Subgroups — #13089366

Order 1440.

718 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊D₁₅)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6817	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#235	C₂	1	1	yes	maximal, characteristic, complemented
4	#10624218	C₂	1	4	yes	maximal, complemented
5	D₃×D₆₀	C₂	1	2	yes	maximal, complemented
6	D₁₂×D₁₅	C₂	1	2	yes	maximal, complemented
7	#10624201	C₂	1	2	yes	maximal, complemented
8	#10624197	C₂	1	2	yes	maximal, complemented
9	#92100	—	3	1	no	maximal
10	#92019	—	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	C₄⋊(C₃⋊D₁₅)	C₂²	1	2	yes	complemented
15	C₄⋊(C₃⋊D₁₅)	C₂²	1	2	yes	complemented
16	C₃×#1220	C₂²	1	2	yes	complemented
17	C₃×D₆₀	C₂²	1	1	yes	characteristic, complemented
18	C₃×C₄×D₁₅	C₂²	1	2	yes	complemented
19	C₃×Dic₃₀	C₂²	1	1	yes	characteristic, 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	#67152	C₂²	1	4	yes	complemented
25	#67151	C₂²	1	4	yes	complemented
26	#67149	C₂²	1	4	yes	complemented
27	#64308	—	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	2	no	—
33	#6817	D₃	1	1	yes	characteristic, complemented
34	C₂×D₆₀	—	3	1	no	—
35	C₂×D₆₀	—	3	1	no	—
36	C₂×D₆₀	—	3	2	no	—
37	C₂×D₆₀	—	6	1	no	—
38	C₅×#235	D₃	1	1	yes	characteristic, complemented
39	C₃×#709	—	3	1	no	—
40	#6790	—	3	4	no	—
41	D₃×D₂₀	—	3	2	no	—
42	D₅×D₁₂	—	3	2	no	—
43	#6773	—	3	2	no	—
44	#6769	—	3	2	no	—
45	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊D₁₅	—	4	2	no	—
47	C₂×C₃⋊D₁₅	C₂³	1	4	yes	—
48	C₂×C₃×D₁₅	C₂³	1	2	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	C₂×C₄⋊(C₃⋊D₃)	—	5	1	no	—
57	C₃×#235	—	5	1	no	—
58	C₃×#235	D₅	1	1	yes	characteristic, complemented
59	#3811	—	5	4	no	—
60	D₃×D₁₂	—	5	2	no	—
61	D₃×D₁₂	—	5	2	no	—
62	#3799	—	5	2	no	—
63	#3799	—	5	2	no	—
64	C₂²×D₁₅	—	3	2	no	—
65	C₂²×D₁₅	—	3	2	no	—
66	C₂²×D₁₅	—	3	4	no	—
67	C₂²×D₁₅	—	6	2	no	—
68	C₂×D₃×D₅	—	3	4	no	—
69	C₃×C₅×Q₈	—	3	1	no	—
70	C₃×C₅×D₄	—	3	2	no	—
71	C₃×C₅×D₄	—	3	1	no	—
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	3	2	no	—
74	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic, complemented
75	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic, complemented
76	#1220	—	3	4	no	—
77	#1220	C₂×D₃	1	2	yes	complemented
78	D₆₀	—	3	4	no	—
79	D₆₀	—	3	2	no	—
80	D₆₀	—	3	2	no	—
81	D₆₀	—	3	2	no	—
82	D₆₀	—	3	2	no	—
83	D₆₀	—	6	2	no	—
84	D₆₀	—	6	2	no	—
85	D₆₀	C₂×D₃	1	1	yes	characteristic, complemented
86	C₄×D₁₅	—	3	4	no	—
87	C₄×D₁₅	C₂×D₃	1	2	yes	complemented
88	Dic₃₀	C₂×D₃	1	1	yes	characteristic, complemented
89	C₅×#67	C₂×D₃	1	2	yes	complemented
90	C₅×D₁₂	C₂×D₃	1	1	yes	characteristic, complemented
91	C₄×C₅×D₃	C₂×D₃	1	2	yes	complemented
92	C₅×Dic₆	C₂×D₃	1	1	yes	characteristic, complemented
93	C₃×C₂²⋊D₅	—	3	2	no	—
94	C₃×D₂₀	—	3	1	no	—
95	C₃×C₄×D₅	—	3	2	no	—
96	C₃×Dic₁₀	—	3	1	no	—
97	#1203	—	3	4	no	—
98	#1202	—	3	4	no	—
99	#1200	—	3	4	no	—
100	#1009	—	15	1	no	—
101	#1009	—	15	1	no	—
102	C₂×C₃²×C₅	—	2	1	no	—
103	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
104	C₃⋊D₁₅	—	2	4	no	—
105	C₃×D₁₅	—	2	2	no	—
106	C₃×C₅×D₃	—	2	2	no	—
107	C₅×#41	—	3	1	no	—
108	#713	—	9	2	no	—
109	D₄×D₅	—	9	2	no	—
110	D₄×D₅	—	9	4	no	—
111	#709	—	3	1	no	—
112	#709	—	9	2	no	—
113	C₂×D₂₀	—	9	2	no	—
114	C₂×D₂₀	—	9	1	no	—
115	C₂²×C₃⋊D₃	—	5	2	no	—
116	C₂×D₃²	—	5	4	no	—
117	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic, complemented
118	C₄⋊(C₃⋊D₃)	—	5	2	no	—
119	C₄⋊(C₃⋊D₃)	—	5	2	no	—
120	C₃×#67	—	5	2	no	—
121	C₃×#67	C₂×D₅	1	2	yes	complemented
122	C₃×D₁₂	—	5	1	no	—
123	C₃×D₁₂	C₂×D₅	1	1	yes	characteristic, complemented
124	C₃×C₄×D₃	—	5	2	no	—
125	C₃×C₄×D₃	C₂×D₅	1	2	yes	complemented
126	C₃×Dic₆	—	5	1	no	—
127	C₃×Dic₆	C₂×D₅	1	1	yes	characteristic, complemented
128	#626	—	5	4	no	—
129	#626	—	5	4	no	—
130	#624	—	5	4	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	—	3	2	no	—
133	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
135	C₂×D₁₅	—	3	4	no	—
136	C₂×D₁₅	—	3	4	no	—
137	C₂×D₁₅	—	6	4	no	—
138	C₂×D₁₅	—	12	4	no	—
139	C₂×D₁₅	—	12	2	no	—
140	C₂×D₁₅	—	12	2	no	—
141	C₂×D₁₅	—	12	4	no	—
142	C₂×D₁₅	C₂²×D₃	1	2	yes	—
143	C₂×C₅×D₃	C₂²×D₃	1	2	yes	—
144	C₂×C₃×D₅	—	3	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	C₂²×D₃	1	2	yes	—
154	C₃×Dic₅	—	3	2	no	—
155	C₅×Dic₃	C₂²×D₃	1	2	yes	—
156	C₃×#41	—	3	1	no	—
157	C₃×#41	—	15	1	no	—
158	#239	—	15	2	no	—
159	#239	—	15	2	no	—
160	D₃×D₄	—	15	2	no	—
161	D₃×D₄	—	15	4	no	—
162	D₃×D₄	—	15	4	no	—
163	D₃×D₄	—	15	2	no	—
164	#235	—	5	1	no	—
165	#235	—	15	2	no	—
166	#235	—	15	2	no	—
167	#235	D₁₅	1	1	yes	characteristic, complemented
168	C₂×D₁₂	—	15	2	no	—
169	C₂×D₁₂	—	15	1	no	—
170	C₂×D₁₂	—	15	1	no	—
171	C₂×D₁₂	—	15	2	no	—
172	C₂×D₁₂	—	30	1	no	—
173	C₃²×C₅	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Hall
174	C₂²×D₅	—	9	4	no	—
175	C₂²×D₅	—	9	2	no	—
176	C₅×Q₈	—	3	1	no	—
177	C₅×D₄	—	3	2	no	—
178	C₅×D₄	—	3	1	no	—
179	C₂×C₄×C₅	—	3	2	no	—
180	C₂×C₄×C₅	D₃²	1	1	yes	characteristic, complemented
181	C₂²⋊D₅	—	3	2	no	—
182	C₂²⋊D₅	—	9	4	no	—
183	D₂₀	—	3	1	no	—
184	D₂₀	—	9	4	no	—
185	D₂₀	—	9	2	no	—
186	D₂₀	—	9	2	no	—
187	C₄×D₅	—	3	2	no	—
188	C₄×D₅	—	9	4	no	—
189	Dic₁₀	—	3	1	no	—
190	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
191	C₂×C₃⋊D₃	—	5	4	no	—
192	C₂×C₃⋊D₃	—	20	2	no	—
193	C₂×C₃×D₃	—	5	2	no	—
194	C₂×C₃×D₃	C₂²×D₅	1	2	yes	—
195	D₃²	—	20	4	no	—
196	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
197	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
198	C₃×Dic₃	—	5	2	no	—
199	C₃×Dic₃	C₂²×D₅	1	2	yes	—
200	H(5,𝔽₂)	—	45	1	no	Sylow
201	C₂×C₃×C₅	—	2	2	no	—
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	—	6	2	no	—
206	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
207	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
208	D₁₅	—	2	2	no	—
209	D₁₅	—	6	4	no	—
210	D₁₅	—	6	4	no	—
211	D₁₅	—	12	4	no	—
212	C₃×D₅	—	6	2	no	—
213	C₅×D₃	—	2	2	no	—
214	C₂²×D₃	—	15	4	no	—
215	C₂²×D₃	—	15	2	no	—
216	C₂²×D₃	—	15	2	no	—
217	C₂²×D₃	—	15	4	no	—
218	C₂²×D₃	—	30	2	no	—
219	C₃×Q₈	—	3	1	no	—
220	C₃×Q₈	—	15	1	no	—
221	C₃×D₄	—	3	2	no	—
222	C₃×D₄	—	3	1	no	—
223	C₃×D₄	—	15	2	no	—
224	C₃×D₄	—	15	1	no	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	—	3	2	no	—
227	C₂×C₃×C₄	—	15	2	no	—
228	C₂×C₃×C₄	D₃×D₅	1	1	yes	characteristic, complemented
229	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic, complemented
230	#67	—	5	2	no	—
231	#67	—	15	4	no	—
232	#67	—	15	4	no	—
233	#67	C₂×D₁₅	1	2	yes	complemented
234	D₁₂	—	5	1	no	—
235	D₁₂	—	15	4	no	—
236	D₁₂	—	15	4	no	—
237	D₁₂	—	15	2	no	—
238	D₁₂	—	15	2	no	—
239	D₁₂	—	15	2	no	—
240	D₁₂	—	15	2	no	—
241	D₁₂	—	30	2	no	—
242	D₁₂	—	30	2	no	—
243	D₁₂	C₂×D₁₅	1	1	yes	characteristic, complemented
244	C₄×D₃	—	5	2	no	—
245	C₄×D₃	—	15	4	no	—
246	C₄×D₃	—	15	4	no	—
247	C₄×D₃	C₂×D₁₅	1	2	yes	complemented
248	Dic₆	—	5	1	no	—
249	Dic₆	C₂×D₁₅	1	1	yes	characteristic, complemented
250	C₂²×C₅	—	3	2	no	—
251	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
252	C₂×D₅	—	3	2	no	—
253	C₂×D₅	—	9	4	no	—
254	C₂×D₅	—	36	4	no	—
255	C₂×D₅	—	36	2	no	—
256	C₄×C₅	—	3	2	no	—
257	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
258	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
259	Dic₅	—	3	2	no	—
260	C₂×C₃²	—	2	1	no	—
261	C₂×C₃²	C₂³×D₅	1	1	yes	characteristic
262	C₃⋊D₃	—	10	4	no	—
263	C₃×D₃	—	2	2	no	—
264	C₃×D₃	—	10	2	no	—
265	#41	—	3	1	no	—
266	#41	—	15	1	no	—
267	#41	—	45	2	no	—
268	#41	—	45	2	no	—
269	C₂×D₄	—	45	2	no	—
270	C₂×D₄	—	45	2	no	—
271	C₂×D₄	—	45	4	no	—
272	C₂×D₄	—	45	1	no	—
273	C₃×C₅	—	2	1	no	—
274	C₃×C₅	#1009	1	1	yes	characteristic, complemented
275	C₃×C₅	#1009	1	1	yes	characteristic, complemented
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	3	2	no	—
278	C₂²×C₃	—	15	2	no	—
279	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
280	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
281	C₂×D₃	—	5	2	no	—
282	C₂×D₃	—	15	4	no	—
283	C₂×D₃	—	15	4	no	—
284	C₂×D₃	—	30	4	no	—
285	C₂×D₃	—	60	4	no	—
286	C₂×D₃	—	60	4	no	—
287	C₂×D₃	—	60	2	no	—
288	C₂×D₃	—	60	2	no	—
289	C₂×D₃	—	60	4	no	—
290	C₂×D₃	C₂²×D₁₅	1	2	yes	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	3	2	no	—
294	C₃×C₄	—	15	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	—	5	2	no	—
300	Dic₃	C₂²×D₁₅	1	2	yes	—
301	C₂×C₅	—	2	1	no	—
302	C₂×C₅	—	6	2	no	—
303	C₂×C₅	C₂²×D₃²	1	1	yes	characteristic
304	D₅	—	6	2	no	—
305	D₅	—	18	4	no	—
306	C₃²	#4149	1	1	yes	characteristic, complemented, Sylow
307	C₂³	—	45	4	no	—
308	C₂³	—	45	2	no	—
309	Q₈	—	3	1	no	—
310	Q₈	—	15	1	no	—
311	D₄	—	3	2	no	—
312	D₄	—	3	1	no	—
313	D₄	—	15	2	no	—
314	D₄	—	15	1	no	—
315	D₄	—	45	4	no	—
316	D₄	—	45	4	no	—
317	D₄	—	45	2	no	—
318	D₄	—	45	2	no	—
319	C₂×C₄	—	3	2	no	—
320	C₂×C₄	—	15	2	no	—
321	C₂×C₄	—	45	4	no	—
322	C₂×C₄	D₃×D₁₅	1	1	yes	characteristic, complemented
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	—	2	1	no	—
326	C₂×C₃	—	2	1	no	—
327	C₂×C₃	—	6	2	no	—
328	C₂×C₃	—	30	2	no	—
329	C₂×C₃	C₂²×D₃×D₅	1	1	yes	characteristic
330	C₂×C₃	C₂³×D₁₅	1	1	yes	characteristic
331	D₃	—	2	2	no	—
332	D₃	—	10	2	no	—
333	D₃	—	30	4	no	—
334	D₃	—	30	4	no	—
335	D₃	—	60	4	no	—
336	C₅	#64308	1	1	yes	characteristic, complemented, Sylow
337	C₂²	—	3	2	no	—
338	C₂²	—	15	2	no	—
339	C₂²	—	45	4	no	—
340	C₂²	—	180	2	no	—
341	C₂²	—	180	4	no	—
342	C₂²	C₂×D₃×D₁₅	1	1	yes	characteristic
343	C₄	—	3	2	no	—
344	C₄	—	15	2	no	—
345	C₄	C₂×D₃×D₁₅	1	1	yes	characteristic
346	C₄	C₂×D₃×D₁₅	1	1	yes	characteristic
347	C₃	—	2	1	no	—
348	C₃	#92019	1	1	yes	characteristic, complemented
349	C₃	#92100	1	1	yes	characteristic, complemented
350	C₂	—	2	1	no	—
351	C₂	—	6	2	no	—
352	C₂	—	30	2	no	—
353	C₂	—	90	4	no	—
354	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. C₂×C₄⋊(C₃⋊D₁₅)	C₂	semidirect
2. C₃×#6817	C₂	semidirect
3. C₃×C₅×#235	C₂	semidirect
4. #10624218	C₂	semidirect
5. D₃×D₆₀	C₂	semidirect
6. D₁₂×D₁₅	C₂	semidirect
7. #10624201	C₂	semidirect
8. #10624197	C₂	semidirect
11. C₂²×C₃⋊D₁₅	C₂²	semidirect
12. C₂×D₃×D₁₅	C₂²	semidirect
13. C₂×C₃²×C₄×C₅	C₂²	semidirect
14. C₄⋊(C₃⋊D₁₅)	C₂²	semidirect
15. C₄⋊(C₃⋊D₁₅)	C₂²	semidirect
16. C₃×#1220	C₂²	semidirect
17. C₃×D₆₀	C₂²	semidirect
18. C₃×C₄×D₁₅	C₂²	semidirect
19. C₃×Dic₃₀	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. #67152	C₂²	semidirect
25. #67151	C₂²	semidirect
26. #67149	C₂²	semidirect
33. #6817	D₃	semidirect
38. C₅×#235	D₃	semidirect
58. C₃×#235	D₅	semidirect
74. C₂×C₃×C₄×C₅	C₂×D₃	semidirect
75. C₂×C₃×C₄×C₅	C₂×D₃	semidirect
77. #1220	C₂×D₃	semidirect
85. D₆₀	C₂×D₃	semidirect
87. C₄×D₁₅	C₂×D₃	semidirect
88. Dic₃₀	C₂×D₃	semidirect
89. C₅×#67	C₂×D₃	semidirect
90. C₅×D₁₂	C₂×D₃	semidirect
91. C₄×C₅×D₃	C₂×D₃	semidirect
92. C₅×Dic₆	C₂×D₃	semidirect
117. C₂×C₃²×C₄	C₂×D₅	semidirect
121. C₃×#67	C₂×D₅	semidirect
123. C₃×D₁₂	C₂×D₅	semidirect
125. C₃×C₄×D₃	C₂×D₅	semidirect
127. C₃×Dic₆	C₂×D₅	semidirect
167. #235	D₁₅	semidirect
173. C₃²×C₅	H(5,𝔽₂)	semidirect
180. C₂×C₄×C₅	D₃²	semidirect
228. C₂×C₃×C₄	D₃×D₅	semidirect
229. C₂×C₃×C₄	C₂×D₁₅	semidirect
233. #67	C₂×D₁₅	semidirect
243. D₁₂	C₂×D₁₅	semidirect
247. C₄×D₃	C₂×D₁₅	semidirect
249. Dic₆	C₂×D₁₅	semidirect
274. C₃×C₅	#1009	semidirect
275. C₃×C₅	#1009	semidirect
306. C₃²	#4149	semidirect
322. C₂×C₄	D₃×D₁₅	semidirect
336. C₅	#64308	semidirect
348. C₃	#92019	semidirect
349. C₃	#92100	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊D₁₅)	2
C₃×#6817	2
C₃×C₅×#235	2
#10624218	2
D₃×D₆₀	2
D₁₂×D₁₅	2
#10624201	2
#10624197	2
#92100	3
#92019	3
#64308	5