Subgroups — C₅×#11335 #11934055

Order 1280.

790 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2911	C₂	1	1	yes	maximal, characteristic
2	C₅×#2296	C₂	1	2	yes	maximal
3	C₅×#1733	C₂	1	2	yes	maximal
4	C₂×C₅×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
5	C₅×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₅×#117	C₂²	1	2	yes	—
9	C₂×C₅×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#428	—	2	4	no	—
12	C₅×#427	—	2	2	no	—
13	C₅×#389	—	2	4	no	—
14	C₅×#385	—	2	4	no	—
15	C₄×C₅×C₄⋊C₄	—	2	4	no	—
16	C₅×#339	—	2	4	no	—
17	C₅×H(3,ℤ/4)	C₄	1	4	yes	complemented
18	C₅×#336	—	2	2	no	—
19	#11335	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	D₄	1	2	yes	—
21	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	1	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×M₁₆	Q₈	1	2	yes	—
29	C₂×C₅×M₁₆	Q₈	1	2	yes	—
30	C₂²×C₅×C₈	—	2	1	no	—
31	C₂²×C₅×C₈	D₄	1	2	yes	—
32	C₅×#117	—	2	2	no	—
33	C₅×#117	—	2	2	no	—
34	C₅×#117	—	2	2	no	—
35	C₅×#117	—	2	2	no	—
36	C₅×#117	—	2	2	no	—
37	C₅×#117	—	2	2	no	—
38	C₅×#117	C₂×C₄	1	4	yes	—
39	C₅×#117	Q₈	1	4	yes	—
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₄⋊C₄	—	2	8	no	—
42	C₂×C₅×C₄⋊C₄	D₄	1	4	yes	—
43	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
45	C₂×C₄²×C₅	—	2	2	no	—
46	C₂×C₄²×C₅	—	2	2	no	—
47	C₂×C₄²×C₅	—	2	2	no	—
48	C₂×C₄²×C₅	—	2	1	no	—
49	C₂×C₄²×C₅	—	2	1	no	—
50	C₂×C₄²×C₅	D₄	1	4	yes	—
51	C₅×#107	—	2	4	no	—
52	C₅×#106	—	2	4	no	—
53	C₅×#95	—	2	8	no	—
54	#2911	C₂×C₅	1	1	yes	characteristic
55	#2296	C₂×C₅	1	2	yes	—
56	#1733	C₂×C₅	1	2	yes	—
57	C₂×H(3,ℤ/4)	C₂×C₅	1	1	yes	characteristic
58	#1728	C₂×C₅	1	1	yes	characteristic
59	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	—	4	4	no	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	2	yes	—
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₂²×C₄×C₅	#41	1	1	yes	characteristic
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	4	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	4	1	no	—
83	C₂×C₅×C₈	—	4	1	no	—
84	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	complemented
85	C₅×C₄⋊C₄	—	2	4	no	—
86	C₅×C₄⋊C₄	—	2	4	no	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	4	16	no	—
90	C₅×C₂²⋊C₄	—	2	4	no	—
91	C₅×C₂²⋊C₄	—	2	4	no	—
92	C₅×C₂²⋊C₄	—	4	2	no	—
93	C₅×C₂²⋊C₄	—	4	2	no	—
94	C₄²×C₅	—	2	4	no	—
95	C₄²×C₅	—	4	1	no	—
96	C₄²×C₅	—	4	2	no	—
97	C₄²×C₅	—	4	2	no	—
98	C₄²×C₅	—	4	1	no	—
99	C₄²×C₅	—	4	2	no	—
100	C₄²×C₅	C₄⋊C₄	1	8	yes	complemented
101	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
102	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
103	C₂×#117	C₂²×C₅	1	2	yes	—
104	C₂×#117	C₂²×C₅	1	2	yes	—
105	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
106	#428	—	2	4	no	—
107	#427	—	2	2	no	—
108	#389	—	2	4	no	—
109	#385	—	2	4	no	—
110	C₄×C₄⋊C₄	—	2	4	no	—
111	#339	—	2	4	no	—
112	H(3,ℤ/4)	C₄×C₅	1	4	yes	complemented
113	#336	—	2	2	no	—
114	C₂³×C₅	—	4	2	no	—
115	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
116	C₂³×C₅	#120	1	2	yes	—
117	C₂³×C₅	#121	1	1	yes	characteristic
118	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	8	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	8	no	—
123	C₂×C₄×C₅	—	4	8	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	8	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅	#120	1	1	yes	characteristic
134	C₂×C₄×C₅	#121	1	2	yes	—
135	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅	#123	1	1	yes	characteristic
140	C₂×C₄×C₅	#123	1	2	yes	—
141	C₂×C₄×C₅	#125	1	2	yes	—
142	C₂×C₄×C₅	#125	1	1	yes	characteristic
143	C₂×C₄×C₅	#127	1	1	yes	characteristic
144	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
146	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
147	C₅×C₈	—	2	4	no	—
148	C₅×C₈	—	4	2	no	—
149	C₂³×C₄	C₅×D₄	1	2	yes	—
150	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
157	C₂×M₁₆	C₅×Q₈	1	2	yes	—
158	C₂×M₁₆	C₅×Q₈	1	2	yes	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	C₅×D₄	1	2	yes	—
161	#117	—	2	2	no	—
162	#117	—	2	2	no	—
163	#117	—	2	2	no	—
164	#117	—	2	2	no	—
165	#117	—	2	2	no	—
166	#117	—	2	2	no	—
167	#117	C₂×C₄×C₅	1	4	yes	—
168	#117	C₅×Q₈	1	4	yes	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	8	no	—
171	C₂×C₄⋊C₄	C₅×D₄	1	4	yes	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₂²⋊C₄	—	2	2	no	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	C₅×D₄	1	4	yes	—
180	#107	—	2	4	no	—
181	#106	—	2	4	no	—
182	#95	—	2	8	no	—
183	C₂²×C₅	—	2	4	no	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	4	1	no	—
186	C₂²×C₅	—	4	2	no	—
187	C₂²×C₅	—	4	1	no	—
188	C₂²×C₅	#385	1	2	yes	—
189	C₂²×C₅	#389	1	1	yes	characteristic
190	C₂²×C₅	#392	1	1	yes	characteristic
191	C₂²×C₅	#395	1	2	yes	—
192	C₂²×C₅	#398	1	1	yes	characteristic
193	C₄×C₅	—	4	8	no	—
194	C₄×C₅	—	4	4	no	—
195	C₄×C₅	—	8	4	no	—
196	C₄×C₅	#384	1	1	yes	characteristic
197	C₄×C₅	#385	1	1	yes	characteristic
198	C₄×C₅	#389	1	2	yes	—
199	C₄×C₅	#395	1	1	yes	characteristic
200	C₄×C₅	#398	1	2	yes	—
201	C₄×C₅	#399	1	1	yes	characteristic
202	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
211	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
212	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
214	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
215	C₂²×C₄	C₅×#41	1	2	yes	—
216	C₂²×C₄	C₅×#41	1	1	yes	characteristic
217	C₂²×C₄	C₅×#41	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	complemented
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	16	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	2	no	—
243	C₄²	C₅×C₄⋊C₄	1	8	yes	complemented
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	—	2	1	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	2	1	no	—
248	C₂×C₅	#1444	1	1	yes	characteristic
249	C₂×C₅	#2010	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
252	C₂³	C₅×#120	1	2	yes	—
253	C₂³	C₅×#121	1	1	yes	characteristic
254	C₂³	C₅×C₂²⋊Q₈	1	2	yes	—
255	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
267	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
268	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
269	C₂×C₄	C₅×#120	1	1	yes	characteristic
270	C₂×C₄	C₅×#121	1	2	yes	—
271	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
272	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
273	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
274	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
275	C₂×C₄	C₅×#123	1	1	yes	characteristic
276	C₂×C₄	C₅×#123	1	2	yes	—
277	C₂×C₄	C₅×#125	1	2	yes	—
278	C₂×C₄	C₅×#125	1	1	yes	characteristic
279	C₂×C₄	C₅×#127	1	1	yes	characteristic
280	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
281	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
282	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
283	C₈	—	2	4	no	—
284	C₈	—	4	2	no	—
285	C₅	#11335	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₅×#385	1	2	yes	—
292	C₂²	C₅×#389	1	1	yes	characteristic
293	C₂²	C₅×#392	1	1	yes	characteristic
294	C₂²	C₅×#395	1	2	yes	—
295	C₂²	C₅×#398	1	1	yes	characteristic
296	C₄	—	4	8	no	—
297	C₄	—	4	4	no	—
298	C₄	—	8	4	no	—
299	C₄	C₅×#384	1	1	yes	characteristic
300	C₄	C₅×#385	1	1	yes	characteristic
301	C₄	C₅×#389	1	2	yes	—
302	C₄	C₅×#395	1	1	yes	characteristic
303	C₄	C₅×#398	1	2	yes	—
304	C₄	C₅×#399	1	1	yes	characteristic
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	1	no	—
309	C₂	C₅×#1444	1	1	yes	characteristic
310	C₂	C₅×#2010	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	#392
prime radical	C₂⁴×C₅	C₂×D₄
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
17. C₅×H(3,ℤ/4)	C₄	semidirect
19. #11335	C₅	direct (N × H)
84. C₂×C₅×C₈	C₄⋊C₄	semidirect
100. C₄²×C₅	C₄⋊C₄	semidirect
112. H(3,ℤ/4)	C₄×C₅	semidirect
227. C₂×C₈	C₅×C₄⋊C₄	semidirect
243. C₄²	C₅×C₄⋊C₄	semidirect
285. C₅	#11335	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2911	2
C₅×#2296	2
C₅×#1733	2
C₂×C₅×H(3,ℤ/4)	2
C₅×#1728	2
#11335	5

Subgroups — C5×#11335 #11934055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11335 #11934055