Subgroups — C₅×#5726 #11756111

Order 960.

584 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#239	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×#895	C₂	1	2	yes	maximal, complemented
5	C₅×#894	C₂	1	4	yes	maximal, complemented
6	C₅×D₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#547	—	3	1	no	maximal, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
11	C₅×#239	—	2	8	no	—
12	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
13	C₂×C₅×D₁₂	C₂²	1	4	yes	—
14	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
17	C₂×C₄×C₅×D₃	C₂²	1	4	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
22	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
24	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
27	#5726	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂×C₅×#41	—	3	2	no	—
29	C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
30	C₅×#123	—	3	4	no	—
31	C₅×C₂²⋊Q₈	—	3	2	no	—
32	C₅×#121	—	3	2	no	—
33	C₄×C₅×D₄	—	3	1	no	—
34	C₄×C₅×D₄	—	3	1	no	—
35	C₂×C₅×C₄⋊C₄	—	3	2	no	—
36	C₂²×C₅×D₃	C₂³	1	2	yes	—
37	C₂²×C₅×D₃	C₂³	1	2	yes	—
38	C₃×C₅×Q₈	—	2	4	no	—
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×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₅×Dic₃	C₂³	1	2	yes	—
44	C₂×C₅×Dic₃	C₂³	1	2	yes	—
45	C₅×D₁₂	—	2	8	no	—
46	C₅×D₁₂	—	2	2	no	—
47	C₅×D₁₂	D₄	1	4	yes	complemented
48	C₄×C₅×D₃	—	2	8	no	—
49	C₄×C₅×D₃	—	2	4	no	—
50	C₄×C₅×D₃	—	2	4	no	—
51	C₂×#239	C₂×C₅	1	2	yes	complemented
52	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
53	#946	C₂×C₅	1	2	yes	complemented
54	#895	C₂×C₅	1	2	yes	complemented
55	#894	C₂×C₅	1	4	yes	complemented
56	D₃×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
57	C₄×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
58	C₄×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
59	C₅×#41	—	6	8	no	—
60	C₂×C₅×Q₈	C₂×D₃	1	2	yes	—
61	C₂×C₅×D₄	—	3	1	no	—
62	C₂×C₅×D₄	—	3	1	no	—
63	C₂×C₅×D₄	—	3	4	no	—
64	C₂²×C₄×C₅	—	3	2	no	—
65	C₂²×C₄×C₅	—	3	4	no	—
66	C₂²×C₄×C₅	—	3	2	no	—
67	C₅×C₄⋊C₄	—	3	1	no	—
68	C₅×C₄⋊C₄	—	3	4	no	—
69	C₅×C₄⋊C₄	—	3	1	no	—
70	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
71	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
72	C₅×C₂²⋊C₄	—	3	2	no	—
73	C₅×C₂²⋊C₄	—	3	2	no	—
74	C₅×C₂²⋊C₄	—	3	4	no	—
75	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
76	#547	—	3	1	no	Sylow
77	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×C₅×D₃	—	2	2	no	—
79	C₂×C₅×D₃	—	2	2	no	—
80	C₂×C₅×D₃	—	2	2	no	—
81	C₂×C₅×D₃	—	2	2	no	—
82	C₂×C₅×D₃	—	2	2	no	—
83	C₂×C₅×D₃	C₂×D₄	1	4	yes	—
84	C₃×C₄×C₅	—	2	4	no	—
85	C₃×C₄×C₅	—	2	1	no	—
86	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₃×C₄×C₅	#41	1	2	yes	complemented
88	C₅×Dic₃	—	2	2	no	—
89	C₅×Dic₃	—	2	2	no	—
90	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
91	#239	—	2	8	no	—
92	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
93	C₂×D₁₂	C₂²×C₅	1	4	yes	—
94	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
96	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
97	C₂×C₄×D₃	C₂²×C₅	1	4	yes	—
98	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
99	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
100	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
101	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
102	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
103	(C₂×D₃)⋊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	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
107	C₂³×C₅	—	3	2	no	—
108	C₂³×C₅	—	3	2	no	—
109	C₅×Q₈	—	2	4	no	—
110	C₅×D₄	—	3	4	no	—
111	C₅×D₄	—	6	8	no	—
112	C₅×D₄	—	6	2	no	—
113	C₂×C₄×C₅	—	3	2	no	—
114	C₂×C₄×C₅	—	3	2	no	—
115	C₂×C₄×C₅	—	6	8	no	—
116	C₂×C₄×C₅	—	6	4	no	—
117	C₂×C₄×C₅	—	6	4	no	—
118	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂²×D₃	1	4	yes	—
120	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
122	C₂×#41	—	3	2	no	—
123	C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
124	#123	—	3	4	no	—
125	C₂²⋊Q₈	—	3	2	no	—
126	#121	—	3	2	no	—
127	C₄×D₄	—	3	1	no	—
128	C₄×D₄	—	3	1	no	—
129	C₂×C₄⋊C₄	—	3	2	no	—
130	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
132	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
133	C₅×D₃	—	2	4	no	—
134	C₅×D₃	—	4	2	no	—
135	C₂²×D₃	C₂³×C₅	1	2	yes	—
136	C₂²×D₃	C₂³×C₅	1	2	yes	—
137	C₃×Q₈	—	2	4	no	—
138	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
139	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
142	C₂×Dic₃	C₂³×C₅	1	2	yes	—
143	C₂×Dic₃	C₂³×C₅	1	2	yes	—
144	D₁₂	—	2	8	no	—
145	D₁₂	—	2	2	no	—
146	D₁₂	C₅×D₄	1	4	yes	complemented
147	C₄×D₃	—	2	8	no	—
148	C₄×D₃	—	2	4	no	—
149	C₄×D₃	—	2	4	no	—
150	C₂²×C₅	—	3	4	no	—
151	C₂²×C₅	—	6	2	no	—
152	C₂²×C₅	—	6	2	no	—
153	C₂²×C₅	—	6	2	no	—
154	C₂²×C₅	—	6	2	no	—
155	C₂²×C₅	—	6	2	no	—
156	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
157	C₄×C₅	—	2	4	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	6	2	no	—
160	C₄×C₅	—	6	2	no	—
161	C₄×C₅	D₃×D₄	1	2	yes	—
162	C₄×C₅	#239	1	2	yes	complemented
163	#41	—	6	8	no	—
164	C₂×Q₈	C₂×C₅×D₃	1	2	yes	—
165	C₂×D₄	—	3	1	no	—
166	C₂×D₄	—	3	1	no	—
167	C₂×D₄	—	3	4	no	—
168	C₂²×C₄	—	3	2	no	—
169	C₂²×C₄	—	3	4	no	—
170	C₂²×C₄	—	3	2	no	—
171	C₄⋊C₄	—	3	1	no	—
172	C₄⋊C₄	—	3	4	no	—
173	C₄⋊C₄	—	3	1	no	—
174	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
175	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
176	C₂²⋊C₄	—	3	2	no	—
177	C₂²⋊C₄	—	3	2	no	—
178	C₂²⋊C₄	—	3	4	no	—
179	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
180	C₃×C₅	#547	1	1	yes	characteristic, complemented, Hall
181	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
182	C₂×D₃	—	2	2	no	—
183	C₂×D₃	—	2	2	no	—
184	C₂×D₃	—	2	2	no	—
185	C₂×D₃	—	2	2	no	—
186	C₂×D₃	—	2	2	no	—
187	C₂×D₃	C₂×C₅×D₄	1	4	yes	—
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
191	C₃×C₄	C₅×#41	1	2	yes	complemented
192	Dic₃	—	2	2	no	—
193	Dic₃	—	2	2	no	—
194	C₂×C₅	—	6	4	no	—
195	C₂×C₅	—	12	2	no	—
196	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
197	C₂×C₅	C₂×#239	1	1	yes	characteristic
198	C₂×C₅	#1007	1	1	yes	characteristic
199	C₂³	—	3	2	no	—
200	C₂³	—	3	2	no	—
201	Q₈	—	2	4	no	—
202	D₄	—	3	4	no	—
203	D₄	—	6	8	no	—
204	D₄	—	6	2	no	—
205	C₂×C₄	—	3	2	no	—
206	C₂×C₄	—	3	2	no	—
207	C₂×C₄	—	6	8	no	—
208	C₂×C₄	—	6	4	no	—
209	C₂×C₄	—	6	4	no	—
210	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₅×D₃	1	4	yes	—
212	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
214	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
215	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
216	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
217	D₃	—	2	4	no	—
218	D₃	—	4	2	no	—
219	C₅	#5726	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	3	4	no	—
221	C₂²	—	6	2	no	—
222	C₂²	—	6	2	no	—
223	C₂²	—	6	2	no	—
224	C₂²	—	6	2	no	—
225	C₂²	—	6	2	no	—
226	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	1	no	—
229	C₄	—	6	2	no	—
230	C₄	—	6	2	no	—
231	C₄	C₅×D₃×D₄	1	2	yes	—
232	C₄	C₅×#239	1	2	yes	complemented
233	C₃	C₅×#547	1	1	yes	characteristic, complemented, Sylow
234	C₂	—	6	4	no	—
235	C₂	—	12	2	no	—
236	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
237	C₂	C₂×C₅×#239	1	1	yes	characteristic
238	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
hypercenter	C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#547
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×#239	C₂	semidirect
2. C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₅×#946	C₂	semidirect
4. C₅×#895	C₂	semidirect
5. C₅×#894	C₂	semidirect
6. C₅×D₃×C₄⋊C₄	C₂	semidirect
7. C₄×C₅×D₁₂	C₂	semidirect
8. C₄×C₅×D₁₂	C₂	semidirect
27. #5726	C₅	direct (N × H)
29. C₅×C₄⋊Q₈	D₃	semidirect
47. C₅×D₁₂	D₄	semidirect
51. C₂×#239	C₂×C₅	semidirect
52. C₃×C₄⋊Q₈	C₂×C₅	semidirect
53. #946	C₂×C₅	semidirect
54. #895	C₂×C₅	semidirect
55. #894	C₂×C₅	semidirect
56. D₃×C₄⋊C₄	C₂×C₅	semidirect
57. C₄×D₁₂	C₂×C₅	semidirect
58. C₄×D₁₂	C₂×C₅	semidirect
87. C₃×C₄×C₅	#41	semidirect
123. C₄⋊Q₈	C₅×D₃	semidirect
146. D₁₂	C₅×D₄	semidirect
162. C₄×C₅	#239	semidirect
180. C₃×C₅	#547	semidirect
191. C₃×C₄	C₅×#41	semidirect
219. C₅	#5726	direct (N × H)
232. C₄	C₅×#239	semidirect
233. C₃	C₅×#547	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#239	2
C₃×C₅×C₄⋊Q₈	2
C₅×#946	2
C₅×#895	2
C₅×#894	2
C₅×D₃×C₄⋊C₄	2
C₄×C₅×D₁₂	2
C₅×#547	3
#5726	5

Subgroups — C5×#5726 #11756111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5726 #11756111