Subgroups — C₅×#5526 #11755911

Order 960.

540 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#235	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#897	C₂	1	4	yes	maximal, complemented
5	C₅×#886	C₂	1	4	yes	maximal, complemented
6	C₅×#875	C₂	1	2	yes	maximal, complemented
7	C₅×#869	C₂	1	2	yes	maximal, complemented
8	C₅×#536	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×#67	C₂²	1	2	yes	—
11	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
12	C₅×#235	—	2	4	no	—
13	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
15	C₂×C₅×Dic₆	—	2	4	no	—
16	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
17	C₂×C₅×Dic₆	C₂²	1	2	yes	—
18	C₂×C₅×Dic₆	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²×C₅	C₂²	1	2	yes	—
22	C₅×(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
23	C₅×C₄⋊Dic₃	C₂²	1	8	yes	—
24	#5526	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×#41	—	3	1	no	—
26	C₂²×C₅×Q₈	—	3	1	no	—
27	C₅×C₄⋊Q₈	—	3	2	no	—
28	C₅×#124	—	3	2	no	—
29	C₅×#123	—	3	4	no	—
30	C₅×C₂²⋊Q₈	—	3	4	no	—
31	C₅×#117	D₃	1	1	yes	characteristic, complemented
32	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₅×D₃	C₂³	1	2	yes	—
34	C₂×C₃×C₄×C₅	D₄	1	4	yes	—
35	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
36	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₅×#67	—	2	4	no	—
39	C₂×C₅×Dic₃	—	2	4	no	—
40	C₂×C₅×Dic₃	C₂³	1	2	yes	—
41	C₂×C₅×Dic₃	C₂³	1	4	yes	—
42	C₅×D₁₂	—	2	2	no	—
43	C₄×C₅×D₃	—	2	4	no	—
44	C₅×Dic₆	—	2	4	no	—
45	C₅×Dic₆	—	2	4	no	—
46	C₅×Dic₆	—	2	2	no	—
47	C₂×#235	C₂×C₅	1	1	yes	characteristic
48	C₂²×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
49	C₃×#117	C₂×C₅	1	1	yes	characteristic, complemented
50	#897	C₂×C₅	1	4	yes	complemented
51	#886	C₂×C₅	1	4	yes	complemented
52	#875	C₂×C₅	1	2	yes	complemented
53	#869	C₂×C₅	1	2	yes	complemented
54	C₅×#41	—	6	4	no	—
55	C₂×C₅×Q₈	—	3	2	no	—
56	C₂×C₅×Q₈	—	3	2	no	—
57	C₂×C₅×Q₈	—	3	1	no	—
58	C₂×C₅×Q₈	—	6	4	no	—
59	C₂×C₅×D₄	—	3	2	no	—
60	C₂×C₅×D₄	—	3	1	no	—
61	C₂²×C₄×C₅	—	3	2	no	—
62	C₂²×C₄×C₅	—	3	2	no	—
63	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
64	C₅×C₄⋊C₄	—	3	8	no	—
65	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
66	C₅×C₂²⋊C₄	—	3	8	no	—
67	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
68	C₄²×C₅	C₂×D₃	1	2	yes	—
69	#536	—	3	1	no	Sylow
70	C₂²×C₃×C₅	—	2	2	no	—
71	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
72	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
73	C₂×C₅×D₃	—	2	2	no	—
74	C₂×C₅×D₃	—	2	4	no	—
75	C₃×C₄×C₅	—	2	4	no	—
76	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₅×Dic₃	—	2	4	no	—
79	C₅×Dic₃	—	2	2	no	—
80	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
81	C₂×#67	C₂²×C₅	1	2	yes	—
82	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
83	#235	—	2	4	no	—
84	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
86	C₂×Dic₆	—	2	4	no	—
87	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
88	C₂×Dic₆	C₂²×C₅	1	2	yes	—
89	C₂×Dic₆	C₂²×C₅	1	2	yes	—
90	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
91	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
92	C₃×C₄²	C₂²×C₅	1	2	yes	—
93	(C₂×D₃)⋊C₄	C₂²×C₅	1	8	yes	—
94	C₄⋊Dic₃	C₂²×C₅	1	8	yes	—
95	C₂³×C₅	—	3	2	no	—
96	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
97	C₅×Q₈	—	6	4	no	—
98	C₅×Q₈	—	6	4	no	—
99	C₅×Q₈	—	6	2	no	—
100	C₅×D₄	—	6	4	no	—
101	C₅×D₄	—	6	2	no	—
102	C₂×C₄×C₅	—	3	4	no	—
103	C₂×C₄×C₅	—	3	2	no	—
104	C₂×C₄×C₅	—	6	4	no	—
105	C₂×C₄×C₅	—	6	4	no	—
106	C₂×C₄×C₅	D₁₂	1	4	yes	—
107	C₂×C₄×C₅	C₂²×D₃	1	4	yes	—
108	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
110	C₂×#41	—	3	1	no	—
111	C₂²×Q₈	—	3	1	no	—
112	C₄⋊Q₈	—	3	2	no	—
113	#124	—	3	2	no	—
114	#123	—	3	4	no	—
115	C₂²⋊Q₈	—	3	4	no	—
116	#117	C₅×D₃	1	1	yes	characteristic, complemented
117	C₂×C₃×C₅	—	2	2	no	—
118	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
120	C₅×D₃	—	4	2	no	—
121	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
122	C₂²×D₃	C₂³×C₅	1	2	yes	—
123	C₂×C₃×C₄	C₅×D₄	1	4	yes	—
124	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
125	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
127	#67	—	2	4	no	—
128	C₂×Dic₃	—	2	4	no	—
129	C₂×Dic₃	C₂³×C₅	1	2	yes	—
130	C₂×Dic₃	C₂³×C₅	1	4	yes	—
131	D₁₂	—	2	2	no	—
132	C₄×D₃	—	2	4	no	—
133	Dic₆	—	2	4	no	—
134	Dic₆	—	2	4	no	—
135	Dic₆	—	2	2	no	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	—	6	2	no	—
138	C₂²×C₅	—	6	4	no	—
139	C₂²×C₅	C₂×D₁₂	1	2	yes	—
140	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
141	C₄×C₅	—	2	4	no	—
142	C₄×C₅	—	6	2	no	—
143	C₄×C₅	—	6	4	no	—
144	C₄×C₅	C₂×D₁₂	1	2	yes	—
145	C₄×C₅	C₂×D₁₂	1	2	yes	—
146	#41	—	6	4	no	—
147	C₂×Q₈	—	3	2	no	—
148	C₂×Q₈	—	3	2	no	—
149	C₂×Q₈	—	3	1	no	—
150	C₂×Q₈	—	6	4	no	—
151	C₂×D₄	—	3	2	no	—
152	C₂×D₄	—	3	1	no	—
153	C₂²×C₄	—	3	2	no	—
154	C₂²×C₄	—	3	2	no	—
155	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
156	C₄⋊C₄	—	3	8	no	—
157	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
158	C₂²⋊C₄	—	3	8	no	—
159	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
160	C₄²	C₂×C₅×D₃	1	2	yes	—
161	C₃×C₅	#536	1	1	yes	characteristic, complemented, Hall
162	C₂²×C₃	—	2	2	no	—
163	C₂²×C₃	C₂×C₅×D₄	1	2	yes	—
164	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
165	C₂×D₃	—	2	2	no	—
166	C₂×D₃	—	2	4	no	—
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
169	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
170	Dic₃	—	2	4	no	—
171	Dic₃	—	2	2	no	—
172	C₂×C₅	—	2	2	no	—
173	C₂×C₅	—	12	2	no	—
174	C₂×C₅	C₂²×D₁₂	1	1	yes	characteristic
175	C₂×C₅	#1010	1	2	yes	—
176	C₂³	—	3	2	no	—
177	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
178	Q₈	—	6	4	no	—
179	Q₈	—	6	4	no	—
180	Q₈	—	6	2	no	—
181	D₄	—	6	4	no	—
182	D₄	—	6	2	no	—
183	C₂×C₄	—	3	4	no	—
184	C₂×C₄	—	3	2	no	—
185	C₂×C₄	—	6	4	no	—
186	C₂×C₄	—	6	4	no	—
187	C₂×C₄	C₅×D₁₂	1	4	yes	—
188	C₂×C₄	C₂²×C₅×D₃	1	4	yes	—
189	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
191	C₂×C₃	—	2	2	no	—
192	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
193	C₂×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
194	D₃	—	4	2	no	—
195	C₅	#5526	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²	—	2	2	no	—
197	C₂²	—	6	2	no	—
198	C₂²	—	6	4	no	—
199	C₂²	C₂×C₅×D₁₂	1	2	yes	—
200	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	—	6	2	no	—
203	C₄	—	6	4	no	—
204	C₄	C₂×C₅×D₁₂	1	2	yes	—
205	C₄	C₂×C₅×D₁₂	1	2	yes	—
206	C₃	C₅×#536	1	1	yes	characteristic, complemented, Sylow
207	C₂	—	2	2	no	—
208	C₂	—	12	2	no	—
209	C₂	C₂²×C₅×D₁₂	1	1	yes	characteristic
210	C₂	C₅×#1010	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#117	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#235	C₂
hypercenter	C₅×#117	D₃
hypocenter	C₃	C₅×#536
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×Dic₆	C₂	semidirect
3. C₃×C₅×#117	C₂	semidirect
4. C₅×#897	C₂	semidirect
5. C₅×#886	C₂	semidirect
6. C₅×#875	C₂	semidirect
7. C₅×#869	C₂	semidirect
24. #5526	C₅	direct (N × H)
31. C₅×#117	D₃	semidirect
48. C₂²×Dic₆	C₂×C₅	semidirect
49. C₃×#117	C₂×C₅	semidirect
50. #897	C₂×C₅	semidirect
51. #886	C₂×C₅	semidirect
52. #875	C₂×C₅	semidirect
53. #869	C₂×C₅	semidirect
116. #117	C₅×D₃	semidirect
161. C₃×C₅	#536	semidirect
195. C₅	#5526	direct (N × H)
206. C₃	C₅×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#235	2
C₂²×C₅×Dic₆	2
C₃×C₅×#117	2
C₅×#897	2
C₅×#886	2
C₅×#875	2
C₅×#869	2
C₅×#536	3
#5526	5

Subgroups — C5×#5526 #11755911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5526 #11755911