Subgroups — C₅×#33600 #11956320

Order 1280.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3050	C₂	1	2	yes	maximal, complemented
4	C₅×#2217	C₂	1	4	yes	maximal, complemented
5	C₅×#2216	C₂	1	2	yes	maximal, complemented
6	C₅×#2215	C₂	1	2	yes	maximal, complemented
7	C₅×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2138	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#135	C₂²	1	2	yes	—
10	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×#525	—	2	2	no	—
12	C₅×#508	—	2	4	no	—
13	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₅×SD₃₂	C₂²	1	2	yes	complemented
15	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₅×C₁₆	—	2	4	no	—
17	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
19	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
20	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
21	C₅×#467	C₂²	1	4	yes	—
22	C₅×#466	C₂²	1	4	yes	—
23	C₅×#465	—	2	4	no	—
24	C₅×#427	C₂²	1	2	yes	—
25	C₅×#416	—	2	2	no	—
26	C₅×#367	C₂²	1	2	yes	complemented
27	C₅×#366	C₂²	1	2	yes	complemented
28	C₅×#358	C₂²	1	4	yes	complemented
29	C₅×#357	C₂²	1	4	yes	complemented
30	#33600	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₅×#41	—	2	2	no	—
32	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
33	C₅×#135	—	2	4	no	—
34	C₅×#135	—	2	4	no	—
35	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
36	C₂×C₅×SD₁₆	—	2	2	no	—
37	C₂×C₅×D₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	—	2	2	no	—
39	C₂²×C₅×C₈	—	2	2	no	—
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	D₄	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	2	yes	—
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×#123	—	4	2	no	—
45	C₅×C₂²⋊Q₈	—	2	4	no	—
46	C₅×#121	—	2	4	no	—
47	C₅×#120	—	4	2	no	—
48	C₄×C₅×D₄	—	4	4	no	—
49	C₅×#117	—	2	2	no	—
50	C₅×Q₃₂	—	2	2	no	—
51	C₅×SD₃₂	—	2	4	no	—
52	C₅×D₁₆	—	2	2	no	—
53	C₂×C₅×C₁₆	—	2	2	no	—
54	C₂×C₅×C₁₆	—	2	4	no	—
55	C₂×C₅×C₁₆	—	2	4	no	—
56	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
57	C₂×C₅×C₁₆	C₂³	1	2	yes	—
58	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
60	C₅×#107	C₂³	1	4	yes	—
61	C₅×#106	—	2	2	no	—
62	C₅×#103	—	2	4	no	—
63	C₅×#102	—	2	4	no	—
64	C₂×#508	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
66	#3050	C₂×C₅	1	2	yes	complemented
67	#2217	C₂×C₅	1	4	yes	complemented
68	#2216	C₂×C₅	1	2	yes	complemented
69	#2215	C₂×C₅	1	2	yes	complemented
70	#2158	C₂×C₅	1	1	yes	characteristic, complemented
71	#2138	C₂×C₅	1	2	yes	complemented
72	C₂⁴×C₅	D₈	1	1	yes	characteristic
73	C₅×#41	—	4	4	no	—
74	C₅×#41	—	4	4	no	—
75	C₂×C₅×Q₈	—	2	2	no	—
76	C₂×C₅×D₄	—	2	2	no	—
77	C₂×C₅×D₄	—	4	4	no	—
78	C₂×C₅×D₄	—	4	2	no	—
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	4	2	no	—
82	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₅	D₈	1	2	yes	—
84	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₅×Q₁₆	—	2	2	no	—
87	C₅×Q₁₆	—	2	2	no	—
88	C₅×SD₁₆	—	4	4	no	—
89	C₅×D₈	—	2	2	no	—
90	C₅×D₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₅×C₈	#41	1	2	yes	—
99	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	2	4	no	—
101	C₅×C₄⋊C₄	—	4	4	no	—
102	C₅×C₂²⋊C₄	—	4	2	no	—
103	C₅×C₂²⋊C₄	—	4	4	no	—
104	C₅×C₂²⋊C₄	—	4	4	no	—
105	C₄²×C₅	—	4	2	no	—
106	C₅×C₁₆	—	2	2	no	—
107	C₅×C₁₆	C₂×D₄	1	2	yes	—
108	C₅×C₁₆	C₂×D₄	1	2	yes	—
109	C₂×#135	C₂²×C₅	1	2	yes	—
110	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
111	#525	—	2	2	no	—
112	#508	—	2	4	no	—
113	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic, complemented
114	C₂×SD₃₂	C₂²×C₅	1	2	yes	complemented
115	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
116	C₂²×C₁₆	—	2	4	no	—
117	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
119	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
120	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
121	#467	C₂²×C₅	1	4	yes	—
122	#466	C₂²×C₅	1	4	yes	—
123	#465	—	2	4	no	—
124	#427	C₂²×C₅	1	2	yes	—
125	#416	—	2	2	no	—
126	#367	C₂²×C₅	1	2	yes	complemented
127	#366	C₂²×C₅	1	2	yes	complemented
128	#358	C₂²×C₅	1	4	yes	complemented
129	#357	C₂²×C₅	1	4	yes	complemented
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	2	4	no	—
132	C₂³×C₅	—	4	2	no	—
133	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
134	C₂³×C₅	C₂×D₈	1	2	yes	—
135	C₅×Q₈	—	4	2	no	—
136	C₅×Q₈	—	4	2	no	—
137	C₅×D₄	—	4	2	no	—
138	C₅×D₄	—	4	2	no	—
139	C₅×D₄	—	8	4	no	—
140	C₅×D₄	—	8	8	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	4	4	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	8	4	no	—
148	C₂×C₄×C₅	#121	1	2	yes	—
149	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₅	#135	1	2	yes	—
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₅×C₈	—	2	2	no	—
154	C₅×C₈	#121	1	2	yes	—
155	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
156	C₅×C₈	C₂×#41	1	1	yes	characteristic
157	C₂×#41	—	2	2	no	—
158	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
159	#135	—	2	4	no	—
160	#135	—	2	4	no	—
161	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	C₂³×C₅	1	2	yes	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
167	C₂²×C₈	C₅×D₄	1	2	yes	—
168	C₂²×C₈	C₂³×C₅	1	2	yes	—
169	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
170	#123	—	4	2	no	—
171	C₂²⋊Q₈	—	2	4	no	—
172	#121	—	2	4	no	—
173	#120	—	4	2	no	—
174	C₄×D₄	—	4	4	no	—
175	#117	—	2	2	no	—
176	Q₃₂	—	2	2	no	—
177	SD₃₂	—	2	4	no	—
178	D₁₆	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	—	2	4	no	—
182	C₂×C₁₆	C₅×D₄	1	4	yes	complemented
183	C₂×C₁₆	C₂³×C₅	1	2	yes	—
184	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
185	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
186	#107	C₂³×C₅	1	4	yes	—
187	#106	—	2	2	no	—
188	#103	—	2	4	no	—
189	#102	—	2	4	no	—
190	C₂²×C₅	—	2	8	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	8	4	no	—
195	C₂²×C₅	—	8	2	no	—
196	C₂²×C₅	#466	1	2	yes	—
197	C₂²×C₅	#508	1	4	yes	complemented
198	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	8	4	no	—
201	C₄×C₅	—	8	2	no	—
202	C₄×C₅	#466	1	2	yes	—
203	C₄×C₅	C₂×#121	1	1	yes	characteristic
204	C₄×C₅	C₂×#135	1	1	yes	characteristic
205	C₂⁴	C₅×D₈	1	1	yes	characteristic
206	#41	—	4	4	no	—
207	#41	—	4	4	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	4	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₅×D₈	1	2	yes	—
216	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	Q₁₆	—	2	2	no	—
220	Q₁₆	—	2	2	no	—
221	SD₁₆	—	4	4	no	—
222	D₈	—	2	2	no	—
223	D₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
229	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
231	C₂×C₈	C₅×#41	1	2	yes	—
232	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₄²	—	4	2	no	—
239	C₁₆	—	2	2	no	—
240	C₁₆	C₂×C₅×D₄	1	2	yes	—
241	C₁₆	C₂×C₅×D₄	1	2	yes	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	—	2	4	no	—
244	C₂×C₅	—	16	2	no	—
245	C₂×C₅	C₂×#466	1	1	yes	characteristic
246	C₂×C₅	C₂×#508	1	2	yes	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
251	C₂³	C₂×C₅×D₈	1	2	yes	—
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	8	4	no	—
257	D₄	—	8	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₅×#121	1	2	yes	—
266	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
267	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
268	C₂×C₄	C₅×#135	1	2	yes	—
269	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	C₅×#121	1	2	yes	—
272	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
273	C₈	C₂×C₅×#41	1	1	yes	characteristic
274	C₅	#33600	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	—	2	8	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	4	no	—
281	C₂²	C₅×#466	1	2	yes	—
282	C₂²	C₅×#508	1	4	yes	complemented
283	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	8	2	no	—
286	C₄	—	8	4	no	—
287	C₄	C₅×#466	1	2	yes	—
288	C₄	C₂×C₅×#121	1	1	yes	characteristic
289	C₄	C₂×C₅×#135	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	2	4	no	—
292	C₂	—	16	2	no	—
293	C₂	C₂×C₅×#466	1	1	yes	characteristic
294	C₂	C₂×C₅×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#508	C₂	semidirect
2. C₂³×C₅×C₁₆	C₂	semidirect
3. C₅×#3050	C₂	semidirect
4. C₅×#2217	C₂	semidirect
5. C₅×#2216	C₂	semidirect
6. C₅×#2215	C₂	semidirect
7. C₅×#2158	C₂	semidirect
8. C₅×#2138	C₂	semidirect
13. C₂×C₅×Q₃₂	C₂²	semidirect
14. C₂×C₅×SD₃₂	C₂²	semidirect
15. C₂×C₅×D₁₆	C₂²	semidirect
26. C₅×#367	C₂²	semidirect
27. C₅×#366	C₂²	semidirect
28. C₅×#358	C₂²	semidirect
29. C₅×#357	C₂²	semidirect
30. #33600	C₅	direct (N × H)
56. C₂×C₅×C₁₆	D₄	semidirect
64. C₂×#508	C₂×C₅	semidirect
65. C₂³×C₁₆	C₂×C₅	semidirect
66. #3050	C₂×C₅	semidirect
67. #2217	C₂×C₅	semidirect
68. #2216	C₂×C₅	semidirect
69. #2215	C₂×C₅	semidirect
70. #2158	C₂×C₅	semidirect
71. #2138	C₂×C₅	semidirect
113. C₂×Q₃₂	C₂²×C₅	semidirect
114. C₂×SD₃₂	C₂²×C₅	semidirect
115. C₂×D₁₆	C₂²×C₅	semidirect
126. #367	C₂²×C₅	semidirect
127. #366	C₂²×C₅	semidirect
128. #358	C₂²×C₅	semidirect
129. #357	C₂²×C₅	semidirect
182. C₂×C₁₆	C₅×D₄	semidirect
197. C₂²×C₅	#508	semidirect
274. C₅	#33600	direct (N × H)
282. C₂²	C₅×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#508	2
C₂³×C₅×C₁₆	2
C₅×#3050	2
C₅×#2217	2
C₅×#2216	2
C₅×#2215	2
C₅×#2158	2
C₅×#2138	2
#33600	5

Subgroups — C5×#33600 #11956320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33600 #11956320