₂×C₄×C₅	#126	1	2	yes	—
131	C₂×C₄×C₅	#126	1	2	yes	—
132	C₂×C₄×C₅	#127	1	1	yes	characteristic
133	C₅×C₈	—	2	4	no	—
134	C₅×C₈	—	4	4	no	—
135	C₅×C₈	—	4	4	no	—
136	C₅×C₈	—	4	2	no	—
137	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
151	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
152	C₂×M₁₆	C₅×D₄	1	4	yes	—
153	C₂×M₁₆	C₅×D₄	1	2	yes	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	C₅×D₄	1	2	yes	—
161	C₂²×C₈	C₅×Q₈	1	2	yes	—
162	#117	—	2	2	no	—
163	#117	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
166	C₂×C₄²	—	2	1	no	—
167	#108	—	2	8	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	2	4	no	—
170	C₂²×C₅	—	4	1	no	—
171	C₂²×C₅	—	4	2	no	—
172	C₂²×C₅	—	4	1	no	—
173	C₂²×C₅	#387	1	2	yes	—
174	C₂²×C₅	#391	1	1	yes	characteristic
175	C₂²×C₅	#398	1	2	yes	—
176	C₂²×C₅	#399	1	1	yes	characteristic
177	C₂²×C₅	#400	1	1	yes	characteristic
178	C₄×C₅	—	8	2	no	—
179	C₄×C₅	#381	1	1	yes	characteristic
180	C₄×C₅	#387	1	2	yes	—
181	C₄×C₅	#390	1	1	yes	characteristic
182	C₄×C₅	#397	1	2	yes	—
183	C₄×C₅	#399	1	1	yes	characteristic
184	C₄×C₅	#400	1	1	yes	characteristic
185	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
191	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
192	C₂²×C₄	C₅×#41	1	2	yes	—
193	C₂²×C₄	C₅×#41	1	2	yes	—
194	C₂²×C₄	C₅×#41	1	2	yes	—
195	C₂²×C₄	C₅×#41	1	1	yes	characteristic
196	C₂²×C₄	C₅×#41	1	1	yes	characteristic
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	4	2	no	—
205	M₁₆	—	4	2	no	—
206	M₁₆	—	4	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
221	C₄⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₄²	—	4	1	no	—
224	C₄²	—	4	1	no	—
225	C₂×C₅	—	2	1	no	—
226	C₂×C₅	—	2	2	no	—
227	C₂×C₅	—	2	1	no	—
228	C₂×C₅	—	2	2	no	—
229	C₂×C₅	#1445	1	1	yes	characteristic
230	C₂×C₅	#2103	1	2	yes	—
231	C₂³	—	4	2	no	—
232	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
233	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
234	C₂³	C₅×C₂²⋊Q₈	1	2	yes	—
235	C₂³	C₅×#125	1	2	yes	—
236	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄	C₅×#117	1	2	yes	—
241	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
242	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
243	C₂×C₄	C₅×#121	1	2	yes	—
244	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
245	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₄	C₅×#123	1	2	yes	—
247	C₂×C₄	C₅×#123	1	1	yes	characteristic
248	C₂×C₄	C₅×#123	1	1	yes	characteristic
249	C₂×C₄	C₅×#123	1	2	yes	—
250	C₂×C₄	C₅×#123	1	2	yes	—
251	C₂×C₄	C₅×#124	1	1	yes	characteristic
252	C₂×C₄	C₅×#124	1	1	yes	characteristic
253	C₂×C₄	C₅×#125	1	2	yes	—
254	C₂×C₄	C₅×#126	1	2	yes	—
255	C₂×C₄	C₅×#126	1	2	yes	—
256	C₂×C₄	C₅×#127	1	1	yes	characteristic
257	C₈	—	2	4	no	—
258	C₈	—	4	4	no	—
259	C₈	—	4	4	no	—
260	C₈	—	4	2	no	—
261	C₅	#11368	1	1	yes	characteristic, direct product factor, Sylow
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₅×#387	1	2	yes	—
268	C₂²	C₅×#391	1	1	yes	characteristic
269	C₂²	C₅×#398	1	2	yes	—
270	C₂²	C₅×#399	1	1	yes	characteristic
271	C₂²	C₅×#400	1	1	yes	characteristic
272	C₄	—	8	2	no	—
273	C₄	C₅×#381	1	1	yes	characteristic
274	C₄	C₅×#387	1	2	yes	—
275	C₄	C₅×#390	1	1	yes	characteristic
276	C₄	C₅×#397	1	2	yes	—
277	C₄	C₅×#399	1	1	yes	characteristic
278	C₄	C₅×#400	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	2	no	—
283	C₂	C₅×#1445	1	1	yes	characteristic
284	C₂	C₅×#2103	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#124
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	#400
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅×#430	C₄	semidirect
18. #11368	C₅	direct (N × H)
101. #430	C₄×C₅	semidirect
261. C₅	#11368	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#430	2
C₂×C₅×#344	2
C₂×C₅×#341	2
C₂×C₅×#338	2
C₅×#1729	2
#11368	5

Subgroups — C5×#11368 #11934088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11368 #11934088