Subgroups — C₅×#19927 #11942647

Order 1280.

606 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×#3384	C₂	1	2	yes	maximal
2	C₅×#3299	C₂	1	2	yes	maximal, complemented
3	C₅×#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1692	C₂	1	2	yes	maximal, complemented
5	C₅×#1683	C₂	1	2	yes	maximal, complemented
6	C₅×#1649	C₂	1	2	yes	maximal, complemented
7	C₅×#1632	C₂	1	2	yes	maximal, complemented
8	C₅×#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×C₈×D₈	C₂	1	1	yes	maximal, characteristic
10	C₅×#554	C₂²	1	2	yes	—
11	C₅×D₄×Q₈	C₂²	1	2	yes	—
12	C₅×#501	—	2	2	no	—
13	C₅×#500	C₂²	1	1	yes	characteristic
14	C₅×#498	C₂²	1	2	yes	—
15	C₅×#494	C₂²	1	1	yes	characteristic
16	C₅×#487	C₂²	1	2	yes	—
17	C₅×#484	—	2	2	no	—
18	C₅×#478	—	2	2	no	—
19	C₅×#477	C₂²	1	2	yes	—
20	C₅×#476	—	2	2	no	—
21	C₅×#476	C₂²	1	2	yes	—
22	C₅×#474	C₂²	1	2	yes	—
23	C₅×#467	—	2	2	no	—
24	C₅×#462	C₂²	1	2	yes	—
25	C₅×#451	—	2	2	no	—
26	C₅×#446	C₂²	1	2	yes	—
27	C₅×#442	—	2	2	no	—
28	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
30	C₅×C₈×D₄	C₂²	1	2	yes	—
31	C₂×C₅×#103	—	2	2	no	—
32	C₅×#335	C₂²	1	1	yes	characteristic
33	C₅×#335	C₂²	1	1	yes	characteristic
34	C₅×#333	C₂²	1	2	yes	—
35	C₅×#332	C₂²	1	2	yes	—
36	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₅×D₄⋊C₈	C₂²	1	2	yes	—
38	C₅×C₈²	C₂²	1	1	yes	characteristic
39	#19927	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₅×Q₈	—	2	2	no	—
41	C₂×C₅×Q₁₆	—	2	2	no	—
42	C₂×C₅×Q₁₆	—	4	1	no	—
43	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
44	C₂×C₅×D₈	D₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	—	2	2	no	—
46	C₅×C₄⋊Q₈	—	2	2	no	—
47	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₅×#125	—	2	2	no	—
50	C₅×C₂²⋊Q₈	—	2	2	no	—
51	C₅×C₂²⋊Q₈	—	2	2	no	—
52	C₅×C₂²⋊Q₈	—	4	2	no	—
53	C₅×C₂²⋊Q₈	—	4	2	no	—
54	C₄×C₅×Q₈	C₂³	1	2	yes	—
55	C₄×C₅×D₄	—	2	2	no	—
56	C₄×C₅×D₄	—	2	2	no	—
57	C₄×C₅×D₄	C₂³	1	2	yes	—
58	C₂×C₅×C₄⋊C₄	—	2	2	no	—
59	C₅×#107	—	2	2	no	—
60	C₅×#107	—	2	2	no	—
61	C₅×#107	—	4	1	no	—
62	C₅×#107	D₄	1	1	yes	characteristic
63	C₅×#107	D₄	1	1	yes	characteristic
64	C₅×#106	—	2	2	no	—
65	C₅×#106	—	2	2	no	—
66	C₅×#106	—	2	2	no	—
67	C₅×C₄⋊C₈	—	2	2	no	—
68	C₅×C₄⋊C₈	C₂³	1	2	yes	—
69	C₅×C₄⋊C₈	C₂³	1	2	yes	—
70	C₅×#103	—	2	2	no	—
71	C₅×#103	—	2	2	no	—
72	C₅×#103	—	2	2	no	—
73	C₅×#103	—	2	2	no	—
74	C₅×#103	D₄	1	2	yes	—
75	C₅×#102	—	2	2	no	—
76	C₅×#102	—	2	2	no	—
77	C₅×#102	D₄	1	2	yes	—
78	C₅×C₂²⋊C₈	—	2	2	no	—
79	C₅×#97	—	2	2	no	—
80	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
83	#3384	C₂×C₅	1	2	yes	—
84	#3299	C₂×C₅	1	2	yes	complemented
85	#1707	C₂×C₅	1	1	yes	characteristic, complemented
86	#1692	C₂×C₅	1	2	yes	complemented
87	#1683	C₂×C₅	1	2	yes	complemented
88	#1649	C₂×C₅	1	2	yes	complemented
89	#1632	C₂×C₅	1	2	yes	complemented
90	#1591	C₂×C₅	1	1	yes	characteristic, complemented
91	C₈×D₈	C₂×C₅	1	1	yes	characteristic
92	C₂×C₅×Q₈	—	2	2	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	—	4	2	no	—
95	C₂×C₅×Q₈	—	4	4	no	—
96	C₂×C₅×Q₈	—	8	2	no	—
97	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
98	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅	—	2	2	no	—
100	C₂²×C₄×C₅	—	4	2	no	—
101	C₂²×C₄×C₅	—	4	2	no	—
102	C₅×Q₁₆	—	4	1	no	—
103	C₅×Q₁₆	—	8	2	no	—
104	C₅×Q₁₆	—	8	2	no	—
105	C₅×D₈	Q₁₆	1	4	yes	complemented
106	C₂×C₅×C₈	—	2	1	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	8	2	no	—
110	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₄⋊C₄	—	4	2	no	—
120	C₅×C₄⋊C₄	—	4	2	no	—
121	C₅×C₄⋊C₄	—	4	2	no	—
122	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
123	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
124	C₅×C₂²⋊C₄	—	2	2	no	—
125	C₅×C₂²⋊C₄	—	4	2	no	—
126	C₅×C₂²⋊C₄	—	4	2	no	—
127	C₄²×C₅	—	2	2	no	—
128	C₄²×C₅	C₂⁴	1	1	yes	characteristic
129	#554	C₂²×C₅	1	2	yes	—
130	D₄×Q₈	C₂²×C₅	1	2	yes	—
131	#501	—	2	2	no	—
132	#500	C₂²×C₅	1	1	yes	characteristic
133	#498	C₂²×C₅	1	2	yes	—
134	#494	C₂²×C₅	1	1	yes	characteristic
135	#487	C₂²×C₅	1	2	yes	—
136	#484	—	2	2	no	—
137	#478	—	2	2	no	—
138	#477	C₂²×C₅	1	2	yes	—
139	#476	—	2	2	no	—
140	#476	C₂²×C₅	1	2	yes	—
141	#474	C₂²×C₅	1	2	yes	—
142	#467	—	2	2	no	—
143	#462	C₂²×C₅	1	2	yes	—
144	#451	—	2	2	no	—
145	#446	C₂²×C₅	1	2	yes	—
146	#442	—	2	2	no	—
147	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
148	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
149	C₈×D₄	C₂²×C₅	1	2	yes	—
150	C₂×#103	—	2	2	no	—
151	#335	C₂²×C₅	1	1	yes	characteristic
152	#335	C₂²×C₅	1	1	yes	characteristic
153	#333	C₂²×C₅	1	2	yes	—
154	#332	C₂²×C₅	1	2	yes	—
155	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
156	D₄⋊C₈	C₂²×C₅	1	2	yes	—
157	C₈²	C₂²×C₅	1	1	yes	characteristic
158	C₂³×C₅	—	2	2	no	—
159	C₅×Q₈	—	2	2	no	—
160	C₅×Q₈	—	2	2	no	—
161	C₅×Q₈	—	4	2	no	—
162	C₅×Q₈	—	4	2	no	—
163	C₅×Q₈	—	8	2	no	—
164	C₅×Q₈	—	8	4	no	—
165	C₅×D₄	—	2	2	no	—
166	C₅×D₄	C₂×Q₁₆	1	4	yes	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	4	2	no	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	8	4	no	—
175	C₂×C₄×C₅	—	16	2	no	—
176	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
179	C₅×C₈	—	2	2	no	—
180	C₅×C₈	—	2	1	no	—
181	C₅×C₈	—	2	1	no	—
182	C₅×C₈	—	2	1	no	—
183	C₅×C₈	—	4	2	no	—
184	C₅×C₈	—	4	2	no	—
185	C₅×C₈	C₂×Q₁₆	1	2	yes	—
186	C₂²×Q₈	—	2	2	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	—	4	1	no	—
189	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
190	C₂×D₈	C₅×D₄	1	1	yes	characteristic
191	C₂²×C₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
194	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
195	#125	—	2	2	no	—
196	C₂²⋊Q₈	—	2	2	no	—
197	C₂²⋊Q₈	—	2	2	no	—
198	C₂²⋊Q₈	—	4	2	no	—
199	C₂²⋊Q₈	—	4	2	no	—
200	C₄×Q₈	C₂³×C₅	1	2	yes	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	C₂³×C₅	1	2	yes	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	#107	—	2	2	no	—
206	#107	—	2	2	no	—
207	#107	—	4	1	no	—
208	#107	C₅×D₄	1	1	yes	characteristic
209	#107	C₅×D₄	1	1	yes	characteristic
210	#106	—	2	2	no	—
211	#106	—	2	2	no	—
212	#106	—	2	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	C₂³×C₅	1	2	yes	—
215	C₄⋊C₈	C₂³×C₅	1	2	yes	—
216	#103	—	2	2	no	—
217	#103	—	2	2	no	—
218	#103	—	2	2	no	—
219	#103	—	2	2	no	—
220	#103	C₅×D₄	1	2	yes	—
221	#102	—	2	2	no	—
222	#102	—	2	2	no	—
223	#102	C₅×D₄	1	2	yes	—
224	C₂²⋊C₈	—	2	2	no	—
225	#97	—	2	2	no	—
226	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
229	C₂²×C₅	—	2	4	no	—
230	C₂²×C₅	—	4	2	no	—
231	C₂²×C₅	—	4	2	no	—
232	C₂²×C₅	D₄²	1	1	yes	characteristic
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	4	2	no	—
235	C₄×C₅	—	4	2	no	—
236	C₄×C₅	—	4	2	no	—
237	C₄×C₅	—	8	2	no	—
238	C₄×C₅	—	8	2	no	—
239	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
240	C₄×C₅	#575	1	1	yes	characteristic
241	C₄×C₅	#576	1	1	yes	characteristic
242	C₄×C₅	#578	1	1	yes	characteristic
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	4	no	—
247	C₂×Q₈	—	8	2	no	—
248	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
249	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	Q₁₆	—	4	1	no	—
254	Q₁₆	—	8	2	no	—
255	Q₁₆	—	8	2	no	—
256	D₈	C₅×Q₁₆	1	4	yes	complemented
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	8	2	no	—
261	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
262	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
263	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
264	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
274	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	C₂⁴×C₅	1	1	yes	characteristic
280	C₂×C₅	—	4	4	no	—
281	C₂×C₅	#3273	1	1	yes	characteristic
282	C₂×C₅	D₄×Q₁₆	1	1	yes	characteristic
283	C₂×C₅	#3291	1	1	yes	characteristic
284	C₂³	—	2	2	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	8	4	no	—
291	D₄	—	2	2	no	—
292	D₄	C₂×C₅×Q₁₆	1	4	yes	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	—	16	2	no	—
302	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
304	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	C₂×C₅×Q₁₆	1	2	yes	—
312	C₅	#19927	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	4	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₅×D₄²	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	C₂²×C₅×Q₁₆	1	1	yes	characteristic
324	C₄	C₅×#575	1	1	yes	characteristic
325	C₄	C₅×#576	1	1	yes	characteristic
326	C₄	C₅×#578	1	1	yes	characteristic
327	C₂	—	4	4	no	—
328	C₂	C₅×#3273	1	1	yes	characteristic
329	C₂	C₅×D₄×Q₁₆	1	1	yes	characteristic
330	C₂	C₅×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂×C₅×D₈	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#3299	C₂	semidirect
3. C₅×#1707	C₂	semidirect
4. C₅×#1692	C₂	semidirect
5. C₅×#1683	C₂	semidirect
6. C₅×#1649	C₂	semidirect
7. C₅×#1632	C₂	semidirect
8. C₅×#1591	C₂	semidirect
39. #19927	C₅	direct (N × H)
84. #3299	C₂×C₅	semidirect
85. #1707	C₂×C₅	semidirect
86. #1692	C₂×C₅	semidirect
87. #1683	C₂×C₅	semidirect
88. #1649	C₂×C₅	semidirect
89. #1632	C₂×C₅	semidirect
90. #1591	C₂×C₅	semidirect
105. C₅×D₈	Q₁₆	semidirect
256. D₈	C₅×Q₁₆	semidirect
312. C₅	#19927	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×#3384	2
C₅×#3299	2
C₅×#1707	2
C₅×#1692	2
C₅×#1683	2
C₅×#1649	2
C₅×#1632	2
C₅×#1591	2
C₅×C₈×D₈	2
#19927	5

Subgroups — C5×#19927 #11942647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19927 #11942647