Subgroups — C₅×#33843 #11956563

Order 1280.

668 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#510	C₂	1	2	yes	maximal, complemented
2	C₅×#3285	C₂	1	2	yes	maximal, complemented
3	C₅×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2220	C₂	1	2	yes	maximal, complemented
5	C₅×#2211	C₂	1	2	yes	maximal, complemented
6	C₅×#2188	C₂	1	2	yes	maximal, complemented
7	C₅×#2186	C₂	1	2	yes	maximal, complemented
8	C₄×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2169	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#135	C₂²	1	2	yes	—
11	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
12	C₅×#548	—	2	2	no	—
13	C₅×#510	—	2	8	no	—
14	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
18	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
19	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₅×M₃₂	C₂²	1	2	yes	—
21	C₅×#495	C₂²	1	2	yes	—
22	C₅×#467	C₂²	1	2	yes	—
23	C₅×#466	C₂²	1	2	yes	—
24	C₅×#464	—	2	2	no	—
25	C₅×#452	—	2	2	no	—
26	C₅×#448	—	2	2	no	—
27	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
28	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
29	C₅×#366	C₂²	1	1	yes	characteristic
30	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₅×#358	C₂²	1	2	yes	—
32	C₅×#358	C₂²	1	2	yes	—
33	C₅×#357	C₂²	1	2	yes	—
34	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
36	#33843	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	2	no	—
38	C₂²×C₅×Q₈	—	2	2	no	—
39	C₅×#135	—	2	4	no	—
40	C₅×#135	—	4	2	no	—
41	C₂×C₅×Q₁₆	—	2	4	no	—
42	C₂×C₅×Q₁₆	—	2	2	no	—
43	C₂×C₅×Q₁₆	—	4	2	no	—
44	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
45	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
46	C₂×C₅×SD₁₆	—	2	2	no	—
47	C₂×C₅×SD₁₆	—	2	2	no	—
48	C₂×C₅×D₈	C₂³	1	2	yes	—
49	C₂²×C₅×C₈	C₂³	1	2	yes	—
50	C₅×#124	—	2	2	no	—
51	C₅×#124	—	4	2	no	—
52	C₅×C₂²⋊Q₈	—	2	2	no	—
53	C₅×C₂²⋊Q₈	—	4	2	no	—
54	C₅×#121	—	2	2	no	—
55	C₅×#121	—	4	2	no	—
56	C₄×C₅×Q₈	—	2	2	no	—
57	C₄×C₅×D₄	—	4	2	no	—
58	C₄×C₅×D₄	D₄	1	1	yes	characteristic
59	C₅×Q₃₂	—	2	4	no	—
60	C₅×Q₃₂	—	4	1	no	—
61	C₅×Q₃₂	D₄	1	4	yes	complemented
62	C₅×SD₃₂	—	2	4	no	—
63	C₅×SD₃₂	—	2	4	no	—
64	C₅×D₁₆	—	4	1	no	—
65	C₅×M₃₂	—	2	4	no	—
66	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₅×C₁₆	C₂³	1	2	yes	—
68	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
69	C₅×#107	C₂³	1	2	yes	—
70	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₅×#103	—	2	2	no	—
72	C₅×#103	—	2	2	no	—
73	C₅×#102	—	2	2	no	—
74	C₅×C₂²⋊C₈	D₄	1	2	yes	—
75	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₂×#510	C₂×C₅	1	2	yes	complemented
77	#3285	C₂×C₅	1	2	yes	complemented
78	#2249	C₂×C₅	1	1	yes	characteristic, complemented
79	#2220	C₂×C₅	1	2	yes	complemented
80	#2211	C₂×C₅	1	2	yes	complemented
81	#2188	C₂×C₅	1	2	yes	complemented
82	#2186	C₂×C₅	1	2	yes	complemented
83	C₄×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
84	#2169	C₂×C₅	1	1	yes	characteristic, complemented
85	C₅×#41	—	4	4	no	—
86	C₅×#41	—	8	2	no	—
87	C₂×C₅×Q₈	—	2	2	no	—
88	C₂×C₅×Q₈	—	2	2	no	—
89	C₂×C₅×Q₈	—	4	4	no	—
90	C₂×C₅×Q₈	—	4	2	no	—
91	C₂×C₅×Q₈	—	8	2	no	—
92	C₂×C₅×D₄	—	2	2	no	—
93	C₂×C₅×D₄	—	4	2	no	—
94	C₂×C₅×D₄	—	4	2	no	—
95	C₂×C₅×D₄	D₈	1	1	yes	characteristic
96	C₂²×C₄×C₅	—	4	2	no	—
97	C₂²×C₄×C₅	—	4	2	no	—
98	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
99	C₅×Q₁₆	—	2	2	no	—
100	C₅×Q₁₆	—	2	2	no	—
101	C₅×Q₁₆	—	2	2	no	—
102	C₅×Q₁₆	—	4	4	no	—
103	C₅×Q₁₆	C₂×D₄	1	4	yes	—
104	C₅×SD₁₆	—	4	4	no	—
105	C₅×SD₁₆	—	8	2	no	—
106	C₅×D₈	—	2	2	no	—
107	C₅×D₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	4	2	no	—
115	C₅×C₄⋊C₄	—	4	2	no	—
116	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
117	C₅×C₂²⋊C₄	—	4	2	no	—
118	C₅×C₂²⋊C₄	—	4	2	no	—
119	C₅×C₂²⋊C₄	—	4	2	no	—
120	C₅×C₂²⋊C₄	—	4	2	no	—
121	C₅×C₂²⋊C₄	D₈	1	2	yes	—
122	C₄²×C₅	—	4	2	no	—
123	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
124	C₅×C₁₆	—	2	2	no	—
125	C₅×C₁₆	—	2	1	no	—
126	C₅×C₁₆	C₂×D₄	1	2	yes	—
127	C₂×#135	C₂²×C₅	1	2	yes	—
128	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
129	#548	—	2	2	no	—
130	#510	—	2	8	no	—
131	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
132	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic
133	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic
134	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
135	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
136	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
137	C₂×M₃₂	C₂²×C₅	1	2	yes	—
138	#495	C₂²×C₅	1	2	yes	—
139	#467	C₂²×C₅	1	2	yes	—
140	#466	C₂²×C₅	1	2	yes	—
141	#464	—	2	2	no	—
142	#452	—	2	2	no	—
143	#448	—	2	2	no	—
144	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
145	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
146	#366	C₂²×C₅	1	1	yes	characteristic
147	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
148	#358	C₂²×C₅	1	2	yes	—
149	#358	C₂²×C₅	1	2	yes	—
150	#357	C₂²×C₅	1	2	yes	—
151	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
152	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
153	C₂³×C₅	—	4	2	no	—
154	C₂³×C₅	C₂×D₈	1	2	yes	—
155	C₅×Q₈	—	2	4	no	—
156	C₅×Q₈	—	4	2	no	—
157	C₅×Q₈	—	4	2	no	—
158	C₅×Q₈	—	4	2	no	—
159	C₅×Q₈	—	8	4	no	—
160	C₅×D₄	—	4	1	no	—
161	C₅×D₄	—	4	2	no	—
162	C₅×D₄	—	4	2	no	—
163	C₅×D₄	—	8	4	no	—
164	C₅×D₄	—	8	4	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	4	2	no	—
168	C₂×C₄×C₅	—	4	2	no	—
169	C₂×C₄×C₅	—	4	2	no	—
170	C₂×C₄×C₅	—	8	4	no	—
171	C₂×C₄×C₅	—	8	4	no	—
172	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
175	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
176	C₅×C₈	—	2	2	no	—
177	C₅×C₈	—	2	1	no	—
178	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
179	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
180	C₂×#41	—	2	2	no	—
181	C₂²×Q₈	—	2	2	no	—
182	#135	—	2	4	no	—
183	#135	—	4	2	no	—
184	C₂×Q₁₆	—	2	4	no	—
185	C₂×Q₁₆	—	2	2	no	—
186	C₂×Q₁₆	—	4	2	no	—
187	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
188	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
189	C₂×SD₁₆	—	2	2	no	—
190	C₂×SD₁₆	—	2	2	no	—
191	C₂×D₈	C₂³×C₅	1	2	yes	—
192	C₂²×C₈	C₂³×C₅	1	2	yes	—
193	#124	—	2	2	no	—
194	#124	—	4	2	no	—
195	C₂²⋊Q₈	—	2	2	no	—
196	C₂²⋊Q₈	—	4	2	no	—
197	#121	—	2	2	no	—
198	#121	—	4	2	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×D₄	—	4	2	no	—
201	C₄×D₄	C₅×D₄	1	1	yes	characteristic
202	Q₃₂	—	2	4	no	—
203	Q₃₂	—	4	1	no	—
204	Q₃₂	C₅×D₄	1	4	yes	complemented
205	SD₃₂	—	2	4	no	—
206	SD₃₂	—	2	4	no	—
207	D₁₆	—	4	1	no	—
208	M₃₂	—	2	4	no	—
209	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
210	C₂×C₁₆	C₂³×C₅	1	2	yes	—
211	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
212	#107	C₂³×C₅	1	2	yes	—
213	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
214	#103	—	2	2	no	—
215	#103	—	2	2	no	—
216	#102	—	2	2	no	—
217	C₂²⋊C₈	C₅×D₄	1	2	yes	—
218	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	—	2	2	no	—
221	C₂²×C₅	—	2	2	no	—
222	C₂²×C₅	—	8	2	no	—
223	C₂²×C₅	—	8	2	no	—
224	C₂²×C₅	—	8	2	no	—
225	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	—	4	4	no	—
230	C₄×C₅	—	8	2	no	—
231	C₄×C₅	—	8	2	no	—
232	C₄×C₅	D₄²	1	1	yes	characteristic
233	C₄×C₅	#576	1	1	yes	characteristic
234	#41	—	4	4	no	—
235	#41	—	8	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	C₅×D₈	1	1	yes	characteristic
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	4	4	no	—
252	Q₁₆	C₂×C₅×D₄	1	4	yes	—
253	SD₁₆	—	4	4	no	—
254	SD₁₆	—	8	2	no	—
255	D₈	—	2	2	no	—
256	D₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
260	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
261	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	C₅×D₈	1	2	yes	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
273	C₁₆	—	2	2	no	—
274	C₁₆	—	2	1	no	—
275	C₁₆	C₂×C₅×D₄	1	2	yes	—
276	C₂×C₅	—	4	2	no	—
277	C₂×C₅	—	16	2	no	—
278	C₂×C₅	D₄×D₈	1	1	yes	characteristic
279	C₂×C₅	#3414	1	1	yes	characteristic
280	C₂×C₅	#3417	1	1	yes	characteristic
281	C₂³	—	4	2	no	—
282	C₂³	C₂×C₅×D₈	1	2	yes	—
283	Q₈	—	2	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	8	4	no	—
288	D₄	—	4	1	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	4	no	—
292	D₄	—	8	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
303	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
307	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
308	C₅	#33843	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	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	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₅×D₄²	1	1	yes	characteristic
323	C₄	C₅×#576	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	16	2	no	—
326	C₂	C₅×D₄×D₈	1	1	yes	characteristic
327	C₂	C₅×#3414	1	1	yes	characteristic
328	C₂	C₅×#3417	1	1	yes	characteristic

	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 ◁ C₄×C₅×D₄ ◁ C₅×C₈×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#510	C₂	semidirect
2. C₅×#3285	C₂	semidirect
3. C₅×#2249	C₂	semidirect
4. C₅×#2220	C₂	semidirect
5. C₅×#2211	C₂	semidirect
6. C₅×#2188	C₂	semidirect
7. C₅×#2186	C₂	semidirect
8. C₄×C₅×Q₃₂	C₂	semidirect
9. C₅×#2169	C₂	semidirect
36. #33843	C₅	direct (N × H)
61. C₅×Q₃₂	D₄	semidirect
76. C₂×#510	C₂×C₅	semidirect
77. #3285	C₂×C₅	semidirect
78. #2249	C₂×C₅	semidirect
79. #2220	C₂×C₅	semidirect
80. #2211	C₂×C₅	semidirect
81. #2188	C₂×C₅	semidirect
82. #2186	C₂×C₅	semidirect
83. C₄×Q₃₂	C₂×C₅	semidirect
84. #2169	C₂×C₅	semidirect
204. Q₃₂	C₅×D₄	semidirect
308. C₅	#33843	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×#510	2
C₅×#3285	2
C₅×#2249	2
C₅×#2220	2
C₅×#2211	2
C₅×#2188	2
C₅×#2186	2
C₄×C₅×Q₃₂	2
C₅×#2169	2
#33843	5

Subgroups — C5×#33843 #11956563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33843 #11956563