Subgroups — C₅×#33656 #11956376

Order 1280.

556 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3054	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2239	C₂	1	4	yes	maximal, complemented
4	C₅×#2238	C₂	1	4	yes	maximal, complemented
5	C₅×#2155	C₂	1	2	yes	maximal, complemented
6	C₅×#2148	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#349	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₅×M₁₆	—	2	1	no	—
10	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₅×#525	—	2	1	no	—
12	C₂×C₅×M₃₂	C₂²	1	2	yes	—
13	C₅×#470	C₂²	1	2	yes	—
14	C₅×#469	C₂²	1	2	yes	—
15	C₅×#468	—	2	2	no	—
16	C₅×#430	—	2	2	no	—
17	C₅×#430	C₂²	1	4	yes	—
18	C₂×C₅×#108	—	2	1	no	—
19	C₂×C₅×#108	C₂²	1	2	yes	—
20	C₅×#427	C₂²	1	1	yes	characteristic
21	C₅×#416	—	2	1	no	—
22	C₅×#365	C₂²	1	8	yes	complemented
23	C₅×#362	C₂²	1	4	yes	complemented
24	C₅×#361	C₂²	1	4	yes	complemented
25	C₅×#349	C₂²	1	4	yes	—
26	#33656	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	—	2	1	no	—
28	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
29	C₅×#135	—	2	2	no	—
30	C₅×#135	—	2	2	no	—
31	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
32	C₂×C₅×SD₁₆	—	2	1	no	—
33	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
34	C₂×C₅×M₁₆	—	2	2	no	—
35	C₂×C₅×M₁₆	—	2	2	no	—
36	C₂×C₅×M₁₆	—	2	4	no	—
37	C₂×C₅×M₁₆	—	2	2	no	—
38	C₂×C₅×M₁₆	—	4	2	no	—
39	C₂×C₅×M₁₆	—	4	1	no	—
40	C₂×C₅×M₁₆	D₄	1	2	yes	—
41	C₂×C₅×M₁₆	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	—	4	1	no	—
43	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₅×#123	—	4	1	no	—
46	C₅×C₂²⋊Q₈	—	2	2	no	—
47	C₅×#121	—	2	2	no	—
48	C₅×#120	—	4	1	no	—
49	C₄×C₅×D₄	—	4	2	no	—
50	C₅×#117	—	2	1	no	—
51	C₅×M₃₂	—	2	2	no	—
52	C₅×M₃₂	C₂³	1	4	yes	—
53	C₂×C₅×C₁₆	—	2	2	no	—
54	C₅×#108	—	2	4	no	—
55	C₅×#108	—	2	2	no	—
56	C₅×#108	C₂³	1	4	yes	—
57	C₅×#107	—	2	1	no	—
58	C₅×#106	C₂³	1	2	yes	—
59	C₅×#103	—	2	2	no	—
60	C₅×#102	—	2	2	no	—
61	#3054	C₂×C₅	1	1	yes	characteristic
62	C₂×#430	C₂×C₅	1	1	yes	characteristic, complemented
63	#2239	C₂×C₅	1	4	yes	complemented
64	#2238	C₂×C₅	1	4	yes	complemented
65	#2155	C₂×C₅	1	2	yes	complemented
66	#2148	C₂×C₅	1	2	yes	complemented
67	C₂×#349	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
69	C₅×#41	—	4	1	no	—
70	C₅×#41	—	4	1	no	—
71	C₅×#41	—	4	1	no	—
72	C₅×#41	—	4	1	no	—
73	C₂×C₅×Q₈	—	2	1	no	—
74	C₂×C₅×D₄	—	2	1	no	—
75	C₂×C₅×D₄	—	4	2	no	—
76	C₂×C₅×D₄	—	4	1	no	—
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	2	1	no	—
79	C₂²×C₄×C₅	—	2	1	no	—
80	C₂²×C₄×C₅	—	2	1	no	—
81	C₂²×C₄×C₅	—	4	1	no	—
82	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
83	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
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×SD₁₆	—	4	2	no	—
88	C₅×D₈	—	2	2	no	—
89	C₅×M₁₆	—	2	4	no	—
90	C₅×M₁₆	—	2	4	no	—
91	C₅×M₁₆	—	2	4	no	—
92	C₅×M₁₆	—	4	4	no	—
93	C₅×M₁₆	—	4	2	no	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	4	4	no	—
97	C₂×C₅×C₈	—	4	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	#41	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	C₅×C₂²⋊C₄	—	4	2	no	—
108	C₄²×C₅	—	4	1	no	—
109	C₅×C₁₆	—	2	4	no	—
110	C₂×#135	C₂²×C₅	1	1	yes	characteristic
111	C₂²×M₁₆	—	2	1	no	—
112	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
113	#525	—	2	1	no	—
114	C₂×M₃₂	C₂²×C₅	1	2	yes	—
115	#470	C₂²×C₅	1	2	yes	—
116	#469	C₂²×C₅	1	2	yes	—
117	#468	—	2	2	no	—
118	#430	—	2	2	no	—
119	#430	C₂²×C₅	1	4	yes	—
120	C₂×#108	—	2	1	no	—
121	C₂×#108	C₂²×C₅	1	2	yes	—
122	#427	C₂²×C₅	1	1	yes	characteristic
123	#416	—	2	1	no	—
124	#365	C₂²×C₅	1	8	yes	complemented
125	#362	C₂²×C₅	1	4	yes	complemented
126	#361	C₂²×C₅	1	4	yes	complemented
127	#349	C₂²×C₅	1	4	yes	—
128	C₂³×C₅	—	2	1	no	—
129	C₂³×C₅	—	2	1	no	—
130	C₂³×C₅	—	4	1	no	—
131	C₂³×C₅	—	8	1	no	—
132	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
133	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
134	C₅×Q₈	—	4	1	no	—
135	C₅×Q₈	—	4	1	no	—
136	C₅×D₄	—	4	1	no	—
137	C₅×D₄	—	4	1	no	—
138	C₅×D₄	—	8	4	no	—
139	C₅×D₄	—	8	1	no	—
140	C₅×D₄	—	8	1	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	—	8	1	no	—
150	C₂×C₄×C₅	—	8	1	no	—
151	C₂×C₄×C₅	#123	1	1	yes	characteristic
152	C₂×C₄×C₅	#123	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
154	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
158	C₅×C₈	—	2	2	no	—
159	C₅×C₈	—	4	4	no	—
160	C₅×C₈	#123	1	2	yes	—
161	C₅×C₈	C₂×#41	1	2	yes	—
162	C₂×#41	—	2	1	no	—
163	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
164	#135	—	2	2	no	—
165	#135	—	2	2	no	—
166	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	4	no	—
173	C₂×M₁₆	—	4	2	no	—
174	C₂×M₁₆	—	4	1	no	—
175	C₂×M₁₆	C₅×D₄	1	2	yes	—
176	C₂×M₁₆	C₂³×C₅	1	2	yes	—
177	C₂²×C₈	—	4	1	no	—
178	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
180	#123	—	4	1	no	—
181	C₂²⋊Q₈	—	2	2	no	—
182	#121	—	2	2	no	—
183	#120	—	4	1	no	—
184	C₄×D₄	—	4	2	no	—
185	#117	—	2	1	no	—
186	M₃₂	—	2	2	no	—
187	M₃₂	C₂³×C₅	1	4	yes	—
188	C₂×C₁₆	—	2	2	no	—
189	#108	—	2	4	no	—
190	#108	—	2	2	no	—
191	#108	C₂³×C₅	1	4	yes	—
192	#107	—	2	1	no	—
193	#106	C₂³×C₅	1	2	yes	—
194	#103	—	2	2	no	—
195	#102	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	4	1	no	—
198	C₂²×C₅	—	4	2	no	—
199	C₂²×C₅	—	4	2	no	—
200	C₂²×C₅	—	8	1	no	—
201	C₂²×C₅	—	8	2	no	—
202	C₂²×C₅	—	8	1	no	—
203	C₂²×C₅	#481	1	1	yes	characteristic
204	C₂²×C₅	#481	1	1	yes	characteristic
205	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	8	2	no	—
208	C₄×C₅	—	8	1	no	—
209	C₄×C₅	#481	1	1	yes	characteristic
210	C₄×C₅	#481	1	1	yes	characteristic
211	C₄×C₅	C₂×#123	1	1	yes	characteristic
212	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
214	#41	—	4	1	no	—
215	#41	—	4	1	no	—
216	#41	—	4	1	no	—
217	#41	—	4	1	no	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
228	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
230	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	Q₁₆	—	2	2	no	—
232	SD₁₆	—	4	2	no	—
233	D₈	—	2	2	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	4	4	no	—
238	M₁₆	—	4	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	4	4	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
244	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂×C₈	C₅×#41	1	2	yes	—
246	C₂×C₈	C₅×#41	1	2	yes	—
247	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₄²	—	4	1	no	—
254	C₁₆	—	2	4	no	—
255	C₂×C₅	—	2	1	no	—
256	C₂×C₅	—	2	1	no	—
257	C₂×C₅	—	2	1	no	—
258	C₂×C₅	—	4	2	no	—
259	C₂×C₅	—	16	1	no	—
260	C₂×C₅	C₂×#481	1	1	yes	characteristic
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	8	1	no	—
265	C₂³	C₂×C₅×SD₁₆	1	2	yes	—
266	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
267	Q₈	—	4	1	no	—
268	Q₈	—	4	1	no	—
269	D₄	—	4	1	no	—
270	D₄	—	4	1	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	1	no	—
273	D₄	—	8	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	8	1	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	C₅×#123	1	1	yes	characteristic
285	C₂×C₄	C₅×#123	1	1	yes	characteristic
286	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
287	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
288	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	4	4	no	—
293	C₈	C₅×#123	1	2	yes	—
294	C₈	C₂×C₅×#41	1	2	yes	—
295	C₅	#33656	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²	—	2	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	1	no	—
303	C₂²	C₅×#481	1	1	yes	characteristic
304	C₂²	C₅×#481	1	1	yes	characteristic
305	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	1	no	—
309	C₄	C₅×#481	1	1	yes	characteristic
310	C₄	C₅×#481	1	1	yes	characteristic
311	C₄	C₂×C₅×#123	1	1	yes	characteristic
312	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	4	2	no	—
317	C₂	—	16	1	no	—
318	C₂	C₂×C₅×#481	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂×#481
prime radical	C₅×#3054	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#430	C₂	semidirect
3. C₅×#2239	C₂	semidirect
4. C₅×#2238	C₂	semidirect
5. C₅×#2155	C₂	semidirect
6. C₅×#2148	C₂	semidirect
7. C₂×C₅×#349	C₂	semidirect
22. C₅×#365	C₂²	semidirect
23. C₅×#362	C₂²	semidirect
24. C₅×#361	C₂²	semidirect
26. #33656	C₅	direct (N × H)
62. C₂×#430	C₂×C₅	semidirect
63. #2239	C₂×C₅	semidirect
64. #2238	C₂×C₅	semidirect
65. #2155	C₂×C₅	semidirect
66. #2148	C₂×C₅	semidirect
67. C₂×#349	C₂×C₅	semidirect
124. #365	C₂²×C₅	semidirect
125. #362	C₂²×C₅	semidirect
126. #361	C₂²×C₅	semidirect
295. C₅	#33656	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3054	2
C₂×C₅×#430	2
C₅×#2239	2
C₅×#2238	2
C₅×#2155	2
C₅×#2148	2
C₂×C₅×#349	2
#33656	5

Subgroups — C5×#33656 #11956376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33656 #11956376