Subgroups — #14285867

Order 1920.

336 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#11747007	C₂	1	1	yes	maximal, characteristic
3	#11746868	C₂	1	1	yes	maximal, characteristic
4	#11746300	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4479	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745869	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745868	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599365	—	3	1	no	maximal, Hall
9	C₂×C₃×C₈×D₅	—	2	1	no	—
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	C₂×#6649	—	2	1	no	—
12	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×#3946	—	2	1	no	—
18	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
19	#90982	—	2	1	no	—
20	#90982	C₄	1	1	yes	characteristic
21	#90982	C₄	1	1	yes	characteristic
22	#90955	C₄	1	1	yes	characteristic
23	#90955	C₄	1	1	yes	characteristic
24	#70422	—	5	1	no	maximal, Hall
25	#65126	—	3	1	no	—
26	D₅×C₄⋊C₈	D₃	1	1	yes	characteristic
27	#64974	—	3	1	no	—
28	C₅×#335	—	3	1	no	—
29	#64699	—	3	1	no	—
30	#64699	—	3	1	no	—
31	#64678	—	3	1	no	—
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₃⋊#168	—	2	1	no	—
38	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₃⋊C₈	—	2	1	no	—
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
45	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
46	C₂×C₃×#168	—	2	1	no	—
47	C₃×C₈×D₅	—	4	2	no	—
48	#6649	—	4	2	no	—
49	C₂×C₃×C₄⋊C₈	—	5	1	no	—
50	#4925	—	5	1	no	—
51	#4924	—	5	1	no	—
52	#4479	—	5	1	no	—
53	#4479	—	5	1	no	—
54	#4479	—	5	1	no	—
55	#4479	D₅	1	1	yes	characteristic, complemented
56	C₂×#676	—	6	1	no	—
57	C₂×C₈×D₅	—	2	1	no	—
58	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
59	C₅×C₄⋊C₈	—	3	1	no	—
60	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
61	C₄×C₅×C₈	—	3	1	no	—
62	#3952	—	3	1	no	—
63	#3952	—	3	1	no	—
64	#3946	—	2	1	no	—
65	#3946	—	3	1	no	—
66	#3946	—	3	1	no	—
67	#3946	—	6	1	no	—
68	C₄⋊#168	—	3	1	no	—
69	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
70	C₄×#168	—	3	1	no	—
71	#1571	—	15	1	no	Sylow
72	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
79	C₃×C₄×D₅	—	2	1	no	—
80	C₃×C₄×D₅	—	2	1	no	—
81	C₃×C₄×D₅	—	4	1	no	—
82	C₃×C₄×D₅	D₈	1	2	yes	—
83	C₃×C₄×D₅	Q₁₆	1	2	yes	—
84	C₃×C₅×C₈	—	4	1	no	—
85	C₃⋊#168	—	2	1	no	—
86	C₃⋊#168	—	2	1	no	—
87	C₃⋊#168	—	4	1	no	—
88	C₃×#168	—	4	1	no	—
89	C₅×C₃⋊C₈	—	2	1	no	—
90	C₅×C₃⋊C₈	—	2	1	no	—
91	C₅×C₃⋊C₈	—	4	1	no	—
92	C₂²×C₃×C₈	—	10	1	no	—
93	C₂×C₃×C₄²	—	5	1	no	—
94	C₂×#208	—	10	1	no	—
95	C₃×C₄⋊C₈	—	5	1	no	—
96	C₃×C₄⋊C₈	—	10	1	no	—
97	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
98	#830	—	5	1	no	—
99	#830	—	5	1	no	—
100	C₄⋊(C₃⋊C₈)	—	5	1	no	—
101	C₄⋊(C₃⋊C₈)	—	5	1	no	—
102	C₄⋊(C₃⋊C₈)	—	5	1	no	—
103	C₄⋊(C₃⋊C₈)	—	10	1	no	—
104	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
105	C₄×C₃⋊C₈	—	5	1	no	—
106	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
107	C₂×C₄×D₅	Dic₆	1	1	yes	characteristic
108	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
109	C₂×C₄×D₅	#67	1	1	yes	characteristic
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	—	3	1	no	—
112	C₂×C₅×C₈	—	3	1	no	—
113	C₂×C₅×C₈	—	6	1	no	—
114	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₄×Dic₅	Dic₆	1	1	yes	characteristic
116	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
117	C₄×Dic₅	#67	1	1	yes	characteristic
118	C₂×#168	—	2	1	no	—
119	C₂×#168	—	3	1	no	—
120	C₂×#168	—	3	1	no	—
121	C₂×#168	—	6	1	no	—
122	#676	—	12	2	no	—
123	C₈×D₅	—	4	2	no	—
124	#431	—	15	1	no	—
125	#423	—	15	1	no	—
126	C₂×C₄⋊C₈	—	5	1	no	—
127	#335	—	3	1	no	—
128	#335	—	15	1	no	—
129	#335	—	15	1	no	—
130	#335	—	15	1	no	—
131	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₃×D₅	—	2	1	no	—
133	C₂×C₃×D₅	—	2	1	no	—
134	C₂×C₃×D₅	#107	1	2	yes	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	#131	1	1	yes	characteristic
137	C₃×C₄×C₅	#131	1	1	yes	characteristic
138	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
140	C₃×Dic₅	—	2	1	no	—
141	C₃×Dic₅	—	2	1	no	—
142	C₃×Dic₅	#107	1	2	yes	—
143	C₂²×C₃×C₄	—	5	1	no	—
144	C₂²×C₃×C₄	—	5	1	no	—
145	C₂²×C₃×C₄	—	5	1	no	—
146	C₂×C₃×C₈	—	2	1	no	—
147	C₂×C₃×C₈	—	10	1	no	—
148	C₂×C₃×C₈	—	20	2	no	—
149	C₃×C₄²	—	5	1	no	—
150	C₃×C₄²	—	5	1	no	—
151	C₃×C₄²	—	5	1	no	—
152	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
153	#208	—	20	2	no	—
154	C₂×C₃⋊C₈	—	2	1	no	—
155	C₂×C₃⋊C₈	—	5	1	no	—
156	C₂×C₃⋊C₈	—	5	1	no	—
157	C₂×C₃⋊C₈	—	10	1	no	—
158	C₂×C₃⋊C₈	C₄×D₅	1	1	yes	characteristic
159	C₂×C₃⋊C₈	C₄×D₅	1	1	yes	characteristic
160	C₂²×D₅	Dic₃⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
164	C₂×Dic₅	Dic₃⋊C₄	1	1	yes	characteristic
165	C₂×Dic₅	Dic₃⋊C₄	1	1	yes	characteristic
166	C₂×Dic₅	Dic₃⋊C₄	1	1	yes	characteristic
167	C₄×D₅	—	2	1	no	—
168	C₄×D₅	—	2	1	no	—
169	C₄×D₅	—	4	1	no	—
170	C₄×D₅	#213	1	2	yes	—
171	C₄×D₅	#216	1	2	yes	—
172	C₅×C₈	—	4	1	no	—
173	C₅×C₈	—	6	1	no	—
174	C₅×C₈	—	6	1	no	—
175	C₅×C₈	—	12	1	no	—
176	#168	—	4	1	no	—
177	#168	—	6	1	no	—
178	#168	—	6	1	no	—
179	#168	—	12	1	no	—
180	C₂×M₁₆	—	30	1	no	—
181	C₂²×C₈	—	10	1	no	—
182	C₂×C₄²	—	5	1	no	—
183	C₄⋊C₈	—	3	1	no	—
184	C₄⋊C₈	—	5	1	no	—
185	C₄⋊C₈	—	10	1	no	—
186	C₄⋊C₈	—	15	1	no	—
187	C₄⋊C₈	—	15	1	no	—
188	C₄⋊C₈	—	15	1	no	—
189	C₄⋊C₈	—	30	1	no	—
190	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
191	C₂²⋊C₈	—	15	1	no	—
192	C₂²⋊C₈	—	15	1	no	—
193	C₄×C₈	—	3	1	no	—
194	C₄×C₈	—	15	1	no	—
195	C₂×C₃×C₅	#424	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
197	C₂×C₃×C₅	#430	1	1	yes	characteristic
198	C₃×D₅	—	2	2	no	—
199	C₂³×C₃	—	5	1	no	—
200	C₂×C₃×C₄	—	5	1	no	—
201	C₂×C₃×C₄	—	5	2	no	—
202	C₂×C₃×C₄	—	5	2	no	—
203	C₂×C₃×C₄	—	5	1	no	—
204	C₂×C₃×C₄	—	5	1	no	—
205	C₂×C₃×C₄	—	10	1	no	—
206	C₂×C₃×C₄	—	10	1	no	—
207	C₂×C₃×C₄	—	20	1	no	—
208	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
210	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
211	C₃×C₈	—	4	1	no	—
212	C₃×C₈	—	20	1	no	—
213	C₃⋊C₈	—	2	1	no	—
214	C₃⋊C₈	—	2	1	no	—
215	C₃⋊C₈	—	4	1	no	—
216	C₃⋊C₈	—	10	1	no	—
217	C₃⋊C₈	—	10	1	no	—
218	C₃⋊C₈	—	20	1	no	—
219	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
220	C₂×D₅	—	2	1	no	—
221	C₂×D₅	—	2	1	no	—
222	C₂×D₅	#807	1	2	yes	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	#901	1	1	yes	characteristic
225	C₄×C₅	#901	1	1	yes	characteristic
226	C₄×C₅	C₂×#213	1	1	yes	characteristic
227	C₄×C₅	C₂×#216	1	1	yes	characteristic
228	Dic₅	—	2	1	no	—
229	Dic₅	—	2	1	no	—
230	Dic₅	#807	1	2	yes	—
231	C₂²×C₄	—	5	1	no	—
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	—	5	1	no	—
234	M₁₆	—	60	2	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	3	1	no	—
237	C₂×C₈	—	3	1	no	—
238	C₂×C₈	—	6	1	no	—
239	C₂×C₈	—	10	1	no	—
240	C₂×C₈	—	15	1	no	—
241	C₂×C₈	—	15	1	no	—
242	C₂×C₈	—	20	2	no	—
243	C₂×C₈	—	30	1	no	—
244	C₄²	—	5	1	no	—
245	C₄²	—	5	1	no	—
246	C₄²	—	5	1	no	—
247	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
248	C₃×C₅	#1571	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	—	5	2	no	—
250	C₂²×C₃	—	10	1	no	—
251	C₂²×C₃	—	10	1	no	—
252	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	5	2	no	—
255	C₃×C₄	—	10	1	no	—
256	C₃×C₄	—	10	1	no	—
257	C₃×C₄	#4047	1	1	yes	characteristic
258	C₃×C₄	#4053	1	1	yes	characteristic
259	C₃×C₄	D₅×D₈	1	1	yes	characteristic
260	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
261	C₂×C₅	C₂×#807	1	1	yes	characteristic
262	C₂×C₅	#5102	1	1	yes	characteristic
263	C₂×C₅	#5124	1	1	yes	characteristic
264	D₅	—	2	2	no	—
265	C₂³	—	5	1	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	5	2	no	—
268	C₂×C₄	—	5	2	no	—
269	C₂×C₄	—	5	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	10	1	no	—
272	C₂×C₄	—	10	1	no	—
273	C₂×C₄	—	20	1	no	—
274	C₂×C₄	D₅×Dic₆	1	1	yes	characteristic
275	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
276	C₂×C₄	D₅×#67	1	1	yes	characteristic
277	C₈	—	4	1	no	—
278	C₈	—	6	1	no	—
279	C₈	—	6	1	no	—
280	C₈	—	12	1	no	—
281	C₈	—	20	1	no	—
282	C₈	—	30	1	no	—
283	C₈	—	30	1	no	—
284	C₈	—	60	1	no	—
285	C₂×C₃	—	10	2	no	—
286	C₂×C₃	#65128	1	1	yes	characteristic
287	C₂×C₃	D₅×#107	1	1	yes	characteristic
288	C₂×C₃	#65182	1	1	yes	characteristic
289	C₅	#70422	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	5	2	no	—
291	C₂²	—	10	1	no	—
292	C₂²	—	10	1	no	—
293	C₂²	D₅×Dic₃⋊C₄	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	5	2	no	—
296	C₄	—	10	1	no	—
297	C₄	—	10	1	no	—
298	C₄	#91265	1	1	yes	characteristic
299	C₄	#91266	1	1	yes	characteristic
300	C₄	D₅×#213	1	1	yes	characteristic
301	C₄	D₅×#216	1	1	yes	characteristic
302	C₃	#10599365	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	10	2	no	—
304	C₂	D₅×#807	1	1	yes	characteristic
305	C₂	#11747647	1	1	yes	characteristic
306	C₂	#11748171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×#67
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₄⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	#1571
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #11746300	C₂	semidirect
5. C₅×#4479	C₂	semidirect
6. #11745869	C₂	semidirect
7. #11745868	C₂	semidirect
55. #4479	D₅	semidirect
248. C₃×C₅	#1571	semidirect
289. C₅	#70422	semidirect
302. C₃	#10599365	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₅×C₄⋊C₈	2
#11747007	2
#11746868	2
#11746300	2
C₅×#4479	2
#11745869	2
#11745868	2
#10599365	3
#70422	5