Subgroups — C₂×#11750795 #14437643

Order 1920.

1056 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91763	C₂	1	1	yes	maximal, characteristic
2	C₂×#91758	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#91105	C₂	1	4	yes	maximal, complemented
5	#11750795	C₂	1	8	yes	maximal, direct product factor
6	C₂×#64983	—	3	1	no	maximal, Hall
7	C₂²×D₆₀	—	2	1	no	—
8	C₂²×D₆₀	—	2	1	no	—
9	C₂²×D₆₀	C₂²	1	2	yes	—
10	C₂²×Dic₃₀	—	2	1	no	—
11	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₅	—	2	1	no	—
14	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
15	#91763	—	2	2	no	—
16	#91763	C₂²	1	4	yes	—
17	#91758	—	2	2	no	—
18	#91758	C₂²	1	4	yes	complemented
19	C₃×C₄×C₅×C₈	C₂²	1	4	yes	complemented
20	#91105	C₂²	1	16	yes	complemented
21	C₂×#4694	—	5	1	no	maximal, Hall
22	C₂×#4020	—	3	1	no	—
23	C₂×#4015	—	3	1	no	—
24	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic, complemented
25	C₂×#3953	—	3	4	no	—
26	#64983	—	3	8	no	—
27	C₂³×D₁₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₅	—	2	2	no	—
32	C₂×D₆₀	—	2	8	no	—
33	C₂×D₆₀	—	2	2	no	—
34	C₂×D₆₀	—	2	4	no	—
35	C₂×D₆₀	—	2	4	no	—
36	C₂×D₆₀	—	4	4	no	—
37	C₂×D₆₀	—	4	4	no	—
38	C₂×D₆₀	C₂³	1	4	yes	—
39	C₂×Dic₃₀	—	2	4	no	—
40	C₂×Dic₃₀	—	4	4	no	—
41	C₂×C₃×C₅×C₈	—	2	4	no	—
42	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	2	yes	—
45	C₄⋊Dic₁₅	—	2	2	no	—
46	C₄⋊Dic₁₅	—	2	4	no	—
47	C₄⋊Dic₁₅	C₂³	1	4	yes	—
48	C₂×#874	—	5	1	no	—
49	C₂×#869	—	5	1	no	—
50	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic, complemented
51	C₂×#821	—	5	4	no	—
52	#4694	—	5	8	no	—
53	C₂²×D₂₀	—	3	2	no	—
54	C₂²×D₂₀	—	6	1	no	—
55	C₂²×D₂₀	—	6	1	no	—
56	C₂²×Dic₁₀	—	6	1	no	—
57	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
58	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
59	C₂×C₄⋊Dic₅	—	3	2	no	—
60	C₂×C₄⋊Dic₅	—	6	1	no	—
61	#4020	—	3	4	no	—
62	#4020	—	6	2	no	—
63	#4015	—	3	4	no	—
64	#4015	—	6	2	no	—
65	C₄×C₅×C₈	C₂×D₃	1	4	yes	complemented
66	#3953	—	3	16	no	—
67	C₂×#486	—	15	1	no	Sylow
68	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂²×D₁₅	—	2	4	no	—
70	C₂²×D₁₅	—	4	4	no	—
71	C₂²×D₁₅	—	4	4	no	—
72	C₂²×D₁₅	—	4	4	no	—
73	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
74	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	#41	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₁₅	—	2	4	no	—
81	C₂×Dic₁₅	—	4	4	no	—
82	D₆₀	—	2	8	no	—
83	D₆₀	—	4	4	no	—
84	D₆₀	—	4	8	no	—
85	D₆₀	—	4	8	no	—
86	Dic₃₀	—	4	8	no	—
87	C₃×C₅×C₈	—	2	4	no	—
88	C₂²×D₁₂	—	5	2	no	—
89	C₂²×D₁₂	—	10	1	no	—
90	C₂²×D₁₂	—	10	1	no	—
91	C₂²×Dic₆	—	10	1	no	—
92	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
93	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
94	C₂×C₄⋊Dic₃	—	5	2	no	—
95	C₂×C₄⋊Dic₃	—	10	1	no	—
96	#874	—	5	4	no	—
97	#874	—	10	2	no	—
98	#869	—	5	4	no	—
99	#869	—	10	2	no	—
100	C₃×C₄×C₈	C₂×D₅	1	4	yes	complemented
101	#821	—	5	16	no	—
102	C₂³×D₅	—	6	2	no	—
103	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
104	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
106	C₂²×Dic₅	—	6	2	no	—
107	C₂×D₂₀	—	3	4	no	—
108	C₂×D₂₀	—	6	8	no	—
109	C₂×D₂₀	—	6	2	no	—
110	C₂×D₂₀	—	6	4	no	—
111	C₂×D₂₀	—	6	4	no	—
112	C₂×D₂₀	—	12	4	no	—
113	C₂×D₂₀	—	12	4	no	—
114	C₂×Dic₁₀	—	6	4	no	—
115	C₂×Dic₁₀	—	12	4	no	—
116	C₂×C₅×C₈	—	2	4	no	—
117	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
118	C₄²×C₅	D₁₂	1	2	yes	—
119	C₄²×C₅	C₂²×D₃	1	2	yes	—
120	C₄⋊Dic₅	—	3	4	no	—
121	C₄⋊Dic₅	—	6	4	no	—
122	C₄⋊Dic₅	—	6	2	no	—
123	C₂×C₄⋊Q₈	—	15	1	no	—
124	C₂×#127	—	15	1	no	—
125	#486	—	15	8	no	—
126	C₂×#102	—	15	4	no	—
127	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic, complemented
128	C₂²×C₃×C₅	#124	1	2	yes	—
129	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
130	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₅	—	4	4	no	—
133	C₂×D₁₅	—	4	4	no	—
134	C₂×D₁₅	—	4	4	no	—
135	C₂×D₁₅	—	8	8	no	—
136	C₃×C₄×C₅	#124	1	2	yes	—
137	C₃×C₄×C₅	C₂×D₈	1	4	yes	—
138	C₃×C₄×C₅	C₂×SD₁₆	1	4	yes	—
139	C₃×C₄×C₅	C₂×#41	1	2	yes	—
140	Dic₁₅	—	4	4	no	—
141	C₂³×D₃	—	10	2	no	—
142	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
143	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
145	C₂²×Dic₃	—	10	2	no	—
146	C₂×D₁₂	—	5	4	no	—
147	C₂×D₁₂	—	10	8	no	—
148	C₂×D₁₂	—	10	4	no	—
149	C₂×D₁₂	—	10	4	no	—
150	C₂×D₁₂	—	10	2	no	—
151	C₂×D₁₂	—	20	4	no	—
152	C₂×D₁₂	—	20	4	no	—
153	C₂×Dic₆	—	10	4	no	—
154	C₂×Dic₆	—	20	4	no	—
155	C₂×C₃×C₈	—	2	4	no	—
156	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
157	C₃×C₄²	D₂₀	1	2	yes	—
158	C₃×C₄²	C₂²×D₅	1	2	yes	—
159	C₄⋊Dic₃	—	5	4	no	—
160	C₄⋊Dic₃	—	10	4	no	—
161	C₄⋊Dic₃	—	10	2	no	—
162	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
163	C₂²×D₅	—	6	4	no	—
164	C₂²×D₅	—	12	4	no	—
165	C₂²×D₅	—	12	4	no	—
166	C₂²×D₅	—	12	4	no	—
167	C₂×C₄×C₅	#204	1	4	yes	—
168	C₂×C₄×C₅	D₂₄	1	4	yes	—
169	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
171	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
172	C₂×C₄×C₅	#235	1	4	yes	—
173	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
174	C₂×Dic₅	—	6	4	no	—
175	C₂×Dic₅	—	12	4	no	—
176	D₂₀	—	6	8	no	—
177	D₂₀	—	12	4	no	—
178	D₂₀	—	12	8	no	—
179	D₂₀	—	12	8	no	—
180	Dic₁₀	—	12	8	no	—
181	C₅×C₈	—	2	4	no	—
182	C₂²×Q₈	—	30	1	no	—
183	C₂²×D₄	—	15	2	no	—
184	C₂²×D₄	—	30	1	no	—
185	C₂²×D₄	—	30	1	no	—
186	C₂²×C₈	C₂×D₁₅	1	2	yes	—
187	C₄⋊Q₈	—	15	4	no	—
188	C₄⋊Q₈	—	30	2	no	—
189	#127	—	15	4	no	—
190	#127	—	30	2	no	—
191	C₂×C₄⋊C₄	—	15	2	no	—
192	C₂×C₄⋊C₄	—	30	1	no	—
193	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
194	#102	—	15	16	no	—
195	C₄×C₈	C₂×D₁₅	1	4	yes	complemented
196	C₂×C₃×C₅	#486	1	4	yes	complemented
197	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
198	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
200	D₁₅	—	8	4	no	—
201	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
202	C₂²×D₃	—	10	4	no	—
203	C₂²×D₃	—	20	4	no	—
204	C₂²×D₃	—	20	4	no	—
205	C₂²×D₃	—	20	4	no	—
206	C₂×C₃×C₄	#677	1	4	yes	—
207	C₂×C₃×C₄	D₄₀	1	4	yes	—
208	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
210	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
211	C₂×C₃×C₄	#709	1	4	yes	—
212	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
213	C₂×Dic₃	—	10	4	no	—
214	C₂×Dic₃	—	20	4	no	—
215	D₁₂	—	10	8	no	—
216	D₁₂	—	20	4	no	—
217	D₁₂	—	20	8	no	—
218	D₁₂	—	20	8	no	—
219	Dic₆	—	20	8	no	—
220	C₃×C₈	—	2	4	no	—
221	C₂²×C₅	#875	1	2	yes	—
222	C₂²×C₅	C₂×#204	1	2	yes	—
223	C₂²×C₅	C₂×D₂₄	1	2	yes	—
224	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
225	C₂×D₅	—	12	4	no	—
226	C₂×D₅	—	12	4	no	—
227	C₂×D₅	—	12	4	no	—
228	C₂×D₅	—	24	8	no	—
229	C₄×C₅	#875	1	2	yes	—
230	C₄×C₅	C₂×#204	1	4	yes	—
231	C₄×C₅	C₂×D₂₄	1	4	yes	—
232	C₄×C₅	C₂×#235	1	2	yes	—
233	Dic₅	—	12	4	no	—
234	C₂⁴	—	30	2	no	—
235	C₂×Q₈	—	30	4	no	—
236	C₂×Q₈	—	60	4	no	—
237	C₂×D₄	—	15	4	no	—
238	C₂×D₄	—	30	4	no	—
239	C₂×D₄	—	30	4	no	—
240	C₂×D₄	—	30	2	no	—
241	C₂×D₄	—	30	8	no	—
242	C₂×D₄	—	60	4	no	—
243	C₂×D₄	—	60	4	no	—
244	C₂²×C₄	—	30	2	no	—
245	C₂²×C₄	D₆₀	1	1	yes	characteristic
246	C₂²×C₄	D₆₀	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	C₂²×D₁₅	1	4	yes	—
250	C₄⋊C₄	—	15	4	no	—
251	C₄⋊C₄	—	30	2	no	—
252	C₄⋊C₄	—	30	4	no	—
253	C₄²	D₆₀	1	2	yes	—
254	C₄²	C₂²×D₁₅	1	2	yes	—
255	C₃×C₅	C₂×#486	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	#4021	1	2	yes	—
257	C₂²×C₃	C₂×#677	1	2	yes	—
258	C₂²×C₃	C₂×D₄₀	1	2	yes	—
259	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
260	C₂×D₃	—	20	4	no	—
261	C₂×D₃	—	20	4	no	—
262	C₂×D₃	—	20	4	no	—
263	C₂×D₃	—	40	8	no	—
264	C₃×C₄	#4021	1	2	yes	—
265	C₃×C₄	C₂×#677	1	4	yes	—
266	C₃×C₄	C₂×D₄₀	1	4	yes	—
267	C₃×C₄	C₂×#709	1	2	yes	—
268	Dic₃	—	20	4	no	—
269	C₂×C₅	#4694	1	4	yes	complemented
270	C₂×C₅	C₂×#875	1	1	yes	characteristic
271	C₂×C₅	C₂²×#204	1	1	yes	characteristic
272	C₂×C₅	C₂²×D₂₄	1	1	yes	characteristic
273	D₅	—	24	4	no	—
274	C₂³	—	30	4	no	—
275	C₂³	—	60	4	no	—
276	C₂³	—	60	4	no	—
277	C₂³	—	60	4	no	—
278	C₂³	C₂×D₆₀	1	1	yes	characteristic
279	Q₈	—	60	8	no	—
280	D₄	—	30	8	no	—
281	D₄	—	60	4	no	—
282	D₄	—	60	8	no	—
283	D₄	—	60	8	no	—
284	C₂×C₄	—	30	4	no	—
285	C₂×C₄	—	60	4	no	—
286	C₂×C₄	#6706	1	4	yes	—
287	C₂×C₄	D₁₂₀	1	4	yes	—
288	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
289	C₂×C₄	C₂×D₆₀	1	2	yes	—
290	C₂×C₄	C₂×D₆₀	1	2	yes	—
291	C₂×C₄	#6817	1	4	yes	—
292	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
293	C₈	—	2	4	no	—
294	C₂×C₃	#64983	1	4	yes	complemented
295	C₂×C₃	C₂×#4021	1	1	yes	characteristic
296	C₂×C₃	C₂²×#677	1	1	yes	characteristic
297	C₂×C₃	C₂²×D₄₀	1	1	yes	characteristic
298	D₃	—	40	4	no	—
299	C₅	C₂×#4694	1	1	yes	characteristic, complemented, Sylow
300	C₂²	—	60	4	no	—
301	C₂²	—	60	4	no	—
302	C₂²	—	60	4	no	—
303	C₂²	—	120	8	no	—
304	C₂²	#91764	1	2	yes	—
305	C₂²	C₂×#6706	1	2	yes	—
306	C₂²	C₂×D₁₂₀	1	2	yes	—
307	C₂²	C₂²×D₆₀	1	1	yes	characteristic
308	C₄	—	60	4	no	—
309	C₄	#91764	1	2	yes	—
310	C₄	C₂×#6706	1	4	yes	—
311	C₄	C₂×D₁₂₀	1	4	yes	—
312	C₄	C₂×#6817	1	2	yes	—
313	C₃	C₂×#64983	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	120	4	no	—
315	C₂	#11750795	1	4	yes	direct product factor
316	C₂	C₂×#91764	1	1	yes	characteristic
317	C₂	C₂²×#6706	1	1	yes	characteristic
318	C₂	C₂²×D₁₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×#91763	C₂
hypercenter	C₂×C₄×C₈	D₁₅
hypocenter	C₃×C₅	C₂×#486
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#91758	C₂	semidirect
3. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
4. C₂×#91105	C₂	semidirect
5. #11750795	C₂	direct (N × H)
18. #91758	C₂²	semidirect
19. C₃×C₄×C₅×C₈	C₂²	semidirect
20. #91105	C₂²	semidirect
24. C₂×C₄×C₅×C₈	D₃	semidirect
50. C₂×C₃×C₄×C₈	D₅	semidirect
65. C₄×C₅×C₈	C₂×D₃	semidirect
100. C₃×C₄×C₈	C₂×D₅	semidirect
127. C₂×C₄×C₈	D₁₅	semidirect
195. C₄×C₈	C₂×D₁₅	semidirect
196. C₂×C₃×C₅	#486	semidirect
255. C₃×C₅	C₂×#486	semidirect
269. C₂×C₅	#4694	semidirect
294. C₂×C₃	#64983	semidirect
299. C₅	C₂×#4694	semidirect
313. C₃	C₂×#64983	semidirect
315. C₂	#11750795	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×#91763	2
C₂×#91758	2
C₂×C₃×C₄×C₅×C₈	2
C₂×#91105	2
#11750795	2
C₂×#64983	3
C₂×#4694	5

Subgroups — C2×#11750795 #14437643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11750795 #14437643