Subgroups — #14285947

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11750936	C₂	1	1	yes	maximal, characteristic
2	D₁₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#11750786	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746301	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746300	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599327	—	3	1	no	maximal, Hall
9	C₂×C₈×D₁₅	—	2	1	no	—
10	C₄²×D₁₅	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#91104	—	2	1	no	—
15	#91104	C₄	1	1	yes	characteristic
16	#91104	C₄	1	1	yes	characteristic
17	#91099	—	2	1	no	—
18	#91098	—	2	1	no	—
19	#91098	C₄	1	1	yes	characteristic
20	#91098	C₄	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
23	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
24	#68534	—	5	1	no	maximal, Hall
25	#65124	—	3	1	no	—
26	D₅×C₄⋊C₈	—	3	1	no	—
27	#64974	—	3	1	no	—
28	C₅×#334	D₃	1	1	yes	characteristic, complemented
29	#64700	—	3	1	no	—
30	#64699	—	3	1	no	—
31	#64678	—	3	1	no	—
32	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
34	C₂×C₄×D₁₅	Q₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	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	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₁₅	D₄	1	1	yes	characteristic
42	C₄×Dic₁₅	Q₈	1	1	yes	characteristic
43	C₂×C₃⋊#168	—	2	1	no	—
44	C₂×C₃⋊#168	—	2	1	no	—
45	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
47	C₈×D₁₅	—	4	2	no	—
48	#4835	—	5	1	no	—
49	D₃×C₄⋊C₈	—	5	1	no	—
50	#4685	—	5	1	no	—
51	C₃×#334	D₅	1	1	yes	characteristic, complemented
52	#4480	—	5	1	no	—
53	#4479	—	5	1	no	—
54	#4458	—	5	1	no	—
55	C₂×C₈×D₅	—	6	1	no	—
56	C₄²×D₅	—	3	1	no	—
57	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
58	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
60	#3952	—	3	1	no	—
61	#3952	—	3	1	no	—
62	#3952	—	6	1	no	—
63	#3947	—	6	1	no	—
64	#3946	—	3	1	no	—
65	#3946	—	3	1	no	—
66	#3946	—	6	1	no	—
67	C₄⋊#168	—	3	1	no	—
68	C₄⋊#168	—	3	1	no	—
69	C₄×#168	—	3	1	no	—
70	#1573	—	15	1	no	Sylow
71	C₂²×D₁₅	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
75	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
76	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
77	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
78	C₄×D₁₅	—	2	2	no	—
79	C₄×D₁₅	—	2	1	no	—
80	C₄×D₁₅	—	2	1	no	—
81	C₄×D₁₅	—	2	1	no	—
82	C₄×D₁₅	—	2	1	no	—
83	C₃×C₅×C₈	—	2	1	no	—
84	C₃×C₅×C₈	—	2	1	no	—
85	C₃×C₅×C₈	—	4	1	no	—
86	C₃×C₅×C₈	—	4	1	no	—
87	C₃⋊#168	—	2	1	no	—
88	C₃⋊#168	—	2	1	no	—
89	C₃⋊#168	—	4	1	no	—
90	C₃⋊#168	—	4	1	no	—
91	C₂×D₃×C₈	—	10	1	no	—
92	C₄²×D₃	—	5	1	no	—
93	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
94	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
95	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
96	(C₂×D₃)⋊C₈	—	5	1	no	—
97	(C₂×D₃)⋊C₈	—	5	1	no	—
98	(C₂×D₃)⋊C₈	—	10	1	no	—
99	#815	—	10	1	no	—
100	Dic₃⋊C₈	—	5	1	no	—
101	Dic₃⋊C₈	—	5	1	no	—
102	Dic₃⋊C₈	—	10	1	no	—
103	C₄⋊(C₃⋊C₈)	—	5	1	no	—
104	C₄⋊(C₃⋊C₈)	—	5	1	no	—
105	C₄×C₃⋊C₈	—	5	1	no	—
106	C₂×C₄×D₅	—	3	1	no	—
107	C₂×C₄×D₅	—	3	1	no	—
108	C₂×C₄×D₅	—	3	1	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
112	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
113	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
114	C₄×Dic₅	—	3	1	no	—
115	C₄×Dic₅	—	3	1	no	—
116	C₄×Dic₅	—	3	1	no	—
117	C₂×#168	—	3	1	no	—
118	C₂×#168	—	3	1	no	—
119	C₂×#168	—	6	1	no	—
120	C₂×#168	—	6	1	no	—
121	C₈×D₅	—	12	2	no	—
122	#432	—	15	1	no	—
123	#431	—	15	1	no	—
124	C₂×C₄⋊C₈	—	15	1	no	—
125	#335	—	15	1	no	—
126	#335	—	15	1	no	—
127	#334	—	15	1	no	—
128	#334	D₁₅	1	1	yes	characteristic, complemented
129	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×D₁₅	—	2	1	no	—
131	C₂×D₁₅	—	2	1	no	—
132	C₂×D₁₅	#108	1	2	yes	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	#131	1	1	yes	characteristic
135	C₃×C₄×C₅	#131	1	1	yes	characteristic
136	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₃×C₄×C₅	#137	1	1	yes	characteristic
138	Dic₁₅	—	2	1	no	—
139	Dic₁₅	—	2	1	no	—
140	Dic₁₅	—	2	1	no	—
141	C₂×C₄×D₃	—	5	1	no	—
142	C₂×C₄×D₃	—	5	1	no	—
143	C₂×C₄×D₃	—	5	1	no	—
144	C₂×C₃×C₈	—	2	1	no	—
145	C₂×C₃×C₈	—	2	1	no	—
146	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
147	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
148	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
149	C₄×Dic₃	—	5	1	no	—
150	C₄×Dic₃	—	5	1	no	—
151	C₄×Dic₃	—	5	1	no	—
152	C₂×C₃⋊C₈	—	5	1	no	—
153	C₂×C₃⋊C₈	—	5	1	no	—
154	C₂×C₃⋊C₈	—	10	1	no	—
155	C₂×C₃⋊C₈	—	10	1	no	—
156	D₃×C₈	—	20	2	no	—
157	C₂²×D₅	—	3	1	no	—
158	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
159	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
161	C₂×Dic₅	—	3	1	no	—
162	C₂×Dic₅	—	3	1	no	—
163	C₂×Dic₅	—	3	1	no	—
164	C₄×D₅	—	6	2	no	—
165	C₄×D₅	—	6	1	no	—
166	C₄×D₅	—	6	1	no	—
167	C₄×D₅	—	6	1	no	—
168	C₄×D₅	—	6	1	no	—
169	C₅×C₈	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	4	1	no	—
172	C₅×C₈	—	4	1	no	—
173	#168	—	6	1	no	—
174	#168	—	6	1	no	—
175	#168	—	12	1	no	—
176	#168	—	12	1	no	—
177	C₂²×C₈	—	30	1	no	—
178	C₂×C₄²	—	15	1	no	—
179	C₄⋊C₈	—	15	1	no	—
180	C₄⋊C₈	—	15	1	no	—
181	C₄⋊C₈	—	15	1	no	—
182	C₄⋊C₈	—	15	1	no	—
183	C₄⋊C₈	—	30	1	no	—
184	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
185	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
186	C₂²⋊C₈	—	15	1	no	—
187	C₂²⋊C₈	—	15	1	no	—
188	C₂²⋊C₈	—	30	1	no	—
189	#97	—	30	1	no	—
190	C₄×C₈	—	15	1	no	—
191	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
192	C₂×C₃×C₅	#424	1	1	yes	characteristic
193	C₂×C₃×C₅	#428	1	1	yes	characteristic
194	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
195	D₁₅	—	2	2	no	—
196	C₂²×D₃	—	5	1	no	—
197	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
198	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
199	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
200	C₂×Dic₃	—	5	1	no	—
201	C₂×Dic₃	—	5	1	no	—
202	C₂×Dic₃	—	5	1	no	—
203	C₄×D₃	—	10	2	no	—
204	C₄×D₃	—	10	1	no	—
205	C₄×D₃	—	10	1	no	—
206	C₄×D₃	—	10	1	no	—
207	C₄×D₃	—	10	1	no	—
208	C₃×C₈	—	2	1	no	—
209	C₃×C₈	—	2	1	no	—
210	C₃×C₈	—	4	1	no	—
211	C₃×C₈	—	4	1	no	—
212	C₃⋊C₈	—	10	1	no	—
213	C₃⋊C₈	—	10	1	no	—
214	C₃⋊C₈	—	20	1	no	—
215	C₃⋊C₈	—	20	1	no	—
216	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂×D₅	—	3	2	no	—
218	C₂×D₅	—	6	1	no	—
219	C₂×D₅	—	6	1	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	#901	1	1	yes	characteristic
222	C₄×C₅	#907	1	1	yes	characteristic
223	C₄×C₅	#914	1	1	yes	characteristic
224	C₄×C₅	#915	1	1	yes	characteristic
225	Dic₅	—	6	1	no	—
226	Dic₅	—	6	1	no	—
227	Dic₅	—	6	1	no	—
228	C₂²×C₄	—	15	1	no	—
229	C₂²×C₄	—	15	1	no	—
230	C₂²×C₄	—	15	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	15	1	no	—
234	C₂×C₈	—	15	1	no	—
235	C₂×C₈	—	30	1	no	—
236	C₂×C₈	—	30	1	no	—
237	C₂×C₈	—	60	2	no	—
238	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
239	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
240	C₄²	—	15	1	no	—
241	C₄²	—	15	1	no	—
242	C₄²	—	15	1	no	—
243	C₄²	C₂²×D₁₅	1	1	yes	characteristic
244	C₃×C₅	#1573	1	1	yes	characteristic, complemented, Hall
245	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
246	C₂×D₃	—	5	2	no	—
247	C₂×D₃	—	10	1	no	—
248	C₂×D₃	—	10	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	#4047	1	1	yes	characteristic
251	C₃×C₄	#4053	1	1	yes	characteristic
252	C₃×C₄	#4060	1	1	yes	characteristic
253	C₃×C₄	#4061	1	1	yes	characteristic
254	Dic₃	—	10	1	no	—
255	Dic₃	—	10	1	no	—
256	Dic₃	—	10	1	no	—
257	C₂×C₅	#4839	1	1	yes	characteristic
258	C₂×C₅	#4861	1	1	yes	characteristic
259	C₂×C₅	D₃×#108	1	1	yes	characteristic
260	D₅	—	6	2	no	—
261	C₂³	—	15	1	no	—
262	C₂×C₄	—	15	1	no	—
263	C₂×C₄	—	15	1	no	—
264	C₂×C₄	—	15	1	no	—
265	C₂×C₄	—	30	2	no	—
266	C₂×C₄	—	30	1	no	—
267	C₂×C₄	—	30	1	no	—
268	C₂×C₄	—	30	1	no	—
269	C₂×C₄	—	30	1	no	—
270	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
271	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
272	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	30	1	no	—
278	C₈	—	30	1	no	—
279	C₈	—	60	1	no	—
280	C₈	—	60	1	no	—
281	C₂×C₃	#65128	1	1	yes	characteristic
282	C₂×C₃	#65150	1	1	yes	characteristic
283	C₂×C₃	D₅×#108	1	1	yes	characteristic
284	D₃	—	10	2	no	—
285	C₅	#68534	1	1	yes	characteristic, complemented, Sylow
286	C₂²	—	15	2	no	—
287	C₂²	—	30	1	no	—
288	C₂²	—	30	1	no	—
289	C₂²	C₄⋊C₄×D₁₅	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	30	1	no	—
292	C₄	—	30	1	no	—
293	C₄	—	30	1	no	—
294	C₄	#91790	1	1	yes	characteristic
295	C₄	#91796	1	1	yes	characteristic
296	C₄	#91803	1	1	yes	characteristic
297	C₄	#91804	1	1	yes	characteristic
298	C₃	#10599327	1	1	yes	characteristic, complemented, Sylow
299	C₂	—	30	2	no	—
300	C₂	#11750940	1	1	yes	characteristic
301	C₂	#11750962	1	1	yes	characteristic
302	C₂	D₁₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#334	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₁₅	C₄⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#334	D₁₅
hypocenter	C₃×C₅	#1573
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#334	C₂	semidirect
5. #11746301	C₂	semidirect
6. #11746300	C₂	semidirect
7. #11746279	C₂	semidirect
28. C₅×#334	D₃	semidirect
51. C₃×#334	D₅	semidirect
128. #334	D₁₅	semidirect
244. C₃×C₅	#1573	semidirect
285. C₅	#68534	semidirect
298. C₃	#10599327	semidirect

Maximal subgroups

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

subgroup	index
#11750936	2
D₁₅×C₄⋊C₈	2
#11750786	2
C₃×C₅×#334	2
#11746301	2
#11746300	2
#11746279	2
#10599327	3
#68534	5