Subgroups — C₅×#83823 #14417339

Order 1920.

1056 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#5798	C₂	1	2	yes	maximal, complemented
3	C₅×#5215	C₂	1	2	yes	maximal, complemented
4	C₅×#5152	C₂	1	4	yes	maximal, complemented
5	C₅×Dic₃×D₈	C₂	1	4	yes	maximal, complemented
6	C₅×#5106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#5101	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2948	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
10	C₂×C₃×C₅×D₈	C₄	1	8	yes	complemented
11	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
12	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
13	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#217	—	2	4	no	—
15	C₅×D₄×Dic₃	—	2	4	no	—
16	C₅×D₄×Dic₃	C₂²	1	8	yes	complemented
17	C₅×#926	C₂²	1	2	yes	—
18	C₅×#926	C₂²	1	1	yes	characteristic
19	C₂×C₅×#208	C₂²	1	1	yes	characteristic
20	C₅×#832	C₂²	1	8	yes	complemented
21	C₅×#818	C₂²	1	2	yes	complemented
22	C₅×#817	C₂²	1	2	yes	complemented
23	C₅×#815	C₂²	1	2	yes	complemented
24	C₅×C₈×Dic₃	C₂²	1	2	yes	complemented
25	#83823	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂²×C₅×D₈	D₃	1	1	yes	characteristic
27	C₅×#518	—	3	2	no	—
28	C₅×#442	—	3	4	no	—
29	C₄×C₅×D₈	—	3	4	no	—
30	C₅×#427	—	3	1	no	—
31	C₅×#418	—	3	2	no	—
32	C₅×#405	—	3	1	no	—
33	C₂⁴×C₃×C₅	—	2	2	no	—
34	C₂×C₃×C₅×D₄	—	2	2	no	—
35	C₂×C₃×C₅×D₄	—	2	4	no	—
36	C₂×C₃×C₅×D₄	C₂×C₄	1	8	yes	complemented
37	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₅×Dic₃	—	2	8	no	—
40	C₃×C₅×D₈	C₂×C₄	1	8	yes	complemented
41	C₃×C₅×D₈	C₂×C₄	1	8	yes	complemented
42	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₅×#217	—	2	2	no	—
46	C₅×#217	—	2	8	no	—
47	C₅×#217	D₄	1	2	yes	—
48	C₅×C₄⋊Dic₃	D₄	1	2	yes	—
49	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
50	C₄×C₅×Dic₃	—	2	2	no	—
51	C₄×C₅×Dic₃	C₂³	1	2	yes	—
52	C₅×#208	—	2	2	no	—
53	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
54	C₂²×C₃×D₈	C₂×C₅	1	1	yes	characteristic
55	#5798	C₂×C₅	1	2	yes	complemented
56	#5215	C₂×C₅	1	2	yes	complemented
57	#5152	C₂×C₅	1	4	yes	complemented
58	Dic₃×D₈	C₂×C₅	1	4	yes	complemented
59	#5106	C₂×C₅	1	1	yes	characteristic, complemented
60	#5101	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
62	C₂×C₅×D₈	Dic₃	1	8	yes	complemented
63	C₂×C₅×D₈	C₂×D₃	1	2	yes	—
64	C₂×C₅×D₈	C₂×D₃	1	2	yes	—
65	C₂×C₅×M₁₆	—	3	1	no	—
66	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₄×C₅×D₄	—	3	8	no	—
68	C₄×C₅×D₄	—	6	4	no	—
69	C₅×#117	—	3	1	no	—
70	C₅×#117	—	3	2	no	—
71	C₂×C₅×C₂²⋊C₄	—	6	4	no	—
72	C₅×#107	—	3	2	no	—
73	C₅×#106	—	3	2	no	—
74	C₅×#102	—	3	8	no	—
75	C₅×#97	—	3	2	no	—
76	C₄×C₅×C₈	—	3	2	no	—
77	#2948	—	3	1	no	Sylow
78	C₂³×C₃×C₅	—	2	4	no	—
79	C₂³×C₃×C₅	—	2	8	no	—
80	C₂³×C₃×C₅	—	8	4	no	—
81	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₅×D₄	—	2	8	no	—
83	C₃×C₅×D₄	—	2	4	no	—
84	C₃×C₅×D₄	C₂²×C₄	1	8	yes	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₂×C₅×Dic₃	—	2	4	no	—
90	C₂×C₅×Dic₃	—	8	8	no	—
91	C₂×C₅×Dic₃	C₂×D₄	1	4	yes	—
92	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₅×C₃⋊C₈	—	2	2	no	—
95	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
96	C₂×C₃×D₈	C₄×C₅	1	8	yes	complemented
97	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
98	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
99	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×#217	—	2	4	no	—
101	D₄×Dic₃	—	2	4	no	—
102	D₄×Dic₃	C₂²×C₅	1	8	yes	complemented
103	#926	C₂²×C₅	1	2	yes	—
104	#926	C₂²×C₅	1	1	yes	characteristic
105	C₂×#208	C₂²×C₅	1	1	yes	characteristic
106	#832	C₂²×C₅	1	8	yes	complemented
107	#818	C₂²×C₅	1	2	yes	complemented
108	#817	C₂²×C₅	1	2	yes	complemented
109	#815	C₂²×C₅	1	2	yes	complemented
110	C₈×Dic₃	C₂²×C₅	1	2	yes	complemented
111	C₂⁴×C₅	—	2	2	no	—
112	C₂×C₅×D₄	—	2	2	no	—
113	C₂×C₅×D₄	—	2	4	no	—
114	C₂×C₅×D₄	C₂×Dic₃	1	8	yes	complemented
115	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
116	C₂²×C₄×C₅	—	6	8	no	—
117	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
118	C₅×D₈	C₂×Dic₃	1	8	yes	complemented
119	C₅×D₈	C₂×Dic₃	1	8	yes	complemented
120	C₅×M₁₆	—	6	2	no	—
121	C₂×C₅×C₈	—	3	2	no	—
122	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	complemented
123	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₅×C₄⋊C₄	—	3	2	no	—
126	C₅×C₄⋊C₄	—	3	4	no	—
127	C₅×C₂²⋊C₄	—	3	2	no	—
128	C₅×C₂²⋊C₄	—	6	8	no	—
129	C₅×C₂²⋊C₄	—	6	2	no	—
130	C₄²×C₅	—	3	2	no	—
131	C₄²×C₅	—	6	2	no	—
132	C₂²×D₈	C₅×D₃	1	1	yes	characteristic
133	#518	—	3	2	no	—
134	#442	—	3	4	no	—
135	C₄×D₈	—	3	4	no	—
136	#427	—	3	1	no	—
137	#418	—	3	2	no	—
138	#405	—	3	1	no	—
139	C₂²×C₃×C₅	—	2	2	no	—
140	C₂²×C₃×C₅	—	2	8	no	—
141	C₂²×C₃×C₅	—	4	8	no	—
142	C₂²×C₃×C₅	—	8	8	no	—
143	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
144	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
146	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
147	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
148	C₅×Dic₃	—	2	4	no	—
149	C₅×Dic₃	—	4	4	no	—
150	C₂⁴×C₃	—	2	2	no	—
151	C₂×C₃×D₄	—	2	2	no	—
152	C₂×C₃×D₄	—	2	4	no	—
153	C₂×C₃×D₄	C₂×C₄×C₅	1	8	yes	complemented
154	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
155	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
156	C₂²×Dic₃	—	2	8	no	—
157	C₃×D₈	C₂×C₄×C₅	1	8	yes	complemented
158	C₃×D₈	C₂×C₄×C₅	1	8	yes	complemented
159	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	complemented
160	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
161	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
162	#217	—	2	2	no	—
163	#217	—	2	8	no	—
164	#217	C₅×D₄	1	2	yes	—
165	C₄⋊Dic₃	C₅×D₄	1	2	yes	—
166	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
167	C₄×Dic₃	—	2	2	no	—
168	C₄×Dic₃	C₂³×C₅	1	2	yes	—
169	#208	—	2	2	no	—
170	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
171	C₂³×C₅	—	2	4	no	—
172	C₂³×C₅	—	2	8	no	—
173	C₂³×C₅	—	8	4	no	—
174	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
175	C₅×D₄	—	2	8	no	—
176	C₅×D₄	—	2	4	no	—
177	C₅×D₄	C₂²×Dic₃	1	8	yes	—
178	C₂×C₄×C₅	—	3	4	no	—
179	C₂×C₄×C₅	—	6	4	no	—
180	C₂×C₄×C₅	—	24	8	no	—
181	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	#237	1	2	yes	—
183	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
184	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
185	C₅×C₈	—	6	2	no	—
186	C₅×C₈	C₂²×Dic₃	1	2	yes	—
187	C₅×C₈	C₂²×Dic₃	1	2	yes	—
188	C₂²×D₄	C₂×C₅×D₃	1	2	yes	—
189	C₂×D₈	C₅×Dic₃	1	8	yes	complemented
190	C₂×D₈	C₂×C₅×D₃	1	2	yes	—
191	C₂×D₈	C₂×C₅×D₃	1	2	yes	—
192	C₂×M₁₆	—	3	1	no	—
193	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
194	C₄×D₄	—	3	8	no	—
195	C₄×D₄	—	6	4	no	—
196	#117	—	3	1	no	—
197	#117	—	3	2	no	—
198	C₂×C₂²⋊C₄	—	6	4	no	—
199	#107	—	3	2	no	—
200	#106	—	3	2	no	—
201	#102	—	3	8	no	—
202	#97	—	3	2	no	—
203	C₄×C₈	—	3	2	no	—
204	C₂×C₃×C₅	—	2	2	no	—
205	C₂×C₃×C₅	—	4	8	no	—
206	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
207	C₂×C₃×C₅	#576	1	2	yes	—
208	C₂³×C₃	—	2	4	no	—
209	C₂³×C₃	—	2	8	no	—
210	C₂³×C₃	—	8	4	no	—
211	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
212	C₃×D₄	—	2	8	no	—
213	C₃×D₄	—	2	4	no	—
214	C₃×D₄	C₂²×C₄×C₅	1	8	yes	—
215	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
216	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₅×#41	1	2	yes	—
218	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
219	C₂×Dic₃	—	2	4	no	—
220	C₂×Dic₃	—	8	8	no	—
221	C₂×Dic₃	C₂×C₅×D₄	1	4	yes	—
222	C₃×C₈	C₂²×C₄×C₅	1	2	yes	—
223	C₃×C₈	C₂²×C₄×C₅	1	2	yes	—
224	C₃⋊C₈	—	2	2	no	—
225	C₂²×C₅	—	2	2	no	—
226	C₂²×C₅	—	2	8	no	—
227	C₂²×C₅	—	4	8	no	—
228	C₂²×C₅	—	8	8	no	—
229	C₂²×C₅	D₄×Dic₃	1	2	yes	—
230	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
231	C₄×C₅	—	6	4	no	—
232	C₄×C₅	—	12	4	no	—
233	C₄×C₅	D₄×Dic₃	1	2	yes	—
234	C₄×C₅	C₂×#237	1	1	yes	characteristic
235	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
236	C₂⁴	—	2	2	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	C₂×C₅×Dic₃	1	8	yes	complemented
240	C₂×D₄	C₂²×C₅×D₃	1	4	yes	—
241	C₂²×C₄	—	6	8	no	—
242	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
243	D₈	C₂×C₅×Dic₃	1	8	yes	complemented
244	D₈	C₂×C₅×Dic₃	1	8	yes	complemented
245	M₁₆	—	6	2	no	—
246	C₂×C₈	—	3	2	no	—
247	C₂×C₈	C₂×C₅×Dic₃	1	4	yes	complemented
248	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
250	C₄⋊C₄	—	3	2	no	—
251	C₄⋊C₄	—	3	4	no	—
252	C₂²⋊C₄	—	3	2	no	—
253	C₂²⋊C₄	—	6	8	no	—
254	C₂²⋊C₄	—	6	2	no	—
255	C₄²	—	3	2	no	—
256	C₄²	—	6	2	no	—
257	C₃×C₅	#2948	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	8	no	—
260	C₂²×C₃	—	4	8	no	—
261	C₂²×C₃	—	8	8	no	—
262	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
263	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
264	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
265	C₃×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
266	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
267	Dic₃	—	2	4	no	—
268	Dic₃	—	4	4	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	4	8	no	—
271	C₂×C₅	#5757	1	2	yes	—
272	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
273	C₂³	—	2	4	no	—
274	C₂³	—	2	8	no	—
275	C₂³	—	8	4	no	—
276	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
277	D₄	—	2	8	no	—
278	D₄	—	2	4	no	—
279	D₄	C₂²×C₅×Dic₃	1	8	yes	—
280	C₂×C₄	—	3	4	no	—
281	C₂×C₄	—	6	4	no	—
282	C₂×C₄	—	24	8	no	—
283	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
284	C₂×C₄	C₅×#237	1	2	yes	—
285	C₂×C₄	C₂²×C₅×Dic₃	1	2	yes	—
286	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
287	C₈	—	6	2	no	—
288	C₈	C₂²×C₅×Dic₃	1	2	yes	—
289	C₈	C₂²×C₅×Dic₃	1	2	yes	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	4	8	no	—
292	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
293	C₂×C₃	C₅×#576	1	2	yes	—
294	C₅	#83823	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	2	8	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	8	8	no	—
299	C₂²	C₅×D₄×Dic₃	1	2	yes	—
300	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
301	C₄	—	6	4	no	—
302	C₄	—	12	4	no	—
303	C₄	C₅×D₄×Dic₃	1	2	yes	—
304	C₄	C₂×C₅×#237	1	1	yes	characteristic
305	C₄	C₂³×C₅×Dic₃	1	1	yes	characteristic
306	C₃	C₅×#2948	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	2	2	no	—
308	C₂	—	4	8	no	—
309	C₂	C₅×#5757	1	2	yes	—
310	C₂	C₂×C₅×D₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting, prime radical	C₂²×C₃×C₅×D₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₂²×C₅×D₈	D₃
hypocenter	C₃	C₅×#2948
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×D₈

Complements

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

N	H ≅ G/N	type
2. C₅×#5798	C₂	semidirect
3. C₅×#5215	C₂	semidirect
4. C₅×#5152	C₂	semidirect
5. C₅×Dic₃×D₈	C₂	semidirect
6. C₅×#5106	C₂	semidirect
7. C₅×#5101	C₂	semidirect
10. C₂×C₃×C₅×D₈	C₄	semidirect
16. C₅×D₄×Dic₃	C₂²	semidirect
20. C₅×#832	C₂²	semidirect
21. C₅×#818	C₂²	semidirect
22. C₅×#817	C₂²	semidirect
23. C₅×#815	C₂²	semidirect
24. C₅×C₈×Dic₃	C₂²	semidirect
25. #83823	C₅	direct (N × H)
36. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
40. C₃×C₅×D₈	C₂×C₄	semidirect
41. C₃×C₅×D₈	C₂×C₄	semidirect
42. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
55. #5798	C₂×C₅	semidirect
56. #5215	C₂×C₅	semidirect
57. #5152	C₂×C₅	semidirect
58. Dic₃×D₈	C₂×C₅	semidirect
59. #5106	C₂×C₅	semidirect
60. #5101	C₂×C₅	semidirect
62. C₂×C₅×D₈	Dic₃	semidirect
96. C₂×C₃×D₈	C₄×C₅	semidirect
102. D₄×Dic₃	C₂²×C₅	semidirect
106. #832	C₂²×C₅	semidirect
107. #818	C₂²×C₅	semidirect
108. #817	C₂²×C₅	semidirect
109. #815	C₂²×C₅	semidirect
110. C₈×Dic₃	C₂²×C₅	semidirect
114. C₂×C₅×D₄	C₂×Dic₃	semidirect
118. C₅×D₈	C₂×Dic₃	semidirect
119. C₅×D₈	C₂×Dic₃	semidirect
122. C₂×C₅×C₈	C₂×Dic₃	semidirect
153. C₂×C₃×D₄	C₂×C₄×C₅	semidirect
157. C₃×D₈	C₂×C₄×C₅	semidirect
158. C₃×D₈	C₂×C₄×C₅	semidirect
159. C₂×C₃×C₈	C₂×C₄×C₅	semidirect
189. C₂×D₈	C₅×Dic₃	semidirect
239. C₂×D₄	C₂×C₅×Dic₃	semidirect
243. D₈	C₂×C₅×Dic₃	semidirect
244. D₈	C₂×C₅×Dic₃	semidirect
247. C₂×C₈	C₂×C₅×Dic₃	semidirect
257. C₃×C₅	#2948	semidirect
294. C₅	#83823	direct (N × H)
306. C₃	C₅×#2948	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×D₈	2
C₅×#5798	2
C₅×#5215	2
C₅×#5152	2
C₅×Dic₃×D₈	2
C₅×#5106	2
C₅×#5101	2
C₅×#2948	3
#83823	5

Subgroups — C5×#83823 #14417339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83823 #14417339