Subgroups — C₅×#19297 #11942017

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3079	C₂	1	2	yes	maximal
3	C₅×#2965	C₂	1	1	yes	maximal, characteristic
4	C₅×#1696	C₂	1	4	yes	maximal, complemented
5	C₅×#1695	C₂	1	4	yes	maximal, complemented
6	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1548	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₄×C₅×#41	C₂²	1	1	yes	characteristic
10	C₅×#500	—	2	1	no	—
11	C₅×#500	—	2	1	no	—
12	C₅×#500	C₂²	1	4	yes	—
13	C₅×#477	C₂²	1	4	yes	—
14	C₅×#476	C₂²	1	4	yes	—
15	C₅×#475	—	2	2	no	—
16	C₅×#474	—	2	2	no	—
17	C₅×#446	C₂²	1	2	yes	—
18	C₅×#436	C₂²	1	2	yes	—
19	C₅×#435	—	2	2	no	—
20	C₅×#431	—	2	1	no	—
21	C₅×#431	C₂²	1	1	yes	characteristic
22	C₅×#428	—	2	2	no	—
23	C₂×C₅×#107	—	2	2	no	—
24	C₅×#423	C₂²	1	2	yes	—
25	C₅×#417	—	2	2	no	—
26	C₄×C₅×M₁₆	—	2	1	no	—
27	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
28	C₅×#334	C₂²	1	4	yes	—
29	C₅×#333	C₂²	1	8	yes	—
30	#19297	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₅×#41	D₄	1	1	yes	characteristic
32	C₂²×C₅×Q₈	—	2	2	no	—
33	C₂×C₅×M₁₆	—	2	2	no	—
34	C₂×C₅×M₁₆	—	2	2	no	—
35	C₂²×C₅×C₈	D₄	1	2	yes	—
36	C₅×C₄⋊Q₈	—	2	2	no	—
37	C₅×C₄⋊Q₈	—	2	2	no	—
38	C₅×C₄⋊Q₈	—	2	2	no	—
39	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
40	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
41	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
42	C₄×C₅×D₄	—	2	1	no	—
43	C₄×C₅×D₄	—	2	1	no	—
44	C₄×C₅×D₄	D₄	1	1	yes	characteristic
45	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
46	C₅×#117	—	2	1	no	—
47	C₅×#117	D₄	1	1	yes	characteristic
48	C₂×C₅×C₄⋊C₄	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	—	2	2	no	—
50	C₂×C₄²×C₅	—	2	1	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×#107	—	2	4	no	—
53	C₅×#107	—	2	4	no	—
54	C₅×#107	—	2	4	no	—
55	C₅×#106	—	2	4	no	—
56	C₅×C₄⋊C₈	—	2	2	no	—
57	C₅×C₄⋊C₈	—	2	2	no	—
58	C₅×C₄⋊C₈	C₂³	1	2	yes	—
59	C₅×C₄⋊C₈	C₂³	1	4	yes	—
60	C₅×#103	—	2	4	no	—
61	C₅×#102	—	2	4	no	—
62	C₅×C₂²⋊C₈	—	2	1	no	—
63	C₅×C₂²⋊C₈	—	2	2	no	—
64	C₅×#97	—	2	2	no	—
65	C₄×C₅×C₈	—	2	1	no	—
66	C₄×C₅×C₈	D₄	1	1	yes	characteristic
67	C₄×C₅×C₈	D₄	1	1	yes	characteristic
68	C₄×C₅×C₈	C₂³	1	2	yes	—
69	C₂×#500	C₂×C₅	1	1	yes	characteristic, complemented
70	#3079	C₂×C₅	1	2	yes	—
71	#2965	C₂×C₅	1	1	yes	characteristic
72	#1696	C₂×C₅	1	4	yes	complemented
73	#1695	C₂×C₅	1	4	yes	complemented
74	#1566	C₂×C₅	1	1	yes	characteristic, complemented
75	#1548	C₂×C₅	1	2	yes	complemented
76	C₅×#41	—	2	4	no	—
77	C₂×C₅×Q₈	—	2	4	no	—
78	C₂×C₅×Q₈	—	4	2	no	—
79	C₂×C₅×Q₈	—	8	4	no	—
80	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₅×D₄	—	2	1	no	—
82	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	—	2	1	no	—
84	C₂²×C₄×C₅	—	2	1	no	—
85	C₂²×C₄×C₅	—	2	1	no	—
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×M₁₆	—	4	4	no	—
91	C₅×M₁₆	—	4	4	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
96	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
97	C₅×C₄⋊C₄	—	2	1	no	—
98	C₅×C₄⋊C₄	—	2	1	no	—
99	C₅×C₄⋊C₄	—	2	4	no	—
100	C₅×C₄⋊C₄	—	2	4	no	—
101	C₅×C₄⋊C₄	—	4	2	no	—
102	C₅×C₄⋊C₄	—	4	2	no	—
103	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
104	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
105	C₅×C₂²⋊C₄	—	2	1	no	—
106	C₅×C₂²⋊C₄	—	2	1	no	—
107	C₅×C₂²⋊C₄	—	2	1	no	—
108	C₄²×C₅	—	2	1	no	—
109	C₄²×C₅	—	2	1	no	—
110	C₄²×C₅	—	2	1	no	—
111	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅	#41	1	1	yes	characteristic
114	C₄²×C₅	C₂⁴	1	1	yes	characteristic
115	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
116	C₄×#41	C₂²×C₅	1	1	yes	characteristic
117	#500	—	2	1	no	—
118	#500	—	2	1	no	—
119	#500	C₂²×C₅	1	4	yes	—
120	#477	C₂²×C₅	1	4	yes	—
121	#476	C₂²×C₅	1	4	yes	—
122	#475	—	2	2	no	—
123	#474	—	2	2	no	—
124	#446	C₂²×C₅	1	2	yes	—
125	#436	C₂²×C₅	1	2	yes	—
126	#435	—	2	2	no	—
127	#431	—	2	1	no	—
128	#431	C₂²×C₅	1	1	yes	characteristic
129	#428	—	2	2	no	—
130	C₂×#107	—	2	2	no	—
131	#423	C₂²×C₅	1	2	yes	—
132	#417	—	2	2	no	—
133	C₄×M₁₆	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
135	#334	C₂²×C₅	1	4	yes	—
136	#333	C₂²×C₅	1	8	yes	—
137	C₂³×C₅	—	2	1	no	—
138	C₂³×C₅	#121	1	1	yes	characteristic
139	C₅×Q₈	—	2	2	no	—
140	C₅×Q₈	—	8	4	no	—
141	C₅×Q₈	—	8	4	no	—
142	C₅×D₄	—	2	2	no	—
143	C₅×D₄	—	4	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	4	4	no	—
153	C₂×C₄×C₅	—	8	4	no	—
154	C₂×C₄×C₅	#121	1	1	yes	characteristic
155	C₂×C₄×C₅	#121	1	1	yes	characteristic
156	C₂×C₄×C₅	#121	1	1	yes	characteristic
157	C₂×C₄×C₅	#137	1	4	yes	—
158	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	—	4	2	no	—
163	C₅×C₈	—	4	4	no	—
164	C₅×C₈	#137	1	4	yes	—
165	C₂×#41	C₅×D₄	1	1	yes	characteristic
166	C₂²×Q₈	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂²×C₈	C₅×D₄	1	2	yes	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
174	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
175	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	C₅×D₄	1	1	yes	characteristic
179	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
180	#117	—	2	1	no	—
181	#117	C₅×D₄	1	1	yes	characteristic
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
186	#107	—	2	4	no	—
187	#107	—	2	4	no	—
188	#107	—	2	4	no	—
189	#106	—	2	4	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	C₂³×C₅	1	2	yes	—
193	C₄⋊C₈	C₂³×C₅	1	4	yes	—
194	#103	—	2	4	no	—
195	#102	—	2	4	no	—
196	C₂²⋊C₈	—	2	1	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	#97	—	2	2	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	C₅×D₄	1	1	yes	characteristic
201	C₄×C₈	C₅×D₄	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₅	1	2	yes	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	—	4	1	no	—
206	C₂²×C₅	—	4	1	no	—
207	C₂²×C₅	—	4	1	no	—
208	C₂²×C₅	#470	1	2	yes	—
209	C₂²×C₅	C₂×#121	1	1	yes	characteristic
210	C₄×C₅	—	2	1	no	—
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	4	1	no	—
213	C₄×C₅	—	4	1	no	—
214	C₄×C₅	—	4	1	no	—
215	C₄×C₅	—	8	4	no	—
216	C₄×C₅	#470	1	2	yes	—
217	C₄×C₅	#472	1	2	yes	—
218	C₄×C₅	C₂×#137	1	2	yes	—
219	C₄×C₅	C₂×#137	1	2	yes	—
220	#41	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	8	4	no	—
224	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₅×#41	1	1	yes	characteristic
234	M₁₆	—	4	4	no	—
235	M₁₆	—	4	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
240	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
248	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
256	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
257	C₄²	C₅×#41	1	1	yes	characteristic
258	C₄²	C₂⁴×C₅	1	1	yes	characteristic
259	C₂×C₅	—	2	2	no	—
260	C₂×C₅	—	8	1	no	—
261	C₂×C₅	#3039	1	1	yes	characteristic
262	C₂×C₅	C₂×#470	1	1	yes	characteristic
263	C₂×C₅	C₂×#472	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	C₂³	C₅×#121	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	—	8	4	no	—
268	Q₈	—	8	4	no	—
269	D₄	—	2	2	no	—
270	D₄	—	4	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	C₅×#121	1	1	yes	characteristic
282	C₂×C₄	C₅×#121	1	1	yes	characteristic
283	C₂×C₄	C₅×#121	1	1	yes	characteristic
284	C₂×C₄	C₅×#137	1	4	yes	—
285	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	4	2	no	—
290	C₈	—	4	4	no	—
291	C₈	C₅×#137	1	4	yes	—
292	C₅	#19297	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₅×#470	1	2	yes	—
299	C₂²	C₂×C₅×#121	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	8	4	no	—
306	C₄	C₅×#470	1	2	yes	—
307	C₄	C₅×#472	1	2	yes	—
308	C₄	C₂×C₅×#137	1	2	yes	—
309	C₄	C₂×C₅×#137	1	2	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	8	1	no	—
312	C₂	C₅×#3039	1	1	yes	characteristic
313	C₂	C₂×C₅×#470	1	1	yes	characteristic
314	C₂	C₂×C₅×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂×C₅×#41	D₄
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
1. C₂×C₅×#500	C₂	semidirect
4. C₅×#1696	C₂	semidirect
5. C₅×#1695	C₂	semidirect
6. C₅×#1566	C₂	semidirect
7. C₅×#1548	C₂	semidirect
30. #19297	C₅	direct (N × H)
69. C₂×#500	C₂×C₅	semidirect
72. #1696	C₂×C₅	semidirect
73. #1695	C₂×C₅	semidirect
74. #1566	C₂×C₅	semidirect
75. #1548	C₂×C₅	semidirect
292. C₅	#19297	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×#500	2
C₅×#3079	2
C₅×#2965	2
C₅×#1696	2
C₅×#1695	2
C₅×#1566	2
C₅×#1548	2
#19297	5

Subgroups — C5×#19297 #11942017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19297 #11942017