Subgroups — C₅×#5730 #11756115

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 216 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	4	yes	maximal, complemented
3	C₅×#897	C₂	1	2	yes	maximal, complemented
4	C₅×#893	C₂	1	2	yes	maximal
5	C₅×D₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₅×#890	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×#554	—	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×C₄×C₅×D₃	C₂²	1	4	yes	—
14	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	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	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
25	C₄×C₅×Dic₃	C₂²	1	2	yes	—
26	#5730	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
28	C₅×#125	—	3	2	no	—
29	C₅×C₂²⋊Q₈	—	3	4	no	—
30	C₅×C₂²⋊Q₈	—	3	2	no	—
31	C₄×C₅×Q₈	—	3	1	no	—
32	C₄×C₅×D₄	—	3	2	no	—
33	C₄×C₅×D₄	—	3	1	no	—
34	C₂×C₅×C₄⋊C₄	—	3	2	no	—
35	C₂²×C₅×D₃	C₂³	1	2	yes	—
36	C₃×C₅×Q₈	—	4	2	no	—
37	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
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	Q₈	1	4	yes	complemented
45	C₄×C₅×D₃	—	2	4	no	—
46	C₄×C₅×D₃	—	4	4	no	—
47	C₅×Dic₆	—	2	2	no	—
48	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
49	#946	C₂×C₅	1	4	yes	complemented
50	#897	C₂×C₅	1	2	yes	complemented
51	#893	C₂×C₅	1	2	yes	—
52	D₃×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
53	#890	C₂×C₅	1	2	yes	complemented
54	C₄×D₁₂	C₂×C₅	1	1	yes	characteristic
55	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂×C₅×Q₈	—	3	1	no	—
57	C₂×C₅×Q₈	C₂×D₃	1	2	yes	—
58	C₂×C₅×D₄	—	3	1	no	—
59	C₂²×C₄×C₅	—	3	2	no	—
60	C₂²×C₄×C₅	—	3	4	no	—
61	C₅×C₄⋊C₄	—	3	1	no	—
62	C₅×C₄⋊C₄	—	3	4	no	—
63	C₅×C₄⋊C₄	—	3	1	no	—
64	C₅×C₄⋊C₄	—	3	2	no	—
65	C₅×C₄⋊C₄	—	3	4	no	—
66	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
67	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
68	C₅×C₂²⋊C₄	—	3	4	no	—
69	C₅×C₂²⋊C₄	—	3	2	no	—
70	C₄²×C₅	—	3	2	no	—
71	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
72	#554	—	3	1	no	Sylow
73	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
74	C₂×C₅×D₃	—	2	2	no	—
75	C₂×C₅×D₃	—	2	2	no	—
76	C₂×C₅×D₃	C₂×Q₈	1	4	yes	—
77	C₃×C₄×C₅	—	2	4	no	—
78	C₃×C₄×C₅	—	2	1	no	—
79	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
80	C₃×C₄×C₅	#41	1	2	yes	—
81	C₅×Dic₃	—	2	2	no	—
82	C₅×Dic₃	—	2	2	no	—
83	C₅×Dic₃	—	2	2	no	—
84	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
85	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
86	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
87	C₂×C₄×D₃	C₂²×C₅	1	4	yes	—
88	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
89	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
90	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
91	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
92	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
93	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
94	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
95	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
96	C₄⋊Dic₃	C₂²×C₅	1	4	yes	—
97	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
98	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
99	C₄×Dic₃	C₂²×C₅	1	2	yes	—
100	C₂³×C₅	—	3	2	no	—
101	C₅×Q₈	—	4	2	no	—
102	C₅×Q₈	—	6	2	no	—
103	C₅×D₄	—	3	4	no	—
104	C₂×C₄×C₅	—	3	2	no	—
105	C₂×C₄×C₅	—	3	2	no	—
106	C₂×C₄×C₅	—	3	2	no	—
107	C₂×C₄×C₅	—	6	4	no	—
108	C₂×C₄×C₅	—	12	4	no	—
109	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²×D₃	1	4	yes	—
111	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
114	#125	—	3	2	no	—
115	C₂²⋊Q₈	—	3	4	no	—
116	C₂²⋊Q₈	—	3	2	no	—
117	C₄×Q₈	—	3	1	no	—
118	C₄×D₄	—	3	2	no	—
119	C₄×D₄	—	3	1	no	—
120	C₂×C₄⋊C₄	—	3	2	no	—
121	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
123	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
124	C₅×D₃	—	2	4	no	—
125	C₂²×D₃	C₂³×C₅	1	2	yes	—
126	C₃×Q₈	—	4	2	no	—
127	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂³×C₅	1	4	yes	—
129	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
131	C₂×Dic₃	C₂³×C₅	1	2	yes	—
132	C₂×Dic₃	C₂³×C₅	1	2	yes	—
133	C₂×Dic₃	C₂³×C₅	1	2	yes	—
134	D₁₂	C₅×Q₈	1	4	yes	complemented
135	C₄×D₃	—	2	4	no	—
136	C₄×D₃	—	4	4	no	—
137	Dic₆	—	2	2	no	—
138	C₂²×C₅	—	3	4	no	—
139	C₂²×C₅	—	6	2	no	—
140	C₂²×C₅	—	6	2	no	—
141	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
142	C₄×C₅	—	2	4	no	—
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	—	6	2	no	—
145	C₄×C₅	—	6	2	no	—
146	C₄×C₅	—	6	2	no	—
147	C₄×C₅	D₃×Q₈	1	2	yes	—
148	C₄×C₅	#239	1	2	yes	—
149	C₂×Q₈	—	3	1	no	—
150	C₂×Q₈	C₂×C₅×D₃	1	2	yes	—
151	C₂×D₄	—	3	1	no	—
152	C₂²×C₄	—	3	2	no	—
153	C₂²×C₄	—	3	4	no	—
154	C₄⋊C₄	—	3	1	no	—
155	C₄⋊C₄	—	3	4	no	—
156	C₄⋊C₄	—	3	1	no	—
157	C₄⋊C₄	—	3	2	no	—
158	C₄⋊C₄	—	3	4	no	—
159	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
160	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
161	C₂²⋊C₄	—	3	4	no	—
162	C₂²⋊C₄	—	3	2	no	—
163	C₄²	—	3	2	no	—
164	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
165	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
166	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
167	C₂×D₃	—	2	2	no	—
168	C₂×D₃	—	2	2	no	—
169	C₂×D₃	C₂×C₅×Q₈	1	4	yes	—
170	C₃×C₄	—	2	4	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
173	C₃×C₄	C₅×#41	1	2	yes	—
174	Dic₃	—	2	2	no	—
175	Dic₃	—	2	2	no	—
176	Dic₃	—	2	2	no	—
177	C₂×C₅	—	6	4	no	—
178	C₂×C₅	#1004	1	1	yes	characteristic
179	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
180	C₂×C₅	C₂×#239	1	1	yes	characteristic
181	C₂³	—	3	2	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	6	2	no	—
184	D₄	—	3	4	no	—
185	C₂×C₄	—	3	2	no	—
186	C₂×C₄	—	3	2	no	—
187	C₂×C₄	—	3	2	no	—
188	C₂×C₄	—	6	4	no	—
189	C₂×C₄	—	12	4	no	—
190	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₅×D₃	1	4	yes	—
192	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
194	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
195	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
196	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
197	D₃	—	2	4	no	—
198	C₅	#5730	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	—	3	4	no	—
200	C₂²	—	6	2	no	—
201	C₂²	—	6	2	no	—
202	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	1	no	—
205	C₄	—	6	2	no	—
206	C₄	—	6	2	no	—
207	C₄	—	6	2	no	—
208	C₄	C₅×D₃×Q₈	1	2	yes	—
209	C₄	C₅×#239	1	2	yes	—
210	C₃	C₅×#554	1	1	yes	characteristic, complemented, Sylow
211	C₂	—	6	4	no	—
212	C₂	C₅×#1004	1	1	yes	characteristic
213	C₂	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
214	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	C₂×C₅×D₁₂	C₂²
hypercenter	C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#554
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
2. C₅×#946	C₂	semidirect
3. C₅×#897	C₂	semidirect
5. C₅×D₃×C₄⋊C₄	C₂	semidirect
6. C₅×#890	C₂	semidirect
8. C₄×C₅×Dic₆	C₂	semidirect
26. #5730	C₅	direct (N × H)
27. C₅×C₄⋊Q₈	D₃	semidirect
44. C₅×D₁₂	Q₈	semidirect
48. C₃×C₄⋊Q₈	C₂×C₅	semidirect
49. #946	C₂×C₅	semidirect
50. #897	C₂×C₅	semidirect
52. D₃×C₄⋊C₄	C₂×C₅	semidirect
53. #890	C₂×C₅	semidirect
55. C₄×Dic₆	C₂×C₅	semidirect
113. C₄⋊Q₈	C₅×D₃	semidirect
134. D₁₂	C₅×Q₈	semidirect
165. C₃×C₅	#554	semidirect
198. C₅	#5730	direct (N × H)
210. C₃	C₅×#554	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×#897	2
C₅×#893	2
C₅×D₃×C₄⋊C₄	2
C₅×#890	2
C₄×C₅×D₁₂	2
C₄×C₅×Dic₆	2
C₅×#554	3
#5730	5

Subgroups — C5×#5730 #11756115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5730 #11756115