Subgroups — #14286293

Order 1920.

352 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	C₄×#6706	C₂	1	1	yes	maximal, characteristic
2	C₈×D₆₀	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₃₀	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
6	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746277	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598869	—	3	1	no	maximal, Hall
9	C₂×#6706	C₄	1	1	yes	characteristic
10	C₂×C₈×D₁₅	—	2	1	no	—
11	C₄×D₆₀	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₀	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	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	#91105	C₄	1	1	yes	characteristic
17	#91104	—	2	1	no	—
18	#91101	C₄	1	1	yes	characteristic
19	#91100	C₄	1	1	yes	characteristic
20	#91098	—	2	1	no	—
21	C₈×Dic₁₅	—	2	1	no	—
22	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
24	#68076	—	5	1	no	maximal, Hall
25	C₄×#677	—	3	1	no	—
26	C₈×D₂₀	—	3	1	no	—
27	C₈×Dic₁₀	—	3	1	no	—
28	C₅×#322	D₃	1	1	yes	characteristic, complemented
29	#64679	—	3	1	no	—
30	#64678	—	3	1	no	—
31	#64676	—	3	1	no	—
32	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₅	—	2	1	no	—
34	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	#6714	—	2	1	no	—
42	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
44	#6712	—	2	1	no	—
45	C₄×Dic₁₅	—	2	1	no	—
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	#6706	C₈	1	4	yes	complemented
49	C₈×D₁₅	—	4	2	no	—
50	C₄×#204	—	5	1	no	—
51	C₈×D₁₂	—	5	1	no	—
52	C₈×Dic₆	—	5	1	no	—
53	C₃×#322	D₅	1	1	yes	characteristic, complemented
54	#4459	—	5	1	no	—
55	#4458	—	5	1	no	—
56	#4456	—	5	1	no	—
57	C₂×#677	—	3	1	no	—
58	C₂×C₈×D₅	—	6	1	no	—
59	C₄×D₂₀	—	3	1	no	—
60	C₄×Dic₁₀	—	3	1	no	—
61	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
64	#3953	—	3	1	no	—
65	#3952	—	6	1	no	—
66	#3949	—	3	1	no	—
67	#3948	—	3	1	no	—
68	#3946	—	6	1	no	—
69	C₈×Dic₅	—	6	1	no	—
70	C₄⋊#168	—	3	1	no	—
71	C₄⋊#168	—	3	1	no	—
72	SD₁₆⋊C₈	—	15	1	no	Sylow
73	C₂²×D₁₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
77	C₂×Dic₁₅	—	2	1	no	—
78	C₂×Dic₁₅	—	2	1	no	—
79	C₂×Dic₁₅	—	2	1	no	—
80	D₆₀	—	2	1	no	—
81	D₆₀	C₂×C₈	1	2	yes	—
82	C₄×D₁₅	—	4	1	no	—
83	C₄×D₁₅	—	4	1	no	—
84	Dic₃₀	—	2	1	no	—
85	Dic₃₀	C₂×C₈	1	2	yes	complemented
86	C₃×C₅×C₈	—	2	1	no	—
87	C₃×C₅×C₈	—	2	1	no	—
88	C₃×C₅×C₈	—	2	1	no	—
89	C₃×C₅×C₈	—	4	1	no	—
90	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
91	C₃⋊#168	—	4	1	no	—
92	C₃⋊#168	—	4	1	no	—
93	C₂×#204	—	5	1	no	—
94	C₂×D₃×C₈	—	10	1	no	—
95	C₄×D₁₂	—	5	1	no	—
96	C₄×Dic₆	—	5	1	no	—
97	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
98	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
99	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
100	#821	—	5	1	no	—
101	(C₂×D₃)⋊C₈	—	10	1	no	—
102	#817	—	5	1	no	—
103	#816	—	5	1	no	—
104	Dic₃⋊C₈	—	10	1	no	—
105	C₈×Dic₃	—	10	1	no	—
106	C₄⋊(C₃⋊C₈)	—	5	1	no	—
107	C₄⋊(C₃⋊C₈)	—	5	1	no	—
108	C₂×D₂₀	—	3	1	no	—
109	C₂×C₄×D₅	—	6	1	no	—
110	C₂×Dic₁₀	—	3	1	no	—
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
113	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
114	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
115	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
116	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
117	#685	—	6	1	no	—
118	C₄⋊Dic₅	—	3	1	no	—
119	C₄⋊Dic₅	—	3	1	no	—
120	#683	—	6	1	no	—
121	C₄×Dic₅	—	6	1	no	—
122	C₂×#168	—	6	1	no	—
123	C₂×#168	—	6	1	no	—
124	#677	—	3	4	no	—
125	C₈×D₅	—	12	2	no	—
126	C₈×Q₈	—	15	1	no	—
127	C₄×SD₁₆	—	15	1	no	—
128	C₈×D₄	—	15	1	no	—
129	#335	—	15	1	no	—
130	Q₈⋊C₈	—	15	1	no	—
131	D₄⋊C₈	—	15	1	no	—
132	#322	D₁₅	1	1	yes	characteristic, complemented
133	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
134	C₂×D₁₅	—	2	2	no	—
135	C₂×D₁₅	—	4	1	no	—
136	C₂×D₁₅	—	4	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
139	C₃×C₄×C₅	#131	1	1	yes	characteristic
140	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₃×C₄×C₅	#137	1	1	yes	characteristic
142	Dic₁₅	—	2	2	no	—
143	Dic₁₅	—	4	1	no	—
144	Dic₁₅	—	4	1	no	—
145	C₂×D₁₂	—	5	1	no	—
146	C₂×C₄×D₃	—	10	1	no	—
147	C₂×Dic₆	—	5	1	no	—
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
150	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
151	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
152	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
153	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
154	(C₂×D₃)⋊C₄	—	10	1	no	—
155	C₄⋊Dic₃	—	5	1	no	—
156	C₄⋊Dic₃	—	5	1	no	—
157	Dic₃⋊C₄	—	10	1	no	—
158	C₄×Dic₃	—	10	1	no	—
159	C₂×C₃⋊C₈	—	10	1	no	—
160	C₂×C₃⋊C₈	—	10	1	no	—
161	#204	—	5	4	no	—
162	D₃×C₈	—	20	2	no	—
163	C₂²×D₅	—	6	1	no	—
164	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
166	C₂×C₄×C₅	#235	1	1	yes	characteristic
167	C₂×Dic₅	—	6	1	no	—
168	C₂×Dic₅	—	6	1	no	—
169	C₂×Dic₅	—	6	1	no	—
170	D₂₀	—	3	2	no	—
171	D₂₀	—	6	1	no	—
172	C₄×D₅	—	12	1	no	—
173	C₄×D₅	—	12	1	no	—
174	Dic₁₀	—	3	2	no	—
175	Dic₁₀	—	6	1	no	—
176	C₅×C₈	—	2	1	no	—
177	C₅×C₈	—	2	1	no	—
178	C₅×C₈	—	2	1	no	—
179	C₅×C₈	—	4	1	no	—
180	C₅×C₈	D₃×C₈	1	2	yes	complemented
181	#168	—	12	1	no	—
182	#168	—	12	1	no	—
183	C₂×SD₁₆	—	15	1	no	—
184	C₂²×C₈	—	30	1	no	—
185	C₄×Q₈	—	15	1	no	—
186	C₄×D₄	—	15	1	no	—
187	#106	—	15	1	no	—
188	C₄⋊C₈	—	15	1	no	—
189	C₄⋊C₈	—	15	1	no	—
190	C₄⋊C₈	—	30	1	no	—
191	#103	—	15	1	no	—
192	#102	—	15	1	no	—
193	C₂²⋊C₈	—	30	1	no	—
194	C₄×C₈	—	30	1	no	—
195	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
196	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
197	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
198	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
199	C₂×C₃×C₅	#440	1	1	yes	characteristic
200	C₂×C₃×C₅	#444	1	1	yes	characteristic
201	D₁₅	—	4	2	no	—
202	C₂²×D₃	—	10	1	no	—
203	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
205	C₂×C₃×C₄	#709	1	1	yes	characteristic
206	C₂×Dic₃	—	10	1	no	—
207	C₂×Dic₃	—	10	1	no	—
208	C₂×Dic₃	—	10	1	no	—
209	D₁₂	—	5	2	no	—
210	D₁₂	—	10	1	no	—
211	C₄×D₃	—	20	1	no	—
212	C₄×D₃	—	20	1	no	—
213	Dic₆	—	5	2	no	—
214	Dic₆	—	10	1	no	—
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	2	1	no	—
217	C₃×C₈	—	2	1	no	—
218	C₃×C₈	—	4	1	no	—
219	C₃×C₈	C₈×D₅	1	2	yes	complemented
220	C₃⋊C₈	—	20	1	no	—
221	C₃⋊C₈	—	20	1	no	—
222	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
223	C₂×D₅	—	6	2	no	—
224	C₂×D₅	—	12	1	no	—
225	C₂×D₅	—	12	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
228	C₄×C₅	#901	1	1	yes	characteristic
229	C₄×C₅	#908	1	1	yes	characteristic
230	C₄×C₅	#909	1	1	yes	characteristic
231	Dic₅	—	6	2	no	—
232	Dic₅	—	12	1	no	—
233	Dic₅	—	12	1	no	—
234	C₂×Q₈	—	15	1	no	—
235	C₂×D₄	—	15	1	no	—
236	C₂²×C₄	—	30	1	no	—
237	SD₁₆	—	15	4	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	30	1	no	—
240	C₂×C₈	—	30	1	no	—
241	C₂×C₈	—	60	2	no	—
242	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
243	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
244	C₂×C₈	D₆₀	1	1	yes	characteristic
245	C₂×C₈	D₆₀	1	1	yes	characteristic
246	C₄⋊C₄	—	15	1	no	—
247	C₄⋊C₄	—	15	1	no	—
248	C₄⋊C₄	—	30	1	no	—
249	C₂²⋊C₄	—	30	1	no	—
250	C₄²	—	30	1	no	—
251	C₄²	C₂²×D₁₅	1	1	yes	characteristic
252	C₃×C₅	SD₁₆⋊C₈	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
254	C₂×D₃	—	10	2	no	—
255	C₂×D₃	—	20	1	no	—
256	C₂×D₃	—	20	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
259	C₃×C₄	#4047	1	1	yes	characteristic
260	C₃×C₄	#4054	1	1	yes	characteristic
261	C₃×C₄	#4055	1	1	yes	characteristic
262	Dic₃	—	10	2	no	—
263	Dic₃	—	20	1	no	—
264	Dic₃	—	20	1	no	—
265	C₂×C₅	C₈×D₁₂	1	1	yes	characteristic
266	C₂×C₅	#4710	1	1	yes	characteristic
267	C₂×C₅	#4717	1	1	yes	characteristic
268	D₅	—	12	2	no	—
269	C₂³	—	30	1	no	—
270	Q₈	—	15	2	no	—
271	Q₈	—	30	1	no	—
272	D₄	—	15	2	no	—
273	D₄	—	30	1	no	—
274	C₂×C₄	—	30	1	no	—
275	C₂×C₄	—	30	1	no	—
276	C₂×C₄	—	30	1	no	—
277	C₂×C₄	—	60	1	no	—
278	C₂×C₄	—	60	1	no	—
279	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
280	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
281	C₂×C₄	#6817	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	4	1	no	—
286	C₈	—	60	1	no	—
287	C₈	—	60	1	no	—
288	C₈	C₈×D₁₅	1	2	yes	complemented
289	C₂×C₃	C₈×D₂₀	1	1	yes	characteristic
290	C₂×C₃	#64999	1	1	yes	characteristic
291	C₂×C₃	#65006	1	1	yes	characteristic
292	D₃	—	20	2	no	—
293	C₅	#68076	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	30	2	no	—
295	C₂²	—	60	1	no	—
296	C₂²	—	60	1	no	—
297	C₂²	C₄×D₆₀	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	30	2	no	—
300	C₄	—	60	1	no	—
301	C₄	—	60	1	no	—
302	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
303	C₄	#91790	1	1	yes	characteristic
304	C₄	#91797	1	1	yes	characteristic
305	C₄	#91798	1	1	yes	characteristic
306	C₃	#10598869	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	60	2	no	—
308	C₂	C₈×D₆₀	1	1	yes	characteristic
309	C₂	#11750811	1	1	yes	characteristic
310	C₂	#11750818	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₀
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×D₆₀	C₂×C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#322	D₁₅
hypocenter	C₃×C₅	SD₁₆⋊C₈
LCS	G ▷ 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
3. C₈×Dic₃₀	C₂	semidirect
4. C₃×C₅×#322	C₂	semidirect
6. #11746279	C₂	semidirect
7. #11746277	C₂	semidirect
28. C₅×#322	D₃	semidirect
48. #6706	C₈	semidirect
53. C₃×#322	D₅	semidirect
85. Dic₃₀	C₂×C₈	semidirect
90. C₃×C₅×C₈	C₂×C₈	semidirect
132. #322	D₁₅	semidirect
180. C₅×C₈	D₃×C₈	semidirect
219. C₃×C₈	C₈×D₅	semidirect
252. C₃×C₅	SD₁₆⋊C₈	semidirect
288. C₈	C₈×D₁₅	semidirect
293. C₅	#68076	semidirect
306. C₃	#10598869	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6706	2
C₈×D₆₀	2
C₈×Dic₃₀	2
C₃×C₅×#322	2
#11746280	2
#11746279	2
#11746277	2
#10598869	3
#68076	5