Subgroups — C₅×#78573 #14417546

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₅×#5516	C₂	1	2	yes	maximal
3	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
4	C₅×#5120	C₂	1	2	yes	maximal
5	C₅×#5106	C₂	1	1	yes	maximal, characteristic
6	C₅×#4716	C₂	1	4	yes	maximal, complemented
7	C₄×C₅×Dic₁₂	C₂	1	4	yes	maximal, complemented
8	C₅×#2950	—	3	1	no	maximal, Hall
9	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	C₅×#926	C₂²	1	2	yes	—
14	C₂×C₅×Dic₁₂	C₄	1	8	yes	complemented
15	C₂×C₅×Dic₁₂	C₂²	1	2	yes	—
16	C₂×C₅×Dic₁₂	C₂²	1	2	yes	—
17	C₅×#887	—	2	4	no	—
18	C₅×#877	—	2	4	no	—
19	C₄×C₅×Dic₆	C₂²	1	8	yes	—
20	C₃×C₅×#97	C₂²	1	2	yes	—
21	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
22	C₅×#818	C₂²	1	2	yes	—
23	C₅×#817	C₂²	1	2	yes	—
24	C₅×#816	C₂²	1	8	yes	—
25	#78573	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂²×C₅×Q₁₆	—	3	1	no	—
27	C₅×#519	—	3	2	no	—
28	C₅×#441	—	3	4	no	—
29	C₄×C₅×Q₁₆	—	3	4	no	—
30	C₅×#427	—	3	1	no	—
31	C₅×#419	—	3	2	no	—
32	C₅×#405	D₃	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂²×C₅×Dic₃	—	2	2	no	—
35	C₂×C₅×Dic₆	—	2	2	no	—
36	C₂×C₅×Dic₆	—	2	4	no	—
37	C₂×C₅×Dic₆	C₂×C₄	1	8	yes	—
38	C₂×C₅×Dic₆	C₂³	1	4	yes	—
39	C₃×C₅×M₁₆	—	2	2	no	—
40	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
45	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
46	C₃×C₄²×C₅	C₂³	1	2	yes	—
47	C₅×#217	—	2	2	no	—
48	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
49	C₅×Dic₃⋊C₄	—	2	8	no	—
50	C₄×C₅×Dic₃	—	2	8	no	—
51	C₄×C₅×Dic₃	—	2	2	no	—
52	C₅×Dic₁₂	C₂×C₄	1	8	yes	—
53	C₅×Dic₁₂	C₂×C₄	1	8	yes	—
54	C₂²×Dic₁₂	C₂×C₅	1	1	yes	characteristic
55	#5516	C₂×C₅	1	2	yes	—
56	C₃×#405	C₂×C₅	1	1	yes	characteristic
57	#5120	C₂×C₅	1	2	yes	—
58	#5106	C₂×C₅	1	1	yes	characteristic
59	#4716	C₂×C₅	1	4	yes	complemented
60	C₄×Dic₁₂	C₂×C₅	1	4	yes	complemented
61	C₂²×C₅×Q₈	—	3	2	no	—
62	C₂×C₅×Q₁₆	—	3	8	no	—
63	C₂×C₅×Q₁₆	—	3	2	no	—
64	C₂×C₅×Q₁₆	—	3	2	no	—
65	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₄×C₅×Q₈	—	3	8	no	—
68	C₄×C₅×Q₈	—	6	4	no	—
69	C₅×#117	—	3	2	no	—
70	C₅×#117	—	6	4	no	—
71	C₅×#117	C₂×D₃	1	1	yes	characteristic
72	C₅×#107	—	3	2	no	—
73	C₅×#106	—	3	2	no	—
74	C₅×#103	—	3	8	no	—
75	C₅×#97	C₂×D₃	1	2	yes	—
76	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
77	#2950	—	3	1	no	Sylow
78	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	#41	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
84	C₂×C₅×Dic₃	—	2	8	no	—
85	C₂×C₅×Dic₃	—	2	4	no	—
86	C₂×C₅×Dic₃	—	2	4	no	—
87	C₅×Dic₆	—	2	8	no	—
88	C₅×Dic₆	—	2	4	no	—
89	C₅×Dic₆	C₂²×C₄	1	8	yes	—
90	C₃×C₅×C₈	—	2	2	no	—
91	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
92	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
94	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
95	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
96	C₃×#117	C₂²×C₅	1	1	yes	characteristic
97	#926	C₂²×C₅	1	2	yes	—
98	C₂×Dic₁₂	C₄×C₅	1	8	yes	complemented
99	C₂×Dic₁₂	C₂²×C₅	1	2	yes	—
100	C₂×Dic₁₂	C₂²×C₅	1	2	yes	—
101	#887	—	2	4	no	—
102	#877	—	2	4	no	—
103	C₄×Dic₆	C₂²×C₅	1	8	yes	—
104	C₃×#97	C₂²×C₅	1	2	yes	—
105	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
106	#818	C₂²×C₅	1	2	yes	—
107	#817	C₂²×C₅	1	2	yes	—
108	#816	C₂²×C₅	1	8	yes	—
109	C₂×C₅×Q₈	—	3	8	no	—
110	C₂×C₅×Q₈	—	3	4	no	—
111	C₂×C₅×Q₈	—	6	2	no	—
112	C₂×C₅×Q₈	—	6	4	no	—
113	C₂²×C₄×C₅	—	6	2	no	—
114	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₅×Q₁₆	—	3	8	no	—
116	C₅×Q₁₆	—	3	8	no	—
117	C₅×M₁₆	—	2	2	no	—
118	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
119	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
120	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
122	C₅×C₄⋊C₄	—	3	4	no	—
123	C₅×C₄⋊C₄	—	6	8	no	—
124	C₅×C₄⋊C₄	D₁₂	1	2	yes	—
125	C₅×C₂²⋊C₄	—	6	2	no	—
126	C₅×C₂²⋊C₄	D₁₂	1	2	yes	—
127	C₄²×C₅	—	6	2	no	—
128	C₄²×C₅	—	6	8	no	—
129	C₄²×C₅	C₂²×D₃	1	2	yes	—
130	C₂²×Q₁₆	—	3	1	no	—
131	#519	—	3	2	no	—
132	#441	—	3	4	no	—
133	C₄×Q₁₆	—	3	4	no	—
134	#427	—	3	1	no	—
135	#419	—	3	2	no	—
136	#405	C₅×D₃	1	1	yes	characteristic
137	C₂²×C₃×C₅	—	2	2	no	—
138	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
139	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₃×C₄×C₅	—	2	4	no	—
141	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
142	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
144	C₅×Dic₃	—	2	8	no	—
145	C₅×Dic₃	—	4	4	no	—
146	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
147	C₂²×Dic₃	—	2	2	no	—
148	C₂×Dic₆	—	2	2	no	—
149	C₂×Dic₆	—	2	4	no	—
150	C₂×Dic₆	C₂×C₄×C₅	1	8	yes	—
151	C₂×Dic₆	C₂³×C₅	1	4	yes	—
152	C₃×M₁₆	—	2	2	no	—
153	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
154	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
155	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
156	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
157	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
158	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
159	C₃×C₄²	C₂³×C₅	1	2	yes	—
160	#217	—	2	2	no	—
161	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
162	Dic₃⋊C₄	—	2	8	no	—
163	C₄×Dic₃	—	2	8	no	—
164	C₄×Dic₃	—	2	2	no	—
165	Dic₁₂	C₂×C₄×C₅	1	8	yes	—
166	Dic₁₂	C₂×C₄×C₅	1	8	yes	—
167	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
168	C₅×Q₈	—	3	8	no	—
169	C₅×Q₈	—	6	8	no	—
170	C₅×Q₈	—	6	4	no	—
171	C₂×C₄×C₅	—	6	8	no	—
172	C₂×C₄×C₅	—	6	4	no	—
173	C₂×C₄×C₅	—	6	4	no	—
174	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
175	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
176	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
177	C₂×C₄×C₅	#235	1	2	yes	—
178	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
179	C₅×C₈	—	2	2	no	—
180	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
181	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
182	C₂²×Q₈	—	3	2	no	—
183	C₂×Q₁₆	—	3	8	no	—
184	C₂×Q₁₆	—	3	2	no	—
185	C₂×Q₁₆	—	3	2	no	—
186	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
188	C₄×Q₈	—	3	8	no	—
189	C₄×Q₈	—	6	4	no	—
190	#117	—	3	2	no	—
191	#117	—	6	4	no	—
192	#117	C₂×C₅×D₃	1	1	yes	characteristic
193	#107	—	3	2	no	—
194	#106	—	3	2	no	—
195	#103	—	3	8	no	—
196	#97	C₂×C₅×D₃	1	2	yes	—
197	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
198	C₂×C₃×C₅	—	2	2	no	—
199	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂×C₃×C₅	#578	1	2	yes	—
201	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
203	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
204	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₅×#41	1	2	yes	—
206	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
207	C₂×Dic₃	—	2	8	no	—
208	C₂×Dic₃	—	2	4	no	—
209	C₂×Dic₃	—	2	4	no	—
210	Dic₆	—	2	8	no	—
211	Dic₆	—	2	4	no	—
212	Dic₆	C₂²×C₄×C₅	1	8	yes	—
213	C₃×C₈	—	2	2	no	—
214	C₃×C₈	C₂²×C₄×C₅	1	2	yes	—
215	C₃×C₈	C₂²×C₄×C₅	1	2	yes	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	C₄×D₁₂	1	2	yes	—
218	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
219	C₄×C₅	—	2	4	no	—
220	C₄×C₅	—	6	8	no	—
221	C₄×C₅	—	12	4	no	—
222	C₄×C₅	C₄×D₁₂	1	2	yes	—
223	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
224	C₄×C₅	C₂×#235	1	1	yes	characteristic
225	C₂×Q₈	—	3	8	no	—
226	C₂×Q₈	—	3	4	no	—
227	C₂×Q₈	—	6	2	no	—
228	C₂×Q₈	—	6	4	no	—
229	C₂²×C₄	—	6	2	no	—
230	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
231	Q₁₆	—	3	8	no	—
232	Q₁₆	—	3	8	no	—
233	M₁₆	—	2	2	no	—
234	C₂×C₈	C₄×C₅×D₃	1	4	yes	—
235	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
236	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
237	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
238	C₄⋊C₄	—	3	4	no	—
239	C₄⋊C₄	—	6	8	no	—
240	C₄⋊C₄	C₅×D₁₂	1	2	yes	—
241	C₂²⋊C₄	—	6	2	no	—
242	C₂²⋊C₄	C₅×D₁₂	1	2	yes	—
243	C₄²	—	6	2	no	—
244	C₄²	—	6	8	no	—
245	C₄²	C₂²×C₅×D₃	1	2	yes	—
246	C₃×C₅	#2950	1	1	yes	characteristic, complemented, Hall
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
249	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
252	C₃×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
253	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
254	Dic₃	—	2	8	no	—
255	Dic₃	—	4	4	no	—
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	C₂×C₄×D₁₂	1	1	yes	characteristic
258	C₂×C₅	#5753	1	2	yes	—
259	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
260	Q₈	—	3	8	no	—
261	Q₈	—	6	8	no	—
262	Q₈	—	6	4	no	—
263	C₂×C₄	—	6	8	no	—
264	C₂×C₄	—	6	4	no	—
265	C₂×C₄	—	6	4	no	—
266	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
267	C₂×C₄	C₂×C₅×D₁₂	1	4	yes	—
268	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
269	C₂×C₄	C₅×#235	1	2	yes	—
270	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	C₂×C₄×C₅×D₃	1	2	yes	—
273	C₈	C₂×C₄×C₅×D₃	1	2	yes	—
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
276	C₂×C₃	C₅×#578	1	2	yes	—
277	C₅	#78573	1	1	yes	characteristic, direct product factor, Sylow
278	C₂²	—	2	2	no	—
279	C₂²	C₄×C₅×D₁₂	1	2	yes	—
280	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	6	8	no	—
283	C₄	—	12	4	no	—
284	C₄	C₄×C₅×D₁₂	1	2	yes	—
285	C₄	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
286	C₄	C₂×C₅×#235	1	1	yes	characteristic
287	C₃	C₅×#2950	1	1	yes	characteristic, complemented, Sylow
288	C₂	—	2	2	no	—
289	C₂	C₂×C₄×C₅×D₁₂	1	1	yes	characteristic
290	C₂	C₅×#5753	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×D₁₂
commutator	C₃×C₄	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#405	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#405	D₃
hypocenter	C₃	C₅×#2950
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405

Complements

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

N	H ≅ G/N	type
6. C₅×#4716	C₂	semidirect
7. C₄×C₅×Dic₁₂	C₂	semidirect
14. C₂×C₅×Dic₁₂	C₄	semidirect
25. #78573	C₅	direct (N × H)
59. #4716	C₂×C₅	semidirect
60. C₄×Dic₁₂	C₂×C₅	semidirect
98. C₂×Dic₁₂	C₄×C₅	semidirect
246. C₃×C₅	#2950	semidirect
277. C₅	#78573	direct (N × H)
287. C₃	C₅×#2950	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×Dic₁₂	2
C₅×#5516	2
C₃×C₅×#405	2
C₅×#5120	2
C₅×#5106	2
C₅×#4716	2
C₄×C₅×Dic₁₂	2
C₅×#2950	3
#78573	5

Subgroups — C5×#78573 #14417546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78573 #14417546