Subgroups — C₅×#77169 #14331622

Order 1920.

1888 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#239	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#893	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#395	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5231	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4991	C₂	1	3	yes	maximal, complemented
6	C₅×#4671	C₂	1	3	yes	maximal, complemented
7	C₅×#4666	C₂	1	3	yes	maximal, complemented
8	C₅×#2388	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#239	—	2	12	no	—
11	C₂²×C₅×D₁₂	C₂²	1	3	yes	—
12	C₂²×C₅×D₁₂	C₂²	1	3	yes	—
13	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅×D₃	C₂²	1	3	yes	—
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
16	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	6	yes	—
17	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	3	yes	—
18	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	3	yes	—
19	C₂×C₄×C₅×Dic₃	C₂²	1	3	yes	—
20	C₅×#893	—	2	12	no	—
21	C₃×C₅×#95	C₂²	1	3	yes	—
22	C₅×C₃⋊#95	C₂²	1	3	yes	—
23	#77169	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂²×C₅×#41	—	3	1	no	—
25	C₂×C₄×C₅×D₄	—	3	3	no	—
26	C₅×#395	D₃	1	1	yes	characteristic, complemented
27	C₅×#394	—	3	3	no	—
28	C₅×#393	—	3	1	no	—
29	C₅×#392	—	3	3	no	—
30	C₄×C₅×C₂²⋊C₄	—	3	3	no	—
31	C₂³×C₅×D₃	C₂³	1	3	yes	—
32	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₅×Q₈	—	2	6	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
37	C₂²×C₅×Dic₃	C₂³	1	3	yes	—
38	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
39	C₅×#239	—	2	32	no	—
40	C₂×C₅×D₁₂	—	2	6	no	—
41	C₂×C₅×D₁₂	—	2	12	no	—
42	C₂×C₅×D₁₂	—	2	6	no	—
43	C₂×C₅×D₁₂	—	2	6	no	—
44	C₂×C₅×D₁₂	—	2	6	no	—
45	C₂×C₄×C₅×D₃	—	2	12	no	—
46	C₂×C₄×C₅×D₃	—	2	6	no	—
47	C₂×C₄×C₅×D₃	D₄	1	12	yes	complemented
48	C₃×C₅×C₄⋊C₄	—	2	6	no	—
49	C₅×(C₂×D₃)⋊C₄	—	2	6	no	—
50	C₅×(C₂×D₃)⋊C₄	—	2	6	no	—
51	C₅×(C₂×D₃)⋊C₄	—	2	12	no	—
52	C₄×C₅×Dic₃	—	2	6	no	—
53	C₂²×#239	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×#893	C₂×C₅	1	3	yes	complemented
55	C₃×#395	C₂×C₅	1	1	yes	characteristic, complemented
56	#5231	C₂×C₅	1	1	yes	characteristic, complemented
57	#4991	C₂×C₅	1	3	yes	complemented
58	#4671	C₂×C₅	1	3	yes	complemented
59	#4666	C₂×C₅	1	3	yes	complemented
60	C₂×C₅×#41	—	6	12	no	—
61	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
62	C₂²×C₅×D₄	—	3	3	no	—
63	C₂²×C₅×D₄	—	3	3	no	—
64	C₂³×C₄×C₅	—	3	3	no	—
65	C₂³×C₄×C₅	—	3	1	no	—
66	C₄×C₅×D₄	—	6	12	no	—
67	C₂×C₅×C₄⋊C₄	C₂×D₃	1	3	yes	—
68	C₂×C₅×C₂²⋊C₄	—	3	6	no	—
69	C₂×C₅×C₂²⋊C₄	—	3	3	no	—
70	C₂×C₅×C₂²⋊C₄	—	3	3	no	—
71	C₂×C₄²×C₅	—	3	3	no	—
72	C₅×#95	—	3	3	no	—
73	C₅×#95	C₂×D₃	1	3	yes	—
74	#2388	—	3	1	no	Sylow
75	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
76	C₂²×C₅×D₃	—	2	3	no	—
77	C₂²×C₅×D₃	—	2	6	no	—
78	C₂²×C₅×D₃	—	2	6	no	—
79	C₂²×C₅×D₃	—	2	3	no	—
80	C₂²×C₅×D₃	—	2	3	no	—
81	C₂²×C₅×D₃	—	2	3	no	—
82	C₂²×C₅×D₃	C₂×D₄	1	6	yes	—
83	C₂²×C₅×D₃	#41	1	2	yes	complemented
84	C₃×C₅×Q₈	—	2	8	no	—
85	C₂×C₃×C₄×C₅	—	2	3	no	—
86	C₂×C₃×C₄×C₅	—	2	6	no	—
87	C₂×C₃×C₄×C₅	—	2	3	no	—
88	C₂×C₃×C₄×C₅	—	2	3	no	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
90	C₂×C₃×C₄×C₅	#41	1	6	yes	complemented
91	C₂×C₅×Dic₃	—	2	3	no	—
92	C₂×C₅×Dic₃	—	2	6	no	—
93	C₂×C₅×Dic₃	C₂×D₄	1	6	yes	—
94	C₅×D₁₂	—	2	24	no	—
95	C₅×D₁₂	—	2	24	no	—
96	C₄×C₅×D₃	—	2	24	no	—
97	C₄×C₅×D₃	—	4	4	no	—
98	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
99	C₂×#239	—	2	12	no	—
100	C₂²×D₁₂	C₂²×C₅	1	3	yes	—
101	C₂²×D₁₂	C₂²×C₅	1	3	yes	—
102	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
103	C₂²×C₄×D₃	C₂²×C₅	1	3	yes	—
104	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
105	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	6	yes	—
106	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	3	yes	—
107	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	3	yes	—
108	C₂×C₄×Dic₃	C₂²×C₅	1	3	yes	—
109	#893	—	2	12	no	—
110	C₃×#95	C₂²×C₅	1	3	yes	—
111	C₃⋊#95	C₂²×C₅	1	3	yes	—
112	C₂⁴×C₅	—	3	3	no	—
113	C₂⁴×C₅	—	3	1	no	—
114	C₅×#41	—	6	32	no	—
115	C₂×C₅×Q₈	—	2	6	no	—
116	C₂×C₅×D₄	—	6	6	no	—
117	C₂×C₅×D₄	—	6	12	no	—
118	C₂×C₅×D₄	—	6	6	no	—
119	C₂×C₅×D₄	—	6	6	no	—
120	C₂×C₅×D₄	—	6	6	no	—
121	C₂²×C₄×C₅	—	3	1	no	—
122	C₂²×C₄×C₅	—	3	12	no	—
123	C₂²×C₄×C₅	—	3	3	no	—
124	C₂²×C₄×C₅	—	6	12	no	—
125	C₂²×C₄×C₅	—	6	6	no	—
126	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
127	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
128	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
129	C₅×C₄⋊C₄	—	2	6	no	—
130	C₅×C₂²⋊C₄	—	6	6	no	—
131	C₅×C₂²⋊C₄	—	6	6	no	—
132	C₅×C₂²⋊C₄	—	6	12	no	—
133	C₄²×C₅	—	6	6	no	—
134	C₂²×#41	—	3	1	no	—
135	C₂×C₄×D₄	—	3	3	no	—
136	#395	C₅×D₃	1	1	yes	characteristic, complemented
137	#394	—	3	3	no	—
138	#393	—	3	1	no	—
139	#392	—	3	3	no	—
140	C₄×C₂²⋊C₄	—	3	3	no	—
141	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
142	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
143	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
144	C₂×C₅×D₃	—	2	6	no	—
145	C₂×C₅×D₃	—	2	6	no	—
146	C₂×C₅×D₃	—	2	6	no	—
147	C₂×C₅×D₃	—	2	6	no	—
148	C₂×C₅×D₃	—	4	1	no	—
149	C₂×C₅×D₃	—	4	6	no	—
150	C₂×C₅×D₃	—	4	3	no	—
151	C₂×C₅×D₃	—	4	6	no	—
152	C₃×C₄×C₅	—	2	6	no	—
153	C₃×C₄×C₅	—	2	6	no	—
154	C₃×C₄×C₅	—	4	1	no	—
155	C₅×Dic₃	—	4	3	no	—
156	C₅×Dic₃	#120	1	4	yes	—
157	C₂³×D₃	C₂³×C₅	1	3	yes	—
158	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
159	C₂×C₃×Q₈	—	2	6	no	—
160	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
161	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
162	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
163	C₂²×Dic₃	C₂³×C₅	1	3	yes	—
164	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
165	#239	—	2	32	no	—
166	C₂×D₁₂	—	2	6	no	—
167	C₂×D₁₂	—	2	12	no	—
168	C₂×D₁₂	—	2	6	no	—
169	C₂×D₁₂	—	2	6	no	—
170	C₂×D₁₂	—	2	6	no	—
171	C₂×C₄×D₃	—	2	12	no	—
172	C₂×C₄×D₃	—	2	6	no	—
173	C₂×C₄×D₃	C₅×D₄	1	12	yes	complemented
174	C₃×C₄⋊C₄	—	2	6	no	—
175	(C₂×D₃)⋊C₄	—	2	6	no	—
176	(C₂×D₃)⋊C₄	—	2	6	no	—
177	(C₂×D₃)⋊C₄	—	2	12	no	—
178	C₄×Dic₃	—	2	6	no	—
179	C₂³×C₅	—	3	6	no	—
180	C₂³×C₅	—	3	2	no	—
181	C₂³×C₅	—	6	3	no	—
182	C₂³×C₅	—	6	6	no	—
183	C₂³×C₅	—	6	6	no	—
184	C₂³×C₅	—	6	3	no	—
185	C₂³×C₅	—	6	3	no	—
186	C₂³×C₅	—	6	3	no	—
187	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
188	C₅×Q₈	—	2	8	no	—
189	C₅×D₄	—	6	24	no	—
190	C₅×D₄	—	6	24	no	—
191	C₂×C₄×C₅	—	2	3	no	—
192	C₂×C₄×C₅	—	2	6	no	—
193	C₂×C₄×C₅	—	2	3	no	—
194	C₂×C₄×C₅	—	2	3	no	—
195	C₂×C₄×C₅	—	3	6	no	—
196	C₂×C₄×C₅	—	6	24	no	—
197	C₂×C₄×C₅	—	6	3	no	—
198	C₂×C₄×C₅	—	6	6	no	—
199	C₂×C₄×C₅	—	12	4	no	—
200	C₂×C₄×C₅	D₃×D₄	1	6	yes	—
201	C₂×C₄×C₅	#239	1	6	yes	complemented
202	C₂×#41	—	6	12	no	—
203	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
204	C₂²×D₄	—	3	3	no	—
205	C₂²×D₄	—	3	3	no	—
206	C₂³×C₄	—	3	3	no	—
207	C₂³×C₄	—	3	1	no	—
208	C₄×D₄	—	6	12	no	—
209	C₂×C₄⋊C₄	C₂×C₅×D₃	1	3	yes	—
210	C₂×C₂²⋊C₄	—	3	3	no	—
211	C₂×C₂²⋊C₄	—	3	3	no	—
212	C₂×C₂²⋊C₄	—	3	6	no	—
213	C₂×C₄²	—	3	3	no	—
214	#95	—	3	3	no	—
215	#95	C₂×C₅×D₃	1	3	yes	—
216	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
217	C₂×C₃×C₅	#525	1	3	yes	—
218	C₂×C₃×C₅	#532	1	3	yes	—
219	C₅×D₃	—	4	6	no	—
220	C₅×D₃	—	4	2	no	—
221	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
222	C₂²×D₃	—	2	3	no	—
223	C₂²×D₃	—	2	6	no	—
224	C₂²×D₃	—	2	6	no	—
225	C₂²×D₃	—	2	3	no	—
226	C₂²×D₃	—	2	3	no	—
227	C₂²×D₃	—	2	3	no	—
228	C₂²×D₃	C₂×C₅×D₄	1	6	yes	—
229	C₂²×D₃	C₅×#41	1	2	yes	complemented
230	C₃×Q₈	—	2	8	no	—
231	C₂×C₃×C₄	—	2	3	no	—
232	C₂×C₃×C₄	—	2	6	no	—
233	C₂×C₃×C₄	—	2	3	no	—
234	C₂×C₃×C₄	—	2	3	no	—
235	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
236	C₂×C₃×C₄	C₅×#41	1	6	yes	complemented
237	C₂×Dic₃	—	2	3	no	—
238	C₂×Dic₃	—	2	6	no	—
239	C₂×Dic₃	C₂×C₅×D₄	1	6	yes	—
240	D₁₂	—	2	24	no	—
241	D₁₂	—	2	24	no	—
242	C₄×D₃	—	2	24	no	—
243	C₄×D₃	—	4	4	no	—
244	C₂²×C₅	—	6	6	no	—
245	C₂²×C₅	—	6	6	no	—
246	C₂²×C₅	—	6	6	no	—
247	C₂²×C₅	—	6	6	no	—
248	C₂²×C₅	—	12	3	no	—
249	C₂²×C₅	—	12	6	no	—
250	C₂²×C₅	—	12	1	no	—
251	C₂²×C₅	—	12	6	no	—
252	C₂²×C₅	C₂×D₃×D₄	1	3	yes	—
253	C₂²×C₅	C₂×#239	1	3	yes	—
254	C₂²×C₅	D₃×#41	1	1	yes	characteristic
255	C₄×C₅	—	2	6	no	—
256	C₄×C₅	—	2	6	no	—
257	C₄×C₅	—	3	4	no	—
258	C₄×C₅	—	4	1	no	—
259	C₄×C₅	—	12	3	no	—
260	C₂⁴	—	3	3	no	—
261	C₂⁴	—	3	1	no	—
262	#41	—	6	32	no	—
263	C₂×Q₈	—	2	6	no	—
264	C₂×D₄	—	6	6	no	—
265	C₂×D₄	—	6	12	no	—
266	C₂×D₄	—	6	6	no	—
267	C₂×D₄	—	6	6	no	—
268	C₂×D₄	—	6	6	no	—
269	C₂²×C₄	—	3	1	no	—
270	C₂²×C₄	—	3	12	no	—
271	C₂²×C₄	—	3	3	no	—
272	C₂²×C₄	—	6	12	no	—
273	C₂²×C₄	—	6	6	no	—
274	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
275	C₂²×C₄	C₂²×C₅×D₃	1	3	yes	—
276	C₂²×C₄	C₂²×C₅×D₃	1	3	yes	—
277	C₄⋊C₄	—	2	6	no	—
278	C₂²⋊C₄	—	6	6	no	—
279	C₂²⋊C₄	—	6	6	no	—
280	C₂²⋊C₄	—	6	12	no	—
281	C₄²	—	6	6	no	—
282	C₃×C₅	#2388	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
284	C₂²×C₃	C₂×C₅×#41	1	3	yes	—
285	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
286	C₂×D₃	—	2	6	no	—
287	C₂×D₃	—	2	6	no	—
288	C₂×D₃	—	2	6	no	—
289	C₂×D₃	—	2	6	no	—
290	C₂×D₃	—	4	1	no	—
291	C₂×D₃	—	4	6	no	—
292	C₂×D₃	—	4	3	no	—
293	C₂×D₃	—	4	6	no	—
294	C₃×C₄	—	2	6	no	—
295	C₃×C₄	—	2	6	no	—
296	C₃×C₄	—	4	1	no	—
297	Dic₃	—	4	3	no	—
298	Dic₃	C₅×#120	1	4	yes	—
299	C₂×C₅	—	12	6	no	—
300	C₂×C₅	—	12	2	no	—
301	C₂×C₅	D₃×#120	1	1	yes	characteristic
302	C₂×C₅	#5627	1	3	yes	—
303	C₂×C₅	#5725	1	3	yes	—
304	C₂³	—	3	6	no	—
305	C₂³	—	3	2	no	—
306	C₂³	—	6	3	no	—
307	C₂³	—	6	6	no	—
308	C₂³	—	6	6	no	—
309	C₂³	—	6	3	no	—
310	C₂³	—	6	3	no	—
311	C₂³	—	6	3	no	—
312	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
313	Q₈	—	2	8	no	—
314	D₄	—	6	24	no	—
315	D₄	—	6	24	no	—
316	C₂×C₄	—	2	3	no	—
317	C₂×C₄	—	2	6	no	—
318	C₂×C₄	—	2	3	no	—
319	C₂×C₄	—	2	3	no	—
320	C₂×C₄	—	3	6	no	—
321	C₂×C₄	—	6	24	no	—
322	C₂×C₄	—	6	3	no	—
323	C₂×C₄	—	6	6	no	—
324	C₂×C₄	—	12	4	no	—
325	C₂×C₄	C₅×D₃×D₄	1	6	yes	—
326	C₂×C₄	C₅×#239	1	6	yes	complemented
327	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
328	C₂×C₃	C₅×#525	1	3	yes	—
329	C₂×C₃	C₅×#532	1	3	yes	—
330	D₃	—	4	6	no	—
331	D₃	—	4	2	no	—
332	C₅	#77169	1	1	yes	characteristic, direct product factor, Sylow
333	C₂²	—	6	6	no	—
334	C₂²	—	6	6	no	—
335	C₂²	—	6	6	no	—
336	C₂²	—	6	6	no	—
337	C₂²	—	12	3	no	—
338	C₂²	—	12	6	no	—
339	C₂²	—	12	1	no	—
340	C₂²	—	12	6	no	—
341	C₂²	C₂×C₅×D₃×D₄	1	3	yes	—
342	C₂²	C₂×C₅×#239	1	3	yes	—
343	C₂²	C₅×D₃×#41	1	1	yes	characteristic
344	C₄	—	2	6	no	—
345	C₄	—	2	6	no	—
346	C₄	—	3	4	no	—
347	C₄	—	4	1	no	—
348	C₄	—	12	3	no	—
349	C₃	C₅×#2388	1	1	yes	characteristic, complemented, Sylow
350	C₂	—	12	6	no	—
351	C₂	—	12	2	no	—
352	C₂	C₅×D₃×#120	1	1	yes	characteristic
353	C₂	C₅×#5627	1	3	yes	—
354	C₂	C₅×#5725	1	3	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator, cosocle	C₂³×C₃	C₂⁴×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#395	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₅×#395	D₃
hypocenter	C₃	C₅×#2388
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#395

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#239	C₂	semidirect
2. C₂×C₅×#893	C₂	semidirect
3. C₃×C₅×#395	C₂	semidirect
4. C₅×#5231	C₂	semidirect
5. C₅×#4991	C₂	semidirect
6. C₅×#4671	C₂	semidirect
7. C₅×#4666	C₂	semidirect
23. #77169	C₅	direct (N × H)
26. C₅×#395	D₃	semidirect
47. C₂×C₄×C₅×D₃	D₄	semidirect
53. C₂²×#239	C₂×C₅	semidirect
54. C₂×#893	C₂×C₅	semidirect
55. C₃×#395	C₂×C₅	semidirect
56. #5231	C₂×C₅	semidirect
57. #4991	C₂×C₅	semidirect
58. #4671	C₂×C₅	semidirect
59. #4666	C₂×C₅	semidirect
83. C₂²×C₅×D₃	#41	semidirect
90. C₂×C₃×C₄×C₅	#41	semidirect
136. #395	C₅×D₃	semidirect
173. C₂×C₄×D₃	C₅×D₄	semidirect
201. C₂×C₄×C₅	#239	semidirect
229. C₂²×D₃	C₅×#41	semidirect
236. C₂×C₃×C₄	C₅×#41	semidirect
282. C₃×C₅	#2388	semidirect
326. C₂×C₄	C₅×#239	semidirect
332. C₅	#77169	direct (N × H)
349. C₃	C₅×#2388	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×#239	2
C₂×C₅×#893	2
C₃×C₅×#395	2
C₅×#5231	2
C₅×#4991	2
C₅×#4671	2
C₅×#4666	2
C₅×#2388	3
#77169	5

Subgroups — C5×#77169 #14331622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#77169 #14331622