Subgroups — C₅×#19204 #11941924

Order 1280.

646 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×#3229	C₂	1	2	yes	maximal
2	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
3	C₅×#2960	C₂	1	1	yes	maximal, characteristic
4	C₅×#1690	C₂	1	4	yes	maximal, complemented
5	C₅×#1670	C₂	1	4	yes	maximal, complemented
6	C₅×#1564	C₂	1	1	yes	maximal, characteristic
7	C₅×#1546	C₂	1	2	yes	maximal
8	C₂²×C₅×Q₁₆	—	2	1	no	—
9	C₂²×C₅×Q₁₆	—	2	1	no	—
10	C₅×#533	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₅×#498	—	2	2	no	—
13	C₅×#494	—	2	1	no	—
14	C₅×#494	—	2	1	no	—
15	C₅×#494	C₂²	1	2	yes	—
16	C₅×#494	C₂²	1	2	yes	—
17	C₅×#487	—	2	2	no	—
18	C₅×#483	—	2	4	no	—
19	C₅×#475	C₂²	1	8	yes	—
20	C₅×#446	—	2	1	no	—
21	C₅×#446	C₂²	1	2	yes	—
22	C₅×C₈×Q₈	—	2	1	no	—
23	C₅×C₈×Q₈	C₂²	1	2	yes	—
24	C₅×#431	C₂²	1	2	yes	—
25	C₅×#424	—	2	1	no	—
26	C₅×#423	C₂²	1	1	yes	characteristic
27	C₅×#405	—	2	1	no	—
28	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₅×#335	C₂²	1	2	yes	—
30	C₅×#334	C₂²	1	2	yes	—
31	C₅×#332	C₂²	1	8	yes	—
32	#19204	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂²×C₅×Q₈	—	2	2	no	—
34	C₂×C₅×Q₁₆	—	2	2	no	—
35	C₂×C₅×Q₁₆	—	2	2	no	—
36	C₂×C₅×Q₁₆	—	2	2	no	—
37	C₂×C₅×Q₁₆	—	2	2	no	—
38	C₂×C₅×Q₁₆	—	4	4	no	—
39	C₂×C₅×Q₁₆	—	4	4	no	—
40	C₂×C₅×M₁₆	—	2	1	no	—
41	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
43	C₅×C₄⋊Q₈	—	2	2	no	—
44	C₅×C₄⋊Q₈	—	2	2	no	—
45	C₅×C₄⋊Q₈	—	2	2	no	—
46	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
48	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
49	C₅×#125	—	2	1	no	—
50	C₅×C₂²⋊Q₈	—	2	1	no	—
51	C₅×C₂²⋊Q₈	D₄	1	2	yes	—
52	C₄×C₅×Q₈	—	2	1	no	—
53	C₄×C₅×Q₈	C₂³	1	2	yes	—
54	C₅×#117	—	2	1	no	—
55	C₂×C₅×C₄⋊C₄	—	2	2	no	—
56	C₂×C₅×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
58	C₅×#106	—	2	8	no	—
59	C₅×C₄⋊C₈	—	2	2	no	—
60	C₅×C₄⋊C₈	—	2	2	no	—
61	C₅×C₄⋊C₈	—	2	1	no	—
62	C₅×C₄⋊C₈	C₂³	1	2	yes	—
63	C₅×C₄⋊C₈	C₂³	1	4	yes	—
64	C₅×#103	—	2	8	no	—
65	C₅×C₂²⋊C₈	—	2	2	no	—
66	C₅×#97	—	2	2	no	—
67	C₄×C₅×C₈	—	2	2	no	—
68	C₄×C₅×C₈	—	2	2	no	—
69	C₄×C₅×C₈	D₄	1	1	yes	characteristic
70	C₄×C₅×C₈	D₄	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	#3229	C₂×C₅	1	2	yes	—
74	C₂×#494	C₂×C₅	1	1	yes	characteristic
75	#2960	C₂×C₅	1	1	yes	characteristic
76	#1690	C₂×C₅	1	4	yes	complemented
77	#1670	C₂×C₅	1	4	yes	complemented
78	#1564	C₂×C₅	1	1	yes	characteristic
79	#1546	C₂×C₅	1	2	yes	—
80	C₂×C₅×Q₈	—	2	1	no	—
81	C₂×C₅×Q₈	—	2	4	no	—
82	C₂×C₅×Q₈	—	4	2	no	—
83	C₂×C₅×Q₈	—	4	4	no	—
84	C₂×C₅×Q₈	—	4	4	no	—
85	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₅	—	4	2	no	—
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	#41	1	1	yes	characteristic
90	C₅×Q₁₆	—	4	4	no	—
91	C₅×Q₁₆	—	4	4	no	—
92	C₅×Q₁₆	—	4	4	no	—
93	C₅×Q₁₆	—	4	4	no	—
94	C₅×M₁₆	—	4	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	SD₁₆	1	4	yes	—
99	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₅×C₄⋊C₄	—	2	1	no	—
104	C₅×C₄⋊C₄	—	2	1	no	—
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	2	4	no	—
109	C₅×C₄⋊C₄	—	2	4	no	—
110	C₅×C₄⋊C₄	—	4	2	no	—
111	C₅×C₄⋊C₄	—	4	2	no	—
112	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
113	C₅×C₂²⋊C₄	—	2	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅	#41	1	1	yes	characteristic
118	C₄²×C₅	C₂⁴	1	1	yes	characteristic
119	C₂²×Q₁₆	—	2	1	no	—
120	C₂²×Q₁₆	—	2	1	no	—
121	#533	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
123	#498	—	2	2	no	—
124	#494	—	2	1	no	—
125	#494	—	2	1	no	—
126	#494	C₂²×C₅	1	2	yes	—
127	#494	C₂²×C₅	1	2	yes	—
128	#487	—	2	2	no	—
129	#483	—	2	4	no	—
130	#475	C₂²×C₅	1	8	yes	—
131	#446	—	2	1	no	—
132	#446	C₂²×C₅	1	2	yes	—
133	C₈×Q₈	—	2	1	no	—
134	C₈×Q₈	C₂²×C₅	1	2	yes	—
135	#431	C₂²×C₅	1	2	yes	—
136	#424	—	2	1	no	—
137	#423	C₂²×C₅	1	1	yes	characteristic
138	#405	—	2	1	no	—
139	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
140	#335	C₂²×C₅	1	2	yes	—
141	#334	C₂²×C₅	1	2	yes	—
142	#332	C₂²×C₅	1	8	yes	—
143	C₂³×C₅	#121	1	1	yes	characteristic
144	C₅×Q₈	—	2	4	no	—
145	C₅×Q₈	—	4	2	no	—
146	C₅×Q₈	—	4	4	no	—
147	C₅×Q₈	—	4	4	no	—
148	C₅×Q₈	—	8	4	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	—	4	4	no	—
154	C₂×C₄×C₅	—	8	4	no	—
155	C₂×C₄×C₅	#121	1	1	yes	characteristic
156	C₂×C₄×C₅	#121	1	1	yes	characteristic
157	C₂×C₄×C₅	#121	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
159	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
160	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
163	C₅×C₈	—	2	1	no	—
164	C₅×C₈	—	2	1	no	—
165	C₅×C₈	—	4	2	no	—
166	C₅×C₈	—	4	4	no	—
167	C₅×C₈	C₂×SD₁₆	1	2	yes	—
168	C₅×C₈	C₂×SD₁₆	1	2	yes	—
169	C₂²×Q₈	—	2	2	no	—
170	C₂×Q₁₆	—	2	2	no	—
171	C₂×Q₁₆	—	2	2	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	—	2	2	no	—
174	C₂×Q₁₆	—	4	4	no	—
175	C₂×Q₁₆	—	4	4	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
178	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	2	2	no	—
182	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
183	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
184	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
185	#125	—	2	1	no	—
186	C₂²⋊Q₈	—	2	1	no	—
187	C₂²⋊Q₈	C₅×D₄	1	2	yes	—
188	C₄×Q₈	—	2	1	no	—
189	C₄×Q₈	C₂³×C₅	1	2	yes	—
190	#117	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
194	#106	—	2	8	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	C₂³×C₅	1	2	yes	—
199	C₄⋊C₈	C₂³×C₅	1	4	yes	—
200	#103	—	2	8	no	—
201	C₂²⋊C₈	—	2	2	no	—
202	#97	—	2	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	2	2	no	—
205	C₄×C₈	C₅×D₄	1	1	yes	characteristic
206	C₄×C₈	C₅×D₄	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	#460	1	2	yes	—
212	C₂²×C₅	C₂×#121	1	1	yes	characteristic
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	4	2	no	—
216	C₄×C₅	—	4	2	no	—
217	C₄×C₅	—	8	4	no	—
218	C₄×C₅	#460	1	2	yes	—
219	C₄×C₅	#473	1	2	yes	—
220	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
221	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₄×C₅	#578	1	2	yes	—
223	C₂×Q₈	—	2	1	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	4	4	no	—
228	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂²×C₄	C₅×#41	1	1	yes	characteristic
233	Q₁₆	—	4	4	no	—
234	Q₁₆	—	4	4	no	—
235	Q₁₆	—	4	4	no	—
236	Q₁₆	—	4	4	no	—
237	M₁₆	—	4	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	C₅×SD₁₆	1	4	yes	—
242	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
243	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
244	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
259	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
260	C₄²	C₅×#41	1	1	yes	characteristic
261	C₄²	C₂⁴×C₅	1	1	yes	characteristic
262	C₂×C₅	—	2	2	no	—
263	C₂×C₅	C₂×#460	1	1	yes	characteristic
264	C₂×C₅	#3059	1	1	yes	characteristic
265	C₂×C₅	C₂×#473	1	1	yes	characteristic
266	C₂³	C₅×#121	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	4	2	no	—
269	Q₈	—	4	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	8	4	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	—	4	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	C₅×#121	1	1	yes	characteristic
279	C₂×C₄	C₅×#121	1	1	yes	characteristic
280	C₂×C₄	C₅×#121	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
282	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
283	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
285	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	4	2	no	—
289	C₈	—	4	4	no	—
290	C₈	C₂×C₅×SD₁₆	1	2	yes	—
291	C₈	C₂×C₅×SD₁₆	1	2	yes	—
292	C₅	#19204	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	C₅×#460	1	2	yes	—
296	C₂²	C₂×C₅×#121	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₅×#460	1	2	yes	—
303	C₄	C₅×#473	1	2	yes	—
304	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
305	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₄	C₅×#578	1	2	yes	—
307	C₂	—	2	2	no	—
308	C₂	C₂×C₅×#460	1	1	yes	characteristic
309	C₂	C₅×#3059	1	1	yes	characteristic
310	C₂	C₂×C₅×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂³×C₅	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1690	C₂	semidirect
5. C₅×#1670	C₂	semidirect
32. #19204	C₅	direct (N × H)
76. #1690	C₂×C₅	semidirect
77. #1670	C₂×C₅	semidirect
292. C₅	#19204	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×#3229	2
C₂×C₅×#494	2
C₅×#2960	2
C₅×#1690	2
C₅×#1670	2
C₅×#1564	2
C₅×#1546	2
#19204	5

Subgroups — C5×#19204 #11941924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19204 #11941924