Subgroups — C₅×D₄×Dic₆ #11755922

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	C₂²×C₅×Dic₆	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
3	C₅×D₄×Dic₃	C₂	1	2	yes	maximal
4	C₅×#924	C₂	1	2	yes	maximal, complemented
5	C₅×#888	C₂	1	2	yes	maximal, complemented
6	C₅×#878	C₂	1	4	yes	maximal, complemented
7	C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×D₄×Q₈	—	3	1	no	maximal, Hall
10	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
12	C₂²×C₅×Dic₃	C₂²	1	4	yes	—
13	C₂×C₅×Dic₆	—	2	8	no	—
14	C₂×C₅×Dic₆	C₂²	1	2	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	1	yes	characteristic
18	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₅×#217	C₂²	1	4	yes	—
21	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
23	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
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	D₄×Dic₆	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	—	3	1	no	—
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	D₃	1	1	yes	characteristic
36	C₂³×C₃×C₅	C₂³	1	2	yes	—
37	C₃×C₅×D₄	Q₈	1	4	yes	complemented
38	C₂×C₃×C₄×C₅	—	2	4	no	—
39	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₅×Dic₃	—	2	8	no	—
44	C₂×C₅×Dic₃	C₂³	1	2	yes	—
45	C₂×C₅×Dic₃	C₂³	1	4	yes	—
46	C₂×C₅×Dic₃	C₂³	1	2	yes	—
47	C₅×Dic₆	—	2	4	no	—
48	C₅×Dic₆	—	2	8	no	—
49	C₅×Dic₆	D₄	1	4	yes	direct product factor
50	C₂²×Dic₆	C₂×C₅	1	2	yes	complemented
51	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic
52	D₄×Dic₃	C₂×C₅	1	2	yes	—
53	#924	C₂×C₅	1	2	yes	complemented
54	#888	C₂×C₅	1	2	yes	complemented
55	#878	C₂×C₅	1	4	yes	complemented
56	#869	C₂×C₅	1	1	yes	characteristic, complemented
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	—	3	2	no	—
61	C₂×C₅×Q₈	—	6	8	no	—
62	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₄×C₅	—	3	4	no	—
64	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
65	C₅×C₄⋊C₄	—	3	2	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	—	3	2	no	—
70	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
71	C₅×C₂²⋊C₄	—	3	4	no	—
72	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
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	—	2	2	no	—
77	C₂²×C₃×C₅	—	2	2	no	—
78	C₂²×C₃×C₅	C₂×Q₈	1	4	yes	complemented
79	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
80	C₃×C₄×C₅	—	2	2	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	4	no	—
85	C₅×Dic₃	—	2	2	no	—
86	C₅×Dic₃	C₂×D₄	1	4	yes	—
87	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
89	C₂²×Dic₃	C₂²×C₅	1	4	yes	—
90	C₂×Dic₆	—	2	8	no	—
91	C₂×Dic₆	C₂²×C₅	1	2	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	1	yes	characteristic
95	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
96	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
97	#217	C₂²×C₅	1	4	yes	—
98	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
99	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
100	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
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	C₂²×D₃	1	2	yes	—
105	C₅×Q₈	—	3	4	no	—
106	C₅×Q₈	—	6	8	no	—
107	C₅×Q₈	—	6	4	no	—
108	C₅×D₄	Dic₆	1	4	yes	direct product factor
109	C₂×C₄×C₅	—	2	4	no	—
110	C₂×C₄×C₅	—	3	2	no	—
111	C₂×C₄×C₅	—	3	2	no	—
112	C₂×C₄×C₅	—	3	4	no	—
113	C₂×C₄×C₅	—	6	8	no	—
114	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×D₃	1	2	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	—	3	1	no	—
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	C₅×D₃	1	1	yes	characteristic
126	C₂×C₃×C₅	—	2	4	no	—
127	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂³×C₃	C₂³×C₅	1	2	yes	—
131	C₃×D₄	C₅×Q₈	1	4	yes	complemented
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
135	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
137	C₂×Dic₃	—	2	8	no	—
138	C₂×Dic₃	C₂³×C₅	1	2	yes	—
139	C₂×Dic₃	C₂³×C₅	1	4	yes	—
140	C₂×Dic₃	C₂³×C₅	1	2	yes	—
141	Dic₆	—	2	4	no	—
142	Dic₆	—	2	8	no	—
143	Dic₆	C₅×D₄	1	4	yes	direct product factor
144	C₂²×C₅	—	2	2	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	C₂×Dic₆	1	4	yes	complemented
147	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
148	C₄×C₅	—	2	2	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	3	4	no	—
151	C₄×C₅	—	6	4	no	—
152	C₄×C₅	—	6	2	no	—
153	C₄×C₅	C₂×Dic₆	1	2	yes	complemented
154	C₄×C₅	D₃×D₄	1	2	yes	—
155	C₂×Q₈	—	3	1	no	—
156	C₂×Q₈	—	3	4	no	—
157	C₂×Q₈	—	3	2	no	—
158	C₂×Q₈	—	6	8	no	—
159	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
160	C₂²×C₄	—	3	4	no	—
161	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
162	C₄⋊C₄	—	3	2	no	—
163	C₄⋊C₄	—	3	2	no	—
164	C₄⋊C₄	—	3	1	no	—
165	C₄⋊C₄	—	3	4	no	—
166	C₄⋊C₄	—	3	2	no	—
167	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
168	C₂²⋊C₄	—	3	4	no	—
169	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
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	—	2	2	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	C₂×C₅×Q₈	1	4	yes	complemented
176	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
177	C₃×C₄	—	2	2	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	4	no	—
182	Dic₃	—	2	2	no	—
183	Dic₃	C₂×C₅×D₄	1	4	yes	—
184	C₂×C₅	—	2	4	no	—
185	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
186	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
187	C₂×C₅	#1010	1	1	yes	characteristic
188	C₂³	C₂²×C₅×D₃	1	2	yes	—
189	Q₈	—	3	4	no	—
190	Q₈	—	6	8	no	—
191	Q₈	—	6	4	no	—
192	D₄	C₅×Dic₆	1	4	yes	direct product factor
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	3	2	no	—
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	—	3	4	no	—
197	C₂×C₄	—	6	8	no	—
198	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
200	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
201	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
202	C₂×C₃	—	2	4	no	—
203	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
204	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
205	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
206	C₅	D₄×Dic₆	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	C₂×C₅×Dic₆	1	4	yes	complemented
210	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	1	no	—
213	C₄	—	3	4	no	—
214	C₄	—	6	4	no	—
215	C₄	—	6	2	no	—
216	C₄	C₂×C₅×Dic₆	1	2	yes	complemented
217	C₄	C₅×D₃×D₄	1	2	yes	—
218	C₃	C₅×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
219	C₂	—	2	4	no	—
220	C₂	C₂²×C₅×Dic₆	1	1	yes	characteristic
221	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
222	C₂	C₅×#1010	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×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×D₄×Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×Dic₆	C₂	semidirect
4. C₅×#924	C₂	semidirect
5. C₅×#888	C₂	semidirect
6. C₅×#878	C₂	semidirect
7. C₅×#869	C₂	semidirect
8. C₄×C₅×Dic₆	C₂	semidirect
27. D₄×Dic₆	C₅	direct (N × H)
37. C₃×C₅×D₄	Q₈	semidirect
49. C₅×Dic₆	D₄	direct (N × H)
50. C₂²×Dic₆	C₂×C₅	semidirect
53. #924	C₂×C₅	semidirect
54. #888	C₂×C₅	semidirect
55. #878	C₂×C₅	semidirect
56. #869	C₂×C₅	semidirect
57. C₄×Dic₆	C₂×C₅	semidirect
78. C₂²×C₃×C₅	C₂×Q₈	semidirect
83. C₃×C₄×C₅	C₂×Q₈	semidirect
108. C₅×D₄	Dic₆	direct (N × H)
131. C₃×D₄	C₅×Q₈	semidirect
143. Dic₆	C₅×D₄	direct (N × H)
146. C₂²×C₅	C₂×Dic₆	semidirect
153. C₄×C₅	C₂×Dic₆	semidirect
172. C₃×C₅	D₄×Q₈	semidirect
175. C₂²×C₃	C₂×C₅×Q₈	semidirect
180. C₃×C₄	C₂×C₅×Q₈	semidirect
192. D₄	C₅×Dic₆	direct (N × H)
206. C₅	D₄×Dic₆	direct (N × H)
209. C₂²	C₂×C₅×Dic₆	semidirect
216. C₄	C₂×C₅×Dic₆	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
C₂²×C₅×Dic₆	2
C₃×C₄×C₅×D₄	2
C₅×D₄×Dic₃	2
C₅×#924	2
C₅×#888	2
C₅×#878	2
C₅×#869	2
C₄×C₅×Dic₆	2
C₅×D₄×Q₈	3
D₄×Dic₆	5

Subgroups — C5×D4×Dic6 #11755922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₄×Dic₆ #11755922