Subgroups — C₅×#71624 #14280927

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	2	yes	maximal
3	C₅×#4528	C₂	1	4	yes	maximal
4	C₅×#1461	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅×Q₈	—	2	2	no	—
7	C₃×C₄×C₅×Q₈	—	2	2	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#830	C₂²	1	4	yes	—
13	C₅×C₄⋊(C₃⋊C₈)	—	2	2	no	—
14	C₄×C₅×C₃⋊C₈	—	2	2	no	—
15	#71624	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₄×C₅×Q₈	D₃	1	1	yes	characteristic
17	C₅×#431	—	3	2	no	—
18	C₅×#324	—	3	4	no	—
19	C₂×C₃×C₅×Q₈	—	2	2	no	—
20	C₂×C₃×C₅×Q₈	—	2	2	no	—
21	C₂×C₃×C₅×Q₈	C₈	1	4	yes	complemented
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₃×C₅×C₄⋊C₄	—	2	2	no	—
28	C₃×C₅×C₄⋊C₄	—	2	4	no	—
29	C₃×C₄²×C₅	—	2	2	no	—
30	C₃×C₄²×C₅	—	2	2	no	—
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₃×C₄²×C₅	D₄	1	2	yes	—
33	C₂×C₅×C₃⋊C₈	—	2	4	no	—
34	C₂×C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic
35	#4925	C₂×C₅	1	2	yes	—
36	#4528	C₂×C₅	1	4	yes	—
37	C₂²×C₅×Q₈	Dic₃	1	1	yes	characteristic
38	C₄×C₅×Q₈	—	2	2	no	—
39	C₄×C₅×Q₈	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	Dic₃	1	1	yes	characteristic
41	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
42	C₂×C₄²×C₅	Dic₃	1	2	yes	—
43	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
44	C₅×C₄⋊C₈	—	6	2	no	—
45	C₅×C₂²⋊C₈	—	3	4	no	—
46	C₄×C₅×C₈	—	6	2	no	—
47	#1461	—	3	1	no	Sylow
48	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	2	8	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	4	1	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
60	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
61	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₅×C₃⋊C₈	—	4	4	no	—
64	C₂²×C₃×Q₈	C₄×C₅	1	1	yes	characteristic
65	C₃×C₄×Q₈	—	2	2	no	—
66	C₃×C₄×Q₈	—	2	2	no	—
67	C₂×C₃×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
68	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
69	C₂×C₃×C₄²	C₄×C₅	1	2	yes	—
70	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
71	#830	C₂²×C₅	1	4	yes	—
72	C₄⋊(C₃⋊C₈)	—	2	2	no	—
73	C₄×C₃⋊C₈	—	2	2	no	—
74	C₂×C₅×Q₈	—	2	2	no	—
75	C₂×C₅×Q₈	—	2	2	no	—
76	C₂×C₅×Q₈	C₃⋊C₈	1	4	yes	complemented
77	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
78	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
80	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
82	C₂×C₅×C₈	—	6	4	no	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₄⋊C₄	—	2	4	no	—
85	C₄²×C₅	—	2	2	no	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	#67	1	2	yes	—
88	C₄²×C₅	#67	1	2	yes	—
89	C₂×C₄×Q₈	C₅×D₃	1	1	yes	characteristic
90	#431	—	3	2	no	—
91	#324	—	3	4	no	—
92	C₂²×C₃×C₅	—	2	1	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
97	C₃×C₄×C₅	—	2	4	no	—
98	C₃×C₄×C₅	—	2	2	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	4	2	no	—
101	C₃×C₄×C₅	C₂²⋊C₈	1	4	yes	—
102	C₃×C₄×C₅	#101	1	2	yes	—
103	C₂×C₃×Q₈	—	2	2	no	—
104	C₂×C₃×Q₈	—	2	2	no	—
105	C₂×C₃×Q₈	C₅×C₈	1	4	yes	complemented
106	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
108	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
109	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄²	—	2	2	no	—
114	C₃×C₄²	—	2	2	no	—
115	C₃×C₄²	C₅×D₄	1	2	yes	—
116	C₃×C₄²	C₅×D₄	1	2	yes	—
117	C₂×C₃⋊C₈	—	2	4	no	—
118	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
119	C₅×Q₈	—	2	8	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	4	1	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
127	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
128	C₂×C₄×C₅	#208	1	2	yes	—
129	C₂×C₄×C₅	#217	1	1	yes	characteristic
130	C₂×C₄×C₅	#217	1	2	yes	—
131	C₂×C₄×C₅	#217	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×#67	1	2	yes	—
133	C₅×C₈	—	12	4	no	—
134	C₂²×Q₈	C₅×Dic₃	1	1	yes	characteristic
135	C₄×Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₂×C₄⋊C₄	C₅×Dic₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
139	C₂×C₄²	C₅×Dic₃	1	2	yes	—
140	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
141	C₄⋊C₈	—	6	2	no	—
142	C₂²⋊C₈	—	3	4	no	—
143	C₄×C₈	—	6	2	no	—
144	C₂×C₃×C₅	—	2	1	no	—
145	C₂×C₃×C₅	—	2	1	no	—
146	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₂×C₃×C₅	#410	1	1	yes	characteristic
148	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
149	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
150	C₃×Q₈	—	2	8	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	4	1	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
159	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
161	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
162	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
163	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
164	C₃⋊C₈	—	4	4	no	—
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
168	C₂²×C₅	C₂×#208	1	1	yes	characteristic
169	C₂²×C₅	C₂×#217	1	1	yes	characteristic
170	C₄×C₅	—	2	4	no	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	—	2	2	no	—
173	C₄×C₅	—	4	2	no	—
174	C₄×C₅	#830	1	4	yes	—
175	C₄×C₅	#836	1	2	yes	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	C₅×C₃⋊C₈	1	4	yes	complemented
179	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
180	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
182	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
184	C₂×C₈	—	6	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	C₅×#67	1	2	yes	—
190	C₄²	C₅×#67	1	2	yes	—
191	C₃×C₅	#1461	1	1	yes	characteristic, complemented, Hall
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
195	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
196	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
197	C₃×C₄	—	2	4	no	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	4	2	no	—
201	C₃×C₄	C₅×C₂²⋊C₈	1	4	yes	—
202	C₃×C₄	C₅×#101	1	2	yes	—
203	C₂×C₅	—	2	1	no	—
204	C₂×C₅	—	2	1	no	—
205	C₂×C₅	C₂×#830	1	1	yes	characteristic
206	C₂×C₅	C₂×#836	1	1	yes	characteristic
207	C₂×C₅	#5240	1	1	yes	characteristic
208	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
209	Q₈	—	2	8	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×C₅×C₃⋊C₈	1	4	yes	—
217	C₂×C₄	C₂×C₅×C₃⋊C₈	1	2	yes	—
218	C₂×C₄	C₅×#208	1	2	yes	—
219	C₂×C₄	C₅×#217	1	1	yes	characteristic
220	C₂×C₄	C₅×#217	1	2	yes	—
221	C₂×C₄	C₅×#217	1	1	yes	characteristic
222	C₂×C₄	C₂×C₅×#67	1	2	yes	—
223	C₈	—	12	4	no	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	—	2	1	no	—
226	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
227	C₂×C₃	C₅×#410	1	1	yes	characteristic
228	C₂×C₃	C₂×C₅×#101	1	1	yes	characteristic
229	C₅	#71624	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	C₂²×C₅×C₃⋊C₈	1	1	yes	characteristic
233	C₂²	C₂×C₅×#208	1	1	yes	characteristic
234	C₂²	C₂×C₅×#217	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	4	2	no	—
239	C₄	C₅×#830	1	4	yes	—
240	C₄	C₅×#836	1	2	yes	—
241	C₃	C₅×#1461	1	1	yes	characteristic, complemented, Sylow
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	C₂×C₅×#830	1	1	yes	characteristic
245	C₂	C₂×C₅×#836	1	1	yes	characteristic
246	C₂	C₅×#5240	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#217
commutator	C₂²×C₃	C₂²×C₅×C₈
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₄×C₅×Q₈	D₃
hypocenter	C₃	C₅×#1461
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
15. #71624	C₅	direct (N × H)
21. C₂×C₃×C₅×Q₈	C₈	semidirect
76. C₂×C₅×Q₈	C₃⋊C₈	semidirect
105. C₂×C₃×Q₈	C₅×C₈	semidirect
178. C₂×Q₈	C₅×C₃⋊C₈	semidirect
191. C₃×C₅	#1461	semidirect
229. C₅	#71624	direct (N × H)
241. C₃	C₅×#1461	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×Q₈	2
C₅×#4925	2
C₅×#4528	2
C₅×#1461	3
#71624	5

Subgroups — C5×#71624 #14280927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71624 #14280927