Subgroups — C₅×#69104 #14315121

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×#67	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#798	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2169	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₄×C₅×#67	C₄	1	1	yes	characteristic
11	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₅×#830	C₄	1	1	yes	characteristic
15	C₅×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
16	C₅×Dic₃⋊C₈	C₄	1	1	yes	characteristic
17	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
18	C₅×#812	—	2	2	no	—
19	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₅×#798	—	2	2	no	—
22	#69104	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×M₃₂	—	3	1	no	—
24	C₂×C₅×M₃₂	—	3	1	no	—
25	C₅×C₈×D₄	—	3	1	no	—
26	C₅×C₄⋊C₁₆	—	3	1	no	—
27	C₅×C₂²⋊C₁₆	—	3	1	no	—
28	C₅×C₂²⋊C₁₆	D₃	1	1	yes	characteristic, complemented
29	C₄×C₅×C₁₆	—	3	1	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×#67	C₈	1	1	yes	characteristic
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	—	2	1	no	—
34	C₂×C₃×C₅×C₈	—	2	1	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₅×#217	C₈	1	1	yes	characteristic
38	C₅×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
39	C₅×Dic₃⋊C₄	C₈	1	1	yes	characteristic
40	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₅×D₃×C₈	—	2	1	no	—
44	C₅×D₃×C₈	—	2	1	no	—
45	C₃×C₅×C₁₆	—	2	1	no	—
46	C₃×C₅×C₁₆	—	2	1	no	—
47	C₅×C₃⋊C₁₆	—	2	1	no	—
48	C₅×C₃⋊C₁₆	D₄	1	2	yes	complemented
49	C₈×#67	C₂×C₅	1	1	yes	characteristic
50	C₂×#812	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×#798	C₂×C₅	1	1	yes	characteristic, complemented
52	C₃×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
53	(C₂×D₃)⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
54	Dic₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
55	Dic₃×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂²×C₅×C₈	—	3	1	no	—
57	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
58	C₄×C₅×D₄	—	3	1	no	—
59	C₅×M₃₂	—	6	2	no	—
60	C₅×M₃₂	—	6	2	no	—
61	C₂×C₅×C₁₆	—	3	1	no	—
62	C₂×C₅×C₁₆	—	3	1	no	—
63	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
64	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
65	C₅×C₄⋊C₈	—	3	1	no	—
66	C₅×C₂²⋊C₈	—	3	1	no	—
67	C₅×C₂²⋊C₈	—	3	1	no	—
68	C₄×C₅×C₈	—	3	1	no	—
69	#2169	—	3	1	no	Sylow
70	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₅×#67	—	2	2	no	—
77	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
80	C₄×C₅×D₃	—	2	1	no	—
81	C₄×C₅×D₃	—	2	1	no	—
82	C₃×C₅×C₈	—	2	1	no	—
83	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₃×C₅×C₈	#41	1	1	yes	characteristic
85	C₅×C₃⋊C₈	—	2	1	no	—
86	C₅×C₃⋊C₈	—	2	1	no	—
87	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
88	C₄×#67	C₄×C₅	1	1	yes	characteristic
89	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
92	#830	C₄×C₅	1	1	yes	characteristic
93	(C₂×D₃)⋊C₈	C₄×C₅	1	1	yes	characteristic
94	Dic₃⋊C₈	C₄×C₅	1	1	yes	characteristic
95	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
96	#812	—	2	2	no	—
97	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
98	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
99	#798	—	2	2	no	—
100	C₂×C₅×D₄	—	3	1	no	—
101	C₂²×C₄×C₅	—	3	1	no	—
102	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
103	C₂×C₅×C₈	—	2	1	no	—
104	C₂×C₅×C₈	—	2	1	no	—
105	C₂×C₅×C₈	—	3	1	no	—
106	C₂×C₅×C₈	—	3	1	no	—
107	C₂×C₅×C₈	—	6	1	no	—
108	C₂×C₅×C₈	—	6	1	no	—
109	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
111	C₅×C₄⋊C₄	—	3	1	no	—
112	C₅×C₂²⋊C₄	—	3	1	no	—
113	C₅×C₂²⋊C₄	—	3	1	no	—
114	C₄²×C₅	—	3	1	no	—
115	C₅×C₁₆	—	2	1	no	—
116	C₅×C₁₆	—	2	1	no	—
117	C₅×C₁₆	—	3	2	no	—
118	C₅×C₁₆	—	6	1	no	—
119	C₂×M₃₂	—	3	1	no	—
120	C₂×M₃₂	—	3	1	no	—
121	C₈×D₄	—	3	1	no	—
122	C₄⋊C₁₆	—	3	1	no	—
123	C₂²⋊C₁₆	—	3	1	no	—
124	C₂²⋊C₁₆	C₅×D₃	1	1	yes	characteristic, complemented
125	C₄×C₁₆	—	3	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₅×D₃	—	2	1	no	—
131	C₂×C₅×D₃	—	2	1	no	—
132	C₂×C₅×D₃	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
135	C₃×C₄×C₅	#131	1	1	yes	characteristic
136	C₅×Dic₃	—	2	1	no	—
137	C₅×Dic₃	—	2	1	no	—
138	C₅×Dic₃	M₃₂	1	2	yes	complemented
139	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂×#67	C₅×C₈	1	1	yes	characteristic
141	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
146	#217	C₅×C₈	1	1	yes	characteristic
147	(C₂×D₃)⋊C₄	C₅×C₈	1	1	yes	characteristic
148	Dic₃⋊C₄	C₅×C₈	1	1	yes	characteristic
149	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
151	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
152	D₃×C₈	—	2	1	no	—
153	D₃×C₈	—	2	1	no	—
154	C₃×C₁₆	—	2	1	no	—
155	C₃×C₁₆	—	2	1	no	—
156	C₃⋊C₁₆	—	2	1	no	—
157	C₃⋊C₁₆	C₅×D₄	1	2	yes	complemented
158	C₂³×C₅	—	3	1	no	—
159	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
160	C₅×D₄	—	6	2	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	3	1	no	—
164	C₂×C₄×C₅	—	3	1	no	—
165	C₂×C₄×C₅	—	3	1	no	—
166	C₂×C₄×C₅	—	6	1	no	—
167	C₂×C₄×C₅	—	6	1	no	—
168	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	6	1	no	—
172	C₅×C₈	—	6	1	no	—
173	C₅×C₈	D₃×D₄	1	1	yes	characteristic
174	C₅×C₈	#237	1	1	yes	characteristic
175	C₂²×C₈	—	3	1	no	—
176	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
177	C₄×D₄	—	3	1	no	—
178	M₃₂	—	6	2	no	—
179	M₃₂	—	6	2	no	—
180	C₂×C₁₆	—	3	1	no	—
181	C₂×C₁₆	—	3	1	no	—
182	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
183	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
184	C₄⋊C₈	—	3	1	no	—
185	C₂²⋊C₈	—	3	1	no	—
186	C₂²⋊C₈	—	3	1	no	—
187	C₄×C₈	—	3	1	no	—
188	C₂×C₃×C₅	—	4	1	no	—
189	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
191	C₂×C₃×C₅	#504	1	1	yes	characteristic
192	C₅×D₃	—	4	1	no	—
193	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
194	C₂²×D₃	C₂×C₅×C₈	1	1	yes	characteristic
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
199	#67	—	2	2	no	—
200	C₂×Dic₃	C₂×C₅×C₈	1	1	yes	characteristic
201	C₂×Dic₃	C₂×C₅×C₈	1	1	yes	characteristic
202	C₂×Dic₃	C₂×C₅×C₈	1	1	yes	characteristic
203	C₄×D₃	—	2	1	no	—
204	C₄×D₃	—	2	1	no	—
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
207	C₃×C₈	C₅×#41	1	1	yes	characteristic
208	C₃⋊C₈	—	2	1	no	—
209	C₃⋊C₈	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	6	1	no	—
214	C₂²×C₅	—	6	1	no	—
215	C₂²×C₅	—	6	1	no	—
216	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	3	2	no	—
219	C₄×C₅	—	6	1	no	—
220	C₄×C₅	—	6	1	no	—
221	C₄×C₅	#67⋊C₄	1	1	yes	characteristic
222	C₄×C₅	#907	1	1	yes	characteristic
223	C₂×D₄	—	3	1	no	—
224	C₂²×C₄	—	3	1	no	—
225	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	3	1	no	—
229	C₂×C₈	—	3	1	no	—
230	C₂×C₈	—	6	1	no	—
231	C₂×C₈	—	6	1	no	—
232	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
234	C₄⋊C₄	—	3	1	no	—
235	C₂²⋊C₄	—	3	1	no	—
236	C₂²⋊C₄	—	3	1	no	—
237	C₄²	—	3	1	no	—
238	C₁₆	—	2	1	no	—
239	C₁₆	—	2	1	no	—
240	C₁₆	—	3	2	no	—
241	C₁₆	—	6	1	no	—
242	C₃×C₅	#2169	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
247	C₂×D₃	—	2	1	no	—
248	C₂×D₃	—	2	1	no	—
249	C₂×D₃	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
252	C₃×C₄	C₅×#131	1	1	yes	characteristic
253	Dic₃	—	2	1	no	—
254	Dic₃	—	2	1	no	—
255	Dic₃	C₅×M₃₂	1	2	yes	complemented
256	C₂×C₅	—	4	1	no	—
257	C₂×C₅	—	12	1	no	—
258	C₂×C₅	#67⋊C₈	1	1	yes	characteristic
259	C₂×C₅	#4901	1	1	yes	characteristic
260	C₂×C₅	D₃×M₃₂	1	1	yes	characteristic
261	C₂³	—	3	1	no	—
262	C₂³	C₅×D₃×C₈	1	1	yes	characteristic
263	D₄	—	6	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	3	1	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	6	1	no	—
270	C₂×C₄	—	6	1	no	—
271	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	6	1	no	—
275	C₈	—	6	1	no	—
276	C₈	C₅×D₃×D₄	1	1	yes	characteristic
277	C₈	C₅×#237	1	1	yes	characteristic
278	C₂×C₃	—	4	1	no	—
279	C₂×C₃	C₅×C₈×D₄	1	1	yes	characteristic
280	C₂×C₃	C₂×C₅×M₃₂	1	1	yes	characteristic
281	C₂×C₃	C₅×#504	1	1	yes	characteristic
282	D₃	—	4	1	no	—
283	C₅	#69104	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	6	1	no	—
288	C₂²	—	6	1	no	—
289	C₂²	—	6	1	no	—
290	C₂²	C₂×C₅×D₃×C₈	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	3	2	no	—
293	C₄	—	6	1	no	—
294	C₄	—	6	1	no	—
295	C₄	C₅×#67⋊C₄	1	1	yes	characteristic
296	C₄	C₅×#907	1	1	yes	characteristic
297	C₃	C₅×#2169	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	4	1	no	—
299	C₂	—	12	1	no	—
300	C₂	C₅×#67⋊C₈	1	1	yes	characteristic
301	C₂	C₅×#4901	1	1	yes	characteristic
302	C₂	C₅×D₃×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×C₂²⋊C₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₅×#67	C₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×C₂²⋊C₁₆	D₃
hypocenter	C₃	C₅×#2169
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₅×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#812	C₂	semidirect
3. C₂×C₅×#798	C₂	semidirect
4. C₃×C₅×C₂²⋊C₁₆	C₂	semidirect
5. C₅×(C₂×D₃)⋊C₁₆	C₂	semidirect
6. C₅×Dic₃⋊C₁₆	C₂	semidirect
7. C₅×Dic₃×C₁₆	C₂	semidirect
22. #69104	C₅	direct (N × H)
28. C₅×C₂²⋊C₁₆	D₃	semidirect
48. C₅×C₃⋊C₁₆	D₄	semidirect
50. C₂×#812	C₂×C₅	semidirect
51. C₂×#798	C₂×C₅	semidirect
52. C₃×C₂²⋊C₁₆	C₂×C₅	semidirect
53. (C₂×D₃)⋊C₁₆	C₂×C₅	semidirect
54. Dic₃⋊C₁₆	C₂×C₅	semidirect
55. Dic₃×C₁₆	C₂×C₅	semidirect
124. C₂²⋊C₁₆	C₅×D₃	semidirect
138. C₅×Dic₃	M₃₂	semidirect
157. C₃⋊C₁₆	C₅×D₄	semidirect
242. C₃×C₅	#2169	semidirect
255. Dic₃	C₅×M₃₂	semidirect
283. C₅	#69104	direct (N × H)
297. C₃	C₅×#2169	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₈×#67	2
C₂×C₅×#812	2
C₂×C₅×#798	2
C₃×C₅×C₂²⋊C₁₆	2
C₅×(C₂×D₃)⋊C₁₆	2
C₅×Dic₃⋊C₁₆	2
C₅×Dic₃×C₁₆	2
C₅×#2169	3
#69104	5

Subgroups — C5×#69104 #14315121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69104 #14315121