Subgroups — #14286981

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₂₀	C₂	1	1	yes	maximal, characteristic
2	#11750789	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746280	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746280	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746278	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598874	—	3	1	no	maximal, Hall
9	C₂×D₁₂₀	C₄	1	1	yes	characteristic
10	C₂×#6705	—	2	1	no	—
11	C₂×C₈×D₁₅	—	2	1	no	—
12	C₄×D₆₀	C₂²	1	1	yes	characteristic
13	C₄×D₆₀	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	#91105	C₄	1	1	yes	characteristic
18	#91105	C₄	1	1	yes	characteristic
19	#91104	—	2	1	no	—
20	#91104	—	2	1	no	—
21	#91102	C₄	1	1	yes	characteristic
22	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
24	#68081	—	5	1	no	maximal, Hall
25	C₄×D₄₀	—	3	1	no	—
26	#64977	—	3	1	no	—
27	C₈×D₂₀	—	3	1	no	—
28	C₅×#322	D₃	1	1	yes	characteristic, complemented
29	#64679	—	3	1	no	—
30	#64679	—	3	1	no	—
31	#64677	—	3	1	no	—
32	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
33	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₅	—	2	1	no	—
35	C₂×C₄×D₁₅	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	#6714	—	2	1	no	—
43	#6714	—	2	1	no	—
44	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
45	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	D₁₂₀	—	2	2	no	—
49	#6705	—	4	2	no	—
50	C₈×D₁₅	—	4	2	no	—
51	C₄×D₂₄	—	5	1	no	—
52	#4688	—	5	1	no	—
53	C₈×D₁₂	—	5	1	no	—
54	C₃×#322	D₅	1	1	yes	characteristic, complemented
55	#4459	—	5	1	no	—
56	#4459	—	5	1	no	—
57	#4457	—	5	1	no	—
58	C₂×D₄₀	—	3	1	no	—
59	C₂×#676	—	6	1	no	—
60	C₂×C₈×D₅	—	6	1	no	—
61	C₄×D₂₀	—	3	1	no	—
62	C₄×D₂₀	—	3	1	no	—
63	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
66	#3953	—	3	1	no	—
67	#3953	—	3	1	no	—
68	#3952	—	6	1	no	—
69	#3952	—	6	1	no	—
70	#3950	—	3	1	no	—
71	C₄⋊#168	—	3	1	no	—
72	C₄⋊#168	—	3	1	no	—
73	#1586	—	15	1	no	Sylow
74	C₂²×D₁₅	—	2	1	no	—
75	C₂²×D₁₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
79	C₂×Dic₁₅	—	2	1	no	—
80	C₂×Dic₁₅	—	2	1	no	—
81	D₆₀	—	2	1	no	—
82	D₆₀	—	2	1	no	—
83	D₆₀	—	2	1	no	—
84	D₆₀	M₁₆	1	2	yes	complemented
85	C₄×D₁₅	—	4	1	no	—
86	C₄×D₁₅	—	4	1	no	—
87	C₄×D₁₅	—	4	1	no	—
88	C₄×D₁₅	—	4	1	no	—
89	C₃×C₅×C₈	—	2	2	no	—
90	C₃×C₅×C₈	—	2	1	no	—
91	C₃×C₅×C₈	—	2	1	no	—
92	C₃×C₅×C₈	—	2	1	no	—
93	C₃×C₅×C₈	—	2	1	no	—
94	C₃⋊#168	—	4	1	no	—
95	C₃⋊#168	—	4	1	no	—
96	C₂×D₂₄	—	5	1	no	—
97	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
98	C₂×D₃×C₈	—	10	1	no	—
99	C₄×D₁₂	—	5	1	no	—
100	C₄×D₁₂	—	5	1	no	—
101	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
102	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
103	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
104	#821	—	5	1	no	—
105	#821	—	5	1	no	—
106	(C₂×D₃)⋊C₈	—	10	1	no	—
107	(C₂×D₃)⋊C₈	—	10	1	no	—
108	#818	—	5	1	no	—
109	C₄⋊(C₃⋊C₈)	—	5	1	no	—
110	C₄⋊(C₃⋊C₈)	—	5	1	no	—
111	C₂×D₂₀	—	3	1	no	—
112	C₂×D₂₀	—	3	1	no	—
113	C₂×C₄×D₅	—	6	1	no	—
114	C₂×C₄×D₅	—	6	1	no	—
115	C₂×C₅×C₈	—	2	1	no	—
116	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
117	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
118	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
119	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
120	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
121	#685	—	6	1	no	—
122	#685	—	6	1	no	—
123	C₄⋊Dic₅	—	3	1	no	—
124	C₄⋊Dic₅	—	3	1	no	—
125	C₂×#168	—	6	1	no	—
126	C₂×#168	—	6	1	no	—
127	D₄₀	—	6	2	no	—
128	#676	—	12	2	no	—
129	C₈×D₅	—	12	2	no	—
130	C₄×D₈	—	15	1	no	—
131	#436	—	15	1	no	—
132	C₈×D₄	—	15	1	no	—
133	#334	—	15	1	no	—
134	D₄⋊C₈	—	15	1	no	—
135	D₄⋊C₈	—	15	1	no	—
136	#322	D₁₅	1	1	yes	characteristic, complemented
137	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×D₁₅	—	2	2	no	—
139	C₂×D₁₅	—	4	1	no	—
140	C₂×D₁₅	—	4	1	no	—
141	C₂×D₁₅	—	4	1	no	—
142	C₂×D₁₅	—	4	1	no	—
143	C₂×D₁₅	—	4	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
146	C₃×C₄×C₅	#131	1	1	yes	characteristic
147	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
149	Dic₁₅	—	4	1	no	—
150	Dic₁₅	—	4	1	no	—
151	C₂×D₁₂	—	5	1	no	—
152	C₂×D₁₂	—	5	1	no	—
153	C₂×C₄×D₃	—	10	1	no	—
154	C₂×C₄×D₃	—	10	1	no	—
155	C₂×C₃×C₈	—	2	1	no	—
156	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
157	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
158	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
159	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
160	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
161	(C₂×D₃)⋊C₄	—	10	1	no	—
162	(C₂×D₃)⋊C₄	—	10	1	no	—
163	C₄⋊Dic₃	—	5	1	no	—
164	C₄⋊Dic₃	—	5	1	no	—
165	C₂×C₃⋊C₈	—	10	1	no	—
166	C₂×C₃⋊C₈	—	10	1	no	—
167	D₂₄	—	10	2	no	—
168	ΓL(1,𝔽₂₅)	—	20	2	no	—
169	D₃×C₈	—	20	2	no	—
170	C₂²×D₅	—	6	1	no	—
171	C₂²×D₅	—	6	1	no	—
172	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
174	C₂×C₄×C₅	#235	1	1	yes	characteristic
175	C₂×Dic₅	—	6	1	no	—
176	C₂×Dic₅	—	6	1	no	—
177	D₂₀	—	3	2	no	—
178	D₂₀	—	6	1	no	—
179	D₂₀	—	6	1	no	—
180	D₂₀	—	6	1	no	—
181	C₄×D₅	—	12	1	no	—
182	C₄×D₅	—	12	1	no	—
183	C₄×D₅	—	12	1	no	—
184	C₄×D₅	—	12	1	no	—
185	C₅×C₈	—	2	2	no	—
186	C₅×C₈	—	2	1	no	—
187	C₅×C₈	—	2	1	no	—
188	C₅×C₈	—	2	1	no	—
189	C₅×C₈	—	2	1	no	—
190	#168	—	12	1	no	—
191	#168	—	12	1	no	—
192	C₂×D₈	—	15	1	no	—
193	C₂×M₁₆	—	30	1	no	—
194	C₂²×C₈	—	30	1	no	—
195	C₄×D₄	—	15	1	no	—
196	C₄×D₄	—	15	1	no	—
197	#107	—	15	1	no	—
198	C₄⋊C₈	—	15	1	no	—
199	C₄⋊C₈	—	15	1	no	—
200	#102	—	15	1	no	—
201	#102	—	15	1	no	—
202	C₂²⋊C₈	—	30	1	no	—
203	C₂²⋊C₈	—	30	1	no	—
204	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
205	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
206	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
207	C₂×C₃×C₅	#435	1	1	yes	characteristic
208	C₂×C₃×C₅	#442	1	1	yes	characteristic
209	C₂×C₃×C₅	#443	1	1	yes	characteristic
210	D₁₅	—	4	2	no	—
211	D₁₅	—	8	1	no	—
212	C₂²×D₃	—	10	1	no	—
213	C₂²×D₃	—	10	1	no	—
214	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
216	C₂×C₃×C₄	#709	1	1	yes	characteristic
217	C₂×Dic₃	—	10	1	no	—
218	C₂×Dic₃	—	10	1	no	—
219	D₁₂	—	5	2	no	—
220	D₁₂	—	10	1	no	—
221	D₁₂	—	10	1	no	—
222	D₁₂	—	10	1	no	—
223	C₄×D₃	—	20	1	no	—
224	C₄×D₃	—	20	1	no	—
225	C₄×D₃	—	20	1	no	—
226	C₄×D₃	—	20	1	no	—
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	1	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃⋊C₈	—	20	1	no	—
233	C₃⋊C₈	—	20	1	no	—
234	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
235	C₂×D₅	—	6	2	no	—
236	C₂×D₅	—	12	1	no	—
237	C₂×D₅	—	12	1	no	—
238	C₂×D₅	—	12	1	no	—
239	C₂×D₅	—	12	1	no	—
240	C₂×D₅	—	12	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
243	C₄×C₅	#907	1	1	yes	characteristic
244	C₄×C₅	#908	1	1	yes	characteristic
245	C₄×C₅	#908	1	1	yes	characteristic
246	Dic₅	—	12	1	no	—
247	Dic₅	—	12	1	no	—
248	C₂×D₄	—	15	1	no	—
249	C₂×D₄	—	15	1	no	—
250	C₂²×C₄	—	30	1	no	—
251	C₂²×C₄	—	30	1	no	—
252	D₈	—	30	2	no	—
253	M₁₆	—	60	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	30	1	no	—
256	C₂×C₈	—	30	1	no	—
257	C₂×C₈	—	60	2	no	—
258	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
259	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
260	C₂×C₈	D₆₀	1	1	yes	characteristic
261	C₂×C₈	D₆₀	1	1	yes	characteristic
262	C₄⋊C₄	—	15	1	no	—
263	C₄⋊C₄	—	15	1	no	—
264	C₂²⋊C₄	—	30	1	no	—
265	C₂²⋊C₄	—	30	1	no	—
266	C₄²	C₂²×D₁₅	1	1	yes	characteristic
267	C₃×C₅	#1586	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
269	C₂×D₃	—	10	2	no	—
270	C₂×D₃	—	20	1	no	—
271	C₂×D₃	—	20	1	no	—
272	C₂×D₃	—	20	1	no	—
273	C₂×D₃	—	20	1	no	—
274	C₂×D₃	—	20	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
277	C₃×C₄	#4053	1	1	yes	characteristic
278	C₃×C₄	#4054	1	1	yes	characteristic
279	C₃×C₄	#4054	1	1	yes	characteristic
280	Dic₃	—	20	1	no	—
281	Dic₃	—	20	1	no	—
282	C₂×C₅	#4700	1	1	yes	characteristic
283	C₂×C₅	#4706	1	1	yes	characteristic
284	C₂×C₅	#4711	1	1	yes	characteristic
285	D₅	—	12	2	no	—
286	D₅	—	24	1	no	—
287	C₂³	—	30	1	no	—
288	C₂³	—	30	1	no	—
289	D₄	—	15	2	no	—
290	D₄	—	30	1	no	—
291	D₄	—	30	1	no	—
292	D₄	—	30	1	no	—
293	C₂×C₄	—	30	1	no	—
294	C₂×C₄	—	30	1	no	—
295	C₂×C₄	—	60	1	no	—
296	C₂×C₄	—	60	1	no	—
297	C₂×C₄	—	60	1	no	—
298	C₂×C₄	—	60	1	no	—
299	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
300	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
301	C₂×C₄	#6817	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	60	1	no	—
308	C₈	—	60	1	no	—
309	C₂×C₃	#64989	1	1	yes	characteristic
310	C₂×C₃	#64995	1	1	yes	characteristic
311	C₂×C₃	#65000	1	1	yes	characteristic
312	D₃	—	20	2	no	—
313	D₃	—	40	1	no	—
314	C₅	#68081	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	30	2	no	—
316	C₂²	—	60	1	no	—
317	C₂²	—	60	1	no	—
318	C₂²	—	60	1	no	—
319	C₂²	—	60	1	no	—
320	C₂²	—	60	1	no	—
321	C₂²	C₄×D₆₀	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	60	1	no	—
324	C₄	—	60	1	no	—
325	C₄	M₁₆×D₁₅	1	1	yes	characteristic
326	C₄	#91796	1	1	yes	characteristic
327	C₄	#91797	1	1	yes	characteristic
328	C₄	#91797	1	1	yes	characteristic
329	C₃	#10598874	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	60	2	no	—
331	C₂	—	120	1	no	—
332	C₂	#11750801	1	1	yes	characteristic
333	C₂	#11750807	1	1	yes	characteristic
334	C₂	#11750812	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₀
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×D₁₂₀	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#322	D₁₅
hypocenter	C₃×C₅	#1586
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

Complements

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

N	H ≅ G/N	type
2. #11750789	C₂	semidirect
3. C₈×D₆₀	C₂	semidirect
4. C₃×C₅×#322	C₂	semidirect
5. #11746280	C₂	semidirect
6. #11746280	C₂	semidirect
7. #11746278	C₂	semidirect
28. C₅×#322	D₃	semidirect
54. C₃×#322	D₅	semidirect
84. D₆₀	M₁₆	semidirect
136. #322	D₁₅	semidirect
267. C₃×C₅	#1586	semidirect
314. C₅	#68081	semidirect
329. C₃	#10598874	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₂₀	2
#11750789	2
C₈×D₆₀	2
C₃×C₅×#322	2
#11746280	2
#11746278	2
#10598874	3
#68081	5