Subgroups — #14286554

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	C₄×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	#11747008	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₂×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64674	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745837	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745837	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601672	—	3	1	no	maximal, Hall
9	C₂×C₅×D₂₄	C₄	1	1	yes	characteristic
10	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
11	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×#6650	—	2	1	no	—
13	C₂×D₃×#168	—	2	1	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
17	C₅×#821	C₄	1	1	yes	characteristic
18	C₅×#821	C₄	1	1	yes	characteristic
19	C₅×#818	C₄	1	1	yes	characteristic
20	C₃×C₄×#168	C₂²	1	1	yes	characteristic
21	C₃×C₄×#168	C₂²	1	1	yes	characteristic
22	(C₂×D₃)⋊#168	—	2	1	no	—
23	(C₂×D₃)⋊#168	—	2	1	no	—
24	#68078	—	5	1	no	maximal, Hall
25	C₄×C₅×D₈	—	3	1	no	—
26	#65301	—	3	1	no	—
27	D₄×#168	—	3	1	no	—
28	D₄⋊#168	—	3	1	no	—
29	D₄⋊#168	—	3	1	no	—
30	#64678	—	3	1	no	—
31	#64674	D₃	1	1	yes	characteristic, complemented
32	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅×D₃	—	2	1	no	—
35	C₂×C₄×C₅×D₃	—	2	1	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₃⋊#168	—	2	1	no	—
40	C₂×C₃⋊#168	—	2	1	no	—
41	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
42	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
43	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₅×D₂₄	—	2	2	no	—
46	C₂×C₃×#168	—	2	1	no	—
47	C₂×C₃×#168	D₄	1	1	yes	characteristic
48	C₂×C₃×#168	D₄	1	1	yes	characteristic
49	#6650	—	4	2	no	—
50	D₃×#168	—	4	2	no	—
51	C₄×D₂₄	D₅	1	1	yes	characteristic
52	#4688	—	5	1	no	—
53	C₈×D₁₂	—	5	1	no	—
54	C₃×#322	—	5	1	no	—
55	#4459	—	5	1	no	—
56	#4459	—	5	1	no	—
57	#4458	—	5	1	no	—
58	C₂×C₅×D₈	—	3	1	no	—
59	C₄×C₅×D₄	—	3	1	no	—
60	C₄×C₅×D₄	—	3	1	no	—
61	C₂×#681	—	6	1	no	—
62	C₂²×#168	—	6	1	no	—
63	C₅×#107	—	3	1	no	—
64	C₅×#102	—	3	1	no	—
65	C₅×#102	—	3	1	no	—
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	C₂×D₃	1	1	yes	characteristic
72	C₄×#168	C₂×D₃	1	1	yes	characteristic
73	#1581	—	15	1	no	Sylow
74	C₂²×C₅×D₃	—	2	1	no	—
75	C₂²×C₅×D₃	—	2	1	no	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
79	C₂×C₅×Dic₃	—	2	1	no	—
80	C₂×C₅×Dic₃	—	2	1	no	—
81	C₅×D₁₂	—	2	1	no	—
82	C₅×D₁₂	—	2	1	no	—
83	C₅×D₁₂	—	2	1	no	—
84	C₅×D₁₂	M₁₆	1	2	yes	complemented
85	C₄×C₅×D₃	—	4	1	no	—
86	C₄×C₅×D₃	—	4	1	no	—
87	C₄×C₅×D₃	—	4	1	no	—
88	C₄×C₅×D₃	—	4	1	no	—
89	C₃×C₅×C₈	—	2	1	no	—
90	C₃×C₅×C₈	—	2	1	no	—
91	C₃⋊#168	—	4	1	no	—
92	C₃⋊#168	—	4	1	no	—
93	C₃×#168	—	2	1	no	—
94	C₃×#168	—	4	1	no	—
95	C₃×#168	D₈	1	2	yes	complemented
96	C₂×D₂₄	Dic₅	1	1	yes	characteristic
97	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
98	C₂×D₃×C₈	—	10	1	no	—
99	C₄×D₁₂	C₂×D₅	1	1	yes	characteristic
100	C₄×D₁₂	C₂×D₅	1	1	yes	characteristic
101	C₃×C₄×C₈	—	5	1	no	—
102	C₃×C₄×C₈	—	5	1	no	—
103	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
104	#821	Dic₅	1	1	yes	characteristic
105	#821	Dic₅	1	1	yes	characteristic
106	(C₂×D₃)⋊C₈	—	10	1	no	—
107	(C₂×D₃)⋊C₈	—	10	1	no	—
108	#818	Dic₅	1	1	yes	characteristic
109	C₄⋊(C₃⋊C₈)	—	5	1	no	—
110	C₄⋊(C₃⋊C₈)	—	5	1	no	—
111	C₂×C₅×D₄	—	3	1	no	—
112	C₂×C₅×D₄	—	3	1	no	—
113	C₂²×C₄×C₅	—	6	1	no	—
114	C₂²×C₄×C₅	—	6	1	no	—
115	C₅×D₈	—	6	2	no	—
116	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
117	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
118	C₅×C₄⋊C₄	—	3	1	no	—
119	C₅×C₄⋊C₄	—	3	1	no	—
120	C₅×C₂²⋊C₄	—	6	1	no	—
121	C₅×C₂²⋊C₄	—	6	1	no	—
122	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
123	#681	—	12	2	no	—
124	C₂×#168	—	2	1	no	—
125	C₂×#168	—	6	1	no	—
126	C₂×#168	—	6	1	no	—
127	C₂×#168	—	12	2	no	—
128	C₂×#168	D₁₂	1	1	yes	characteristic
129	C₂×#168	D₁₂	1	1	yes	characteristic
130	C₄×D₈	—	3	1	no	—
131	#436	—	15	1	no	—
132	C₈×D₄	—	15	1	no	—
133	#335	—	15	1	no	—
134	D₄⋊C₈	—	15	1	no	—
135	D₄⋊C₈	—	15	1	no	—
136	#322	—	5	1	no	—
137	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₅×D₃	—	2	2	no	—
139	C₂×C₅×D₃	—	4	1	no	—
140	C₂×C₅×D₃	—	4	1	no	—
141	C₂×C₅×D₃	—	4	1	no	—
142	C₂×C₅×D₃	—	4	1	no	—
143	C₂×C₅×D₃	—	4	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
146	C₃×C₄×C₅	#131	1	1	yes	characteristic
147	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
148	C₃×C₄×C₅	#135	1	1	yes	characteristic
149	C₅×Dic₃	—	4	1	no	—
150	C₅×Dic₃	—	4	1	no	—
151	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
152	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
153	C₂×C₄×D₃	—	2	1	no	—
154	C₂×C₄×D₃	—	2	1	no	—
155	C₂×C₃×C₈	—	5	1	no	—
156	C₂×C₃×C₈	—	5	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	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
161	(C₂×D₃)⋊C₄	—	2	1	no	—
162	(C₂×D₃)⋊C₄	—	2	1	no	—
163	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
164	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
165	C₂×C₃⋊C₈	—	10	1	no	—
166	C₂×C₃⋊C₈	—	10	1	no	—
167	D₂₄	—	2	2	no	—
168	ΓL(1,𝔽₂₅)	—	20	2	no	—
169	D₃×C₈	—	20	2	no	—
170	C₂³×C₅	—	6	1	no	—
171	C₂³×C₅	—	6	1	no	—
172	C₅×D₄	—	3	2	no	—
173	C₅×D₄	—	6	1	no	—
174	C₅×D₄	—	6	1	no	—
175	C₅×D₄	—	6	1	no	—
176	C₂×C₄×C₅	—	6	1	no	—
177	C₂×C₄×C₅	—	6	1	no	—
178	C₂×C₄×C₅	—	12	1	no	—
179	C₂×C₄×C₅	—	12	1	no	—
180	C₂×C₄×C₅	—	12	1	no	—
181	C₂×C₄×C₅	—	12	1	no	—
182	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
184	C₂×C₄×C₅	#235	1	1	yes	characteristic
185	C₅×C₈	—	2	1	no	—
186	C₅×C₈	—	2	1	no	—
187	#168	—	2	1	no	—
188	#168	—	4	1	no	—
189	#168	—	12	1	no	—
190	#168	—	12	1	no	—
191	#168	D₂₄	1	2	yes	complemented
192	C₂×D₈	—	3	1	no	—
193	C₂×M₁₆	—	30	1	no	—
194	C₂²×C₈	—	30	1	no	—
195	C₄×D₄	—	3	1	no	—
196	C₄×D₄	—	3	1	no	—
197	#107	—	3	1	no	—
198	C₄⋊C₈	—	15	1	no	—
199	C₄⋊C₈	—	15	1	no	—
200	#102	—	3	1	no	—
201	#102	—	3	1	no	—
202	C₂²⋊C₈	—	30	1	no	—
203	C₂²⋊C₈	—	30	1	no	—
204	C₄×C₈	—	5	1	no	—
205	C₄×C₈	—	5	1	no	—
206	C₄×C₈	D₃×D₅	1	1	yes	characteristic
207	C₂×C₃×C₅	#435	1	1	yes	characteristic
208	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
209	C₂×C₃×C₅	#444	1	1	yes	characteristic
210	C₅×D₃	—	4	2	no	—
211	C₅×D₃	—	8	1	no	—
212	C₂²×D₃	—	2	1	no	—
213	C₂²×D₃	—	2	1	no	—
214	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
215	C₂×C₃×C₄	#711	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
217	C₂×Dic₃	—	2	1	no	—
218	C₂×Dic₃	—	2	1	no	—
219	D₁₂	—	2	1	no	—
220	D₁₂	—	2	1	no	—
221	D₁₂	—	2	1	no	—
222	D₁₂	#681	1	2	yes	complemented
223	C₄×D₃	—	4	1	no	—
224	C₄×D₃	—	4	1	no	—
225	C₄×D₃	—	4	1	no	—
226	C₄×D₃	—	4	1	no	—
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	5	2	no	—
230	C₃×C₈	—	10	1	no	—
231	C₃×C₈	—	20	1	no	—
232	C₃⋊C₈	—	20	1	no	—
233	C₃⋊C₈	—	20	1	no	—
234	C₂²×C₅	—	6	2	no	—
235	C₂²×C₅	—	12	1	no	—
236	C₂²×C₅	—	12	1	no	—
237	C₂²×C₅	—	12	1	no	—
238	C₂²×C₅	—	12	1	no	—
239	C₂²×C₅	—	12	1	no	—
240	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	12	1	no	—
243	C₄×C₅	—	12	1	no	—
244	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
245	C₄×C₅	#904	1	1	yes	characteristic
246	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
247	C₄×C₅	#907	1	1	yes	characteristic
248	C₂×D₄	—	3	1	no	—
249	C₂×D₄	—	3	1	no	—
250	C₂²×C₄	—	6	1	no	—
251	C₂²×C₄	—	6	1	no	—
252	D₈	—	6	2	no	—
253	M₁₆	—	60	2	no	—
254	C₂×C₈	—	5	1	no	—
255	C₂×C₈	—	5	1	no	—
256	C₂×C₈	—	10	1	no	—
257	C₂×C₈	—	30	1	no	—
258	C₂×C₈	—	30	1	no	—
259	C₂×C₈	—	60	2	no	—
260	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
261	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
262	C₄⋊C₄	—	3	1	no	—
263	C₄⋊C₄	—	3	1	no	—
264	C₂²⋊C₄	—	6	1	no	—
265	C₂²⋊C₄	—	6	1	no	—
266	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
267	C₃×C₅	#1581	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
269	C₂×D₃	—	2	2	no	—
270	C₂×D₃	—	4	1	no	—
271	C₂×D₃	—	4	1	no	—
272	C₂×D₃	—	4	1	no	—
273	C₂×D₃	—	4	1	no	—
274	C₂×D₃	—	4	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	D₅×D₈	1	1	yes	characteristic
277	C₃×C₄	#4058	1	1	yes	characteristic
278	C₃×C₄	C₂×#681	1	1	yes	characteristic
279	C₃×C₄	#4094	1	1	yes	characteristic
280	Dic₃	—	4	1	no	—
281	Dic₃	—	4	1	no	—
282	C₂×C₅	—	12	2	no	—
283	C₂×C₅	—	24	1	no	—
284	C₂×C₅	C₄×D₂₄	1	1	yes	characteristic
285	C₂×C₅	#4706	1	1	yes	characteristic
286	C₂×C₅	#4717	1	1	yes	characteristic
287	C₂³	—	6	1	no	—
288	C₂³	—	6	1	no	—
289	D₄	—	3	2	no	—
290	D₄	—	6	1	no	—
291	D₄	—	6	1	no	—
292	D₄	—	6	1	no	—
293	C₂×C₄	—	6	1	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	—	12	1	no	—
296	C₂×C₄	—	12	1	no	—
297	C₂×C₄	—	12	1	no	—
298	C₂×C₄	—	12	1	no	—
299	C₂×C₄	#6772	1	1	yes	characteristic
300	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
301	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	5	2	no	—
305	C₈	—	10	1	no	—
306	C₈	—	20	1	no	—
307	C₈	—	60	1	no	—
308	C₈	—	60	1	no	—
309	C₂×C₃	#65300	1	1	yes	characteristic
310	C₂×C₃	D₈×Dic₅	1	1	yes	characteristic
311	C₂×C₃	#65486	1	1	yes	characteristic
312	D₃	—	4	2	no	—
313	D₃	—	8	1	no	—
314	C₅	#68078	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	6	2	no	—
316	C₂²	—	12	1	no	—
317	C₂²	—	12	1	no	—
318	C₂²	—	12	1	no	—
319	C₂²	—	12	1	no	—
320	C₂²	—	12	1	no	—
321	C₂²	Dic₅×D₁₂	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	12	1	no	—
324	C₄	—	12	1	no	—
325	C₄	D₅×D₂₄	1	1	yes	characteristic
326	C₄	#91270	1	1	yes	characteristic
327	C₄	D₃×#681	1	1	yes	characteristic
328	C₄	#91288	1	1	yes	characteristic
329	C₃	#10601672	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	12	2	no	—
331	C₂	—	24	1	no	—
332	C₂	#11746987	1	1	yes	characteristic
333	C₂	Dic₅×D₂₄	1	1	yes	characteristic
334	C₂	#11748063	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×D₁₂
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×D₂₄	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1581
LCS	G ▷ C₂×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
2. #11747008	C₂	semidirect
3. D₁₂×#168	C₂	semidirect
4. #11746279	C₂	semidirect
5. C₃×#64674	C₂	semidirect
6. #11745837	C₂	semidirect
7. #11745837	C₂	semidirect
31. #64674	D₃	semidirect
84. C₅×D₁₂	M₁₆	semidirect
95. C₃×#168	D₈	semidirect
191. #168	D₂₄	semidirect
222. D₁₂	#681	semidirect
267. C₃×C₅	#1581	semidirect
314. C₅	#68078	semidirect
329. C₃	#10601672	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₂₄	2
#11747008	2
D₁₂×#168	2
#11746279	2
C₃×#64674	2
#11745837	2
#10601672	3
#68078	5