Subgroups — C₅×#21698 #11944418

Order 1280.

1014 conjugacy classes of subgroups.

The subgroup lattice has 294 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
2	C₂×C₅×#441	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×Q₁₆	C₂	1	2	yes	maximal
4	C₅×#2327	C₂	1	2	yes	maximal
5	C₅×#1971	C₂	1	2	yes	maximal
6	C₅×#1969	C₂	1	2	yes	maximal
7	C₅×#1960	C₂	1	2	yes	maximal
8	C₅×#1846	C₂	1	1	yes	maximal, characteristic
9	C₅×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
11	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	—
14	C₅×#494	C₄	1	8	yes	complemented
15	C₅×#441	—	2	8	no	—
16	C₄×C₅×Q₁₆	—	2	8	no	—
17	C₂×C₅×#107	C₂²	1	2	yes	—
18	C₂×C₅×#106	C₂²	1	2	yes	—
19	C₂×C₅×#103	C₂²	1	4	yes	—
20	C₂×C₅×#103	C₂²	1	4	yes	—
21	C₂×C₅×#97	C₂²	1	1	yes	characteristic
22	C₂×C₅×#97	C₂²	1	1	yes	characteristic
23	C₂×C₅×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	C₅×#391	C₂²	1	4	yes	—
28	C₅×#387	—	2	8	no	—
29	C₅×#382	C₂²	1	2	yes	—
30	C₅×#376	C₂²	1	1	yes	characteristic
31	#21698	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅×Q₈	C₂³	1	4	yes	—
33	C₂×C₅×Q₁₆	—	2	4	no	—
34	C₂×C₅×Q₁₆	—	2	4	no	—
35	C₂×C₅×Q₁₆	—	2	4	no	—
36	C₂×C₅×Q₁₆	—	2	4	no	—
37	C₂×C₅×Q₁₆	C₂×C₄	1	8	yes	—
38	C₂×C₅×Q₁₆	C₂×C₄	1	8	yes	—
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₅×C₄⋊Q₈	—	2	4	no	—
44	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	—
45	C₄×C₅×Q₈	—	2	8	no	—
46	C₄×C₅×Q₈	—	2	8	no	—
47	C₂×C₅×C₄⋊C₄	—	2	4	no	—
48	C₂×C₅×C₄⋊C₄	—	2	4	no	—
49	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
50	C₂×C₄²×C₅	—	2	4	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₅×#107	—	2	4	no	—
55	C₅×#106	—	2	4	no	—
56	C₅×#103	—	2	8	no	—
57	C₅×#103	—	2	8	no	—
58	C₅×#97	—	2	2	no	—
59	C₅×#97	—	2	2	no	—
60	C₅×#97	D₄	1	4	yes	—
61	C₄×C₅×C₈	—	2	2	no	—
62	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
63	C₄×C₅×C₈	D₄	1	4	yes	—
64	C₅×#95	—	2	4	no	—
65	C₅×#95	—	2	8	no	—
66	C₅×#95	D₄	1	4	yes	—
67	C₂×#494	C₂×C₅	1	1	yes	characteristic
68	C₂×#441	C₂×C₅	1	2	yes	—
69	C₂×C₄×Q₁₆	C₂×C₅	1	2	yes	—
70	#2327	C₂×C₅	1	2	yes	—
71	#1971	C₂×C₅	1	2	yes	—
72	#1969	C₂×C₅	1	2	yes	—
73	#1960	C₂×C₅	1	2	yes	—
74	#1846	C₂×C₅	1	1	yes	characteristic
75	#1726	C₂×C₅	1	1	yes	characteristic
76	C₂×C₅×Q₈	—	2	4	no	—
77	C₂×C₅×Q₈	—	2	4	no	—
78	C₂×C₅×Q₈	—	2	4	no	—
79	C₂×C₅×Q₈	—	2	4	no	—
80	C₂×C₅×Q₈	—	2	4	no	—
81	C₂×C₅×Q₈	C₂²×C₄	1	8	yes	—
82	C₂²×C₄×C₅	—	2	4	no	—
83	C₂²×C₄×C₅	—	2	4	no	—
84	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₅×Q₁₆	—	2	16	no	—
89	C₅×Q₁₆	—	2	16	no	—
90	C₂×C₅×C₈	—	2	1	no	—
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
97	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₅×C₈	#41	1	2	yes	—
101	C₂×C₅×C₈	#41	1	2	yes	—
102	C₅×C₄⋊C₄	—	2	16	no	—
103	C₅×C₄⋊C₄	—	2	4	no	—
104	C₅×C₄⋊C₄	—	2	4	no	—
105	C₅×C₄⋊C₄	—	4	8	no	—
106	C₄²×C₅	—	2	1	no	—
107	C₄²×C₅	—	2	1	no	—
108	C₄²×C₅	—	4	8	no	—
109	C₄²×C₅	C₂²×C₄	1	2	yes	—
110	C₄²×C₅	C₂×D₄	1	2	yes	—
111	C₄²×C₅	C₂×D₄	1	2	yes	—
112	C₄²×C₅	#41	1	2	yes	—
113	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
114	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
115	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
116	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	—
117	#494	C₄×C₅	1	8	yes	complemented
118	#441	—	2	8	no	—
119	C₄×Q₁₆	—	2	8	no	—
120	C₂×#107	C₂²×C₅	1	2	yes	—
121	C₂×#106	C₂²×C₅	1	2	yes	—
122	C₂×#103	C₂²×C₅	1	4	yes	—
123	C₂×#103	C₂²×C₅	1	4	yes	—
124	C₂×#97	C₂²×C₅	1	1	yes	characteristic
125	C₂×#97	C₂²×C₅	1	1	yes	characteristic
126	C₂×#97	C₂²×C₅	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
130	#391	C₂²×C₅	1	4	yes	—
131	#387	—	2	8	no	—
132	#382	C₂²×C₅	1	2	yes	—
133	#376	C₂²×C₅	1	1	yes	characteristic
134	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₅×Q₈	—	2	8	no	—
136	C₅×Q₈	—	2	8	no	—
137	C₅×Q₈	—	4	4	no	—
138	C₅×Q₈	—	4	4	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	8	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	4	4	no	—
145	C₂×C₄×C₅	—	4	4	no	—
146	C₂×C₄×C₅	—	4	4	no	—
147	C₂×C₄×C₅	—	4	8	no	—
148	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
149	C₂×C₄×C₅	#127	1	4	yes	—
150	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
156	C₅×C₈	—	2	2	no	—
157	C₅×C₈	—	2	2	no	—
158	C₅×C₈	C₄×D₄	1	4	yes	—
159	C₅×C₈	C₄×D₄	1	4	yes	—
160	C₂²×Q₈	C₂³×C₅	1	4	yes	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	2	4	no	—
163	C₂×Q₁₆	—	2	4	no	—
164	C₂×Q₁₆	—	2	4	no	—
165	C₂×Q₁₆	C₂×C₄×C₅	1	8	yes	—
166	C₂×Q₁₆	C₂×C₄×C₅	1	8	yes	—
167	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
171	C₄⋊Q₈	—	2	4	no	—
172	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	—
173	C₄×Q₈	—	2	8	no	—
174	C₄×Q₈	—	2	8	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
178	C₂×C₄²	—	2	4	no	—
179	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
182	#107	—	2	4	no	—
183	#106	—	2	4	no	—
184	#103	—	2	8	no	—
185	#103	—	2	8	no	—
186	#97	—	2	2	no	—
187	#97	—	2	2	no	—
188	#97	C₅×D₄	1	4	yes	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
191	C₄×C₈	C₅×D₄	1	4	yes	—
192	#95	—	2	4	no	—
193	#95	—	2	8	no	—
194	#95	C₅×D₄	1	4	yes	—
195	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂²×C₅	C₂×#127	1	1	yes	characteristic
197	C₂²×C₅	#532	1	1	yes	characteristic
198	C₂²×C₅	#578	1	4	yes	—
199	C₄×C₅	—	2	4	no	—
200	C₄×C₅	—	2	4	no	—
201	C₄×C₅	—	4	8	no	—
202	C₄×C₅	—	4	2	no	—
203	C₄×C₅	—	8	4	no	—
204	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
205	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
206	C₄×C₅	#532	1	1	yes	characteristic
207	C₄×C₅	#532	1	1	yes	characteristic
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	C₂²×C₄×C₅	1	8	yes	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
217	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
220	Q₁₆	—	2	16	no	—
221	Q₁₆	—	2	16	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
228	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
229	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
230	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
231	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
232	C₂×C₈	C₅×#41	1	2	yes	—
233	C₂×C₈	C₅×#41	1	2	yes	—
234	C₄⋊C₄	—	2	16	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	8	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	4	8	no	—
241	C₄²	C₂²×C₄×C₅	1	2	yes	—
242	C₄²	C₂×C₅×D₄	1	2	yes	—
243	C₄²	C₂×C₅×D₄	1	2	yes	—
244	C₄²	C₅×#41	1	2	yes	—
245	C₂×C₅	C₄×#127	1	1	yes	characteristic
246	C₂×C₅	#2950	1	2	yes	—
247	C₂×C₅	#3156	1	2	yes	—
248	C₂×C₅	#3177	1	2	yes	—
249	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
250	Q₈	—	2	8	no	—
251	Q₈	—	2	8	no	—
252	Q₈	—	4	4	no	—
253	Q₈	—	4	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
264	C₂×C₄	C₅×#127	1	4	yes	—
265	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	2	no	—
273	C₈	C₄×C₅×D₄	1	4	yes	—
274	C₈	C₄×C₅×D₄	1	4	yes	—
275	C₅	#21698	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
277	C₂²	C₂×C₅×#127	1	1	yes	characteristic
278	C₂²	C₅×#532	1	1	yes	characteristic
279	C₂²	C₅×#578	1	4	yes	—
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	4	8	no	—
283	C₄	—	4	2	no	—
284	C₄	—	8	4	no	—
285	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
286	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
287	C₄	C₅×#532	1	1	yes	characteristic
288	C₄	C₅×#532	1	1	yes	characteristic
289	C₂	C₄×C₅×#127	1	1	yes	characteristic
290	C₂	C₅×#2950	1	2	yes	—
291	C₂	C₅×#3156	1	2	yes	—
292	C₂	C₅×#3177	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
14. C₅×#494	C₄	semidirect
31. #21698	C₅	direct (N × H)
117. #494	C₄×C₅	semidirect
275. C₅	#21698	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×#441	2
C₂×C₄×C₅×Q₁₆	2
C₅×#2327	2
C₅×#1971	2
C₅×#1969	2
C₅×#1960	2
C₅×#1846	2
C₅×#1726	2
#21698	5

Subgroups — C5×#21698 #11944418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21698 #11944418