Subgroups — C₂×#11747815 #14416522

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×C₃×D₄₀	C₂	1	1	yes	maximal, characteristic
2	C₂×#91434	C₂	1	2	yes	maximal, complemented
3	C₂×#91099	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#817	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90964	C₂	1	2	yes	maximal, complemented
6	#11747815	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65158	—	3	1	no	maximal, Hall
8	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
9	C₂²×D₅×Dic₃	—	2	2	no	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×D₄₀	C₄	1	8	yes	complemented
15	C₂×C₃×D₄₀	C₂²	1	4	yes	—
16	C₂×#6665	—	2	2	no	—
17	#91434	—	2	4	no	—
18	#91434	C₂²	1	8	yes	complemented
19	#91099	C₂²	1	4	yes	complemented
20	C₅×#817	C₂²	1	4	yes	complemented
21	#90964	C₂²	1	8	yes	complemented
22	C₂×#5152	—	5	1	no	maximal, Hall
23	C₂²×D₄₀	D₃	1	1	yes	characteristic
24	C₂×#4039	—	3	2	no	—
25	C₂×C₅×#106	—	3	1	no	—
26	C₂×#3947	—	3	1	no	—
27	C₂×#3941	—	3	2	no	—
28	#65158	—	3	8	no	—
29	C₂³×C₃×D₅	—	2	2	no	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₅	D₄	1	2	yes	—
32	C₂²×C₅×Dic₃	—	2	2	no	—
33	C₂×C₃×D₂₀	—	2	2	no	—
34	C₂×C₃×D₂₀	—	2	4	no	—
35	C₂×C₃×D₂₀	C₂×C₄	1	8	yes	complemented
36	C₂×C₃×D₂₀	C₂³	1	4	yes	—
37	C₂×D₅×Dic₃	—	2	8	no	—
38	C₂×D₅×Dic₃	—	4	8	no	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
40	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
41	C₄×Dic₁₅	D₄	1	2	yes	—
42	C₄×Dic₁₅	C₂³	1	2	yes	—
43	C₂×C₃⋊#168	—	2	2	no	—
44	C₂×C₃⋊#168	C₂³	1	2	yes	—
45	C₅×C₄⋊Dic₃	—	2	2	no	—
46	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
47	C₃×D₄₀	C₂×C₄	1	16	yes	complemented
48	#6665	—	2	8	no	—
49	C₂²×C₃×D₈	—	5	1	no	—
50	C₂×D₄×Dic₃	—	5	2	no	—
51	C₂×#832	—	5	2	no	—
52	#5152	—	5	8	no	—
53	C₂×#817	D₅	1	1	yes	characteristic, complemented
54	C₂×#815	—	5	1	no	—
55	C₂²×D₂₀	C₂×D₃	1	2	yes	—
56	C₂²×C₄×D₅	—	6	2	no	—
57	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₅×C₄⋊C₄	—	3	2	no	—
59	C₂×#685	—	6	2	no	—
60	C₂×C₄×Dic₅	—	3	1	no	—
61	C₂²×#168	—	3	1	no	—
62	C₂×D₄₀	Dic₃	1	8	yes	complemented
63	C₂×D₄₀	C₂×D₃	1	4	yes	—
64	#4039	—	3	8	no	—
65	#4039	—	6	4	no	—
66	C₅×#106	—	3	4	no	—
67	#3947	—	3	4	no	—
68	#3941	—	3	8	no	—
69	C₂×#442	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₃×D₅	—	2	8	no	—
72	C₂²×C₃×D₅	—	2	4	no	—
73	C₂²×C₃×D₅	—	4	4	no	—
74	C₂²×C₃×D₅	—	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	C₂×Dic₁₅	—	2	4	no	—
80	C₂×Dic₁₅	C₂×D₄	1	4	yes	—
81	C₂×C₅×Dic₃	—	2	4	no	—
82	C₂×C₅×Dic₃	—	4	4	no	—
83	C₃×D₂₀	—	2	8	no	—
84	C₃×D₂₀	—	2	4	no	—
85	C₃×D₂₀	C₂²×C₄	1	8	yes	complemented
86	D₅×Dic₃	—	4	16	no	—
87	C₃×C₅×C₈	C₂²×C₄	1	4	yes	complemented
88	C₃⋊#168	—	2	2	no	—
89	C₂²×C₃×D₄	—	5	2	no	—
90	C₂³×Dic₃	—	10	2	no	—
91	C₂×C₃×D₈	—	5	8	no	—
92	C₂×C₃×D₈	—	5	4	no	—
93	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
94	C₂×#217	—	10	2	no	—
95	D₄×Dic₃	—	5	8	no	—
96	D₄×Dic₃	—	10	4	no	—
97	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
98	C₂×C₄×Dic₃	—	5	1	no	—
99	C₂²×C₃⋊C₈	—	5	1	no	—
100	#832	—	5	8	no	—
101	#817	C₂×D₅	1	4	yes	complemented
102	#815	—	5	4	no	—
103	C₂³×D₅	—	2	2	no	—
104	C₂²×C₄×C₅	—	6	2	no	—
105	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
106	C₂²×Dic₅	—	3	2	no	—
107	C₂×D₂₀	—	2	2	no	—
108	C₂×D₂₀	—	2	4	no	—
109	C₂×D₂₀	C₂×Dic₃	1	8	yes	complemented
110	C₂×D₂₀	C₂²×D₃	1	4	yes	—
111	C₂×C₄×D₅	—	6	8	no	—
112	C₂×C₄×D₅	—	12	8	no	—
113	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	complemented
114	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
115	C₅×C₄⋊C₄	—	3	4	no	—
116	C₅×C₄⋊C₄	—	6	2	no	—
117	#685	—	6	8	no	—
118	C₄×Dic₅	—	3	2	no	—
119	C₄×Dic₅	—	3	2	no	—
120	C₂×#168	—	3	2	no	—
121	C₂×#168	—	6	2	no	—
122	D₄₀	C₂×Dic₃	1	16	yes	complemented
123	C₂²×D₈	—	5	1	no	—
124	C₂×C₄×D₄	—	15	2	no	—
125	#442	—	15	8	no	—
126	C₂×#106	—	3	1	no	—
127	C₂×#102	—	15	2	no	—
128	C₂×#97	—	15	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×C₃×D₅	—	2	8	no	—
133	C₂×C₃×D₅	—	4	4	no	—
134	C₂×C₃×D₅	—	4	4	no	—
135	C₂×C₃×D₅	—	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	Dic₁₅	—	2	4	no	—
140	C₅×Dic₃	—	4	4	no	—
141	C₂⁴×C₃	—	10	2	no	—
142	C₂×C₃×D₄	—	5	8	no	—
143	C₂×C₃×D₄	—	5	4	no	—
144	C₂×C₃×D₄	—	10	2	no	—
145	C₂×C₃×D₄	—	10	4	no	—
146	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₂²×Dic₃	—	2	2	no	—
148	C₂²×Dic₃	—	5	2	no	—
149	C₂²×Dic₃	—	10	8	no	—
150	C₂²×Dic₃	—	20	8	no	—
151	C₃×D₈	—	5	16	no	—
152	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
153	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
154	#217	—	10	8	no	—
155	C₄⋊Dic₃	—	2	2	no	—
156	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
157	C₄×Dic₃	—	5	2	no	—
158	C₄×Dic₃	—	5	2	no	—
159	C₂×C₃⋊C₈	—	5	2	no	—
160	C₂×C₃⋊C₈	—	10	2	no	—
161	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
162	C₂²×D₅	—	2	8	no	—
163	C₂²×D₅	—	2	4	no	—
164	C₂²×D₅	—	4	4	no	—
165	C₂²×D₅	—	4	4	no	—
166	C₂×C₄×C₅	—	6	4	no	—
167	C₂×C₄×C₅	—	12	4	no	—
168	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅	#237	1	2	yes	—
170	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
171	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
172	C₂×Dic₅	—	3	4	no	—
173	C₂×Dic₅	—	6	4	no	—
174	D₂₀	—	2	8	no	—
175	D₂₀	—	2	4	no	—
176	D₂₀	C₂²×Dic₃	1	8	yes	complemented
177	C₄×D₅	—	12	16	no	—
178	C₅×C₈	C₂²×Dic₃	1	4	yes	complemented
179	#168	—	6	2	no	—
180	C₂²×D₄	—	5	2	no	—
181	C₂³×C₄	—	30	2	no	—
182	C₂×D₈	—	5	8	no	—
183	C₂×D₈	—	5	4	no	—
184	C₂²×C₈	—	15	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	—	3	2	no	—
189	C₂×C₂²⋊C₄	—	30	2	no	—
190	C₂×C₄²	—	15	1	no	—
191	#106	—	3	4	no	—
192	#102	—	15	8	no	—
193	#97	—	15	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	C₃×D₅	—	4	8	no	—
198	C₂³×C₃	—	10	4	no	—
199	C₂³×C₃	—	10	8	no	—
200	C₂³×C₃	—	20	4	no	—
201	C₂³×C₃	—	20	4	no	—
202	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
203	C₃×D₄	—	5	8	no	—
204	C₃×D₄	—	10	4	no	—
205	C₃×D₄	—	10	8	no	—
206	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
207	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
208	C₂×C₃×C₄	#713	1	2	yes	—
209	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
210	C₂×Dic₃	—	2	4	no	—
211	C₂×Dic₃	—	4	4	no	—
212	C₂×Dic₃	—	5	4	no	—
213	C₂×Dic₃	—	10	4	no	—
214	C₂×Dic₃	—	20	16	no	—
215	C₃×C₈	C₂×C₄×D₅	1	4	yes	complemented
216	C₃⋊C₈	—	10	2	no	—
217	C₂²×C₅	#911	1	4	yes	—
218	C₂²×C₅	D₄×Dic₃	1	2	yes	—
219	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
220	C₂×D₅	—	2	8	no	—
221	C₂×D₅	—	4	16	no	—
222	C₂×D₅	—	4	4	no	—
223	C₂×D₅	—	4	4	no	—
224	C₄×C₅	—	12	4	no	—
225	C₄×C₅	D₄×Dic₃	1	2	yes	—
226	C₄×C₅	C₂×#237	1	1	yes	characteristic
227	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
228	Dic₅	—	6	4	no	—
229	C₂⁴	—	10	2	no	—
230	C₂×D₄	—	5	8	no	—
231	C₂×D₄	—	5	4	no	—
232	C₂×D₄	—	10	4	no	—
233	C₂×D₄	—	10	2	no	—
234	C₂²×C₄	—	6	2	no	—
235	C₂²×C₄	—	15	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	—	5	16	no	—
240	C₂×C₈	—	15	2	no	—
241	C₂×C₈	—	30	2	no	—
242	C₂×C₈	D₅×Dic₃	1	4	yes	complemented
243	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
244	C₄⋊C₄	—	3	4	no	—
245	C₄⋊C₄	—	6	2	no	—
246	C₂²⋊C₄	—	30	8	no	—
247	C₄²	—	15	2	no	—
248	C₄²	—	15	2	no	—
249	C₃×C₅	C₂×#442	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	10	8	no	—
251	C₂²×C₃	—	20	4	no	—
252	C₂²×C₃	—	20	4	no	—
253	C₂²×C₃	—	20	16	no	—
254	C₂²×C₃	#4039	1	2	yes	—
255	C₂²×C₃	#4060	1	4	yes	—
256	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
257	C₃×C₄	#4039	1	2	yes	—
258	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
259	C₃×C₄	C₂×#713	1	1	yes	characteristic
260	Dic₃	—	4	4	no	—
261	Dic₃	—	10	4	no	—
262	C₂×C₅	#5152	1	4	yes	complemented
263	C₂×C₅	C₂×#911	1	2	yes	—
264	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
265	D₅	—	4	8	no	—
266	C₂³	—	10	8	no	—
267	C₂³	—	10	4	no	—
268	C₂³	—	20	4	no	—
269	C₂³	—	20	4	no	—
270	C₂³	#6777	1	1	yes	characteristic
271	D₄	—	5	8	no	—
272	D₄	—	10	8	no	—
273	D₄	—	10	4	no	—
274	C₂×C₄	—	6	4	no	—
275	C₂×C₄	—	12	4	no	—
276	C₂×C₄	—	15	4	no	—
277	C₂×C₄	—	30	4	no	—
278	C₂×C₄	—	60	16	no	—
279	C₂×C₄	#6766	1	2	yes	—
280	C₂×C₄	#6777	1	1	yes	characteristic
281	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
282	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
283	C₈	—	30	2	no	—
284	C₈	C₂×D₅×Dic₃	1	4	yes	complemented
285	C₂×C₃	—	20	8	no	—
286	C₂×C₃	#65158	1	4	yes	complemented
287	C₂×C₃	C₂×#4039	1	1	yes	characteristic
288	C₂×C₃	C₂×#4060	1	2	yes	—
289	C₅	C₂×#5152	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	10	8	no	—
291	C₂²	—	20	16	no	—
292	C₂²	—	20	4	no	—
293	C₂²	—	20	4	no	—
294	C₂²	#91258	1	4	yes	—
295	C₂²	#91434	1	2	yes	—
296	C₂²	C₂×#6777	1	1	yes	characteristic
297	C₄	—	12	4	no	—
298	C₄	—	30	4	no	—
299	C₄	#91434	1	2	yes	—
300	C₄	C₂×#6766	1	1	yes	characteristic
301	C₄	C₂²×D₅×Dic₃	1	1	yes	characteristic
302	C₃	C₂×#65158	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	20	8	no	—
304	C₂	#11747815	1	4	yes	direct product factor
305	C₂	C₂×#91258	1	2	yes	—
306	C₂	C₂×#91434	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6777
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×C₃×D₄₀	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×#91434	C₂	semidirect
3. C₂×#91099	C₂	semidirect
4. C₂×C₅×#817	C₂	semidirect
5. C₂×#90964	C₂	semidirect
6. #11747815	C₂	direct (N × H)
14. C₂×C₃×D₄₀	C₄	semidirect
18. #91434	C₂²	semidirect
19. #91099	C₂²	semidirect
20. C₅×#817	C₂²	semidirect
21. #90964	C₂²	semidirect
35. C₂×C₃×D₂₀	C₂×C₄	semidirect
39. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
47. C₃×D₄₀	C₂×C₄	semidirect
53. C₂×#817	D₅	semidirect
62. C₂×D₄₀	Dic₃	semidirect
85. C₃×D₂₀	C₂²×C₄	semidirect
87. C₃×C₅×C₈	C₂²×C₄	semidirect
101. #817	C₂×D₅	semidirect
109. C₂×D₂₀	C₂×Dic₃	semidirect
113. C₂×C₅×C₈	C₂×Dic₃	semidirect
122. D₄₀	C₂×Dic₃	semidirect
152. C₂×C₃×C₈	C₄×D₅	semidirect
176. D₂₀	C₂²×Dic₃	semidirect
178. C₅×C₈	C₂²×Dic₃	semidirect
194. C₂×C₃×C₅	#442	semidirect
215. C₃×C₈	C₂×C₄×D₅	semidirect
242. C₂×C₈	D₅×Dic₃	semidirect
249. C₃×C₅	C₂×#442	semidirect
262. C₂×C₅	#5152	semidirect
284. C₈	C₂×D₅×Dic₃	semidirect
286. C₂×C₃	#65158	semidirect
289. C₅	C₂×#5152	semidirect
302. C₃	C₂×#65158	semidirect
304. C₂	#11747815	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×C₃×D₄₀	2
C₂×#91434	2
C₂×#91099	2
C₂×C₅×#817	2
C₂×#90964	2
#11747815	2
C₂×#65158	3
C₂×#5152	5

Subgroups — C2×#11747815 #14416522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11747815 #14416522