Subgroups — C₅×#29799 #11952519

Order 1280.

866 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#462	C₂	1	2	yes	maximal
2	C₅×#2868	C₂	1	1	yes	maximal, characteristic
3	C₅×#2295	C₂	1	1	yes	maximal, characteristic
4	C₅×#2043	C₂	1	4	yes	maximal
5	C₅×#2016	C₂	1	4	yes	maximal
6	C₅×#1981	C₂	1	2	yes	maximal
7	C₅×#1831	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
11	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₅×#462	—	2	8	no	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₅×#103	C₂²	1	4	yes	—
15	C₂×C₅×#103	C₂²	1	8	yes	—
16	C₅×#391	C₂²	1	2	yes	—
17	C₅×#389	C₂²	1	2	yes	—
18	C₅×#387	—	2	2	no	—
19	C₅×#384	—	2	1	no	—
20	C₅×#384	—	2	2	no	—
21	C₅×#384	—	2	1	no	—
22	C₄×C₅×C₄⋊C₄	—	2	1	no	—
23	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₅×#376	—	2	1	no	—
25	C₄³×C₅	C₂²	1	1	yes	characteristic
26	C₅×#336	C₂²	1	4	yes	—
27	#29799	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₅×Q₈	C₂³	1	4	yes	—
29	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
30	C₂×C₅×Q₁₆	—	2	8	no	—
31	C₂×C₅×Q₁₆	—	4	8	no	—
32	C₂²×C₅×C₈	C₂³	1	4	yes	—
33	C₅×C₄⋊Q₈	—	2	4	no	—
34	C₅×C₄⋊Q₈	—	2	4	no	—
35	C₅×C₄⋊Q₈	—	4	8	no	—
36	C₄×C₅×Q₈	—	2	2	no	—
37	C₄×C₅×Q₈	D₄	1	4	yes	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	—	2	4	no	—
41	C₂×C₅×C₄⋊C₄	—	2	4	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅	—	2	1	no	—
47	C₂×C₄²×C₅	—	2	1	no	—
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	D₄	1	2	yes	—
50	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂³	1	2	yes	—
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₅×C₄⋊C₈	—	2	4	no	—
54	C₅×#103	—	2	8	no	—
55	C₅×#103	—	4	8	no	—
56	C₅×#95	—	2	2	no	—
57	C₅×#95	—	2	2	no	—
58	C₅×#95	—	2	2	no	—
59	C₅×#95	—	2	1	no	—
60	C₂×#462	C₂×C₅	1	2	yes	—
61	#2868	C₂×C₅	1	1	yes	characteristic
62	#2295	C₂×C₅	1	1	yes	characteristic
63	#2043	C₂×C₅	1	4	yes	—
64	#2016	C₂×C₅	1	4	yes	—
65	#1981	C₂×C₅	1	2	yes	—
66	#1831	C₂×C₅	1	1	yes	characteristic
67	C₂×C₅×Q₈	—	2	1	no	—
68	C₂×C₅×Q₈	—	2	4	no	—
69	C₂×C₅×Q₈	—	2	4	no	—
70	C₂×C₅×Q₈	—	2	4	no	—
71	C₂×C₅×Q₈	—	2	8	no	—
72	C₂×C₅×Q₈	—	2	8	no	—
73	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	2	4	no	—
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₅×Q₁₆	—	4	16	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	8	no	—
86	C₅×C₄⋊C₄	—	2	4	no	—
87	C₅×C₄⋊C₄	—	2	1	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	4	8	no	—
90	C₅×C₄⋊C₄	—	4	8	no	—
91	C₅×C₄⋊C₄	—	4	4	no	—
92	C₅×C₄⋊C₄	—	4	4	no	—
93	C₅×C₄⋊C₄	—	4	4	no	—
94	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
95	C₄²×C₅	—	2	4	no	—
96	C₄²×C₅	—	2	4	no	—
97	C₄²×C₅	—	2	2	no	—
98	C₄²×C₅	—	2	4	no	—
99	C₄²×C₅	—	4	2	no	—
100	C₄²×C₅	Q₁₆	1	4	yes	complemented
101	C₄²×C₅	C₂×D₄	1	2	yes	—
102	C₄²×C₅	#41	1	2	yes	—
103	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
104	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
105	C₂×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
106	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
107	#462	—	2	8	no	—
108	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
109	C₂×#103	C₂²×C₅	1	4	yes	—
110	C₂×#103	C₂²×C₅	1	8	yes	—
111	#391	C₂²×C₅	1	2	yes	—
112	#389	C₂²×C₅	1	2	yes	—
113	#387	—	2	2	no	—
114	#384	—	2	1	no	—
115	#384	—	2	2	no	—
116	#384	—	2	1	no	—
117	C₄×C₄⋊C₄	—	2	1	no	—
118	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
119	#376	—	2	1	no	—
120	C₄³	C₂²×C₅	1	1	yes	characteristic
121	#336	C₂²×C₅	1	4	yes	—
122	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
123	C₅×Q₈	—	2	4	no	—
124	C₅×Q₈	—	4	2	no	—
125	C₅×Q₈	—	4	4	no	—
126	C₅×Q₈	—	4	4	no	—
127	C₅×Q₈	—	4	8	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	4	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	4	8	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	#121	1	2	yes	—
143	C₂×C₄×C₅	#121	1	2	yes	—
144	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
145	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
146	C₂×C₄×C₅	#137	1	4	yes	—
147	C₂×C₄×C₅	#137	1	2	yes	—
148	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
151	C₅×C₈	—	4	4	no	—
152	C₂²×Q₈	C₂³×C₅	1	4	yes	—
153	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
154	C₂×Q₁₆	—	2	8	no	—
155	C₂×Q₁₆	—	4	8	no	—
156	C₂²×C₈	C₂³×C₅	1	4	yes	—
157	C₄⋊Q₈	—	2	4	no	—
158	C₄⋊Q₈	—	2	4	no	—
159	C₄⋊Q₈	—	4	8	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₄×Q₈	C₅×D₄	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
169	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
170	C₂×C₄²	—	2	1	no	—
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
173	C₂×C₄²	C₅×D₄	1	2	yes	—
174	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₅	1	2	yes	—
176	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
177	C₄⋊C₈	—	2	4	no	—
178	#103	—	2	8	no	—
179	#103	—	4	8	no	—
180	#95	—	2	2	no	—
181	#95	—	2	2	no	—
182	#95	—	2	2	no	—
183	#95	—	2	1	no	—
184	C₂²×C₅	C₂×#121	1	1	yes	characteristic
185	C₂²×C₅	#535	1	2	yes	—
186	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
187	C₂²×C₅	C₂×#137	1	2	yes	—
188	C₂²×C₅	C₂×#137	1	1	yes	characteristic
189	C₄×C₅	—	2	4	no	—
190	C₄×C₅	—	2	4	no	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	4	2	no	—
193	C₄×C₅	—	4	2	no	—
194	C₄×C₅	—	8	4	no	—
195	C₄×C₅	—	8	2	no	—
196	C₄×C₅	#462	1	4	yes	—
197	C₄×C₅	#535	1	2	yes	—
198	C₄×C₅	#540	1	2	yes	—
199	C₂×Q₈	—	2	1	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	8	no	—
204	C₂×Q₈	—	2	8	no	—
205	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
212	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
213	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
214	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
215	Q₁₆	—	4	16	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	8	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	1	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	8	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	4	2	no	—
232	C₄²	C₅×Q₁₆	1	4	yes	complemented
233	C₄²	C₂×C₅×D₄	1	2	yes	—
234	C₄²	C₅×#41	1	2	yes	—
235	C₂×C₅	#2619	1	1	yes	characteristic
236	C₂×C₅	C₂×#462	1	1	yes	characteristic
237	C₂×C₅	#3039	1	1	yes	characteristic
238	C₂×C₅	#3209	1	2	yes	—
239	C₂×C₅	#3218	1	2	yes	—
240	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	4	2	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	4	4	no	—
245	Q₈	—	4	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	C₅×#121	1	2	yes	—
261	C₂×C₄	C₅×#121	1	2	yes	—
262	C₂×C₄	C₂×C₅×Q₁₆	1	4	yes	—
263	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
264	C₂×C₄	C₅×#137	1	4	yes	—
265	C₂×C₄	C₅×#137	1	2	yes	—
266	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
268	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
269	C₈	—	4	4	no	—
270	C₅	#29799	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	C₂×C₅×#121	1	1	yes	characteristic
272	C₂²	C₅×#535	1	2	yes	—
273	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
274	C₂²	C₂×C₅×#137	1	1	yes	characteristic
275	C₂²	C₂×C₅×#137	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	4	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	2	no	—
281	C₄	—	8	4	no	—
282	C₄	—	8	2	no	—
283	C₄	C₅×#462	1	4	yes	—
284	C₄	C₅×#535	1	2	yes	—
285	C₄	C₅×#540	1	2	yes	—
286	C₂	C₅×#2619	1	1	yes	characteristic
287	C₂	C₂×C₅×#462	1	1	yes	characteristic
288	C₂	C₅×#3039	1	1	yes	characteristic
289	C₂	C₅×#3209	1	2	yes	—
290	C₂	C₅×#3218	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
27. #29799	C₅	direct (N × H)
100. C₄²×C₅	Q₁₆	semidirect
232. C₄²	C₅×Q₁₆	semidirect
270. C₅	#29799	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×#462	2
C₅×#2868	2
C₅×#2295	2
C₅×#2043	2
C₅×#2016	2
C₅×#1981	2
C₅×#1831	2
#29799	5

Subgroups — C5×#29799 #11952519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#29799 #11952519