Subgroups — C₅×#5585 #11755970

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#947	C₂	1	2	yes	maximal, complemented
3	C₅×#894	C₂	1	4	yes	maximal, complemented
4	C₅×#893	C₂	1	2	yes	maximal, complemented
5	C₅×#875	C₂	1	2	yes	maximal, complemented
6	C₅×#874	C₂	1	1	yes	maximal, characteristic
7	C₄×C₅×D₁₂	C₂	1	2	yes	maximal, complemented
8	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#559	—	3	1	no	maximal, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×D₁₂	C₂²	1	2	yes	—
12	C₂×C₅×D₁₂	C₂²	1	2	yes	—
13	C₂×C₅×D₁₂	C₂²	1	2	yes	—
14	C₂×C₄×C₅×D₃	C₂²	1	4	yes	—
15	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₃×C₄²×C₅	C₂²	1	2	yes	—
20	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
21	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
22	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
23	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₄×C₅×Dic₃	C₂²	1	2	yes	—
26	#5585	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#127	—	3	1	no	—
28	C₅×#124	—	3	2	no	—
29	C₅×#124	—	3	2	no	—
30	C₅×#123	—	3	4	no	—
31	C₄×C₅×Q₈	—	3	1	no	—
32	C₄×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
33	C₄×C₅×D₄	—	3	2	no	—
34	C₄×C₅×D₄	—	3	2	no	—
35	C₂²×C₅×D₃	C₂³	1	4	yes	—
36	C₃×C₅×Q₈	—	2	2	no	—
37	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
39	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₅×Dic₃	C₂³	1	2	yes	—
43	C₂×C₅×Dic₃	C₂³	1	2	yes	—
44	C₅×D₁₂	—	2	4	no	—
45	C₅×D₁₂	—	2	4	no	—
46	C₅×D₁₂	—	4	2	no	—
47	C₄×C₅×D₃	—	4	4	no	—
48	C₅×Dic₆	—	2	2	no	—
49	C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
50	#947	C₂×C₅	1	2	yes	complemented
51	#894	C₂×C₅	1	4	yes	complemented
52	#893	C₂×C₅	1	2	yes	complemented
53	#875	C₂×C₅	1	2	yes	complemented
54	#874	C₂×C₅	1	1	yes	characteristic
55	C₄×D₁₂	C₂×C₅	1	2	yes	complemented
56	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×Q₈	—	3	1	no	—
58	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
59	C₂×C₅×D₄	—	3	2	no	—
60	C₂×C₅×D₄	—	3	2	no	—
61	C₂×C₅×D₄	—	3	2	no	—
62	C₂²×C₄×C₅	—	3	4	no	—
63	C₅×C₄⋊C₄	—	3	1	no	—
64	C₅×C₄⋊C₄	—	3	2	no	—
65	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
66	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
67	C₅×C₂²⋊C₄	—	3	4	no	—
68	C₅×C₂²⋊C₄	—	3	4	no	—
69	C₅×C₂²⋊C₄	—	3	4	no	—
70	C₄²×C₅	—	3	2	no	—
71	C₄²×C₅	C₂×D₃	1	2	yes	—
72	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
73	#559	—	3	1	no	Sylow
74	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×C₅×D₃	—	2	4	no	—
76	C₂×C₅×D₃	—	2	4	no	—
77	C₂×C₅×D₃	—	2	4	no	—
78	C₃×C₄×C₅	—	2	2	no	—
79	C₃×C₄×C₅	—	2	2	no	—
80	C₃×C₄×C₅	—	2	1	no	—
81	C₃×C₄×C₅	#41	1	2	yes	—
82	C₃×C₄×C₅	#41	1	2	yes	—
83	C₅×Dic₃	—	2	2	no	—
84	C₅×Dic₃	—	2	2	no	—
85	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
86	C₂×D₁₂	C₂²×C₅	1	2	yes	—
87	C₂×D₁₂	C₂²×C₅	1	2	yes	—
88	C₂×D₁₂	C₂²×C₅	1	2	yes	—
89	C₂×C₄×D₃	C₂²×C₅	1	4	yes	—
90	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
91	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
92	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
93	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
94	C₃×C₄²	C₂²×C₅	1	2	yes	—
95	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
96	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
97	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
98	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
99	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
100	C₄×Dic₃	C₂²×C₅	1	2	yes	—
101	C₂³×C₅	—	3	4	no	—
102	C₅×Q₈	—	2	2	no	—
103	C₅×Q₈	—	6	2	no	—
104	C₅×D₄	—	6	4	no	—
105	C₅×D₄	—	6	4	no	—
106	C₅×D₄	—	12	2	no	—
107	C₂×C₄×C₅	—	3	2	no	—
108	C₂×C₄×C₅	—	3	2	no	—
109	C₂×C₄×C₅	—	12	4	no	—
110	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
111	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
112	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	#127	—	3	1	no	—
116	#124	—	3	2	no	—
117	#124	—	3	2	no	—
118	#123	—	3	4	no	—
119	C₄×Q₈	—	3	1	no	—
120	C₄×Q₈	C₅×D₃	1	1	yes	characteristic, complemented
121	C₄×D₄	—	3	2	no	—
122	C₄×D₄	—	3	2	no	—
123	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
124	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
125	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
126	C₅×D₃	—	4	4	no	—
127	C₂²×D₃	C₂³×C₅	1	4	yes	—
128	C₃×Q₈	—	2	2	no	—
129	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
131	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
132	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
134	C₂×Dic₃	C₂³×C₅	1	2	yes	—
135	C₂×Dic₃	C₂³×C₅	1	2	yes	—
136	D₁₂	—	2	4	no	—
137	D₁₂	—	2	4	no	—
138	D₁₂	—	4	2	no	—
139	C₄×D₃	—	4	4	no	—
140	Dic₆	—	2	2	no	—
141	C₂²×C₅	—	6	4	no	—
142	C₂²×C₅	—	6	4	no	—
143	C₂²×C₅	—	6	4	no	—
144	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	6	2	no	—
149	C₄×C₅	—	6	2	no	—
150	C₄×C₅	#235	1	2	yes	—
151	C₄×C₅	#239	1	2	yes	—
152	C₂×Q₈	—	3	1	no	—
153	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
154	C₂×D₄	—	3	2	no	—
155	C₂×D₄	—	3	2	no	—
156	C₂×D₄	—	3	2	no	—
157	C₂²×C₄	—	3	4	no	—
158	C₄⋊C₄	—	3	1	no	—
159	C₄⋊C₄	—	3	2	no	—
160	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
161	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
162	C₂²⋊C₄	—	3	4	no	—
163	C₂²⋊C₄	—	3	4	no	—
164	C₂²⋊C₄	—	3	4	no	—
165	C₄²	—	3	2	no	—
166	C₄²	C₂×C₅×D₃	1	2	yes	—
167	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
168	C₃×C₅	#559	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
170	C₂×D₃	—	2	4	no	—
171	C₂×D₃	—	2	4	no	—
172	C₂×D₃	—	2	4	no	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	C₅×#41	1	2	yes	—
177	C₃×C₄	C₅×#41	1	2	yes	—
178	Dic₃	—	2	2	no	—
179	Dic₃	—	2	2	no	—
180	C₂×C₅	—	12	4	no	—
181	C₂×C₅	C₂×#235	1	1	yes	characteristic
182	C₂×C₅	C₂×#239	1	1	yes	characteristic
183	C₂×C₅	#1009	1	1	yes	characteristic
184	C₂³	—	3	4	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	6	2	no	—
187	D₄	—	6	4	no	—
188	D₄	—	6	4	no	—
189	D₄	—	12	2	no	—
190	C₂×C₄	—	3	2	no	—
191	C₂×C₄	—	3	2	no	—
192	C₂×C₄	—	12	4	no	—
193	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
194	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
195	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
198	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
199	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
200	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
201	D₃	—	4	4	no	—
202	C₅	#5585	1	1	yes	characteristic, direct product factor, Sylow
203	C₂²	—	6	4	no	—
204	C₂²	—	6	4	no	—
205	C₂²	—	6	4	no	—
206	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	6	2	no	—
211	C₄	—	6	2	no	—
212	C₄	C₅×#235	1	2	yes	—
213	C₄	C₅×#239	1	2	yes	—
214	C₃	C₅×#559	1	1	yes	characteristic, complemented, Sylow
215	C₂	—	12	4	no	—
216	C₂	C₂×C₅×#235	1	1	yes	characteristic
217	C₂	C₂×C₅×#239	1	1	yes	characteristic
218	C₂	C₅×#1009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₅×#874	C₂
hypercenter	C₄×C₅×Q₈	D₃
hypocenter	C₃	C₅×#559
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×Q₈	C₂	semidirect
2. C₅×#947	C₂	semidirect
3. C₅×#894	C₂	semidirect
4. C₅×#893	C₂	semidirect
5. C₅×#875	C₂	semidirect
7. C₄×C₅×D₁₂	C₂	semidirect
8. C₄×C₅×Dic₆	C₂	semidirect
26. #5585	C₅	direct (N × H)
32. C₄×C₅×Q₈	D₃	semidirect
49. C₃×C₄×Q₈	C₂×C₅	semidirect
50. #947	C₂×C₅	semidirect
51. #894	C₂×C₅	semidirect
52. #893	C₂×C₅	semidirect
53. #875	C₂×C₅	semidirect
55. C₄×D₁₂	C₂×C₅	semidirect
56. C₄×Dic₆	C₂×C₅	semidirect
120. C₄×Q₈	C₅×D₃	semidirect
168. C₃×C₅	#559	semidirect
202. C₅	#5585	direct (N × H)
214. C₃	C₅×#559	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×Q₈	2
C₅×#947	2
C₅×#894	2
C₅×#893	2
C₅×#875	2
C₅×#874	2
C₄×C₅×D₁₂	2
C₄×C₅×Dic₆	2
C₅×#559	3
#5585	5

Subgroups — C5×#5585 #11755970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5585 #11755970