Subgroups — #14417393

Order 1920.

1056 conjugacy classes of subgroups.

The subgroup lattice has 312 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	#11754488	C₂	1	2	yes	maximal, complemented
3	#11751202	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5106	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748366	C₂	1	2	yes	maximal, complemented
6	#11747815	C₂	1	4	yes	maximal, complemented
7	#11747809	C₂	1	4	yes	maximal, complemented
8	#10610898	—	3	1	no	maximal, Hall
9	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	#91815	C₂²	1	1	yes	characteristic
12	C₂×#6710	C₂²	1	1	yes	characteristic
13	C₅×#926	C₂²	1	2	yes	—
14	C₂×C₃×D₄₀	C₄	1	8	yes	complemented
15	C₂×C₃×D₄₀	C₂²	1	2	yes	—
16	C₂×C₃×D₄₀	C₂²	1	2	yes	—
17	D₅×#217	—	2	4	no	—
18	#91434	C₂²	1	8	yes	complemented
19	Dic₃×D₂₀	—	2	4	no	—
20	#91099	C₂²	1	2	yes	complemented
21	C₈×Dic₁₅	C₂²	1	2	yes	complemented
22	C₅×#818	C₂²	1	2	yes	complemented
23	C₅×#817	C₂²	1	2	yes	complemented
24	#90964	C₂²	1	8	yes	complemented
25	#83823	—	5	1	no	maximal, Hall
26	C₂²×D₄₀	D₃	1	1	yes	characteristic
27	#65855	—	3	2	no	—
28	C₅×#427	—	3	1	no	—
29	#65390	—	3	1	no	—
30	#65290	—	3	2	no	—
31	#65161	—	3	4	no	—
32	#65158	—	3	4	no	—
33	C₂³×C₃×D₅	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃×D₂₀	—	2	2	no	—
36	C₂×C₃×D₂₀	—	2	4	no	—
37	C₂×C₃×D₂₀	C₂×C₄	1	8	yes	complemented
38	C₂×C₃×D₂₀	C₂³	1	4	yes	—
39	C₂×D₅×Dic₃	—	2	8	no	—
40	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
43	#6719	D₄	1	2	yes	—
44	C₄⋊Dic₁₅	D₄	1	2	yes	—
45	C₄×Dic₁₅	C₂³	1	2	yes	—
46	#6710	—	2	2	no	—
47	C₂×C₃⋊#168	C₂³	1	2	yes	—
48	C₅×#217	—	2	2	no	—
49	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
50	C₄×C₅×Dic₃	—	2	2	no	—
51	C₃×D₄₀	C₂×C₄	1	8	yes	complemented
52	C₃×D₄₀	C₂×C₄	1	8	yes	complemented
53	#6665	—	2	8	no	—
54	C₂²×C₃×D₈	—	5	1	no	—
55	#5798	—	5	2	no	—
56	#5215	—	5	2	no	—
57	#5152	—	5	4	no	—
58	Dic₃×D₈	—	5	4	no	—
59	#5106	D₅	1	1	yes	characteristic, complemented
60	#5101	—	5	1	no	—
61	C₂²×D₂₀	C₂×D₃	1	2	yes	—
62	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	C₅×#117	—	3	2	no	—
64	#4072	—	3	1	no	—
65	C₂×#681	—	3	1	no	—
66	C₂×D₄₀	Dic₃	1	8	yes	complemented
67	C₂×D₄₀	C₂×D₃	1	2	yes	—
68	C₂×D₄₀	C₂×D₃	1	2	yes	—
69	#4039	—	3	8	no	—
70	D₅×C₂²⋊C₄	—	6	4	no	—
71	C₄×D₂₀	—	6	4	no	—
72	C₅×#107	—	3	2	no	—
73	C₅×#106	—	3	2	no	—
74	#3947	—	3	2	no	—
75	C₈×Dic₅	—	3	2	no	—
76	#3941	—	3	8	no	—
77	#2948	—	15	1	no	Sylow
78	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₃×D₅	—	2	4	no	—
80	C₂²×C₃×D₅	—	2	8	no	—
81	C₂²×C₃×D₅	—	8	4	no	—
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	#41	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₁₅	C₂×D₄	1	4	yes	—
87	C₂×C₅×Dic₃	—	2	4	no	—
88	C₃×D₂₀	—	2	8	no	—
89	C₃×D₂₀	—	2	4	no	—
90	C₃×D₂₀	C₂²×C₄	1	8	yes	—
91	D₅×Dic₃	—	8	8	no	—
92	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₃⋊#168	—	2	2	no	—
95	C₂²×C₃×D₄	—	5	2	no	—
96	C₂×C₃×D₈	—	5	8	no	—
97	C₂×C₃×D₈	—	5	2	no	—
98	C₂×C₃×D₈	—	5	2	no	—
99	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
100	C₂×#217	—	10	4	no	—
101	D₄×Dic₃	—	5	8	no	—
102	D₄×Dic₃	—	10	4	no	—
103	#926	—	5	1	no	—
104	#926	C₂×D₅	1	2	yes	—
105	C₂×#208	—	5	1	no	—
106	#832	—	5	8	no	—
107	#818	C₂×D₅	1	2	yes	complemented
108	#817	C₂×D₅	1	2	yes	complemented
109	#815	—	5	2	no	—
110	C₈×Dic₃	—	5	2	no	—
111	C₂³×D₅	—	2	2	no	—
112	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₂×D₂₀	—	2	2	no	—
114	C₂×D₂₀	—	2	4	no	—
115	C₂×D₂₀	C₂×Dic₃	1	8	yes	complemented
116	C₂×D₂₀	C₂²×D₃	1	4	yes	—
117	C₂×C₄×D₅	—	6	8	no	—
118	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	complemented
119	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
121	C₅×C₄⋊C₄	—	3	4	no	—
122	C₅×C₂²⋊C₄	—	6	2	no	—
123	C₄²×C₅	—	6	2	no	—
124	C₂²⋊Dic₅	—	3	2	no	—
125	#685	—	6	8	no	—
126	C₄⋊Dic₅	—	3	2	no	—
127	C₄×Dic₅	—	3	2	no	—
128	#681	—	6	2	no	—
129	C₂×#168	—	3	2	no	—
130	D₄₀	C₂×Dic₃	1	8	yes	complemented
131	D₄₀	C₂×Dic₃	1	8	yes	complemented
132	C₂²×D₈	—	5	1	no	—
133	#518	—	15	2	no	—
134	#442	—	15	4	no	—
135	C₄×D₈	—	15	4	no	—
136	#427	—	3	1	no	—
137	#418	—	15	2	no	—
138	#405	—	15	1	no	—
139	C₂²×C₃×C₅	—	2	2	no	—
140	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
141	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₃×D₅	—	2	8	no	—
143	C₂×C₃×D₅	—	4	8	no	—
144	C₂×C₃×D₅	—	8	8	no	—
145	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
146	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
147	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
148	Dic₁₅	—	2	4	no	—
149	C₅×Dic₃	—	4	4	no	—
150	C₂⁴×C₃	—	10	2	no	—
151	C₂×C₃×D₄	—	5	8	no	—
152	C₂×C₃×D₄	—	5	4	no	—
153	C₂×C₃×D₄	—	10	2	no	—
154	C₂×C₃×D₄	—	10	4	no	—
155	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
156	C₂²×Dic₃	—	10	8	no	—
157	C₃×D₈	—	5	8	no	—
158	C₃×D₈	—	5	8	no	—
159	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
160	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
161	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
162	#217	—	2	2	no	—
163	#217	—	5	2	no	—
164	#217	—	10	8	no	—
165	C₄⋊Dic₃	—	5	2	no	—
166	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
167	C₄×Dic₃	—	2	2	no	—
168	C₄×Dic₃	—	5	2	no	—
169	#208	—	10	2	no	—
170	C₂×C₃⋊C₈	—	5	2	no	—
171	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
172	C₂²×D₅	—	2	4	no	—
173	C₂²×D₅	—	2	8	no	—
174	C₂²×D₅	—	8	4	no	—
175	C₂×C₄×C₅	—	6	4	no	—
176	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	#237	1	2	yes	—
178	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
179	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
180	C₂×Dic₅	—	3	4	no	—
181	D₂₀	—	2	8	no	—
182	D₂₀	—	2	4	no	—
183	D₂₀	C₂²×Dic₃	1	8	yes	—
184	C₄×D₅	—	24	8	no	—
185	C₅×C₈	C₂²×Dic₃	1	2	yes	—
186	C₅×C₈	C₂²×Dic₃	1	2	yes	—
187	#168	—	6	2	no	—
188	C₂²×D₄	—	5	2	no	—
189	C₂×D₈	—	5	8	no	—
190	C₂×D₈	—	5	2	no	—
191	C₂×D₈	—	5	2	no	—
192	C₂×M₁₆	—	15	1	no	—
193	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
194	C₄×D₄	—	15	8	no	—
195	C₄×D₄	—	30	4	no	—
196	#117	—	3	2	no	—
197	#117	—	15	1	no	—
198	C₂×C₂²⋊C₄	—	30	4	no	—
199	#107	—	3	2	no	—
200	#106	—	3	2	no	—
201	#102	—	15	8	no	—
202	#97	—	15	2	no	—
203	C₄×C₈	—	15	2	no	—
204	C₂×C₃×C₅	—	2	2	no	—
205	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂×C₃×C₅	#576	1	2	yes	—
207	C₃×D₅	—	4	8	no	—
208	C₂³×C₃	—	10	4	no	—
209	C₂³×C₃	—	10	8	no	—
210	C₂³×C₃	—	40	4	no	—
211	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
212	C₃×D₄	—	5	8	no	—
213	C₃×D₄	—	10	4	no	—
214	C₃×D₄	—	10	8	no	—
215	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
216	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
217	C₂×C₃×C₄	#713	1	2	yes	—
218	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
219	C₂×Dic₃	—	2	4	no	—
220	C₂×Dic₃	—	5	4	no	—
221	C₂×Dic₃	—	40	8	no	—
222	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
223	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
224	C₃⋊C₈	—	10	2	no	—
225	C₂²×C₅	—	2	2	no	—
226	C₂²×C₅	D₄×Dic₃	1	2	yes	—
227	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
228	C₂×D₅	—	2	8	no	—
229	C₂×D₅	—	4	8	no	—
230	C₂×D₅	—	8	8	no	—
231	C₄×C₅	—	12	4	no	—
232	C₄×C₅	D₄×Dic₃	1	2	yes	—
233	C₄×C₅	C₂×#237	1	1	yes	characteristic
234	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
235	Dic₅	—	6	4	no	—
236	C₂⁴	—	10	2	no	—
237	C₂×D₄	—	5	8	no	—
238	C₂×D₄	—	5	4	no	—
239	C₂×D₄	—	10	4	no	—
240	C₂×D₄	—	10	2	no	—
241	C₂²×C₄	—	30	8	no	—
242	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
243	D₈	—	5	8	no	—
244	D₈	—	5	8	no	—
245	M₁₆	—	30	2	no	—
246	C₂×C₈	—	15	2	no	—
247	C₂×C₈	D₅×Dic₃	1	4	yes	complemented
248	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
249	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
250	C₄⋊C₄	—	3	4	no	—
251	C₄⋊C₄	—	15	2	no	—
252	C₂²⋊C₄	—	6	2	no	—
253	C₂²⋊C₄	—	15	2	no	—
254	C₂²⋊C₄	—	30	8	no	—
255	C₄²	—	6	2	no	—
256	C₄²	—	15	2	no	—
257	C₃×C₅	#2948	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	10	8	no	—
260	C₂²×C₃	—	20	8	no	—
261	C₂²×C₃	—	40	8	no	—
262	C₂²×C₃	#4039	1	2	yes	—
263	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
264	C₃×C₄	#4039	1	2	yes	—
265	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
266	C₃×C₄	C₂×#713	1	1	yes	characteristic
267	Dic₃	—	4	4	no	—
268	Dic₃	—	10	4	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	#5757	1	2	yes	—
271	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
272	D₅	—	4	8	no	—
273	C₂³	—	10	4	no	—
274	C₂³	—	10	8	no	—
275	C₂³	—	40	4	no	—
276	C₂³	#6777	1	1	yes	characteristic
277	D₄	—	5	8	no	—
278	D₄	—	10	8	no	—
279	D₄	—	10	4	no	—
280	C₂×C₄	—	6	4	no	—
281	C₂×C₄	—	15	4	no	—
282	C₂×C₄	—	120	8	no	—
283	C₂×C₄	#6766	1	2	yes	—
284	C₂×C₄	#6777	1	1	yes	characteristic
285	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
286	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
287	C₈	—	30	2	no	—
288	C₈	C₂×D₅×Dic₃	1	2	yes	—
289	C₈	C₂×D₅×Dic₃	1	2	yes	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	20	8	no	—
292	C₂×C₃	C₂×#4039	1	1	yes	characteristic
293	C₂×C₃	#66103	1	2	yes	—
294	C₅	#83823	1	1	yes	characteristic, complemented, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	10	8	no	—
297	C₂²	—	20	8	no	—
298	C₂²	—	40	8	no	—
299	C₂²	#91434	1	2	yes	—
300	C₂²	C₂×#6777	1	1	yes	characteristic
301	C₄	—	12	4	no	—
302	C₄	—	30	4	no	—
303	C₄	#91434	1	2	yes	—
304	C₄	C₂×#6766	1	1	yes	characteristic
305	C₄	C₂²×D₅×Dic₃	1	1	yes	characteristic
306	C₃	#10610898	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	2	2	no	—
308	C₂	—	20	8	no	—
309	C₂	#11753981	1	2	yes	—
310	C₂	C₂×#91434	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6777
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₃×D₄₀	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2948
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #11754488	C₂	semidirect
3. #11751202	C₂	semidirect
4. C₅×#5106	C₂	semidirect
5. #11748366	C₂	semidirect
6. #11747815	C₂	semidirect
7. #11747809	C₂	semidirect
14. C₂×C₃×D₄₀	C₄	semidirect
18. #91434	C₂²	semidirect
20. #91099	C₂²	semidirect
21. C₈×Dic₁₅	C₂²	semidirect
22. C₅×#818	C₂²	semidirect
23. C₅×#817	C₂²	semidirect
24. #90964	C₂²	semidirect
37. C₂×C₃×D₂₀	C₂×C₄	semidirect
40. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
51. C₃×D₄₀	C₂×C₄	semidirect
52. C₃×D₄₀	C₂×C₄	semidirect
59. #5106	D₅	semidirect
66. C₂×D₄₀	Dic₃	semidirect
107. #818	C₂×D₅	semidirect
108. #817	C₂×D₅	semidirect
115. C₂×D₂₀	C₂×Dic₃	semidirect
118. C₂×C₅×C₈	C₂×Dic₃	semidirect
130. D₄₀	C₂×Dic₃	semidirect
131. D₄₀	C₂×Dic₃	semidirect
159. C₂×C₃×C₈	C₄×D₅	semidirect
247. C₂×C₈	D₅×Dic₃	semidirect
257. C₃×C₅	#2948	semidirect
294. C₅	#83823	semidirect
306. C₃	#10610898	semidirect

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
#11754488	2
#11751202	2
C₅×#5106	2
#11748366	2
#11747815	2
#11747809	2
#10610898	3
#83823	5