Subgroups — C₂×#11747796 #14416521

Order 1920.

1216 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂×#91428	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#815	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3949	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90968	C₂	1	2	yes	maximal, complemented
6	#11747796	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65000	—	3	1	no	maximal, Hall
8	C₂²×D₆₀	C₂²	1	2	yes	—
9	C₂²×#1200	—	2	2	no	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×D₁₂₀	C₄	1	8	yes	complemented
12	C₂×D₁₂₀	C₂²	1	4	yes	—
13	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
16	C₂×#6667	—	2	2	no	—
17	#91428	—	2	4	no	—
18	#91428	C₂²	1	8	yes	complemented
19	C₅×#815	C₂²	1	4	yes	complemented
20	C₃×#3949	C₂²	1	4	yes	complemented
21	#90968	C₂²	1	8	yes	complemented
22	C₂×#4869	—	5	1	no	maximal, Hall
23	C₂²×D₄₀	—	3	1	no	—
24	C₂×C₄×D₂₀	—	3	2	no	—
25	C₂×C₅×#97	—	3	1	no	—
26	C₂×#3953	—	3	2	no	—
27	C₂×#3949	D₃	1	1	yes	characteristic, complemented
28	#65000	—	3	8	no	—
29	C₂³×D₁₅	—	2	2	no	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂×D₆₀	—	2	2	no	—
32	C₂×D₆₀	—	2	4	no	—
33	C₂×D₆₀	C₂×C₄	1	8	yes	complemented
34	C₂×D₆₀	C₂³	1	4	yes	—
35	C₂²×C₅×Dic₃	D₄	1	2	yes	—
36	C₂²×C₃×Dic₅	—	2	2	no	—
37	C₂×#1200	—	2	8	no	—
38	C₂×#1200	—	4	8	no	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
40	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
41	D₁₂₀	C₂×C₄	1	16	yes	complemented
42	C₄×C₅×Dic₃	D₄	1	2	yes	—
43	C₄×C₅×Dic₃	C₂³	1	2	yes	—
44	C₂×C₅×C₃⋊C₈	—	2	2	no	—
45	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
46	C₃×C₄⋊Dic₅	—	2	2	no	—
47	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
48	#6667	—	2	8	no	—
49	C₂²×D₂₄	—	5	1	no	—
50	C₂×#893	—	5	2	no	—
51	C₂×C₃×#106	—	5	1	no	—
52	C₂×#815	D₅	1	1	yes	characteristic, complemented
53	C₂×#809	—	5	2	no	—
54	#4869	—	5	8	no	—
55	C₂²×D₂₀	—	3	2	no	—
56	C₂²×C₄×D₅	—	6	2	no	—
57	C₂²×C₅×C₈	—	3	1	no	—
58	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂×C₄²×C₅	—	3	1	no	—
60	C₂×#685	—	6	2	no	—
61	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
62	C₂×D₄₀	—	3	8	no	—
63	C₂×D₄₀	—	3	4	no	—
64	C₄×D₂₀	—	3	8	no	—
65	C₄×D₂₀	—	6	4	no	—
66	C₅×#97	—	3	4	no	—
67	#3953	—	3	8	no	—
68	#3949	C₂×D₃	1	4	yes	complemented
69	C₂×#442	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×D₁₅	—	2	8	no	—
72	C₂²×D₁₅	—	2	4	no	—
73	C₂²×D₁₅	—	4	4	no	—
74	C₂²×D₁₅	—	4	4	no	—
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	#41	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	D₆₀	—	2	8	no	—
80	D₆₀	—	2	4	no	—
81	D₆₀	C₂²×C₄	1	8	yes	complemented
82	C₂×C₅×Dic₃	—	2	4	no	—
83	C₂×C₅×Dic₃	C₂×D₄	1	4	yes	—
84	C₂×C₃×Dic₅	—	2	4	no	—
85	C₂×C₃×Dic₅	—	4	4	no	—
86	#1200	—	4	16	no	—
87	C₃×C₅×C₈	C₂²×C₄	1	4	yes	complemented
88	C₅×C₃⋊C₈	—	2	2	no	—
89	C₂²×D₁₂	—	5	2	no	—
90	C₂²×C₄×D₃	—	10	2	no	—
91	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
92	C₂×C₃×C₄⋊C₄	—	5	2	no	—
93	C₂×(C₂×D₃)⋊C₄	—	10	2	no	—
94	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
95	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
96	C₂×D₂₄	—	5	8	no	—
97	C₂×D₂₄	—	5	4	no	—
98	#893	—	5	8	no	—
99	#893	—	10	4	no	—
100	C₃×#106	—	5	4	no	—
101	#815	C₂×D₅	1	4	yes	complemented
102	#809	—	5	8	no	—
103	C₂³×D₅	—	6	2	no	—
104	C₂²×C₄×C₅	—	3	2	no	—
105	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
106	C₂²×Dic₅	—	2	2	no	—
107	C₂×D₂₀	—	3	8	no	—
108	C₂×D₂₀	—	3	4	no	—
109	C₂×D₂₀	—	6	2	no	—
110	C₂×D₂₀	—	6	4	no	—
111	C₂×C₄×D₅	—	6	8	no	—
112	C₂×C₄×D₅	—	12	8	no	—
113	C₂×C₅×C₈	—	3	2	no	—
114	C₂×C₅×C₈	—	6	2	no	—
115	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
116	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
117	C₄²×C₅	—	3	2	no	—
118	C₄²×C₅	—	3	2	no	—
119	#685	—	6	8	no	—
120	C₄⋊Dic₅	—	2	2	no	—
121	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
122	D₄₀	—	3	16	no	—
123	C₂²×D₈	—	15	1	no	—
124	C₂×C₄×D₄	—	15	2	no	—
125	#442	—	15	8	no	—
126	C₂×#106	—	5	1	no	—
127	C₂×#102	—	15	2	no	—
128	C₂×#97	—	3	1	no	—
129	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
130	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₅	—	2	8	no	—
133	C₂×D₁₅	—	4	4	no	—
134	C₂×D₁₅	—	4	4	no	—
135	C₂×D₁₅	—	4	16	no	—
136	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
138	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
139	C₃×Dic₅	—	4	4	no	—
140	C₅×Dic₃	—	2	4	no	—
141	C₂³×D₃	—	10	2	no	—
142	C₂²×C₃×C₄	—	10	2	no	—
143	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
144	C₂²×Dic₃	D₂₀	1	2	yes	—
145	C₂×D₁₂	—	5	8	no	—
146	C₂×D₁₂	—	5	4	no	—
147	C₂×D₁₂	—	10	2	no	—
148	C₂×D₁₂	—	10	4	no	—
149	C₂×C₄×D₃	—	10	8	no	—
150	C₂×C₄×D₃	—	20	8	no	—
151	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
152	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
153	C₃×C₄⋊C₄	—	5	4	no	—
154	C₃×C₄⋊C₄	—	10	2	no	—
155	(C₂×D₃)⋊C₄	—	10	8	no	—
156	C₄×Dic₃	D₂₀	1	2	yes	—
157	C₄×Dic₃	C₂²×D₅	1	2	yes	—
158	C₂×C₃⋊C₈	—	2	2	no	—
159	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
160	D₂₄	—	5	16	no	—
161	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
162	C₂²×D₅	—	6	8	no	—
163	C₂²×D₅	—	6	4	no	—
164	C₂²×D₅	—	12	4	no	—
165	C₂²×D₅	—	12	4	no	—
166	C₂×C₄×C₅	—	3	4	no	—
167	C₂×C₄×C₅	—	6	4	no	—
168	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
169	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	#239	1	2	yes	—
171	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
172	C₂×Dic₅	—	2	4	no	—
173	C₂×Dic₅	—	4	4	no	—
174	D₂₀	—	3	8	no	—
175	D₂₀	—	6	8	no	—
176	D₂₀	—	6	4	no	—
177	C₄×D₅	—	12	16	no	—
178	C₅×C₈	—	6	2	no	—
179	C₅×C₈	C₂×C₄×D₃	1	4	yes	complemented
180	C₂²×D₄	—	15	2	no	—
181	C₂³×C₄	—	30	2	no	—
182	C₂×D₈	—	15	8	no	—
183	C₂×D₈	—	15	4	no	—
184	C₂²×C₈	—	3	1	no	—
185	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
186	C₄×D₄	—	15	8	no	—
187	C₄×D₄	—	30	4	no	—
188	C₂×C₄⋊C₄	—	5	2	no	—
189	C₂×C₂²⋊C₄	—	30	2	no	—
190	C₂×C₄²	—	3	1	no	—
191	#106	—	5	4	no	—
192	#102	—	15	8	no	—
193	#97	—	3	4	no	—
194	C₂×C₃×C₅	#442	1	4	yes	complemented
195	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
197	D₁₅	—	4	8	no	—
198	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
199	C₂²×D₃	—	10	4	no	—
200	C₂²×D₃	—	10	8	no	—
201	C₂²×D₃	—	20	4	no	—
202	C₂²×D₃	—	20	4	no	—
203	C₂×C₃×C₄	—	10	4	no	—
204	C₂×C₃×C₄	—	20	4	no	—
205	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
206	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
207	C₂×C₃×C₄	#709	1	2	yes	—
208	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
209	C₂×Dic₃	—	2	4	no	—
210	C₂×Dic₃	C₂×D₂₀	1	4	yes	—
211	D₁₂	—	5	8	no	—
212	D₁₂	—	10	4	no	—
213	D₁₂	—	10	8	no	—
214	C₄×D₃	—	20	16	no	—
215	C₃×C₈	C₂×C₄×D₅	1	4	yes	complemented
216	C₃⋊C₈	—	2	2	no	—
217	C₂²×C₅	#893	1	2	yes	—
218	C₂²×C₅	#914	1	4	yes	—
219	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
220	C₂×D₅	—	6	8	no	—
221	C₂×D₅	—	12	16	no	—
222	C₂×D₅	—	12	4	no	—
223	C₂×D₅	—	12	4	no	—
224	C₄×C₅	—	6	4	no	—
225	C₄×C₅	#893	1	2	yes	—
226	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
227	C₄×C₅	C₂×#239	1	1	yes	characteristic
228	Dic₅	—	4	4	no	—
229	C₂⁴	—	30	2	no	—
230	C₂×D₄	—	15	8	no	—
231	C₂×D₄	—	15	4	no	—
232	C₂×D₄	—	30	4	no	—
233	C₂×D₄	—	30	2	no	—
234	C₂²×C₄	—	3	2	no	—
235	C₂²×C₄	—	10	2	no	—
236	C₂²×C₄	—	30	8	no	—
237	C₂²×C₄	—	60	8	no	—
238	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
239	D₈	—	15	16	no	—
240	C₂×C₈	—	3	2	no	—
241	C₂×C₈	—	6	2	no	—
242	C₂×C₈	#1200	1	4	yes	complemented
243	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
244	C₄⋊C₄	—	5	4	no	—
245	C₄⋊C₄	—	10	2	no	—
246	C₂²⋊C₄	—	30	8	no	—
247	C₄²	—	3	2	no	—
248	C₄²	—	3	2	no	—
249	C₃×C₅	C₂×#442	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	C₄×D₂₀	1	2	yes	—
251	C₂²×C₃	#4054	1	4	yes	—
252	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
253	C₂×D₃	—	10	8	no	—
254	C₂×D₃	—	20	4	no	—
255	C₂×D₃	—	20	4	no	—
256	C₂×D₃	—	20	16	no	—
257	C₃×C₄	—	20	4	no	—
258	C₃×C₄	C₄×D₂₀	1	2	yes	—
259	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
260	C₃×C₄	C₂×#709	1	1	yes	characteristic
261	Dic₃	—	2	4	no	—
262	C₂×C₅	#4869	1	4	yes	complemented
263	C₂×C₅	C₂×#893	1	1	yes	characteristic
264	C₂×C₅	C₂×#914	1	2	yes	—
265	D₅	—	12	8	no	—
266	C₂³	—	30	8	no	—
267	C₂³	—	30	4	no	—
268	C₂³	—	60	4	no	—
269	C₂³	—	60	4	no	—
270	C₂³	D₃×D₂₀	1	1	yes	characteristic
271	D₄	—	15	8	no	—
272	D₄	—	30	8	no	—
273	D₄	—	30	4	no	—
274	C₂×C₄	—	3	4	no	—
275	C₂×C₄	—	6	4	no	—
276	C₂×C₄	—	10	4	no	—
277	C₂×C₄	—	20	4	no	—
278	C₂×C₄	—	60	16	no	—
279	C₂×C₄	#6769	1	2	yes	—
280	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
281	C₂×C₄	C₂×#1200	1	2	yes	—
282	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
283	C₈	—	6	2	no	—
284	C₈	C₂×#1200	1	4	yes	complemented
285	C₂×C₃	#65000	1	4	yes	complemented
286	C₂×C₃	C₂×C₄×D₂₀	1	1	yes	characteristic
287	C₂×C₃	C₂×#4054	1	2	yes	—
288	D₃	—	20	8	no	—
289	C₅	C₂×#4869	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	30	8	no	—
291	C₂²	—	60	16	no	—
292	C₂²	—	60	4	no	—
293	C₂²	—	60	4	no	—
294	C₂²	#91253	1	4	yes	—
295	C₂²	#91428	1	2	yes	—
296	C₂²	C₂×D₃×D₂₀	1	1	yes	characteristic
297	C₄	—	6	4	no	—
298	C₄	—	20	4	no	—
299	C₄	#91428	1	2	yes	—
300	C₄	C₂×#6769	1	1	yes	characteristic
301	C₄	C₂²×#1200	1	1	yes	characteristic
302	C₃	C₂×#65000	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	60	8	no	—
304	C₂	#11747796	1	4	yes	direct product factor
305	C₂	C₂×#91253	1	2	yes	—
306	C₂	C₂×#91428	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃×D₂₀
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×D₁₂₀	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#442
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#91428	C₂	semidirect
3. C₂×C₅×#815	C₂	semidirect
4. C₂×C₃×#3949	C₂	semidirect
5. C₂×#90968	C₂	semidirect
6. #11747796	C₂	direct (N × H)
11. C₂×D₁₂₀	C₄	semidirect
18. #91428	C₂²	semidirect
19. C₅×#815	C₂²	semidirect
20. C₃×#3949	C₂²	semidirect
21. #90968	C₂²	semidirect
27. C₂×#3949	D₃	semidirect
33. C₂×D₆₀	C₂×C₄	semidirect
39. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
41. D₁₂₀	C₂×C₄	semidirect
52. C₂×#815	D₅	semidirect
68. #3949	C₂×D₃	semidirect
81. D₆₀	C₂²×C₄	semidirect
87. C₃×C₅×C₈	C₂²×C₄	semidirect
101. #815	C₂×D₅	semidirect
115. C₂×C₅×C₈	C₄×D₃	semidirect
151. C₂×C₃×C₈	C₄×D₅	semidirect
179. C₅×C₈	C₂×C₄×D₃	semidirect
194. C₂×C₃×C₅	#442	semidirect
215. C₃×C₈	C₂×C₄×D₅	semidirect
242. C₂×C₈	#1200	semidirect
249. C₃×C₅	C₂×#442	semidirect
262. C₂×C₅	#4869	semidirect
284. C₈	C₂×#1200	semidirect
285. C₂×C₃	#65000	semidirect
289. C₅	C₂×#4869	semidirect
302. C₃	C₂×#65000	semidirect
304. C₂	#11747796	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₂₀	2
C₂×#91428	2
C₂×C₅×#815	2
C₂×C₃×#3949	2
C₂×#90968	2
#11747796	2
C₂×#65000	3
C₂×#4869	5

Subgroups — C2×#11747796 #14416521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11747796 #14416521