Subgroups — C₅×#33457 #11956177

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₅×#2950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2948	C₂	1	1	yes	maximal, characteristic
4	C₅×#2175	C₂	1	4	yes	maximal, complemented
5	C₄×C₅×SD₃₂	C₂	1	4	yes	maximal, complemented
6	C₅×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2141	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2140	C₂	1	1	yes	maximal, characteristic
9	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
12	C₅×#519	—	2	1	no	—
13	C₅×#518	—	2	1	no	—
14	C₂×C₅×SD₃₂	C₄	1	8	yes	complemented
15	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₅×#442	—	2	2	no	—
20	C₅×#441	—	2	2	no	—
21	C₄×C₅×Q₁₆	C₂²	1	4	yes	complemented
22	C₄×C₅×D₈	C₂²	1	4	yes	—
23	C₅×#427	C₂²	1	1	yes	characteristic
24	C₅×#427	C₂²	1	1	yes	characteristic
25	C₅×#419	—	2	1	no	—
26	C₅×#418	—	2	1	no	—
27	C₅×#405	C₂²	1	1	yes	characteristic
28	C₅×#367	C₂²	1	2	yes	complemented
29	C₅×#366	C₂²	1	2	yes	complemented
30	C₅×#358	C₂²	1	4	yes	complemented
31	C₅×#357	C₂²	1	4	yes	—
32	C₅×#346	C₂²	1	2	yes	complemented
33	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
34	#33457	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂²×C₅×Q₈	—	2	1	no	—
36	C₂²×C₅×D₄	—	2	1	no	—
37	C₂×C₅×Q₁₆	—	2	1	no	—
38	C₂×C₅×Q₁₆	—	2	2	no	—
39	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	complemented
40	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
41	C₂×C₅×D₈	—	2	1	no	—
42	C₂×C₅×D₈	—	2	2	no	—
43	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
44	C₂×C₅×D₈	C₂³	1	2	yes	—
45	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₄×C₅×Q₈	—	2	4	no	—
48	C₄×C₅×Q₈	—	4	2	no	—
49	C₄×C₅×D₄	—	2	4	no	—
50	C₄×C₅×D₄	—	4	2	no	—
51	C₅×#117	—	2	1	no	—
52	C₅×#117	—	2	1	no	—
53	C₅×#117	—	4	2	no	—
54	C₅×#117	D₄	1	1	yes	characteristic
55	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
56	C₅×SD₃₂	C₂×C₄	1	8	yes	complemented
57	C₅×SD₃₂	C₂×C₄	1	8	yes	complemented
58	C₅×M₃₂	—	2	2	no	—
59	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
60	C₂×C₅×C₁₆	C₂³	1	2	yes	—
61	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
63	C₅×#107	C₂³	1	2	yes	—
64	C₅×#107	C₂³	1	2	yes	—
65	C₅×#106	—	2	1	no	—
66	C₅×#106	—	2	1	no	—
67	C₅×#103	—	2	4	no	—
68	C₅×#102	—	2	4	no	—
69	C₅×#97	D₄	1	2	yes	—
70	C₄×C₅×C₈	C₂³	1	2	yes	—
71	C₂²×SD₃₂	C₂×C₅	1	1	yes	characteristic
72	#2950	C₂×C₅	1	1	yes	characteristic, complemented
73	#2948	C₂×C₅	1	1	yes	characteristic
74	#2175	C₂×C₅	1	4	yes	complemented
75	C₄×SD₃₂	C₂×C₅	1	4	yes	complemented
76	#2158	C₂×C₅	1	1	yes	characteristic, complemented
77	#2141	C₂×C₅	1	1	yes	characteristic, complemented
78	#2140	C₂×C₅	1	1	yes	characteristic
79	#2108	C₂×C₅	1	1	yes	characteristic, complemented
80	C₂⁴×C₅	—	4	1	no	—
81	C₂×C₅×Q₈	—	2	4	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×Q₈	—	4	2	no	—
85	C₂×C₅×D₄	—	2	4	no	—
86	C₂×C₅×D₄	—	2	2	no	—
87	C₂×C₅×D₄	—	4	1	no	—
88	C₂×C₅×D₄	—	4	2	no	—
89	C₂²×C₄×C₅	—	4	1	no	—
90	C₂²×C₄×C₅	—	4	4	no	—
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₅×Q₁₆	—	2	4	no	—
93	C₅×Q₁₆	—	2	2	no	—
94	C₅×Q₁₆	C₂²×C₄	1	4	yes	—
95	C₅×D₈	—	2	4	no	—
96	C₅×D₈	—	2	2	no	—
97	C₅×D₈	C₂²×C₄	1	4	yes	—
98	C₅×M₁₆	—	2	2	no	—
99	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	#41	1	2	yes	—
103	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	4	4	no	—
107	C₅×C₄⋊C₄	D₈	1	2	yes	—
108	C₅×C₂²⋊C₄	—	4	1	no	—
109	C₅×C₂²⋊C₄	—	4	1	no	—
110	C₅×C₂²⋊C₄	—	4	4	no	—
111	C₅×C₂²⋊C₄	D₈	1	2	yes	—
112	C₄²×C₅	—	4	1	no	—
113	C₄²×C₅	—	4	1	no	—
114	C₄²×C₅	—	4	4	no	—
115	C₄²×C₅	C₂×D₄	1	2	yes	—
116	C₅×C₁₆	—	2	2	no	—
117	C₅×C₁₆	C₂²×C₄	1	2	yes	—
118	C₅×C₁₆	C₂²×C₄	1	2	yes	—
119	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
120	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
121	#519	—	2	1	no	—
122	#518	—	2	1	no	—
123	C₂×SD₃₂	C₄×C₅	1	8	yes	complemented
124	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
125	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
126	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
127	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
128	#442	—	2	2	no	—
129	#441	—	2	2	no	—
130	C₄×Q₁₆	C₂²×C₅	1	4	yes	complemented
131	C₄×D₈	C₂²×C₅	1	4	yes	—
132	#427	C₂²×C₅	1	1	yes	characteristic
133	#427	C₂²×C₅	1	1	yes	characteristic
134	#419	—	2	1	no	—
135	#418	—	2	1	no	—
136	#405	C₂²×C₅	1	1	yes	characteristic
137	#367	C₂²×C₅	1	2	yes	complemented
138	#366	C₂²×C₅	1	2	yes	complemented
139	#358	C₂²×C₅	1	4	yes	complemented
140	#357	C₂²×C₅	1	4	yes	—
141	#346	C₂²×C₅	1	2	yes	complemented
142	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
143	C₂³×C₅	—	4	2	no	—
144	C₂³×C₅	—	4	4	no	—
145	C₂³×C₅	—	16	2	no	—
146	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
147	C₅×Q₈	—	2	4	no	—
148	C₅×Q₈	—	4	2	no	—
149	C₅×Q₈	—	4	4	no	—
150	C₅×D₄	—	2	4	no	—
151	C₅×D₄	—	4	2	no	—
152	C₅×D₄	—	4	4	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	—	4	4	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	16	4	no	—
158	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
159	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
160	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₅	#135	1	2	yes	—
162	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₅×C₈	—	2	2	no	—
164	C₅×C₈	C₄×D₄	1	2	yes	—
165	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
166	C₅×C₈	C₂×#41	1	1	yes	characteristic
167	C₂²×Q₈	—	2	1	no	—
168	C₂²×D₄	—	2	1	no	—
169	C₂×Q₁₆	—	2	1	no	—
170	C₂×Q₁₆	—	2	2	no	—
171	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	complemented
172	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
176	C₂×D₈	C₂³×C₅	1	2	yes	—
177	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
179	C₄×Q₈	—	2	4	no	—
180	C₄×Q₈	—	4	2	no	—
181	C₄×D₄	—	2	4	no	—
182	C₄×D₄	—	4	2	no	—
183	#117	—	2	1	no	—
184	#117	—	2	1	no	—
185	#117	—	4	2	no	—
186	#117	C₅×D₄	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	—	4	2	no	—
188	SD₃₂	C₂×C₄×C₅	1	8	yes	complemented
189	SD₃₂	C₂×C₄×C₅	1	8	yes	complemented
190	M₃₂	—	2	2	no	—
191	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
192	C₂×C₁₆	C₂³×C₅	1	2	yes	—
193	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
194	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
195	#107	C₂³×C₅	1	2	yes	—
196	#107	C₂³×C₅	1	2	yes	—
197	#106	—	2	1	no	—
198	#106	—	2	1	no	—
199	#103	—	2	4	no	—
200	#102	—	2	4	no	—
201	#97	C₅×D₄	1	2	yes	—
202	C₄×C₈	C₂³×C₅	1	2	yes	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	4	4	no	—
205	C₂²×C₅	—	8	4	no	—
206	C₂²×C₅	—	16	4	no	—
207	C₂²×C₅	C₄×D₈	1	2	yes	—
208	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
209	C₄×C₅	—	2	4	no	—
210	C₄×C₅	—	4	4	no	—
211	C₄×C₅	—	8	2	no	—
212	C₄×C₅	—	8	2	no	—
213	C₄×C₅	C₄×D₈	1	2	yes	—
214	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
215	C₄×C₅	C₂×#135	1	1	yes	characteristic
216	C₂⁴	—	4	1	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
228	Q₁₆	—	2	4	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	C₂²×C₄×C₅	1	4	yes	—
231	D₈	—	2	4	no	—
232	D₈	—	2	2	no	—
233	D₈	C₂²×C₄×C₅	1	4	yes	—
234	M₁₆	—	2	2	no	—
235	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
236	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
237	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂×C₈	C₅×#41	1	2	yes	—
239	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	C₅×D₈	1	2	yes	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	4	1	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₂²⋊C₄	C₅×D₈	1	2	yes	—
248	C₄²	—	4	1	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	4	4	no	—
251	C₄²	C₂×C₅×D₄	1	2	yes	—
252	C₁₆	—	2	2	no	—
253	C₁₆	C₂²×C₄×C₅	1	2	yes	—
254	C₁₆	C₂²×C₄×C₅	1	2	yes	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	—	8	4	no	—
257	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
258	C₂×C₅	#3416	1	2	yes	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	16	2	no	—
262	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
263	Q₈	—	2	4	no	—
264	Q₈	—	4	4	no	—
265	Q₈	—	4	2	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	16	4	no	—
274	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
275	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
276	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
277	C₂×C₄	C₅×#135	1	2	yes	—
278	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	C₄×C₅×D₄	1	2	yes	—
281	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
282	C₈	C₂×C₅×#41	1	1	yes	characteristic
283	C₅	#33457	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	16	4	no	—
288	C₂²	C₄×C₅×D₈	1	2	yes	—
289	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	4	4	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	C₄×C₅×D₈	1	2	yes	—
295	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
296	C₄	C₂×C₅×#135	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	8	4	no	—
299	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic
300	C₂	C₅×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2950	C₂	semidirect
4. C₅×#2175	C₂	semidirect
5. C₄×C₅×SD₃₂	C₂	semidirect
6. C₅×#2158	C₂	semidirect
7. C₅×#2141	C₂	semidirect
9. C₅×#2108	C₂	semidirect
14. C₂×C₅×SD₃₂	C₄	semidirect
21. C₄×C₅×Q₁₆	C₂²	semidirect
28. C₅×#367	C₂²	semidirect
29. C₅×#366	C₂²	semidirect
30. C₅×#358	C₂²	semidirect
32. C₅×#346	C₂²	semidirect
33. C₄×C₅×C₁₆	C₂²	semidirect
34. #33457	C₅	direct (N × H)
39. C₂×C₅×Q₁₆	C₂×C₄	semidirect
56. C₅×SD₃₂	C₂×C₄	semidirect
57. C₅×SD₃₂	C₂×C₄	semidirect
59. C₂×C₅×C₁₆	C₂×C₄	semidirect
72. #2950	C₂×C₅	semidirect
74. #2175	C₂×C₅	semidirect
75. C₄×SD₃₂	C₂×C₅	semidirect
76. #2158	C₂×C₅	semidirect
77. #2141	C₂×C₅	semidirect
79. #2108	C₂×C₅	semidirect
123. C₂×SD₃₂	C₄×C₅	semidirect
130. C₄×Q₁₆	C₂²×C₅	semidirect
137. #367	C₂²×C₅	semidirect
138. #366	C₂²×C₅	semidirect
139. #358	C₂²×C₅	semidirect
141. #346	C₂²×C₅	semidirect
142. C₄×C₁₆	C₂²×C₅	semidirect
171. C₂×Q₁₆	C₂×C₄×C₅	semidirect
188. SD₃₂	C₂×C₄×C₅	semidirect
189. SD₃₂	C₂×C₄×C₅	semidirect
191. C₂×C₁₆	C₂×C₄×C₅	semidirect
283. C₅	#33457	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×SD₃₂	2
C₅×#2950	2
C₅×#2948	2
C₅×#2175	2
C₄×C₅×SD₃₂	2
C₅×#2158	2
C₅×#2141	2
C₅×#2140	2
C₅×#2108	2
#33457	5

Subgroups — C5×#33457 #11956177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33457 #11956177