Subgroups — C₅×#11366 #11934086

Order 1280.

634 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#343	C₂	1	2	yes	maximal
3	C₂×C₅×#341	C₂	1	1	yes	maximal, characteristic
4	C₅×#1736	C₂	1	2	yes	maximal
5	C₅×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₅×#106	—	2	2	no	—
12	C₂×C₅×#106	—	2	2	no	—
13	C₂×C₅×#106	C₄	1	4	yes	complemented
14	C₅×#343	—	2	4	no	—
15	C₅×#341	—	2	2	no	—
16	C₅×#340	—	2	4	no	—
17	C₅×#336	—	2	2	no	—
18	#11366	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₅	—	2	2	no	—
20	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	2	1	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	D₄	1	2	yes	—
30	C₂×C₅×M₁₆	Q₈	1	4	yes	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	—	2	2	no	—
33	C₂²×C₅×C₈	—	2	1	no	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₅×C₈	D₄	1	2	yes	—
37	C₂²×C₅×C₈	D₄	1	2	yes	—
38	C₅×#117	—	2	2	no	—
39	C₅×#117	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
46	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
47	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
48	C₂×C₄²×C₅	—	2	1	no	—
49	C₅×#106	—	2	8	no	—
50	C₂²×#106	C₂×C₅	1	1	yes	characteristic
51	C₂×#343	C₂×C₅	1	2	yes	—
52	C₂×#341	C₂×C₅	1	1	yes	characteristic
53	#1736	C₂×C₅	1	2	yes	—
54	#1729	C₂×C₅	1	1	yes	characteristic
55	C₂⁴×C₅	#41	1	1	yes	characteristic
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	4	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	—	4	2	no	—
60	C₂²×C₄×C₅	—	8	4	no	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₅	#41	1	2	yes	—
68	C₂²×C₄×C₅	#41	1	2	yes	—
69	C₂²×C₄×C₅	#41	1	1	yes	characteristic
70	C₂²×C₄×C₅	#41	1	2	yes	—
71	C₅×M₁₆	—	2	4	no	—
72	C₅×M₁₆	—	2	4	no	—
73	C₅×M₁₆	—	2	4	no	—
74	C₅×M₁₆	—	4	2	no	—
75	C₅×M₁₆	—	4	2	no	—
76	C₅×M₁₆	—	4	2	no	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	4	1	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₂×C₅×C₈	—	4	1	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	2	4	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	4	2	no	—
92	C₅×C₄⋊C₄	—	4	2	no	—
93	C₅×C₂²⋊C₄	—	4	2	no	—
94	C₄²×C₅	—	4	1	no	—
95	C₄²×C₅	—	4	1	no	—
96	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
97	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
98	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
99	C₂×#117	C₂²×C₅	1	1	yes	characteristic
100	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
101	C₂×#106	—	2	2	no	—
102	C₂×#106	—	2	2	no	—
103	C₂×#106	C₄×C₅	1	4	yes	complemented
104	#343	—	2	4	no	—
105	#341	—	2	2	no	—
106	#340	—	2	4	no	—
107	#336	—	2	2	no	—
108	C₂³×C₅	—	2	4	no	—
109	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
110	C₂³×C₅	#121	1	1	yes	characteristic
111	C₂³×C₅	#124	1	1	yes	characteristic
112	C₂³×C₅	#125	1	2	yes	—
113	C₂³×C₅	#126	1	2	yes	—
114	C₂×C₄×C₅	—	4	4	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	4	4	no	—
117	C₂×C₄×C₅	—	8	4	no	—
118	C₂×C₄×C₅	—	8	4	no	—
119	C₂×C₄×C₅	—	8	4	no	—
120	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	#117	1	2	yes	—
122	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₅	#121	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₅	#123	1	1	yes	characteristic
129	C₂×C₄×C₅	#123	1	1	yes	characteristic
130	C₂×C₄×C₅	#123	1	2	yes	—
131	C₂×C₄×C₅	#123	1	2	yes	—
132	C₂×C₄×C₅	#123	1	2	yes	—
133	C₂×C₄×C₅	#124	1	1	yes	characteristic
134	C₂×C₄×C₅	#125	1	2	yes	—
135	C₂×C₄×C₅	#126	1	2	yes	—
136	C₂×C₄×C₅	#127	1	1	yes	characteristic
137	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
138	C₅×C₈	—	2	4	no	—
139	C₅×C₈	—	4	2	no	—
140	C₅×C₈	—	4	4	no	—
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
143	C₂×M₁₆	—	2	1	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	—	2	2	no	—
151	C₂×M₁₆	C₅×D₄	1	2	yes	—
152	C₂×M₁₆	C₅×Q₈	1	4	yes	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
158	C₂²×C₈	C₅×D₄	1	2	yes	—
159	C₂²×C₈	C₅×D₄	1	2	yes	—
160	#117	—	2	2	no	—
161	#117	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
168	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
169	C₂×C₂²⋊C₄	—	2	1	no	—
170	C₂×C₄²	—	2	1	no	—
171	#106	—	2	8	no	—
172	C₂²×C₅	—	2	4	no	—
173	C₂²×C₅	—	2	4	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	#387	1	2	yes	—
177	C₂²×C₅	#390	1	1	yes	characteristic
178	C₂²×C₅	#397	1	1	yes	characteristic
179	C₂²×C₅	#398	1	1	yes	characteristic
180	C₂²×C₅	#400	1	2	yes	—
181	C₂²×C₅	#471	1	4	yes	—
182	C₄×C₅	—	8	2	no	—
183	C₄×C₅	—	8	4	no	—
184	C₄×C₅	#381	1	1	yes	characteristic
185	C₄×C₅	#387	1	2	yes	—
186	C₄×C₅	#391	1	1	yes	characteristic
187	C₄×C₅	#397	1	1	yes	characteristic
188	C₄×C₅	#398	1	1	yes	characteristic
189	C₄×C₅	#399	1	2	yes	—
190	C₂⁴	C₅×#41	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	8	4	no	—
196	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
202	C₂²×C₄	C₅×#41	1	2	yes	—
203	C₂²×C₄	C₅×#41	1	2	yes	—
204	C₂²×C₄	C₅×#41	1	1	yes	characteristic
205	C₂²×C₄	C₅×#41	1	2	yes	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₄²	—	4	1	no	—
230	C₄²	—	4	1	no	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	#1445	1	1	yes	characteristic
234	C₂×C₅	#1977	1	2	yes	—
235	C₂×C₅	#2079	1	2	yes	—
236	C₂×C₅	#2090	1	2	yes	—
237	C₂³	—	2	4	no	—
238	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
239	C₂³	C₅×#121	1	1	yes	characteristic
240	C₂³	C₅×#124	1	1	yes	characteristic
241	C₂³	C₅×#125	1	2	yes	—
242	C₂³	C₅×#126	1	2	yes	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₅×#117	1	2	yes	—
251	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
252	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
253	C₂×C₄	C₅×#121	1	1	yes	characteristic
254	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
257	C₂×C₄	C₅×#123	1	1	yes	characteristic
258	C₂×C₄	C₅×#123	1	1	yes	characteristic
259	C₂×C₄	C₅×#123	1	2	yes	—
260	C₂×C₄	C₅×#123	1	2	yes	—
261	C₂×C₄	C₅×#123	1	2	yes	—
262	C₂×C₄	C₅×#124	1	1	yes	characteristic
263	C₂×C₄	C₅×#125	1	2	yes	—
264	C₂×C₄	C₅×#126	1	2	yes	—
265	C₂×C₄	C₅×#127	1	1	yes	characteristic
266	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	—	4	2	no	—
269	C₈	—	4	4	no	—
270	C₅	#11366	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	C₅×#387	1	2	yes	—
276	C₂²	C₅×#390	1	1	yes	characteristic
277	C₂²	C₅×#397	1	1	yes	characteristic
278	C₂²	C₅×#398	1	1	yes	characteristic
279	C₂²	C₅×#400	1	2	yes	—
280	C₂²	C₅×#471	1	4	yes	—
281	C₄	—	8	2	no	—
282	C₄	—	8	4	no	—
283	C₄	C₅×#381	1	1	yes	characteristic
284	C₄	C₅×#387	1	2	yes	—
285	C₄	C₅×#391	1	1	yes	characteristic
286	C₄	C₅×#397	1	1	yes	characteristic
287	C₄	C₅×#398	1	1	yes	characteristic
288	C₄	C₅×#399	1	2	yes	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	C₅×#1445	1	1	yes	characteristic
292	C₂	C₅×#1977	1	2	yes	—
293	C₂	C₅×#2079	1	2	yes	—
294	C₂	C₅×#2090	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	#121
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×C₅×#106	C₄	semidirect
18. #11366	C₅	direct (N × H)
103. C₂×#106	C₄×C₅	semidirect
270. C₅	#11366	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#106	2
C₂×C₅×#343	2
C₂×C₅×#341	2
C₅×#1736	2
C₅×#1729	2
#11366	5

Subgroups — C5×#11366 #11934086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11366 #11934086