Subgroups — #14285615

Order 1920.

360 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	C₄×C₅×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	D₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
3	#11746853	C₂	1	1	yes	maximal, characteristic
4	#11746278	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64678	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745869	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745866	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600643	—	3	1	no	maximal, Hall
9	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
10	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
11	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
12	C₂×D₃×#168	—	2	1	no	—
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
16	C₅×#815	C₄	1	1	yes	characteristic
17	C₅×#815	C₄	1	1	yes	characteristic
18	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
21	Dic₃⋊#168	—	2	1	no	—
22	(C₂×D₃)⋊#168	—	2	1	no	—
23	Dic₃×#168	—	2	1	no	—
24	#68568	—	5	1	no	maximal, Hall
25	C₄×C₅×M₁₆	—	3	1	no	—
26	#65214	—	3	1	no	—
27	C₂×C₄⋊#168	—	3	1	no	—
28	#64678	—	3	1	no	—
29	#64678	D₃	1	1	yes	characteristic, complemented
30	#64677	—	3	1	no	—
31	#64677	—	3	1	no	—
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
34	C₂×C₄×C₅×D₃	Q₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊#168	—	2	1	no	—
39	C₂×C₃⋊#168	—	2	1	no	—
40	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
42	C₄×C₅×Dic₃	Q₈	1	1	yes	characteristic
43	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
46	C₅×ΓL(1,𝔽₂₅)	C₈	1	4	yes	complemented
47	C₂×C₃×#168	—	2	1	no	—
48	C₂×C₃×#168	—	2	1	no	—
49	D₃×#168	—	4	2	no	—
50	#4833	—	5	1	no	—
51	D₃×C₄⋊C₈	—	5	1	no	—
52	C₄×ΓL(1,𝔽₂₅)	D₅	1	1	yes	characteristic
53	C₃×#335	—	5	1	no	—
54	#4480	—	5	1	no	—
55	#4479	—	5	1	no	—
56	#4457	—	5	1	no	—
57	C₂×C₅×M₁₆	—	3	1	no	—
58	C₂×C₅×M₁₆	—	3	1	no	—
59	C₂×C₄²×C₅	—	3	1	no	—
60	C₂²×#168	—	6	1	no	—
61	C₅×#97	—	3	1	no	—
62	C₅×#97	—	3	1	no	—
63	C₄×C₅×C₈	—	3	1	no	—
64	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
65	C₂²⋊#168	—	6	1	no	—
66	C₄⋊#168	—	3	1	no	—
67	C₄⋊#168	—	3	1	no	—
68	C₄⋊#168	—	6	1	no	—
69	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
70	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
71	C₄×#168	—	6	1	no	—
72	#1565	—	15	1	no	Sylow
73	C₂²×C₅×D₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₅×D₃	—	2	1	no	—
81	C₄×C₅×D₃	—	2	1	no	—
82	C₄×C₅×D₃	—	4	1	no	—
83	C₄×C₅×D₃	C₂×C₈	1	2	yes	—
84	C₄×C₅×D₃	M₁₆	1	2	yes	—
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
87	C₃⋊#168	—	4	1	no	—
88	C₃⋊#168	—	4	1	no	—
89	C₃×#168	—	4	1	no	—
90	C₃×#168	—	4	1	no	—
91	C₅×C₃⋊C₈	—	2	1	no	—
92	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	complemented
93	C₂×ΓL(1,𝔽₂₅)	Dic₅	1	1	yes	characteristic
94	C₂×ΓL(1,𝔽₂₅)	Dic₅	1	1	yes	characteristic
95	C₂×D₃×C₈	—	10	1	no	—
96	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	—	5	1	no	—
98	C₃×C₄⋊C₈	—	5	1	no	—
99	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
100	(C₂×D₃)⋊C₈	—	10	1	no	—
101	#815	Dic₅	1	1	yes	characteristic
102	#815	Dic₅	1	1	yes	characteristic
103	Dic₃⋊C₈	—	10	1	no	—
104	C₈×Dic₃	—	10	1	no	—
105	C₄⋊(C₃⋊C₈)	—	5	1	no	—
106	C₄⋊(C₃⋊C₈)	—	5	1	no	—
107	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
108	C₂²×C₄×C₅	—	3	1	no	—
109	C₂²×C₄×C₅	—	3	1	no	—
110	C₂²×C₄×C₅	—	3	1	no	—
111	C₅×M₁₆	—	3	4	no	—
112	C₅×M₁₆	—	6	2	no	—
113	C₂×C₅×C₈	—	3	1	no	—
114	C₂×C₅×C₈	—	3	1	no	—
115	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
116	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
117	C₄²×C₅	—	3	1	no	—
118	C₄²×C₅	—	3	1	no	—
119	C₄²×C₅	—	3	1	no	—
120	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
121	C₂×#168	—	2	1	no	—
122	C₂×#168	—	2	1	no	—
123	C₂×#168	—	6	1	no	—
124	C₂×#168	—	6	1	no	—
125	C₂×#168	—	12	2	no	—
126	#431	—	15	1	no	—
127	C₂×C₄⋊C₈	—	15	1	no	—
128	C₄×M₁₆	—	3	1	no	—
129	#335	—	5	1	no	—
130	#335	—	15	1	no	—
131	#334	—	15	1	no	—
132	#334	—	15	1	no	—
133	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₅×D₃	—	2	1	no	—
135	C₂×C₅×D₃	—	2	1	no	—
136	C₂×C₅×D₃	C₄⋊C₈	1	2	yes	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
140	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₃×C₄×C₅	#137	1	1	yes	characteristic
142	C₅×Dic₃	—	2	1	no	—
143	C₅×Dic₃	—	2	1	no	—
144	C₅×Dic₃	C₄⋊C₈	1	2	yes	—
145	C₂×C₄×D₃	Dic₁₀	1	1	yes	characteristic
146	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
147	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
148	C₂×C₃×C₈	—	10	1	no	—
149	C₂×C₃×C₈	—	10	1	no	—
150	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
151	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
152	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
153	C₄×Dic₃	Dic₁₀	1	1	yes	characteristic
154	C₄×Dic₃	D₂₀	1	1	yes	characteristic
155	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
156	C₂×C₃⋊C₈	—	10	1	no	—
157	C₂×C₃⋊C₈	—	10	1	no	—
158	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
159	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
160	ΓL(1,𝔽₂₅)	—	2	2	no	—
161	ΓL(1,𝔽₂₅)	#168	1	4	yes	complemented
162	D₃×C₈	—	20	2	no	—
163	C₂³×C₅	—	3	1	no	—
164	C₂×C₄×C₅	—	3	1	no	—
165	C₂×C₄×C₅	—	3	2	no	—
166	C₂×C₄×C₅	—	3	2	no	—
167	C₂×C₄×C₅	—	3	1	no	—
168	C₂×C₄×C₅	—	3	1	no	—
169	C₂×C₄×C₅	—	6	1	no	—
170	C₂×C₄×C₅	—	6	1	no	—
171	C₂×C₄×C₅	—	12	1	no	—
172	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	—	3	2	no	—
177	C₅×C₈	—	6	1	no	—
178	C₅×C₈	D₃×C₈	1	2	yes	complemented
179	#168	—	4	1	no	—
180	#168	—	4	1	no	—
181	#168	—	12	1	no	—
182	#168	—	12	1	no	—
183	C₂×M₁₆	—	3	1	no	—
184	C₂×M₁₆	—	3	1	no	—
185	C₂²×C₈	—	30	1	no	—
186	C₂×C₄²	—	3	1	no	—
187	C₄⋊C₈	—	5	1	no	—
188	C₄⋊C₈	—	5	1	no	—
189	C₄⋊C₈	—	15	1	no	—
190	C₄⋊C₈	—	15	1	no	—
191	C₄⋊C₈	—	30	1	no	—
192	C₂²⋊C₈	—	30	1	no	—
193	#97	—	3	1	no	—
194	#97	—	3	1	no	—
195	C₄×C₈	—	3	1	no	—
196	C₄×C₈	—	30	1	no	—
197	C₄×C₈	D₃×D₅	1	1	yes	characteristic
198	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
199	C₂×C₃×C₅	#428	1	1	yes	characteristic
200	C₂×C₃×C₅	#430	1	1	yes	characteristic
201	C₅×D₃	—	2	2	no	—
202	C₂²×D₃	C₄⋊Dic₅	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
206	C₂×Dic₃	C₄⋊Dic₅	1	1	yes	characteristic
207	C₂×Dic₃	C₄⋊Dic₅	1	1	yes	characteristic
208	C₂×Dic₃	C₄⋊Dic₅	1	1	yes	characteristic
209	C₄×D₃	—	2	1	no	—
210	C₄×D₃	—	2	1	no	—
211	C₄×D₃	—	4	1	no	—
212	C₄×D₃	C₂×#168	1	2	yes	—
213	C₄×D₃	#681	1	2	yes	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	20	1	no	—
216	C₃×C₈	—	20	1	no	—
217	C₃×C₈	C₂×#168	1	2	yes	complemented
218	C₃⋊C₈	—	2	1	no	—
219	C₃⋊C₈	—	20	1	no	—
220	C₃⋊C₈	—	20	1	no	—
221	C₃⋊C₈	C₂×#168	1	2	yes	complemented
222	C₂²×C₅	—	3	2	no	—
223	C₂²×C₅	—	6	1	no	—
224	C₂²×C₅	—	6	1	no	—
225	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	3	2	no	—
228	C₄×C₅	—	6	1	no	—
229	C₄×C₅	—	6	1	no	—
230	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
231	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
232	C₄×C₅	#911	1	1	yes	characteristic
233	C₄×C₅	#918	1	1	yes	characteristic
234	C₂²×C₄	—	3	1	no	—
235	C₂²×C₄	—	3	1	no	—
236	C₂²×C₄	—	3	1	no	—
237	M₁₆	—	3	4	no	—
238	M₁₆	—	6	2	no	—
239	C₂×C₈	—	3	1	no	—
240	C₂×C₈	—	3	1	no	—
241	C₂×C₈	—	10	1	no	—
242	C₂×C₈	—	10	1	no	—
243	C₂×C₈	—	30	1	no	—
244	C₂×C₈	—	30	1	no	—
245	C₂×C₈	—	60	2	no	—
246	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
247	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
248	C₄²	—	3	1	no	—
249	C₄²	—	3	1	no	—
250	C₄²	—	3	1	no	—
251	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
252	C₃×C₅	#1565	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	C₄⋊#168	1	2	yes	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	#4054	1	1	yes	characteristic
259	C₃×C₄	#4055	1	1	yes	characteristic
260	C₃×C₄	C₂²×#168	1	1	yes	characteristic
261	C₃×C₄	C₂×#681	1	1	yes	characteristic
262	Dic₃	—	2	1	no	—
263	Dic₃	—	2	1	no	—
264	Dic₃	C₄⋊#168	1	2	yes	—
265	C₂×C₅	—	6	2	no	—
266	C₂×C₅	D₃×C₄⋊C₈	1	1	yes	characteristic
267	C₂×C₅	#4881	1	1	yes	characteristic
268	C₂×C₅	#4893	1	1	yes	characteristic
269	C₂³	—	3	1	no	—
270	C₂×C₄	—	3	1	no	—
271	C₂×C₄	—	3	2	no	—
272	C₂×C₄	—	3	2	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	3	1	no	—
275	C₂×C₄	—	6	1	no	—
276	C₂×C₄	—	6	1	no	—
277	C₂×C₄	—	12	1	no	—
278	C₂×C₄	D₃×Dic₁₀	1	1	yes	characteristic
279	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
280	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	3	2	no	—
283	C₈	—	6	1	no	—
284	C₈	—	20	1	no	—
285	C₈	—	20	1	no	—
286	C₈	—	60	1	no	—
287	C₈	—	60	1	no	—
288	C₈	D₃×#168	1	2	yes	complemented
289	C₂×C₃	C₂×C₄⋊#168	1	1	yes	characteristic
290	C₂×C₃	#65410	1	1	yes	characteristic
291	C₂×C₃	#65414	1	1	yes	characteristic
292	D₃	—	2	2	no	—
293	C₅	#68568	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	3	2	no	—
295	C₂²	—	6	1	no	—
296	C₂²	—	6	1	no	—
297	C₂²	D₃×C₄⋊Dic₅	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	3	2	no	—
300	C₄	—	6	1	no	—
301	C₄	—	6	1	no	—
302	C₄	#91254	1	1	yes	characteristic
303	C₄	#91263	1	1	yes	characteristic
304	C₄	C₂×D₃×#168	1	1	yes	characteristic
305	C₄	D₃×#681	1	1	yes	characteristic
306	C₃	#10600643	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	6	2	no	—
308	C₂	D₃×C₄⋊#168	1	1	yes	characteristic
309	C₂	#11747630	1	1	yes	characteristic
310	C₂	#11748059	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₂₀
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₅×D₃	C₄⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1565
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #11746278	C₂	semidirect
5. C₃×#64678	C₂	semidirect
6. #11745869	C₂	semidirect
7. #11745866	C₂	semidirect
29. #64678	D₃	semidirect
46. C₅×ΓL(1,𝔽₂₅)	C₈	semidirect
86. C₃×C₅×C₈	C₂×C₈	semidirect
92. C₅×C₃⋊C₈	C₂×C₈	semidirect
161. ΓL(1,𝔽₂₅)	#168	semidirect
178. C₅×C₈	D₃×C₈	semidirect
217. C₃×C₈	C₂×#168	semidirect
221. C₃⋊C₈	C₂×#168	semidirect
252. C₃×C₅	#1565	semidirect
288. C₈	D₃×#168	semidirect
293. C₅	#68568	semidirect
306. C₃	#10600643	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×ΓL(1,𝔽₂₅)	2
D₃×C₄⋊#168	2
#11746853	2
#11746278	2
C₃×#64678	2
#11745869	2
#11745866	2
#10600643	3
#68568	5