Subgroups — #14304772

Order 1920.

460 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11751214	C₂	1	1	yes	maximal, characteristic
3	#11751211	C₂	1	1	yes	maximal, characteristic
4	#11751203	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#91089	C₂	1	1	yes	maximal, characteristic
6	#11746373	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746344	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600586	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	#91815	C₂²	1	1	yes	characteristic
14	C₂×#6710	C₂²	1	1	yes	characteristic
15	C₂×#6710	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	—	2	2	no	—
17	#91105	C₄	1	2	yes	complemented
18	#91104	—	2	2	no	—
19	#91100	C₄	1	2	yes	complemented
20	#91098	—	2	2	no	—
21	#91089	—	2	2	no	—
22	#69793	—	5	1	no	maximal, Hall
23	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic, complemented
24	#65402	—	3	1	no	—
25	#65399	—	3	1	no	—
26	#65391	—	3	1	no	—
27	C₂×#3937	—	3	1	no	—
28	#64772	—	3	1	no	—
29	#64743	—	3	1	no	—
30	C₂²×C₃×C₄×C₅	—	2	1	no	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×#1220	—	2	1	no	—
33	#6817	—	2	1	no	—
34	#6817	—	2	1	no	—
35	#6817	—	2	1	no	—
36	#6817	—	2	1	no	—
37	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₅	—	2	1	no	—
39	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₂×C₃×C₅×C₈	—	2	1	no	—
44	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
46	C₃×C₄²×C₅	—	2	1	no	—
47	C₃×C₄²×C₅	D₄	1	2	yes	—
48	#6719	—	2	1	no	—
49	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₅	—	2	1	no	—
51	#6710	—	2	1	no	—
52	#6710	—	2	1	no	—
53	#6710	—	2	1	no	—
54	#6710	—	2	1	no	—
55	C₂×C₃⋊#168	—	2	1	no	—
56	C₂×C₃⋊#168	—	2	1	no	—
57	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic, complemented
58	#5113	—	5	1	no	—
59	#5110	—	5	1	no	—
60	#5102	—	5	1	no	—
61	C₂×#805	—	5	1	no	—
62	#4552	—	5	1	no	—
63	#4523	—	5	1	no	—
64	C₂×#709	—	3	1	no	—
65	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
68	#4072	—	3	1	no	—
69	C₂×#681	—	3	1	no	—
70	C₂×#681	—	3	1	no	—
71	C₄×C₅×C₈	—	2	2	no	—
72	#3953	—	3	2	no	—
73	#3952	—	6	2	no	—
74	#3948	—	3	2	no	—
75	#3946	—	6	2	no	—
76	#3937	—	6	2	no	—
77	#1937	—	15	1	no	Sylow
78	C₂³×C₃×C₅	#41	1	1	yes	characteristic
79	C₂²×D₁₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	2	4	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
88	#1220	—	4	1	no	—
89	#1220	—	4	1	no	—
90	C₂×Dic₁₅	—	2	2	no	—
91	C₂×Dic₁₅	—	2	1	no	—
92	D₆₀	—	2	1	no	—
93	D₆₀	—	2	1	no	—
94	C₄×D₁₅	—	4	1	no	—
95	C₄×D₁₅	—	4	1	no	—
96	Dic₃₀	—	2	1	no	—
97	Dic₃₀	—	2	1	no	—
98	C₃×C₅×C₈	—	2	2	no	—
99	C₃×C₅×C₈	—	2	2	no	—
100	C₃⋊#168	—	4	1	no	—
101	C₃⋊#168	—	4	1	no	—
102	C₂×#235	—	5	1	no	—
103	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
104	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
105	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
106	#926	—	5	1	no	—
107	C₂×#208	—	5	1	no	—
108	C₂×#208	—	5	1	no	—
109	C₃×C₄×C₈	—	2	2	no	—
110	#821	—	5	2	no	—
111	(C₂×D₃)⋊C₈	—	10	2	no	—
112	#816	—	5	2	no	—
113	Dic₃⋊C₈	—	10	2	no	—
114	#805	—	10	2	no	—
115	C₂²×C₄×C₅	—	2	1	no	—
116	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₂²⋊D₅	—	6	1	no	—
118	#709	—	6	1	no	—
119	#709	—	6	1	no	—
120	#709	—	6	1	no	—
121	#709	—	6	1	no	—
122	C₂×D₂₀	—	3	1	no	—
123	C₂×C₄×D₅	—	6	1	no	—
124	C₂×Dic₁₀	—	3	1	no	—
125	C₂×C₅×C₈	—	2	1	no	—
126	C₂×C₅×C₈	—	2	1	no	—
127	C₂×C₅×C₈	—	2	1	no	—
128	C₂×C₅×C₈	—	2	1	no	—
129	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
130	C₂×C₅×C₈	D₁₂	1	2	yes	—
131	C₄²×C₅	—	2	1	no	—
132	C₄²×C₅	#67	1	2	yes	—
133	C₂²⋊Dic₅	—	6	1	no	—
134	C₄⋊Dic₅	—	3	2	no	—
135	C₄×Dic₅	—	6	1	no	—
136	#681	—	6	1	no	—
137	#681	—	6	1	no	—
138	#681	—	6	1	no	—
139	#681	—	6	1	no	—
140	C₂×#168	—	6	1	no	—
141	C₂×#168	—	6	1	no	—
142	#424	—	15	1	no	—
143	C₂×#104	—	15	1	no	—
144	#416	—	15	1	no	—
145	#408	—	15	1	no	—
146	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic, complemented
147	#341	—	15	1	no	—
148	#339	—	15	1	no	—
149	C₂²×C₃×C₅	—	2	2	no	—
150	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
151	C₂²×C₃×C₅	#123	1	1	yes	characteristic
152	C₂²×C₃×C₅	#126	1	1	yes	characteristic
153	C₂×D₁₅	—	4	1	no	—
154	C₂×D₁₅	—	4	2	no	—
155	C₃×C₄×C₅	—	2	4	no	—
156	C₃×C₄×C₅	#117	1	1	yes	characteristic
157	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
158	C₃×C₄×C₅	#121	1	1	yes	characteristic
159	C₃×C₄×C₅	#124	1	1	yes	characteristic
160	Dic₁₅	—	4	1	no	—
161	Dic₁₅	—	4	2	no	—
162	C₂²×C₃×C₄	—	2	1	no	—
163	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
164	C₂×#67	—	10	1	no	—
165	#235	—	10	1	no	—
166	#235	—	10	1	no	—
167	#235	—	10	1	no	—
168	#235	—	10	1	no	—
169	C₂×D₁₂	—	5	1	no	—
170	C₂×C₄×D₃	—	10	1	no	—
171	C₂×Dic₆	—	5	1	no	—
172	C₂×C₃×C₈	—	2	1	no	—
173	C₂×C₃×C₈	—	2	1	no	—
174	C₂×C₃×C₈	—	2	1	no	—
175	C₂×C₃×C₈	—	2	1	no	—
176	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
177	C₂×C₃×C₈	D₂₀	1	2	yes	—
178	C₃×C₄²	—	2	1	no	—
179	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
180	#217	—	10	1	no	—
181	C₄⋊Dic₃	—	5	2	no	—
182	C₄×Dic₃	—	10	1	no	—
183	#208	—	10	1	no	—
184	#208	—	10	1	no	—
185	#208	—	10	1	no	—
186	#208	—	10	1	no	—
187	C₂×C₃⋊C₈	—	10	1	no	—
188	C₂×C₃⋊C₈	—	10	1	no	—
189	C₂³×C₅	#235	1	1	yes	characteristic
190	C₂²×D₅	—	6	1	no	—
191	C₂×C₄×C₅	—	2	4	no	—
192	C₂×C₄×C₅	—	2	2	no	—
193	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
195	C₂×C₄×C₅	#235	1	1	yes	characteristic
196	C₂×C₄×C₅	#235	1	1	yes	characteristic
197	C₂×C₄×C₅	#235	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
199	C₂²⋊D₅	—	12	1	no	—
200	C₂²⋊D₅	—	12	1	no	—
201	C₂×Dic₅	—	6	2	no	—
202	C₂×Dic₅	—	6	1	no	—
203	D₂₀	—	6	1	no	—
204	D₂₀	—	6	1	no	—
205	C₄×D₅	—	12	1	no	—
206	C₄×D₅	—	12	1	no	—
207	Dic₁₀	—	6	1	no	—
208	Dic₁₀	—	6	1	no	—
209	C₅×C₈	—	2	2	no	—
210	C₅×C₈	—	2	2	no	—
211	#168	—	12	1	no	—
212	#168	—	12	1	no	—
213	C₂×#41	—	15	1	no	—
214	C₂×M₁₆	—	15	1	no	—
215	C₂×M₁₆	—	15	1	no	—
216	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
217	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
218	#117	—	15	1	no	—
219	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
220	C₄⋊C₈	—	30	2	no	—
221	#104	—	30	2	no	—
222	#103	—	15	2	no	—
223	#102	—	15	2	no	—
224	C₂²⋊C₈	—	30	2	no	—
225	C₄×C₈	—	2	2	no	—
226	C₂×C₃×C₅	—	2	1	no	—
227	C₂×C₃×C₅	—	2	1	no	—
228	C₂×C₃×C₅	#388	1	1	yes	characteristic
229	C₂×C₃×C₅	#443	1	2	yes	—
230	D₁₅	—	8	1	no	—
231	C₂³×C₃	#709	1	1	yes	characteristic
232	C₂²×D₃	—	10	1	no	—
233	C₂×C₃×C₄	—	2	4	no	—
234	C₂×C₃×C₄	—	2	2	no	—
235	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
237	C₂×C₃×C₄	#709	1	1	yes	characteristic
238	C₂×C₃×C₄	#709	1	1	yes	characteristic
239	C₂×C₃×C₄	#709	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
241	#67	—	20	1	no	—
242	#67	—	20	1	no	—
243	C₂×Dic₃	—	10	2	no	—
244	C₂×Dic₃	—	10	1	no	—
245	D₁₂	—	10	1	no	—
246	D₁₂	—	10	1	no	—
247	C₄×D₃	—	20	1	no	—
248	C₄×D₃	—	20	1	no	—
249	Dic₆	—	10	1	no	—
250	Dic₆	—	10	1	no	—
251	C₃×C₈	—	2	2	no	—
252	C₃×C₈	—	2	2	no	—
253	C₃⋊C₈	—	20	1	no	—
254	C₃⋊C₈	—	20	1	no	—
255	C₂²×C₅	—	2	2	no	—
256	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
257	C₂²×C₅	#876	1	1	yes	characteristic
258	C₂²×C₅	#929	1	1	yes	characteristic
259	C₂×D₅	—	12	1	no	—
260	C₂×D₅	—	12	2	no	—
261	C₄×C₅	—	2	4	no	—
262	C₄×C₅	#872	1	1	yes	characteristic
263	C₄×C₅	#875	1	1	yes	characteristic
264	C₄×C₅	C₄×#67	1	1	yes	characteristic
265	C₄×C₅	#930	1	1	yes	characteristic
266	Dic₅	—	12	1	no	—
267	Dic₅	—	12	2	no	—
268	#41	—	30	1	no	—
269	#41	—	30	1	no	—
270	#41	—	30	1	no	—
271	#41	—	30	1	no	—
272	C₂×Q₈	—	15	1	no	—
273	C₂×D₄	—	15	1	no	—
274	C₂×D₄	—	30	1	no	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	30	1	no	—
277	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
278	M₁₆	—	30	1	no	—
279	M₁₆	—	30	1	no	—
280	M₁₆	—	30	1	no	—
281	M₁₆	—	30	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	30	1	no	—
287	C₂×C₈	—	30	1	no	—
288	C₂×C₈	C₄×D₁₅	1	2	yes	—
289	C₂×C₈	D₆₀	1	2	yes	—
290	C₄⋊C₄	—	15	2	no	—
291	C₂²⋊C₄	—	30	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	30	1	no	—
294	C₄²	#1220	1	2	yes	—
295	C₃×C₅	#1937	1	1	yes	characteristic, complemented, Hall
296	C₂²×C₃	—	2	2	no	—
297	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
298	C₂²×C₃	#4022	1	1	yes	characteristic
299	C₂²×C₃	#4075	1	1	yes	characteristic
300	C₂×D₃	—	20	1	no	—
301	C₂×D₃	—	20	2	no	—
302	C₃×C₄	—	2	4	no	—
303	C₃×C₄	#4018	1	1	yes	characteristic
304	C₃×C₄	#4021	1	1	yes	characteristic
305	C₃×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
306	C₃×C₄	#4076	1	1	yes	characteristic
307	Dic₃	—	20	1	no	—
308	Dic₃	—	20	2	no	—
309	C₂×C₅	—	2	1	no	—
310	C₂×C₅	—	2	1	no	—
311	C₂×C₅	#4700	1	2	yes	—
312	C₂×C₅	#4940	1	1	yes	characteristic
313	D₅	—	24	1	no	—
314	C₂³	—	30	1	no	—
315	C₂³	#6817	1	1	yes	characteristic
316	Q₈	—	30	1	no	—
317	Q₈	—	30	1	no	—
318	D₄	—	30	1	no	—
319	D₄	—	30	1	no	—
320	D₄	—	60	1	no	—
321	D₄	—	60	1	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	30	2	no	—
325	C₂×C₄	—	30	1	no	—
326	C₂×C₄	—	60	1	no	—
327	C₂×C₄	—	60	1	no	—
328	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
329	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
330	C₂×C₄	#6817	1	1	yes	characteristic
331	C₂×C₄	#6817	1	1	yes	characteristic
332	C₂×C₄	#6817	1	1	yes	characteristic
333	C₂×C₄	C₂×#1220	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	2	2	no	—
336	C₈	—	60	1	no	—
337	C₈	—	60	1	no	—
338	C₂×C₃	—	2	1	no	—
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	#64989	1	2	yes	—
341	C₂×C₃	#65229	1	1	yes	characteristic
342	D₃	—	40	1	no	—
343	C₅	#69793	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	2	2	no	—
345	C₂²	—	60	1	no	—
346	C₂²	—	60	2	no	—
347	C₂²	C₄×D₆₀	1	1	yes	characteristic
348	C₂²	#91765	1	1	yes	characteristic
349	C₂²	#91818	1	1	yes	characteristic
350	C₄	—	2	4	no	—
351	C₄	—	60	1	no	—
352	C₄	—	60	2	no	—
353	C₄	#91761	1	1	yes	characteristic
354	C₄	#91764	1	1	yes	characteristic
355	C₄	C₄×#1220	1	1	yes	characteristic
356	C₄	#91819	1	1	yes	characteristic
357	C₃	#10600586	1	1	yes	characteristic, complemented, Sylow
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	120	1	no	—
361	C₂	#11750801	1	2	yes	—
362	C₂	#11751041	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₀
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×#6817	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄×C₈	D₁₅
hypocenter	C₃×C₅	#1937
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
4. #11751203	C₂	semidirect
6. #11746373	C₂	semidirect
7. #11746344	C₂	semidirect
17. #91105	C₄	semidirect
19. #91100	C₄	semidirect
23. C₂×C₄×C₅×C₈	D₃	semidirect
57. C₂×C₃×C₄×C₈	D₅	semidirect
146. C₂×C₄×C₈	D₁₅	semidirect
295. C₃×C₅	#1937	semidirect
343. C₅	#69793	semidirect
357. C₃	#10600586	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×C₈	2
#11751214	2
#11751211	2
#11751203	2
C₂×#91089	2
#11746373	2
#11746344	2
#10600586	3
#69793	5