Subgroups — C₂×#11756512 #14510135

Order 1920.

2816 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#92101	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#11756512	C₂	1	16	yes	maximal, direct product factor
4	C₂×#91798	C₂	1	6	yes	maximal, complemented
5	C₂²×Dic₆₀	C₂	1	3	yes	maximal, complemented
6	C₂×#91793	C₂	1	3	yes	maximal, complemented
7	C₂×#66087	—	3	1	no	maximal, Hall
8	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
9	#92101	—	2	8	no	—
10	#92101	C₂²	1	16	yes	complemented
11	C₂×Q₈×D₁₅	—	2	2	no	—
12	C₂×#6819	—	2	6	no	—
13	C₂×#6817	C₂²	1	6	yes	—
14	C₂²×Dic₃₀	C₂²	1	6	yes	—
15	C₃×C₅×#131	C₂²	1	8	yes	complemented
16	C₂×C₃×C₅×M₁₆	C₂²	1	3	yes	—
17	C₂²×C₃×C₅×C₈	C₂²	1	3	yes	—
18	#91798	C₂²	1	48	yes	complemented
19	C₂×Dic₆₀	C₂²	1	3	yes	—
20	C₂×Dic₆₀	C₂²	1	6	yes	—
21	C₂×Dic₆₀	C₂²	1	24	yes	complemented
22	#91793	C₂²	1	24	yes	complemented
23	C₂×D₁₂₀	C₂²	1	1	yes	characteristic
24	C₂×#6706	C₂²	1	6	yes	—
25	C₂×#5753	—	5	1	no	maximal, Hall
26	C₂×#4150	—	3	2	no	—
27	C₂×C₅×#131	D₃	1	1	yes	characteristic, complemented
28	#66087	—	3	16	no	—
29	C₂×#4055	—	3	6	no	—
30	C₂²×Dic₂₀	—	3	3	no	—
31	C₂×#4050	—	3	3	no	—
32	C₃×C₅×#41	D₄	1	4	yes	—
33	C₃×C₅×#41	C₂³	1	4	yes	—
34	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×D₄	D₄	1	3	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
37	C₂×#1220	—	2	6	no	—
38	C₂²×Dic₁₅	—	2	6	no	—
39	Q₈×D₁₅	—	2	16	no	—
40	#6819	—	2	48	no	—
41	#6817	—	2	12	no	—
42	#6817	C₂³	1	24	yes	—
43	C₂×D₆₀	C₂³	1	2	yes	—
44	C₂×C₄×D₁₅	—	2	6	no	—
45	C₂×Dic₃₀	—	2	6	no	—
46	C₂×Dic₃₀	—	2	12	no	—
47	C₂×Dic₃₀	C₂³	1	6	yes	—
48	C₂×Dic₃₀	C₂³	1	24	yes	—
49	C₃×C₅×M₁₆	C₂³	1	12	yes	—
50	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
51	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₅×C₈	C₂³	1	12	yes	—
53	Dic₆₀	C₂³	1	12	yes	—
54	Dic₆₀	C₂³	1	24	yes	—
55	D₁₂₀	C₂³	1	4	yes	—
56	#6706	C₂³	1	24	yes	—
57	C₂×#1010	—	5	2	no	—
58	C₂×C₃×#131	D₅	1	1	yes	characteristic, complemented
59	#5753	—	5	16	no	—
60	C₂×#909	—	5	6	no	—
61	C₂²×Dic₁₂	—	5	3	no	—
62	C₂×#904	—	5	3	no	—
63	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
64	#4150	—	3	16	no	—
65	#4150	—	6	8	no	—
66	C₂×Q₈×D₅	—	6	2	no	—
67	C₂×#711	—	6	6	no	—
68	C₂×#709	—	3	6	no	—
69	C₂²×Dic₁₀	—	3	6	no	—
70	C₅×#131	C₂×D₃	1	8	yes	complemented
71	C₂×C₅×M₁₆	C₂×D₃	1	3	yes	—
72	C₂²×C₅×C₈	C₂×D₃	1	3	yes	—
73	#4055	—	3	48	no	—
74	C₂×Dic₂₀	—	3	3	no	—
75	C₂×Dic₂₀	—	3	6	no	—
76	C₂×Dic₂₀	—	3	24	no	—
77	#4050	—	3	24	no	—
78	C₂×D₄₀	—	3	1	no	—
79	C₂×#677	—	3	6	no	—
80	C₂×#578	—	15	1	no	Sylow
81	C₂³×C₃×C₅	C₂×D₄	1	3	yes	—
82	C₂²×D₁₅	—	2	2	no	—
83	C₃×C₅×Q₈	C₂×D₄	1	4	yes	—
84	C₃×C₅×D₄	C₂×D₄	1	12	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂⁴	1	6	yes	—
89	#1220	—	2	24	no	—
90	C₂×Dic₁₅	—	2	6	no	—
91	C₂×Dic₁₅	—	2	24	no	—
92	D₆₀	—	2	2	no	—
93	D₆₀	C₂⁴	1	4	yes	—
94	C₄×D₁₅	—	2	24	no	—
95	Dic₃₀	—	2	6	no	—
96	Dic₃₀	—	2	24	no	—
97	Dic₃₀	C₂⁴	1	12	yes	—
98	C₃×C₅×C₈	C₂⁴	1	6	yes	—
99	C₃×C₅×C₈	C₂⁴	1	2	yes	—
100	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
101	#1010	—	5	16	no	—
102	#1010	—	10	8	no	—
103	C₂×D₃×Q₈	—	10	2	no	—
104	C₂×#237	—	10	6	no	—
105	C₂×#235	—	5	6	no	—
106	C₂²×Dic₆	—	5	6	no	—
107	C₃×#131	C₂×D₅	1	8	yes	complemented
108	C₂×C₃×M₁₆	C₂×D₅	1	3	yes	—
109	C₂²×C₃×C₈	C₂×D₅	1	3	yes	—
110	#909	—	5	48	no	—
111	C₂×Dic₁₂	—	5	3	no	—
112	C₂×Dic₁₂	—	5	6	no	—
113	C₂×Dic₁₂	—	5	24	no	—
114	#904	—	5	24	no	—
115	C₂×D₂₄	—	5	1	no	—
116	C₂×#204	—	5	6	no	—
117	C₅×#41	D₁₂	1	4	yes	—
118	C₅×#41	C₂²×D₃	1	4	yes	—
119	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
120	C₂×C₅×D₄	D₁₂	1	3	yes	—
121	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
122	C₂×C₂²⋊D₅	—	6	6	no	—
123	C₂²×Dic₅	—	6	6	no	—
124	Q₈×D₅	—	6	16	no	—
125	#711	—	6	48	no	—
126	#709	—	3	24	no	—
127	#709	—	6	12	no	—
128	C₂×D₂₀	—	3	2	no	—
129	C₂×C₄×D₅	—	6	6	no	—
130	C₂×Dic₁₀	—	3	6	no	—
131	C₂×Dic₁₀	—	3	24	no	—
132	C₂×Dic₁₀	—	6	6	no	—
133	C₂×Dic₁₀	—	6	12	no	—
134	C₅×M₁₆	C₂²×D₃	1	12	yes	—
135	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₅×C₈	C₂²×D₃	1	3	yes	—
137	C₂×C₅×C₈	C₂²×D₃	1	12	yes	—
138	Dic₂₀	—	3	12	no	—
139	Dic₂₀	—	3	24	no	—
140	D₄₀	—	3	4	no	—
141	#677	—	3	24	no	—
142	C₂×2₋¹⁺⁴	—	15	2	no	—
143	#578	—	15	16	no	—
144	C₂×#137	—	15	6	no	—
145	C₂×#135	—	15	3	no	—
146	C₂²×Q₁₆	—	15	3	no	—
147	C₂×#131	D₁₅	1	1	yes	characteristic, complemented
148	C₂²×C₃×C₅	—	2	6	no	—
149	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
150	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₅	—	2	4	no	—
152	C₂×D₁₅	—	4	4	no	—
153	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₃×C₄×C₅	C₂²×D₄	1	6	yes	—
155	C₃×C₄×C₅	C₂⁵	1	1	yes	characteristic
156	Dic₁₅	—	2	12	no	—
157	C₃×#41	D₂₀	1	4	yes	—
158	C₃×#41	C₂²×D₅	1	4	yes	—
159	C₂×C₃×Q₈	D₂₀	1	1	yes	characteristic
160	C₂×C₃×D₄	D₂₀	1	3	yes	—
161	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
162	C₂×#67	—	10	6	no	—
163	C₂²×Dic₃	—	10	6	no	—
164	D₃×Q₈	—	10	16	no	—
165	#237	—	10	48	no	—
166	#235	—	5	24	no	—
167	#235	—	10	12	no	—
168	C₂×D₁₂	—	5	2	no	—
169	C₂×C₄×D₃	—	10	6	no	—
170	C₂×Dic₆	—	5	6	no	—
171	C₂×Dic₆	—	5	24	no	—
172	C₂×Dic₆	—	10	6	no	—
173	C₂×Dic₆	—	10	12	no	—
174	C₃×M₁₆	C₂²×D₅	1	12	yes	—
175	C₂×C₃×C₈	C₂²×D₅	1	3	yes	—
176	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₃×C₈	C₂²×D₅	1	12	yes	—
178	Dic₁₂	—	5	12	no	—
179	Dic₁₂	—	5	24	no	—
180	D₂₄	—	5	4	no	—
181	#204	—	5	24	no	—
182	C₂³×C₅	C₂×D₁₂	1	3	yes	—
183	C₂²×D₅	—	6	2	no	—
184	C₅×Q₈	C₂×D₁₂	1	4	yes	—
185	C₅×D₄	C₂×D₁₂	1	12	yes	—
186	C₂×C₄×C₅	C₂×D₁₂	1	3	yes	—
187	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
188	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂³×D₃	1	6	yes	—
190	C₂²⋊D₅	—	6	24	no	—
191	C₂×Dic₅	—	6	6	no	—
192	C₂×Dic₅	—	6	24	no	—
193	D₂₀	—	3	4	no	—
194	D₂₀	—	6	2	no	—
195	C₄×D₅	—	6	24	no	—
196	Dic₁₀	—	3	12	no	—
197	Dic₁₀	—	6	6	no	—
198	Dic₁₀	—	6	24	no	—
199	C₅×C₈	C₂³×D₃	1	6	yes	—
200	C₅×C₈	C₂³×D₃	1	2	yes	—
201	2₋¹⁺⁴	—	15	16	no	—
202	2₋¹⁺⁴	—	30	8	no	—
203	C₂×#41	—	15	6	no	—
204	C₂×#41	—	30	6	no	—
205	C₂×#41	C₂×D₁₅	1	1	yes	characteristic
206	C₂²×Q₈	—	15	6	no	—
207	C₂²×Q₈	—	30	2	no	—
208	#137	—	15	48	no	—
209	#135	—	15	24	no	—
210	C₂×Q₁₆	—	15	3	no	—
211	C₂×Q₁₆	—	15	6	no	—
212	C₂×Q₁₆	—	15	24	no	—
213	C₂×SD₁₆	—	15	6	no	—
214	C₂×D₈	—	15	1	no	—
215	#131	C₂×D₁₅	1	8	yes	complemented
216	C₂×M₁₆	C₂×D₁₅	1	3	yes	—
217	C₂²×C₈	C₂×D₁₅	1	3	yes	—
218	C₂×C₃×C₅	—	2	6	no	—
219	C₂×C₃×C₅	#578	1	2	yes	complemented
220	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
221	D₁₅	—	4	4	no	—
222	C₂³×C₃	C₂×D₂₀	1	3	yes	—
223	C₂²×D₃	—	10	2	no	—
224	C₃×Q₈	C₂×D₂₀	1	4	yes	—
225	C₃×D₄	C₂×D₂₀	1	12	yes	—
226	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
227	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
228	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂³×D₅	1	6	yes	—
230	#67	—	10	24	no	—
231	C₂×Dic₃	—	10	6	no	—
232	C₂×Dic₃	—	10	24	no	—
233	D₁₂	—	5	4	no	—
234	D₁₂	—	10	2	no	—
235	C₄×D₃	—	10	24	no	—
236	Dic₆	—	5	12	no	—
237	Dic₆	—	10	6	no	—
238	Dic₆	—	10	24	no	—
239	C₃×C₈	C₂³×D₅	1	6	yes	—
240	C₃×C₈	C₂³×D₅	1	2	yes	—
241	C₂²×C₅	—	2	6	no	—
242	C₂²×C₅	C₂²×D₁₂	1	6	yes	—
243	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
244	C₂×D₅	—	6	4	no	—
245	C₂×D₅	—	12	4	no	—
246	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
247	C₄×C₅	C₂²×D₁₂	1	6	yes	—
248	C₄×C₅	C₂⁴×D₃	1	1	yes	characteristic
249	Dic₅	—	6	12	no	—
250	#41	—	15	24	no	—
251	#41	—	30	48	no	—
252	#41	—	30	12	no	—
253	#41	D₆₀	1	4	yes	—
254	#41	C₂²×D₁₅	1	4	yes	—
255	C₂×Q₈	—	15	6	no	—
256	C₂×Q₈	—	15	24	no	—
257	C₂×Q₈	—	30	16	no	—
258	C₂×Q₈	—	30	6	no	—
259	C₂×Q₈	—	30	12	no	—
260	C₂×Q₈	D₆₀	1	1	yes	characteristic
261	C₂×D₄	—	15	2	no	—
262	C₂×D₄	—	30	6	no	—
263	C₂×D₄	D₆₀	1	3	yes	—
264	C₂²×C₄	—	30	6	no	—
265	C₂²×C₄	—	30	6	no	—
266	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
267	Q₁₆	—	15	12	no	—
268	Q₁₆	—	15	24	no	—
269	SD₁₆	—	15	24	no	—
270	D₈	—	15	4	no	—
271	M₁₆	C₂²×D₁₅	1	12	yes	—
272	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₁₅	1	3	yes	—
274	C₂×C₈	C₂²×D₁₅	1	12	yes	—
275	C₃×C₅	C₂×#578	1	1	yes	characteristic, complemented, Hall
276	C₂²×C₃	—	2	6	no	—
277	C₂²×C₃	C₂²×D₂₀	1	6	yes	—
278	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
279	C₂×D₃	—	10	4	no	—
280	C₂×D₃	—	20	4	no	—
281	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
282	C₃×C₄	C₂²×D₂₀	1	6	yes	—
283	C₃×C₄	C₂⁴×D₅	1	1	yes	characteristic
284	Dic₃	—	10	12	no	—
285	C₂×C₅	—	2	6	no	—
286	C₂×C₅	#5753	1	2	yes	complemented
287	C₂×C₅	C₂³×D₁₂	1	1	yes	characteristic
288	D₅	—	12	4	no	—
289	C₂³	—	30	2	no	—
290	C₂³	C₂×D₆₀	1	3	yes	—
291	Q₈	—	15	12	no	—
292	Q₈	—	30	24	no	—
293	Q₈	—	30	6	no	—
294	Q₈	C₂×D₆₀	1	4	yes	—
295	D₄	—	15	4	no	—
296	D₄	—	30	24	no	—
297	D₄	—	30	2	no	—
298	D₄	C₂×D₆₀	1	12	yes	—
299	C₂×C₄	—	30	24	no	—
300	C₂×C₄	—	30	6	no	—
301	C₂×C₄	—	30	24	no	—
302	C₂×C₄	C₂×D₆₀	1	6	yes	—
303	C₂×C₄	C₂×D₆₀	1	3	yes	—
304	C₂×C₄	C₂³×D₁₅	1	6	yes	—
305	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
306	C₈	C₂³×D₁₅	1	2	yes	—
307	C₈	C₂³×D₁₅	1	6	yes	—
308	C₂×C₃	—	2	6	no	—
309	C₂×C₃	#66087	1	2	yes	complemented
310	C₂×C₃	C₂³×D₂₀	1	1	yes	characteristic
311	D₃	—	20	4	no	—
312	C₅	C₂×#5753	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	6	no	—
314	C₂²	—	30	4	no	—
315	C₂²	—	60	4	no	—
316	C₂²	C₂²×D₆₀	1	6	yes	—
317	C₂²	C₂²×D₆₀	1	1	yes	characteristic
318	C₄	—	30	12	no	—
319	C₄	C₂²×D₆₀	1	6	yes	—
320	C₄	C₂²×D₆₀	1	1	yes	characteristic
321	C₄	C₂⁴×D₁₅	1	1	yes	characteristic
322	C₃	C₂×#66087	1	1	yes	characteristic, complemented, Sylow
323	C₂	—	2	6	no	—
324	C₂	—	60	4	no	—
325	C₂	#11756512	1	2	yes	direct product factor
326	C₂	C₂³×D₆₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₀
commutator, cosocle	C₃×C₄×C₅	C₂⁵
Frattini	C₄	C₂⁴×D₁₅
Fitting	C₂×C₃×C₅×#131	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×#131	D₁₅
hypocenter	C₃×C₅	C₂×#578
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#92101	C₂	semidirect
2. C₂×C₃×C₅×#131	C₂	semidirect
3. #11756512	C₂	direct (N × H)
4. C₂×#91798	C₂	semidirect
5. C₂²×Dic₆₀	C₂	semidirect
6. C₂×#91793	C₂	semidirect
10. #92101	C₂²	semidirect
15. C₃×C₅×#131	C₂²	semidirect
18. #91798	C₂²	semidirect
21. C₂×Dic₆₀	C₂²	semidirect
22. #91793	C₂²	semidirect
27. C₂×C₅×#131	D₃	semidirect
58. C₂×C₃×#131	D₅	semidirect
70. C₅×#131	C₂×D₃	semidirect
107. C₃×#131	C₂×D₅	semidirect
147. C₂×#131	D₁₅	semidirect
215. #131	C₂×D₁₅	semidirect
219. C₂×C₃×C₅	#578	semidirect
275. C₃×C₅	C₂×#578	semidirect
286. C₂×C₅	#5753	semidirect
309. C₂×C₃	#66087	semidirect
312. C₅	C₂×#5753	semidirect
322. C₃	C₂×#66087	semidirect
325. C₂	#11756512	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#92101	2
C₂×C₃×C₅×#131	2
#11756512	2
C₂×#91798	2
C₂²×Dic₆₀	2
C₂×#91793	2
C₂×#66087	3
C₂×#5753	5

Subgroups — C2×#11756512 #14510135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11756512 #14510135