Subgroups — C₅×#33611 #11956331

Order 1280.

674 conjugacy classes of subgroups.

The subgroup lattice has 294 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	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3061	C₂	1	2	yes	maximal, complemented
4	C₅×#2219	C₂	1	2	yes	maximal, complemented
5	C₅×#2218	C₂	1	2	yes	maximal, complemented
6	C₅×#2217	C₂	1	4	yes	maximal, complemented
7	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2139	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#135	C₂²	1	2	yes	—
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	C₅×#537	—	2	2	no	—
12	C₅×#508	—	2	4	no	—
13	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
16	C₅×#504	—	2	4	no	—
17	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₅×M₃₂	C₂²	1	2	yes	—
19	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
20	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
21	C₅×#468	—	2	4	no	—
22	C₅×#467	C₂²	1	4	yes	—
23	C₅×#466	C₂²	1	4	yes	—
24	C₂×C₅×#107	C₂²	1	2	yes	—
25	C₅×#417	—	2	2	no	—
26	C₅×#367	C₂²	1	2	yes	—
27	C₅×#367	C₂²	1	2	yes	—
28	C₅×#358	C₂²	1	4	yes	—
29	C₅×#357	C₂²	1	4	yes	—
30	#33611	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₅×#41	—	2	2	no	—
32	C₂×C₅×#41	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	C₅×#131	—	2	2	no	—
39	C₅×#131	—	2	2	no	—
40	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
41	C₂×C₅×M₁₆	D₄	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	2	yes	—
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×C₂²⋊Q₈	—	2	4	no	—
45	C₅×C₂²⋊Q₈	—	4	2	no	—
46	C₅×#121	—	2	4	no	—
47	C₅×#121	—	4	2	no	—
48	C₅×#121	—	4	4	no	—
49	C₂×C₅×C₄⋊C₄	—	2	2	no	—
50	C₅×Q₃₂	—	2	2	no	—
51	C₅×SD₃₂	—	2	4	no	—
52	C₅×D₁₆	—	2	2	no	—
53	C₅×M₃₂	—	2	2	no	—
54	C₅×M₃₂	—	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	—	2	2	no	—
61	C₅×#107	C₂³	1	4	yes	—
62	C₅×#103	—	2	4	no	—
63	C₅×#102	—	2	4	no	—
64	C₂×#508	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂×#504	C₂×C₅	1	1	yes	characteristic, complemented
66	#3061	C₂×C₅	1	2	yes	complemented
67	#2219	C₂×C₅	1	2	yes	complemented
68	#2218	C₂×C₅	1	2	yes	complemented
69	#2217	C₂×C₅	1	4	yes	complemented
70	C₂×#367	C₂×C₅	1	1	yes	characteristic, complemented
71	#2139	C₂×C₅	1	2	yes	complemented
72	C₅×#41	—	2	2	no	—
73	C₅×#41	—	2	2	no	—
74	C₅×#41	—	4	4	no	—
75	C₅×#41	—	4	4	no	—
76	C₂×C₅×Q₈	—	2	2	no	—
77	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
78	C₂×C₅×D₄	—	2	2	no	—
79	C₂×C₅×D₄	—	4	2	no	—
80	C₂×C₅×D₄	—	4	4	no	—
81	C₂×C₅×D₄	D₈	1	2	yes	—
82	C₂×C₅×D₄	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₅	—	4	2	no	—
84	C₂²×C₄×C₅	—	4	2	no	—
85	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₅×Q₁₆	—	2	2	no	—
88	C₅×Q₁₆	—	2	2	no	—
89	C₅×SD₁₆	—	4	4	no	—
90	C₅×D₈	—	2	2	no	—
91	C₅×D₈	—	2	2	no	—
92	C₅×M₁₆	—	2	2	no	—
93	C₅×M₁₆	—	2	4	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
97	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₅×C₈	#41	1	2	yes	—
100	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	4	2	no	—
103	C₅×C₄⋊C₄	—	4	4	no	—
104	C₅×C₂²⋊C₄	—	4	4	no	—
105	C₅×C₂²⋊C₄	—	4	4	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	C₂×#131	C₂²×C₅	1	1	yes	characteristic
111	#537	—	2	2	no	—
112	#508	—	2	4	no	—
113	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic
114	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
115	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
116	#504	—	2	4	no	—
117	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
118	C₂×M₃₂	C₂²×C₅	1	2	yes	—
119	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
120	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
121	#468	—	2	4	no	—
122	#467	C₂²×C₅	1	4	yes	—
123	#466	C₂²×C₅	1	4	yes	—
124	C₂×#107	C₂²×C₅	1	2	yes	—
125	#417	—	2	2	no	—
126	#367	C₂²×C₅	1	2	yes	—
127	#367	C₂²×C₅	1	2	yes	—
128	#358	C₂²×C₅	1	4	yes	—
129	#357	C₂²×C₅	1	4	yes	—
130	C₂³×C₅	—	4	2	no	—
131	C₂³×C₅	C₂×D₈	1	2	yes	—
132	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
133	C₅×Q₈	—	2	2	no	—
134	C₅×Q₈	—	4	2	no	—
135	C₅×Q₈	—	4	2	no	—
136	C₅×D₄	—	2	4	no	—
137	C₅×D₄	—	2	2	no	—
138	C₅×D₄	—	4	2	no	—
139	C₅×D₄	—	4	2	no	—
140	C₅×D₄	—	8	4	no	—
141	C₅×D₄	—	16	4	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	4	4	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	8	4	no	—
147	C₂×C₄×C₅	—	16	2	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	C₂×#41	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	#131	—	2	2	no	—
165	#131	—	2	2	no	—
166	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
167	C₂×M₁₆	C₅×D₄	1	2	yes	—
168	C₂²×C₈	C₂³×C₅	1	2	yes	—
169	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
170	C₂²⋊Q₈	—	2	4	no	—
171	C₂²⋊Q₈	—	4	2	no	—
172	#121	—	2	4	no	—
173	#121	—	4	2	no	—
174	#121	—	4	4	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	Q₃₂	—	2	2	no	—
177	SD₃₂	—	2	4	no	—
178	D₁₆	—	2	2	no	—
179	M₃₂	—	2	2	no	—
180	M₃₂	—	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	—	2	2	no	—
187	#107	C₂³×C₅	1	4	yes	—
188	#103	—	2	4	no	—
189	#102	—	2	4	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	4	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	8	2	no	—
194	C₂²×C₅	—	8	2	no	—
195	C₂²×C₅	—	8	2	no	—
196	C₂²×C₅	#466	1	2	yes	—
197	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	8	4	no	—
200	C₄×C₅	—	8	2	no	—
201	C₄×C₅	#466	1	2	yes	—
202	C₄×C₅	C₂×#121	1	1	yes	characteristic
203	C₄×C₅	C₂×#135	1	1	yes	characteristic
204	#41	—	2	2	no	—
205	#41	—	2	2	no	—
206	#41	—	4	4	no	—
207	#41	—	4	4	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	C₅×D₈	1	2	yes	—
214	C₂×D₄	C₅×D₈	1	1	yes	characteristic
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	4	2	no	—
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	M₁₆	—	2	2	no	—
225	M₁₆	—	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	2	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₁₆	—	2	2	no	—
239	C₁₆	C₂×C₅×D₄	1	2	yes	—
240	C₁₆	C₂×C₅×D₄	1	2	yes	—
241	C₂×C₅	—	2	2	no	—
242	C₂×C₅	—	4	2	no	—
243	C₂×C₅	—	16	2	no	—
244	C₂×C₅	C₂×#466	1	1	yes	characteristic
245	C₂×C₅	#3416	1	2	yes	—
246	C₂³	—	4	2	no	—
247	C₂³	C₂×C₅×D₈	1	2	yes	—
248	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
249	Q₈	—	2	2	no	—
250	Q₈	—	4	2	no	—
251	Q₈	—	4	2	no	—
252	D₄	—	2	4	no	—
253	D₄	—	2	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	8	4	no	—
257	D₄	—	16	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	—	16	2	no	—
264	C₂×C₄	C₅×#121	1	2	yes	—
265	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
266	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
267	C₂×C₄	C₅×#135	1	2	yes	—
268	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	C₅×#121	1	2	yes	—
271	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
272	C₈	C₂×C₅×#41	1	1	yes	characteristic
273	C₅	#33611	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	C₅×#466	1	2	yes	—
281	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	C₅×#466	1	2	yes	—
286	C₄	C₂×C₅×#121	1	1	yes	characteristic
287	C₄	C₂×C₅×#135	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	4	2	no	—
290	C₂	—	16	2	no	—
291	C₂	C₂×C₅×#466	1	1	yes	characteristic
292	C₂	C₅×#3416	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×C₅×#41 ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#508	C₂	semidirect
2. C₂×C₅×#504	C₂	semidirect
3. C₅×#3061	C₂	semidirect
4. C₅×#2219	C₂	semidirect
5. C₅×#2218	C₂	semidirect
6. C₅×#2217	C₂	semidirect
7. C₂×C₅×#367	C₂	semidirect
8. C₅×#2139	C₂	semidirect
30. #33611	C₅	direct (N × H)
56. C₂×C₅×C₁₆	D₄	semidirect
64. C₂×#508	C₂×C₅	semidirect
65. C₂×#504	C₂×C₅	semidirect
66. #3061	C₂×C₅	semidirect
67. #2219	C₂×C₅	semidirect
68. #2218	C₂×C₅	semidirect
69. #2217	C₂×C₅	semidirect
70. C₂×#367	C₂×C₅	semidirect
71. #2139	C₂×C₅	semidirect
182. C₂×C₁₆	C₅×D₄	semidirect
273. C₅	#33611	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×#508	2
C₂×C₅×#504	2
C₅×#3061	2
C₅×#2219	2
C₅×#2218	2
C₅×#2217	2
C₂×C₅×#367	2
C₅×#2139	2
#33611	5

Subgroups — C5×#33611 #11956331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33611 #11956331