Subgroups — C₅×#5236 #11750735

Order 960.

308 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₅×#926	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#837	C₂	1	4	yes	maximal, complemented
5	C₅×#421	—	3	1	no	maximal, Hall
6	C₃×C₅×#41	—	2	1	no	—
7	C₃×C₅×#41	—	2	1	no	—
8	C₃×C₅×#41	C₄	1	2	yes	complemented
9	C₃×C₅×#41	C₂²	1	2	yes	—
10	C₂×C₃×C₅×Q₈	C₄	1	1	yes	characteristic, complemented
11	C₂×C₃×C₅×D₄	—	2	1	no	—
12	C₂×C₃×C₅×D₄	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₃×C₄×C₅	—	2	1	no	—
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₅×#217	—	2	1	no	—
16	C₅×C₄⋊Dic₃	—	2	1	no	—
17	C₄×C₅×Dic₃	C₂²	1	2	yes	complemented
18	C₅×#208	—	2	1	no	—
19	C₅×#208	C₂²	1	2	yes	complemented
20	C₂×C₅×C₃⋊C₈	—	2	1	no	—
21	#5236	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#41	D₃	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	3	1	no	—
24	C₅×#117	—	3	1	no	—
25	C₅×#104	—	3	4	no	—
26	C₂³×C₃×C₅	—	2	1	no	—
27	C₂³×C₃×C₅	D₄	1	1	yes	characteristic
28	C₃×C₅×Q₈	—	2	1	no	—
29	C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
30	C₃×C₅×D₄	—	2	2	no	—
31	C₃×C₅×D₄	—	2	2	no	—
32	C₃×C₅×D₄	—	2	1	no	—
33	C₃×C₅×D₄	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×C₅	—	2	1	no	—
35	C₂×C₃×C₄×C₅	—	2	2	no	—
36	C₂×C₃×C₄×C₅	—	2	2	no	—
37	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₅×Dic₃	—	2	2	no	—
44	C₅×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃×#41	C₂×C₅	1	1	yes	characteristic
46	#926	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂×#208	C₂×C₅	1	1	yes	characteristic, complemented
48	#837	C₂×C₅	1	4	yes	complemented
49	C₅×#41	—	2	1	no	—
50	C₅×#41	—	2	1	no	—
51	C₅×#41	Dic₃	1	2	yes	complemented
52	C₅×#41	C₂×D₃	1	2	yes	—
53	C₂×C₅×Q₈	Dic₃	1	1	yes	characteristic, complemented
54	C₂×C₅×D₄	—	2	1	no	—
55	C₂×C₅×D₄	Dic₃	1	1	yes	characteristic, complemented
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
58	C₅×M₁₆	—	3	2	no	—
59	C₅×M₁₆	—	6	1	no	—
60	C₂×C₅×C₈	—	6	1	no	—
61	C₅×C₄⋊C₄	—	6	1	no	—
62	C₅×C₂²⋊C₄	—	6	1	no	—
63	C₄²×C₅	—	3	2	no	—
64	#421	—	3	1	no	Sylow
65	C₂²×C₃×C₅	—	2	2	no	—
66	C₂²×C₃×C₅	—	4	1	no	—
67	C₂²×C₃×C₅	—	4	2	no	—
68	C₂²×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₃×C₄×C₅	—	2	2	no	—
72	C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₅×Dic₃	—	4	2	no	—
77	C₃×#41	—	2	1	no	—
78	C₃×#41	—	2	1	no	—
79	C₃×#41	C₄×C₅	1	2	yes	complemented
80	C₃×#41	C₂²×C₅	1	2	yes	—
81	C₂×C₃×Q₈	C₄×C₅	1	1	yes	characteristic, complemented
82	C₂×C₃×D₄	—	2	1	no	—
83	C₂×C₃×D₄	C₄×C₅	1	1	yes	characteristic, complemented
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
86	#217	—	2	1	no	—
87	C₄⋊Dic₃	—	2	1	no	—
88	C₄×Dic₃	C₂²×C₅	1	2	yes	complemented
89	#208	—	2	1	no	—
90	#208	C₂²×C₅	1	2	yes	complemented
91	C₂×C₃⋊C₈	—	2	1	no	—
92	C₂³×C₅	—	2	1	no	—
93	C₂³×C₅	#67	1	1	yes	characteristic
94	C₅×Q₈	—	2	1	no	—
95	C₅×Q₈	C₂×Dic₃	1	2	yes	—
96	C₅×D₄	—	2	2	no	—
97	C₅×D₄	—	2	2	no	—
98	C₅×D₄	—	2	1	no	—
99	C₅×D₄	C₂×Dic₃	1	2	yes	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	6	2	no	—
104	C₂×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
106	C₂×C₄×C₅	#67	1	1	yes	characteristic
107	C₂×C₄×C₅	#67	1	1	yes	characteristic
108	C₂×C₄×C₅	#67	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
110	C₅×C₈	—	6	2	no	—
111	C₂×#41	C₅×D₃	1	1	yes	characteristic
112	C₂×M₁₆	—	3	1	no	—
113	#117	—	3	1	no	—
114	#104	—	3	4	no	—
115	C₂×C₃×C₅	—	2	1	no	—
116	C₂×C₃×C₅	—	2	1	no	—
117	C₂×C₃×C₅	—	2	1	no	—
118	C₂×C₃×C₅	—	4	2	no	—
119	C₂×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×C₃	—	2	1	no	—
121	C₂³×C₃	C₅×D₄	1	1	yes	characteristic
122	C₃×Q₈	—	2	1	no	—
123	C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
124	C₃×D₄	—	2	2	no	—
125	C₃×D₄	—	2	2	no	—
126	C₃×D₄	—	2	1	no	—
127	C₃×D₄	C₂×C₄×C₅	1	2	yes	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
133	C₂×C₃×C₄	C₅×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₅×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₅×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
137	C₂×Dic₃	—	2	2	no	—
138	C₃⋊C₈	—	2	2	no	—
139	C₂²×C₅	—	2	2	no	—
140	C₂²×C₅	—	4	1	no	—
141	C₂²×C₅	—	4	2	no	—
142	C₂²×C₅	#217	1	1	yes	characteristic
143	C₂²×C₅	#217	1	1	yes	characteristic
144	C₂²×C₅	C₂×#67	1	1	yes	characteristic
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	12	2	no	—
147	C₄×C₅	#217	1	1	yes	characteristic
148	C₄×C₅	#217	1	1	yes	characteristic
149	C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
150	C₄×C₅	C₂×#67	1	1	yes	characteristic
151	#41	—	2	1	no	—
152	#41	—	2	1	no	—
153	#41	C₅×Dic₃	1	2	yes	complemented
154	#41	C₂×C₅×D₃	1	2	yes	—
155	C₂×Q₈	C₅×Dic₃	1	1	yes	characteristic, complemented
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	C₅×Dic₃	1	1	yes	characteristic, complemented
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
160	M₁₆	—	3	2	no	—
161	M₁₆	—	6	1	no	—
162	C₂×C₈	—	6	1	no	—
163	C₄⋊C₄	—	6	1	no	—
164	C₂²⋊C₄	—	6	1	no	—
165	C₄²	—	3	2	no	—
166	C₃×C₅	#421	1	1	yes	characteristic, complemented, Hall
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	4	1	no	—
169	C₂²×C₃	—	4	2	no	—
170	C₂²×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₃	C₂×C₅×D₄	1	1	yes	characteristic
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
176	C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
177	C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
178	Dic₃	—	4	2	no	—
179	C₂×C₅	—	2	1	no	—
180	C₂×C₅	—	2	1	no	—
181	C₂×C₅	—	2	1	no	—
182	C₂×C₅	—	4	2	no	—
183	C₂×C₅	C₂×#217	1	1	yes	characteristic
184	C₂³	—	2	1	no	—
185	C₂³	C₅×#67	1	1	yes	characteristic
186	Q₈	—	2	1	no	—
187	Q₈	C₂×C₅×Dic₃	1	2	yes	—
188	D₄	—	2	2	no	—
189	D₄	—	2	2	no	—
190	D₄	—	2	1	no	—
191	D₄	C₂×C₅×Dic₃	1	2	yes	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	6	2	no	—
196	C₂×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
197	C₂×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
198	C₂×C₄	C₅×#67	1	1	yes	characteristic
199	C₂×C₄	C₅×#67	1	1	yes	characteristic
200	C₂×C₄	C₅×#67	1	1	yes	characteristic
201	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
202	C₈	—	6	2	no	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	—	2	1	no	—
206	C₂×C₃	—	4	2	no	—
207	C₂×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₅	#5236	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²	—	2	2	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	2	no	—
212	C₂²	C₅×#217	1	1	yes	characteristic
213	C₂²	C₅×#217	1	1	yes	characteristic
214	C₂²	C₂×C₅×#67	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	12	2	no	—
217	C₄	C₅×#217	1	1	yes	characteristic
218	C₄	C₅×#217	1	1	yes	characteristic
219	C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
220	C₄	C₂×C₅×#67	1	1	yes	characteristic
221	C₃	C₅×#421	1	1	yes	characteristic, complemented, Sylow
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂×C₅×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×#67
commutator	C₃×C₄	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₂×C₃×C₅×#41	C₂
socle	C₂×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#41	D₃
hypocenter	C₃	C₅×#421
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₅×#926	C₂	semidirect
3. C₂×C₅×#208	C₂	semidirect
4. C₅×#837	C₂	semidirect
8. C₃×C₅×#41	C₄	semidirect
10. C₂×C₃×C₅×Q₈	C₄	semidirect
12. C₂×C₃×C₅×D₄	C₄	semidirect
17. C₄×C₅×Dic₃	C₂²	semidirect
19. C₅×#208	C₂²	semidirect
21. #5236	C₅	direct (N × H)
46. #926	C₂×C₅	semidirect
47. C₂×#208	C₂×C₅	semidirect
48. #837	C₂×C₅	semidirect
51. C₅×#41	Dic₃	semidirect
53. C₂×C₅×Q₈	Dic₃	semidirect
55. C₂×C₅×D₄	Dic₃	semidirect
79. C₃×#41	C₄×C₅	semidirect
81. C₂×C₃×Q₈	C₄×C₅	semidirect
83. C₂×C₃×D₄	C₄×C₅	semidirect
88. C₄×Dic₃	C₂²×C₅	semidirect
90. #208	C₂²×C₅	semidirect
153. #41	C₅×Dic₃	semidirect
155. C₂×Q₈	C₅×Dic₃	semidirect
157. C₂×D₄	C₅×Dic₃	semidirect
166. C₃×C₅	#421	semidirect
208. C₅	#5236	direct (N × H)
221. C₃	C₅×#421	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×#41	2
C₅×#926	2
C₂×C₅×#208	2
C₅×#837	2
C₅×#421	3
#5236	5

Subgroups — C5×#5236 #11750735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5236 #11750735