Subgroups — C₅×#21718 #11944438

Order 1280.

774 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#2399	C₂	1	1	yes	maximal, characteristic
4	C₅×#1985	C₂	1	1	yes	maximal, characteristic
5	C₅×#1985	C₂	1	1	yes	maximal, characteristic
6	C₅×#1939	C₂	1	4	yes	maximal
7	C₅×#1843	C₂	1	1	yes	maximal, characteristic
8	C₅×#1771	C₂	1	4	yes	maximal
9	C₅×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₅×#446	—	2	4	no	—
13	C₅×C₈×Q₈	—	2	4	no	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₅×#97	C₂²	1	2	yes	—
20	C₂×C₅×#97	C₂²	1	1	yes	characteristic
21	C₂×C₅×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×#398	C₄	1	2	yes	—
26	C₅×#398	C₄	1	2	yes	—
27	C₅×#395	C₄	1	2	yes	—
28	C₅×#395	C₄	1	2	yes	—
29	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
30	C₅×#376	C₂²	1	1	yes	characteristic
31	C₅×#336	C₂²	1	4	yes	—
32	C₅×#336	C₂²	1	4	yes	—
33	#21718	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₄×C₅×Q₈	—	2	2	no	—
42	C₄×C₅×Q₈	—	2	2	no	—
43	C₄×C₅×Q₈	—	2	2	no	—
44	C₄×C₅×Q₈	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	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	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₅	C₂³	1	2	yes	—
52	C₂×C₄²×C₅	C₂³	1	2	yes	—
53	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂³	1	2	yes	—
55	C₅×C₄⋊C₈	—	2	4	no	—
56	C₅×C₄⋊C₈	—	2	4	no	—
57	C₅×C₄⋊C₈	—	2	2	no	—
58	C₅×C₄⋊C₈	—	2	2	no	—
59	C₅×C₄⋊C₈	—	2	8	no	—
60	C₅×#97	—	2	4	no	—
61	C₅×#97	—	2	2	no	—
62	C₅×#97	Q₈	1	8	yes	complemented
63	C₄×C₅×C₈	—	2	4	no	—
64	C₄×C₅×C₈	—	2	4	no	—
65	C₄×C₅×C₈	D₄	1	4	yes	—
66	C₅×#95	C₂×C₄	1	2	yes	—
67	C₅×#95	C₂×C₄	1	2	yes	—
68	C₅×#95	C₂×C₄	1	4	yes	—
69	C₅×#95	C₂×C₄	1	4	yes	—
70	C₂×#446	C₂×C₅	1	1	yes	characteristic
71	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
72	#2399	C₂×C₅	1	1	yes	characteristic
73	#1985	C₂×C₅	1	1	yes	characteristic
74	#1985	C₂×C₅	1	1	yes	characteristic
75	#1939	C₂×C₅	1	4	yes	—
76	#1843	C₂×C₅	1	1	yes	characteristic
77	#1771	C₂×C₅	1	4	yes	—
78	#1726	C₂×C₅	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₂×C₅×Q₈	—	2	2	no	—
81	C₂×C₅×Q₈	—	2	4	no	—
82	C₂×C₅×Q₈	—	4	2	no	—
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	4	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	4	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
101	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	4	no	—
106	C₅×C₄⋊C₄	—	2	8	no	—
107	C₅×C₄⋊C₄	—	4	4	no	—
108	C₄²×C₅	—	2	2	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	C₂×D₄	1	2	yes	—
113	C₄²×C₅	C₂×Q₈	1	4	yes	—
114	C₄²×C₅	#41	1	2	yes	—
115	C₄²×C₅	#41	1	4	yes	—
116	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
118	#446	—	2	4	no	—
119	C₈×Q₈	—	2	4	no	—
120	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
123	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
124	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
125	C₂×#97	C₂²×C₅	1	2	yes	—
126	C₂×#97	C₂²×C₅	1	1	yes	characteristic
127	C₂×#97	C₂²×C₅	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
130	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
131	#398	C₄×C₅	1	2	yes	—
132	#398	C₄×C₅	1	2	yes	—
133	#395	C₄×C₅	1	2	yes	—
134	#395	C₄×C₅	1	2	yes	—
135	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
136	#376	C₂²×C₅	1	1	yes	characteristic
137	#336	C₂²×C₅	1	4	yes	—
138	#336	C₂²×C₅	1	4	yes	—
139	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
140	C₅×Q₈	—	2	8	no	—
141	C₅×Q₈	—	4	4	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	4	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	4	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
156	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
157	C₂×C₄×C₅	C₄×Q₈	1	8	yes	—
158	C₂×C₄×C₅	#131	1	4	yes	—
159	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
161	C₂×C₄×C₅	C₂×#41	1	2	yes	—
162	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
163	C₅×C₈	—	2	2	no	—
164	C₅×C₈	—	2	4	no	—
165	C₅×C₈	—	4	2	no	—
166	C₅×C₈	—	4	2	no	—
167	C₅×C₈	C₄⋊Q₈	1	4	yes	—
168	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₅	1	2	yes	—
171	C₂²×C₈	C₂³×C₅	1	2	yes	—
172	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₅	1	2	yes	—
174	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
175	C₄×Q₈	—	2	2	no	—
176	C₄×Q₈	—	2	2	no	—
177	C₄×Q₈	—	2	2	no	—
178	C₄×Q₈	—	2	2	no	—
179	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
180	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
184	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
185	C₂×C₄²	C₂³×C₅	1	2	yes	—
186	C₂×C₄²	C₂³×C₅	1	2	yes	—
187	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₅	1	2	yes	—
189	C₄⋊C₈	—	2	4	no	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	2	8	no	—
194	#97	—	2	4	no	—
195	#97	—	2	2	no	—
196	#97	C₅×Q₈	1	8	yes	complemented
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	—	2	4	no	—
199	C₄×C₈	C₅×D₄	1	4	yes	—
200	#95	C₂×C₄×C₅	1	2	yes	—
201	#95	C₂×C₄×C₅	1	2	yes	—
202	#95	C₂×C₄×C₅	1	4	yes	—
203	#95	C₂×C₄×C₅	1	4	yes	—
204	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂²×C₅	C₂×C₄×Q₈	1	2	yes	—
206	C₂²×C₅	C₂×#131	1	2	yes	—
207	C₂²×C₅	#568	1	2	yes	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	4	no	—
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	—	2	8	no	—
212	C₄×C₅	—	4	2	no	—
213	C₄×C₅	—	4	2	no	—
214	C₄×C₅	—	4	2	no	—
215	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
216	C₄×C₅	#532	1	1	yes	characteristic
217	C₄×C₅	#533	1	2	yes	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
228	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
238	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
239	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
240	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	C₂×C₅×D₄	1	2	yes	—
252	C₄²	C₂×C₅×Q₈	1	4	yes	—
253	C₄²	C₅×#41	1	2	yes	—
254	C₄²	C₅×#41	1	4	yes	—
255	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
256	C₂×C₅	#2929	1	1	yes	characteristic
257	C₂×C₅	#2930	1	1	yes	characteristic
258	C₂×C₅	#2962	1	4	yes	—
259	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
260	Q₈	—	2	8	no	—
261	Q₈	—	4	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
276	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
277	C₂×C₄	C₄×C₅×Q₈	1	8	yes	—
278	C₂×C₄	C₅×#131	1	4	yes	—
279	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
281	C₂×C₄	C₂×C₅×#41	1	2	yes	—
282	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	2	4	no	—
285	C₈	—	4	2	no	—
286	C₈	—	4	2	no	—
287	C₈	C₅×C₄⋊Q₈	1	4	yes	—
288	C₅	#21718	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
290	C₂²	C₂×C₄×C₅×Q₈	1	2	yes	—
291	C₂²	C₂×C₅×#131	1	2	yes	—
292	C₂²	C₅×#568	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	8	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
301	C₄	C₅×#532	1	1	yes	characteristic
302	C₄	C₅×#533	1	2	yes	—
303	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
304	C₂	C₅×#2929	1	1	yes	characteristic
305	C₂	C₅×#2930	1	1	yes	characteristic
306	C₂	C₅×#2962	1	4	yes	—

	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
33. #21718	C₅	direct (N × H)
62. C₅×#97	Q₈	semidirect
196. #97	C₅×Q₈	semidirect
288. C₅	#21718	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×#446	2
C₂×C₅×C₈×Q₈	2
C₅×#2399	2
C₅×#1985	2
C₅×#1939	2
C₅×#1843	2
C₅×#1771	2
C₅×#1726	2
#21718	5

Subgroups — C5×#21718 #11944438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21718 #11944438