Subgroups — C₅×#35346 #11958066

Order 1280.

7198 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×#120	C₂	1	3	yes	maximal, complemented
3	C₅×#2404	C₂	1	12	yes	maximal, complemented
4	C₅×#2403	C₂	1	12	yes	maximal, complemented
5	C₅×#2308	C₂	1	3	yes	maximal, complemented
6	C₂⁶×C₅	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₅×H(5,𝔽₂)	—	2	24	no	—
8	C₂²×C₅×#41	C₂²	1	6	yes	complemented
9	C₂³×C₅×D₄	C₂²	1	3	yes	—
10	C₂³×C₅×D₄	C₂²	1	6	yes	complemented
11	C₂³×C₅×D₄	C₂²	1	6	yes	complemented
12	C₂×C₅×#120	—	2	24	no	—
13	C₂×C₅×#120	C₂²	1	24	yes	complemented
14	C₂×C₅×#120	C₂²	1	8	yes	complemented
15	C₂²×C₅×C₂²⋊C₄	C₂²	1	3	yes	complemented
16	C₂²×C₅×C₂²⋊C₄	C₂²	1	6	yes	complemented
17	C₅×#393	C₂²	1	8	yes	complemented
18	C₅×#392	C₂²	1	24	yes	complemented
19	C₅×#386	C₂²	1	24	yes	complemented
20	C₅×#386	C₂²	1	24	yes	complemented
21	C₅×#381	C₂²	1	6	yes	complemented
22	C₅×#379	C₂²	1	6	yes	complemented
23	#35346	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁵×C₅	—	2	12	no	—
25	C₂⁵×C₅	—	4	8	no	—
26	C₂⁵×C₅	C₂³	1	6	yes	—
27	C₂⁵×C₅	C₂³	1	3	yes	—
28	C₂⁵×C₅	C₂³	1	1	yes	characteristic
29	C₂⁵×C₅	C₂³	1	2	yes	complemented
30	C₅×H(5,𝔽₂)	—	2	128	no	—
31	C₂×C₅×#41	—	2	48	no	—
32	C₂×C₅×#41	—	2	24	no	—
33	C₂²×C₅×Q₈	C₂³	1	2	yes	complemented
34	C₂²×C₅×D₄	—	2	24	no	—
35	C₂²×C₅×D₄	—	2	24	no	—
36	C₂²×C₅×D₄	—	2	48	no	—
37	C₂²×C₅×D₄	—	2	24	no	—
38	C₂²×C₅×D₄	—	4	24	no	—
39	C₂²×C₅×D₄	D₄	1	24	yes	complemented
40	C₂²×C₅×D₄	C₂³	1	24	yes	—
41	C₂²×C₅×D₄	C₂³	1	12	yes	—
42	C₂²×C₅×D₄	C₂³	1	6	yes	complemented
43	C₂³×C₄×C₅	C₂³	1	3	yes	—
44	C₂³×C₄×C₅	C₂³	1	6	yes	—
45	C₂³×C₄×C₅	C₂³	1	6	yes	complemented
46	C₅×#120	—	2	96	no	—
47	C₅×#120	—	4	16	no	—
48	C₂×C₅×C₂²⋊C₄	—	2	12	no	—
49	C₂×C₅×C₂²⋊C₄	—	2	24	no	—
50	C₂×C₅×C₂²⋊C₄	C₂³	1	12	yes	—
51	C₂×C₅×C₂²⋊C₄	C₂³	1	24	yes	—
52	C₂×C₅×C₂²⋊C₄	C₂³	1	24	yes	complemented
53	C₅×#95	C₂³	1	24	yes	complemented
54	C₂²×H(5,𝔽₂)	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂²×#120	C₂×C₅	1	3	yes	complemented
56	#2404	C₂×C₅	1	12	yes	complemented
57	#2403	C₂×C₅	1	12	yes	complemented
58	#2308	C₂×C₅	1	3	yes	complemented
59	C₂⁴×C₅	—	2	12	no	—
60	C₂⁴×C₅	—	2	24	no	—
61	C₂⁴×C₅	—	2	12	no	—
62	C₂⁴×C₅	—	2	12	no	—
63	C₂⁴×C₅	—	2	24	no	—
64	C₂⁴×C₅	—	4	4	no	—
65	C₂⁴×C₅	—	4	12	no	—
66	C₂⁴×C₅	—	4	12	no	—
67	C₂⁴×C₅	—	4	8	no	—
68	C₂⁴×C₅	—	4	96	no	—
69	C₂⁴×C₅	—	4	16	no	—
70	C₂⁴×C₅	C₂×D₄	1	12	yes	—
71	C₂⁴×C₅	C₂×D₄	1	12	yes	complemented
72	C₂⁴×C₅	C₂⁴	1	6	yes	—
73	C₂⁴×C₅	C₂⁴	1	4	yes	—
74	C₂⁴×C₅	C₂⁴	1	3	yes	—
75	C₅×#41	—	2	192	no	—
76	C₂×C₅×Q₈	—	2	12	no	—
77	C₂×C₅×D₄	—	2	192	no	—
78	C₂×C₅×D₄	—	2	96	no	—
79	C₂×C₅×D₄	—	2	24	no	—
80	C₂×C₅×D₄	—	2	24	no	—
81	C₂×C₅×D₄	—	2	24	no	—
82	C₂×C₅×D₄	—	2	48	no	—
83	C₂×C₅×D₄	—	2	48	no	—
84	C₂×C₅×D₄	—	2	12	no	—
85	C₂×C₅×D₄	—	4	24	no	—
86	C₂×C₅×D₄	—	4	96	no	—
87	C₂×C₅×D₄	C₂×D₄	1	48	yes	complemented
88	C₂²×C₄×C₅	—	2	12	no	—
89	C₂²×C₄×C₅	—	2	12	no	—
90	C₂²×C₄×C₅	—	2	24	no	—
91	C₂²×C₄×C₅	—	2	12	no	—
92	C₂²×C₄×C₅	—	4	12	no	—
93	C₂²×C₄×C₅	C₂×D₄	1	12	yes	complemented
94	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
95	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
96	C₅×C₂²⋊C₄	—	2	24	no	—
97	C₅×C₂²⋊C₄	—	2	48	no	—
98	C₅×C₂²⋊C₄	—	4	24	no	—
99	C₂⁶	C₂²×C₅	1	1	yes	characteristic, complemented
100	C₂×H(5,𝔽₂)	—	2	24	no	—
101	C₂²×#41	C₂²×C₅	1	6	yes	complemented
102	C₂³×D₄	C₂²×C₅	1	3	yes	—
103	C₂³×D₄	C₂²×C₅	1	6	yes	complemented
104	C₂³×D₄	C₂²×C₅	1	6	yes	complemented
105	C₂×#120	—	2	24	no	—
106	C₂×#120	C₂²×C₅	1	24	yes	complemented
107	C₂×#120	C₂²×C₅	1	8	yes	complemented
108	C₂²×C₂²⋊C₄	C₂²×C₅	1	3	yes	complemented
109	C₂²×C₂²⋊C₄	C₂²×C₅	1	6	yes	complemented
110	#393	C₂²×C₅	1	8	yes	complemented
111	#392	C₂²×C₅	1	24	yes	complemented
112	#386	C₂²×C₅	1	24	yes	complemented
113	#386	C₂²×C₅	1	24	yes	complemented
114	#381	C₂²×C₅	1	6	yes	complemented
115	#379	C₂²×C₅	1	6	yes	complemented
116	C₂³×C₅	—	2	12	no	—
117	C₂³×C₅	—	2	48	no	—
118	C₂³×C₅	—	2	16	no	—
119	C₂³×C₅	—	2	12	no	—
120	C₂³×C₅	—	2	12	no	—
121	C₂³×C₅	—	2	24	no	—
122	C₂³×C₅	—	2	12	no	—
123	C₂³×C₅	—	2	24	no	—
124	C₂³×C₅	—	4	24	no	—
125	C₂³×C₅	—	4	24	no	—
126	C₂³×C₅	—	4	96	no	—
127	C₂³×C₅	—	4	24	no	—
128	C₂³×C₅	—	4	128	no	—
129	C₂³×C₅	—	4	24	no	—
130	C₂³×C₅	—	8	24	no	—
131	C₂³×C₅	—	8	24	no	—
132	C₂³×C₅	#120	1	16	yes	complemented
133	C₂³×C₅	C₂²×D₄	1	12	yes	—
134	C₂³×C₅	C₂²×D₄	1	12	yes	—
135	C₂³×C₅	C₂²×D₄	1	6	yes	—
136	C₂³×C₅	H(5,𝔽₂)	1	8	yes	complemented
137	C₂³×C₅	C₂⁵	1	1	yes	characteristic
138	C₅×Q₈	—	2	16	no	—
139	C₅×D₄	—	2	48	no	—
140	C₅×D₄	—	2	96	no	—
141	C₅×D₄	—	4	96	no	—
142	C₂×C₄×C₅	—	2	48	no	—
143	C₂×C₄×C₅	—	2	24	no	—
144	C₂×C₄×C₅	—	2	12	no	—
145	C₂×C₄×C₅	—	2	12	no	—
146	C₂×C₄×C₅	—	2	24	no	—
147	C₂×C₄×C₅	—	4	24	no	—
148	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
149	C₂⁵	—	2	12	no	—
150	C₂⁵	—	4	8	no	—
151	C₂⁵	C₂³×C₅	1	6	yes	—
152	C₂⁵	C₂³×C₅	1	3	yes	—
153	C₂⁵	C₂³×C₅	1	1	yes	characteristic
154	C₂⁵	C₂³×C₅	1	2	yes	complemented
155	H(5,𝔽₂)	—	2	128	no	—
156	C₂×#41	—	2	48	no	—
157	C₂×#41	—	2	24	no	—
158	C₂²×Q₈	C₂³×C₅	1	2	yes	complemented
159	C₂²×D₄	—	2	24	no	—
160	C₂²×D₄	—	2	24	no	—
161	C₂²×D₄	—	2	48	no	—
162	C₂²×D₄	—	2	24	no	—
163	C₂²×D₄	—	4	24	no	—
164	C₂²×D₄	C₅×D₄	1	24	yes	complemented
165	C₂²×D₄	C₂³×C₅	1	24	yes	—
166	C₂²×D₄	C₂³×C₅	1	12	yes	—
167	C₂²×D₄	C₂³×C₅	1	6	yes	complemented
168	C₂³×C₄	C₂³×C₅	1	3	yes	—
169	C₂³×C₄	C₂³×C₅	1	6	yes	—
170	C₂³×C₄	C₂³×C₅	1	6	yes	complemented
171	#120	—	2	96	no	—
172	#120	—	4	16	no	—
173	C₂×C₂²⋊C₄	—	2	12	no	—
174	C₂×C₂²⋊C₄	—	2	24	no	—
175	C₂×C₂²⋊C₄	C₂³×C₅	1	12	yes	—
176	C₂×C₂²⋊C₄	C₂³×C₅	1	24	yes	—
177	C₂×C₂²⋊C₄	C₂³×C₅	1	24	yes	complemented
178	#95	C₂³×C₅	1	24	yes	complemented
179	C₂²×C₅	—	2	12	no	—
180	C₂²×C₅	—	2	24	no	—
181	C₂²×C₅	—	2	12	no	—
182	C₂²×C₅	—	2	24	no	—
183	C₂²×C₅	—	2	24	no	—
184	C₂²×C₅	—	4	16	no	—
185	C₂²×C₅	—	4	96	no	—
186	C₂²×C₅	—	4	4	no	—
187	C₂²×C₅	—	4	12	no	—
188	C₂²×C₅	—	4	24	no	—
189	C₂²×C₅	—	8	24	no	—
190	C₂²×C₅	—	8	16	no	—
191	C₂²×C₅	C₂×#120	1	12	yes	—
192	C₂²×C₅	C₂³×D₄	1	3	yes	—
193	C₂²×C₅	C₂×H(5,𝔽₂)	1	4	yes	—
194	C₄×C₅	—	2	12	no	—
195	C₄×C₅	—	4	12	no	—
196	C₂⁴	—	2	12	no	—
197	C₂⁴	—	2	24	no	—
198	C₂⁴	—	2	12	no	—
199	C₂⁴	—	2	12	no	—
200	C₂⁴	—	2	24	no	—
201	C₂⁴	—	4	4	no	—
202	C₂⁴	—	4	12	no	—
203	C₂⁴	—	4	12	no	—
204	C₂⁴	—	4	8	no	—
205	C₂⁴	—	4	96	no	—
206	C₂⁴	—	4	16	no	—
207	C₂⁴	C₂×C₅×D₄	1	12	yes	—
208	C₂⁴	C₂×C₅×D₄	1	12	yes	complemented
209	C₂⁴	C₂⁴×C₅	1	6	yes	—
210	C₂⁴	C₂⁴×C₅	1	4	yes	—
211	C₂⁴	C₂⁴×C₅	1	3	yes	—
212	#41	—	2	192	no	—
213	C₂×Q₈	—	2	12	no	—
214	C₂×D₄	—	2	192	no	—
215	C₂×D₄	—	2	96	no	—
216	C₂×D₄	—	2	24	no	—
217	C₂×D₄	—	2	24	no	—
218	C₂×D₄	—	2	24	no	—
219	C₂×D₄	—	2	48	no	—
220	C₂×D₄	—	2	48	no	—
221	C₂×D₄	—	2	12	no	—
222	C₂×D₄	—	4	24	no	—
223	C₂×D₄	—	4	96	no	—
224	C₂×D₄	C₂×C₅×D₄	1	48	yes	complemented
225	C₂²×C₄	—	2	12	no	—
226	C₂²×C₄	—	2	12	no	—
227	C₂²×C₄	—	2	24	no	—
228	C₂²×C₄	—	2	12	no	—
229	C₂²×C₄	—	4	12	no	—
230	C₂²×C₄	C₂×C₅×D₄	1	12	yes	complemented
231	C₂²×C₄	C₂⁴×C₅	1	12	yes	—
232	C₂²×C₄	C₂⁴×C₅	1	6	yes	—
233	C₂²⋊C₄	—	2	24	no	—
234	C₂²⋊C₄	—	2	48	no	—
235	C₂²⋊C₄	—	4	24	no	—
236	C₂×C₅	—	2	12	no	—
237	C₂×C₅	—	4	8	no	—
238	C₂×C₅	—	4	12	no	—
239	C₂×C₅	C₂²×#120	1	1	yes	characteristic
240	C₂×C₅	#3484	1	6	yes	—
241	C₂³	—	2	12	no	—
242	C₂³	—	2	48	no	—
243	C₂³	—	2	16	no	—
244	C₂³	—	2	12	no	—
245	C₂³	—	2	12	no	—
246	C₂³	—	2	24	no	—
247	C₂³	—	2	12	no	—
248	C₂³	—	2	24	no	—
249	C₂³	—	4	24	no	—
250	C₂³	—	4	24	no	—
251	C₂³	—	4	96	no	—
252	C₂³	—	4	24	no	—
253	C₂³	—	4	128	no	—
254	C₂³	—	4	24	no	—
255	C₂³	—	8	24	no	—
256	C₂³	—	8	24	no	—
257	C₂³	C₅×#120	1	16	yes	complemented
258	C₂³	C₂²×C₅×D₄	1	12	yes	—
259	C₂³	C₂²×C₅×D₄	1	12	yes	—
260	C₂³	C₂²×C₅×D₄	1	6	yes	—
261	C₂³	C₅×H(5,𝔽₂)	1	8	yes	complemented
262	C₂³	C₂⁵×C₅	1	1	yes	characteristic
263	Q₈	—	2	16	no	—
264	D₄	—	2	48	no	—
265	D₄	—	2	96	no	—
266	D₄	—	4	96	no	—
267	C₂×C₄	—	2	48	no	—
268	C₂×C₄	—	2	24	no	—
269	C₂×C₄	—	2	12	no	—
270	C₂×C₄	—	2	12	no	—
271	C₂×C₄	—	2	24	no	—
272	C₂×C₄	—	4	24	no	—
273	C₂×C₄	C₂²×C₅×D₄	1	12	yes	—
274	C₅	#35346	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	—	2	12	no	—
276	C₂²	—	2	24	no	—
277	C₂²	—	2	12	no	—
278	C₂²	—	2	24	no	—
279	C₂²	—	2	24	no	—
280	C₂²	—	4	16	no	—
281	C₂²	—	4	96	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	12	no	—
284	C₂²	—	4	24	no	—
285	C₂²	—	8	24	no	—
286	C₂²	—	8	16	no	—
287	C₂²	C₂×C₅×#120	1	12	yes	—
288	C₂²	C₂³×C₅×D₄	1	3	yes	—
289	C₂²	C₂×C₅×H(5,𝔽₂)	1	4	yes	—
290	C₄	—	2	12	no	—
291	C₄	—	4	12	no	—
292	C₂	—	2	12	no	—
293	C₂	—	4	8	no	—
294	C₂	—	4	12	no	—
295	C₂	C₂²×C₅×#120	1	1	yes	characteristic
296	C₂	C₅×#3484	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×H(5,𝔽₂)	C₂	semidirect
2. C₂²×C₅×#120	C₂	semidirect
3. C₅×#2404	C₂	semidirect
4. C₅×#2403	C₂	semidirect
5. C₅×#2308	C₂	semidirect
6. C₂⁶×C₅	C₂²	semidirect
8. C₂²×C₅×#41	C₂²	semidirect
10. C₂³×C₅×D₄	C₂²	semidirect
11. C₂³×C₅×D₄	C₂²	semidirect
13. C₂×C₅×#120	C₂²	semidirect
14. C₂×C₅×#120	C₂²	semidirect
15. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
16. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
17. C₅×#393	C₂²	semidirect
18. C₅×#392	C₂²	semidirect
19. C₅×#386	C₂²	semidirect
20. C₅×#386	C₂²	semidirect
21. C₅×#381	C₂²	semidirect
22. C₅×#379	C₂²	semidirect
23. #35346	C₅	direct (N × H)
29. C₂⁵×C₅	C₂³	semidirect
33. C₂²×C₅×Q₈	C₂³	semidirect
39. C₂²×C₅×D₄	D₄	semidirect
42. C₂²×C₅×D₄	C₂³	semidirect
45. C₂³×C₄×C₅	C₂³	semidirect
52. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
53. C₅×#95	C₂³	semidirect
54. C₂²×H(5,𝔽₂)	C₂×C₅	semidirect
55. C₂²×#120	C₂×C₅	semidirect
56. #2404	C₂×C₅	semidirect
57. #2403	C₂×C₅	semidirect
58. #2308	C₂×C₅	semidirect
71. C₂⁴×C₅	C₂×D₄	semidirect
87. C₂×C₅×D₄	C₂×D₄	semidirect
93. C₂²×C₄×C₅	C₂×D₄	semidirect
99. C₂⁶	C₂²×C₅	semidirect
101. C₂²×#41	C₂²×C₅	semidirect
103. C₂³×D₄	C₂²×C₅	semidirect
104. C₂³×D₄	C₂²×C₅	semidirect
106. C₂×#120	C₂²×C₅	semidirect
107. C₂×#120	C₂²×C₅	semidirect
108. C₂²×C₂²⋊C₄	C₂²×C₅	semidirect
109. C₂²×C₂²⋊C₄	C₂²×C₅	semidirect
110. #393	C₂²×C₅	semidirect
111. #392	C₂²×C₅	semidirect
112. #386	C₂²×C₅	semidirect
113. #386	C₂²×C₅	semidirect
114. #381	C₂²×C₅	semidirect
115. #379	C₂²×C₅	semidirect
132. C₂³×C₅	#120	semidirect
136. C₂³×C₅	H(5,𝔽₂)	semidirect
154. C₂⁵	C₂³×C₅	semidirect
158. C₂²×Q₈	C₂³×C₅	semidirect
164. C₂²×D₄	C₅×D₄	semidirect
167. C₂²×D₄	C₂³×C₅	semidirect
170. C₂³×C₄	C₂³×C₅	semidirect
177. C₂×C₂²⋊C₄	C₂³×C₅	semidirect
178. #95	C₂³×C₅	semidirect
208. C₂⁴	C₂×C₅×D₄	semidirect
224. C₂×D₄	C₂×C₅×D₄	semidirect
230. C₂²×C₄	C₂×C₅×D₄	semidirect
257. C₂³	C₅×#120	semidirect
261. C₂³	C₅×H(5,𝔽₂)	semidirect
274. C₅	#35346	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×H(5,𝔽₂)	2
C₂²×C₅×#120	2
C₅×#2404	2
C₅×#2403	2
C₅×#2308	2
#35346	5

Subgroups — C5×#35346 #11958066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#35346 #11958066