Subgroups — C₅×#33794 #11956514

Order 1280.

504 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×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2240	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2234	C₂	1	4	yes	maximal, complemented
8	C₅×#2228	C₂	1	4	yes	maximal, complemented
9	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#533	—	2	1	no	—
11	C₅×#532	—	2	1	no	—
12	C₅×#500	C₂²	1	1	yes	characteristic
13	C₅×#500	C₂²	1	1	yes	characteristic
14	C₅×#500	C₂²	1	1	yes	characteristic
15	C₅×#499	—	2	1	no	—
16	C₅×#499	C₂²	1	2	yes	—
17	C₅×#498	—	2	1	no	—
18	C₅×#493	C₂²	1	1	yes	characteristic
19	C₅×#486	—	2	1	no	—
20	C₅×#466	—	2	1	no	—
21	C₅×#466	C₂²	1	2	yes	—
22	C₄×C₅×D₈	—	2	1	no	—
23	C₄×C₅×D₈	C₂²	1	2	yes	—
24	C₅×#427	—	2	1	no	—
25	C₅×#427	C₂²	1	2	yes	—
26	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	C₅×#367	C₂²	1	4	yes	—
28	C₅×#366	C₂²	1	4	yes	—
29	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₅×#357	C₂²	1	4	yes	—
31	C₅×#357	C₂²	1	4	yes	—
32	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
35	#33794	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×#41	—	4	1	no	—
37	C₂×C₅×D₈	—	2	1	no	—
38	C₂×C₅×D₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊Q₈	—	2	1	no	—
42	C₅×C₄⋊Q₈	—	2	1	no	—
43	C₅×C₄⋊Q₈	—	2	1	no	—
44	C₅×#127	—	2	1	no	—
45	C₅×#125	—	2	2	no	—
46	C₅×#124	—	4	1	no	—
47	C₅×C₂²⋊Q₈	—	4	1	no	—
48	C₅×C₂²⋊Q₈	—	4	1	no	—
49	C₅×#121	—	2	2	no	—
50	C₅×#121	—	4	1	no	—
51	C₄×C₅×Q₈	—	4	1	no	—
52	C₄×C₅×Q₈	—	4	1	no	—
53	C₄×C₅×D₄	—	2	2	no	—
54	C₄×C₅×D₄	—	4	1	no	—
55	C₅×#117	—	2	2	no	—
56	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
57	C₂×C₅×C₁₆	C₂³	1	2	yes	—
58	C₂×C₅×C₁₆	C₂³	1	2	yes	—
59	C₅×#107	—	2	1	no	—
60	C₅×#107	—	2	1	no	—
61	C₅×#107	—	2	1	no	—
62	C₅×#107	C₂³	1	2	yes	—
63	C₅×#107	C₂³	1	2	yes	—
64	C₅×#107	C₂³	1	2	yes	—
65	C₅×#106	—	2	2	no	—
66	C₅×#106	—	2	2	no	—
67	C₅×#102	—	2	2	no	—
68	C₅×#102	—	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	#3169	C₂×C₅	1	1	yes	characteristic
74	#3160	C₂×C₅	1	1	yes	characteristic, complemented
75	#2254	C₂×C₅	1	1	yes	characteristic, complemented
76	#2252	C₂×C₅	1	1	yes	characteristic, complemented
77	#2240	C₂×C₅	1	1	yes	characteristic, complemented
78	#2240	C₂×C₅	1	1	yes	characteristic, complemented
79	#2234	C₂×C₅	1	4	yes	complemented
80	#2228	C₂×C₅	1	4	yes	complemented
81	#2162	C₂×C₅	1	1	yes	characteristic, complemented
82	C₅×#41	—	8	4	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×Q₈	—	4	1	no	—
85	C₂×C₅×Q₈	—	4	1	no	—
86	C₂×C₅×D₄	—	2	2	no	—
87	C₂×C₅×D₄	—	4	2	no	—
88	C₂×C₅×D₄	—	4	1	no	—
89	C₂×C₅×D₄	—	4	1	no	—
90	C₂²×C₄×C₅	—	4	2	no	—
91	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₅×D₈	—	2	4	no	—
95	C₅×D₈	—	4	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	4	1	no	—
106	C₅×C₄⋊C₄	—	4	2	no	—
107	C₅×C₄⋊C₄	—	4	2	no	—
108	C₅×C₄⋊C₄	—	4	1	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₂²⋊C₄	—	4	2	no	—
111	C₅×C₂²⋊C₄	—	4	2	no	—
112	C₅×C₂²⋊C₄	—	4	2	no	—
113	C₄²×C₅	—	4	2	no	—
114	C₄²×C₅	D₈	1	1	yes	characteristic
115	C₄²×C₅	D₈	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
118	C₅×C₁₆	—	2	2	no	—
119	C₅×C₁₆	—	2	2	no	—
120	#533	—	2	1	no	—
121	#532	—	2	1	no	—
122	#500	C₂²×C₅	1	1	yes	characteristic
123	#500	C₂²×C₅	1	1	yes	characteristic
124	#500	C₂²×C₅	1	1	yes	characteristic
125	#499	—	2	1	no	—
126	#499	C₂²×C₅	1	2	yes	—
127	#498	—	2	1	no	—
128	#493	C₂²×C₅	1	1	yes	characteristic
129	#486	—	2	1	no	—
130	#466	—	2	1	no	—
131	#466	C₂²×C₅	1	2	yes	—
132	C₄×D₈	—	2	1	no	—
133	C₄×D₈	C₂²×C₅	1	2	yes	—
134	#427	—	2	1	no	—
135	#427	C₂²×C₅	1	2	yes	—
136	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
137	#367	C₂²×C₅	1	4	yes	—
138	#366	C₂²×C₅	1	4	yes	—
139	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
140	#357	C₂²×C₅	1	4	yes	—
141	#357	C₂²×C₅	1	4	yes	—
142	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
143	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
144	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
145	C₂³×C₅	—	4	2	no	—
146	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
147	C₅×Q₈	—	8	2	no	—
148	C₅×Q₈	—	8	2	no	—
149	C₅×Q₈	—	8	2	no	—
150	C₅×D₄	—	4	2	no	—
151	C₅×D₄	—	4	2	no	—
152	C₅×D₄	—	8	4	no	—
153	C₅×D₄	—	8	2	no	—
154	C₅×D₄	—	8	2	no	—
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	4	2	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	8	4	no	—
163	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₅×C₈	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	#137	1	2	yes	—
172	C₅×C₈	2₋¹⁺⁴	1	2	yes	—
173	C₂×#41	—	4	1	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	C₂³×C₅	1	2	yes	—
176	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	1	no	—
179	C₄⋊Q₈	—	2	1	no	—
180	C₄⋊Q₈	—	2	1	no	—
181	#127	—	2	1	no	—
182	#125	—	2	2	no	—
183	#124	—	4	1	no	—
184	C₂²⋊Q₈	—	4	1	no	—
185	C₂²⋊Q₈	—	4	1	no	—
186	#121	—	2	2	no	—
187	#121	—	4	1	no	—
188	C₄×Q₈	—	4	1	no	—
189	C₄×Q₈	—	4	1	no	—
190	C₄×D₄	—	2	2	no	—
191	C₄×D₄	—	4	1	no	—
192	#117	—	2	2	no	—
193	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
194	C₂×C₁₆	C₂³×C₅	1	2	yes	—
195	C₂×C₁₆	C₂³×C₅	1	2	yes	—
196	#107	—	2	1	no	—
197	#107	—	2	1	no	—
198	#107	—	2	1	no	—
199	#107	C₂³×C₅	1	2	yes	—
200	#107	C₂³×C₅	1	2	yes	—
201	#107	C₂³×C₅	1	2	yes	—
202	#106	—	2	2	no	—
203	#106	—	2	2	no	—
204	#102	—	2	2	no	—
205	#102	—	2	2	no	—
206	C₄×C₈	C₅×D₄	1	1	yes	characteristic
207	C₄×C₈	C₅×D₄	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
209	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	8	2	no	—
214	C₂²×C₅	—	8	2	no	—
215	C₂²×C₅	—	8	2	no	—
216	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	8	2	no	—
221	C₄×C₅	—	8	2	no	—
222	C₄×C₅	—	8	2	no	—
223	C₄×C₅	#508	1	2	yes	—
224	C₄×C₅	#509	1	2	yes	—
225	C₄×C₅	#536	1	1	yes	characteristic
226	C₄×C₅	C₂×#137	1	1	yes	characteristic
227	#41	—	8	4	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
237	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
239	D₈	—	2	4	no	—
240	D₈	—	4	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
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	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	C₅×D₈	1	1	yes	characteristic
260	C₄²	C₅×D₈	1	1	yes	characteristic
261	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
262	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
263	C₁₆	—	2	2	no	—
264	C₁₆	—	2	2	no	—
265	C₂×C₅	—	4	1	no	—
266	C₂×C₅	—	16	2	no	—
267	C₂×C₅	#3226	1	1	yes	characteristic
268	C₂×C₅	C₂×#508	1	1	yes	characteristic
269	C₂×C₅	C₂×#509	1	1	yes	characteristic
270	C₂³	—	4	2	no	—
271	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
272	Q₈	—	8	2	no	—
273	Q₈	—	8	2	no	—
274	Q₈	—	8	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	2	no	—
279	D₄	—	8	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
289	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
290	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
291	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
292	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	C₅×#137	1	2	yes	—
297	C₈	C₅×2₋¹⁺⁴	1	2	yes	—
298	C₅	#33794	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	C₅×#508	1	2	yes	—
313	C₄	C₅×#509	1	2	yes	—
314	C₄	C₅×#536	1	1	yes	characteristic
315	C₄	C₂×C₅×#137	1	1	yes	characteristic
316	C₂	—	4	1	no	—
317	C₂	—	16	2	no	—
318	C₂	C₅×#3226	1	1	yes	characteristic
319	C₂	C₂×C₅×#508	1	1	yes	characteristic
320	C₂	C₂×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#3160	C₂	semidirect
3. C₅×#2254	C₂	semidirect
4. C₅×#2252	C₂	semidirect
5. C₅×#2240	C₂	semidirect
6. C₅×#2240	C₂	semidirect
7. C₅×#2234	C₂	semidirect
8. C₅×#2228	C₂	semidirect
9. C₅×#2162	C₂	semidirect
35. #33794	C₅	direct (N × H)
74. #3160	C₂×C₅	semidirect
75. #2254	C₂×C₅	semidirect
76. #2252	C₂×C₅	semidirect
77. #2240	C₂×C₅	semidirect
78. #2240	C₂×C₅	semidirect
79. #2234	C₂×C₅	semidirect
80. #2228	C₂×C₅	semidirect
81. #2162	C₂×C₅	semidirect
298. C₅	#33794	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×#3169	2
C₅×#3160	2
C₅×#2254	2
C₅×#2252	2
C₅×#2240	2
C₅×#2234	2
C₅×#2228	2
C₅×#2162	2
#33794	5

Subgroups — C5×#33794 #11956514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33794 #11956514