Subgroups — C₅×#5732 #11756117

Order 960.

424 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	C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#947	C₂	1	2	yes	maximal
3	C₅×Q₈×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₅×#898	C₂	1	4	yes	maximal, complemented
5	C₅×#897	C₂	1	2	yes	maximal, complemented
6	C₅×#892	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×#556	—	3	1	no	maximal, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
11	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
13	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
18	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
19	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
20	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊Dic₃	C₂²	1	4	yes	—
23	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
24	C₄×C₅×Dic₃	C₂²	1	2	yes	—
25	C₄×C₅×Dic₃	C₂²	1	4	yes	—
26	#5732	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
28	C₅×#126	—	3	4	no	—
29	C₅×#124	—	3	2	no	—
30	C₅×C₂²⋊Q₈	—	3	2	no	—
31	C₄×C₅×Q₈	—	3	2	no	—
32	C₄×C₅×Q₈	—	3	1	no	—
33	C₄×C₅×D₄	—	3	1	no	—
34	C₅×#117	—	3	2	no	—
35	C₂²×C₅×D₃	C₂³	1	2	yes	—
36	C₃×C₅×Q₈	—	2	4	no	—
37	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
38	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₅×Dic₃	C₂³	1	2	yes	—
42	C₂×C₅×Dic₃	C₂³	1	2	yes	—
43	C₂×C₅×Dic₃	C₂³	1	2	yes	—
44	C₅×D₁₂	—	2	2	no	—
45	C₄×C₅×D₃	—	2	4	no	—
46	C₅×Dic₆	—	2	2	no	—
47	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
48	#947	C₂×C₅	1	2	yes	—
49	Q₈×Dic₃	C₂×C₅	1	2	yes	complemented
50	#898	C₂×C₅	1	4	yes	complemented
51	#897	C₂×C₅	1	2	yes	complemented
52	#892	C₂×C₅	1	2	yes	complemented
53	C₄×D₁₂	C₂×C₅	1	1	yes	characteristic
54	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×C₅×Q₈	—	3	1	no	—
56	C₂×C₅×Q₈	C₂×D₃	1	2	yes	—
57	C₂×C₅×D₄	—	3	1	no	—
58	C₂²×C₄×C₅	—	3	2	no	—
59	C₅×C₄⋊C₄	—	3	1	no	—
60	C₅×C₄⋊C₄	—	3	1	no	—
61	C₅×C₄⋊C₄	—	3	4	no	—
62	C₅×C₄⋊C₄	—	3	2	no	—
63	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
64	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
65	C₅×C₂²⋊C₄	—	3	4	no	—
66	C₅×C₂²⋊C₄	—	3	2	no	—
67	C₅×C₂²⋊C₄	—	3	4	no	—
68	C₄²×C₅	—	3	4	no	—
69	C₄²×C₅	—	3	2	no	—
70	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
71	#556	—	3	1	no	Sylow
72	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
73	C₂×C₅×D₃	—	2	2	no	—
74	C₂×C₅×D₃	—	2	2	no	—
75	C₂×C₅×D₃	—	2	2	no	—
76	C₃×C₄×C₅	—	2	4	no	—
77	C₃×C₄×C₅	—	2	1	no	—
78	C₃×C₄×C₅	#41	1	2	yes	—
79	C₃×C₄×C₅	#41	1	2	yes	—
80	C₅×Dic₃	—	2	2	no	—
81	C₅×Dic₃	—	2	2	no	—
82	C₅×Dic₃	—	2	2	no	—
83	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
84	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
86	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
88	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
89	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
90	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
91	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
92	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
93	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
94	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
95	C₄⋊Dic₃	C₂²×C₅	1	4	yes	—
96	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
97	C₄×Dic₃	C₂²×C₅	1	2	yes	—
98	C₄×Dic₃	C₂²×C₅	1	4	yes	—
99	C₂³×C₅	—	3	2	no	—
100	C₅×Q₈	—	2	4	no	—
101	C₅×Q₈	—	6	2	no	—
102	C₅×D₄	—	6	2	no	—
103	C₂×C₄×C₅	—	3	2	no	—
104	C₂×C₄×C₅	—	3	2	no	—
105	C₂×C₄×C₅	—	3	2	no	—
106	C₂×C₄×C₅	—	6	4	no	—
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×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
111	C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
112	#126	—	3	4	no	—
113	#124	—	3	2	no	—
114	C₂²⋊Q₈	—	3	2	no	—
115	C₄×Q₈	—	3	2	no	—
116	C₄×Q₈	—	3	1	no	—
117	C₄×D₄	—	3	1	no	—
118	#117	—	3	2	no	—
119	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
120	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
121	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
122	C₅×D₃	—	4	2	no	—
123	C₂²×D₃	C₂³×C₅	1	2	yes	—
124	C₃×Q₈	—	2	4	no	—
125	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
126	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
129	C₂×Dic₃	C₂³×C₅	1	2	yes	—
130	C₂×Dic₃	C₂³×C₅	1	2	yes	—
131	C₂×Dic₃	C₂³×C₅	1	2	yes	—
132	D₁₂	—	2	2	no	—
133	C₄×D₃	—	2	4	no	—
134	Dic₆	—	2	2	no	—
135	C₂²×C₅	—	6	2	no	—
136	C₂²×C₅	—	6	2	no	—
137	C₂²×C₅	—	6	2	no	—
138	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
139	C₄×C₅	—	2	4	no	—
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	—	6	2	no	—
142	C₄×C₅	—	6	2	no	—
143	C₄×C₅	—	6	2	no	—
144	C₄×C₅	#237	1	2	yes	—
145	C₄×C₅	#239	1	2	yes	—
146	C₂×Q₈	—	3	1	no	—
147	C₂×Q₈	C₂×C₅×D₃	1	2	yes	—
148	C₂×D₄	—	3	1	no	—
149	C₂²×C₄	—	3	2	no	—
150	C₄⋊C₄	—	3	1	no	—
151	C₄⋊C₄	—	3	1	no	—
152	C₄⋊C₄	—	3	4	no	—
153	C₄⋊C₄	—	3	2	no	—
154	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
155	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
156	C₂²⋊C₄	—	3	4	no	—
157	C₂²⋊C₄	—	3	2	no	—
158	C₂²⋊C₄	—	3	4	no	—
159	C₄²	—	3	4	no	—
160	C₄²	—	3	2	no	—
161	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
162	C₃×C₅	#556	1	1	yes	characteristic, complemented, Hall
163	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
164	C₂×D₃	—	2	2	no	—
165	C₂×D₃	—	2	2	no	—
166	C₂×D₃	—	2	2	no	—
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	C₅×#41	1	2	yes	—
170	C₃×C₄	C₅×#41	1	2	yes	—
171	Dic₃	—	2	2	no	—
172	Dic₃	—	2	2	no	—
173	Dic₃	—	2	2	no	—
174	C₂×C₅	—	12	2	no	—
175	C₂×C₅	C₂×#237	1	1	yes	characteristic
176	C₂×C₅	C₂×#239	1	1	yes	characteristic
177	C₂×C₅	#1007	1	1	yes	characteristic
178	C₂³	—	3	2	no	—
179	Q₈	—	2	4	no	—
180	Q₈	—	6	2	no	—
181	D₄	—	6	2	no	—
182	C₂×C₄	—	3	2	no	—
183	C₂×C₄	—	3	2	no	—
184	C₂×C₄	—	3	2	no	—
185	C₂×C₄	—	6	4	no	—
186	C₂×C₄	C₂²×C₅×D₃	1	4	yes	—
187	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
189	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
190	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
191	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
192	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
193	D₃	—	4	2	no	—
194	C₅	#5732	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	—	6	2	no	—
196	C₂²	—	6	2	no	—
197	C₂²	—	6	2	no	—
198	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	1	no	—
201	C₄	—	6	2	no	—
202	C₄	—	6	2	no	—
203	C₄	—	6	2	no	—
204	C₄	C₅×#237	1	2	yes	—
205	C₄	C₅×#239	1	2	yes	—
206	C₃	C₅×#556	1	1	yes	characteristic, complemented, Sylow
207	C₂	—	12	2	no	—
208	C₂	C₂×C₅×#237	1	1	yes	characteristic
209	C₂	C₂×C₅×#239	1	1	yes	characteristic
210	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×#556
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. C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₅×Q₈×Dic₃	C₂	semidirect
4. C₅×#898	C₂	semidirect
5. C₅×#897	C₂	semidirect
6. C₅×#892	C₂	semidirect
8. C₄×C₅×Dic₆	C₂	semidirect
26. #5732	C₅	direct (N × H)
27. C₅×C₄⋊Q₈	D₃	semidirect
47. C₃×C₄⋊Q₈	C₂×C₅	semidirect
49. Q₈×Dic₃	C₂×C₅	semidirect
50. #898	C₂×C₅	semidirect
51. #897	C₂×C₅	semidirect
52. #892	C₂×C₅	semidirect
54. C₄×Dic₆	C₂×C₅	semidirect
111. C₄⋊Q₈	C₅×D₃	semidirect
162. C₃×C₅	#556	semidirect
194. C₅	#5732	direct (N × H)
206. C₃	C₅×#556	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₄⋊Q₈	2
C₅×#947	2
C₅×Q₈×Dic₃	2
C₅×#898	2
C₅×#897	2
C₅×#892	2
C₄×C₅×D₁₂	2
C₄×C₅×Dic₆	2
C₅×#556	3
#5732	5

Subgroups — C5×#5732 #11756117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5732 #11756117