Subgroups — C₅×#12955 #11935675

Order 1280.

408 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	C₂×C₅×#444	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#360	C₂	1	1	yes	maximal, characteristic
4	C₅×#2139	C₂	1	1	yes	maximal, characteristic
5	C₅×#1943	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂×C₅×#131	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₅×#444	—	2	2	no	—
13	C₅×#444	—	2	2	no	—
14	C₂×C₅×#108	C₂²	1	1	yes	characteristic
15	C₂×C₅×#107	C₂²	1	1	yes	characteristic
16	C₂×C₅×#106	C₂²	1	1	yes	characteristic
17	C₂×C₅×#104	—	2	1	no	—
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₅×#389	—	2	1	no	—
20	C₅×#365	—	2	2	no	—
21	C₅×#360	—	2	2	no	—
22	C₅×#358	C₄	1	2	yes	complemented
23	C₅×#357	C₄	1	2	yes	complemented
24	C₅×#340	—	2	1	no	—
25	#12955	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	—	2	1	no	—
27	C₅×#135	—	2	1	no	—
28	C₅×#135	—	2	1	no	—
29	C₅×#135	D₄	1	2	yes	—
30	C₅×#135	Q₈	1	2	yes	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×SD₁₆	—	2	1	no	—
33	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
34	C₅×#131	—	2	4	no	—
35	C₅×#131	—	4	2	no	—
36	C₂×C₅×M₁₆	—	2	1	no	—
37	C₂×C₅×M₁₆	—	4	1	no	—
38	C₂²×C₅×C₈	—	4	1	no	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₂×C₅×C₄⋊C₄	—	2	1	no	—
43	C₂×C₅×C₄⋊C₄	—	4	1	no	—
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₅×M₃₂	—	2	1	no	—
46	C₅×M₃₂	—	2	1	no	—
47	C₅×M₃₂	—	2	1	no	—
48	C₅×M₃₂	—	2	1	no	—
49	C₂×C₅×C₁₆	—	2	1	no	—
50	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
51	C₅×#108	—	2	1	no	—
52	C₅×#108	D₄	1	2	yes	—
53	C₅×#107	—	2	1	no	—
54	C₅×#107	C₂×C₄	1	2	yes	—
55	C₅×#106	—	2	1	no	—
56	C₅×#106	—	2	1	no	—
57	C₅×#104	—	2	4	no	—
58	C₅×#97	—	2	1	no	—
59	C₅×#97	—	2	1	no	—
60	C₅×#95	—	4	1	no	—
61	C₂×#444	C₂×C₅	1	1	yes	characteristic
62	C₂×#365	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂×#360	C₂×C₅	1	1	yes	characteristic
64	#2139	C₂×C₅	1	1	yes	characteristic
65	#1943	C₂×C₅	1	1	yes	characteristic, complemented
66	#1387	C₂×C₅	1	1	yes	characteristic, complemented
67	#1375	C₂×C₅	1	1	yes	characteristic, complemented
68	C₅×#41	—	2	2	no	—
69	C₅×#41	—	2	2	no	—
70	C₅×#41	—	4	1	no	—
71	C₅×#41	—	4	1	no	—
72	C₂×C₅×Q₈	—	2	1	no	—
73	C₂×C₅×D₄	—	2	1	no	—
74	C₂×C₅×D₄	—	4	1	no	—
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	—	4	1	no	—
77	C₂²×C₄×C₅	—	4	1	no	—
78	C₂²×C₄×C₅	—	4	1	no	—
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₅×Q₁₆	—	2	1	no	—
81	C₅×Q₁₆	C₄⋊C₄	1	2	yes	complemented
82	C₅×SD₁₆	—	2	2	no	—
83	C₅×SD₁₆	—	2	2	no	—
84	C₅×D₈	—	2	1	no	—
85	C₅×D₈	C₄⋊C₄	1	2	yes	complemented
86	C₅×M₁₆	—	2	2	no	—
87	C₅×M₁₆	—	4	4	no	—
88	C₅×M₁₆	—	4	1	no	—
89	C₂×C₅×C₈	—	2	1	no	—
90	C₂×C₅×C₈	—	2	1	no	—
91	C₂×C₅×C₈	—	4	4	no	—
92	C₂×C₅×C₈	—	4	1	no	—
93	C₂×C₅×C₈	SD₁₆	1	2	yes	—
94	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₅×C₈	#41	1	1	yes	characteristic
99	C₂×C₅×C₈	#41	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	4	1	no	—
102	C₅×C₄⋊C₄	—	4	1	no	—
103	C₅×C₄⋊C₄	—	4	1	no	—
104	C₅×C₄⋊C₄	—	8	2	no	—
105	C₄²×C₅	—	2	1	no	—
106	C₄²×C₅	—	2	1	no	—
107	C₅×C₁₆	—	2	2	no	—
108	C₅×C₁₆	—	4	1	no	—
109	C₂×#135	C₂²×C₅	1	1	yes	characteristic
110	C₂×#131	—	2	1	no	—
111	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
112	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
113	#444	—	2	2	no	—
114	#444	—	2	2	no	—
115	C₂×#108	C₂²×C₅	1	1	yes	characteristic
116	C₂×#107	C₂²×C₅	1	1	yes	characteristic
117	C₂×#106	C₂²×C₅	1	1	yes	characteristic
118	C₂×#104	—	2	1	no	—
119	C₂×#97	C₂²×C₅	1	1	yes	characteristic
120	#389	—	2	1	no	—
121	#365	—	2	2	no	—
122	#360	—	2	2	no	—
123	#358	C₄×C₅	1	2	yes	complemented
124	#357	C₄×C₅	1	2	yes	complemented
125	#340	—	2	1	no	—
126	C₂³×C₅	—	4	1	no	—
127	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₅×Q₈	—	2	2	no	—
129	C₅×Q₈	—	4	1	no	—
130	C₅×D₄	—	2	2	no	—
131	C₅×D₄	—	4	2	no	—
132	C₅×D₄	—	4	2	no	—
133	C₅×D₄	—	4	1	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	4	1	no	—
141	C₂×C₄×C₅	—	8	1	no	—
142	C₂×C₄×C₅	—	8	1	no	—
143	C₂×C₄×C₅	—	8	1	no	—
144	C₂×C₄×C₅	—	8	1	no	—
145	C₂×C₄×C₅	—	8	1	no	—
146	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	#120	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₄×C₅	#135	1	1	yes	characteristic
151	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₅×C₈	—	2	2	no	—
153	C₅×C₈	—	4	2	no	—
154	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₅×C₈	C₄×D₄	1	1	yes	characteristic
156	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
157	C₅×C₈	#123	1	1	yes	characteristic
158	C₂×#41	—	2	1	no	—
159	#135	—	2	1	no	—
160	#135	—	2	1	no	—
161	#135	C₅×D₄	1	2	yes	—
162	#135	C₅×Q₈	1	2	yes	—
163	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂×SD₁₆	—	2	1	no	—
165	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
166	#131	—	2	4	no	—
167	#131	—	4	2	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	4	1	no	—
170	C₂²×C₈	—	4	1	no	—
171	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	4	1	no	—
176	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
183	#108	—	2	1	no	—
184	#108	C₅×D₄	1	2	yes	—
185	#107	—	2	1	no	—
186	#107	C₂×C₄×C₅	1	2	yes	—
187	#106	—	2	1	no	—
188	#106	—	2	1	no	—
189	#104	—	2	4	no	—
190	#97	—	2	1	no	—
191	#97	—	2	1	no	—
192	#95	—	4	1	no	—
193	C₂²×C₅	—	2	1	no	—
194	C₂²×C₅	—	2	1	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	—	8	1	no	—
197	C₂²×C₅	—	8	1	no	—
198	C₂²×C₅	#442	1	1	yes	characteristic
199	C₂²×C₅	#449	1	1	yes	characteristic
200	C₂²×C₅	#476	1	1	yes	characteristic
201	C₄×C₅	—	4	2	no	—
202	C₄×C₅	—	8	1	no	—
203	C₄×C₅	—	8	2	no	—
204	C₄×C₅	—	16	1	no	—
205	C₄×C₅	#385	1	1	yes	characteristic
206	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
207	C₄×C₅	#450	1	1	yes	characteristic
208	C₄×C₅	#478	1	1	yes	characteristic
209	#41	—	2	2	no	—
210	#41	—	2	2	no	—
211	#41	—	4	1	no	—
212	#41	—	4	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	Q₁₆	—	2	1	no	—
222	Q₁₆	C₅×C₄⋊C₄	1	2	yes	complemented
223	SD₁₆	—	2	2	no	—
224	SD₁₆	—	2	2	no	—
225	D₈	—	2	1	no	—
226	D₈	C₅×C₄⋊C₄	1	2	yes	complemented
227	M₁₆	—	2	2	no	—
228	M₁₆	—	4	4	no	—
229	M₁₆	—	4	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	4	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	C₅×SD₁₆	1	2	yes	—
235	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
236	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
239	C₂×C₈	C₅×#41	1	1	yes	characteristic
240	C₂×C₈	C₅×#41	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	—	8	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	4	1	no	—
250	C₂×C₅	—	2	1	no	—
251	C₂×C₅	—	2	1	no	—
252	C₂×C₅	—	8	2	no	—
253	C₂×C₅	#1864	1	1	yes	characteristic
254	C₂×C₅	#2180	1	1	yes	characteristic
255	C₂×C₅	AGL(1,ℤ/16)	1	1	yes	characteristic
256	C₂³	—	4	1	no	—
257	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	Q₈	—	4	1	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
277	C₂×C₄	C₅×#120	1	1	yes	characteristic
278	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
280	C₂×C₄	C₅×#135	1	1	yes	characteristic
281	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	2	no	—
284	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
285	C₈	C₄×C₅×D₄	1	1	yes	characteristic
286	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
287	C₈	C₅×#123	1	1	yes	characteristic
288	C₅	#12955	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	8	1	no	—
293	C₂²	—	8	1	no	—
294	C₂²	C₅×#442	1	1	yes	characteristic
295	C₂²	C₅×#449	1	1	yes	characteristic
296	C₂²	C₅×#476	1	1	yes	characteristic
297	C₄	—	4	2	no	—
298	C₄	—	8	1	no	—
299	C₄	—	8	2	no	—
300	C₄	—	16	1	no	—
301	C₄	C₅×#385	1	1	yes	characteristic
302	C₄	C₄×C₅×SD₁₆	1	1	yes	characteristic
303	C₄	C₅×#450	1	1	yes	characteristic
304	C₄	C₅×#478	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	8	2	no	—
308	C₂	C₅×#1864	1	1	yes	characteristic
309	C₂	C₅×#2180	1	1	yes	characteristic
310	C₂	C₅×AGL(1,ℤ/16)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#442
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#365	C₂	semidirect
5. C₅×#1943	C₂	semidirect
6. C₅×#1387	C₂	semidirect
7. C₅×#1375	C₂	semidirect
22. C₅×#358	C₄	semidirect
23. C₅×#357	C₄	semidirect
25. #12955	C₅	direct (N × H)
62. C₂×#365	C₂×C₅	semidirect
65. #1943	C₂×C₅	semidirect
66. #1387	C₂×C₅	semidirect
67. #1375	C₂×C₅	semidirect
81. C₅×Q₁₆	C₄⋊C₄	semidirect
85. C₅×D₈	C₄⋊C₄	semidirect
123. #358	C₄×C₅	semidirect
124. #357	C₄×C₅	semidirect
222. Q₁₆	C₅×C₄⋊C₄	semidirect
226. D₈	C₅×C₄⋊C₄	semidirect
288. C₅	#12955	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×#444	2
C₂×C₅×#365	2
C₂×C₅×#360	2
C₅×#2139	2
C₅×#1943	2
C₅×#1387	2
C₅×#1375	2
#12955	5

Subgroups — C5×#12955 #11935675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12955 #11935675