Subgroups — #14284036

Order 1920.

528 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91764	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751027	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746285	C₂	1	4	yes	maximal, complemented
5	#10600653	—	3	1	no	maximal, Hall
6	C₂²×D₆₀	C₄	1	1	yes	characteristic
7	C₂²×Dic₃₀	C₄	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	—	2	1	no	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×#6714	—	2	1	no	—
11	C₂×#6714	—	2	1	no	—
12	#91764	—	2	1	no	—
13	#91764	—	2	1	no	—
14	#91764	C₄	1	2	yes	—
15	#91764	C₂²	1	2	yes	—
16	C₃×C₅×#97	C₂²	1	2	yes	complemented
17	C₃×C₄×C₅×C₈	—	2	1	no	—
18	#91114	—	2	1	no	—
19	C₄⋊(C₃⋊#168)	—	2	1	no	—
20	#91087	C₂²	1	2	yes	complemented
21	#69860	—	5	1	no	maximal, Hall
22	C₂×#4021	—	3	1	no	—
23	C₄×C₅×M₁₆	D₃	1	1	yes	characteristic, complemented
24	#65215	—	3	1	no	—
25	#64684	—	3	4	no	—
26	C₂³×D₁₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×Dic₁₅	—	2	1	no	—
31	C₂×D₆₀	—	2	1	no	—
32	C₂×D₆₀	—	4	2	no	—
33	C₂×D₆₀	C₂×C₄	1	2	yes	—
34	C₂×Dic₃₀	—	2	1	no	—
35	C₂×Dic₃₀	—	4	2	no	—
36	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
37	C₃×C₅×M₁₆	—	2	4	no	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	#6714	—	2	2	no	—
44	#6714	—	2	2	no	—
45	#6714	—	2	2	no	—
46	#6714	—	2	2	no	—
47	C₂×C₃⋊#168	—	2	2	no	—
48	C₂×#875	—	5	1	no	—
49	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
50	#4926	—	5	1	no	—
51	#4464	—	5	4	no	—
52	C₂²×D₂₀	—	3	1	no	—
53	C₂²×Dic₁₀	—	3	1	no	—
54	C₂×C₅×M₁₆	—	2	1	no	—
55	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
56	C₂×#685	—	6	1	no	—
57	C₂×#685	—	6	1	no	—
58	#4021	—	3	2	no	—
59	#4021	—	3	2	no	—
60	#4021	—	6	1	no	—
61	#4021	—	6	1	no	—
62	C₅×#97	C₂×D₃	1	2	yes	complemented
63	C₄×C₅×C₈	—	2	1	no	—
64	C₂²⋊#168	—	6	1	no	—
65	C₄⋊#168	—	6	1	no	—
66	#3935	—	3	2	no	—
67	#1527	—	15	1	no	Sylow
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×D₁₅	—	2	2	no	—
70	C₂²×D₁₅	—	4	2	no	—
71	C₂²×D₁₅	—	4	2	no	—
72	C₂²×D₁₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	4	1	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₁₅	—	2	2	no	—
83	C₂×Dic₁₅	—	4	2	no	—
84	D₆₀	—	4	2	no	—
85	D₆₀	—	4	2	no	—
86	Dic₃₀	—	4	2	no	—
87	Dic₃₀	—	4	2	no	—
88	C₃×C₅×C₈	—	2	4	no	—
89	C₃⋊#168	—	4	2	no	—
90	C₂²×D₁₂	—	5	1	no	—
91	C₂²×Dic₆	—	5	1	no	—
92	C₂×C₃×M₁₆	—	2	1	no	—
93	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
94	C₂×(C₂×D₃)⋊C₄	—	10	1	no	—
95	C₂×(C₂×D₃)⋊C₄	—	10	1	no	—
96	#875	—	5	2	no	—
97	#875	—	5	2	no	—
98	#875	—	10	1	no	—
99	#875	—	10	1	no	—
100	C₃×#97	C₂×D₅	1	2	yes	complemented
101	C₃×C₄×C₈	—	2	1	no	—
102	#830	—	10	1	no	—
103	C₄⋊(C₃⋊C₈)	—	10	1	no	—
104	#803	—	5	2	no	—
105	C₂³×D₅	—	6	1	no	—
106	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
107	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
108	C₂²×C₄×C₅	#67	1	1	yes	characteristic
109	C₂²×Dic₅	—	6	1	no	—
110	C₂×D₂₀	—	3	2	no	—
111	C₂×D₂₀	—	6	1	no	—
112	C₂×D₂₀	—	12	2	no	—
113	C₂×Dic₁₀	—	3	2	no	—
114	C₂×Dic₁₀	—	6	1	no	—
115	C₂×Dic₁₀	—	12	2	no	—
116	C₅×M₁₆	—	2	4	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
119	C₄²×C₅	D₁₂	1	1	yes	characteristic
120	C₄²×C₅	#67	1	1	yes	characteristic
121	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
122	#685	—	6	2	no	—
123	#685	—	6	2	no	—
124	#685	—	6	2	no	—
125	#685	—	6	2	no	—
126	C₂×#168	—	6	2	no	—
127	C₂×#124	—	15	1	no	—
128	#432	—	15	1	no	—
129	C₄×M₁₆	D₁₅	1	1	yes	characteristic, complemented
130	#329	—	15	4	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	—	2	1	no	—
133	C₂²×C₃×C₅	#104	1	2	yes	—
134	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×D₁₅	—	4	2	no	—
136	C₂×D₁₅	—	4	2	no	—
137	C₂×D₁₅	—	4	2	no	—
138	C₂×D₁₅	—	8	4	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	#100	1	2	yes	—
145	Dic₁₅	—	4	2	no	—
146	C₂³×D₃	—	10	1	no	—
147	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
148	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
150	C₂²×Dic₃	—	10	1	no	—
151	C₂×D₁₂	—	5	2	no	—
152	C₂×D₁₂	—	10	1	no	—
153	C₂×D₁₂	—	20	2	no	—
154	C₂×Dic₆	—	5	2	no	—
155	C₂×Dic₆	—	10	1	no	—
156	C₂×Dic₆	—	20	2	no	—
157	C₃×M₁₆	—	2	4	no	—
158	C₂×C₃×C₈	—	2	2	no	—
159	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
160	C₃×C₄²	D₂₀	1	1	yes	characteristic
161	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
162	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
163	(C₂×D₃)⋊C₄	—	10	2	no	—
164	(C₂×D₃)⋊C₄	—	10	2	no	—
165	(C₂×D₃)⋊C₄	—	10	2	no	—
166	(C₂×D₃)⋊C₄	—	10	2	no	—
167	C₂×C₃⋊C₈	—	10	2	no	—
168	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
169	C₂²×D₅	—	6	2	no	—
170	C₂²×D₅	—	12	2	no	—
171	C₂²×D₅	—	12	2	no	—
172	C₂²×D₅	—	12	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
182	C₂×Dic₅	—	6	2	no	—
183	C₂×Dic₅	—	12	2	no	—
184	D₂₀	—	12	2	no	—
185	D₂₀	—	12	2	no	—
186	Dic₁₀	—	12	2	no	—
187	Dic₁₀	—	12	2	no	—
188	C₅×C₈	—	2	4	no	—
189	#168	—	12	2	no	—
190	C₂²×Q₈	—	15	1	no	—
191	C₂²×D₄	—	15	1	no	—
192	C₂×M₁₆	—	2	1	no	—
193	#124	—	15	2	no	—
194	#124	—	15	2	no	—
195	#124	—	30	1	no	—
196	#124	—	30	1	no	—
197	C₂×C₂²⋊C₄	—	30	1	no	—
198	C₂×C₂²⋊C₄	—	30	1	no	—
199	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
200	C₄⋊C₈	—	30	1	no	—
201	C₂²⋊C₈	—	30	1	no	—
202	#97	—	15	2	no	—
203	#97	C₂×D₁₅	1	2	yes	complemented
204	C₄×C₈	—	2	1	no	—
205	C₂×C₃×C₅	—	2	2	no	—
206	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
207	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
208	C₂×C₃×C₅	#421	1	1	yes	characteristic
209	D₁₅	—	8	2	no	—
210	C₂³×C₃	#685	1	1	yes	characteristic
211	C₂²×D₃	—	10	2	no	—
212	C₂²×D₃	—	20	2	no	—
213	C₂²×D₃	—	20	2	no	—
214	C₂²×D₃	—	20	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	4	1	no	—
218	C₂×C₃×C₄	#685	1	1	yes	characteristic
219	C₂×C₃×C₄	#685	1	1	yes	characteristic
220	C₂×C₃×C₄	#685	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
224	C₂×Dic₃	—	10	2	no	—
225	C₂×Dic₃	—	20	2	no	—
226	D₁₂	—	20	2	no	—
227	D₁₂	—	20	2	no	—
228	Dic₆	—	20	2	no	—
229	Dic₆	—	20	2	no	—
230	C₃×C₈	—	2	4	no	—
231	C₃⋊C₈	—	20	2	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	#805	1	2	yes	—
235	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
236	C₂×D₅	—	12	2	no	—
237	C₂×D₅	—	12	2	no	—
238	C₂×D₅	—	12	2	no	—
239	C₂×D₅	—	24	4	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	#823	1	2	yes	—
246	Dic₅	—	12	2	no	—
247	C₂⁴	—	30	1	no	—
248	C₂×Q₈	—	15	2	no	—
249	C₂×Q₈	—	30	1	no	—
250	C₂×Q₈	—	60	2	no	—
251	C₂×D₄	—	15	2	no	—
252	C₂×D₄	—	30	1	no	—
253	C₂×D₄	—	60	2	no	—
254	C₂²×C₄	—	30	1	no	—
255	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
256	C₂²×C₄	D₆₀	1	1	yes	characteristic
257	C₂²×C₄	#1220	1	1	yes	characteristic
258	M₁₆	—	2	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	30	2	no	—
261	C₂²⋊C₄	—	30	2	no	—
262	C₂²⋊C₄	—	30	2	no	—
263	C₂²⋊C₄	—	30	2	no	—
264	C₂²⋊C₄	—	30	2	no	—
265	C₄²	C₄×D₁₅	1	1	yes	characteristic
266	C₄²	D₆₀	1	1	yes	characteristic
267	C₄²	#1220	1	1	yes	characteristic
268	C₄²	C₂²×D₁₅	1	1	yes	characteristic
269	C₃×C₅	#1527	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	#3937	1	2	yes	—
273	C₂²×C₃	C₂×#685	1	1	yes	characteristic
274	C₂×D₃	—	20	2	no	—
275	C₂×D₃	—	20	2	no	—
276	C₂×D₃	—	20	2	no	—
277	C₂×D₃	—	40	4	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	#3955	1	2	yes	—
284	Dic₃	—	20	2	no	—
285	C₂×C₅	—	2	2	no	—
286	C₂×C₅	C₂×#805	1	1	yes	characteristic
287	C₂×C₅	C₂×#823	1	1	yes	characteristic
288	C₂×C₅	#5139	1	1	yes	characteristic
289	D₅	—	24	2	no	—
290	C₂³	—	30	2	no	—
291	C₂³	—	60	2	no	—
292	C₂³	—	60	2	no	—
293	C₂³	—	60	2	no	—
294	C₂³	#6714	1	1	yes	characteristic
295	Q₈	—	60	2	no	—
296	Q₈	—	60	2	no	—
297	D₄	—	60	2	no	—
298	D₄	—	60	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	30	2	no	—
303	C₂×C₄	—	60	2	no	—
304	C₂×C₄	#6714	1	1	yes	characteristic
305	C₂×C₄	#6714	1	1	yes	characteristic
306	C₂×C₄	#6714	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
308	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
309	C₂×C₄	C₂×#1220	1	1	yes	characteristic
310	C₈	—	2	4	no	—
311	C₈	—	60	2	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	C₂×#3937	1	1	yes	characteristic
314	C₂×C₃	C₂×#3955	1	1	yes	characteristic
315	C₂×C₃	#65428	1	1	yes	characteristic
316	D₃	—	40	2	no	—
317	C₅	#69860	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	60	2	no	—
321	C₂²	—	60	2	no	—
322	C₂²	—	60	2	no	—
323	C₂²	—	120	4	no	—
324	C₂²	#91089	1	2	yes	—
325	C₂²	C₂×#6714	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	60	2	no	—
332	C₄	#91107	1	2	yes	—
333	C₃	#10600653	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	2	no	—
335	C₂	—	120	2	no	—
336	C₂	C₂×#91089	1	1	yes	characteristic
337	C₂	C₂×#91107	1	1	yes	characteristic
338	C₂	#11751240	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6714
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₆₀	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×M₁₆	D₁₅
hypocenter	C₃×C₅	#1527
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×M₁₆	C₂	semidirect
3. #11751027	C₂	semidirect
4. #11746285	C₂	semidirect
16. C₃×C₅×#97	C₂²	semidirect
20. #91087	C₂²	semidirect
23. C₄×C₅×M₁₆	D₃	semidirect
49. C₃×C₄×M₁₆	D₅	semidirect
62. C₅×#97	C₂×D₃	semidirect
100. C₃×#97	C₂×D₅	semidirect
129. C₄×M₁₆	D₁₅	semidirect
203. #97	C₂×D₁₅	semidirect
269. C₃×C₅	#1527	semidirect
317. C₅	#69860	semidirect
333. C₃	#10600653	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91764	2
C₃×C₄×C₅×M₁₆	2
#11751027	2
#11746285	2
#10600653	3
#69860	5