Subgroups — C₅×#72132 #14297014

Order 1920.

1184 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#835	C₂	1	4	yes	maximal, complemented
3	C₅×#5212	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#830	C₂	1	1	yes	maximal, characteristic
5	C₅×#1787	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	—	2	8	no	—
7	C₂²×C₃×C₅×Q₈	—	2	2	no	—
8	C₂²×C₃×C₅×Q₈	—	2	1	no	—
9	C₂²×C₃×C₅×Q₈	C₄	1	4	yes	complemented
10	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
11	C₂³×C₃×C₄×C₅	—	2	1	no	—
12	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×#217	—	2	1	no	—
14	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
16	C₅×#835	—	2	8	no	—
17	C₅×C₃⋊#95	—	2	1	no	—
18	C₅×#830	—	2	2	no	—
19	#72132	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂³×C₅×Q₈	D₃	1	1	yes	characteristic
21	C₂×C₅×#103	—	3	4	no	—
22	C₂×C₅×C₂²⋊C₈	—	3	1	no	—
23	C₅×#380	—	3	1	no	—
24	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×Q₈	—	2	16	no	—
26	C₂×C₃×C₅×Q₈	—	2	8	no	—
27	C₂×C₃×C₅×Q₈	—	2	16	no	—
28	C₂×C₃×C₅×Q₈	—	2	4	no	—
29	C₂×C₃×C₅×Q₈	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	—	2	4	no	—
31	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₅×Q₈	D₄	1	8	yes	—
33	C₂²×C₃×C₄×C₅	—	2	8	no	—
34	C₂²×C₃×C₄×C₅	—	2	4	no	—
35	C₂²×C₃×C₄×C₅	—	2	2	no	—
36	C₂²×C₃×C₄×C₅	—	2	4	no	—
37	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
39	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂²×C₅×Dic₃	—	2	2	no	—
42	C₅×#217	—	4	2	no	—
43	C₅×C₄⋊Dic₃	—	2	4	no	—
44	C₂×C₅×C₃⋊C₈	—	2	2	no	—
45	C₂×C₅×C₃⋊C₈	—	2	4	no	—
46	C₂³×C₃×Q₈	C₂×C₅	1	1	yes	characteristic
47	C₂×#835	C₂×C₅	1	4	yes	complemented
48	#5212	C₂×C₅	1	1	yes	characteristic
49	C₂×#830	C₂×C₅	1	1	yes	characteristic
50	C₂²×C₅×Q₈	—	2	8	no	—
51	C₂²×C₅×Q₈	—	2	2	no	—
52	C₂²×C₅×Q₈	—	2	1	no	—
53	C₂²×C₅×Q₈	Dic₃	1	4	yes	complemented
54	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
55	C₂³×C₄×C₅	—	2	1	no	—
56	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₅×C₈	—	3	2	no	—
58	C₂×C₅×C₄⋊C₄	—	3	2	no	—
59	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
60	C₅×#103	—	6	8	no	—
61	C₅×C₂²⋊C₈	—	6	2	no	—
62	C₅×#95	—	6	1	no	—
63	#1787	—	3	1	no	Sylow
64	C₂³×C₃×C₅	—	2	2	no	—
65	C₂³×C₃×C₅	—	2	2	no	—
66	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂³×C₃×C₅	SD₁₆	1	2	yes	—
68	C₂³×C₃×C₅	Q₁₆	1	2	yes	—
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₃×C₅×Q₈	—	2	16	no	—
71	C₃×C₅×Q₈	—	2	4	no	—
72	C₃×C₅×Q₈	—	2	8	no	—
73	C₃×C₅×Q₈	C₂²⋊C₄	1	8	yes	complemented
74	C₂×C₃×C₄×C₅	—	2	16	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	8	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	4	no	—
79	C₂×C₃×C₄×C₅	—	2	4	no	—
80	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×Dic₃	—	4	2	no	—
85	C₂×C₅×Dic₃	—	4	2	no	—
86	C₂×C₅×Dic₃	—	4	2	no	—
87	C₅×C₃⋊C₈	—	4	2	no	—
88	C₂²×C₃×Q₈	—	2	8	no	—
89	C₂²×C₃×Q₈	—	2	2	no	—
90	C₂²×C₃×Q₈	—	2	1	no	—
91	C₂²×C₃×Q₈	C₄×C₅	1	4	yes	complemented
92	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
93	C₂³×C₃×C₄	—	2	1	no	—
94	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
95	C₂×#217	—	2	1	no	—
96	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
97	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
98	#835	—	2	8	no	—
99	C₃⋊#95	—	2	1	no	—
100	#830	—	2	2	no	—
101	C₂⁴×C₅	#67	1	1	yes	characteristic
102	C₂×C₅×Q₈	—	2	16	no	—
103	C₂×C₅×Q₈	—	2	8	no	—
104	C₂×C₅×Q₈	—	2	16	no	—
105	C₂×C₅×Q₈	—	2	4	no	—
106	C₂×C₅×Q₈	—	2	2	no	—
107	C₂×C₅×Q₈	—	2	4	no	—
108	C₂×C₅×Q₈	C₂×Dic₃	1	4	yes	—
109	C₂×C₅×Q₈	#67	1	8	yes	—
110	C₂²×C₄×C₅	—	2	8	no	—
111	C₂²×C₄×C₅	—	2	4	no	—
112	C₂²×C₄×C₅	—	2	2	no	—
113	C₂²×C₄×C₅	—	2	4	no	—
114	C₂²×C₄×C₅	—	6	2	no	—
115	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
116	C₂²×C₄×C₅	#67	1	1	yes	characteristic
117	C₂²×C₄×C₅	#67	1	2	yes	—
118	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₅×C₈	—	6	2	no	—
120	C₂×C₅×C₈	—	6	4	no	—
121	C₅×C₄⋊C₄	—	6	4	no	—
122	C₅×C₂²⋊C₄	—	12	2	no	—
123	C₂³×Q₈	C₅×D₃	1	1	yes	characteristic
124	C₂×#103	—	3	4	no	—
125	C₂×C₂²⋊C₈	—	3	1	no	—
126	#380	—	3	1	no	—
127	C₂²×C₃×C₅	—	2	2	no	—
128	C₂²×C₃×C₅	—	2	2	no	—
129	C₂²×C₃×C₅	—	2	4	no	—
130	C₂²×C₃×C₅	—	2	4	no	—
131	C₂²×C₃×C₅	#103	1	4	yes	complemented
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₃×C₅	#120	1	2	yes	—
134	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
135	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
136	C₂²×C₃×C₅	#137	1	2	yes	—
137	C₃×C₄×C₅	—	2	8	no	—
138	C₃×C₄×C₅	—	2	2	no	—
139	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
140	C₃×C₄×C₅	#120	1	2	yes	—
141	C₅×Dic₃	—	8	2	no	—
142	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
143	C₂×C₃×Q₈	—	2	16	no	—
144	C₂×C₃×Q₈	—	2	8	no	—
145	C₂×C₃×Q₈	—	2	16	no	—
146	C₂×C₃×Q₈	—	2	4	no	—
147	C₂×C₃×Q₈	—	2	2	no	—
148	C₂×C₃×Q₈	—	2	4	no	—
149	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
150	C₂×C₃×Q₈	C₅×D₄	1	8	yes	—
151	C₂²×C₃×C₄	—	2	8	no	—
152	C₂²×C₃×C₄	—	2	4	no	—
153	C₂²×C₃×C₄	—	2	2	no	—
154	C₂²×C₃×C₄	—	2	4	no	—
155	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
156	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
158	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
159	C₂²×Dic₃	—	2	2	no	—
160	#217	—	4	2	no	—
161	C₄⋊Dic₃	—	2	4	no	—
162	C₂×C₃⋊C₈	—	2	2	no	—
163	C₂×C₃⋊C₈	—	2	4	no	—
164	C₂³×C₅	—	2	2	no	—
165	C₂³×C₅	—	2	2	no	—
166	C₂³×C₅	#215	1	2	yes	—
167	C₂³×C₅	#216	1	2	yes	—
168	C₂³×C₅	#217	1	2	yes	—
169	C₂³×C₅	C₂×#67	1	1	yes	characteristic
170	C₅×Q₈	—	2	16	no	—
171	C₅×Q₈	—	2	4	no	—
172	C₅×Q₈	—	2	8	no	—
173	C₅×Q₈	#217	1	8	yes	complemented
174	C₂×C₄×C₅	—	2	16	no	—
175	C₂×C₄×C₅	—	2	4	no	—
176	C₂×C₄×C₅	—	2	8	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	4	no	—
179	C₂×C₄×C₅	—	2	4	no	—
180	C₂×C₄×C₅	—	12	2	no	—
181	C₂×C₄×C₅	—	12	2	no	—
182	C₂×C₄×C₅	—	12	2	no	—
183	C₂×C₄×C₅	#217	1	2	yes	—
184	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×#67	1	4	yes	—
186	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
187	C₅×C₈	—	12	2	no	—
188	C₂²×Q₈	—	2	8	no	—
189	C₂²×Q₈	—	2	2	no	—
190	C₂²×Q₈	—	2	1	no	—
191	C₂²×Q₈	C₅×Dic₃	1	4	yes	complemented
192	C₂²×Q₈	C₂×C₅×D₃	1	2	yes	—
193	C₂³×C₄	—	2	1	no	—
194	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
195	C₂²×C₈	—	3	2	no	—
196	C₂×C₄⋊C₄	—	3	2	no	—
197	C₂×C₂²⋊C₄	—	6	1	no	—
198	#103	—	6	8	no	—
199	C₂²⋊C₈	—	6	2	no	—
200	#95	—	6	1	no	—
201	C₂×C₃×C₅	—	2	4	no	—
202	C₂×C₃×C₅	#379	1	1	yes	characteristic
203	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
204	C₂×C₃×C₅	#419	1	1	yes	characteristic
205	C₂×C₃×C₅	#448	1	2	yes	—
206	C₂×C₃×C₅	#451	1	2	yes	—
207	C₂³×C₃	—	2	2	no	—
208	C₂³×C₃	—	2	2	no	—
209	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
210	C₂³×C₃	C₅×SD₁₆	1	2	yes	—
211	C₂³×C₃	C₅×Q₁₆	1	2	yes	—
212	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
213	C₃×Q₈	—	2	16	no	—
214	C₃×Q₈	—	2	4	no	—
215	C₃×Q₈	—	2	8	no	—
216	C₃×Q₈	C₅×C₂²⋊C₄	1	8	yes	complemented
217	C₂×C₃×C₄	—	2	16	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	2	8	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	4	no	—
222	C₂×C₃×C₄	—	2	4	no	—
223	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
224	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
226	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂×Dic₃	—	4	2	no	—
228	C₂×Dic₃	—	4	2	no	—
229	C₂×Dic₃	—	4	2	no	—
230	C₃⋊C₈	—	4	2	no	—
231	C₂²×C₅	—	2	2	no	—
232	C₂²×C₅	—	2	4	no	—
233	C₂²×C₅	—	2	4	no	—
234	C₂²×C₅	—	2	2	no	—
235	C₂²×C₅	#835	1	4	yes	complemented
236	C₂²×C₅	C₂×#215	1	1	yes	characteristic
237	C₂²×C₅	#942	1	2	yes	—
238	C₂²×C₅	C₂×#216	1	1	yes	characteristic
239	C₂²×C₅	C₂×#217	1	1	yes	characteristic
240	C₂²×C₅	#953	1	2	yes	—
241	C₄×C₅	—	2	8	no	—
242	C₄×C₅	—	2	2	no	—
243	C₄×C₅	—	24	2	no	—
244	C₄×C₅	C₂×#217	1	2	yes	—
245	C₄×C₅	#953	1	2	yes	—
246	C₂⁴	C₅×#67	1	1	yes	characteristic
247	C₂×Q₈	—	2	16	no	—
248	C₂×Q₈	—	2	8	no	—
249	C₂×Q₈	—	2	16	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	C₂×C₅×Dic₃	1	4	yes	—
254	C₂×Q₈	C₅×#67	1	8	yes	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	6	2	no	—
260	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
261	C₂²×C₄	C₅×#67	1	1	yes	characteristic
262	C₂²×C₄	C₅×#67	1	2	yes	—
263	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
264	C₂×C₈	—	6	2	no	—
265	C₂×C₈	—	6	4	no	—
266	C₄⋊C₄	—	6	4	no	—
267	C₂²⋊C₄	—	12	2	no	—
268	C₃×C₅	#1787	1	1	yes	characteristic, complemented, Hall
269	C₂²×C₃	—	2	2	no	—
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	2	4	no	—
272	C₂²×C₃	—	2	4	no	—
273	C₂²×C₃	C₅×#103	1	4	yes	complemented
274	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
275	C₂²×C₃	C₅×#120	1	2	yes	—
276	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
277	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
278	C₂²×C₃	C₅×#137	1	2	yes	—
279	C₃×C₄	—	2	8	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
282	C₃×C₄	C₅×#120	1	2	yes	—
283	Dic₃	—	8	2	no	—
284	C₂×C₅	—	2	4	no	—
285	C₂×C₅	C₂×#835	1	1	yes	characteristic
286	C₂×C₅	#5225	1	1	yes	characteristic
287	C₂×C₅	#5227	1	2	yes	—
288	C₂×C₅	#5228	1	2	yes	—
289	C₂×C₅	#5247	1	1	yes	characteristic
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	C₅×#215	1	2	yes	—
293	C₂³	C₅×#216	1	2	yes	—
294	C₂³	C₅×#217	1	2	yes	—
295	C₂³	C₂×C₅×#67	1	1	yes	characteristic
296	Q₈	—	2	16	no	—
297	Q₈	—	2	4	no	—
298	Q₈	—	2	8	no	—
299	Q₈	C₅×#217	1	8	yes	complemented
300	C₂×C₄	—	2	16	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	12	2	no	—
307	C₂×C₄	—	12	2	no	—
308	C₂×C₄	—	12	2	no	—
309	C₂×C₄	C₅×#217	1	2	yes	—
310	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
311	C₂×C₄	C₂×C₅×#67	1	4	yes	—
312	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
313	C₈	—	12	2	no	—
314	C₂×C₃	—	2	4	no	—
315	C₂×C₃	C₅×#379	1	1	yes	characteristic
316	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
317	C₂×C₃	C₅×#419	1	1	yes	characteristic
318	C₂×C₃	C₅×#448	1	2	yes	—
319	C₂×C₃	C₅×#451	1	2	yes	—
320	C₅	#72132	1	1	yes	characteristic, direct product factor, Sylow
321	C₂²	—	2	2	no	—
322	C₂²	—	2	4	no	—
323	C₂²	—	2	4	no	—
324	C₂²	—	2	2	no	—
325	C₂²	C₅×#835	1	4	yes	complemented
326	C₂²	C₂×C₅×#215	1	1	yes	characteristic
327	C₂²	C₅×#942	1	2	yes	—
328	C₂²	C₂×C₅×#216	1	1	yes	characteristic
329	C₂²	C₂×C₅×#217	1	1	yes	characteristic
330	C₂²	C₅×#953	1	2	yes	—
331	C₄	—	2	8	no	—
332	C₄	—	2	2	no	—
333	C₄	—	24	2	no	—
334	C₄	C₂×C₅×#217	1	2	yes	—
335	C₄	C₅×#953	1	2	yes	—
336	C₃	C₅×#1787	1	1	yes	characteristic, complemented, Sylow
337	C₂	—	2	4	no	—
338	C₂	C₂×C₅×#835	1	1	yes	characteristic
339	C₂	C₅×#5225	1	1	yes	characteristic
340	C₂	C₅×#5227	1	2	yes	—
341	C₂	C₅×#5228	1	2	yes	—
342	C₂	C₅×#5247	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂³×C₃×C₅×Q₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂³×C₅×Q₈	D₃
hypocenter	C₃	C₅×#1787
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#835	C₂	semidirect
9. C₂²×C₃×C₅×Q₈	C₄	semidirect
19. #72132	C₅	direct (N × H)
47. C₂×#835	C₂×C₅	semidirect
53. C₂²×C₅×Q₈	Dic₃	semidirect
73. C₃×C₅×Q₈	C₂²⋊C₄	semidirect
91. C₂²×C₃×Q₈	C₄×C₅	semidirect
131. C₂²×C₃×C₅	#103	semidirect
173. C₅×Q₈	#217	semidirect
191. C₂²×Q₈	C₅×Dic₃	semidirect
216. C₃×Q₈	C₅×C₂²⋊C₄	semidirect
235. C₂²×C₅	#835	semidirect
268. C₃×C₅	#1787	semidirect
273. C₂²×C₃	C₅×#103	semidirect
299. Q₈	C₅×#217	semidirect
320. C₅	#72132	direct (N × H)
325. C₂²	C₅×#835	semidirect
336. C₃	C₅×#1787	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×Q₈	2
C₂×C₅×#835	2
C₅×#5212	2
C₂×C₅×#830	2
C₅×#1787	3
#72132	5

Subgroups — C5×#72132 #14297014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#72132 #14297014