Subgroups — C₂×#11747075 #14438757

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×#91763	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6658	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#11747075	C₂	1	8	yes	maximal, direct product factor
6	C₂×#64982	—	3	1	no	maximal, Hall
7	C₂²×D₆₀	—	2	2	no	—
8	C₂²×D₆₀	C₂²	1	2	yes	—
9	C₂²×C₃×Dic₁₀	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	#91763	—	2	2	no	—
12	#91763	C₂²	1	4	yes	—
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
15	C₃×#4015	—	2	2	no	—
16	C₃×#4015	C₂²	1	4	yes	complemented
17	C₂×#6658	—	2	16	no	—
18	C₂×#6658	C₂²	1	16	yes	complemented
19	C₄×C₅×C₃⋊C₈	C₂²	1	4	yes	complemented
20	C₂×#5085	—	5	1	no	maximal, Hall
21	C₂²×#677	—	3	4	no	—
22	C₂×#4020	—	3	1	no	—
23	C₂×#4015	D₃	1	1	yes	characteristic, complemented
24	C₂×C₄×C₅×C₈	—	3	1	no	—
25	#64982	—	3	8	no	—
26	C₂³×D₁₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×D₆₀	—	2	8	no	—
30	C₂×D₆₀	—	2	2	no	—
31	C₂×D₆₀	—	2	8	no	—
32	C₂×D₆₀	—	4	8	no	—
33	C₂×D₆₀	C₂³	1	4	yes	—
34	C₂²×C₃×Dic₅	—	2	2	no	—
35	C₂×C₃×Dic₁₀	—	2	8	no	—
36	C₂×C₃×Dic₁₀	—	2	2	no	—
37	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	2	yes	—
40	C₂×C₅×C₃⋊C₈	D₄	1	8	yes	complemented
41	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
42	C₃×C₄⋊Dic₅	—	2	8	no	—
43	#6658	—	2	32	no	—
44	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
45	C₂²×#215	—	5	4	no	—
46	C₂×#874	—	5	1	no	—
47	#5085	—	5	8	no	—
48	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
49	C₂²×D₂₀	—	3	2	no	—
50	C₂²×D₂₀	—	6	2	no	—
51	C₂²×Dic₁₀	C₂×D₃	1	2	yes	—
52	C₂²×C₅×C₈	—	3	2	no	—
53	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
54	C₂×C₄⋊Dic₅	—	2	2	no	—
55	C₂×#677	—	3	16	no	—
56	C₂×#677	—	6	16	no	—
57	#4020	—	3	4	no	—
58	#4020	—	6	2	no	—
59	#4015	—	2	2	no	—
60	#4015	C₂×D₃	1	4	yes	complemented
61	C₄×C₅×C₈	—	3	4	no	—
62	C₂×#492	—	15	1	no	Sylow
63	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×D₁₅	—	2	4	no	—
65	C₂²×D₁₅	—	4	4	no	—
66	C₂²×D₁₅	—	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	D₆₀	—	2	8	no	—
73	D₆₀	—	4	4	no	—
74	D₆₀	—	4	16	no	—
75	C₂×C₃×Dic₅	—	2	4	no	—
76	C₂×C₃×Dic₅	—	4	4	no	—
77	C₃×Dic₁₀	—	2	8	no	—
78	C₃×Dic₁₀	—	4	4	no	—
79	C₅×C₃⋊C₈	C₂×D₄	1	8	yes	complemented
80	C₂²×C₃×Q₈	—	5	2	no	—
81	C₂²×D₁₂	—	5	2	no	—
82	C₂²×D₁₂	—	10	2	no	—
83	C₃×C₄⋊Q₈	—	5	4	no	—
84	C₃×C₄⋊Q₈	—	10	2	no	—
85	C₂×C₃×C₄⋊C₄	—	10	2	no	—
86	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
87	C₂×#215	—	5	16	no	—
88	C₂×#215	—	10	16	no	—
89	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
90	#874	—	5	4	no	—
91	#874	—	10	2	no	—
92	C₄×C₃⋊C₈	C₂×D₅	1	4	yes	complemented
93	C₂³×D₅	—	6	2	no	—
94	C₂²×C₄×C₅	#67	1	2	yes	—
95	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
96	C₂²×Dic₅	—	2	2	no	—
97	C₂×D₂₀	—	3	4	no	—
98	C₂×D₂₀	—	6	8	no	—
99	C₂×D₂₀	—	6	2	no	—
100	C₂×D₂₀	—	6	8	no	—
101	C₂×D₂₀	—	12	8	no	—
102	C₂×Dic₁₀	—	2	8	no	—
103	C₂×Dic₁₀	—	2	2	no	—
104	C₂×Dic₁₀	C₂²×D₃	1	4	yes	—
105	C₂×C₅×C₈	—	3	8	no	—
106	C₂×C₅×C₈	—	3	4	no	—
107	C₄²×C₅	#67	1	2	yes	—
108	C₄²×C₅	C₂²×D₃	1	2	yes	—
109	C₄⋊Dic₅	—	2	8	no	—
110	#677	—	6	32	no	—
111	C₂²×SD₁₆	—	15	4	no	—
112	C₂×C₄⋊Q₈	—	5	1	no	—
113	C₂×#127	—	15	1	no	—
114	#492	—	15	8	no	—
115	C₂×C₄×C₈	—	3	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×D₁₅	—	4	4	no	—
120	C₂×D₁₅	—	4	8	no	—
121	C₂×D₁₅	—	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	C₃×Dic₅	—	4	4	no	—
126	C₂³×D₃	—	10	2	no	—
127	C₂×C₃×Q₈	—	5	4	no	—
128	C₂×C₃×Q₈	—	10	2	no	—
129	C₂×C₃×Q₈	—	10	8	no	—
130	C₂²×C₃×C₄	—	10	2	no	—
131	C₂²×C₃×C₄	D₂₀	1	2	yes	—
132	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
133	C₂×D₁₂	—	5	4	no	—
134	C₂×D₁₂	—	10	8	no	—
135	C₂×D₁₂	—	10	8	no	—
136	C₂×D₁₂	—	10	2	no	—
137	C₂×D₁₂	—	20	8	no	—
138	C₃×C₄⋊C₄	—	10	8	no	—
139	C₃×C₄²	D₂₀	1	2	yes	—
140	C₃×C₄²	C₂²×D₅	1	2	yes	—
141	#215	—	10	32	no	—
142	C₂×C₃⋊C₈	D₂₀	1	8	yes	complemented
143	C₂×C₃⋊C₈	C₂²×D₅	1	4	yes	—
144	C₂³×C₅	C₂×#67	1	1	yes	characteristic
145	C₂²×D₅	—	6	4	no	—
146	C₂²×D₅	—	12	4	no	—
147	C₂²×D₅	—	12	8	no	—
148	C₂×C₄×C₅	#215	1	8	yes	complemented
149	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
150	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×#67	1	4	yes	—
152	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
153	C₂×Dic₅	—	2	4	no	—
154	C₂×Dic₅	—	4	4	no	—
155	D₂₀	—	6	8	no	—
156	D₂₀	—	12	4	no	—
157	D₂₀	—	12	16	no	—
158	Dic₁₀	—	2	8	no	—
159	Dic₁₀	—	4	4	no	—
160	C₅×C₈	—	3	8	no	—
161	C₂²×Q₈	—	5	2	no	—
162	C₂²×D₄	—	15	2	no	—
163	C₂²×D₄	—	30	2	no	—
164	C₂×SD₁₆	—	15	16	no	—
165	C₂×SD₁₆	—	30	16	no	—
166	C₂²×C₈	—	3	2	no	—
167	C₄⋊Q₈	—	5	4	no	—
168	C₄⋊Q₈	—	10	2	no	—
169	#127	—	15	4	no	—
170	#127	—	30	2	no	—
171	C₂×C₄⋊C₄	—	10	2	no	—
172	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
173	C₄×C₈	—	3	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	D₁₅	—	8	4	no	—
178	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
179	C₂²×D₃	—	10	4	no	—
180	C₂²×D₃	—	20	4	no	—
181	C₂²×D₃	—	20	8	no	—
182	C₃×Q₈	—	10	8	no	—
183	C₃×Q₈	—	20	4	no	—
184	C₂×C₃×C₄	—	10	4	no	—
185	C₂×C₃×C₄	—	20	4	no	—
186	C₂×C₃×C₄	#677	1	8	yes	complemented
187	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
188	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
190	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
191	D₁₂	—	10	8	no	—
192	D₁₂	—	20	4	no	—
193	D₁₂	—	20	16	no	—
194	C₃⋊C₈	C₂×D₂₀	1	8	yes	complemented
195	C₂²×C₅	#940	1	2	yes	—
196	C₂²×C₅	C₂×#215	1	4	yes	—
197	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
198	C₂×D₅	—	12	4	no	—
199	C₂×D₅	—	12	8	no	—
200	C₂×D₅	—	24	8	no	—
201	C₄×C₅	#940	1	2	yes	—
202	C₄×C₅	C₂×#215	1	8	yes	complemented
203	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
204	Dic₅	—	4	4	no	—
205	C₂⁴	—	30	2	no	—
206	C₂×Q₈	—	5	4	no	—
207	C₂×Q₈	—	10	2	no	—
208	C₂×Q₈	—	10	8	no	—
209	C₂×D₄	—	15	4	no	—
210	C₂×D₄	—	30	8	no	—
211	C₂×D₄	—	30	8	no	—
212	C₂×D₄	—	30	2	no	—
213	C₂×D₄	—	60	8	no	—
214	C₂²×C₄	—	10	2	no	—
215	C₂²×C₄	#1202	1	2	yes	—
216	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
217	SD₁₆	—	30	32	no	—
218	C₂×C₈	—	3	8	no	—
219	C₂×C₈	—	3	4	no	—
220	C₄⋊C₄	—	10	8	no	—
221	C₄²	#1202	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	#4020	1	2	yes	—
225	C₂²×C₃	C₂×#677	1	4	yes	—
226	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
227	C₂×D₃	—	20	4	no	—
228	C₂×D₃	—	20	8	no	—
229	C₂×D₃	—	40	8	no	—
230	C₃×C₄	—	20	4	no	—
231	C₃×C₄	#4020	1	2	yes	—
232	C₃×C₄	C₂×#677	1	8	yes	complemented
233	C₃×C₄	C₂²×D₂₀	1	2	yes	—
234	C₂×C₅	#5085	1	4	yes	complemented
235	C₂×C₅	C₂×#940	1	1	yes	characteristic
236	C₂×C₅	C₂²×#215	1	2	yes	—
237	D₅	—	24	4	no	—
238	C₂³	—	30	4	no	—
239	C₂³	—	60	4	no	—
240	C₂³	—	60	8	no	—
241	C₂³	C₂×#1202	1	1	yes	characteristic
242	Q₈	—	10	8	no	—
243	Q₈	—	20	4	no	—
244	D₄	—	30	8	no	—
245	D₄	—	60	4	no	—
246	D₄	—	60	16	no	—
247	C₂×C₄	—	10	4	no	—
248	C₂×C₄	—	20	4	no	—
249	C₂×C₄	#6658	1	8	yes	complemented
250	C₂×C₄	D₃×D₂₀	1	4	yes	—
251	C₂×C₄	C₂×#1202	1	1	yes	characteristic
252	C₂×C₄	C₂×#1202	1	4	yes	—
253	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
254	C₈	—	3	8	no	—
255	C₂×C₃	#64982	1	4	yes	complemented
256	C₂×C₃	C₂×#4020	1	1	yes	characteristic
257	C₂×C₃	C₂²×#677	1	2	yes	—
258	D₃	—	40	4	no	—
259	C₅	C₂×#5085	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	60	4	no	—
261	C₂²	—	60	8	no	—
262	C₂²	—	120	8	no	—
263	C₂²	C₂×#6658	1	4	yes	—
264	C₂²	#91458	1	2	yes	—
265	C₂²	C₂²×#1202	1	1	yes	characteristic
266	C₄	—	20	4	no	—
267	C₄	C₂×#6658	1	8	yes	complemented
268	C₄	#91458	1	2	yes	—
269	C₄	C₂×D₃×D₂₀	1	2	yes	—
270	C₃	C₂×#64982	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	120	4	no	—
272	C₂	#11747075	1	4	yes	direct product factor
273	C₂	C₂²×#6658	1	2	yes	—
274	C₂	C₂×#91458	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1202
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×#91763	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
2. C₂×C₃×#4015	C₂	semidirect
3. C₂²×#6658	C₂	semidirect
4. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
5. #11747075	C₂	direct (N × H)
16. C₃×#4015	C₂²	semidirect
18. C₂×#6658	C₂²	semidirect
19. C₄×C₅×C₃⋊C₈	C₂²	semidirect
23. C₂×#4015	D₃	semidirect
40. C₂×C₅×C₃⋊C₈	D₄	semidirect
48. C₂×C₄×C₃⋊C₈	D₅	semidirect
60. #4015	C₂×D₃	semidirect
67. C₂×C₃×C₄×C₅	SD₁₆	semidirect
79. C₅×C₃⋊C₈	C₂×D₄	semidirect
92. C₄×C₃⋊C₈	C₂×D₅	semidirect
123. C₃×C₄×C₅	C₂×SD₁₆	semidirect
142. C₂×C₃⋊C₈	D₂₀	semidirect
148. C₂×C₄×C₅	#215	semidirect
174. C₂×C₃×C₅	#492	semidirect
186. C₂×C₃×C₄	#677	semidirect
194. C₃⋊C₈	C₂×D₂₀	semidirect
202. C₄×C₅	C₂×#215	semidirect
223. C₃×C₅	C₂×#492	semidirect
232. C₃×C₄	C₂×#677	semidirect
234. C₂×C₅	#5085	semidirect
249. C₂×C₄	#6658	semidirect
255. C₂×C₃	#64982	semidirect
259. C₅	C₂×#5085	semidirect
267. C₄	C₂×#6658	semidirect
270. C₃	C₂×#64982	semidirect
272. C₂	#11747075	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×C₃×#4015	2
C₂²×#6658	2
C₂×C₄×C₅×C₃⋊C₈	2
#11747075	2
C₂×#64982	3
C₂×#5085	5

Subgroups — C2×#11747075 #14438757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11747075 #14438757