Subgroups — C₅×#5727 #11756112

Order 960.

560 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₃×Q₈	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#946	C₂	1	2	yes	maximal, complemented
4	C₅×#896	C₂	1	4	yes	maximal, complemented
5	C₅×#893	C₂	1	2	yes	maximal
6	C₅×#888	C₂	1	2	yes	maximal, complemented
7	C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
8	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×D₄×Q₈	—	3	1	no	maximal, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
11	C₅×D₃×Q₈	—	2	8	no	—
12	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
14	C₂×C₄×C₅×D₃	C₂²	1	4	yes	—
15	C₂×C₅×Dic₆	C₂²	1	4	yes	—
16	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
21	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
22	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
26	C₄×C₅×Dic₃	C₂²	1	2	yes	—
27	#5727	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂²×C₅×Q₈	—	3	2	no	—
29	C₅×C₄⋊Q₈	—	3	2	no	—
30	C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
31	C₅×C₂²⋊Q₈	—	3	2	no	—
32	C₅×C₂²⋊Q₈	—	3	4	no	—
33	C₄×C₅×Q₈	—	3	1	no	—
34	C₄×C₅×D₄	—	3	2	no	—
35	C₄×C₅×D₄	—	3	1	no	—
36	C₂²×C₅×D₃	C₂³	1	2	yes	—
37	C₃×C₅×Q₈	—	2	4	no	—
38	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₂³	1	4	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	C₂×C₅×Dic₃	C₂³	1	2	yes	—
45	C₅×D₁₂	Q₈	1	4	yes	complemented
46	C₄×C₅×D₃	—	2	4	no	—
47	C₄×C₅×D₃	—	2	8	no	—
48	C₅×Dic₆	—	2	8	no	—
49	C₅×Dic₆	D₄	1	4	yes	complemented
50	C₂×D₃×Q₈	C₂×C₅	1	2	yes	complemented
51	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
52	#946	C₂×C₅	1	2	yes	complemented
53	#896	C₂×C₅	1	4	yes	complemented
54	#893	C₂×C₅	1	2	yes	—
55	#888	C₂×C₅	1	2	yes	complemented
56	C₄×D₁₂	C₂×C₅	1	1	yes	characteristic
57	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₅×Q₈	—	3	1	no	—
59	C₂×C₅×Q₈	—	3	4	no	—
60	C₂×C₅×Q₈	—	6	8	no	—
61	C₂×C₅×Q₈	C₂×D₃	1	2	yes	—
62	C₂×C₅×D₄	—	3	1	no	—
63	C₂²×C₄×C₅	—	3	4	no	—
64	C₂²×C₄×C₅	—	3	2	no	—
65	C₅×C₄⋊C₄	—	3	1	no	—
66	C₅×C₄⋊C₄	—	3	2	no	—
67	C₅×C₄⋊C₄	—	3	1	no	—
68	C₅×C₄⋊C₄	—	3	4	no	—
69	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
70	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
71	C₅×C₂²⋊C₄	—	3	4	no	—
72	C₅×C₂²⋊C₄	—	3	2	no	—
73	C₄²×C₅	—	3	2	no	—
74	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
75	D₄×Q₈	—	3	1	no	Sylow
76	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
77	C₂×C₅×D₃	—	2	2	no	—
78	C₂×C₅×D₃	—	2	2	no	—
79	C₂×C₅×D₃	C₂×Q₈	1	4	yes	complemented
80	C₃×C₄×C₅	—	2	4	no	—
81	C₃×C₄×C₅	—	2	1	no	—
82	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₃×C₄×C₅	C₂×Q₈	1	2	yes	complemented
84	C₅×Dic₃	—	2	2	no	—
85	C₅×Dic₃	—	2	2	no	—
86	C₅×Dic₃	C₂×D₄	1	4	yes	—
87	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
88	D₃×Q₈	—	2	8	no	—
89	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
91	C₂×C₄×D₃	C₂²×C₅	1	4	yes	—
92	C₂×Dic₆	C₂²×C₅	1	4	yes	—
93	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
94	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
95	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
96	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
97	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
98	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
99	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
100	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
101	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
102	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
103	C₄×Dic₃	C₂²×C₅	1	2	yes	—
104	C₂³×C₅	—	3	2	no	—
105	C₅×Q₈	—	2	4	no	—
106	C₅×Q₈	—	3	4	no	—
107	C₅×Q₈	—	6	8	no	—
108	C₅×D₄	—	3	4	no	—
109	C₂×C₄×C₅	—	3	2	no	—
110	C₂×C₄×C₅	—	3	2	no	—
111	C₂×C₄×C₅	—	3	2	no	—
112	C₂×C₄×C₅	—	6	8	no	—
113	C₂×C₄×C₅	—	6	4	no	—
114	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×D₃	1	4	yes	—
116	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
118	C₂²×Q₈	—	3	2	no	—
119	C₄⋊Q₈	—	3	2	no	—
120	C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
121	C₂²⋊Q₈	—	3	2	no	—
122	C₂²⋊Q₈	—	3	4	no	—
123	C₄×Q₈	—	3	1	no	—
124	C₄×D₄	—	3	2	no	—
125	C₄×D₄	—	3	1	no	—
126	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
129	C₅×D₃	—	2	4	no	—
130	C₂²×D₃	C₂³×C₅	1	2	yes	—
131	C₃×Q₈	—	2	4	no	—
132	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
134	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
136	C₂×Dic₃	C₂³×C₅	1	2	yes	—
137	C₂×Dic₃	C₂³×C₅	1	2	yes	—
138	C₂×Dic₃	C₂³×C₅	1	2	yes	—
139	D₁₂	C₅×Q₈	1	4	yes	complemented
140	C₄×D₃	—	2	4	no	—
141	C₄×D₃	—	2	8	no	—
142	Dic₆	—	2	8	no	—
143	Dic₆	C₅×D₄	1	4	yes	complemented
144	C₂²×C₅	—	3	4	no	—
145	C₂²×C₅	—	6	2	no	—
146	C₂²×C₅	—	6	2	no	—
147	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
148	C₄×C₅	—	2	4	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	3	4	no	—
151	C₄×C₅	—	6	2	no	—
152	C₄×C₅	—	6	2	no	—
153	C₄×C₅	D₃×D₄	1	2	yes	—
154	C₄×C₅	D₃×Q₈	1	2	yes	complemented
155	C₂×Q₈	—	3	1	no	—
156	C₂×Q₈	—	3	4	no	—
157	C₂×Q₈	—	6	8	no	—
158	C₂×Q₈	C₂×C₅×D₃	1	2	yes	—
159	C₂×D₄	—	3	1	no	—
160	C₂²×C₄	—	3	4	no	—
161	C₂²×C₄	—	3	2	no	—
162	C₄⋊C₄	—	3	1	no	—
163	C₄⋊C₄	—	3	2	no	—
164	C₄⋊C₄	—	3	1	no	—
165	C₄⋊C₄	—	3	4	no	—
166	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
167	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
168	C₂²⋊C₄	—	3	4	no	—
169	C₂²⋊C₄	—	3	2	no	—
170	C₄²	—	3	2	no	—
171	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
172	C₃×C₅	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
173	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
174	C₂×D₃	—	2	2	no	—
175	C₂×D₃	—	2	2	no	—
176	C₂×D₃	C₂×C₅×Q₈	1	4	yes	complemented
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
180	C₃×C₄	C₂×C₅×Q₈	1	2	yes	complemented
181	Dic₃	—	2	2	no	—
182	Dic₃	—	2	2	no	—
183	Dic₃	C₂×C₅×D₄	1	4	yes	—
184	C₂×C₅	—	6	4	no	—
185	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
186	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
187	C₂×C₅	#1007	1	1	yes	characteristic
188	C₂³	—	3	2	no	—
189	Q₈	—	2	4	no	—
190	Q₈	—	3	4	no	—
191	Q₈	—	6	8	no	—
192	D₄	—	3	4	no	—
193	C₂×C₄	—	3	2	no	—
194	C₂×C₄	—	3	2	no	—
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	—	6	8	no	—
197	C₂×C₄	—	6	4	no	—
198	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₅×D₃	1	4	yes	—
200	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
201	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
202	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
203	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
204	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
205	D₃	—	2	4	no	—
206	C₅	#5727	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	3	4	no	—
208	C₂²	—	6	2	no	—
209	C₂²	—	6	2	no	—
210	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	1	no	—
213	C₄	—	3	4	no	—
214	C₄	—	6	2	no	—
215	C₄	—	6	2	no	—
216	C₄	C₅×D₃×D₄	1	2	yes	—
217	C₄	C₅×D₃×Q₈	1	2	yes	complemented
218	C₃	C₅×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
219	C₂	—	6	4	no	—
220	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
221	C₂	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
222	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	C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₅×D₁₂	C₂²
hypercenter	C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×D₄×Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×D₃×Q₈	C₂	semidirect
2. C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₅×#946	C₂	semidirect
4. C₅×#896	C₂	semidirect
6. C₅×#888	C₂	semidirect
8. C₄×C₅×Dic₆	C₂	semidirect
27. #5727	C₅	direct (N × H)
30. C₅×C₄⋊Q₈	D₃	semidirect
45. C₅×D₁₂	Q₈	semidirect
49. C₅×Dic₆	D₄	semidirect
50. C₂×D₃×Q₈	C₂×C₅	semidirect
51. C₃×C₄⋊Q₈	C₂×C₅	semidirect
52. #946	C₂×C₅	semidirect
53. #896	C₂×C₅	semidirect
55. #888	C₂×C₅	semidirect
57. C₄×Dic₆	C₂×C₅	semidirect
79. C₂×C₅×D₃	C₂×Q₈	semidirect
83. C₃×C₄×C₅	C₂×Q₈	semidirect
120. C₄⋊Q₈	C₅×D₃	semidirect
139. D₁₂	C₅×Q₈	semidirect
143. Dic₆	C₅×D₄	semidirect
154. C₄×C₅	D₃×Q₈	semidirect
172. C₃×C₅	D₄×Q₈	semidirect
176. C₂×D₃	C₂×C₅×Q₈	semidirect
180. C₃×C₄	C₂×C₅×Q₈	semidirect
206. C₅	#5727	direct (N × H)
217. C₄	C₅×D₃×Q₈	semidirect
218. C₃	C₅×D₄×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₃×Q₈	2
C₃×C₅×C₄⋊Q₈	2
C₅×#946	2
C₅×#896	2
C₅×#893	2
C₅×#888	2
C₄×C₅×D₁₂	2
C₄×C₅×Dic₆	2
C₅×D₄×Q₈	3
#5727	5

Subgroups — C5×#5727 #11756112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5727 #11756112