Subgroups — C₂×#11751005 #14478274

Order 1920.

736 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91793	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₁₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751005	C₂	1	8	yes	maximal, direct product factor
5	C₂×#91084	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₄₀	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#65193	—	3	1	no	maximal, Hall
8	C₂×#6817	—	2	2	no	—
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₀	C₂²	1	2	yes	—
11	#91793	—	2	4	no	—
12	#91793	C₂²	1	8	yes	complemented
13	C₂×D₁₂₀	C₂²	1	2	yes	—
14	C₂×#6706	—	2	2	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₅×C₁₆	C₂²	1	4	yes	complemented
18	Dic₁₂₀	C₂²	1	4	yes	complemented
19	#91084	C₂²	1	8	yes	complemented
20	D₂₄₀	C₂²	1	4	yes	complemented
21	C₂×#4904	—	5	1	no	maximal, Hall
22	C₂×#4050	—	3	2	no	—
23	C₂²×C₅×C₁₆	D₃	1	1	yes	characteristic, complemented
24	C₂×Dic₄₀	—	3	1	no	—
25	#65193	—	3	8	no	—
26	C₂×#3932	—	3	2	no	—
27	C₂×D₈₀	—	3	1	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×#1220	—	4	2	no	—
30	#6817	—	2	8	no	—
31	#6817	—	4	4	no	—
32	C₂×D₆₀	—	2	2	no	—
33	C₂×C₄×D₁₅	—	4	2	no	—
34	C₂×Dic₃₀	—	2	2	no	—
35	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
39	Dic₆₀	—	2	2	no	—
40	Dic₆₀	C₂³	1	4	yes	—
41	D₁₂₀	—	2	2	no	—
42	D₁₂₀	C₂³	1	4	yes	—
43	#6706	—	2	8	no	—
44	C₃×C₅×C₁₆	C₂³	1	2	yes	—
45	C₃×C₅×C₁₆	C₂³	1	2	yes	—
46	C₂×#904	—	5	2	no	—
47	C₂²×C₃×C₁₆	D₅	1	1	yes	characteristic, complemented
48	C₂×Dic₂₄	—	5	1	no	—
49	#4904	—	5	8	no	—
50	C₂×#800	—	5	2	no	—
51	C₂×D₄₈	—	5	1	no	—
52	C₂×#709	—	6	2	no	—
53	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×Dic₂₀	—	3	2	no	—
55	#4050	—	3	8	no	—
56	#4050	—	6	4	no	—
57	C₂×D₄₀	—	3	2	no	—
58	C₂×#677	—	6	2	no	—
59	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
60	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
61	C₂×C₅×C₁₆	C₂×D₃	1	4	yes	complemented
62	Dic₄₀	—	3	4	no	—
63	#3932	—	3	8	no	—
64	D₈₀	—	3	4	no	—
65	C₂×#508	—	15	1	no	Sylow
66	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
67	C₂²×D₁₅	—	4	2	no	—
68	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
72	#1220	—	4	8	no	—
73	C₂×Dic₁₅	—	4	2	no	—
74	D₆₀	—	2	4	no	—
75	D₆₀	—	4	2	no	—
76	C₄×D₁₅	—	4	8	no	—
77	Dic₃₀	—	2	4	no	—
78	Dic₃₀	—	4	2	no	—
79	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
80	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
82	C₂×#235	—	10	2	no	—
83	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
84	C₂×Dic₁₂	—	5	2	no	—
85	#904	—	5	8	no	—
86	#904	—	10	4	no	—
87	C₂×D₂₄	—	5	2	no	—
88	C₂×#204	—	10	2	no	—
89	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
90	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
91	C₂×C₃×C₁₆	C₂×D₅	1	4	yes	complemented
92	Dic₂₄	—	5	4	no	—
93	#800	—	5	8	no	—
94	D₄₈	—	5	4	no	—
95	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
96	C₂×C₂²⋊D₅	—	12	2	no	—
97	#709	—	6	8	no	—
98	#709	—	12	4	no	—
99	C₂×D₂₀	—	6	2	no	—
100	C₂×C₄×D₅	—	12	2	no	—
101	C₂×Dic₁₀	—	6	2	no	—
102	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
103	C₂×C₅×C₈	D₁₂	1	2	yes	—
104	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
106	Dic₂₀	—	3	4	no	—
107	Dic₂₀	—	6	2	no	—
108	D₄₀	—	3	4	no	—
109	D₄₀	—	6	2	no	—
110	#677	—	6	8	no	—
111	C₅×C₁₆	C₂²×D₃	1	2	yes	—
112	C₅×C₁₆	C₂²×D₃	1	2	yes	—
113	C₂×#135	—	15	2	no	—
114	#508	—	15	8	no	—
115	C₂×Q₃₂	—	15	1	no	—
116	C₂×SD₃₂	—	15	2	no	—
117	C₂×D₁₆	—	15	1	no	—
118	C₂²×C₁₆	D₁₅	1	1	yes	characteristic, complemented
119	C₂²×C₃×C₅	—	2	2	no	—
120	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
121	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
122	C₂×D₁₅	—	4	4	no	—
123	C₂×D₁₅	—	8	4	no	—
124	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
125	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
126	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
127	Dic₁₅	—	4	4	no	—
128	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
129	C₂×#67	—	20	2	no	—
130	#235	—	10	8	no	—
131	#235	—	20	4	no	—
132	C₂×D₁₂	—	10	2	no	—
133	C₂×C₄×D₃	—	20	2	no	—
134	C₂×Dic₆	—	10	2	no	—
135	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
136	C₂×C₃×C₈	D₂₀	1	2	yes	—
137	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
139	Dic₁₂	—	5	4	no	—
140	Dic₁₂	—	10	2	no	—
141	D₂₄	—	5	4	no	—
142	D₂₄	—	10	2	no	—
143	#204	—	10	8	no	—
144	C₃×C₁₆	C₂²×D₅	1	2	yes	—
145	C₃×C₁₆	C₂²×D₅	1	2	yes	—
146	C₂³×C₅	D₂₄	1	1	yes	characteristic
147	C₂²×D₅	—	12	2	no	—
148	C₂×C₄×C₅	D₂₄	1	2	yes	—
149	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
152	C₂²⋊D₅	—	12	8	no	—
153	C₂×Dic₅	—	12	2	no	—
154	D₂₀	—	6	4	no	—
155	D₂₀	—	12	2	no	—
156	C₄×D₅	—	12	8	no	—
157	Dic₁₀	—	6	4	no	—
158	Dic₁₀	—	12	2	no	—
159	C₅×C₈	C₂×D₁₂	1	2	yes	—
160	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
161	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
162	C₂×#41	—	30	2	no	—
163	#135	—	15	8	no	—
164	#135	—	30	4	no	—
165	C₂×Q₁₆	—	15	2	no	—
166	C₂×SD₁₆	—	30	2	no	—
167	C₂×D₈	—	15	2	no	—
168	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
169	Q₃₂	—	15	4	no	—
170	SD₃₂	—	15	8	no	—
171	D₁₆	—	15	4	no	—
172	C₂×C₁₆	C₂×D₁₅	1	1	yes	characteristic
173	C₂×C₁₆	C₂×D₁₅	1	1	yes	characteristic
174	C₂×C₁₆	C₂×D₁₅	1	4	yes	complemented
175	C₂×C₃×C₅	—	2	2	no	—
176	C₂×C₃×C₅	#508	1	2	yes	complemented
177	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
178	D₁₅	—	8	4	no	—
179	C₂³×C₃	D₄₀	1	1	yes	characteristic
180	C₂²×D₃	—	20	2	no	—
181	C₂×C₃×C₄	D₄₀	1	2	yes	—
182	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
185	#67	—	20	8	no	—
186	C₂×Dic₃	—	20	2	no	—
187	D₁₂	—	10	4	no	—
188	D₁₂	—	20	2	no	—
189	C₄×D₃	—	20	8	no	—
190	Dic₆	—	10	4	no	—
191	Dic₆	—	20	2	no	—
192	C₃×C₈	C₂×D₂₀	1	2	yes	—
193	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
194	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	C₂×D₂₄	1	2	yes	—
197	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
198	C₂×D₅	—	12	4	no	—
199	C₂×D₅	—	24	4	no	—
200	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
201	C₄×C₅	C₂×D₂₄	1	2	yes	—
202	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
203	Dic₅	—	12	4	no	—
204	#41	—	30	8	no	—
205	#41	—	60	4	no	—
206	C₂×Q₈	—	30	2	no	—
207	C₂×D₄	—	30	2	no	—
208	C₂×D₄	—	60	2	no	—
209	C₂²×C₄	—	60	2	no	—
210	C₂²×C₄	D₆₀	1	1	yes	characteristic
211	Q₁₆	—	15	4	no	—
212	Q₁₆	—	30	2	no	—
213	SD₁₆	—	30	8	no	—
214	D₈	—	15	4	no	—
215	D₈	—	30	2	no	—
216	C₂×C₈	D₆₀	1	1	yes	characteristic
217	C₂×C₈	D₆₀	1	2	yes	—
218	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
219	C₂×C₈	C₂²×D₁₅	1	2	yes	—
220	C₁₆	C₂²×D₁₅	1	2	yes	—
221	C₁₆	C₂²×D₁₅	1	2	yes	—
222	C₃×C₅	C₂×#508	1	1	yes	characteristic, complemented, Hall
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	C₂×D₄₀	1	2	yes	—
225	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
226	C₂×D₃	—	20	4	no	—
227	C₂×D₃	—	40	4	no	—
228	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
229	C₃×C₄	C₂×D₄₀	1	2	yes	—
230	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
231	Dic₃	—	20	4	no	—
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	#4904	1	2	yes	complemented
234	C₂×C₅	C₂²×D₂₄	1	1	yes	characteristic
235	D₅	—	24	4	no	—
236	C₂³	—	60	2	no	—
237	C₂³	D₁₂₀	1	1	yes	characteristic
238	Q₈	—	30	4	no	—
239	Q₈	—	60	2	no	—
240	D₄	—	30	4	no	—
241	D₄	—	60	8	no	—
242	D₄	—	60	2	no	—
243	C₂×C₄	—	60	2	no	—
244	C₂×C₄	—	60	8	no	—
245	C₂×C₄	D₁₂₀	1	2	yes	—
246	C₂×C₄	D₁₂₀	1	1	yes	characteristic
247	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
248	C₂×C₄	C₂×D₆₀	1	2	yes	—
249	C₈	C₂×D₆₀	1	1	yes	characteristic
250	C₈	C₂×D₆₀	1	2	yes	—
251	C₈	C₂³×D₁₅	1	1	yes	characteristic
252	C₂×C₃	—	2	2	no	—
253	C₂×C₃	#65193	1	2	yes	complemented
254	C₂×C₃	C₂²×D₄₀	1	1	yes	characteristic
255	D₃	—	40	4	no	—
256	C₅	C₂×#4904	1	1	yes	characteristic, complemented, Sylow
257	C₂²	—	2	2	no	—
258	C₂²	—	60	4	no	—
259	C₂²	—	120	4	no	—
260	C₂²	C₂×D₁₂₀	1	2	yes	—
261	C₂²	C₂×D₁₂₀	1	1	yes	characteristic
262	C₄	—	60	4	no	—
263	C₄	C₂×D₁₂₀	1	2	yes	—
264	C₄	C₂×D₁₂₀	1	1	yes	characteristic
265	C₄	C₂²×D₆₀	1	1	yes	characteristic
266	C₃	C₂×#65193	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	2	2	no	—
268	C₂	—	120	4	no	—
269	C₂	#11751005	1	2	yes	direct product factor
270	C₂	C₂²×D₁₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₂₀
commutator, cosocle	C₃×C₅×C₈	C₂⁴
Frattini	C₈	C₂³×D₁₅
Fitting	C₂²×C₃×C₅×C₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂²×C₁₆	D₁₅
hypocenter	C₃×C₅	C₂×#508
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#91793	C₂	semidirect
2. C₂²×C₃×C₅×C₁₆	C₂	semidirect
3. C₂×Dic₁₂₀	C₂	semidirect
4. #11751005	C₂	direct (N × H)
5. C₂×#91084	C₂	semidirect
6. C₂×D₂₄₀	C₂	semidirect
12. #91793	C₂²	semidirect
17. C₂×C₃×C₅×C₁₆	C₂²	semidirect
18. Dic₁₂₀	C₂²	semidirect
19. #91084	C₂²	semidirect
20. D₂₄₀	C₂²	semidirect
23. C₂²×C₅×C₁₆	D₃	semidirect
47. C₂²×C₃×C₁₆	D₅	semidirect
61. C₂×C₅×C₁₆	C₂×D₃	semidirect
91. C₂×C₃×C₁₆	C₂×D₅	semidirect
118. C₂²×C₁₆	D₁₅	semidirect
174. C₂×C₁₆	C₂×D₁₅	semidirect
176. C₂×C₃×C₅	#508	semidirect
222. C₃×C₅	C₂×#508	semidirect
233. C₂×C₅	#4904	semidirect
253. C₂×C₃	#65193	semidirect
256. C₅	C₂×#4904	semidirect
266. C₃	C₂×#65193	semidirect
269. C₂	#11751005	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×#91793	2
C₂²×C₃×C₅×C₁₆	2
C₂×Dic₁₂₀	2
#11751005	2
C₂×#91084	2
C₂×D₂₄₀	2
C₂×#65193	3
C₂×#4904	5

Subgroups — C2×#11751005 #14478274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751005 #14478274