Subgroups — C₅×#67692 #14279152

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#408	C₂	1	1	yes	maximal, characteristic
2	C₅×#5240	C₂	1	1	yes	maximal, characteristic
3	C₅×#5237	C₂	1	1	yes	maximal, characteristic
4	C₅×#1395	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic, complemented
7	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic, complemented
8	C₅×#926	—	2	1	no	—
9	C₂×C₅×#208	—	2	1	no	—
10	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
11	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
12	C₅×#836	C₄	1	2	yes	complemented
13	C₅×#834	—	2	1	no	—
14	C₅×#834	C₄	1	2	yes	—
15	C₅×#833	—	2	1	no	—
16	#67692	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#413	—	3	1	no	—
18	C₅×#410	—	3	1	no	—
19	C₅×#408	D₃	1	1	yes	characteristic
20	C₃×C₅×#41	—	4	1	no	—
21	C₃×C₅×#41	—	4	1	no	—
22	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	Q₈	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₃×C₅×M₁₆	—	4	1	no	—
30	C₂×C₃×C₅×C₈	—	2	1	no	—
31	C₂×C₃×C₅×C₈	—	2	1	no	—
32	C₂×C₃×C₅×C₈	—	4	1	no	—
33	C₅×#217	—	2	2	no	—
34	C₅×C₄⋊Dic₃	—	4	1	no	—
35	C₄×C₅×Dic₃	—	4	1	no	—
36	C₅×#208	—	2	2	no	—
37	C₅×#208	—	4	1	no	—
38	C₂×C₅×C₃⋊C₈	—	4	1	no	—
39	C₃×#408	C₂×C₅	1	1	yes	characteristic
40	#5240	C₂×C₅	1	1	yes	characteristic
41	#5237	C₂×C₅	1	1	yes	characteristic
42	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
43	C₂×C₅×M₁₆	—	6	1	no	—
44	C₂×C₅×M₁₆	Dic₃	1	1	yes	characteristic, complemented
45	C₂²×C₅×C₈	Dic₃	1	1	yes	characteristic, complemented
46	C₅×#117	—	6	1	no	—
47	C₅×#101	—	3	2	no	—
48	C₅×#100	—	6	1	no	—
49	C₅×#99	—	3	2	no	—
50	C₅×#99	—	6	1	no	—
51	C₅×C₂²⋊C₈	—	2	1	no	—
52	C₅×C₂²⋊C₈	—	2	1	no	—
53	#1395	—	3	1	no	Sylow
54	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₃×C₅×Q₈	—	4	1	no	—
58	C₃×C₅×D₄	—	4	1	no	—
59	C₃×C₅×D₄	—	4	1	no	—
60	C₃×C₅×D₄	—	4	1	no	—
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	—	2	1	no	—
63	C₂×C₃×C₄×C₅	—	2	1	no	—
64	C₂×C₃×C₄×C₅	—	2	1	no	—
65	C₂×C₃×C₄×C₅	—	2	1	no	—
66	C₂×C₃×C₄×C₅	—	2	1	no	—
67	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₅×Dic₃	—	4	2	no	—
72	C₃×C₅×C₈	—	4	1	no	—
73	C₃×C₅×C₈	—	4	1	no	—
74	C₅×C₃⋊C₈	—	4	2	no	—
75	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic, complemented
77	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic, complemented
78	#926	—	2	1	no	—
79	C₂×#208	—	2	1	no	—
80	C₃×C₂²⋊C₈	—	2	1	no	—
81	C₃×C₂²⋊C₈	—	2	1	no	—
82	#836	C₄×C₅	1	2	yes	complemented
83	#834	—	2	1	no	—
84	#834	C₄×C₅	1	2	yes	—
85	#833	—	2	1	no	—
86	C₅×#41	—	4	1	no	—
87	C₅×#41	—	4	1	no	—
88	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
89	C₂×C₅×D₄	Dic₆	1	1	yes	characteristic
90	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
91	C₂×C₅×D₄	D₁₂	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
93	C₂²×C₄×C₅	#67	1	1	yes	characteristic
94	C₂²×C₄×C₅	#67	1	1	yes	characteristic
95	C₅×M₁₆	—	4	1	no	—
96	C₅×M₁₆	—	6	2	no	—
97	C₅×M₁₆	—	12	1	no	—
98	C₂×C₅×C₈	—	2	1	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	—	12	1	no	—
102	C₅×C₄⋊C₄	—	12	1	no	—
103	C₅×C₂²⋊C₄	—	6	2	no	—
104	C₄²×C₅	—	12	1	no	—
105	#413	—	3	1	no	—
106	#410	—	3	1	no	—
107	#408	C₅×D₃	1	1	yes	characteristic
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	—	2	1	no	—
111	C₂²×C₃×C₅	—	4	1	no	—
112	C₂²×C₃×C₅	—	4	1	no	—
113	C₂²×C₃×C₅	—	4	1	no	—
114	C₂²×C₃×C₅	#95	1	1	yes	characteristic
115	C₃×C₄×C₅	—	2	1	no	—
116	C₃×C₄×C₅	—	2	1	no	—
117	C₃×C₄×C₅	—	2	1	no	—
118	C₃×C₄×C₅	#99	1	1	yes	characteristic
119	C₃×C₄×C₅	#99	1	1	yes	characteristic
120	C₅×Dic₃	—	8	2	no	—
121	C₃×#41	—	4	1	no	—
122	C₃×#41	—	4	1	no	—
123	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
124	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
126	C₂×C₃×D₄	C₅×Q₈	1	1	yes	characteristic
127	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
130	C₃×M₁₆	—	4	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	4	1	no	—
134	#217	—	2	2	no	—
135	C₄⋊Dic₃	—	4	1	no	—
136	C₄×Dic₃	—	4	1	no	—
137	#208	—	2	2	no	—
138	#208	—	4	1	no	—
139	C₂×C₃⋊C₈	—	4	1	no	—
140	C₂³×C₅	Dic₃⋊C₄	1	1	yes	characteristic
141	C₂³×C₅	C₄⋊Dic₃	1	1	yes	characteristic
142	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
143	C₅×Q₈	—	4	1	no	—
144	C₅×D₄	—	4	1	no	—
145	C₅×D₄	—	4	1	no	—
146	C₅×D₄	—	4	1	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	12	2	no	—
154	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
155	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	#217	1	1	yes	characteristic
158	C₅×C₈	—	4	1	no	—
159	C₅×C₈	—	4	1	no	—
160	C₅×C₈	—	12	2	no	—
161	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
162	C₂×M₁₆	—	6	1	no	—
163	C₂×M₁₆	C₅×Dic₃	1	1	yes	characteristic, complemented
164	C₂²×C₈	C₅×Dic₃	1	1	yes	characteristic, complemented
165	#117	—	6	1	no	—
166	#101	—	3	2	no	—
167	#100	—	6	1	no	—
168	#99	—	3	2	no	—
169	#99	—	6	1	no	—
170	C₂²⋊C₈	—	2	1	no	—
171	C₂²⋊C₈	—	2	1	no	—
172	C₂×C₃×C₅	—	2	1	no	—
173	C₂×C₃×C₅	—	4	1	no	—
174	C₂×C₃×C₅	—	4	1	no	—
175	C₂×C₃×C₅	—	4	1	no	—
176	C₂×C₃×C₅	#342	1	1	yes	characteristic
177	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
178	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
179	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
180	C₃×Q₈	—	4	1	no	—
181	C₃×D₄	—	4	1	no	—
182	C₃×D₄	—	4	1	no	—
183	C₃×D₄	—	4	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
191	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₂×Dic₃	—	4	2	no	—
195	C₃×C₈	—	4	1	no	—
196	C₃×C₈	—	4	1	no	—
197	C₃⋊C₈	—	4	2	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	4	1	no	—
202	C₂²×C₅	—	4	1	no	—
203	C₂²×C₅	—	4	1	no	—
204	C₂²×C₅	C₃⋊#95	1	1	yes	characteristic
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	24	2	no	—
209	C₄×C₅	#834	1	1	yes	characteristic
210	C₄×C₅	#834	1	1	yes	characteristic
211	#41	—	4	1	no	—
212	#41	—	4	1	no	—
213	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
214	C₂×D₄	C₅×Dic₆	1	1	yes	characteristic
215	C₂×D₄	C₄×C₅×D₃	1	1	yes	characteristic
216	C₂×D₄	C₅×D₁₂	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
218	C₂²×C₄	C₅×#67	1	1	yes	characteristic
219	C₂²×C₄	C₅×#67	1	1	yes	characteristic
220	M₁₆	—	4	1	no	—
221	M₁₆	—	6	2	no	—
222	M₁₆	—	12	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	12	1	no	—
227	C₄⋊C₄	—	12	1	no	—
228	C₂²⋊C₄	—	6	2	no	—
229	C₄²	—	12	1	no	—
230	C₃×C₅	#1395	1	1	yes	characteristic, complemented, Hall
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	4	1	no	—
235	C₂²×C₃	—	4	1	no	—
236	C₂²×C₃	—	4	1	no	—
237	C₂²×C₃	C₅×#95	1	1	yes	characteristic
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	C₅×#99	1	1	yes	characteristic
242	C₃×C₄	C₅×#99	1	1	yes	characteristic
243	Dic₃	—	8	2	no	—
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	4	1	no	—
246	C₂×C₅	—	4	1	no	—
247	C₂×C₅	—	4	1	no	—
248	C₂×C₅	#4526	1	1	yes	characteristic
249	C₂³	C₅×Dic₃⋊C₄	1	1	yes	characteristic
250	C₂³	C₅×C₄⋊Dic₃	1	1	yes	characteristic
251	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
252	Q₈	—	4	1	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	12	2	no	—
263	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
264	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₅×#217	1	1	yes	characteristic
267	C₈	—	4	1	no	—
268	C₈	—	4	1	no	—
269	C₈	—	12	2	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	4	1	no	—
272	C₂×C₃	—	4	1	no	—
273	C₂×C₃	—	4	1	no	—
274	C₂×C₃	C₅×#342	1	1	yes	characteristic
275	C₅	#67692	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	C₅×C₃⋊#95	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	24	2	no	—
287	C₄	C₅×#834	1	1	yes	characteristic
288	C₄	C₅×#834	1	1	yes	characteristic
289	C₃	C₅×#1395	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	2	1	no	—
291	C₂	—	4	1	no	—
292	C₂	—	4	1	no	—
293	C₂	—	4	1	no	—
294	C₂	C₅×#4526	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#834
commutator	C₂×C₃×C₄	C₄²×C₅
Frattini	C₂×#41	C₂×C₅×D₃
Fitting	C₃×C₅×#408	C₂
socle	C₂×C₃×C₅	#342
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂×C₃×#41	C₂²×C₅
hypercenter	C₅×#408	D₃
hypocenter	C₃	C₅×#1395
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ C₅×#408

Complements

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

N	H ≅ G/N	type
6. C₂×C₃×C₅×M₁₆	C₄	semidirect
7. C₂²×C₃×C₅×C₈	C₄	semidirect
12. C₅×#836	C₄	semidirect
16. #67692	C₅	direct (N × H)
44. C₂×C₅×M₁₆	Dic₃	semidirect
45. C₂²×C₅×C₈	Dic₃	semidirect
76. C₂×C₃×M₁₆	C₄×C₅	semidirect
77. C₂²×C₃×C₈	C₄×C₅	semidirect
82. #836	C₄×C₅	semidirect
163. C₂×M₁₆	C₅×Dic₃	semidirect
164. C₂²×C₈	C₅×Dic₃	semidirect
230. C₃×C₅	#1395	semidirect
275. C₅	#67692	direct (N × H)
289. C₃	C₅×#1395	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#408	2
C₅×#5240	2
C₅×#5237	2
C₅×#1395	3
#67692	5

Subgroups — C5×#67692 #14279152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#67692 #14279152