Subgroups — #14285617

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	C₃×C₄×#676	C₂	1	1	yes	maximal, characteristic
2	D₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#11746869	C₂	1	1	yes	maximal, characteristic
4	#11746278	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745867	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745864	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599361	—	3	1	no	maximal, Hall
9	C₂×C₃×#676	C₄	1	1	yes	characteristic
10	C₂×C₃×#676	C₄	1	1	yes	characteristic
11	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
12	C₂×D₅×C₃⋊C₈	—	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×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃×#3947	C₄	1	1	yes	characteristic
19	C₃×#3947	C₄	1	1	yes	characteristic
20	C₃×C₄×#168	C₂²	1	1	yes	characteristic
21	#90982	—	2	1	no	—
22	#90955	—	2	1	no	—
23	Dic₅×C₃⋊C₈	—	2	1	no	—
24	#69850	—	5	1	no	maximal, Hall
25	#65122	—	3	1	no	—
26	D₅×C₄⋊C₈	—	3	1	no	—
27	C₄×#676	D₃	1	1	yes	characteristic
28	C₅×#335	—	3	1	no	—
29	#64700	—	3	1	no	—
30	#64699	—	3	1	no	—
31	#64677	—	3	1	no	—
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	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	C₂×C₅×C₃⋊C₈	—	2	1	no	—
41	C₂×C₅×C₃⋊C₈	—	2	1	no	—
42	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
44	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
45	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₃×#676	—	2	2	no	—
48	C₃×#676	C₈	1	4	yes	complemented
49	D₅×C₃⋊C₈	—	4	2	no	—
50	C₃×C₄×M₁₆	—	5	1	no	—
51	#4925	—	5	1	no	—
52	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
53	#4458	—	5	1	no	—
54	#4458	D₅	1	1	yes	characteristic, complemented
55	#4457	—	5	1	no	—
56	#4457	—	5	1	no	—
57	C₂×#676	Dic₃	1	1	yes	characteristic
58	C₂×#676	Dic₃	1	1	yes	characteristic
59	C₂×C₈×D₅	—	6	1	no	—
60	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
61	C₅×C₄⋊C₈	—	3	1	no	—
62	C₅×C₄⋊C₈	—	3	1	no	—
63	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
64	#3952	—	6	1	no	—
65	#3947	Dic₃	1	1	yes	characteristic
66	#3947	Dic₃	1	1	yes	characteristic
67	#3946	—	6	1	no	—
68	C₈×Dic₅	—	6	1	no	—
69	C₄⋊#168	—	3	1	no	—
70	C₄⋊#168	—	3	1	no	—
71	C₄×#168	C₂×D₃	1	1	yes	characteristic
72	#1565	—	15	1	no	Sylow
73	C₂²×C₃×D₅	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×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
80	C₃×C₄×D₅	—	2	1	no	—
81	C₃×C₄×D₅	—	2	1	no	—
82	C₃×C₄×D₅	—	4	1	no	—
83	C₃×C₄×D₅	C₂×C₈	1	2	yes	—
84	C₃×C₄×D₅	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	—	2	1	no	—
90	C₃×#168	C₂×C₈	1	2	yes	complemented
91	C₅×C₃⋊C₈	—	4	1	no	—
92	C₅×C₃⋊C₈	—	4	1	no	—
93	C₂×C₃×M₁₆	—	5	1	no	—
94	C₂×C₃×M₁₆	—	5	1	no	—
95	C₂×C₃×C₄²	—	5	1	no	—
96	C₂²×C₃⋊C₈	—	10	1	no	—
97	C₃×#97	—	5	1	no	—
98	C₃×#97	—	5	1	no	—
99	C₃×C₄×C₈	—	5	1	no	—
100	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
101	#830	—	10	1	no	—
102	C₄⋊(C₃⋊C₈)	—	5	1	no	—
103	C₄⋊(C₃⋊C₈)	—	5	1	no	—
104	C₄⋊(C₃⋊C₈)	—	10	1	no	—
105	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
106	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
107	C₄×C₃⋊C₈	—	10	1	no	—
108	C₂×C₄×D₅	Dic₆	1	1	yes	characteristic
109	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
110	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₅×C₈	—	6	1	no	—
112	C₂×C₅×C₈	—	6	1	no	—
113	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
115	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₄×Dic₅	Dic₆	1	1	yes	characteristic
117	C₄×Dic₅	D₁₂	1	1	yes	characteristic
118	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
119	C₂×#168	—	6	1	no	—
120	C₂×#168	—	6	1	no	—
121	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
122	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
123	#676	—	2	2	no	—
124	#676	C₃⋊C₈	1	4	yes	complemented
125	C₈×D₅	—	12	2	no	—
126	#431	—	15	1	no	—
127	C₂×C₄⋊C₈	—	15	1	no	—
128	C₄×M₁₆	—	5	1	no	—
129	#335	—	3	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×C₃×D₅	—	2	1	no	—
135	C₂×C₃×D₅	—	2	1	no	—
136	C₂×C₃×D₅	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	C₃×Dic₅	—	2	1	no	—
143	C₃×Dic₅	—	2	1	no	—
144	C₃×Dic₅	C₄⋊C₈	1	2	yes	—
145	C₂²×C₃×C₄	—	5	1	no	—
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	5	1	no	—
148	C₃×M₁₆	—	5	4	no	—
149	C₃×M₁₆	—	10	2	no	—
150	C₂×C₃×C₈	—	5	1	no	—
151	C₂×C₃×C₈	—	5	1	no	—
152	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
153	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
154	C₃×C₄²	—	5	1	no	—
155	C₃×C₄²	—	5	1	no	—
156	C₃×C₄²	—	5	1	no	—
157	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
158	C₂×C₃⋊C₈	—	2	1	no	—
159	C₂×C₃⋊C₈	—	2	1	no	—
160	C₂×C₃⋊C₈	—	10	1	no	—
161	C₂×C₃⋊C₈	—	10	1	no	—
162	C₂×C₃⋊C₈	—	20	2	no	—
163	C₂²×D₅	C₄⋊Dic₃	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
167	C₂×Dic₅	C₄⋊Dic₃	1	1	yes	characteristic
168	C₂×Dic₅	C₄⋊Dic₃	1	1	yes	characteristic
169	C₂×Dic₅	C₄⋊Dic₃	1	1	yes	characteristic
170	C₄×D₅	—	2	1	no	—
171	C₄×D₅	—	2	1	no	—
172	C₄×D₅	—	4	1	no	—
173	C₄×D₅	C₂×C₃⋊C₈	1	2	yes	—
174	C₄×D₅	#208	1	2	yes	—
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	—	12	1	no	—
177	C₅×C₈	—	12	1	no	—
178	C₅×C₈	C₂×C₃⋊C₈	1	2	yes	complemented
179	#168	—	2	1	no	—
180	#168	—	12	1	no	—
181	#168	—	12	1	no	—
182	#168	C₂×C₃⋊C₈	1	2	yes	complemented
183	C₂×M₁₆	—	5	1	no	—
184	C₂×M₁₆	—	5	1	no	—
185	C₂²×C₈	—	30	1	no	—
186	C₂×C₄²	—	5	1	no	—
187	C₄⋊C₈	—	3	1	no	—
188	C₄⋊C₈	—	3	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	—	5	1	no	—
194	#97	—	5	1	no	—
195	C₄×C₈	—	5	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	C₃×D₅	—	2	2	no	—
202	C₂³×C₃	—	5	1	no	—
203	C₂×C₃×C₄	—	5	1	no	—
204	C₂×C₃×C₄	—	5	2	no	—
205	C₂×C₃×C₄	—	5	2	no	—
206	C₂×C₃×C₄	—	5	1	no	—
207	C₂×C₃×C₄	—	5	1	no	—
208	C₂×C₃×C₄	—	10	1	no	—
209	C₂×C₃×C₄	—	10	1	no	—
210	C₂×C₃×C₄	—	20	1	no	—
211	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
212	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
213	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	5	2	no	—
216	C₃×C₈	—	10	1	no	—
217	C₃×C₈	C₈×D₅	1	2	yes	complemented
218	C₃⋊C₈	—	4	1	no	—
219	C₃⋊C₈	—	4	1	no	—
220	C₃⋊C₈	—	20	1	no	—
221	C₃⋊C₈	—	20	1	no	—
222	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
223	C₂×D₅	—	2	1	no	—
224	C₂×D₅	—	2	1	no	—
225	C₂×D₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	#908	1	1	yes	characteristic
228	C₄×C₅	#909	1	1	yes	characteristic
229	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
230	C₄×C₅	C₂×#208	1	1	yes	characteristic
231	Dic₅	—	2	1	no	—
232	Dic₅	—	2	1	no	—
233	Dic₅	C₄⋊(C₃⋊C₈)	1	2	yes	—
234	C₂²×C₄	—	5	1	no	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	M₁₆	—	5	4	no	—
238	M₁₆	—	10	2	no	—
239	C₂×C₈	—	5	1	no	—
240	C₂×C₈	—	5	1	no	—
241	C₂×C₈	—	6	1	no	—
242	C₂×C₈	—	6	1	no	—
243	C₂×C₈	—	30	1	no	—
244	C₂×C₈	—	30	1	no	—
245	C₂×C₈	—	60	2	no	—
246	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
247	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
248	C₄²	—	5	1	no	—
249	C₄²	—	5	1	no	—
250	C₄²	—	5	1	no	—
251	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
252	C₃×C₅	#1565	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	5	2	no	—
254	C₂²×C₃	—	10	1	no	—
255	C₂²×C₃	—	10	1	no	—
256	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	5	2	no	—
259	C₃×C₄	—	10	1	no	—
260	C₃×C₄	—	10	1	no	—
261	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
262	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
263	C₃×C₄	#4057	1	1	yes	characteristic
264	C₃×C₄	#4064	1	1	yes	characteristic
265	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
266	C₂×C₅	#5121	1	1	yes	characteristic
267	C₂×C₅	#5125	1	1	yes	characteristic
268	D₅	—	2	2	no	—
269	C₂³	—	5	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	2	no	—
272	C₂×C₄	—	5	2	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	10	1	no	—
276	C₂×C₄	—	10	1	no	—
277	C₂×C₄	—	20	1	no	—
278	C₂×C₄	D₅×Dic₆	1	1	yes	characteristic
279	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
280	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	5	2	no	—
283	C₈	—	10	1	no	—
284	C₈	—	12	1	no	—
285	C₈	—	12	1	no	—
286	C₈	—	60	1	no	—
287	C₈	—	60	1	no	—
288	C₈	D₅×C₃⋊C₈	1	2	yes	complemented
289	C₂×C₃	—	10	2	no	—
290	C₂×C₃	D₅×C₄⋊C₈	1	1	yes	characteristic
291	C₂×C₃	#65170	1	1	yes	characteristic
292	C₂×C₃	#65182	1	1	yes	characteristic
293	C₅	#69850	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	5	2	no	—
295	C₂²	—	10	1	no	—
296	C₂²	—	10	1	no	—
297	C₂²	D₅×C₄⋊Dic₃	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	5	2	no	—
300	C₄	—	10	1	no	—
301	C₄	—	10	1	no	—
302	C₄	#91250	1	1	yes	characteristic
303	C₄	#91261	1	1	yes	characteristic
304	C₄	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
305	C₄	D₅×#208	1	1	yes	characteristic
306	C₃	#10599361	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	10	2	no	—
308	C₂	D₅×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
309	C₂	#11747619	1	1	yes	characteristic
310	C₂	#11748057	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×D₁₂
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×C₃×D₅	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. C₅×#4458	C₂	semidirect
6. #11745867	C₂	semidirect
7. #11745864	C₂	semidirect
48. C₃×#676	C₈	semidirect
54. #4458	D₅	semidirect
86. C₃×C₅×C₈	C₂×C₈	semidirect
90. C₃×#168	C₂×C₈	semidirect
124. #676	C₃⋊C₈	semidirect
178. C₅×C₈	C₂×C₃⋊C₈	semidirect
182. #168	C₂×C₃⋊C₈	semidirect
217. C₃×C₈	C₈×D₅	semidirect
252. C₃×C₅	#1565	semidirect
288. C₈	D₅×C₃⋊C₈	semidirect
293. C₅	#69850	semidirect
306. C₃	#10599361	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#676	2
D₅×C₄⋊(C₃⋊C₈)	2
#11746869	2
#11746278	2
C₅×#4458	2
#11745867	2
#11745864	2
#10599361	3
#69850	5