Subgroups — #14418174

Order 1920.

916 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756507	C₂	1	1	yes	maximal, characteristic
2	#11754130	C₂	1	2	yes	maximal, complemented
3	C₅×Dic₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65414	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90980	C₂	1	2	yes	maximal, complemented
6	#11748069	C₂	1	4	yes	maximal, complemented
7	#11748068	C₂	1	4	yes	maximal, complemented
8	#10610272	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	#91798	C₄	1	4	yes	complemented
12	#91797	C₄	1	4	yes	complemented
13	#91793	C₂²	1	4	yes	—
14	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₅×#208	C₂²	1	1	yes	characteristic
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	#91292	—	2	4	no	—
18	#91291	C₂²	1	8	yes	complemented
19	#91290	—	2	4	no	—
20	C₅×#815	C₂²	1	2	yes	complemented
21	C₅×C₈×Dic₃	C₂²	1	2	yes	complemented
22	C₃×#3951	C₂²	1	4	yes	complemented
23	#90980	C₂²	1	4	yes	complemented
24	#90980	C₂²	1	4	yes	complemented
25	#79258	—	5	1	no	maximal, Hall
26	#66082	—	3	1	no	—
27	#66079	—	3	2	no	—
28	C₄×C₅×M₁₆	—	3	1	no	—
29	#65414	D₃	1	1	yes	characteristic, complemented
30	C₂×#3937	—	3	2	no	—
31	#65006	—	3	4	no	—
32	#64989	—	3	4	no	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×#1220	—	2	2	no	—
35	#6817	—	2	2	no	—
36	#6817	—	2	2	no	—
37	#6817	C₂×C₄	1	4	yes	—
38	#6817	C₂³	1	4	yes	—
39	C₂×D₆₀	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₅	—	2	2	no	—
41	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
42	C₂²×C₅×Dic₃	—	2	1	no	—
43	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	Dic₆₀	C₂×C₄	1	4	yes	complemented
46	D₁₂₀	C₂×C₄	1	4	yes	complemented
47	#6706	C₂×C₄	1	8	yes	complemented
48	C₄×C₅×Dic₃	—	2	1	no	—
49	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
50	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
51	C₅×#208	D₄	1	4	yes	—
52	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
53	C₃×#681	—	2	2	no	—
54	C₃×#681	C₂³	1	4	yes	—
55	C₂×C₃×#168	—	2	2	no	—
56	#6651	—	2	8	no	—
57	#6648	—	2	8	no	—
58	#5748	—	5	1	no	—
59	#5745	—	5	2	no	—
60	C₃×#430	—	5	1	no	—
61	C₂×#825	—	5	2	no	—
62	Dic₃×M₁₆	D₅	1	1	yes	characteristic, complemented
63	#4895	—	5	4	no	—
64	#4894	—	5	4	no	—
65	C₂×#709	—	3	2	no	—
66	C₂×C₅×M₁₆	—	3	1	no	—
67	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
68	C₂×C₄²×C₅	—	3	1	no	—
69	C₂×#681	C₂×D₃	1	2	yes	—
70	#4055	—	3	4	no	—
71	#4054	—	3	4	no	—
72	#4053	—	6	4	no	—
73	D₅×M₁₆	—	6	4	no	—
74	#4050	—	3	4	no	—
75	#4047	—	3	8	no	—
76	C₅×#97	—	3	2	no	—
77	C₄×C₅×C₈	—	3	2	no	—
78	#3951	C₂×D₃	1	4	yes	complemented
79	#3937	—	3	4	no	—
80	#3937	—	3	4	no	—
81	#2956	—	15	1	no	Sylow
82	C₂³×C₃×C₅	#41	1	1	yes	characteristic
83	C₂²×D₁₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	#1220	—	2	4	no	—
89	#1220	—	2	4	no	—
90	C₂×Dic₁₅	—	2	2	no	—
91	D₆₀	—	2	2	no	—
92	D₆₀	C₂²×C₄	1	4	yes	—
93	C₄×D₁₅	—	2	4	no	—
94	C₄×D₁₅	—	2	4	no	—
95	Dic₃₀	—	2	2	no	—
96	Dic₃₀	C₂²×C₄	1	4	yes	—
97	C₂×C₅×Dic₃	—	2	1	no	—
98	C₂×C₅×Dic₃	—	2	1	no	—
99	C₂×C₅×Dic₃	—	4	2	no	—
100	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
101	C₃×#168	—	2	4	no	—
102	C₅×C₃⋊C₈	C₂×D₄	1	4	yes	—
103	C₂×#235	—	5	2	no	—
104	C₂×C₃×M₁₆	—	5	2	no	—
105	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
106	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
107	C₂×#208	C₂×D₅	1	1	yes	characteristic
108	#909	—	5	4	no	—
109	#908	—	5	4	no	—
110	#907	—	5	8	no	—
111	D₃×M₁₆	—	10	4	no	—
112	#904	—	5	4	no	—
113	#901	—	10	4	no	—
114	C₃×#108	—	5	4	no	—
115	#825	—	5	4	no	—
116	#825	—	5	4	no	—
117	#815	C₂×D₅	1	2	yes	complemented
118	C₈×Dic₃	C₂×D₅	1	2	yes	complemented
119	C₂²×C₄×C₅	—	6	1	no	—
120	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
121	C₂×C₂²⋊D₅	—	6	2	no	—
122	#709	—	3	4	no	—
123	#709	—	3	4	no	—
124	#709	—	6	2	no	—
125	#709	—	6	2	no	—
126	C₂×D₂₀	—	3	2	no	—
127	C₂×C₄×D₅	—	6	2	no	—
128	C₂×Dic₁₀	—	3	2	no	—
129	C₅×M₁₆	—	3	4	no	—
130	C₅×M₁₆	C₄×D₃	1	4	yes	—
131	C₂×C₅×C₈	—	3	2	no	—
132	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
133	C₄²×C₅	—	3	1	no	—
134	C₄²×C₅	—	3	1	no	—
135	C₄²×C₅	—	6	1	no	—
136	#681	—	2	2	no	—
137	#681	C₂²×D₃	1	4	yes	—
138	C₂×#168	—	2	2	no	—
139	Dic₂₀	—	3	4	no	—
140	D₄₀	—	3	4	no	—
141	#677	—	3	8	no	—
142	#676	—	6	8	no	—
143	C₈×D₅	—	6	8	no	—
144	#575	—	15	1	no	—
145	#568	—	15	2	no	—
146	#444	—	15	4	no	—
147	#443	—	15	4	no	—
148	#430	—	5	1	no	—
149	C₂×#104	—	15	2	no	—
150	C₄×M₁₆	—	3	1	no	—
151	C₂²×C₃×C₅	—	4	1	no	—
152	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
153	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
154	C₂×D₁₅	—	2	4	no	—
155	C₂×D₁₅	—	8	2	no	—
156	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
157	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
158	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	Dic₁₅	—	2	4	no	—
160	C₅×Dic₃	—	2	2	no	—
161	C₅×Dic₃	—	4	1	no	—
162	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
163	C₂×#67	—	10	2	no	—
164	C₂²×Dic₃	—	2	1	no	—
165	#235	—	5	4	no	—
166	#235	—	5	4	no	—
167	#235	—	10	2	no	—
168	#235	—	10	2	no	—
169	C₂×D₁₂	—	5	2	no	—
170	C₂×C₄×D₃	—	10	2	no	—
171	C₂×Dic₆	—	5	2	no	—
172	C₃×M₁₆	—	5	4	no	—
173	C₃×M₁₆	—	10	2	no	—
174	C₃×M₁₆	C₄×D₅	1	4	yes	—
175	C₂×C₃×C₈	—	10	2	no	—
176	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
177	C₄×Dic₃	—	2	1	no	—
178	C₄×Dic₃	C₂²×D₅	1	1	yes	characteristic
179	C₄×Dic₃	C₂²×D₅	1	1	yes	characteristic
180	#208	D₂₀	1	4	yes	—
181	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
182	Dic₁₂	—	5	4	no	—
183	D₂₄	—	5	4	no	—
184	#204	—	5	8	no	—
185	ΓL(1,𝔽₂₅)	—	10	8	no	—
186	D₃×C₈	—	10	8	no	—
187	C₂³×C₅	#239	1	1	yes	characteristic
188	C₂²×D₅	—	6	2	no	—
189	C₂×C₄×C₅	—	6	1	no	—
190	C₂×C₄×C₅	—	6	1	no	—
191	C₂×C₄×C₅	—	12	2	no	—
192	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
193	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
194	C₂×C₄×C₅	#239	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₂²⋊D₅	—	6	4	no	—
197	C₂²⋊D₅	—	6	4	no	—
198	C₂×Dic₅	—	6	2	no	—
199	D₂₀	—	3	4	no	—
200	D₂₀	—	6	2	no	—
201	C₄×D₅	—	6	4	no	—
202	C₄×D₅	—	6	4	no	—
203	Dic₁₀	—	3	4	no	—
204	Dic₁₀	—	6	2	no	—
205	C₅×C₈	—	3	4	no	—
206	C₅×C₈	C₂×C₄×D₃	1	4	yes	—
207	#168	—	2	4	no	—
208	C₂×#41	—	15	2	no	—
209	#137	—	15	4	no	—
210	AGL(1,ℤ/8)	—	15	4	no	—
211	#135	—	15	4	no	—
212	#131	—	15	8	no	—
213	#131	—	30	4	no	—
214	C₂×M₁₆	—	3	1	no	—
215	C₂×M₁₆	—	5	2	no	—
216	C₂×M₁₆	—	30	4	no	—
217	C₂×M₁₆	D₃×D₅	1	1	yes	characteristic
218	C₂×C₄²	—	3	1	no	—
219	#108	—	5	4	no	—
220	#104	—	15	4	no	—
221	#104	—	15	4	no	—
222	#97	—	3	2	no	—
223	C₄×C₈	—	3	2	no	—
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	2	2	no	—
226	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	D₁₅	—	4	4	no	—
228	C₂³×C₃	#709	1	1	yes	characteristic
229	C₂²×D₃	—	10	2	no	—
230	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
231	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
232	C₂×C₃×C₄	#709	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
234	#67	—	10	4	no	—
235	#67	—	10	4	no	—
236	C₂×Dic₃	—	2	1	no	—
237	C₂×Dic₃	—	2	1	no	—
238	C₂×Dic₃	—	4	2	no	—
239	C₂×Dic₃	—	10	2	no	—
240	D₁₂	—	5	4	no	—
241	D₁₂	—	10	2	no	—
242	C₄×D₃	—	10	4	no	—
243	C₄×D₃	—	10	4	no	—
244	Dic₆	—	5	4	no	—
245	Dic₆	—	10	2	no	—
246	C₃×C₈	—	10	4	no	—
247	C₃×C₈	C₂×C₄×D₅	1	4	yes	—
248	C₃⋊C₈	C₂×D₂₀	1	4	yes	—
249	C₂²×C₅	—	4	1	no	—
250	C₂²×C₅	#893	1	2	yes	—
251	C₂²×C₅	C₂×#239	1	1	yes	characteristic
252	C₂×D₅	—	6	4	no	—
253	C₂×D₅	—	24	2	no	—
254	C₄×C₅	—	6	2	no	—
255	C₄×C₅	—	12	1	no	—
256	C₄×C₅	#893	1	2	yes	—
257	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
258	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
259	Dic₅	—	6	4	no	—
260	#41	—	15	4	no	—
261	#41	—	15	4	no	—
262	#41	—	30	2	no	—
263	#41	—	30	2	no	—
264	C₂×Q₈	—	15	2	no	—
265	C₂×D₄	—	15	2	no	—
266	C₂×D₄	—	30	2	no	—
267	C₂²×C₄	—	6	1	no	—
268	C₂²×C₄	—	30	2	no	—
269	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
270	Q₁₆	—	15	4	no	—
271	SD₁₆	—	15	8	no	—
272	D₈	—	15	4	no	—
273	M₁₆	—	3	4	no	—
274	M₁₆	—	5	4	no	—
275	M₁₆	—	10	2	no	—
276	M₁₆	—	30	8	no	—
277	M₁₆	#1200	1	4	yes	—
278	C₂×C₈	—	3	2	no	—
279	C₂×C₈	—	10	2	no	—
280	C₂×C₈	—	30	8	no	—
281	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
282	C₄²	—	3	1	no	—
283	C₄²	—	3	1	no	—
284	C₄²	—	6	1	no	—
285	C₃×C₅	#2956	1	1	yes	characteristic, complemented, Hall
286	C₂²×C₃	—	4	1	no	—
287	C₂²×C₃	C₄×D₂₀	1	2	yes	—
288	C₂²×C₃	C₂×#709	1	1	yes	characteristic
289	C₂×D₃	—	10	4	no	—
290	C₂×D₃	—	40	2	no	—
291	C₃×C₄	C₄×D₂₀	1	2	yes	—
292	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
293	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
294	Dic₃	—	2	2	no	—
295	Dic₃	—	4	1	no	—
296	Dic₃	—	10	4	no	—
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	C₂×#893	1	1	yes	characteristic
300	D₅	—	12	4	no	—
301	C₂³	—	30	2	no	—
302	C₂³	#6769	1	1	yes	characteristic
303	Q₈	—	15	4	no	—
304	Q₈	—	30	2	no	—
305	D₄	—	15	4	no	—
306	D₄	—	30	4	no	—
307	D₄	—	30	4	no	—
308	D₄	—	30	2	no	—
309	C₂×C₄	—	6	1	no	—
310	C₂×C₄	—	6	1	no	—
311	C₂×C₄	—	12	2	no	—
312	C₂×C₄	—	30	2	no	—
313	C₂×C₄	—	30	4	no	—
314	C₂×C₄	—	30	4	no	—
315	C₂×C₄	#6769	1	1	yes	characteristic
316	C₂×C₄	D₃×D₂₀	1	2	yes	—
317	C₂×C₄	C₂×#1200	1	2	yes	—
318	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
319	C₈	—	3	4	no	—
320	C₈	—	10	4	no	—
321	C₈	C₂×#1200	1	4	yes	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	C₂×C₄×D₂₀	1	1	yes	characteristic
325	D₃	—	20	4	no	—
326	C₅	#79258	1	1	yes	characteristic, complemented, Sylow
327	C₂²	—	4	1	no	—
328	C₂²	—	30	4	no	—
329	C₂²	—	120	2	no	—
330	C₂²	#91428	1	2	yes	—
331	C₂²	C₂×#6769	1	1	yes	characteristic
332	C₄	—	6	2	no	—
333	C₄	—	12	1	no	—
334	C₄	—	30	4	no	—
335	C₄	#91428	1	2	yes	—
336	C₄	C₂×D₃×D₂₀	1	1	yes	characteristic
337	C₄	C₂²×#1200	1	1	yes	characteristic
338	C₃	#10610272	1	1	yes	characteristic, complemented, Sylow
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	—	60	4	no	—
342	C₂	C₂×#91428	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₃×D₂₀
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂²
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	#11756507	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×M₁₆	D₃×D₅
hypocenter	C₃×C₅	#2956
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. #11754130	C₂	semidirect
3. C₅×Dic₃×M₁₆	C₂	semidirect
4. C₃×#65414	C₂	semidirect
5. C₂×#90980	C₂	semidirect
6. #11748069	C₂	semidirect
7. #11748068	C₂	semidirect
11. #91798	C₄	semidirect
12. #91797	C₄	semidirect
18. #91291	C₂²	semidirect
20. C₅×#815	C₂²	semidirect
21. C₅×C₈×Dic₃	C₂²	semidirect
22. C₃×#3951	C₂²	semidirect
23. #90980	C₂²	semidirect
24. #90980	C₂²	semidirect
29. #65414	D₃	semidirect
45. Dic₆₀	C₂×C₄	semidirect
46. D₁₂₀	C₂×C₄	semidirect
47. #6706	C₂×C₄	semidirect
62. Dic₃×M₁₆	D₅	semidirect
78. #3951	C₂×D₃	semidirect
117. #815	C₂×D₅	semidirect
118. C₈×Dic₃	C₂×D₅	semidirect
285. C₃×C₅	#2956	semidirect
326. C₅	#79258	semidirect
338. C₃	#10610272	semidirect

Maximal subgroups

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

subgroup	index
#11756507	2
#11754130	2
C₅×Dic₃×M₁₆	2
C₃×#65414	2
C₂×#90980	2
#11748069	2
#11748068	2
#10610272	3
#79258	5