Subgroups — C₅×#29808 #11952528

Order 1280.

914 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
3	C₅×#2857	C₂	1	2	yes	maximal
4	C₅×#2056	C₂	1	4	yes	maximal, complemented
5	C₅×#2001	C₂	1	4	yes	maximal
6	C₅×#1956	C₂	1	2	yes	maximal, complemented
7	C₅×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#494	—	2	4	no	—
13	C₅×#431	—	2	4	no	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×#103	C₂²	1	8	yes	—
16	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#391	C₂²	1	4	yes	—
20	C₅×#389	—	2	2	no	—
21	C₅×#389	—	2	2	no	—
22	C₅×#384	C₂²	1	2	yes	—
23	C₅×#380	—	2	2	no	—
24	C₅×#380	—	2	2	no	—
25	C₅×#340	C₂²	1	4	yes	—
26	#29808	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₅×Q₈	C₂³	1	4	yes	—
28	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₅×Q₁₆	—	2	8	no	—
32	C₂×C₅×Q₁₆	—	4	8	no	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₅×C₄⋊Q₈	—	2	4	no	—
36	C₅×C₄⋊Q₈	—	2	4	no	—
37	C₅×C₂²⋊Q₈	—	4	8	no	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	4	no	—
43	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
44	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₄²×C₅	—	2	2	no	—
46	C₂×C₄²×C₅	—	2	2	no	—
47	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₅×C₄⋊C₈	—	2	4	no	—
52	C₅×#103	—	4	8	no	—
53	C₅×C₂²⋊C₈	—	2	4	no	—
54	C₄×C₅×C₈	—	2	2	no	—
55	C₄×C₅×C₈	D₄	1	4	yes	—
56	C₅×#95	—	2	4	no	—
57	C₅×#95	—	2	4	no	—
58	C₂×#494	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×#431	C₂×C₅	1	1	yes	characteristic
60	#2857	C₂×C₅	1	2	yes	—
61	#2056	C₂×C₅	1	4	yes	complemented
62	#2001	C₂×C₅	1	4	yes	—
63	#1956	C₂×C₅	1	2	yes	complemented
64	#1834	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₅×Q₈	—	2	4	no	—
67	C₂×C₅×Q₈	—	2	4	no	—
68	C₂×C₅×Q₈	—	2	8	no	—
69	C₂×C₅×Q₈	—	2	8	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	4	no	—
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	—	2	4	no	—
77	C₂²×C₄×C₅	—	2	4	no	—
78	C₂²×C₄×C₅	—	2	4	no	—
79	C₂²×C₄×C₅	Q₁₆	1	4	yes	complemented
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₅×Q₁₆	—	4	16	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
91	C₅×C₄⋊C₄	—	2	8	no	—
92	C₅×C₄⋊C₄	—	4	8	no	—
93	C₅×C₄⋊C₄	—	4	4	no	—
94	C₅×C₄⋊C₄	—	4	4	no	—
95	C₅×C₄⋊C₄	—	4	4	no	—
96	C₅×C₄⋊C₄	—	4	4	no	—
97	C₅×C₂²⋊C₄	—	4	8	no	—
98	C₄²×C₅	—	2	2	no	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	—	2	1	no	—
101	C₄²×C₅	—	2	1	no	—
102	C₄²×C₅	C₂×D₄	1	2	yes	—
103	C₄²×C₅	#41	1	2	yes	—
104	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
105	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
106	C₂×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
107	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
108	#494	—	2	4	no	—
109	#431	—	2	4	no	—
110	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
111	C₂×#103	C₂²×C₅	1	8	yes	—
112	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	#391	C₂²×C₅	1	4	yes	—
116	#389	—	2	2	no	—
117	#389	—	2	2	no	—
118	#384	C₂²×C₅	1	2	yes	—
119	#380	—	2	2	no	—
120	#380	—	2	2	no	—
121	#340	C₂²×C₅	1	4	yes	—
122	C₂³×C₅	—	2	1	no	—
123	C₂³×C₅	—	2	2	no	—
124	C₂³×C₅	—	2	1	no	—
125	C₂³×C₅	—	2	2	no	—
126	C₂³×C₅	C₂×Q₁₆	1	2	yes	—
127	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₅×Q₈	—	4	8	no	—
129	C₅×Q₈	—	4	8	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	8	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	4	4	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	—	4	4	no	—
142	C₂×C₄×C₅	—	4	4	no	—
143	C₂×C₄×C₅	—	4	4	no	—
144	C₂×C₄×C₅	—	4	4	no	—
145	C₂×C₄×C₅	—	4	4	no	—
146	C₂×C₄×C₅	#121	1	2	yes	—
147	C₂×C₄×C₅	#121	1	2	yes	—
148	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
149	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
150	C₂×C₄×C₅	#135	1	2	yes	—
151	C₂×C₄×C₅	#137	1	4	yes	—
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
155	C₅×C₈	—	2	4	no	—
156	C₅×C₈	—	4	2	no	—
157	C₂²×Q₈	C₂³×C₅	1	4	yes	—
158	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
159	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
160	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
161	C₂×Q₁₆	—	2	8	no	—
162	C₂×Q₁₆	—	4	8	no	—
163	C₂²×C₈	C₂³×C₅	1	2	yes	—
164	C₂²×C₈	C₂³×C₅	1	2	yes	—
165	C₄⋊Q₈	—	2	4	no	—
166	C₄⋊Q₈	—	2	4	no	—
167	C₂²⋊Q₈	—	4	8	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
174	C₂×C₂²⋊C₄	—	2	4	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
178	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
181	C₄⋊C₈	—	2	4	no	—
182	#103	—	4	8	no	—
183	C₂²⋊C₈	—	2	4	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₅×D₄	1	4	yes	—
186	#95	—	2	4	no	—
187	#95	—	2	4	no	—
188	C₂²×C₅	—	2	4	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	4	1	no	—
191	C₂²×C₅	—	4	2	no	—
192	C₂²×C₅	—	4	1	no	—
193	C₂²×C₅	C₂×#121	1	1	yes	characteristic
194	C₂²×C₅	#535	1	2	yes	—
195	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
196	C₂²×C₅	C₂×#135	1	1	yes	characteristic
197	C₂²×C₅	C₂×#137	1	2	yes	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	4	1	no	—
200	C₄×C₅	—	4	1	no	—
201	C₄×C₅	—	8	4	no	—
202	C₄×C₅	—	8	4	no	—
203	C₄×C₅	#467	1	4	yes	—
204	C₄×C₅	#470	1	4	yes	—
205	C₄×C₅	#535	1	2	yes	—
206	C₄×C₅	#540	1	2	yes	—
207	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	8	no	—
211	C₂×Q₈	—	2	8	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₅×Q₁₆	1	4	yes	complemented
222	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
228	Q₁₆	—	4	16	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	4	8	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₂²⋊C₄	—	4	8	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	C₂×C₅×D₄	1	2	yes	—
245	C₄²	C₅×#41	1	2	yes	—
246	C₂×C₅	—	4	2	no	—
247	C₂×C₅	#2619	1	1	yes	characteristic
248	C₂×C₅	C₂×#467	1	1	yes	characteristic
249	C₂×C₅	C₂×#470	1	1	yes	characteristic
250	C₂×C₅	#3209	1	2	yes	—
251	C₂×C₅	#3210	1	2	yes	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	2	no	—
256	C₂³	C₂×C₅×Q₁₆	1	2	yes	—
257	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
258	Q₈	—	4	8	no	—
259	Q₈	—	4	8	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	C₅×#121	1	2	yes	—
277	C₂×C₄	C₅×#121	1	2	yes	—
278	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
279	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
280	C₂×C₄	C₅×#135	1	2	yes	—
281	C₂×C₄	C₅×#137	1	4	yes	—
282	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
284	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
285	C₈	—	2	4	no	—
286	C₈	—	4	2	no	—
287	C₅	#29808	1	1	yes	characteristic, direct product factor, Sylow
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₂×C₅×#121	1	1	yes	characteristic
294	C₂²	C₅×#535	1	2	yes	—
295	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
296	C₂²	C₂×C₅×#135	1	1	yes	characteristic
297	C₂²	C₂×C₅×#137	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	8	4	no	—
302	C₄	—	8	4	no	—
303	C₄	C₅×#467	1	4	yes	—
304	C₄	C₅×#470	1	4	yes	—
305	C₄	C₅×#535	1	2	yes	—
306	C₄	C₅×#540	1	2	yes	—
307	C₂	—	4	2	no	—
308	C₂	C₅×#2619	1	1	yes	characteristic
309	C₂	C₂×C₅×#467	1	1	yes	characteristic
310	C₂	C₂×C₅×#470	1	1	yes	characteristic
311	C₂	C₅×#3209	1	2	yes	—
312	C₂	C₅×#3210	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#494	C₂	semidirect
4. C₅×#2056	C₂	semidirect
6. C₅×#1956	C₂	semidirect
7. C₅×#1834	C₂	semidirect
26. #29808	C₅	direct (N × H)
58. C₂×#494	C₂×C₅	semidirect
61. #2056	C₂×C₅	semidirect
63. #1956	C₂×C₅	semidirect
64. #1834	C₂×C₅	semidirect
79. C₂²×C₄×C₅	Q₁₆	semidirect
221. C₂²×C₄	C₅×Q₁₆	semidirect
287. C₅	#29808	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×#494	2
C₂×C₅×#431	2
C₅×#2857	2
C₅×#2056	2
C₅×#2001	2
C₅×#1956	2
C₅×#1834	2
#29808	5

Subgroups — C5×#29808 #11952528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#29808 #11952528