Subgroups — #13089364

Order 1440.

704 conjugacy classes of subgroups.

The subgroup lattice has 358 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×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#235	C₂	1	1	yes	maximal, characteristic, complemented
4	#67×D₁₅	C₂	1	4	yes	maximal, complemented
5	#10624205	C₂	1	2	yes	maximal, complemented
6	D₃×D₆₀	C₂	1	2	yes	maximal, complemented
7	#10624194	C₂	1	2	yes	maximal, complemented
8	#10624193	C₂	1	2	yes	maximal, complemented
9	#92100	—	3	1	no	maximal
10	#92017	—	3	1	no	maximal
11	C₂²×C₃×D₁₅	C₂²	1	2	yes	complemented
12	C₂×D₃×D₁₅	C₂²	1	4	yes	complemented
13	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
14	#67184	C₂²	1	2	yes	complemented
15	C₄⋊(C₃⋊D₁₅)	C₂²	1	1	yes	characteristic, complemented
16	C₄×C₃⋊D₁₅	C₂²	1	2	yes	complemented
17	#67180	C₂²	1	1	yes	characteristic, complemented
18	C₃×D₆₀	C₂²	1	2	yes	complemented
19	C₃×D₆₀	C₂²	1	2	yes	complemented
20	C₃×C₅×#67	C₂²	1	2	yes	complemented
21	C₃×C₅×D₁₂	C₂²	1	1	yes	characteristic, complemented
22	C₃×C₄×C₅×D₃	C₂²	1	2	yes	complemented
23	C₃×C₅×Dic₆	C₂²	1	1	yes	characteristic, complemented
24	#67151	C₂²	1	4	yes	complemented
25	#67150	C₂²	1	4	yes	complemented
26	Dic₃×D₁₅	C₂²	1	4	yes	complemented
27	#64307	—	5	1	no	maximal, Hall
28	C₃×C₅×#41	—	3	1	no	—
29	#6821	—	3	2	no	—
30	D₄×D₁₅	—	3	4	no	—
31	D₄×D₁₅	—	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×D₆₀	—	3	2	no	—
37	C₂×D₆₀	D₃	1	1	yes	characteristic, complemented
38	C₅×#235	D₃	1	1	yes	characteristic, complemented
39	C₂×C₃×D₂₀	—	3	1	no	—
40	D₅×#67	—	3	4	no	—
41	#6777	—	3	2	no	—
42	D₃×D₂₀	—	3	2	no	—
43	#6766	—	3	2	no	—
44	#6765	—	3	2	no	—
45	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊D₁₅	C₂³	1	2	yes	—
47	C₂×C₃×D₁₅	—	4	2	no	—
48	C₂×C₃×D₁₅	C₂³	1	4	yes	—
49	C₂×C₃×C₅×D₃	C₂³	1	2	yes	—
50	D₃×D₁₅	—	4	4	no	—
51	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₃⋊Dic₁₅	C₂³	1	2	yes	—
54	C₃×C₅×Dic₃	C₂³	1	2	yes	—
55	#4149	—	9	1	no	Hall
56	#3828	—	5	1	no	—
57	C₃×#235	D₅	1	1	yes	characteristic, complemented
58	C₂×C₃×D₁₂	—	5	1	no	—
59	D₃×#67	—	5	4	no	—
60	#3802	—	5	2	no	—
61	D₃×D₁₂	—	5	2	no	—
62	#3796	—	5	2	no	—
63	#3795	—	5	2	no	—
64	C₂²×D₁₅	—	3	4	no	—
65	C₂²×D₁₅	C₂×D₃	1	2	yes	complemented
66	C₂²×C₃×D₅	—	3	2	no	—
67	C₂×D₃×D₅	—	3	4	no	—
68	C₃×C₅×Q₈	—	3	1	no	—
69	C₃×C₅×D₄	—	3	2	no	—
70	C₃×C₅×D₄	—	3	1	no	—
71	C₂×C₃×C₄×C₅	—	2	1	no	—
72	C₂×C₃×C₄×C₅	—	3	2	no	—
73	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic, complemented
74	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic, complemented
75	#1220	—	3	2	no	—
76	#1220	—	3	2	no	—
77	#1220	—	3	4	no	—
78	#1220	—	6	2	no	—
79	D₆₀	—	3	4	no	—
80	D₆₀	—	3	1	no	—
81	D₆₀	—	3	1	no	—
82	D₆₀	—	6	1	no	—
83	D₆₀	C₂×D₃	1	2	yes	complemented
84	D₆₀	C₂×D₃	1	2	yes	complemented
85	C₄×D₁₅	—	3	4	no	—
86	C₄×D₁₅	—	3	2	no	—
87	C₄×D₁₅	—	3	2	no	—
88	C₄×D₁₅	—	6	2	no	—
89	Dic₃₀	—	3	1	no	—
90	Dic₃₀	—	3	1	no	—
91	Dic₃₀	—	6	1	no	—
92	C₅×#67	C₂×D₃	1	2	yes	complemented
93	C₅×D₁₂	C₂×D₃	1	1	yes	characteristic, complemented
94	C₄×C₅×D₃	C₂×D₃	1	2	yes	complemented
95	C₅×Dic₆	C₂×D₃	1	1	yes	characteristic, complemented
96	C₃×D₂₀	—	3	2	no	—
97	C₃×D₂₀	—	3	2	no	—
98	#1202	—	3	4	no	—
99	#1201	—	3	4	no	—
100	D₅×Dic₃	—	3	4	no	—
101	#1009	—	15	1	no	—
102	#1004	—	15	1	no	—
103	C₂×C₃²×C₅	—	2	1	no	—
104	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
105	C₃⋊D₁₅	—	2	2	no	—
106	C₃×D₁₅	—	2	4	no	—
107	C₃×C₅×D₃	—	2	2	no	—
108	C₅×#41	—	3	1	no	—
109	#713	—	9	2	no	—
110	D₄×D₅	—	9	2	no	—
111	D₄×D₅	—	9	4	no	—
112	#709	—	9	2	no	—
113	#709	—	9	1	no	—
114	C₂×D₂₀	—	3	1	no	—
115	C₂×D₂₀	—	9	2	no	—
116	C₂²×C₃×D₃	—	5	2	no	—
117	C₂×D₃²	—	5	4	no	—
118	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic, complemented
119	#638	—	5	2	no	—
120	C₄⋊(C₃⋊D₃)	—	5	1	no	—
121	C₄×C₃⋊D₃	—	5	2	no	—
122	#634	—	5	1	no	—
123	C₃×#67	C₂×D₅	1	2	yes	complemented
124	C₃×D₁₂	—	5	2	no	—
125	C₃×D₁₂	—	5	2	no	—
126	C₃×D₁₂	C₂×D₅	1	1	yes	characteristic, complemented
127	C₃×C₄×D₃	C₂×D₅	1	2	yes	complemented
128	C₃×Dic₆	C₂×D₅	1	1	yes	characteristic, complemented
129	#626	—	5	4	no	—
130	#625	—	5	4	no	—
131	D₃×Dic₃	—	5	4	no	—
132	C₂²×C₃×C₅	—	2	1	no	—
133	C₂²×C₃×C₅	—	3	2	no	—
134	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
135	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
136	C₂×D₁₅	—	3	2	no	—
137	C₂×D₁₅	—	3	2	no	—
138	C₂×D₁₅	—	4	2	no	—
139	C₂×D₁₅	—	6	2	no	—
140	C₂×D₁₅	—	12	4	no	—
141	C₂×D₁₅	C₂²×D₃	1	4	yes	—
142	C₂×C₅×D₃	C₂²×D₃	1	2	yes	—
143	C₂×C₃×D₅	—	3	4	no	—
144	C₂×C₃×D₅	—	12	2	no	—
145	D₃×D₅	—	12	4	no	—
146	C₃×C₄×C₅	—	2	1	no	—
147	C₃×C₄×C₅	—	2	1	no	—
148	C₃×C₄×C₅	—	3	2	no	—
149	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
150	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
152	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
153	Dic₁₅	—	3	2	no	—
154	Dic₁₅	—	3	2	no	—
155	Dic₁₅	—	6	2	no	—
156	C₅×Dic₃	C₂²×D₃	1	2	yes	—
157	C₃×#41	—	3	1	no	—
158	C₂×C₃×D₄	—	15	1	no	—
159	C₂×#67	—	15	4	no	—
160	#239	—	15	2	no	—
161	#237	—	15	2	no	—
162	#237	—	15	2	no	—
163	D₃×D₄	—	15	2	no	—
164	D₃×D₄	—	15	4	no	—
165	D₃×D₄	—	15	2	no	—
166	D₃×D₄	—	15	2	no	—
167	#235	—	15	2	no	—
168	#235	—	15	1	no	—
169	#235	—	15	1	no	—
170	#235	—	30	1	no	—
171	#235	D₁₅	1	1	yes	characteristic, complemented
172	C₂×D₁₂	—	5	1	no	—
173	C₂×D₁₂	—	15	2	no	—
174	C₃²×C₅	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Hall
175	C₂²×D₅	—	3	2	no	—
176	C₂²×D₅	—	9	4	no	—
177	C₅×Q₈	—	3	1	no	—
178	C₅×D₄	—	3	2	no	—
179	C₅×D₄	—	3	1	no	—
180	C₂×C₄×C₅	—	3	2	no	—
181	C₂×C₄×C₅	D₃²	1	1	yes	characteristic, complemented
182	C₂²⋊D₅	—	9	4	no	—
183	C₂²⋊D₅	—	9	2	no	—
184	D₂₀	—	3	2	no	—
185	D₂₀	—	3	2	no	—
186	D₂₀	—	9	4	no	—
187	D₂₀	—	9	1	no	—
188	C₄×D₅	—	9	4	no	—
189	C₄×D₅	—	9	2	no	—
190	Dic₁₀	—	9	1	no	—
191	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
192	C₂×C₃⋊D₃	—	5	2	no	—
193	C₂×C₃×D₃	—	5	4	no	—
194	C₂×C₃×D₃	—	20	2	no	—
195	C₂×C₃×D₃	C₂²×D₅	1	2	yes	—
196	D₃²	—	20	4	no	—
197	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
198	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
199	C₃⋊Dic₃	—	5	2	no	—
200	C₃×Dic₃	C₂²×D₅	1	2	yes	—
201	H(5,𝔽₂)	—	45	1	no	Sylow
202	C₂×C₃×C₅	—	2	1	no	—
203	C₂×C₃×C₅	—	2	1	no	—
204	C₂×C₃×C₅	—	2	1	no	—
205	C₂×C₃×C₅	—	4	1	no	—
206	C₂×C₃×C₅	—	6	2	no	—
207	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
208	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
209	D₁₅	—	2	4	no	—
210	D₁₅	—	6	2	no	—
211	D₁₅	—	6	2	no	—
212	D₁₅	—	12	2	no	—
213	C₃×D₅	—	6	4	no	—
214	C₅×D₃	—	2	2	no	—
215	C₂³×C₃	—	15	2	no	—
216	C₂²×D₃	—	5	2	no	—
217	C₂²×D₃	—	15	4	no	—
218	C₂²×D₃	—	15	4	no	—
219	C₃×Q₈	—	3	1	no	—
220	C₃×D₄	—	3	2	no	—
221	C₃×D₄	—	3	1	no	—
222	C₃×D₄	—	15	2	no	—
223	C₃×D₄	—	15	2	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	3	2	no	—
226	C₂×C₃×C₄	D₃×D₅	1	1	yes	characteristic, complemented
227	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic, complemented
228	#67	—	15	4	no	—
229	#67	—	15	2	no	—
230	#67	—	15	2	no	—
231	#67	—	15	4	no	—
232	#67	—	15	4	no	—
233	#67	—	30	2	no	—
234	#67	C₂×D₁₅	1	2	yes	complemented
235	C₂×Dic₃	—	15	4	no	—
236	D₁₂	—	5	2	no	—
237	D₁₂	—	5	2	no	—
238	D₁₂	—	15	4	no	—
239	D₁₂	—	15	1	no	—
240	D₁₂	—	15	1	no	—
241	D₁₂	—	30	1	no	—
242	D₁₂	C₂×D₁₅	1	1	yes	characteristic, complemented
243	C₄×D₃	—	15	4	no	—
244	C₄×D₃	—	15	2	no	—
245	C₄×D₃	—	15	2	no	—
246	C₄×D₃	—	30	2	no	—
247	C₄×D₃	C₂×D₁₅	1	2	yes	complemented
248	Dic₆	—	15	1	no	—
249	Dic₆	—	15	1	no	—
250	Dic₆	—	30	1	no	—
251	Dic₆	C₂×D₁₅	1	1	yes	characteristic, complemented
252	C₂²×C₅	—	3	2	no	—
253	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
254	C₂×D₅	—	3	4	no	—
255	C₂×D₅	—	9	2	no	—
256	C₂×D₅	—	12	2	no	—
257	C₂×D₅	—	36	4	no	—
258	C₄×C₅	—	3	2	no	—
259	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
260	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
261	Dic₅	—	9	2	no	—
262	C₂×C₃²	—	2	1	no	—
263	C₂×C₃²	C₂³×D₅	1	1	yes	characteristic
264	C₃⋊D₃	—	10	2	no	—
265	C₃×D₃	—	2	2	no	—
266	C₃×D₃	—	10	4	no	—
267	#41	—	3	1	no	—
268	#41	—	45	2	no	—
269	#41	—	45	2	no	—
270	#41	—	45	1	no	—
271	C₂×D₄	—	15	1	no	—
272	C₂×D₄	—	45	2	no	—
273	C₂×D₄	—	45	2	no	—
274	C₂×D₄	—	45	4	no	—
275	C₃×C₅	—	2	1	no	—
276	C₃×C₅	#1004	1	1	yes	characteristic, complemented
277	C₃×C₅	#1009	1	1	yes	characteristic, complemented
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	3	2	no	—
280	C₂²×C₃	—	15	4	no	—
281	C₂²×C₃	—	60	2	no	—
282	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
283	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
284	C₂×D₃	—	5	4	no	—
285	C₂×D₃	—	15	2	no	—
286	C₂×D₃	—	15	2	no	—
287	C₂×D₃	—	20	2	no	—
288	C₂×D₃	—	30	2	no	—
289	C₂×D₃	—	60	4	no	—
290	C₂×D₃	—	60	4	no	—
291	C₂×D₃	C₂²×D₁₅	1	2	yes	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	3	2	no	—
295	C₃×C₄	C₂×D₃×D₅	1	1	yes	characteristic
296	C₃×C₄	C₂×D₃×D₅	1	1	yes	characteristic
297	C₃×C₄	C₂²×D₁₅	1	1	yes	characteristic
298	C₃×C₄	C₂²×D₁₅	1	1	yes	characteristic
299	Dic₃	—	15	2	no	—
300	Dic₃	—	15	2	no	—
301	Dic₃	—	30	2	no	—
302	Dic₃	C₂²×D₁₅	1	2	yes	—
303	C₂×C₅	—	2	1	no	—
304	C₂×C₅	—	6	2	no	—
305	C₂×C₅	C₂²×D₃²	1	1	yes	characteristic
306	D₅	—	6	4	no	—
307	D₅	—	18	2	no	—
308	C₃²	#4149	1	1	yes	characteristic, complemented, Sylow
309	C₂³	—	15	2	no	—
310	C₂³	—	45	4	no	—
311	Q₈	—	3	1	no	—
312	Q₈	—	45	1	no	—
313	D₄	—	3	2	no	—
314	D₄	—	3	1	no	—
315	D₄	—	15	2	no	—
316	D₄	—	15	2	no	—
317	D₄	—	45	4	no	—
318	D₄	—	45	4	no	—
319	D₄	—	45	2	no	—
320	D₄	—	45	1	no	—
321	C₂×C₄	—	3	2	no	—
322	C₂×C₄	—	45	4	no	—
323	C₂×C₄	—	45	2	no	—
324	C₂×C₄	D₃×D₁₅	1	1	yes	characteristic, complemented
325	C₂×C₃	—	2	1	no	—
326	C₂×C₃	—	2	1	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	4	1	no	—
329	C₂×C₃	—	6	2	no	—
330	C₂×C₃	—	30	4	no	—
331	C₂×C₃	C₂²×D₃×D₅	1	1	yes	characteristic
332	C₂×C₃	C₂³×D₁₅	1	1	yes	characteristic
333	D₃	—	2	2	no	—
334	D₃	—	10	4	no	—
335	D₃	—	30	2	no	—
336	D₃	—	30	2	no	—
337	D₃	—	60	2	no	—
338	C₅	#64307	1	1	yes	characteristic, complemented, Sylow
339	C₂²	—	3	2	no	—
340	C₂²	—	15	4	no	—
341	C₂²	—	45	2	no	—
342	C₂²	—	60	2	no	—
343	C₂²	—	180	4	no	—
344	C₂²	C₂×D₃×D₁₅	1	1	yes	characteristic
345	C₄	—	3	2	no	—
346	C₄	—	45	2	no	—
347	C₄	C₂×D₃×D₁₅	1	1	yes	characteristic
348	C₄	C₂×D₃×D₁₅	1	1	yes	characteristic
349	C₃	—	2	1	no	—
350	C₃	#92017	1	1	yes	characteristic, complemented
351	C₃	#92100	1	1	yes	characteristic, complemented
352	C₂	—	2	1	no	—
353	C₂	—	6	2	no	—
354	C₂	—	30	4	no	—
355	C₂	—	90	2	no	—
356	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	H(5,𝔽₂)
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×D₆₀	C₂	semidirect
3. C₃×C₅×#235	C₂	semidirect
4. #67×D₁₅	C₂	semidirect
5. #10624205	C₂	semidirect
6. D₃×D₆₀	C₂	semidirect
7. #10624194	C₂	semidirect
8. #10624193	C₂	semidirect
11. C₂²×C₃×D₁₅	C₂²	semidirect
12. C₂×D₃×D₁₅	C₂²	semidirect
13. C₂×C₃²×C₄×C₅	C₂²	semidirect
14. #67184	C₂²	semidirect
15. C₄⋊(C₃⋊D₁₅)	C₂²	semidirect
16. C₄×C₃⋊D₁₅	C₂²	semidirect
17. #67180	C₂²	semidirect
18. C₃×D₆₀	C₂²	semidirect
19. C₃×D₆₀	C₂²	semidirect
20. C₃×C₅×#67	C₂²	semidirect
21. C₃×C₅×D₁₂	C₂²	semidirect
22. C₃×C₄×C₅×D₃	C₂²	semidirect
23. C₃×C₅×Dic₆	C₂²	semidirect
24. #67151	C₂²	semidirect
25. #67150	C₂²	semidirect
26. Dic₃×D₁₅	C₂²	semidirect
37. C₂×D₆₀	D₃	semidirect
38. C₅×#235	D₃	semidirect
57. C₃×#235	D₅	semidirect
65. C₂²×D₁₅	C₂×D₃	semidirect
73. C₂×C₃×C₄×C₅	C₂×D₃	semidirect
74. C₂×C₃×C₄×C₅	C₂×D₃	semidirect
83. D₆₀	C₂×D₃	semidirect
84. D₆₀	C₂×D₃	semidirect
92. C₅×#67	C₂×D₃	semidirect
93. C₅×D₁₂	C₂×D₃	semidirect
94. C₄×C₅×D₃	C₂×D₃	semidirect
95. C₅×Dic₆	C₂×D₃	semidirect
118. C₂×C₃²×C₄	C₂×D₅	semidirect
123. C₃×#67	C₂×D₅	semidirect
126. C₃×D₁₂	C₂×D₅	semidirect
127. C₃×C₄×D₃	C₂×D₅	semidirect
128. C₃×Dic₆	C₂×D₅	semidirect
171. #235	D₁₅	semidirect
174. C₃²×C₅	H(5,𝔽₂)	semidirect
181. C₂×C₄×C₅	D₃²	semidirect
226. C₂×C₃×C₄	D₃×D₅	semidirect
227. C₂×C₃×C₄	C₂×D₁₅	semidirect
234. #67	C₂×D₁₅	semidirect
242. D₁₂	C₂×D₁₅	semidirect
247. C₄×D₃	C₂×D₁₅	semidirect
251. Dic₆	C₂×D₁₅	semidirect
276. C₃×C₅	#1004	semidirect
277. C₃×C₅	#1009	semidirect
308. C₃²	#4149	semidirect
324. C₂×C₄	D₃×D₁₅	semidirect
338. C₅	#64307	semidirect
350. C₃	#92017	semidirect
351. C₃	#92100	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×D₆₀	2
C₃×C₅×#235	2
#67×D₁₅	2
#10624205	2
D₃×D₆₀	2
#10624194	2
#10624193	2
#92100	3
#92017	3
#64307	5