Subgroups — #13089273

Order 1440.

984 conjugacy classes of subgroups.

The subgroup lattice has 330 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	C₅×#3828	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#709	C₂	1	1	yes	maximal, characteristic, complemented
4	#10624173	C₂	1	2	yes	maximal, complemented
5	#10624163	C₂	1	2	yes	maximal, complemented
6	#10624160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₅×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
8	#10624154	C₂	1	2	yes	maximal, complemented
9	#10624153	C₂	1	1	yes	maximal, characteristic, complemented
10	#10624151	C₂	1	1	yes	maximal, characteristic, complemented
11	#10624149	C₂	1	1	yes	maximal, characteristic, complemented
12	#92016	—	3	4	no	maximal
13	C₂²×C₃⋊D₁₅	C₂²	1	1	yes	characteristic
14	C₂×D₅×C₃⋊D₃	C₂²	1	2	yes	—
15	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃⋊Dic₁₅	C₂²	1	1	yes	characteristic, complemented
17	C₄×C₃⋊D₁₅	C₂²	1	1	yes	characteristic
18	C₄×C₃⋊D₁₅	C₂²	1	1	yes	characteristic
19	C₄×C₃⋊D₁₅	C₂²	1	1	yes	characteristic, complemented
20	C₄×C₃⋊D₁₅	C₂²	1	1	yes	characteristic, complemented
21	C₅×#638	C₂²	1	2	yes	complemented
22	C₅×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic, complemented
23	C₄×C₅×C₃⋊D₃	C₂²	1	2	yes	complemented
24	C₅×#634	C₂²	1	1	yes	characteristic, complemented
25	C₃²×C₂²⋊D₅	C₂²	1	2	yes	complemented
26	C₃²×D₂₀	C₂²	1	1	yes	characteristic, complemented
27	C₃²×C₄×D₅	C₂²	1	2	yes	complemented
28	C₃²×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
29	#67141	C₂²	1	2	yes	complemented
30	#67140	C₂²	1	2	yes	complemented
31	#67139	C₂²	1	2	yes	complemented
32	#67138	C₂²	1	2	yes	complemented
33	#67137	C₂²	1	2	yes	—
34	C₃⋊D₃×Dic₅	C₂²	1	2	yes	complemented
35	D₅×C₃⋊Dic₃	C₂²	1	2	yes	complemented
36	#41×C₃⋊D₃	—	5	1	no	maximal, Hall
37	C₂×C₄×D₁₅	—	3	4	no	—
38	C₅×#235	—	3	4	no	—
39	C₃×#709	D₃	1	4	yes	complemented
40	#6790	—	3	8	no	—
41	#6780	—	3	8	no	—
42	#6777	—	3	4	no	—
43	C₄×D₃×D₅	—	3	8	no	—
44	#6771	—	3	8	no	—
45	D₁₅⋊Q₈	—	3	4	no	—
46	#6768	—	3	4	no	—
47	#6766	—	3	4	no	—
48	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₃⋊D₁₅	—	2	2	no	—
50	C₂×C₃⋊D₁₅	C₂³	1	1	yes	characteristic
51	C₂×C₃⋊D₁₅	C₂³	1	1	yes	characteristic
52	C₂×C₅×C₃⋊D₃	C₂³	1	2	yes	—
53	C₂×C₃²×D₅	C₂³	1	2	yes	—
54	D₅×C₃⋊D₃	—	2	4	no	—
55	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₃⋊Dic₁₅	C₂³	1	1	yes	characteristic
58	C₃⋊Dic₁₅	C₂³	1	1	yes	characteristic
59	C₅×C₃⋊Dic₃	C₂³	1	2	yes	—
60	C₃²×Dic₅	C₂³	1	2	yes	—
61	D₅×#41	—	9	1	no	Hall
62	C₃²×#41	—	5	1	no	—
63	#3832	—	5	1	no	—
64	Q₈×C₃⋊D₃	—	5	1	no	—
65	#3830	—	5	2	no	—
66	#3830	—	5	1	no	—
67	D₄×C₃⋊D₃	—	5	2	no	—
68	D₄×C₃⋊D₃	—	5	1	no	—
69	#3828	—	5	2	no	—
70	#3828	D₅	1	1	yes	characteristic, complemented
71	C₂×C₄×C₃⋊D₃	—	5	1	no	—
72	C₂×C₄×C₃⋊D₃	—	5	2	no	—
73	C₂²×D₁₅	—	3	4	no	—
74	C₂×D₃×D₅	—	3	8	no	—
75	C₂×C₃×C₄×C₅	C₂×D₃	1	4	yes	complemented
76	C₂×Dic₁₅	—	3	4	no	—
77	C₄×D₁₅	—	3	4	no	—
78	C₄×D₁₅	—	3	4	no	—
79	C₄×D₁₅	—	3	4	no	—
80	C₄×D₁₅	—	3	4	no	—
81	C₅×#67	—	3	8	no	—
82	C₅×D₁₂	—	3	4	no	—
83	C₄×C₅×D₃	—	3	8	no	—
84	C₅×Dic₆	—	3	4	no	—
85	C₃×C₂²⋊D₅	C₂×D₃	1	8	yes	complemented
86	C₃×D₂₀	C₂×D₃	1	4	yes	complemented
87	C₃×C₄×D₅	C₂×D₃	1	8	yes	complemented
88	C₃×Dic₁₀	C₂×D₃	1	4	yes	complemented
89	#1204	—	3	8	no	—
90	#1203	—	3	8	no	—
91	#1202	—	3	8	no	—
92	#1201	—	3	8	no	—
93	#1200	—	3	8	no	—
94	D₃×Dic₅	—	3	8	no	—
95	D₅×Dic₃	—	3	8	no	—
96	D₃×#41	—	15	4	no	—
97	C₂×C₃²×C₅	—	2	1	no	—
98	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
99	C₃⋊D₁₅	—	2	1	no	—
100	C₃⋊D₁₅	#41	1	2	yes	complemented
101	C₅×C₃⋊D₃	—	2	2	no	—
102	C₃²×D₅	—	2	2	no	—
103	C₅×#41	—	9	1	no	—
104	#713	—	9	1	no	—
105	Q₈×D₅	—	9	1	no	—
106	#711	—	9	2	no	—
107	#711	—	9	1	no	—
108	D₄×D₅	—	9	2	no	—
109	D₄×D₅	—	9	1	no	—
110	#709	—	9	2	no	—
111	#709	C₃⋊D₃	1	1	yes	characteristic, complemented
112	C₂×C₄×D₅	—	9	1	no	—
113	C₂×C₄×D₅	—	9	2	no	—
114	C₂²×C₃⋊D₃	—	5	1	no	—
115	C₂²×C₃⋊D₃	—	5	2	no	—
116	C₃²×Q₈	—	5	1	no	—
117	C₃²×D₄	—	5	2	no	—
118	C₃²×D₄	—	5	1	no	—
119	C₂×C₃²×C₄	—	5	2	no	—
120	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic, complemented
121	#638	—	5	2	no	—
122	#638	—	5	2	no	—
123	#638	C₂×D₅	1	2	yes	complemented
124	C₂×C₃⋊Dic₃	—	5	1	no	—
125	C₂×C₃⋊Dic₃	—	5	2	no	—
126	C₄⋊(C₃⋊D₃)	—	5	2	no	—
127	C₄⋊(C₃⋊D₃)	C₂×D₅	1	1	yes	characteristic, complemented
128	C₄×C₃⋊D₃	—	5	2	no	—
129	C₄×C₃⋊D₃	—	5	1	no	—
130	C₄×C₃⋊D₃	—	5	1	no	—
131	C₄×C₃⋊D₃	—	5	1	no	—
132	C₄×C₃⋊D₃	—	5	1	no	—
133	C₄×C₃⋊D₃	—	5	2	no	—
134	C₄×C₃⋊D₃	C₂×D₅	1	2	yes	complemented
135	#634	—	5	2	no	—
136	#634	C₂×D₅	1	1	yes	characteristic, complemented
137	C₂²×C₃×C₅	C₂²×D₃	1	4	yes	—
138	C₂×D₁₅	—	3	4	no	—
139	C₂×D₁₅	—	3	4	no	—
140	C₂×D₁₅	—	6	8	no	—
141	C₂×C₅×D₃	—	3	8	no	—
142	C₂×C₃×D₅	C₂²×D₃	1	8	yes	—
143	D₃×D₅	—	6	16	no	—
144	C₃×C₄×C₅	C₂²×D₃	1	4	yes	—
145	C₃×C₄×C₅	C₂²×D₃	1	4	yes	—
146	Dic₁₅	—	3	4	no	—
147	Dic₁₅	—	3	4	no	—
148	C₃×Dic₅	C₂²×D₃	1	8	yes	—
149	C₅×Dic₃	—	3	8	no	—
150	C₃×#41	—	5	4	no	—
151	#239	—	15	4	no	—
152	D₃×Q₈	—	15	4	no	—
153	#237	—	15	4	no	—
154	#237	—	15	8	no	—
155	D₃×D₄	—	15	4	no	—
156	D₃×D₄	—	15	8	no	—
157	#235	—	3	4	no	—
158	#235	—	15	8	no	—
159	C₂×C₄×D₃	—	15	8	no	—
160	C₂×C₄×D₃	—	15	4	no	—
161	C₃²×C₅	C₂×#41	1	1	yes	characteristic, complemented, Hall
162	C₂²×D₅	—	9	1	no	—
163	C₂²×D₅	—	9	2	no	—
164	C₅×Q₈	—	9	1	no	—
165	C₅×D₄	—	9	2	no	—
166	C₅×D₄	—	9	1	no	—
167	C₂×C₄×C₅	—	9	2	no	—
168	C₂×C₄×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
169	C₂²⋊D₅	—	9	2	no	—
170	C₂²⋊D₅	—	9	2	no	—
171	C₂²⋊D₅	C₂×C₃⋊D₃	1	2	yes	complemented
172	C₂×Dic₅	—	9	1	no	—
173	C₂×Dic₅	—	9	2	no	—
174	D₂₀	—	9	2	no	—
175	D₂₀	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
176	C₄×D₅	—	9	2	no	—
177	C₄×D₅	—	9	1	no	—
178	C₄×D₅	—	9	1	no	—
179	C₄×D₅	—	9	1	no	—
180	C₄×D₅	—	9	1	no	—
181	C₄×D₅	—	9	2	no	—
182	C₄×D₅	C₂×C₃⋊D₃	1	2	yes	complemented
183	Dic₁₀	—	9	2	no	—
184	Dic₁₀	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
185	C₂²×C₃²	—	5	2	no	—
186	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
187	C₂×C₃⋊D₃	—	5	1	no	—
188	C₂×C₃⋊D₃	—	5	1	no	—
189	C₂×C₃⋊D₃	—	10	4	no	—
190	C₂×C₃⋊D₃	—	10	2	no	—
191	C₂×C₃⋊D₃	C₂²×D₅	1	2	yes	—
192	C₃²×C₄	—	5	2	no	—
193	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
194	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
195	C₃⋊Dic₃	—	5	1	no	—
196	C₃⋊Dic₃	—	5	1	no	—
197	C₃⋊Dic₃	C₂²×D₅	1	2	yes	—
198	C₂×#41	—	45	1	no	Sylow
199	C₂×C₃×C₅	—	2	4	no	—
200	C₂×C₃×C₅	C₂³×D₃	1	4	yes	—
201	D₁₅	—	3	8	no	—
202	D₁₅	—	6	4	no	—
203	C₃×D₅	—	2	8	no	—
204	C₅×D₃	—	6	8	no	—
205	C₂²×D₃	—	15	8	no	—
206	C₂²×D₃	—	15	4	no	—
207	C₃×Q₈	—	5	4	no	—
208	C₃×D₄	—	5	8	no	—
209	C₃×D₄	—	5	4	no	—
210	C₂×C₃×C₄	—	5	8	no	—
211	C₂×C₃×C₄	D₃×D₅	1	4	yes	complemented
212	#67	—	3	8	no	—
213	#67	—	15	8	no	—
214	#67	—	15	8	no	—
215	C₂×Dic₃	—	15	8	no	—
216	C₂×Dic₃	—	15	4	no	—
217	D₁₂	—	3	4	no	—
218	D₁₂	—	15	8	no	—
219	C₄×D₃	—	3	8	no	—
220	C₄×D₃	—	15	8	no	—
221	C₄×D₃	—	15	8	no	—
222	C₄×D₃	—	15	4	no	—
223	C₄×D₃	—	15	4	no	—
224	C₄×D₃	—	15	4	no	—
225	C₄×D₃	—	15	4	no	—
226	Dic₆	—	3	4	no	—
227	Dic₆	—	15	8	no	—
228	C₂²×C₅	—	9	2	no	—
229	C₂²×C₅	C₂²×C₃⋊D₃	1	1	yes	characteristic
230	C₂×D₅	—	9	1	no	—
231	C₂×D₅	—	9	1	no	—
232	C₂×D₅	—	18	4	no	—
233	C₂×D₅	—	18	2	no	—
234	C₂×D₅	C₂²×C₃⋊D₃	1	2	yes	—
235	C₄×C₅	—	9	2	no	—
236	C₄×C₅	C₂²×C₃⋊D₃	1	1	yes	characteristic
237	C₄×C₅	C₂²×C₃⋊D₃	1	1	yes	characteristic
238	Dic₅	—	9	1	no	—
239	Dic₅	—	9	1	no	—
240	Dic₅	C₂²×C₃⋊D₃	1	2	yes	—
241	C₂×C₃²	—	2	1	no	—
242	C₂×C₃²	—	10	2	no	—
243	C₂×C₃²	C₂³×D₅	1	1	yes	characteristic
244	C₃⋊D₃	—	2	2	no	—
245	C₃⋊D₃	—	5	2	no	—
246	C₃⋊D₃	—	10	1	no	—
247	#41	—	5	1	no	—
248	#41	—	9	1	no	—
249	#41	—	45	2	no	—
250	#41	—	45	2	no	—
251	#41	—	45	1	no	—
252	#41	—	45	1	no	—
253	C₂×Q₈	—	45	1	no	—
254	C₂×D₄	—	45	2	no	—
255	C₂×D₄	—	45	1	no	—
256	C₂²×C₄	—	45	2	no	—
257	C₂²×C₄	—	45	1	no	—
258	C₃×C₅	D₃×#41	1	4	yes	complemented
259	C₂²×C₃	—	5	8	no	—
260	C₂²×C₃	C₂×D₃×D₅	1	4	yes	—
261	C₂×D₃	—	3	8	no	—
262	C₂×D₃	—	15	4	no	—
263	C₂×D₃	—	15	4	no	—
264	C₂×D₃	—	30	16	no	—
265	C₂×D₃	—	30	8	no	—
266	C₃×C₄	—	5	8	no	—
267	C₃×C₄	C₂×D₃×D₅	1	4	yes	—
268	C₃×C₄	C₂×D₃×D₅	1	4	yes	—
269	Dic₃	—	3	8	no	—
270	Dic₃	—	15	4	no	—
271	Dic₃	—	15	4	no	—
272	C₂×C₅	—	2	1	no	—
273	C₂×C₅	—	18	2	no	—
274	C₂×C₅	C₂³×C₃⋊D₃	1	1	yes	characteristic
275	D₅	—	2	2	no	—
276	D₅	—	9	2	no	—
277	D₅	—	18	1	no	—
278	C₃²	D₅×#41	1	1	yes	characteristic, complemented, Sylow
279	C₂³	—	45	2	no	—
280	C₂³	—	45	1	no	—
281	Q₈	—	5	1	no	—
282	Q₈	—	9	1	no	—
283	Q₈	—	45	2	no	—
284	D₄	—	5	2	no	—
285	D₄	—	5	1	no	—
286	D₄	—	9	2	no	—
287	D₄	—	9	1	no	—
288	D₄	—	45	2	no	—
289	D₄	—	45	2	no	—
290	D₄	—	45	2	no	—
291	C₂×C₄	—	5	2	no	—
292	C₂×C₄	—	9	2	no	—
293	C₂×C₄	—	45	2	no	—
294	C₂×C₄	—	45	2	no	—
295	C₂×C₄	—	45	2	no	—
296	C₂×C₄	—	45	1	no	—
297	C₂×C₄	—	45	1	no	—
298	C₂×C₄	—	45	1	no	—
299	C₂×C₄	—	45	1	no	—
300	C₂×C₄	—	45	1	no	—
301	C₂×C₄	D₅×C₃⋊D₃	1	1	yes	characteristic, complemented
302	C₂×C₃	—	2	4	no	—
303	C₂×C₃	—	10	8	no	—
304	C₂×C₃	C₂²×D₃×D₅	1	4	yes	—
305	D₃	—	6	8	no	—
306	D₃	—	15	8	no	—
307	D₃	—	30	4	no	—
308	C₅	#41×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
309	C₂²	—	5	2	no	—
310	C₂²	—	9	2	no	—
311	C₂²	—	45	1	no	—
312	C₂²	—	45	1	no	—
313	C₂²	—	90	2	no	—
314	C₂²	—	90	4	no	—
315	C₂²	C₂×D₅×C₃⋊D₃	1	1	yes	characteristic
316	C₄	—	5	2	no	—
317	C₄	—	9	2	no	—
318	C₄	—	45	1	no	—
319	C₄	—	45	1	no	—
320	C₄	C₂×D₅×C₃⋊D₃	1	1	yes	characteristic
321	C₄	C₂×D₅×C₃⋊D₃	1	1	yes	characteristic
322	C₃	#92016	1	4	yes	complemented
323	C₂	—	2	1	no	—
324	C₂	—	10	2	no	—
325	C₂	—	18	2	no	—
326	C₂	—	45	2	no	—
327	C₂	—	90	1	no	—
328	C₂	C₂²×D₅×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₅×C₃⋊D₃
commutator, socle, cosocle	C₂×C₃²×C₅	C₂⁴
Frattini	C₂	C₂²×D₅×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₅	C₂²
hypercenter	C₂×C₄	D₅×C₃⋊D₃
hypocenter	C₃²×C₅	C₂×#41
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₄ ◁ 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. C₅×#3828	C₂	semidirect
3. C₃²×#709	C₂	semidirect
4. #10624173	C₂	semidirect
5. #10624163	C₂	semidirect
6. #10624160	C₂	semidirect
7. C₄×D₅×C₃⋊D₃	C₂	semidirect
8. #10624154	C₂	semidirect
9. #10624153	C₂	semidirect
10. #10624151	C₂	semidirect
11. #10624149	C₂	semidirect
15. C₂×C₃²×C₄×C₅	C₂²	semidirect
16. C₂×C₃⋊Dic₁₅	C₂²	semidirect
19. C₄×C₃⋊D₁₅	C₂²	semidirect
20. C₄×C₃⋊D₁₅	C₂²	semidirect
21. C₅×#638	C₂²	semidirect
22. C₅×C₄⋊(C₃⋊D₃)	C₂²	semidirect
23. C₄×C₅×C₃⋊D₃	C₂²	semidirect
24. C₅×#634	C₂²	semidirect
25. C₃²×C₂²⋊D₅	C₂²	semidirect
26. C₃²×D₂₀	C₂²	semidirect
27. C₃²×C₄×D₅	C₂²	semidirect
28. C₃²×Dic₁₀	C₂²	semidirect
29. #67141	C₂²	semidirect
30. #67140	C₂²	semidirect
31. #67139	C₂²	semidirect
32. #67138	C₂²	semidirect
34. C₃⋊D₃×Dic₅	C₂²	semidirect
35. D₅×C₃⋊Dic₃	C₂²	semidirect
39. C₃×#709	D₃	semidirect
70. #3828	D₅	semidirect
75. C₂×C₃×C₄×C₅	C₂×D₃	semidirect
85. C₃×C₂²⋊D₅	C₂×D₃	semidirect
86. C₃×D₂₀	C₂×D₃	semidirect
87. C₃×C₄×D₅	C₂×D₃	semidirect
88. C₃×Dic₁₀	C₂×D₃	semidirect
100. C₃⋊D₁₅	#41	semidirect
111. #709	C₃⋊D₃	semidirect
120. C₂×C₃²×C₄	C₂×D₅	semidirect
123. #638	C₂×D₅	semidirect
127. C₄⋊(C₃⋊D₃)	C₂×D₅	semidirect
134. C₄×C₃⋊D₃	C₂×D₅	semidirect
136. #634	C₂×D₅	semidirect
161. C₃²×C₅	C₂×#41	semidirect
168. C₂×C₄×C₅	C₂×C₃⋊D₃	semidirect
171. C₂²⋊D₅	C₂×C₃⋊D₃	semidirect
175. D₂₀	C₂×C₃⋊D₃	semidirect
182. C₄×D₅	C₂×C₃⋊D₃	semidirect
184. Dic₁₀	C₂×C₃⋊D₃	semidirect
211. C₂×C₃×C₄	D₃×D₅	semidirect
258. C₃×C₅	D₃×#41	semidirect
278. C₃²	D₅×#41	semidirect
301. C₂×C₄	D₅×C₃⋊D₃	semidirect
308. C₅	#41×C₃⋊D₃	semidirect
322. C₃	#92016	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
C₅×#3828	2
C₃²×#709	2
#10624173	2
#10624163	2
#10624160	2
C₄×D₅×C₃⋊D₃	2
#10624154	2
#10624153	2
#10624151	2
#10624149	2
#92016	3
#41×C₃⋊D₃	5