Subgroups — C₂²×#90968 #14412269

Order 1920.

2144 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₆₀	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90968	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#3953	—	3	1	no	maximal, Hall
6	C₂⁴×D₁₅	—	2	1	no	—
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×D₆₀	—	2	4	no	—
9	C₂²×D₆₀	—	2	3	no	—
10	C₂²×D₆₀	C₄	1	8	yes	complemented
11	C₂²×D₆₀	C₂²	1	6	yes	—
12	C₂³×C₃×Dic₅	—	2	1	no	—
13	C₂²×C₅×C₃⋊C₈	—	2	4	no	—
14	C₂²×C₅×C₃⋊C₈	C₂²	1	6	yes	complemented
15	C₂×C₃×C₄⋊Dic₅	—	2	3	no	—
16	C₂×C₃×C₄⋊Dic₅	C₂²	1	6	yes	complemented
17	#90968	C₂²	1	16	yes	direct product factor
18	C₂²×#809	—	5	1	no	maximal, Hall
19	C₂³×D₂₀	—	3	1	no	—
20	C₂³×C₅×C₈	—	3	1	no	—
21	C₂²×C₄⋊Dic₅	D₃	1	1	yes	characteristic, complemented
22	C₂×#3953	—	3	12	no	—
23	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
24	C₂³×D₁₅	—	2	8	no	—
25	C₂³×D₁₅	—	2	6	no	—
26	C₂³×D₁₅	—	4	4	no	—
27	C₂³×D₁₅	—	4	4	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
29	C₂²×C₃×C₄×C₅	D₄	1	3	yes	—
30	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
32	C₂×D₆₀	—	2	24	no	—
33	C₂×D₆₀	—	2	6	no	—
34	C₂×D₆₀	—	2	12	no	—
35	C₂×D₆₀	C₂×C₄	1	24	yes	complemented
36	C₂×D₆₀	C₂³	1	4	yes	—
37	C₂²×C₃×Dic₅	—	2	6	no	—
38	C₂²×C₃×Dic₅	—	4	4	no	—
39	C₂×C₅×C₃⋊C₈	—	2	12	no	—
40	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	complemented
41	C₃×C₄⋊Dic₅	—	2	6	no	—
42	C₃×C₄⋊Dic₅	C₂³	1	4	yes	complemented
43	C₂³×D₁₂	—	5	1	no	—
44	C₂²×C₃×C₄⋊C₄	—	5	1	no	—
45	C₂³×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
46	C₂×#809	—	5	12	no	—
47	C₂⁴×D₅	—	6	1	no	—
48	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
49	C₂³×Dic₅	—	2	1	no	—
50	C₂²×D₂₀	—	3	8	no	—
51	C₂²×D₂₀	—	3	6	no	—
52	C₂²×D₂₀	—	6	4	no	—
53	C₂²×D₂₀	—	6	3	no	—
54	C₂²×C₅×C₈	—	3	6	no	—
55	C₂²×C₅×C₈	—	6	4	no	—
56	C₂×C₄⋊Dic₅	—	2	3	no	—
57	C₂×C₄⋊Dic₅	C₂×D₃	1	6	yes	complemented
58	#3953	—	3	16	no	—
59	C₂²×#102	—	15	1	no	Sylow
60	C₂³×C₃×C₅	C₂²⋊C₄	1	4	yes	—
61	C₂³×C₃×C₅	D₈	1	4	yes	—
62	C₂³×C₃×C₅	SD₁₆	1	4	yes	—
63	C₂³×C₃×C₅	C₂×D₄	1	3	yes	—
64	C₂²×D₁₅	—	2	24	no	—
65	C₂²×D₁₅	—	2	4	no	—
66	C₂²×D₁₅	—	4	32	no	—
67	C₂²×D₁₅	—	4	12	no	—
68	C₂²×D₁₅	—	4	12	no	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	12	yes	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
71	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
72	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
73	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
74	D₆₀	—	2	24	no	—
75	D₆₀	—	2	4	no	—
76	D₆₀	C₂²×C₄	1	8	yes	complemented
77	C₂×C₃×Dic₅	—	2	4	no	—
78	C₂×C₃×Dic₅	—	4	12	no	—
79	C₅×C₃⋊C₈	—	2	4	no	—
80	C₂⁴×D₃	—	10	1	no	—
81	C₂³×C₃×C₄	—	10	1	no	—
82	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
83	C₂²×D₁₂	—	5	8	no	—
84	C₂²×D₁₂	—	5	6	no	—
85	C₂²×D₁₂	—	10	3	no	—
86	C₂²×D₁₂	—	10	4	no	—
87	C₂×C₃×C₄⋊C₄	—	5	6	no	—
88	C₂×C₃×C₄⋊C₄	—	10	3	no	—
89	C₂²×C₃⋊C₈	—	2	4	no	—
90	C₂²×C₃⋊C₈	C₂×D₅	1	6	yes	complemented
91	#809	—	5	16	no	—
92	C₂⁴×C₅	#67	1	1	yes	characteristic
93	C₂³×D₅	—	6	8	no	—
94	C₂³×D₅	—	6	6	no	—
95	C₂³×D₅	—	12	4	no	—
96	C₂³×D₅	—	12	4	no	—
97	C₂²×C₄×C₅	C₄×D₃	1	4	yes	—
98	C₂²×C₄×C₅	D₁₂	1	4	yes	—
99	C₂²×C₄×C₅	#67	1	3	yes	—
100	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
101	C₂²×Dic₅	—	2	6	no	—
102	C₂²×Dic₅	—	4	4	no	—
103	C₂×D₂₀	—	3	24	no	—
104	C₂×D₂₀	—	3	4	no	—
105	C₂×D₂₀	—	6	24	no	—
106	C₂×D₂₀	—	6	12	no	—
107	C₂×D₂₀	—	6	6	no	—
108	C₂×C₅×C₈	—	3	4	no	—
109	C₂×C₅×C₈	—	6	12	no	—
110	C₄⋊Dic₅	—	2	6	no	—
111	C₄⋊Dic₅	C₂²×D₃	1	4	yes	complemented
112	C₂³×D₄	—	15	1	no	—
113	C₂³×C₈	—	3	1	no	—
114	C₂²×C₄⋊C₄	—	5	1	no	—
115	C₂×#102	—	15	12	no	—
116	C₂²×C₃×C₅	#102	1	16	yes	complemented
117	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	6	yes	—
118	C₂²×C₃×C₅	C₂×D₈	1	6	yes	—
119	C₂²×C₃×C₅	C₂×SD₁₆	1	6	yes	—
120	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₅	—	2	8	no	—
122	C₂×D₁₅	—	4	4	no	—
123	C₂×D₁₅	—	4	4	no	—
124	C₂×D₁₅	—	4	48	no	—
125	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
126	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
127	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₃×Dic₅	—	4	4	no	—
129	C₂⁴×C₃	D₂₀	1	1	yes	characteristic
130	C₂³×D₃	—	10	8	no	—
131	C₂³×D₃	—	10	6	no	—
132	C₂³×D₃	—	20	4	no	—
133	C₂³×D₃	—	20	4	no	—
134	C₂²×C₃×C₄	—	10	6	no	—
135	C₂²×C₃×C₄	—	20	4	no	—
136	C₂²×C₃×C₄	C₄×D₅	1	4	yes	—
137	C₂²×C₃×C₄	D₂₀	1	3	yes	—
138	C₂²×C₃×C₄	C₂²⋊D₅	1	4	yes	—
139	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
140	C₂×D₁₂	—	5	24	no	—
141	C₂×D₁₂	—	5	4	no	—
142	C₂×D₁₂	—	10	24	no	—
143	C₂×D₁₂	—	10	6	no	—
144	C₂×D₁₂	—	10	12	no	—
145	C₃×C₄⋊C₄	—	5	4	no	—
146	C₃×C₄⋊C₄	—	10	6	no	—
147	C₂×C₃⋊C₈	—	2	12	no	—
148	C₂×C₃⋊C₈	C₂²×D₅	1	4	yes	complemented
149	C₂³×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
150	C₂³×C₅	#213	1	4	yes	—
151	C₂³×C₅	#215	1	4	yes	—
152	C₂³×C₅	C₂×#67	1	3	yes	—
153	C₂²×D₅	—	6	24	no	—
154	C₂²×D₅	—	6	4	no	—
155	C₂²×D₅	—	12	32	no	—
156	C₂²×D₅	—	12	12	no	—
157	C₂²×D₅	—	12	12	no	—
158	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	12	yes	—
159	C₂×C₄×C₅	C₂×C₄×D₃	1	6	yes	—
160	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
161	C₂×C₄×C₅	C₂×#67	1	3	yes	—
162	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
163	C₂×Dic₅	—	2	4	no	—
164	C₂×Dic₅	—	4	12	no	—
165	D₂₀	—	3	8	no	—
166	D₂₀	—	6	24	no	—
167	D₂₀	—	6	4	no	—
168	C₅×C₈	—	6	4	no	—
169	C₂⁵	—	30	1	no	—
170	C₂²×D₄	—	15	8	no	—
171	C₂²×D₄	—	15	6	no	—
172	C₂²×D₄	—	30	3	no	—
173	C₂²×D₄	—	30	4	no	—
174	C₂³×C₄	—	10	1	no	—
175	C₂³×C₄	D₃×D₅	1	1	yes	characteristic
176	C₂²×C₈	—	3	6	no	—
177	C₂²×C₈	—	6	4	no	—
178	C₂×C₄⋊C₄	—	5	6	no	—
179	C₂×C₄⋊C₄	—	10	3	no	—
180	#102	—	15	16	no	—
181	C₂×C₃×C₅	C₂×#102	1	12	yes	complemented
182	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
184	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
185	D₁₅	—	4	8	no	—
186	C₂³×C₃	#677	1	4	yes	—
187	C₂³×C₃	D₄₀	1	4	yes	—
188	C₂³×C₃	#685	1	4	yes	—
189	C₂³×C₃	C₂×D₂₀	1	3	yes	—
190	C₂²×D₃	—	10	4	no	—
191	C₂²×D₃	—	10	24	no	—
192	C₂²×D₃	—	20	32	no	—
193	C₂²×D₃	—	20	12	no	—
194	C₂²×D₃	—	20	12	no	—
195	C₂×C₃×C₄	—	10	4	no	—
196	C₂×C₃×C₄	—	20	12	no	—
197	C₂×C₃×C₄	#685	1	12	yes	—
198	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
199	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
200	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	6	yes	—
201	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
202	D₁₂	—	5	8	no	—
203	D₁₂	—	10	4	no	—
204	D₁₂	—	10	24	no	—
205	C₃⋊C₈	—	2	4	no	—
206	C₂²×C₅	#809	1	16	yes	complemented
207	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
208	C₂²×C₅	C₂×#213	1	6	yes	—
209	C₂²×C₅	C₂×#215	1	6	yes	—
210	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
211	C₂×D₅	—	6	8	no	—
212	C₂×D₅	—	12	48	no	—
213	C₂×D₅	—	12	4	no	—
214	C₂×D₅	—	12	4	no	—
215	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
216	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
217	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
218	Dic₅	—	4	4	no	—
219	C₂⁴	—	30	8	no	—
220	C₂⁴	—	30	6	no	—
221	C₂⁴	—	60	4	no	—
222	C₂⁴	—	60	4	no	—
223	C₂⁴	#1202	1	1	yes	characteristic
224	C₂×D₄	—	15	24	no	—
225	C₂×D₄	—	15	4	no	—
226	C₂×D₄	—	30	24	no	—
227	C₂×D₄	—	30	6	no	—
228	C₂×D₄	—	30	12	no	—
229	C₂²×C₄	—	10	6	no	—
230	C₂²×C₄	—	20	4	no	—
231	C₂²×C₄	#1200	1	4	yes	—
232	C₂²×C₄	#1202	1	3	yes	—
233	C₂²×C₄	#1203	1	4	yes	—
234	C₂²×C₄	C₂×D₃×D₅	1	3	yes	—
235	C₂×C₈	—	3	4	no	—
236	C₂×C₈	—	6	12	no	—
237	C₄⋊C₄	—	5	4	no	—
238	C₄⋊C₄	—	10	6	no	—
239	C₃×C₅	C₂²×#102	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	#3953	1	16	yes	complemented
241	C₂²×C₃	C₂×#677	1	6	yes	—
242	C₂²×C₃	C₂×D₄₀	1	6	yes	—
243	C₂²×C₃	C₂×#685	1	6	yes	—
244	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
245	C₂×D₃	—	10	8	no	—
246	C₂×D₃	—	20	4	no	—
247	C₂×D₃	—	20	4	no	—
248	C₂×D₃	—	20	48	no	—
249	C₃×C₄	—	20	4	no	—
250	C₃×C₄	C₂×#685	1	6	yes	—
251	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
252	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
253	C₂×C₅	C₂×#809	1	12	yes	complemented
254	C₂×C₅	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic
255	C₂×C₅	C₂²×#213	1	1	yes	characteristic
256	C₂×C₅	C₂²×#215	1	1	yes	characteristic
257	D₅	—	12	8	no	—
258	C₂³	—	30	4	no	—
259	C₂³	—	30	24	no	—
260	C₂³	—	60	32	no	—
261	C₂³	—	60	12	no	—
262	C₂³	—	60	12	no	—
263	C₂³	#6653	1	4	yes	—
264	C₂³	#6658	1	4	yes	—
265	C₂³	#6667	1	4	yes	—
266	C₂³	C₂×#1202	1	3	yes	—
267	D₄	—	15	8	no	—
268	D₄	—	30	4	no	—
269	D₄	—	30	24	no	—
270	C₂×C₄	—	10	4	no	—
271	C₂×C₄	—	20	12	no	—
272	C₂×C₄	#6667	1	12	yes	—
273	C₂×C₄	C₂×#1200	1	6	yes	—
274	C₂×C₄	C₂×#1202	1	3	yes	—
275	C₂×C₄	C₂×#1203	1	6	yes	—
276	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
277	C₈	—	6	4	no	—
278	C₂×C₃	C₂×#3953	1	12	yes	complemented
279	C₂×C₃	C₂²×#677	1	1	yes	characteristic
280	C₂×C₃	C₂²×D₄₀	1	1	yes	characteristic
281	C₂×C₃	C₂²×#685	1	1	yes	characteristic
282	D₃	—	20	8	no	—
283	C₅	C₂²×#809	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	30	8	no	—
285	C₂²	—	60	48	no	—
286	C₂²	—	60	4	no	—
287	C₂²	—	60	4	no	—
288	C₂²	#90968	1	16	yes	direct product factor
289	C₂²	C₂×#6653	1	6	yes	—
290	C₂²	C₂×#6658	1	6	yes	—
291	C₂²	C₂×#6667	1	6	yes	—
292	C₂²	C₂²×#1202	1	1	yes	characteristic
293	C₄	—	20	4	no	—
294	C₄	C₂×#6667	1	6	yes	—
295	C₄	C₂²×#1200	1	1	yes	characteristic
296	C₄	C₂²×#1203	1	1	yes	characteristic
297	C₃	C₂²×#3953	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	60	8	no	—
299	C₂	C₂×#90968	1	12	yes	direct product factor
300	C₂	C₂²×#6653	1	1	yes	characteristic
301	C₂	C₂²×#6658	1	1	yes	characteristic
302	C₂	C₂²×#6667	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#1202
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂³×C₃×C₄×C₅	C₂²
socle	C₂⁴×C₃×C₅	D₄
prime radical	C₂³×D₆₀	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂³×C₄	D₃×D₅
hypocenter	C₃×C₅	C₂²×#102
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₅×C₃⋊C₈	C₂	semidirect
3. C₂²×C₃×C₄⋊Dic₅	C₂	semidirect
4. C₂×#90968	C₂	direct (N × H)
10. C₂²×D₆₀	C₄	semidirect
14. C₂²×C₅×C₃⋊C₈	C₂²	semidirect
16. C₂×C₃×C₄⋊Dic₅	C₂²	semidirect
17. #90968	C₂²	direct (N × H)
21. C₂²×C₄⋊Dic₅	D₃	semidirect
35. C₂×D₆₀	C₂×C₄	semidirect
40. C₂×C₅×C₃⋊C₈	C₂³	semidirect
42. C₃×C₄⋊Dic₅	C₂³	semidirect
45. C₂³×C₃⋊C₈	D₅	semidirect
57. C₂×C₄⋊Dic₅	C₂×D₃	semidirect
76. D₆₀	C₂²×C₄	semidirect
90. C₂²×C₃⋊C₈	C₂×D₅	semidirect
111. C₄⋊Dic₅	C₂²×D₃	semidirect
116. C₂²×C₃×C₅	#102	semidirect
148. C₂×C₃⋊C₈	C₂²×D₅	semidirect
181. C₂×C₃×C₅	C₂×#102	semidirect
206. C₂²×C₅	#809	semidirect
239. C₃×C₅	C₂²×#102	semidirect
240. C₂²×C₃	#3953	semidirect
253. C₂×C₅	C₂×#809	semidirect
278. C₂×C₃	C₂×#3953	semidirect
283. C₅	C₂²×#809	semidirect
288. C₂²	#90968	direct (N × H)
297. C₃	C₂²×#3953	semidirect
299. C₂	C₂×#90968	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₆₀	2
C₂³×C₅×C₃⋊C₈	2
C₂²×C₃×C₄⋊Dic₅	2
C₂×#90968	2
C₂²×#3953	3
C₂²×#809	5

Subgroups — C22×#90968 #14412269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#90968 #14412269