Subgroups — C₅×#22314 #11945034

Order 1280.

630 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2399	C₂	1	1	yes	maximal, characteristic
2	C₅×#1985	C₂	1	2	yes	maximal
3	C₅×#1984	C₂	1	2	yes	maximal
4	C₅×#1939	C₂	1	2	yes	maximal
5	C₅×#1917	C₂	1	2	yes	maximal
6	C₅×#1916	C₂	1	2	yes	maximal
7	C₅×#1832	C₂	1	1	yes	maximal, characteristic
8	C₅×#1771	C₂	1	2	yes	maximal
9	C₅×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₅×#97	C₂²	1	2	yes	—
15	C₂×C₅×#97	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₅×#398	C₄	1	2	yes	—
19	C₅×#398	C₄	1	2	yes	—
20	C₅×#395	C₄	1	2	yes	—
21	C₅×#395	C₄	1	2	yes	—
22	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₅×#376	C₂²	1	1	yes	characteristic
26	C₅×#336	C₂²	1	2	yes	—
27	C₅×#336	C₂²	1	2	yes	—
28	C₅×#336	C₂²	1	2	yes	—
29	C₅×#336	C₂²	1	1	yes	characteristic
30	C₅×#336	C₂²	1	1	yes	characteristic
31	C₅×#336	C₂²	1	2	yes	—
32	C₅×#336	C₂²	1	2	yes	—
33	C₅×#336	C₂²	1	2	yes	—
34	C₅×#336	C₂²	1	2	yes	—
35	#22314	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₄×C₅×Q₈	—	4	2	no	—
42	C₄×C₅×Q₈	—	4	2	no	—
43	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂³	1	2	yes	—
53	C₂×C₄²×C₅	C₂³	1	2	yes	—
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₅×C₄⋊C₈	—	2	4	no	—
56	C₅×C₄⋊C₈	—	4	2	no	—
57	C₅×C₄⋊C₈	—	4	2	no	—
58	C₅×#97	—	4	2	no	—
59	C₅×#97	Q₈	1	4	yes	complemented
60	C₄×C₅×C₈	—	2	2	no	—
61	C₄×C₅×C₈	—	4	2	no	—
62	C₅×#95	C₂×C₄	1	2	yes	—
63	C₅×#95	C₂×C₄	1	2	yes	—
64	C₅×#95	C₂×C₄	1	1	yes	characteristic
65	C₅×#95	C₂×C₄	1	2	yes	—
66	C₅×#95	C₂×C₄	1	1	yes	characteristic
67	C₅×#95	C₂×C₄	1	1	yes	characteristic
68	C₅×#95	C₂×C₄	1	1	yes	characteristic
69	C₅×#95	C₂×C₄	1	2	yes	—
70	#2399	C₂×C₅	1	1	yes	characteristic
71	#1985	C₂×C₅	1	2	yes	—
72	#1984	C₂×C₅	1	2	yes	—
73	#1939	C₂×C₅	1	2	yes	—
74	#1917	C₂×C₅	1	2	yes	—
75	#1916	C₂×C₅	1	2	yes	—
76	#1832	C₂×C₅	1	1	yes	characteristic
77	#1771	C₂×C₅	1	2	yes	—
78	#1746	C₂×C₅	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	2	4	no	—
80	C₂×C₅×Q₈	—	4	2	no	—
81	C₂×C₅×Q₈	—	4	2	no	—
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
103	C₅×C₄⋊C₄	—	2	4	no	—
104	C₅×C₄⋊C₄	—	2	4	no	—
105	C₅×C₄⋊C₄	—	4	2	no	—
106	C₅×C₄⋊C₄	—	4	2	no	—
107	C₅×C₄⋊C₄	—	4	2	no	—
108	C₅×C₄⋊C₄	—	4	2	no	—
109	C₅×C₄⋊C₄	—	4	2	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	2	2	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	—	2	2	no	—
116	C₄²×C₅	C₂×Q₈	1	2	yes	—
117	C₄²×C₅	#41	1	2	yes	—
118	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
119	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
120	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
121	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
122	C₂×#97	C₂²×C₅	1	2	yes	—
123	C₂×#97	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
126	#398	C₄×C₅	1	2	yes	—
127	#398	C₄×C₅	1	2	yes	—
128	#395	C₄×C₅	1	2	yes	—
129	#395	C₄×C₅	1	2	yes	—
130	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
131	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
132	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
133	#376	C₂²×C₅	1	1	yes	characteristic
134	#336	C₂²×C₅	1	2	yes	—
135	#336	C₂²×C₅	1	1	yes	characteristic
136	#336	C₂²×C₅	1	1	yes	characteristic
137	#336	C₂²×C₅	1	2	yes	—
138	#336	C₂²×C₅	1	2	yes	—
139	#336	C₂²×C₅	1	2	yes	—
140	#336	C₂²×C₅	1	2	yes	—
141	#336	C₂²×C₅	1	2	yes	—
142	#336	C₂²×C₅	1	2	yes	—
143	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
144	C₅×Q₈	—	4	8	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
166	C₂×C₄×C₅	#131	1	4	yes	—
167	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
170	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
171	C₅×C₈	—	2	4	no	—
172	C₅×C₈	—	4	2	no	—
173	C₅×C₈	—	4	2	no	—
174	C₅×C₈	—	4	2	no	—
175	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
176	C₂²×C₈	C₂³×C₅	1	2	yes	—
177	C₂²×C₈	C₂³×C₅	1	2	yes	—
178	C₂²×C₈	C₂³×C₅	1	2	yes	—
179	C₂²×C₈	C₂³×C₅	1	2	yes	—
180	C₄×Q₈	—	4	2	no	—
181	C₄×Q₈	—	4	2	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
189	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₅	1	2	yes	—
191	C₂×C₄²	C₂³×C₅	1	2	yes	—
192	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	4	2	no	—
196	C₄⋊C₈	—	4	2	no	—
197	#97	—	4	2	no	—
198	#97	C₅×Q₈	1	4	yes	complemented
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	—	4	2	no	—
201	#95	C₂×C₄×C₅	1	2	yes	—
202	#95	C₂×C₄×C₅	1	2	yes	—
203	#95	C₂×C₄×C₅	1	1	yes	characteristic
204	#95	C₂×C₄×C₅	1	1	yes	characteristic
205	#95	C₂×C₄×C₅	1	2	yes	—
206	#95	C₂×C₄×C₅	1	1	yes	characteristic
207	#95	C₂×C₄×C₅	1	1	yes	characteristic
208	#95	C₂×C₄×C₅	1	2	yes	—
209	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
210	C₂²×C₅	#518	1	1	yes	characteristic
211	C₂²×C₅	#519	1	1	yes	characteristic
212	C₂²×C₅	C₂×#131	1	2	yes	—
213	C₂²×C₅	#568	1	2	yes	—
214	C₄×C₅	—	2	4	no	—
215	C₄×C₅	—	2	4	no	—
216	C₄×C₅	—	4	2	no	—
217	C₄×C₅	—	4	2	no	—
218	C₄×C₅	—	4	2	no	—
219	C₄×C₅	—	4	2	no	—
220	C₄×C₅	—	4	1	no	—
221	C₄×C₅	—	4	1	no	—
222	C₄×C₅	#542	1	2	yes	—
223	C₄×C₅	#544	1	2	yes	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	4	2	no	—
227	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
234	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
235	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂×C₅×Q₈	1	2	yes	—
262	C₄²	C₅×#41	1	2	yes	—
263	C₂×C₅	#2331	1	1	yes	characteristic
264	C₂×C₅	#2962	1	2	yes	—
265	C₂×C₅	#2976	1	2	yes	—
266	C₂×C₅	#2986	1	1	yes	characteristic
267	C₂×C₅	#2987	1	1	yes	characteristic
268	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
269	Q₈	—	4	8	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
291	C₂×C₄	C₅×#131	1	4	yes	—
292	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
293	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
294	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
295	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
296	C₈	—	2	4	no	—
297	C₈	—	4	2	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₅	#22314	1	1	yes	characteristic, direct product factor, Sylow
301	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
302	C₂²	C₅×#518	1	1	yes	characteristic
303	C₂²	C₅×#519	1	1	yes	characteristic
304	C₂²	C₂×C₅×#131	1	2	yes	—
305	C₂²	C₅×#568	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	2	4	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	C₅×#542	1	2	yes	—
315	C₄	C₅×#544	1	2	yes	—
316	C₂	C₅×#2331	1	1	yes	characteristic
317	C₂	C₅×#2962	1	2	yes	—
318	C₂	C₅×#2976	1	2	yes	—
319	C₂	C₅×#2986	1	1	yes	characteristic
320	C₂	C₅×#2987	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
35. #22314	C₅	direct (N × H)
59. C₅×#97	Q₈	semidirect
198. #97	C₅×Q₈	semidirect
300. C₅	#22314	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×#2399	2
C₅×#1985	2
C₅×#1984	2
C₅×#1939	2
C₅×#1917	2
C₅×#1916	2
C₅×#1832	2
C₅×#1771	2
C₅×#1746	2
#22314	5

Subgroups — C5×#22314 #11945034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22314 #11945034