Subgroups — #13089351

Order 1440.

624 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624283	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#235	C₂	1	1	yes	maximal, characteristic, complemented
4	#10624210	C₂	1	4	yes	maximal, complemented
5	#10624198	C₂	1	2	yes	maximal, complemented
6	#10624197	C₂	1	2	yes	maximal, complemented
7	#10624196	C₂	1	2	yes	maximal, complemented
8	D₃×Dic₃₀	C₂	1	2	yes	maximal, complemented
9	#92101	—	3	1	no	maximal
10	#92004	—	3	1	no	maximal
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	#67184	C₂²	1	2	yes	—
13	C₄⋊(C₃⋊D₁₅)	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊D₁₅	C₂²	1	2	yes	—
15	#67180	C₂²	1	1	yes	characteristic
16	C₂×C₃×Dic₁₅	C₂²	1	2	yes	—
17	C₃×Dic₃₀	C₂²	1	2	yes	—
18	C₃×Dic₃₀	C₂²	1	2	yes	—
19	C₃×C₅×#67	C₂²	1	2	yes	—
20	C₃×C₅×D₁₂	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₅×D₃	C₂²	1	2	yes	—
22	C₃×C₅×Dic₆	C₂²	1	1	yes	characteristic
23	#67153	C₂²	1	4	yes	—
24	#67152	C₂²	1	4	yes	—
25	#67149	C₂²	1	4	yes	—
26	D₃×Dic₁₅	C₂²	1	4	yes	—
27	#64297	—	5	1	no	maximal, Hall
28	C₃×C₅×#41	—	3	1	no	—
29	Q₈×D₁₅	—	3	2	no	—
30	#6819	—	3	4	no	—
31	#6819	—	3	2	no	—
32	#6817	—	3	1	no	—
33	#6817	—	3	1	no	—
34	#6817	—	3	2	no	—
35	#6817	—	6	1	no	—
36	C₂×Dic₃₀	—	3	2	no	—
37	C₂×Dic₃₀	D₃	1	1	yes	characteristic, complemented
38	C₅×#235	D₃	1	1	yes	characteristic, complemented
39	C₂×C₃×Dic₁₀	—	3	1	no	—
40	#6782	—	3	4	no	—
41	D₁₅⋊Q₈	—	3	2	no	—
42	#6769	—	3	2	no	—
43	#6768	—	3	2	no	—
44	D₃×Dic₁₀	—	3	2	no	—
45	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊D₁₅	C₂³	1	2	yes	—
47	C₂×C₃×C₅×D₃	C₂³	1	2	yes	—
48	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₃⋊Dic₁₅	C₂³	1	2	yes	—
51	C₃×Dic₁₅	C₂³	1	4	yes	—
52	C₃×C₅×Dic₃	C₂³	1	2	yes	—
53	#4150	—	9	1	no	Hall
54	#3828	—	5	1	no	—
55	C₃×#235	D₅	1	1	yes	characteristic, complemented
56	C₂×C₃×Dic₆	—	5	1	no	—
57	#3804	—	5	4	no	—
58	#3799	—	5	2	no	—
59	#3797	—	5	2	no	—
60	#3796	—	5	2	no	—
61	D₃×Dic₆	—	5	2	no	—
62	C₃×C₅×Q₈	—	3	1	no	—
63	C₃×C₅×D₄	—	3	2	no	—
64	C₃×C₅×D₄	—	3	1	no	—
65	C₂×C₃×C₄×C₅	—	2	1	no	—
66	C₂×C₃×C₄×C₅	—	3	2	no	—
67	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
69	#1220	—	3	2	no	—
70	#1220	—	3	2	no	—
71	#1220	—	3	4	no	—
72	#1220	—	6	2	no	—
73	C₂×Dic₁₅	—	3	4	no	—
74	C₂×Dic₁₅	C₂×D₃	1	2	yes	—
75	D₆₀	—	3	1	no	—
76	D₆₀	—	3	1	no	—
77	D₆₀	—	6	1	no	—
78	C₄×D₁₅	—	3	2	no	—
79	C₄×D₁₅	—	3	2	no	—
80	C₄×D₁₅	—	3	4	no	—
81	C₄×D₁₅	—	6	2	no	—
82	Dic₃₀	—	3	1	no	—
83	Dic₃₀	—	3	1	no	—
84	Dic₃₀	—	3	4	no	—
85	Dic₃₀	—	6	1	no	—
86	Dic₃₀	C₂×D₃	1	2	yes	—
87	Dic₃₀	C₂×D₃	1	2	yes	—
88	C₅×#67	C₂×D₃	1	2	yes	—
89	C₅×D₁₂	C₂×D₃	1	1	yes	characteristic
90	C₄×C₅×D₃	C₂×D₃	1	2	yes	—
91	C₅×Dic₆	C₂×D₃	1	1	yes	characteristic
92	C₂×C₃×Dic₅	—	3	2	no	—
93	C₃×Dic₁₀	—	3	2	no	—
94	C₃×Dic₁₀	—	3	2	no	—
95	#1204	—	3	4	no	—
96	#1203	—	3	4	no	—
97	#1200	—	3	4	no	—
98	D₃×Dic₅	—	3	4	no	—
99	#1010	—	15	1	no	—
100	#1007	—	15	1	no	—
101	C₂×C₃²×C₅	—	2	1	no	—
102	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
103	C₃⋊D₁₅	—	2	2	no	—
104	C₃×C₅×D₃	—	2	2	no	—
105	C₅×#41	—	3	1	no	—
106	Q₈×D₅	—	9	2	no	—
107	#711	—	9	2	no	—
108	#711	—	9	4	no	—
109	#709	—	9	2	no	—
110	#709	—	9	1	no	—
111	C₂×Dic₁₀	—	3	1	no	—
112	C₂×Dic₁₀	—	9	2	no	—
113	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
114	#638	—	5	2	no	—
115	C₄⋊(C₃⋊D₃)	—	5	1	no	—
116	C₄×C₃⋊D₃	—	5	2	no	—
117	#634	—	5	1	no	—
118	C₃×#67	C₂×D₅	1	2	yes	—
119	C₂×C₃×Dic₃	—	5	2	no	—
120	C₃×D₁₂	C₂×D₅	1	1	yes	characteristic
121	C₃×C₄×D₃	C₂×D₅	1	2	yes	—
122	C₃×Dic₆	—	5	2	no	—
123	C₃×Dic₆	—	5	2	no	—
124	C₃×Dic₆	C₂×D₅	1	1	yes	characteristic
125	#627	—	5	4	no	—
126	#626	—	5	4	no	—
127	#624	—	5	4	no	—
128	D₃×Dic₃	—	5	4	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	3	2	no	—
131	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
133	C₂×D₁₅	—	3	2	no	—
134	C₂×D₁₅	—	3	2	no	—
135	C₂×D₁₅	—	6	2	no	—
136	C₂×C₅×D₃	C₂²×D₃	1	2	yes	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	3	2	no	—
140	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
144	Dic₁₅	—	3	2	no	—
145	Dic₁₅	—	3	2	no	—
146	Dic₁₅	—	6	2	no	—
147	Dic₁₅	C₂²×D₃	1	4	yes	—
148	C₃×Dic₅	—	3	4	no	—
149	C₅×Dic₃	C₂²×D₃	1	2	yes	—
150	C₃×#41	—	3	1	no	—
151	C₂×C₃×Q₈	—	15	1	no	—
152	#239	—	15	2	no	—
153	#239	—	15	2	no	—
154	D₃×Q₈	—	15	2	no	—
155	D₃×Q₈	—	15	2	no	—
156	D₃×Q₈	—	15	2	no	—
157	#237	—	15	2	no	—
158	#237	—	15	4	no	—
159	#235	—	15	2	no	—
160	#235	—	15	1	no	—
161	#235	—	15	1	no	—
162	#235	—	15	4	no	—
163	#235	—	30	1	no	—
164	#235	D₁₅	1	1	yes	characteristic, complemented
165	C₂×Dic₆	—	5	1	no	—
166	C₂×Dic₆	—	15	2	no	—
167	C₃²×C₅	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
168	C₅×Q₈	—	3	1	no	—
169	C₅×D₄	—	3	2	no	—
170	C₅×D₄	—	3	1	no	—
171	C₂×C₄×C₅	—	3	2	no	—
172	C₂×C₄×C₅	D₃²	1	1	yes	characteristic
173	C₂²⋊D₅	—	9	4	no	—
174	C₂²⋊D₅	—	9	2	no	—
175	C₂×Dic₅	—	3	2	no	—
176	C₂×Dic₅	—	9	4	no	—
177	D₂₀	—	9	1	no	—
178	C₄×D₅	—	9	4	no	—
179	C₄×D₅	—	9	2	no	—
180	Dic₁₀	—	3	2	no	—
181	Dic₁₀	—	3	2	no	—
182	Dic₁₀	—	9	4	no	—
183	Dic₁₀	—	9	1	no	—
184	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
185	C₂×C₃⋊D₃	—	5	2	no	—
186	C₂×C₃×D₃	C₂²×D₅	1	2	yes	—
187	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
188	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
189	C₃⋊Dic₃	—	5	2	no	—
190	C₃×Dic₃	—	5	4	no	—
191	C₃×Dic₃	C₂²×D₅	1	2	yes	—
192	2₋¹⁺⁴	—	45	1	no	Sylow
193	C₂×C₃×C₅	—	2	1	no	—
194	C₂×C₃×C₅	—	2	1	no	—
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	—	4	1	no	—
197	C₂×C₃×C₅	—	6	2	no	—
198	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
200	D₁₅	—	6	2	no	—
201	D₁₅	—	6	2	no	—
202	D₁₅	—	12	2	no	—
203	C₅×D₃	—	2	2	no	—
204	C₃×Q₈	—	3	1	no	—
205	C₃×Q₈	—	15	2	no	—
206	C₃×Q₈	—	15	2	no	—
207	C₃×D₄	—	3	2	no	—
208	C₃×D₄	—	3	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	3	2	no	—
211	C₂×C₃×C₄	—	15	2	no	—
212	C₂×C₃×C₄	D₃×D₅	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
214	#67	—	15	4	no	—
215	#67	—	15	2	no	—
216	#67	—	15	2	no	—
217	#67	—	30	2	no	—
218	#67	C₂×D₁₅	1	2	yes	—
219	C₂×Dic₃	—	5	2	no	—
220	C₂×Dic₃	—	15	4	no	—
221	D₁₂	—	15	4	no	—
222	D₁₂	—	15	1	no	—
223	D₁₂	—	15	1	no	—
224	D₁₂	—	30	1	no	—
225	D₁₂	C₂×D₁₅	1	1	yes	characteristic
226	C₄×D₃	—	15	4	no	—
227	C₄×D₃	—	15	4	no	—
228	C₄×D₃	—	15	2	no	—
229	C₄×D₃	—	15	2	no	—
230	C₄×D₃	—	15	4	no	—
231	C₄×D₃	—	30	2	no	—
232	C₄×D₃	C₂×D₁₅	1	2	yes	—
233	Dic₆	—	5	2	no	—
234	Dic₆	—	5	2	no	—
235	Dic₆	—	15	4	no	—
236	Dic₆	—	15	1	no	—
237	Dic₆	—	15	1	no	—
238	Dic₆	—	15	4	no	—
239	Dic₆	—	30	1	no	—
240	Dic₆	C₂×D₁₅	1	1	yes	characteristic
241	C₂²×C₅	—	3	2	no	—
242	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
243	C₂×D₅	—	9	2	no	—
244	C₄×C₅	—	3	2	no	—
245	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
246	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
247	Dic₅	—	3	4	no	—
248	Dic₅	—	9	2	no	—
249	C₂×C₃²	—	2	1	no	—
250	C₂×C₃²	C₂³×D₅	1	1	yes	characteristic
251	C₃⋊D₃	—	10	2	no	—
252	C₃×D₃	—	2	2	no	—
253	#41	—	3	1	no	—
254	#41	—	45	2	no	—
255	#41	—	45	4	no	—
256	#41	—	45	2	no	—
257	#41	—	45	1	no	—
258	C₂×Q₈	—	15	1	no	—
259	C₂×Q₈	—	45	2	no	—
260	C₂×Q₈	—	45	2	no	—
261	C₃×C₅	—	2	1	no	—
262	C₃×C₅	#1007	1	1	yes	characteristic, complemented
263	C₃×C₅	#1010	1	1	yes	characteristic, complemented
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	3	2	no	—
266	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
267	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
268	C₂×D₃	—	15	2	no	—
269	C₂×D₃	—	15	2	no	—
270	C₂×D₃	—	30	2	no	—
271	C₂×D₃	C₂²×D₁₅	1	2	yes	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	3	2	no	—
275	C₃×C₄	—	15	4	no	—
276	C₃×C₄	C₂×D₃×D₅	1	1	yes	characteristic
277	C₃×C₄	C₂×D₃×D₅	1	1	yes	characteristic
278	C₃×C₄	C₂²×D₁₅	1	1	yes	characteristic
279	C₃×C₄	C₂²×D₁₅	1	1	yes	characteristic
280	Dic₃	—	5	4	no	—
281	Dic₃	—	15	2	no	—
282	Dic₃	—	15	2	no	—
283	Dic₃	—	30	2	no	—
284	Dic₃	C₂²×D₁₅	1	2	yes	—
285	C₂×C₅	—	2	1	no	—
286	C₂×C₅	—	6	2	no	—
287	C₂×C₅	C₂²×D₃²	1	1	yes	characteristic
288	D₅	—	18	2	no	—
289	C₃²	#4150	1	1	yes	characteristic, complemented, Sylow
290	Q₈	—	3	1	no	—
291	Q₈	—	15	2	no	—
292	Q₈	—	15	2	no	—
293	Q₈	—	45	4	no	—
294	Q₈	—	45	1	no	—
295	D₄	—	3	2	no	—
296	D₄	—	3	1	no	—
297	D₄	—	45	4	no	—
298	D₄	—	45	2	no	—
299	D₄	—	45	1	no	—
300	C₂×C₄	—	3	2	no	—
301	C₂×C₄	—	15	2	no	—
302	C₂×C₄	—	45	4	no	—
303	C₂×C₄	—	45	4	no	—
304	C₂×C₄	—	45	2	no	—
305	C₂×C₄	D₃×D₁₅	1	1	yes	characteristic
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	4	1	no	—
310	C₂×C₃	—	6	2	no	—
311	C₂×C₃	C₂²×D₃×D₅	1	1	yes	characteristic
312	C₂×C₃	C₂³×D₁₅	1	1	yes	characteristic
313	D₃	—	2	2	no	—
314	D₃	—	30	2	no	—
315	D₃	—	30	2	no	—
316	D₃	—	60	2	no	—
317	C₅	#64297	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	3	2	no	—
319	C₂²	—	45	2	no	—
320	C₂²	C₂×D₃×D₁₅	1	1	yes	characteristic
321	C₄	—	3	2	no	—
322	C₄	—	15	4	no	—
323	C₄	—	45	2	no	—
324	C₄	C₂×D₃×D₁₅	1	1	yes	characteristic
325	C₄	C₂×D₃×D₁₅	1	1	yes	characteristic
326	C₃	—	2	1	no	—
327	C₃	#92004	1	1	yes	characteristic, complemented
328	C₃	#92101	1	1	yes	characteristic, complemented
329	C₂	—	2	1	no	—
330	C₂	—	6	2	no	—
331	C₂	—	90	2	no	—
332	C₂	C₂²×D₃×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃×D₁₅
commutator, socle, cosocle	C₂×C₃²×C₅	C₂⁴
Fitting	C₂×C₃²×C₄×C₅	C₂²
hypercenter	C₂×C₄	D₃×D₁₅
hypocenter	C₃²×C₅	2₋¹⁺⁴
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. #10624283	C₂	semidirect
2. C₂×C₃×Dic₃₀	C₂	semidirect
3. C₃×C₅×#235	C₂	semidirect
4. #10624210	C₂	semidirect
5. #10624198	C₂	semidirect
6. #10624197	C₂	semidirect
7. #10624196	C₂	semidirect
8. D₃×Dic₃₀	C₂	semidirect
37. C₂×Dic₃₀	D₃	semidirect
38. C₅×#235	D₃	semidirect
55. C₃×#235	D₅	semidirect
164. #235	D₁₅	semidirect
167. C₃²×C₅	2₋¹⁺⁴	semidirect
262. C₃×C₅	#1007	semidirect
263. C₃×C₅	#1010	semidirect
289. C₃²	#4150	semidirect
317. C₅	#64297	semidirect
327. C₃	#92004	semidirect
328. C₃	#92101	semidirect

Maximal subgroups

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

subgroup	index
#10624283	2
C₂×C₃×Dic₃₀	2
C₃×C₅×#235	2
#10624210	2
#10624198	2
#10624197	2
#10624196	2
D₃×Dic₃₀	2
#92101	3
#92004	3
#64297	5