Subgroups — C₅×#5728 #11756113

Order 960.

444 conjugacy classes of subgroups.

The subgroup lattice has 232 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×#946	C₂	1	2	yes	maximal
3	C₅×Q₈×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₅×#897	C₂	1	2	yes	maximal
5	C₅×#892	C₂	1	2	yes	maximal
6	C₅×#890	C₂	1	2	yes	maximal, complemented
7	C₅×#887	C₂	1	2	yes	maximal, complemented
8	C₄²×C₅×D₃	C₂	1	1	yes	maximal, characteristic
9	C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#533	—	3	1	no	maximal, Hall
11	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
12	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₆	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
20	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
21	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
22	C₅×C₄⋊Dic₃	C₂²	1	4	yes	—
23	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
24	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
25	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
26	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
27	C₄×C₅×Dic₃	C₂²	1	4	yes	—
28	#5728	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₅×C₄⋊Q₈	—	3	1	no	—
30	C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
31	C₅×#125	—	3	2	no	—
32	C₅×C₂²⋊Q₈	—	3	2	no	—
33	C₅×C₂²⋊Q₈	—	3	2	no	—
34	C₄×C₅×Q₈	—	3	2	no	—
35	C₄×C₅×Q₈	—	3	2	no	—
36	C₅×#117	—	3	2	no	—
37	C₂×C₄²×C₅	—	3	1	no	—
38	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
39	C₃×C₅×Q₈	—	2	4	no	—
40	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
45	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
46	C₂×C₅×Dic₃	C₂³	1	4	yes	—
47	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
48	C₄×C₅×D₃	—	2	2	no	—
49	C₄×C₅×D₃	—	2	2	no	—
50	C₄×C₅×D₃	Q₈	1	4	yes	complemented
51	C₅×Dic₆	—	2	4	no	—
52	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
53	#946	C₂×C₅	1	2	yes	—
54	Q₈×Dic₃	C₂×C₅	1	2	yes	complemented
55	#897	C₂×C₅	1	2	yes	—
56	#892	C₂×C₅	1	2	yes	—
57	#890	C₂×C₅	1	2	yes	complemented
58	#887	C₂×C₅	1	2	yes	complemented
59	C₄²×D₃	C₂×C₅	1	1	yes	characteristic
60	#869	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×C₅×Q₈	—	3	2	no	—
62	C₂×C₅×Q₈	C₂×D₃	1	2	yes	—
63	C₂²×C₄×C₅	—	3	1	no	—
64	C₂²×C₄×C₅	—	3	1	no	—
65	C₂²×C₄×C₅	—	3	1	no	—
66	C₅×C₄⋊C₄	—	3	2	no	—
67	C₅×C₄⋊C₄	—	3	4	no	—
68	C₅×C₄⋊C₄	—	3	4	no	—
69	C₅×C₄⋊C₄	—	3	2	no	—
70	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
71	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
72	C₅×C₂²⋊C₄	—	3	4	no	—
73	C₄²×C₅	—	3	1	no	—
74	C₄²×C₅	—	3	1	no	—
75	C₄²×C₅	—	3	1	no	—
76	C₄²×C₅	—	3	4	no	—
77	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
78	#533	—	3	1	no	Sylow
79	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×D₃	—	2	1	no	—
81	C₂×C₅×D₃	—	2	1	no	—
82	C₂×C₅×D₃	C₂×Q₈	1	2	yes	—
83	C₃×C₄×C₅	—	2	4	no	—
84	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
85	C₃×C₄×C₅	#41	1	2	yes	—
86	C₃×C₄×C₅	#41	1	2	yes	—
87	C₅×Dic₃	—	2	4	no	—
88	C₅×Dic₃	—	2	1	no	—
89	C₅×Dic₃	—	2	1	no	—
90	C₅×Dic₃	C₂×Q₈	1	2	yes	—
91	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
92	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
95	C₂×Dic₆	C₂²×C₅	1	2	yes	—
96	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
97	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
98	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
99	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
100	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
101	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
102	C₄⋊Dic₃	C₂²×C₅	1	4	yes	—
103	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
104	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
105	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
106	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
107	C₄×Dic₃	C₂²×C₅	1	4	yes	—
108	C₂³×C₅	—	3	1	no	—
109	C₅×Q₈	—	2	4	no	—
110	C₅×Q₈	—	6	4	no	—
111	C₂×C₄×C₅	—	3	1	no	—
112	C₂×C₄×C₅	—	3	4	no	—
113	C₂×C₄×C₅	—	3	1	no	—
114	C₂×C₄×C₅	—	3	1	no	—
115	C₂×C₄×C₅	—	3	4	no	—
116	C₂×C₄×C₅	—	6	2	no	—
117	C₂×C₄×C₅	—	6	2	no	—
118	C₂×C₄×C₅	C₂²×D₃	1	4	yes	—
119	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
122	C₄⋊Q₈	—	3	1	no	—
123	C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
124	#125	—	3	2	no	—
125	C₂²⋊Q₈	—	3	2	no	—
126	C₂²⋊Q₈	—	3	2	no	—
127	C₄×Q₈	—	3	2	no	—
128	C₄×Q₈	—	3	2	no	—
129	#117	—	3	2	no	—
130	C₂×C₄²	—	3	1	no	—
131	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
133	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
134	C₅×D₃	—	2	2	no	—
135	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
136	C₃×Q₈	—	2	4	no	—
137	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
138	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
141	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
142	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
143	C₂×Dic₃	C₂³×C₅	1	4	yes	—
144	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
145	C₄×D₃	—	2	2	no	—
146	C₄×D₃	—	2	2	no	—
147	C₄×D₃	C₅×Q₈	1	4	yes	complemented
148	Dic₆	—	2	4	no	—
149	C₂²×C₅	—	3	2	no	—
150	C₂²×C₅	—	6	1	no	—
151	C₂²×C₅	—	6	1	no	—
152	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
153	C₄×C₅	—	2	4	no	—
154	C₄×C₅	—	3	2	no	—
155	C₄×C₅	—	6	1	no	—
156	C₄×C₅	—	6	1	no	—
157	C₄×C₅	—	6	4	no	—
158	C₄×C₅	#237	1	2	yes	—
159	C₄×C₅	D₃×Q₈	1	2	yes	—
160	C₄×C₅	#239	1	2	yes	—
161	C₂×Q₈	—	3	2	no	—
162	C₂×Q₈	C₂×C₅×D₃	1	2	yes	—
163	C₂²×C₄	—	3	1	no	—
164	C₂²×C₄	—	3	1	no	—
165	C₂²×C₄	—	3	1	no	—
166	C₄⋊C₄	—	3	2	no	—
167	C₄⋊C₄	—	3	4	no	—
168	C₄⋊C₄	—	3	4	no	—
169	C₄⋊C₄	—	3	2	no	—
170	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
171	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
172	C₂²⋊C₄	—	3	4	no	—
173	C₄²	—	3	1	no	—
174	C₄²	—	3	1	no	—
175	C₄²	—	3	1	no	—
176	C₄²	—	3	4	no	—
177	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
178	C₃×C₅	#533	1	1	yes	characteristic, complemented, Hall
179	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
180	C₂×D₃	—	2	1	no	—
181	C₂×D₃	—	2	1	no	—
182	C₂×D₃	C₂×C₅×Q₈	1	2	yes	—
183	C₃×C₄	—	2	4	no	—
184	C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
185	C₃×C₄	C₅×#41	1	2	yes	—
186	C₃×C₄	C₅×#41	1	2	yes	—
187	Dic₃	—	2	4	no	—
188	Dic₃	—	2	1	no	—
189	Dic₃	—	2	1	no	—
190	Dic₃	C₂×C₅×Q₈	1	2	yes	—
191	C₂×C₅	—	6	2	no	—
192	C₂×C₅	C₂×#237	1	1	yes	characteristic
193	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
194	C₂×C₅	C₂×#239	1	1	yes	characteristic
195	C₂³	—	3	1	no	—
196	Q₈	—	2	4	no	—
197	Q₈	—	6	4	no	—
198	C₂×C₄	—	3	1	no	—
199	C₂×C₄	—	3	4	no	—
200	C₂×C₄	—	3	1	no	—
201	C₂×C₄	—	3	1	no	—
202	C₂×C₄	—	3	4	no	—
203	C₂×C₄	—	6	2	no	—
204	C₂×C₄	—	6	2	no	—
205	C₂×C₄	C₂²×C₅×D₃	1	4	yes	—
206	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
209	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
210	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
211	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
212	D₃	—	2	2	no	—
213	C₅	#5728	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	3	2	no	—
215	C₂²	—	6	1	no	—
216	C₂²	—	6	1	no	—
217	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	3	2	no	—
220	C₄	—	6	1	no	—
221	C₄	—	6	1	no	—
222	C₄	—	6	4	no	—
223	C₄	C₅×#237	1	2	yes	—
224	C₄	C₅×D₃×Q₈	1	2	yes	—
225	C₄	C₅×#239	1	2	yes	—
226	C₃	C₅×#533	1	1	yes	characteristic, complemented, Sylow
227	C₂	—	6	2	no	—
228	C₂	C₂×C₅×#237	1	1	yes	characteristic
229	C₂	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
230	C₂	C₂×C₅×#239	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	C₂²×C₅×D₃	C₂³
hypercenter	C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#533
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
6. C₅×#890	C₂	semidirect
7. C₅×#887	C₂	semidirect
9. C₅×#869	C₂	semidirect
28. #5728	C₅	direct (N × H)
30. C₅×C₄⋊Q₈	D₃	semidirect
50. C₄×C₅×D₃	Q₈	semidirect
52. C₃×C₄⋊Q₈	C₂×C₅	semidirect
54. Q₈×Dic₃	C₂×C₅	semidirect
57. #890	C₂×C₅	semidirect
58. #887	C₂×C₅	semidirect
60. #869	C₂×C₅	semidirect
123. C₄⋊Q₈	C₅×D₃	semidirect
147. C₄×D₃	C₅×Q₈	semidirect
178. C₃×C₅	#533	semidirect
213. C₅	#5728	direct (N × H)
226. C₃	C₅×#533	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×#946	2
C₅×Q₈×Dic₃	2
C₅×#897	2
C₅×#892	2
C₅×#890	2
C₅×#887	2
C₄²×C₅×D₃	2
C₅×#869	2
C₅×#533	3
#5728	5

Subgroups — C5×#5728 #11756113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5728 #11756113