Subgroups — C₅×#5515 #11755900

Order 960.

436 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#930	C₂	1	1	yes	maximal, characteristic
3	C₅×#929	C₂	1	2	yes	maximal, complemented
4	C₄×C₅×#67	C₂	1	2	yes	maximal, complemented
5	C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#898	C₂	1	2	yes	maximal, complemented
7	C₅×#896	C₂	1	2	yes	maximal, complemented
8	C₅×#894	C₂	1	2	yes	maximal, complemented
9	C₅×#889	C₂	1	2	yes	maximal, complemented
10	C₅×#539	—	3	1	no	maximal, Hall
11	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×#67	C₂²	1	2	yes	—
15	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
17	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₅×#217	C₂²	1	2	yes	—
21	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
22	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
31	C₄×C₅×Dic₃	C₂²	1	2	yes	—
32	#5515	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₅×#126	—	3	2	no	—
34	C₅×#125	—	3	2	no	—
35	C₅×#123	—	3	2	no	—
36	C₅×#123	—	3	2	no	—
37	C₅×C₂²⋊Q₈	—	3	1	no	—
38	C₅×C₂²⋊Q₈	—	3	2	no	—
39	C₅×#121	—	3	1	no	—
40	C₄×C₅×D₄	—	3	2	no	—
41	C₂×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
42	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
43	C₂²×C₅×D₃	C₂³	1	2	yes	—
44	C₂×C₃×C₄×C₅	—	2	2	no	—
45	C₂×C₃×C₄×C₅	—	4	1	no	—
46	C₂×C₃×C₄×C₅	—	4	1	no	—
47	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₅×#67	—	2	4	no	—
52	C₂×C₅×Dic₃	C₂³	1	2	yes	—
53	C₂×C₅×Dic₃	C₂³	1	2	yes	—
54	C₂×C₅×Dic₃	C₂³	1	2	yes	—
55	C₅×D₁₂	—	4	1	no	—
56	C₄×C₅×D₃	—	4	2	no	—
57	C₅×Dic₆	—	4	1	no	—
58	C₂×C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
59	#930	C₂×C₅	1	1	yes	characteristic
60	#929	C₂×C₅	1	2	yes	complemented
61	C₄×#67	C₂×C₅	1	2	yes	complemented
62	#924	C₂×C₅	1	1	yes	characteristic, complemented
63	#898	C₂×C₅	1	2	yes	complemented
64	#896	C₂×C₅	1	2	yes	complemented
65	#894	C₂×C₅	1	2	yes	complemented
66	#889	C₂×C₅	1	2	yes	complemented
67	C₂×C₅×Q₈	—	3	1	no	—
68	C₂×C₅×D₄	—	3	2	no	—
69	C₂×C₅×D₄	—	3	1	no	—
70	C₂²×C₄×C₅	—	3	2	no	—
71	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
74	C₅×C₄⋊C₄	—	3	2	no	—
75	C₅×C₄⋊C₄	—	3	1	no	—
76	C₅×C₄⋊C₄	—	3	2	no	—
77	C₅×C₄⋊C₄	—	3	2	no	—
78	C₅×C₄⋊C₄	—	3	1	no	—
79	C₅×C₄⋊C₄	—	3	2	no	—
80	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
81	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
82	C₅×C₂²⋊C₄	—	3	2	no	—
83	C₅×C₂²⋊C₄	—	3	2	no	—
84	C₅×C₂²⋊C₄	—	3	2	no	—
85	C₅×C₂²⋊C₄	—	3	2	no	—
86	C₅×C₂²⋊C₄	—	3	2	no	—
87	C₄²×C₅	—	3	2	no	—
88	#539	—	3	1	no	Sylow
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	—	2	1	no	—
91	C₂²×C₃×C₅	#41	1	2	yes	—
92	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
93	C₂×C₅×D₃	—	2	2	no	—
94	C₂×C₅×D₃	—	2	2	no	—
95	C₂×C₅×D₃	—	2	2	no	—
96	C₃×C₄×C₅	—	2	2	no	—
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	—	2	2	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₅×Dic₃	—	2	2	no	—
101	C₅×Dic₃	—	2	2	no	—
102	C₅×Dic₃	—	2	2	no	—
103	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
106	C₂×#67	C₂²×C₅	1	2	yes	—
107	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
109	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
110	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
111	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
112	#217	C₂²×C₅	1	2	yes	—
113	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
114	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
115	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
116	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
117	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
118	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
119	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
120	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
121	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
122	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
123	C₄×Dic₃	C₂²×C₅	1	2	yes	—
124	C₂³×C₅	—	3	2	no	—
125	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
126	C₅×Q₈	—	12	1	no	—
127	C₅×D₄	—	6	4	no	—
128	C₅×D₄	—	12	1	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	3	2	no	—
131	C₂×C₄×C₅	—	3	2	no	—
132	C₂×C₄×C₅	—	3	2	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	12	2	no	—
136	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
137	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
138	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
140	#126	—	3	2	no	—
141	#125	—	3	2	no	—
142	#123	—	3	2	no	—
143	#123	—	3	2	no	—
144	C₂²⋊Q₈	—	3	1	no	—
145	C₂²⋊Q₈	—	3	2	no	—
146	#121	—	3	1	no	—
147	C₄×D₄	—	3	2	no	—
148	C₂×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
149	C₂×C₃×C₅	—	2	2	no	—
150	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
151	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
153	C₅×D₃	—	4	2	no	—
154	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
155	C₂²×D₃	C₂³×C₅	1	2	yes	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	4	1	no	—
158	C₂×C₃×C₄	—	4	1	no	—
159	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
160	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
162	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
163	#67	—	2	4	no	—
164	C₂×Dic₃	C₂³×C₅	1	2	yes	—
165	C₂×Dic₃	C₂³×C₅	1	2	yes	—
166	C₂×Dic₃	C₂³×C₅	1	2	yes	—
167	D₁₂	—	4	1	no	—
168	C₄×D₃	—	4	2	no	—
169	Dic₆	—	4	1	no	—
170	C₂²×C₅	—	2	1	no	—
171	C₂²×C₅	—	2	1	no	—
172	C₂²×C₅	—	6	2	no	—
173	C₂²×C₅	—	6	2	no	—
174	C₂²×C₅	—	6	2	no	—
175	C₂²×C₅	#235	1	2	yes	—
176	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	6	2	no	—
182	C₄×C₅	—	6	2	no	—
183	C₄×C₅	—	6	2	no	—
184	C₂×Q₈	—	3	1	no	—
185	C₂×D₄	—	3	2	no	—
186	C₂×D₄	—	3	1	no	—
187	C₂²×C₄	—	3	2	no	—
188	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
191	C₄⋊C₄	—	3	2	no	—
192	C₄⋊C₄	—	3	1	no	—
193	C₄⋊C₄	—	3	2	no	—
194	C₄⋊C₄	—	3	2	no	—
195	C₄⋊C₄	—	3	1	no	—
196	C₄⋊C₄	—	3	2	no	—
197	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
198	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
199	C₂²⋊C₄	—	3	2	no	—
200	C₂²⋊C₄	—	3	2	no	—
201	C₂²⋊C₄	—	3	2	no	—
202	C₂²⋊C₄	—	3	2	no	—
203	C₂²⋊C₄	—	3	2	no	—
204	C₄²	—	3	2	no	—
205	C₃×C₅	#539	1	1	yes	characteristic, complemented, Hall
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	C₅×#41	1	2	yes	—
209	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
210	C₂×D₃	—	2	2	no	—
211	C₂×D₃	—	2	2	no	—
212	C₂×D₃	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	2	1	no	—
217	Dic₃	—	2	2	no	—
218	Dic₃	—	2	2	no	—
219	Dic₃	—	2	2	no	—
220	C₂×C₅	—	2	2	no	—
221	C₂×C₅	—	12	2	no	—
222	C₂×C₅	C₂×#235	1	1	yes	characteristic
223	C₂×C₅	#1004	1	1	yes	characteristic
224	C₂×C₅	#1007	1	1	yes	characteristic
225	C₂³	—	3	2	no	—
226	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
227	Q₈	—	12	1	no	—
228	D₄	—	6	4	no	—
229	D₄	—	12	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	3	2	no	—
232	C₂×C₄	—	3	2	no	—
233	C₂×C₄	—	3	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	12	2	no	—
237	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
238	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
239	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
243	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
244	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
245	D₃	—	4	2	no	—
246	C₅	#5515	1	1	yes	characteristic, direct product factor, Sylow
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	6	2	no	—
250	C₂²	—	6	2	no	—
251	C₂²	—	6	2	no	—
252	C₂²	C₅×#235	1	2	yes	—
253	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	6	2	no	—
259	C₄	—	6	2	no	—
260	C₄	—	6	2	no	—
261	C₃	C₅×#539	1	1	yes	characteristic, complemented, Sylow
262	C₂	—	2	2	no	—
263	C₂	—	12	2	no	—
264	C₂	C₂×C₅×#235	1	1	yes	characteristic
265	C₂	C₅×#1004	1	1	yes	characteristic
266	C₂	C₅×#1007	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×C₄⋊C₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₅×#930	C₂
hypercenter	C₂×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₅×#539
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊C₄	C₂	semidirect
3. C₅×#929	C₂	semidirect
4. C₄×C₅×#67	C₂	semidirect
5. C₅×#924	C₂	semidirect
6. C₅×#898	C₂	semidirect
7. C₅×#896	C₂	semidirect
8. C₅×#894	C₂	semidirect
9. C₅×#889	C₂	semidirect
32. #5515	C₅	direct (N × H)
41. C₂×C₅×C₄⋊C₄	D₃	semidirect
58. C₂×C₃×C₄⋊C₄	C₂×C₅	semidirect
60. #929	C₂×C₅	semidirect
61. C₄×#67	C₂×C₅	semidirect
62. #924	C₂×C₅	semidirect
63. #898	C₂×C₅	semidirect
64. #896	C₂×C₅	semidirect
65. #894	C₂×C₅	semidirect
66. #889	C₂×C₅	semidirect
148. C₂×C₄⋊C₄	C₅×D₃	semidirect
205. C₃×C₅	#539	semidirect
246. C₅	#5515	direct (N × H)
261. C₃	C₅×#539	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊C₄	2
C₅×#930	2
C₅×#929	2
C₄×C₅×#67	2
C₅×#924	2
C₅×#898	2
C₅×#896	2
C₅×#894	2
C₅×#889	2
C₅×#539	3
#5515	5

Subgroups — C5×#5515 #11755900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5515 #11755900