Subgroups — C₅×#33582 #11956302

Order 1280.

806 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×#509	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2207	C₂	1	4	yes	maximal, complemented
5	C₅×#2206	C₂	1	4	yes	maximal, complemented
6	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2140	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₅×D₈	—	2	1	no	—
10	C₂²×C₅×D₈	C₂²	1	2	yes	—
11	C₂×C₅×#127	—	2	1	no	—
12	C₄×C₅×#41	—	2	1	no	—
13	C₅×#509	—	2	8	no	—
14	C₂×C₅×SD₃₂	C₂²	1	4	yes	complemented
15	C₂×C₅×D₁₆	C₂²	1	4	yes	complemented
16	C₂×C₅×M₃₂	C₂²	1	2	yes	—
17	C₅×#493	—	2	2	no	—
18	C₅×#493	C₂²	1	4	yes	—
19	C₄×C₅×Q₁₆	C₂²	1	2	yes	complemented
20	C₄×C₅×SD₁₆	—	2	2	no	—
21	C₄×C₅×D₈	C₂²	1	2	yes	complemented
22	C₅×#427	C₂²	1	1	yes	characteristic
23	C₅×#416	—	2	1	no	—
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
26	C₅×#357	C₂²	1	8	yes	complemented
27	#33582	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	—	2	1	no	—
29	C₂²×C₅×D₄	—	2	2	no	—
30	C₂²×C₅×D₄	—	4	2	no	—
31	C₅×#135	—	2	2	no	—
32	C₅×#135	D₄	1	4	yes	complemented
33	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
34	C₂×C₅×SD₁₆	—	2	1	no	—
35	C₂×C₅×D₈	—	2	4	no	—
36	C₂×C₅×D₈	—	2	2	no	—
37	C₂×C₅×D₈	—	2	4	no	—
38	C₂×C₅×D₈	—	2	4	no	—
39	C₂×C₅×D₈	—	4	4	no	—
40	C₂×C₅×D₈	C₂³	1	4	yes	—
41	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
42	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×#127	—	2	4	no	—
45	C₅×#127	—	4	1	no	—
46	C₅×#127	—	4	1	no	—
47	C₄×C₅×Q₈	—	2	2	no	—
48	C₄×C₅×D₄	—	2	2	no	—
49	C₄×C₅×D₄	—	4	2	no	—
50	C₅×#117	—	2	1	no	—
51	C₅×#117	—	4	1	no	—
52	C₂×C₄²×C₅	—	4	1	no	—
53	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
54	C₅×SD₃₂	—	2	8	no	—
55	C₅×D₁₆	—	2	8	no	—
56	C₅×M₃₂	—	2	4	no	—
57	C₂×C₅×C₁₆	C₂³	1	4	yes	—
58	C₅×#107	C₂³	1	2	yes	—
59	C₅×#106	—	2	1	no	—
60	C₅×#103	—	2	2	no	—
61	C₅×#102	—	2	2	no	—
62	C₄×C₅×C₈	D₄	1	2	yes	—
63	C₄×C₅×C₈	C₂³	1	2	yes	—
64	C₂×#509	C₂×C₅	1	2	yes	complemented
65	C₂×#493	C₂×C₅	1	1	yes	characteristic, complemented
66	C₄×#135	C₂×C₅	1	1	yes	characteristic, complemented
67	#2207	C₂×C₅	1	4	yes	complemented
68	#2206	C₂×C₅	1	4	yes	complemented
69	#2150	C₂×C₅	1	1	yes	characteristic, complemented
70	#2140	C₂×C₅	1	2	yes	complemented
71	C₂⁴×C₅	—	4	2	no	—
72	C₅×#41	—	2	4	no	—
73	C₅×#41	—	4	2	no	—
74	C₂×C₅×Q₈	—	2	1	no	—
75	C₂×C₅×D₄	—	2	1	no	—
76	C₂×C₅×D₄	—	2	4	no	—
77	C₂×C₅×D₄	—	4	1	no	—
78	C₂×C₅×D₄	—	4	4	no	—
79	C₂×C₅×D₄	—	4	4	no	—
80	C₂×C₅×D₄	—	4	4	no	—
81	C₂×C₅×D₄	—	4	2	no	—
82	C₂×C₅×D₄	—	4	4	no	—
83	C₂×C₅×D₄	—	8	8	no	—
84	C₂²×C₄×C₅	—	4	1	no	—
85	C₂²×C₄×C₅	—	4	2	no	—
86	C₂²×C₄×C₅	D₈	1	2	yes	—
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₅×Q₁₆	—	2	1	no	—
89	C₅×Q₁₆	C₂×D₄	1	2	yes	complemented
90	C₅×SD₁₆	—	2	4	no	—
91	C₅×D₈	—	2	4	no	—
92	C₅×D₈	—	2	1	no	—
93	C₅×D₈	—	2	4	no	—
94	C₅×D₈	—	4	8	no	—
95	C₅×D₈	—	4	4	no	—
96	C₅×D₈	C₂×D₄	1	2	yes	complemented
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
101	C₂×C₅×C₈	#41	1	2	yes	—
102	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	4	2	no	—
105	C₅×C₂²⋊C₄	—	4	1	no	—
106	C₅×C₂²⋊C₄	—	4	2	no	—
107	C₄²×C₅	—	4	1	no	—
108	C₄²×C₅	—	4	2	no	—
109	C₄²×C₅	D₈	1	1	yes	characteristic
110	C₄²×C₅	D₈	1	1	yes	characteristic
111	C₄²×C₅	C₂×D₄	1	2	yes	—
112	C₅×C₁₆	—	2	4	no	—
113	C₂×#135	C₂²×C₅	1	1	yes	characteristic
114	C₂²×D₈	—	2	1	no	—
115	C₂²×D₈	C₂²×C₅	1	2	yes	—
116	C₂×#127	—	2	1	no	—
117	C₄×#41	—	2	1	no	—
118	#509	—	2	8	no	—
119	C₂×SD₃₂	C₂²×C₅	1	4	yes	complemented
120	C₂×D₁₆	C₂²×C₅	1	4	yes	complemented
121	C₂×M₃₂	C₂²×C₅	1	2	yes	—
122	#493	—	2	2	no	—
123	#493	C₂²×C₅	1	4	yes	—
124	C₄×Q₁₆	C₂²×C₅	1	2	yes	complemented
125	C₄×SD₁₆	—	2	2	no	—
126	C₄×D₈	C₂²×C₅	1	2	yes	complemented
127	#427	C₂²×C₅	1	1	yes	characteristic
128	#416	—	2	1	no	—
129	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
130	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
131	#357	C₂²×C₅	1	8	yes	complemented
132	C₂³×C₅	—	4	1	no	—
133	C₂³×C₅	—	4	4	no	—
134	C₂³×C₅	—	8	8	no	—
135	C₂³×C₅	—	8	4	no	—
136	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
137	C₅×Q₈	—	2	2	no	—
138	C₅×Q₈	—	4	1	no	—
139	C₅×D₄	—	2	2	no	—
140	C₅×D₄	—	4	4	no	—
141	C₅×D₄	—	4	4	no	—
142	C₅×D₄	—	4	1	no	—
143	C₅×D₄	—	4	4	no	—
144	C₅×D₄	—	8	8	no	—
145	C₅×D₄	—	8	4	no	—
146	C₅×D₄	—	8	8	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	—	4	4	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	8	4	no	—
152	C₂×C₄×C₅	#121	1	2	yes	—
153	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
157	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
158	C₅×C₈	—	2	2	no	—
159	C₅×C₈	#121	1	2	yes	—
160	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
161	C₅×C₈	C₂×#41	1	1	yes	characteristic
162	C₂×#41	—	2	1	no	—
163	C₂²×D₄	—	2	2	no	—
164	C₂²×D₄	—	4	2	no	—
165	#135	—	2	2	no	—
166	#135	C₅×D₄	1	4	yes	complemented
167	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
168	C₂×SD₁₆	—	2	1	no	—
169	C₂×D₈	—	2	4	no	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	—	2	2	no	—
172	C₂×D₈	—	2	4	no	—
173	C₂×D₈	—	4	4	no	—
174	C₂×D₈	C₂³×C₅	1	4	yes	—
175	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
176	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
178	#127	—	2	4	no	—
179	#127	—	4	1	no	—
180	#127	—	4	1	no	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	4	2	no	—
184	#117	—	2	1	no	—
185	#117	—	4	1	no	—
186	C₂×C₄²	—	4	1	no	—
187	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
188	SD₃₂	—	2	8	no	—
189	D₁₆	—	2	8	no	—
190	M₃₂	—	2	4	no	—
191	C₂×C₁₆	C₂³×C₅	1	4	yes	—
192	#107	C₂³×C₅	1	2	yes	—
193	#106	—	2	1	no	—
194	#103	—	2	2	no	—
195	#102	—	2	2	no	—
196	C₄×C₈	C₅×D₄	1	2	yes	—
197	C₄×C₈	C₂³×C₅	1	2	yes	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	4	2	no	—
200	C₂²×C₅	—	8	2	no	—
201	C₂²×C₅	—	8	8	no	—
202	C₂²×C₅	—	8	4	no	—
203	C₂²×C₅	—	16	8	no	—
204	C₂²×C₅	#459	1	2	yes	—
205	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	4	2	no	—
208	C₄×C₅	—	8	2	no	—
209	C₄×C₅	#459	1	2	yes	—
210	C₄×C₅	#509	1	4	yes	complemented
211	C₄×C₅	C₂×#121	1	1	yes	characteristic
212	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₂⁴	—	4	2	no	—
214	#41	—	2	4	no	—
215	#41	—	4	2	no	—
216	C₂×Q₈	—	2	1	no	—
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	—	4	4	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	8	8	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	C₅×D₈	1	2	yes	—
229	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
230	Q₁₆	—	2	1	no	—
231	Q₁₆	C₂×C₅×D₄	1	2	yes	complemented
232	SD₁₆	—	2	4	no	—
233	D₈	—	2	4	no	—
234	D₈	—	2	1	no	—
235	D₈	—	2	4	no	—
236	D₈	—	4	8	no	—
237	D₈	—	4	4	no	—
238	D₈	C₂×C₅×D₄	1	2	yes	complemented
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
241	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
243	C₂×C₈	C₅×#41	1	2	yes	—
244	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	4	2	no	—
251	C₄²	C₅×D₈	1	1	yes	characteristic
252	C₄²	C₅×D₈	1	1	yes	characteristic
253	C₄²	C₂×C₅×D₄	1	2	yes	—
254	C₁₆	—	2	4	no	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	—	8	2	no	—
257	C₂×C₅	—	16	4	no	—
258	C₂×C₅	C₂×#459	1	1	yes	characteristic
259	C₂×C₅	C₂×#509	1	2	yes	—
260	C₂³	—	4	1	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	8	8	no	—
263	C₂³	—	8	4	no	—
264	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
265	Q₈	—	2	2	no	—
266	Q₈	—	4	1	no	—
267	D₄	—	2	2	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	1	no	—
271	D₄	—	4	4	no	—
272	D₄	—	8	8	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	4	no	—
280	C₂×C₄	C₅×#121	1	2	yes	—
281	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
282	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
284	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
285	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	C₅×#121	1	2	yes	—
288	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
289	C₈	C₂×C₅×#41	1	1	yes	characteristic
290	C₅	#33582	1	1	yes	characteristic, direct product factor, Sylow
291	C₂²	—	2	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	8	no	—
296	C₂²	—	16	8	no	—
297	C₂²	C₅×#459	1	2	yes	—
298	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	2	no	—
302	C₄	C₅×#459	1	2	yes	—
303	C₄	C₅×#509	1	4	yes	complemented
304	C₄	C₂×C₅×#121	1	1	yes	characteristic
305	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	8	2	no	—
308	C₂	—	16	4	no	—
309	C₂	C₂×C₅×#459	1	1	yes	characteristic
310	C₂	C₂×C₅×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
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
1. C₂×C₅×#509	C₂	semidirect
2. C₂×C₅×#493	C₂	semidirect
3. C₄×C₅×#135	C₂	semidirect
4. C₅×#2207	C₂	semidirect
5. C₅×#2206	C₂	semidirect
6. C₅×#2150	C₂	semidirect
7. C₅×#2140	C₂	semidirect
14. C₂×C₅×SD₃₂	C₂²	semidirect
15. C₂×C₅×D₁₆	C₂²	semidirect
19. C₄×C₅×Q₁₆	C₂²	semidirect
21. C₄×C₅×D₈	C₂²	semidirect
25. C₅×C₄⋊C₁₆	C₂²	semidirect
26. C₅×#357	C₂²	semidirect
27. #33582	C₅	direct (N × H)
32. C₅×#135	D₄	semidirect
64. C₂×#509	C₂×C₅	semidirect
65. C₂×#493	C₂×C₅	semidirect
66. C₄×#135	C₂×C₅	semidirect
67. #2207	C₂×C₅	semidirect
68. #2206	C₂×C₅	semidirect
69. #2150	C₂×C₅	semidirect
70. #2140	C₂×C₅	semidirect
89. C₅×Q₁₆	C₂×D₄	semidirect
96. C₅×D₈	C₂×D₄	semidirect
119. C₂×SD₃₂	C₂²×C₅	semidirect
120. C₂×D₁₆	C₂²×C₅	semidirect
124. C₄×Q₁₆	C₂²×C₅	semidirect
126. C₄×D₈	C₂²×C₅	semidirect
130. C₄⋊C₁₆	C₂²×C₅	semidirect
131. #357	C₂²×C₅	semidirect
166. #135	C₅×D₄	semidirect
210. C₄×C₅	#509	semidirect
231. Q₁₆	C₂×C₅×D₄	semidirect
238. D₈	C₂×C₅×D₄	semidirect
290. C₅	#33582	direct (N × H)
303. C₄	C₅×#509	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#509	2
C₂×C₅×#493	2
C₄×C₅×#135	2
C₅×#2207	2
C₅×#2206	2
C₅×#2150	2
C₅×#2140	2
#33582	5

Subgroups — C5×#33582 #11956302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33582 #11956302