Subgroups — C₅×#33820 #11956540

Order 1280.

528 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3169	C₂	1	1	yes	maximal, characteristic
3	C₅×#2227	C₂	1	2	yes	maximal, complemented
4	C₅×#2224	C₂	1	2	yes	maximal, complemented
5	C₅×#2220	C₂	1	4	yes	maximal, complemented
6	C₅×#2208	C₂	1	2	yes	maximal, complemented
7	C₅×#2207	C₂	1	2	yes	maximal, complemented
8	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#533	—	2	1	no	—
10	C₅×#532	—	2	1	no	—
11	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
12	C₂×C₅×M₃₂	C₂²	1	2	yes	—
13	C₅×#500	C₂²	1	1	yes	characteristic
14	C₅×#500	C₂²	1	1	yes	characteristic
15	C₅×#494	C₂²	1	1	yes	characteristic
16	C₅×#493	C₂²	1	1	yes	characteristic
17	C₅×#487	—	2	1	no	—
18	C₅×#486	—	2	1	no	—
19	C₅×#467	—	2	1	no	—
20	C₅×#467	C₂²	1	2	yes	—
21	C₅×#466	—	2	1	no	—
22	C₅×#466	C₂²	1	2	yes	—
23	C₄×C₅×Q₁₆	—	2	1	no	—
24	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
25	C₄×C₅×D₈	—	2	1	no	—
26	C₄×C₅×D₈	C₂²	1	2	yes	—
27	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
28	C₅×#366	C₂²	1	4	yes	—
29	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₅×#358	C₂²	1	4	yes	—
32	C₅×#357	C₂²	1	4	yes	—
33	#33820	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅×#41	—	4	1	no	—
35	C₂×C₅×Q₁₆	—	2	1	no	—
36	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
37	C₂×C₅×D₈	—	2	1	no	—
38	C₂×C₅×D₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊Q₈	—	2	1	no	—
42	C₅×C₄⋊Q₈	—	2	1	no	—
43	C₅×C₄⋊Q₈	—	2	1	no	—
44	C₅×#127	—	2	1	no	—
45	C₅×#125	—	4	1	no	—
46	C₅×#124	—	4	1	no	—
47	C₅×C₂²⋊Q₈	—	2	2	no	—
48	C₅×C₂²⋊Q₈	—	4	1	no	—
49	C₅×#121	—	2	2	no	—
50	C₅×#121	—	4	1	no	—
51	C₄×C₅×Q₈	—	2	2	no	—
52	C₄×C₅×Q₈	—	4	1	no	—
53	C₄×C₅×D₄	—	2	2	no	—
54	C₄×C₅×D₄	—	4	1	no	—
55	C₅×#117	—	4	1	no	—
56	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
57	C₅×SD₃₂	—	4	4	no	—
58	C₅×M₃₂	—	4	2	no	—
59	C₂×C₅×C₁₆	C₂³	1	4	yes	—
60	C₅×#107	—	2	1	no	—
61	C₅×#107	—	2	1	no	—
62	C₅×#107	C₂³	1	2	yes	—
63	C₅×#107	C₂³	1	2	yes	—
64	C₅×#103	—	2	2	no	—
65	C₅×#103	—	2	2	no	—
66	C₅×#102	—	2	2	no	—
67	C₅×#102	—	2	2	no	—
68	C₄×C₅×C₈	D₄	1	1	yes	characteristic
69	C₄×C₅×C₈	D₄	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	#3170	C₂×C₅	1	1	yes	characteristic, complemented
73	#3169	C₂×C₅	1	1	yes	characteristic
74	#2227	C₂×C₅	1	2	yes	complemented
75	#2224	C₂×C₅	1	2	yes	complemented
76	#2220	C₂×C₅	1	4	yes	complemented
77	#2208	C₂×C₅	1	2	yes	complemented
78	#2207	C₂×C₅	1	2	yes	complemented
79	#2150	C₂×C₅	1	1	yes	characteristic, complemented
80	C₅×#41	—	8	4	no	—
81	C₂×C₅×Q₈	—	2	2	no	—
82	C₂×C₅×Q₈	—	4	1	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×D₄	—	2	2	no	—
85	C₂×C₅×D₄	—	4	2	no	—
86	C₂×C₅×D₄	—	4	1	no	—
87	C₂×C₅×D₄	—	4	1	no	—
88	C₂²×C₄×C₅	—	4	2	no	—
89	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₅×Q₁₆	—	2	2	no	—
93	C₅×Q₁₆	—	2	2	no	—
94	C₅×Q₁₆	—	4	2	no	—
95	C₅×D₈	—	2	2	no	—
96	C₅×D₈	—	2	2	no	—
97	C₅×D₈	—	4	2	no	—
98	C₂×C₅×C₈	—	2	1	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	4	1	no	—
108	C₅×C₄⋊C₄	—	4	1	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₄⋊C₄	—	4	2	no	—
111	C₅×C₄⋊C₄	—	4	2	no	—
112	C₅×C₄⋊C₄	—	4	1	no	—
113	C₅×C₂²⋊C₄	—	4	2	no	—
114	C₅×C₂²⋊C₄	—	4	2	no	—
115	C₅×C₂²⋊C₄	—	4	2	no	—
116	C₄²×C₅	—	4	2	no	—
117	C₄²×C₅	D₈	1	1	yes	characteristic
118	C₄²×C₅	D₈	1	1	yes	characteristic
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	C₅×C₁₆	—	2	4	no	—
122	#533	—	2	1	no	—
123	#532	—	2	1	no	—
124	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
125	C₂×M₃₂	C₂²×C₅	1	2	yes	—
126	#500	C₂²×C₅	1	1	yes	characteristic
127	#500	C₂²×C₅	1	1	yes	characteristic
128	#494	C₂²×C₅	1	1	yes	characteristic
129	#493	C₂²×C₅	1	1	yes	characteristic
130	#487	—	2	1	no	—
131	#486	—	2	1	no	—
132	#467	—	2	1	no	—
133	#467	C₂²×C₅	1	2	yes	—
134	#466	—	2	1	no	—
135	#466	C₂²×C₅	1	2	yes	—
136	C₄×Q₁₆	—	2	1	no	—
137	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
138	C₄×D₈	—	2	1	no	—
139	C₄×D₈	C₂²×C₅	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
141	#366	C₂²×C₅	1	4	yes	—
142	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
143	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
144	#358	C₂²×C₅	1	4	yes	—
145	#357	C₂²×C₅	1	4	yes	—
146	C₂³×C₅	—	4	2	no	—
147	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
148	C₅×Q₈	—	4	2	no	—
149	C₅×Q₈	—	4	2	no	—
150	C₅×Q₈	—	8	2	no	—
151	C₅×Q₈	—	8	2	no	—
152	C₅×D₄	—	4	2	no	—
153	C₅×D₄	—	4	2	no	—
154	C₅×D₄	—	8	2	no	—
155	C₅×D₄	—	8	4	no	—
156	C₅×D₄	—	8	2	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	8	4	no	—
165	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	—	2	1	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
175	C₅×C₈	2₋¹⁺⁴	1	1	yes	characteristic
176	C₂×#41	—	4	1	no	—
177	C₂×Q₁₆	—	2	1	no	—
178	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
179	C₂×D₈	—	2	1	no	—
180	C₂×D₈	C₂³×C₅	1	2	yes	—
181	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	#127	—	2	1	no	—
187	#125	—	4	1	no	—
188	#124	—	4	1	no	—
189	C₂²⋊Q₈	—	2	2	no	—
190	C₂²⋊Q₈	—	4	1	no	—
191	#121	—	2	2	no	—
192	#121	—	4	1	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×Q₈	—	4	1	no	—
195	C₄×D₄	—	2	2	no	—
196	C₄×D₄	—	4	1	no	—
197	#117	—	4	1	no	—
198	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
199	SD₃₂	—	4	4	no	—
200	M₃₂	—	4	2	no	—
201	C₂×C₁₆	C₂³×C₅	1	4	yes	—
202	#107	—	2	1	no	—
203	#107	—	2	1	no	—
204	#107	C₂³×C₅	1	2	yes	—
205	#107	C₂³×C₅	1	2	yes	—
206	#103	—	2	2	no	—
207	#103	—	2	2	no	—
208	#102	—	2	2	no	—
209	#102	—	2	2	no	—
210	C₄×C₈	C₅×D₄	1	1	yes	characteristic
211	C₄×C₈	C₅×D₄	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
213	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	8	2	no	—
218	C₂²×C₅	—	8	2	no	—
219	C₂²×C₅	—	8	2	no	—
220	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	8	2	no	—
225	C₄×C₅	—	8	2	no	—
226	C₄×C₅	—	8	2	no	—
227	C₄×C₅	#509	1	2	yes	—
228	C₄×C₅	#510	1	2	yes	—
229	C₄×C₅	#537	1	1	yes	characteristic
230	C₄×C₅	#575	1	1	yes	characteristic
231	#41	—	8	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
241	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	4	2	no	—
246	D₈	—	2	2	no	—
247	D₈	—	2	2	no	—
248	D₈	—	4	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
253	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
254	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
255	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	C₅×D₈	1	1	yes	characteristic
269	C₄²	C₅×D₈	1	1	yes	characteristic
270	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
271	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
272	C₁₆	—	2	4	no	—
273	C₂×C₅	—	4	1	no	—
274	C₂×C₅	—	16	2	no	—
275	C₂×C₅	#3245	1	1	yes	characteristic
276	C₂×C₅	C₂×#509	1	1	yes	characteristic
277	C₂×C₅	C₂×#510	1	1	yes	characteristic
278	C₂³	—	4	2	no	—
279	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
280	Q₈	—	4	2	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	8	2	no	—
283	Q₈	—	8	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	8	2	no	—
287	D₄	—	8	4	no	—
288	D₄	—	8	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	4	no	—
297	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
298	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
299	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
300	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
307	C₈	C₅×2₋¹⁺⁴	1	1	yes	characteristic
308	C₅	#33820	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₅×#509	1	2	yes	—
323	C₄	C₅×#510	1	2	yes	—
324	C₄	C₅×#537	1	1	yes	characteristic
325	C₄	C₅×#575	1	1	yes	characteristic
326	C₂	—	4	1	no	—
327	C₂	—	16	2	no	—
328	C₂	C₅×#3245	1	1	yes	characteristic
329	C₂	C₂×C₅×#509	1	1	yes	characteristic
330	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#3169	C₂
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. C₅×#3170	C₂	semidirect
3. C₅×#2227	C₂	semidirect
4. C₅×#2224	C₂	semidirect
5. C₅×#2220	C₂	semidirect
6. C₅×#2208	C₂	semidirect
7. C₅×#2207	C₂	semidirect
8. C₅×#2150	C₂	semidirect
33. #33820	C₅	direct (N × H)
72. #3170	C₂×C₅	semidirect
74. #2227	C₂×C₅	semidirect
75. #2224	C₂×C₅	semidirect
76. #2220	C₂×C₅	semidirect
77. #2208	C₂×C₅	semidirect
78. #2207	C₂×C₅	semidirect
79. #2150	C₂×C₅	semidirect
308. C₅	#33820	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×#3170	2
C₅×#3169	2
C₅×#2227	2
C₅×#2224	2
C₅×#2220	2
C₅×#2208	2
C₅×#2207	2
C₅×#2150	2
#33820	5

Subgroups — C5×#33820 #11956540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33820 #11956540