Subgroups — C₅×#12634 #11935354

Order 1280.

452 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	C₅×#2932	C₂	1	1	yes	maximal, characteristic
2	C₅×#2132	C₂	1	1	yes	maximal, characteristic
3	C₅×#2126	C₂	1	1	yes	maximal, characteristic
4	C₅×#1757	C₂	1	2	yes	maximal, complemented
5	C₅×#1395	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₄	1	1	yes	characteristic, complemented
7	C₂×C₅×#131	C₂²	1	1	yes	characteristic
8	C₅×#532	C₂²	1	1	yes	characteristic
9	C₅×#435	—	2	2	no	—
10	C₅×#435	—	2	2	no	—
11	C₅×#423	C₄	1	1	yes	characteristic, complemented
12	C₅×#413	C₂²	1	2	yes	—
13	C₅×#410	C₂²	1	2	yes	—
14	C₅×#408	C₄	1	1	yes	characteristic, complemented
15	C₅×#408	C₂²	1	1	yes	characteristic
16	C₅×#405	—	2	2	no	—
17	C₂×C₅×#97	C₄	1	1	yes	characteristic, complemented
18	C₅×#356	C₄	1	2	yes	complemented
19	C₅×#355	C₄	1	2	yes	complemented
20	C₅×#354	C₄	1	2	yes	—
21	C₅×#353	C₄	1	2	yes	—
22	C₅×#344	—	2	2	no	—
23	#12634	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×#41	C₂³	1	1	yes	characteristic
26	C₅×#131	—	4	2	no	—
27	C₅×#131	—	4	2	no	—
28	C₂×C₅×M₁₆	—	2	1	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	D₄	1	2	yes	—
33	C₂²×C₅×C₈	—	2	1	no	—
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	D₄	1	2	yes	—
37	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₅×#127	C₂×C₄	1	1	yes	characteristic
39	C₅×#124	C₂×C₄	1	1	yes	characteristic
40	C₅×#124	C₂×C₄	1	1	yes	characteristic
41	C₅×#121	—	2	2	no	—
42	C₄×C₅×D₄	—	2	2	no	—
43	C₅×#117	—	2	2	no	—
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×#101	C₂×C₄	1	4	yes	—
47	C₅×#100	—	2	2	no	—
48	C₅×#99	—	2	2	no	—
49	C₅×#99	C₂×C₄	1	4	yes	—
50	C₅×C₂²⋊C₈	—	2	2	no	—
51	C₅×C₂²⋊C₈	—	2	2	no	—
52	C₅×#97	—	2	1	no	—
53	C₅×#97	—	2	1	no	—
54	C₅×#97	—	4	2	no	—
55	C₄×C₅×C₈	—	4	2	no	—
56	#2932	C₂×C₅	1	1	yes	characteristic
57	#2132	C₂×C₅	1	1	yes	characteristic
58	#2126	C₂×C₅	1	1	yes	characteristic
59	#1757	C₂×C₅	1	2	yes	complemented
60	#1395	C₂×C₅	1	2	yes	complemented
61	C₅×#41	—	4	1	no	—
62	C₅×#41	—	4	1	no	—
63	C₅×#41	—	4	1	no	—
64	C₅×#41	—	4	1	no	—
65	C₂×C₅×Q₈	C₄²	1	1	yes	characteristic
66	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₅×D₄	—	2	1	no	—
68	C₂×C₅×D₄	—	2	2	no	—
69	C₂×C₅×D₄	C₄²	1	1	yes	characteristic
70	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₅×D₄	#41	1	2	yes	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	—	4	1	no	—
78	C₅×M₁₆	—	4	2	no	—
79	C₅×M₁₆	—	4	2	no	—
80	C₅×M₁₆	—	4	2	no	—
81	C₅×M₁₆	—	4	1	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	1	no	—
90	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
92	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	—	4	2	no	—
95	C₅×C₂²⋊C₄	—	2	2	no	—
96	C₅×C₂²⋊C₄	—	2	2	no	—
97	C₅×C₂²⋊C₄	—	2	4	no	—
98	C₄²×C₅	—	2	1	no	—
99	C₄²×C₅	—	4	2	no	—
100	C₄²×C₅	C₄²	1	1	yes	characteristic
101	C₄²×C₅	C₄²	1	1	yes	characteristic
102	C₂×#131	C₄×C₅	1	1	yes	characteristic, complemented
103	C₂×#131	C₂²×C₅	1	1	yes	characteristic
104	#532	C₂²×C₅	1	1	yes	characteristic
105	#435	—	2	2	no	—
106	#435	—	2	2	no	—
107	#423	C₄×C₅	1	1	yes	characteristic, complemented
108	#413	C₂²×C₅	1	2	yes	—
109	#410	C₂²×C₅	1	2	yes	—
110	#408	C₄×C₅	1	1	yes	characteristic, complemented
111	#408	C₂²×C₅	1	1	yes	characteristic
112	#405	—	2	2	no	—
113	C₂×#97	C₄×C₅	1	1	yes	characteristic, complemented
114	#356	C₄×C₅	1	2	yes	complemented
115	#355	C₄×C₅	1	2	yes	complemented
116	#354	C₄×C₅	1	2	yes	—
117	#353	C₄×C₅	1	2	yes	—
118	#344	—	2	2	no	—
119	C₂³×C₅	—	2	1	no	—
120	C₂³×C₅	#117	1	1	yes	characteristic
121	C₂³×C₅	C₄×D₄	1	2	yes	—
122	C₅×Q₈	—	4	1	no	—
123	C₅×Q₈	—	4	1	no	—
124	C₅×D₄	—	4	1	no	—
125	C₅×D₄	—	4	2	no	—
126	C₅×D₄	—	4	1	no	—
127	C₅×D₄	—	4	2	no	—
128	C₅×D₄	—	8	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	4	4	no	—
139	C₂×C₄×C₅	—	8	1	no	—
140	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	—	4	4	no	—
146	C₅×C₈	—	4	1	no	—
147	C₅×C₈	—	4	1	no	—
148	C₅×C₈	#99	1	2	yes	—
149	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂×#41	C₂³×C₅	1	1	yes	characteristic
151	#131	—	4	2	no	—
152	#131	—	4	2	no	—
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂×M₁₆	C₅×D₄	1	2	yes	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
161	C₂²×C₈	C₅×D₄	1	2	yes	—
162	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
163	#127	C₂×C₄×C₅	1	1	yes	characteristic
164	#124	C₂×C₄×C₅	1	1	yes	characteristic
165	#124	C₂×C₄×C₅	1	1	yes	characteristic
166	#121	—	2	2	no	—
167	C₄×D₄	—	2	2	no	—
168	#117	—	2	2	no	—
169	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	#101	C₂×C₄×C₅	1	4	yes	—
172	#100	—	2	2	no	—
173	#99	—	2	2	no	—
174	#99	C₂×C₄×C₅	1	4	yes	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	#97	—	2	1	no	—
178	#97	—	2	1	no	—
179	#97	—	4	2	no	—
180	C₄×C₈	—	4	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	1	no	—
183	C₂²×C₅	—	4	1	no	—
184	C₂²×C₅	—	4	2	no	—
185	C₂²×C₅	—	4	2	no	—
186	C₂²×C₅	—	4	1	no	—
187	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	4	1	no	—
191	C₄×C₅	—	4	1	no	—
192	C₄×C₅	—	4	1	no	—
193	C₄×C₅	—	8	4	no	—
194	C₄×C₅	C₂×#99	1	1	yes	characteristic
195	C₄×C₅	#410	1	1	yes	characteristic
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	#41	—	4	1	no	—
199	#41	—	4	1	no	—
200	C₂×Q₈	C₄²×C₅	1	1	yes	characteristic
201	C₂×Q₈	C₂²×C₄×C₅	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	C₄²×C₅	1	1	yes	characteristic
205	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₂×D₄	C₅×#41	1	2	yes	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
227	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	4	2	no	—
235	C₄²	C₄²×C₅	1	1	yes	characteristic
236	C₄²	C₄²×C₅	1	1	yes	characteristic
237	C₂×C₅	—	2	1	no	—
238	C₂×C₅	—	4	2	no	—
239	C₂×C₅	—	4	1	no	—
240	C₂×C₅	—	8	1	no	—
241	C₂×C₅	C₄×#99	1	1	yes	characteristic
242	C₂³	—	2	1	no	—
243	C₂³	C₅×#117	1	1	yes	characteristic
244	C₂³	C₄×C₅×D₄	1	2	yes	—
245	Q₈	—	4	1	no	—
246	Q₈	—	4	1	no	—
247	D₄	—	4	1	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	1	no	—
250	D₄	—	4	2	no	—
251	D₄	—	8	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
264	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₈	—	4	4	no	—
269	C₈	—	4	1	no	—
270	C₈	—	4	1	no	—
271	C₈	C₅×#99	1	2	yes	—
272	C₅	#12634	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	—	2	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	8	4	no	—
286	C₄	C₂×C₅×#99	1	1	yes	characteristic
287	C₄	C₅×#410	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	4	2	no	—
290	C₂	—	4	1	no	—
291	C₂	—	8	1	no	—
292	C₂	C₄×C₅×#99	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×#99
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
socle	C₂×C₅	C₄×#99
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1757	C₂	semidirect
5. C₅×#1395	C₂	semidirect
6. C₂×C₅×#131	C₄	semidirect
11. C₅×#423	C₄	semidirect
14. C₅×#408	C₄	semidirect
17. C₂×C₅×#97	C₄	semidirect
18. C₅×#356	C₄	semidirect
19. C₅×#355	C₄	semidirect
23. #12634	C₅	direct (N × H)
59. #1757	C₂×C₅	semidirect
60. #1395	C₂×C₅	semidirect
102. C₂×#131	C₄×C₅	semidirect
107. #423	C₄×C₅	semidirect
110. #408	C₄×C₅	semidirect
113. C₂×#97	C₄×C₅	semidirect
114. #356	C₄×C₅	semidirect
115. #355	C₄×C₅	semidirect
272. C₅	#12634	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×#2932	2
C₅×#2132	2
C₅×#2126	2
C₅×#1757	2
C₅×#1395	2
#12634	5

Subgroups — C5×#12634 #11935354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12634 #11935354