Subgroups — #14286978

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	#11751113	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₃⋊#168)	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₃⋊#168)	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746278	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746275	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601673	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×D₈	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂×#6710	—	2	1	no	—
13	C₂²×C₃⋊#168	—	2	1	no	—
14	C₃×C₅×#107	C₄	1	1	yes	characteristic
15	C₃×C₅×#102	C₄	1	1	yes	characteristic
16	C₃×C₅×#102	C₄	1	1	yes	characteristic
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	#91114	—	2	1	no	—
19	#91114	—	2	1	no	—
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
23	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
24	#70880	—	5	1	no	maximal, Hall
25	C₄×C₅×D₈	D₃	1	1	yes	characteristic
26	#65301	—	3	1	no	—
27	D₄×#168	—	3	1	no	—
28	D₄⋊#168	—	3	1	no	—
29	D₄⋊#168	—	3	1	no	—
30	#64677	—	3	1	no	—
31	#64674	—	3	1	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₃×C₅×D₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
42	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	#6710	—	4	2	no	—
45	C₂×C₃⋊#168	—	2	1	no	—
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₂×C₃⋊#168	—	4	2	no	—
49	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
50	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
51	C₃×C₄×D₈	D₅	1	1	yes	characteristic
52	#5012	—	5	1	no	—
53	D₄×C₃⋊C₈	—	5	1	no	—
54	D₄⋊(C₃⋊C₈)	—	5	1	no	—
55	D₄⋊(C₃⋊C₈)	—	5	1	no	—
56	#4457	—	5	1	no	—
57	#4454	—	5	1	no	—
58	C₂×C₅×D₈	Dic₃	1	1	yes	characteristic
59	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
60	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
61	C₂×#681	—	6	1	no	—
62	C₂²×#168	—	6	1	no	—
63	C₅×#107	Dic₃	1	1	yes	characteristic
64	C₅×#102	Dic₃	1	1	yes	characteristic
65	C₅×#102	Dic₃	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₂²⋊#168	—	6	1	no	—
68	C₂²⋊#168	—	6	1	no	—
69	C₄⋊#168	—	3	1	no	—
70	C₄⋊#168	—	3	1	no	—
71	C₄×#168	—	3	1	no	—
72	C₄×#168	—	3	1	no	—
73	#1586	—	15	1	no	Sylow
74	C₂³×C₃×C₅	—	2	1	no	—
75	C₂³×C₃×C₅	—	2	1	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	—	2	1	no	—
78	C₃×C₅×D₄	—	2	1	no	—
79	C₃×C₅×D₄	M₁₆	1	2	yes	complemented
80	C₂×C₃×C₄×C₅	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	—	4	1	no	—
85	C₂×C₃×C₄×C₅	—	4	1	no	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
89	C₃×C₅×C₈	—	2	1	no	—
90	C₃×C₅×C₈	—	2	1	no	—
91	C₃⋊#168	—	2	2	no	—
92	C₃⋊#168	—	2	1	no	—
93	C₃⋊#168	—	2	1	no	—
94	C₃⋊#168	—	4	1	no	—
95	C₃⋊#168	—	4	1	no	—
96	C₂×C₃×D₈	Dic₅	1	1	yes	characteristic
97	C₃×C₄×D₄	C₂×D₅	1	1	yes	characteristic
98	C₃×C₄×D₄	C₂×D₅	1	1	yes	characteristic
99	C₂×#208	—	10	1	no	—
100	C₂²×C₃⋊C₈	—	10	1	no	—
101	C₃×#107	Dic₅	1	1	yes	characteristic
102	C₃×#102	Dic₅	1	1	yes	characteristic
103	C₃×#102	Dic₅	1	1	yes	characteristic
104	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
105	#830	—	10	1	no	—
106	#830	—	10	1	no	—
107	C₄⋊(C₃⋊C₈)	—	5	1	no	—
108	C₄⋊(C₃⋊C₈)	—	5	1	no	—
109	C₄×C₃⋊C₈	—	5	1	no	—
110	C₄×C₃⋊C₈	—	5	1	no	—
111	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
112	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
113	C₂²×C₄×C₅	—	2	1	no	—
114	C₂²×C₄×C₅	—	2	1	no	—
115	C₅×D₈	—	2	2	no	—
116	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
118	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
119	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
120	C₅×C₂²⋊C₄	—	2	1	no	—
121	C₅×C₂²⋊C₄	—	2	1	no	—
122	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
123	#681	—	12	2	no	—
124	C₂×#168	—	3	1	no	—
125	C₂×#168	—	3	1	no	—
126	C₂×#168	—	6	1	no	—
127	C₂×#168	—	6	1	no	—
128	C₂×#168	—	6	1	no	—
129	C₂×#168	—	12	2	no	—
130	C₄×D₈	D₁₅	1	1	yes	characteristic
131	#436	—	15	1	no	—
132	C₈×D₄	—	15	1	no	—
133	#334	—	15	1	no	—
134	D₄⋊C₈	—	15	1	no	—
135	D₄⋊C₈	—	15	1	no	—
136	#322	—	15	1	no	—
137	C₂²×C₃×C₅	—	2	2	no	—
138	C₂²×C₃×C₅	—	4	1	no	—
139	C₂²×C₃×C₅	—	4	1	no	—
140	C₂²×C₃×C₅	—	4	1	no	—
141	C₂²×C₃×C₅	—	4	1	no	—
142	C₂²×C₃×C₅	—	4	1	no	—
143	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	4	1	no	—
146	C₃×C₄×C₅	—	4	1	no	—
147	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
148	C₃×C₄×C₅	#131	1	1	yes	characteristic
149	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
152	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
153	C₂²×C₃×C₄	—	2	1	no	—
154	C₂²×C₃×C₄	—	2	1	no	—
155	C₃×D₈	—	2	2	no	—
156	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
157	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
158	C₃×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
159	C₃×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
160	C₃×C₂²⋊C₄	—	2	1	no	—
161	C₃×C₂²⋊C₄	—	2	1	no	—
162	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
163	#208	—	20	2	no	—
164	C₂×C₃⋊C₈	—	5	1	no	—
165	C₂×C₃⋊C₈	—	5	1	no	—
166	C₂×C₃⋊C₈	—	10	1	no	—
167	C₂×C₃⋊C₈	—	10	1	no	—
168	C₂×C₃⋊C₈	—	10	1	no	—
169	C₂×C₃⋊C₈	—	20	2	no	—
170	C₂³×C₅	—	2	1	no	—
171	C₂³×C₅	—	2	1	no	—
172	C₅×D₄	—	2	1	no	—
173	C₅×D₄	—	2	1	no	—
174	C₅×D₄	—	2	1	no	—
175	C₅×D₄	#208	1	2	yes	complemented
176	C₂×C₄×C₅	—	2	1	no	—
177	C₂×C₄×C₅	—	2	1	no	—
178	C₂×C₄×C₅	—	4	1	no	—
179	C₂×C₄×C₅	—	4	1	no	—
180	C₂×C₄×C₅	—	4	1	no	—
181	C₂×C₄×C₅	—	4	1	no	—
182	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅	#237	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
185	C₅×C₈	—	2	1	no	—
186	C₅×C₈	—	2	1	no	—
187	#168	—	6	2	no	—
188	#168	—	6	1	no	—
189	#168	—	6	1	no	—
190	#168	—	12	1	no	—
191	#168	—	12	1	no	—
192	C₂×D₈	Dic₁₅	1	1	yes	characteristic
193	C₂×M₁₆	—	30	1	no	—
194	C₂²×C₈	—	30	1	no	—
195	C₄×D₄	C₂×D₁₅	1	1	yes	characteristic
196	C₄×D₄	C₂×D₁₅	1	1	yes	characteristic
197	#107	Dic₁₅	1	1	yes	characteristic
198	C₄⋊C₈	—	15	1	no	—
199	C₄⋊C₈	—	15	1	no	—
200	#102	Dic₁₅	1	1	yes	characteristic
201	#102	Dic₁₅	1	1	yes	characteristic
202	C₂²⋊C₈	—	30	1	no	—
203	C₂²⋊C₈	—	30	1	no	—
204	C₄×C₈	—	15	1	no	—
205	C₄×C₈	—	15	1	no	—
206	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
207	C₂×C₃×C₅	—	4	2	no	—
208	C₂×C₃×C₅	—	8	1	no	—
209	C₂×C₃×C₅	#435	1	1	yes	characteristic
210	C₂×C₃×C₅	#442	1	1	yes	characteristic
211	C₂×C₃×C₅	#443	1	1	yes	characteristic
212	C₂³×C₃	—	2	1	no	—
213	C₂³×C₃	—	2	1	no	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	—	2	1	no	—
216	C₃×D₄	—	2	1	no	—
217	C₃×D₄	#681	1	2	yes	complemented
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	4	1	no	—
221	C₂×C₃×C₄	—	4	1	no	—
222	C₂×C₃×C₄	—	4	1	no	—
223	C₂×C₃×C₄	—	4	1	no	—
224	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	#711	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃⋊C₈	—	10	2	no	—
230	C₃⋊C₈	—	10	1	no	—
231	C₃⋊C₈	—	10	1	no	—
232	C₃⋊C₈	—	20	1	no	—
233	C₃⋊C₈	—	20	1	no	—
234	C₂²×C₅	—	2	2	no	—
235	C₂²×C₅	—	4	1	no	—
236	C₂²×C₅	—	4	1	no	—
237	C₂²×C₅	—	4	1	no	—
238	C₂²×C₅	—	4	1	no	—
239	C₂²×C₅	—	4	1	no	—
240	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	1	no	—
243	C₄×C₅	—	4	1	no	—
244	C₄×C₅	#911	1	1	yes	characteristic
245	C₄×C₅	#911	1	1	yes	characteristic
246	C₄×C₅	C₂×#208	1	1	yes	characteristic
247	C₄×C₅	#948	1	1	yes	characteristic
248	C₂×D₄	C₂×Dic₁₅	1	1	yes	characteristic
249	C₂×D₄	C₂×Dic₁₅	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	D₈	—	2	2	no	—
253	M₁₆	—	60	2	no	—
254	C₂×C₈	—	15	1	no	—
255	C₂×C₈	—	15	1	no	—
256	C₂×C₈	—	30	1	no	—
257	C₂×C₈	—	30	1	no	—
258	C₂×C₈	—	30	1	no	—
259	C₂×C₈	—	60	2	no	—
260	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
261	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
262	C₄⋊C₄	C₂×Dic₁₅	1	1	yes	characteristic
263	C₄⋊C₄	C₂×Dic₁₅	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₄²	C₂²×D₁₅	1	1	yes	characteristic
267	C₃×C₅	#1586	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	4	1	no	—
270	C₂²×C₃	—	4	1	no	—
271	C₂²×C₃	—	4	1	no	—
272	C₂²×C₃	—	4	1	no	—
273	C₂²×C₃	—	4	1	no	—
274	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	4	1	no	—
277	C₃×C₄	—	4	1	no	—
278	C₃×C₄	#4057	1	1	yes	characteristic
279	C₃×C₄	#4057	1	1	yes	characteristic
280	C₃×C₄	C₂×#681	1	1	yes	characteristic
281	C₃×C₄	#4094	1	1	yes	characteristic
282	C₂×C₅	—	4	2	no	—
283	C₂×C₅	—	8	1	no	—
284	C₂×C₅	#5011	1	1	yes	characteristic
285	C₂×C₅	#5152	1	1	yes	characteristic
286	C₂×C₅	#5196	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	#6710	1	2	yes	complemented
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
300	C₂×C₄	#6819	1	1	yes	characteristic
301	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	30	2	no	—
305	C₈	—	30	1	no	—
306	C₈	—	30	1	no	—
307	C₈	—	60	1	no	—
308	C₈	—	60	1	no	—
309	C₂×C₃	—	4	2	no	—
310	C₂×C₃	—	8	1	no	—
311	C₂×C₃	#65300	1	1	yes	characteristic
312	C₂×C₃	#65441	1	1	yes	characteristic
313	C₂×C₃	#65485	1	1	yes	characteristic
314	C₅	#70880	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	D₄×Dic₁₅	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	#91800	1	1	yes	characteristic
326	C₄	#91800	1	1	yes	characteristic
327	C₄	C₂×#6710	1	1	yes	characteristic
328	C₄	#91837	1	1	yes	characteristic
329	C₃	#10601673	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	#11751112	1	1	yes	characteristic
333	C₂	#11751253	1	1	yes	characteristic
334	C₂	#11751297	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₄×C₅×D₈	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×C₅×D₈	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×D₈	D₁₅
hypocenter	C₃×C₅	#1586
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #11751113	C₂	semidirect
3. D₄×C₃⋊#168	C₂	semidirect
4. D₄⋊(C₃⋊#168)	C₂	semidirect
5. D₄⋊(C₃⋊#168)	C₂	semidirect
6. #11746278	C₂	semidirect
7. #11746275	C₂	semidirect
79. C₃×C₅×D₄	M₁₆	semidirect
175. C₅×D₄	#208	semidirect
217. C₃×D₄	#681	semidirect
267. C₃×C₅	#1586	semidirect
292. D₄	#6710	semidirect
314. C₅	#70880	semidirect
329. C₃	#10601673	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₈	2
#11751113	2
D₄×C₃⋊#168	2
D₄⋊(C₃⋊#168)	2
#11746278	2
#11746275	2
#10601673	3
#70880	5