Subgroups — C₂×#11747066 #14438754

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91758	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#874	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6659	C₂	1	4	yes	maximal, complemented
4	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#11747066	C₂	1	8	yes	maximal, direct product factor
6	C₂×#65365	—	3	1	no	maximal, Hall
7	C₂²×Dic₃₀	C₂²	1	2	yes	—
8	C₂²×C₅×D₁₂	—	2	2	no	—
9	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₅	—	2	2	no	—
12	#91758	—	2	2	no	—
13	#91758	C₂²	1	4	yes	complemented
14	C₅×#874	—	2	2	no	—
15	C₅×#874	C₂²	1	4	yes	—
16	C₂²×C₃×#168	C₂²	1	2	yes	—
17	C₂×#6659	—	2	16	no	—
18	C₂×#6659	C₂²	1	16	yes	complemented
19	C₃×C₄×#168	C₂²	1	4	yes	complemented
20	C₂×#4693	—	5	1	no	maximal, Hall
21	C₂×C₅×#127	—	3	1	no	—
22	C₂²×#687	—	3	4	no	—
23	C₂×#4015	—	3	1	no	—
24	#65365	—	3	8	no	—
25	C₂×C₄×#168	D₃	1	1	yes	characteristic, complemented
26	C₂³×C₅×D₃	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₅	—	2	2	no	—
30	C₂×Dic₃₀	—	2	8	no	—
31	C₂×Dic₃₀	—	2	2	no	—
32	C₂×Dic₃₀	C₂³	1	4	yes	—
33	C₂×C₅×D₁₂	—	2	8	no	—
34	C₂×C₅×D₁₂	—	2	2	no	—
35	C₂×C₅×D₁₂	—	2	8	no	—
36	C₂×C₅×D₁₂	—	4	8	no	—
37	C₂×C₅×D₁₂	C₂³	1	4	yes	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	2	yes	—
40	C₄⋊Dic₁₅	—	2	8	no	—
41	C₂×C₃×#168	D₄	1	8	yes	complemented
42	C₂×C₃×#168	C₂³	1	4	yes	—
43	#6659	—	2	32	no	—
44	C₂²×#204	—	5	4	no	—
45	C₂×#874	D₅	1	1	yes	characteristic
46	C₂×#869	—	5	1	no	—
47	C₂×C₃×C₄×C₈	—	5	1	no	—
48	#4693	—	5	8	no	—
49	C₂²×C₅×D₄	—	3	2	no	—
50	C₂²×C₅×D₄	—	6	2	no	—
51	C₂²×Dic₁₀	—	3	2	no	—
52	C₅×#127	—	3	4	no	—
53	C₅×#127	—	6	2	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂×#687	—	3	16	no	—
56	C₂×#687	—	6	16	no	—
57	C₂×C₄⋊Dic₅	—	6	2	no	—
58	C₂²×#168	C₂×D₃	1	2	yes	—
59	#4015	—	3	4	no	—
60	#4015	—	6	2	no	—
61	C₄×#168	C₂×D₃	1	4	yes	complemented
62	C₂×#492	—	15	1	no	Sylow
63	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₅×D₃	—	2	4	no	—
65	C₂²×C₅×D₃	—	4	4	no	—
66	C₂²×C₅×D₃	—	4	8	no	—
67	C₂×C₃×C₄×C₅	SD₁₆	1	8	yes	complemented
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
71	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₁₅	—	2	4	no	—
73	C₂×Dic₁₅	—	4	4	no	—
74	Dic₃₀	—	2	8	no	—
75	Dic₃₀	—	4	4	no	—
76	C₅×D₁₂	—	2	8	no	—
77	C₅×D₁₂	—	4	4	no	—
78	C₅×D₁₂	—	4	16	no	—
79	C₃×#168	C₂×D₄	1	8	yes	complemented
80	C₂²×D₁₂	—	2	2	no	—
81	C₂²×D₁₂	C₂×D₅	1	2	yes	—
82	C₂²×Dic₆	—	5	2	no	—
83	C₂²×C₃×C₈	—	5	2	no	—
84	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
85	C₂×C₄⋊Dic₃	—	10	2	no	—
86	C₂×#204	—	5	16	no	—
87	C₂×#204	—	10	16	no	—
88	#874	—	2	2	no	—
89	#874	C₂×D₅	1	4	yes	—
90	#869	—	5	4	no	—
91	#869	—	10	2	no	—
92	C₃×C₄×C₈	—	5	4	no	—
93	C₂⁴×C₅	—	6	2	no	—
94	C₂×C₅×D₄	—	3	4	no	—
95	C₂×C₅×D₄	—	6	8	no	—
96	C₂×C₅×D₄	—	6	8	no	—
97	C₂×C₅×D₄	—	6	2	no	—
98	C₂×C₅×D₄	—	12	8	no	—
99	C₂²×C₄×C₅	D₁₂	1	2	yes	—
100	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
101	C₂²×Dic₅	—	6	2	no	—
102	C₂×Dic₁₀	—	3	4	no	—
103	C₂×Dic₁₀	—	6	8	no	—
104	C₂×Dic₁₀	—	6	2	no	—
105	C₄²×C₅	D₁₂	1	2	yes	—
106	C₄²×C₅	C₂²×D₃	1	2	yes	—
107	#687	—	6	32	no	—
108	C₄⋊Dic₅	—	6	8	no	—
109	C₂×#168	D₁₂	1	8	yes	complemented
110	C₂×#168	C₂²×D₃	1	4	yes	—
111	C₂²×SD₁₆	—	15	4	no	—
112	C₂×C₄⋊Q₈	—	15	1	no	—
113	C₂×#127	—	3	1	no	—
114	#492	—	15	8	no	—
115	C₂×C₄×C₈	—	5	1	no	—
116	C₂²×C₃×C₅	#127	1	2	yes	—
117	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
118	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₅×D₃	—	4	4	no	—
120	C₂×C₅×D₃	—	4	8	no	—
121	C₂×C₅×D₃	—	8	8	no	—
122	C₃×C₄×C₅	#127	1	2	yes	—
123	C₃×C₄×C₅	C₂×SD₁₆	1	8	yes	complemented
124	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
125	Dic₁₅	—	4	4	no	—
126	C₂³×D₃	—	2	2	no	—
127	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
128	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
129	C₂²×Dic₃	—	10	2	no	—
130	C₂×D₁₂	—	2	8	no	—
131	C₂×D₁₂	—	2	2	no	—
132	C₂×D₁₂	—	2	8	no	—
133	C₂×D₁₂	—	4	8	no	—
134	C₂×D₁₂	C₂²×D₅	1	4	yes	—
135	C₂×Dic₆	—	5	4	no	—
136	C₂×Dic₆	—	10	8	no	—
137	C₂×Dic₆	—	10	2	no	—
138	C₂×C₃×C₈	—	5	8	no	—
139	C₂×C₃×C₈	—	5	4	no	—
140	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
141	C₃×C₄²	C₂²×D₅	1	2	yes	—
142	C₄⋊Dic₃	—	10	8	no	—
143	#204	—	10	32	no	—
144	C₂³×C₅	—	6	4	no	—
145	C₂³×C₅	—	12	4	no	—
146	C₂³×C₅	—	12	8	no	—
147	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
148	C₅×D₄	—	6	8	no	—
149	C₅×D₄	—	12	4	no	—
150	C₅×D₄	—	12	16	no	—
151	C₂×C₄×C₅	#204	1	8	yes	complemented
152	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
153	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
155	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
156	C₂×Dic₅	—	6	4	no	—
157	C₂×Dic₅	—	12	4	no	—
158	Dic₁₀	—	6	8	no	—
159	Dic₁₀	—	12	4	no	—
160	#168	C₂×D₁₂	1	8	yes	complemented
161	C₂²×Q₈	—	15	2	no	—
162	C₂²×D₄	—	3	2	no	—
163	C₂²×D₄	—	6	2	no	—
164	C₂×SD₁₆	—	15	16	no	—
165	C₂×SD₁₆	—	30	16	no	—
166	C₂²×C₈	—	5	2	no	—
167	C₄⋊Q₈	—	15	4	no	—
168	C₄⋊Q₈	—	30	2	no	—
169	#127	—	3	4	no	—
170	#127	—	6	2	no	—
171	C₂×C₄⋊C₄	—	30	2	no	—
172	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
173	C₄×C₈	—	5	4	no	—
174	C₂×C₃×C₅	#492	1	4	yes	complemented
175	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
176	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
177	C₅×D₃	—	8	4	no	—
178	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
179	C₂²×D₃	—	2	4	no	—
180	C₂²×D₃	—	4	8	no	—
181	C₂²×D₃	—	4	4	no	—
182	C₂×C₃×C₄	#687	1	8	yes	complemented
183	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
184	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
185	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
187	C₂×Dic₃	—	10	4	no	—
188	C₂×Dic₃	—	20	4	no	—
189	D₁₂	—	2	8	no	—
190	D₁₂	—	4	16	no	—
191	D₁₂	—	4	4	no	—
192	Dic₆	—	10	8	no	—
193	Dic₆	—	20	4	no	—
194	C₃×C₈	—	5	8	no	—
195	C₂²×C₅	—	12	4	no	—
196	C₂²×C₅	—	12	8	no	—
197	C₂²×C₅	—	24	8	no	—
198	C₂²×C₅	#874	1	2	yes	—
199	C₂²×C₅	C₂×#204	1	4	yes	—
200	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
201	C₄×C₅	#874	1	2	yes	—
202	C₄×C₅	C₂×#204	1	8	yes	complemented
203	C₄×C₅	C₂²×D₁₂	1	2	yes	—
204	Dic₅	—	12	4	no	—
205	C₂⁴	—	6	2	no	—
206	C₂×Q₈	—	15	4	no	—
207	C₂×Q₈	—	30	2	no	—
208	C₂×Q₈	—	30	8	no	—
209	C₂×D₄	—	3	4	no	—
210	C₂×D₄	—	6	8	no	—
211	C₂×D₄	—	6	8	no	—
212	C₂×D₄	—	6	2	no	—
213	C₂×D₄	—	12	8	no	—
214	C₂²×C₄	—	30	2	no	—
215	C₂²×C₄	#1203	1	2	yes	—
216	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
217	SD₁₆	—	30	32	no	—
218	C₂×C₈	—	5	8	no	—
219	C₂×C₈	—	5	4	no	—
220	C₄⋊C₄	—	30	8	no	—
221	C₄²	#1203	1	2	yes	—
222	C₄²	C₂×D₃×D₅	1	2	yes	—
223	C₃×C₅	C₂×#492	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	C₂×#687	1	4	yes	—
225	C₂²×C₃	#4086	1	2	yes	—
226	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
227	C₂×D₃	—	4	8	no	—
228	C₂×D₃	—	4	4	no	—
229	C₂×D₃	—	8	8	no	—
230	C₃×C₄	C₂×#687	1	8	yes	complemented
231	C₃×C₄	#4086	1	2	yes	—
232	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
233	Dic₃	—	20	4	no	—
234	C₂×C₅	—	24	4	no	—
235	C₂×C₅	#4693	1	4	yes	complemented
236	C₂×C₅	C₂×#874	1	1	yes	characteristic
237	C₂×C₅	C₂²×#204	1	2	yes	—
238	C₂³	—	6	4	no	—
239	C₂³	—	12	4	no	—
240	C₂³	—	12	8	no	—
241	C₂³	C₂×#1203	1	1	yes	characteristic
242	Q₈	—	30	8	no	—
243	Q₈	—	60	4	no	—
244	D₄	—	6	8	no	—
245	D₄	—	12	4	no	—
246	D₄	—	12	16	no	—
247	C₂×C₄	—	30	4	no	—
248	C₂×C₄	—	60	4	no	—
249	C₂×C₄	#6659	1	8	yes	complemented
250	C₂×C₄	D₅×D₁₂	1	4	yes	—
251	C₂×C₄	C₂×#1203	1	1	yes	characteristic
252	C₂×C₄	C₂×#1203	1	4	yes	—
253	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
254	C₈	—	5	8	no	—
255	C₂×C₃	#65365	1	4	yes	complemented
256	C₂×C₃	C₂²×#687	1	2	yes	—
257	C₂×C₃	C₂×#4086	1	1	yes	characteristic
258	D₃	—	8	4	no	—
259	C₅	C₂×#4693	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	12	4	no	—
261	C₂²	—	12	8	no	—
262	C₂²	—	24	8	no	—
263	C₂²	C₂×#6659	1	4	yes	—
264	C₂²	#91451	1	2	yes	—
265	C₂²	C₂²×#1203	1	1	yes	characteristic
266	C₄	—	60	4	no	—
267	C₄	C₂×#6659	1	8	yes	complemented
268	C₄	#91451	1	2	yes	—
269	C₄	C₂×D₅×D₁₂	1	2	yes	—
270	C₃	C₂×#65365	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	24	4	no	—
272	C₂	#11747066	1	4	yes	direct product factor
273	C₂	C₂²×#6659	1	2	yes	—
274	C₂	C₂×#91451	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1203
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×#874	C₂
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₂×#492
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×#91758	C₂	semidirect
3. C₂²×#6659	C₂	semidirect
4. C₂×C₃×C₄×#168	C₂	semidirect
5. #11747066	C₂	direct (N × H)
13. #91758	C₂²	semidirect
18. C₂×#6659	C₂²	semidirect
19. C₃×C₄×#168	C₂²	semidirect
25. C₂×C₄×#168	D₃	semidirect
41. C₂×C₃×#168	D₄	semidirect
61. C₄×#168	C₂×D₃	semidirect
67. C₂×C₃×C₄×C₅	SD₁₆	semidirect
79. C₃×#168	C₂×D₄	semidirect
109. C₂×#168	D₁₂	semidirect
123. C₃×C₄×C₅	C₂×SD₁₆	semidirect
151. C₂×C₄×C₅	#204	semidirect
160. #168	C₂×D₁₂	semidirect
174. C₂×C₃×C₅	#492	semidirect
182. C₂×C₃×C₄	#687	semidirect
202. C₄×C₅	C₂×#204	semidirect
223. C₃×C₅	C₂×#492	semidirect
230. C₃×C₄	C₂×#687	semidirect
235. C₂×C₅	#4693	semidirect
249. C₂×C₄	#6659	semidirect
255. C₂×C₃	#65365	semidirect
259. C₅	C₂×#4693	semidirect
267. C₄	C₂×#6659	semidirect
270. C₃	C₂×#65365	semidirect
272. C₂	#11747066	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×#91758	2
C₂×C₅×#874	2
C₂²×#6659	2
C₂×C₃×C₄×#168	2
#11747066	2
C₂×#65365	3
C₂×#4693	5

Subgroups — C2×#11747066 #14438754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11747066 #14438754