Subgroups — #14283207

Order 1920.

484 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756238	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746291	C₂	1	2	yes	maximal
5	#11746288	C₂	1	2	yes	maximal, complemented
6	#10601626	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	#91819	C₂²	1	1	yes	characteristic
9	#91818	—	2	1	no	—
10	C₄×#1220	—	2	1	no	—
11	C₂×C₄⋊Dic₁₅	—	2	1	no	—
12	#91813	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₅	—	2	1	no	—
14	#91765	—	2	1	no	—
15	#91764	C₄	1	1	yes	characteristic, complemented
16	C₄×D₆₀	C₄	1	1	yes	characteristic, complemented
17	#91761	—	2	1	no	—
18	#91759	C₄	1	1	yes	characteristic, complemented
19	C₄×Dic₃₀	C₄	1	1	yes	characteristic, complemented
20	C₃×C₅×C₄⋊C₈	—	2	1	no	—
21	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₃×C₅×#97	—	2	1	no	—
23	#91115	C₂²	1	2	yes	—
24	#70833	—	5	1	no	maximal, Hall
25	#65813	—	3	1	no	—
26	C₅×#432	D₃	1	1	yes	characteristic, complemented
27	C₄×C₄⋊Dic₅	—	3	1	no	—
28	#64690	—	3	2	no	—
29	#64687	—	3	2	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×#1220	—	2	1	no	—
34	C₂²×Dic₁₅	—	2	2	no	—
35	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₅	—	2	1	no	—
37	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	#6719	—	2	1	no	—
44	#6714	—	2	1	no	—
45	#6714	—	2	1	no	—
46	#6714	—	2	1	no	—
47	#6714	—	2	1	no	—
48	C₄⋊Dic₁₅	—	2	4	no	—
49	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
51	#6712	—	2	1	no	—
52	#6712	—	2	1	no	—
53	#6712	—	2	1	no	—
54	#6712	—	2	1	no	—
55	C₄×Dic₁₅	—	2	1	no	—
56	C₄×Dic₁₅	—	4	2	no	—
57	#5479	—	5	1	no	—
58	C₃×#432	D₅	1	1	yes	characteristic, complemented
59	C₄×C₄⋊Dic₃	—	5	1	no	—
60	#4470	—	5	2	no	—
61	#4467	—	5	2	no	—
62	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
63	#4076	—	3	1	no	—
64	#4075	—	6	1	no	—
65	C₄×C₂²⋊D₅	—	6	1	no	—
66	C₂×C₄⋊Dic₅	—	6	1	no	—
67	#4070	—	3	1	no	—
68	C₂×C₄×Dic₅	—	6	1	no	—
69	#4022	—	6	1	no	—
70	#4021	—	3	1	no	—
71	C₄×D₂₀	—	3	1	no	—
72	#4018	—	6	1	no	—
73	#4016	—	3	1	no	—
74	C₄×Dic₁₀	—	3	1	no	—
75	C₅×C₄⋊C₈	—	2	1	no	—
76	C₅×C₂²⋊C₈	C₂×D₃	1	2	yes	—
77	C₅×#97	—	2	1	no	—
78	#3963	—	3	2	no	—
79	#1509	—	15	1	no	Sylow
80	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₁₅	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	2	1	no	—
83	C₂×C₃×C₄×C₅	—	2	1	no	—
84	C₂×C₃×C₄×C₅	—	2	2	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	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	#1220	—	4	1	no	—
93	#1220	—	4	1	no	—
94	C₂×Dic₁₅	—	2	4	no	—
95	C₂×Dic₁₅	—	2	1	no	—
96	C₂×Dic₁₅	—	2	1	no	—
97	C₂×Dic₁₅	—	2	1	no	—
98	C₂×Dic₁₅	—	4	2	no	—
99	C₂×Dic₁₅	—	4	2	no	—
100	D₆₀	—	4	1	no	—
101	C₄×D₁₅	—	4	2	no	—
102	Dic₃₀	—	4	1	no	—
103	C₃×C₅×C₈	—	4	2	no	—
104	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
105	#930	—	5	1	no	—
106	#929	—	10	1	no	—
107	C₄×#67	—	10	1	no	—
108	C₂×C₄⋊Dic₃	—	10	1	no	—
109	#924	—	5	1	no	—
110	C₂×C₄×Dic₃	—	10	1	no	—
111	#876	—	10	1	no	—
112	#875	—	5	1	no	—
113	C₄×D₁₂	—	5	1	no	—
114	#872	—	10	1	no	—
115	#870	—	5	1	no	—
116	C₄×Dic₆	—	5	1	no	—
117	C₃×C₄⋊C₈	—	2	1	no	—
118	C₃×C₂²⋊C₈	C₂×D₅	1	2	yes	—
119	C₃×#97	—	2	1	no	—
120	C₃⋊#95	—	5	2	no	—
121	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
122	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
123	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₂²⋊D₅	—	6	1	no	—
125	C₂²×Dic₅	—	6	2	no	—
126	C₂×D₂₀	—	3	1	no	—
127	C₂×C₄×D₅	—	6	1	no	—
128	C₂×Dic₁₀	—	3	1	no	—
129	C₂×C₅×C₈	—	2	2	no	—
130	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
131	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
132	C₄²×C₅	#67	1	1	yes	characteristic
133	C₄²×C₅	#67	1	1	yes	characteristic
134	C₂²⋊Dic₅	—	6	1	no	—
135	#685	—	6	1	no	—
136	#685	—	6	1	no	—
137	#685	—	6	1	no	—
138	#685	—	6	1	no	—
139	C₄⋊Dic₅	—	3	1	no	—
140	C₄⋊Dic₅	—	3	1	no	—
141	C₄⋊Dic₅	—	6	4	no	—
142	#683	—	6	1	no	—
143	#683	—	6	1	no	—
144	#683	—	6	1	no	—
145	#683	—	6	1	no	—
146	C₄×Dic₅	—	6	1	no	—
147	C₄×Dic₅	—	12	2	no	—
148	#529	—	15	1	no	—
149	#432	D₁₅	1	1	yes	characteristic, complemented
150	C₄×C₄⋊C₄	—	15	1	no	—
151	#328	—	15	2	no	—
152	#327	—	15	2	no	—
153	C₂²×C₃×C₅	—	2	1	no	—
154	C₂²×C₃×C₅	—	2	1	no	—
155	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂²×C₃×C₅	#137	1	1	yes	characteristic
158	C₂×D₁₅	—	4	1	no	—
159	C₂×D₁₅	—	4	1	no	—
160	C₂×D₁₅	—	4	1	no	—
161	C₃×C₄×C₅	—	2	1	no	—
162	C₃×C₄×C₅	—	2	1	no	—
163	C₃×C₄×C₅	—	2	1	no	—
164	C₃×C₄×C₅	—	2	1	no	—
165	C₃×C₄×C₅	—	2	1	no	—
166	C₃×C₄×C₅	#104	1	2	yes	—
167	Dic₁₅	—	4	4	no	—
168	Dic₁₅	—	4	1	no	—
169	Dic₁₅	—	4	1	no	—
170	Dic₁₅	—	4	1	no	—
171	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
172	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
173	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×#67	—	10	1	no	—
175	C₂²×Dic₃	—	10	2	no	—
176	C₂×D₁₂	—	5	1	no	—
177	C₂×C₄×D₃	—	10	1	no	—
178	C₂×Dic₆	—	5	1	no	—
179	C₂×C₃×C₈	—	2	2	no	—
180	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
181	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
182	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
183	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
184	#217	—	10	1	no	—
185	(C₂×D₃)⋊C₄	—	10	1	no	—
186	(C₂×D₃)⋊C₄	—	10	1	no	—
187	(C₂×D₃)⋊C₄	—	10	1	no	—
188	(C₂×D₃)⋊C₄	—	10	1	no	—
189	C₄⋊Dic₃	—	5	1	no	—
190	C₄⋊Dic₃	—	5	1	no	—
191	C₄⋊Dic₃	—	10	4	no	—
192	Dic₃⋊C₄	—	10	1	no	—
193	Dic₃⋊C₄	—	10	1	no	—
194	Dic₃⋊C₄	—	10	1	no	—
195	Dic₃⋊C₄	—	10	1	no	—
196	C₄×Dic₃	—	10	1	no	—
197	C₄×Dic₃	—	20	2	no	—
198	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
199	C₂²×D₅	—	6	1	no	—
200	C₂×C₄×C₅	—	2	1	no	—
201	C₂×C₄×C₅	—	2	1	no	—
202	C₂×C₄×C₅	—	2	2	no	—
203	C₂×C₄×C₅	—	4	1	no	—
204	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
205	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
206	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
207	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
208	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
209	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
210	C₂²⋊D₅	—	12	1	no	—
211	C₂²⋊D₅	—	12	1	no	—
212	C₂×Dic₅	—	6	4	no	—
213	C₂×Dic₅	—	6	1	no	—
214	C₂×Dic₅	—	6	1	no	—
215	C₂×Dic₅	—	6	1	no	—
216	C₂×Dic₅	—	12	2	no	—
217	C₂×Dic₅	—	12	2	no	—
218	D₂₀	—	12	1	no	—
219	C₄×D₅	—	12	2	no	—
220	Dic₁₀	—	12	1	no	—
221	C₅×C₈	—	4	2	no	—
222	#126	—	30	1	no	—
223	#125	—	15	1	no	—
224	#124	—	15	1	no	—
225	#123	—	30	1	no	—
226	C₂²⋊Q₈	—	15	1	no	—
227	#121	—	15	1	no	—
228	C₄×Q₈	—	15	1	no	—
229	C₄×D₄	—	15	1	no	—
230	C₄×D₄	—	30	1	no	—
231	#117	—	30	1	no	—
232	C₂×C₄⋊C₄	—	30	1	no	—
233	C₂×C₄²	—	30	1	no	—
234	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
235	C₄⋊C₈	—	2	1	no	—
236	C₂²⋊C₈	C₂×D₁₅	1	2	yes	—
237	#97	—	2	1	no	—
238	#95	—	15	2	no	—
239	C₂×C₃×C₅	—	2	1	no	—
240	C₂×C₃×C₅	—	2	1	no	—
241	C₂×C₃×C₅	#410	1	1	yes	characteristic
242	C₂×C₃×C₅	#417	1	1	yes	characteristic
243	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
244	D₁₅	—	8	1	no	—
245	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
246	C₂²×D₃	—	10	1	no	—
247	C₂×C₃×C₄	—	2	1	no	—
248	C₂×C₃×C₄	—	2	1	no	—
249	C₂×C₃×C₄	—	2	2	no	—
250	C₂×C₃×C₄	—	4	1	no	—
251	C₂×C₃×C₄	#685	1	1	yes	characteristic
252	C₂×C₃×C₄	#685	1	1	yes	characteristic
253	C₂×C₃×C₄	#685	1	1	yes	characteristic
254	C₂×C₃×C₄	#685	1	1	yes	characteristic
255	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
256	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
257	#67	—	20	1	no	—
258	#67	—	20	1	no	—
259	C₂×Dic₃	—	10	4	no	—
260	C₂×Dic₃	—	10	1	no	—
261	C₂×Dic₃	—	10	1	no	—
262	C₂×Dic₃	—	10	1	no	—
263	C₂×Dic₃	—	20	2	no	—
264	C₂×Dic₃	—	20	2	no	—
265	D₁₂	—	20	1	no	—
266	C₄×D₃	—	20	2	no	—
267	Dic₆	—	20	1	no	—
268	C₃×C₈	—	4	2	no	—
269	C₂²×C₅	—	2	1	no	—
270	C₂²×C₅	—	2	1	no	—
271	C₂²×C₅	#908	1	1	yes	characteristic
272	C₂²×C₅	#909	1	1	yes	characteristic
273	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
274	C₂×D₅	—	12	1	no	—
275	C₂×D₅	—	12	1	no	—
276	C₂×D₅	—	12	1	no	—
277	C₄×C₅	—	2	1	no	—
278	C₄×C₅	—	2	1	no	—
279	C₄×C₅	—	2	1	no	—
280	C₄×C₅	—	2	1	no	—
281	C₄×C₅	—	2	1	no	—
282	C₄×C₅	#825	1	2	yes	—
283	Dic₅	—	12	4	no	—
284	Dic₅	—	12	1	no	—
285	Dic₅	—	12	1	no	—
286	Dic₅	—	12	1	no	—
287	C₂×Q₈	—	15	1	no	—
288	C₂×D₄	—	15	1	no	—
289	C₂×D₄	—	30	1	no	—
290	C₂²×C₄	—	30	2	no	—
291	C₂²×C₄	—	30	1	no	—
292	C₂²×C₄	D₆₀	1	1	yes	characteristic
293	C₂²×C₄	D₆₀	1	1	yes	characteristic
294	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
295	C₂×C₈	—	2	2	no	—
296	C₄⋊C₄	—	15	1	no	—
297	C₄⋊C₄	—	15	1	no	—
298	C₄⋊C₄	—	30	4	no	—
299	C₄⋊C₄	—	30	1	no	—
300	C₄⋊C₄	—	30	1	no	—
301	C₄⋊C₄	—	30	1	no	—
302	C₄⋊C₄	—	30	1	no	—
303	C₂²⋊C₄	—	30	1	no	—
304	C₂²⋊C₄	—	30	1	no	—
305	C₂²⋊C₄	—	30	1	no	—
306	C₂²⋊C₄	—	30	1	no	—
307	C₂²⋊C₄	—	30	1	no	—
308	C₄²	—	30	1	no	—
309	C₄²	—	60	2	no	—
310	C₄²	C₄×D₁₅	1	1	yes	characteristic
311	C₄²	C₄×D₁₅	1	1	yes	characteristic
312	C₄²	#1220	1	1	yes	characteristic
313	C₄²	#1220	1	1	yes	characteristic
314	C₃×C₅	#1509	1	1	yes	characteristic, complemented, Hall
315	C₂²×C₃	—	2	1	no	—
316	C₂²×C₃	—	2	1	no	—
317	C₂²×C₃	#4054	1	1	yes	characteristic
318	C₂²×C₃	#4055	1	1	yes	characteristic
319	C₂²×C₃	C₂×#685	1	1	yes	characteristic
320	C₂×D₃	—	20	1	no	—
321	C₂×D₃	—	20	1	no	—
322	C₂×D₃	—	20	1	no	—
323	C₃×C₄	—	2	1	no	—
324	C₃×C₄	—	2	1	no	—
325	C₃×C₄	—	2	1	no	—
326	C₃×C₄	—	2	1	no	—
327	C₃×C₄	—	2	1	no	—
328	C₃×C₄	#3957	1	2	yes	—
329	Dic₃	—	20	4	no	—
330	Dic₃	—	20	1	no	—
331	Dic₃	—	20	1	no	—
332	Dic₃	—	20	1	no	—
333	C₂×C₅	—	2	1	no	—
334	C₂×C₅	—	2	1	no	—
335	C₂×C₅	#5006	1	1	yes	characteristic
336	C₂×C₅	#5133	1	1	yes	characteristic
337	C₂×C₅	C₂×#825	1	1	yes	characteristic
338	D₅	—	24	1	no	—
339	C₂³	—	30	1	no	—
340	C₂³	C₂×D₆₀	1	1	yes	characteristic
341	Q₈	—	60	1	no	—
342	D₄	—	60	1	no	—
343	D₄	—	60	1	no	—
344	D₄	—	60	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	30	4	no	—
350	C₂×C₄	—	30	1	no	—
351	C₂×C₄	—	30	1	no	—
352	C₂×C₄	—	30	1	no	—
353	C₂×C₄	—	60	2	no	—
354	C₂×C₄	—	60	2	no	—
355	C₂×C₄	—	60	2	no	—
356	C₂×C₄	#6714	1	1	yes	characteristic
357	C₂×C₄	#6714	1	1	yes	characteristic
358	C₂×C₄	#6714	1	1	yes	characteristic
359	C₂×C₄	#6714	1	1	yes	characteristic
360	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
361	C₂×C₄	C₂×#1220	1	1	yes	characteristic
362	C₈	—	4	2	no	—
363	C₂×C₃	—	2	1	no	—
364	C₂×C₃	—	2	1	no	—
365	C₂×C₃	#65295	1	1	yes	characteristic
366	C₂×C₃	#65422	1	1	yes	characteristic
367	C₂×C₃	C₂×#3957	1	1	yes	characteristic
368	D₃	—	40	1	no	—
369	C₅	#70833	1	1	yes	characteristic, complemented, Sylow
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	60	1	no	—
373	C₂²	—	60	1	no	—
374	C₂²	—	60	1	no	—
375	C₂²	#91797	1	1	yes	characteristic
376	C₂²	#91798	1	1	yes	characteristic
377	C₂²	C₂×#6714	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	60	4	no	—
384	C₄	—	60	1	no	—
385	C₄	—	60	1	no	—
386	C₄	—	60	1	no	—
387	C₄	#91109	1	2	yes	—
388	C₃	#10601626	1	1	yes	characteristic, complemented, Sylow
389	C₂	—	2	1	no	—
390	C₂	—	2	1	no	—
391	C₂	—	120	1	no	—
392	C₂	#11751107	1	1	yes	characteristic
393	C₂	#11751234	1	1	yes	characteristic
394	C₂	C₂×#91109	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6714
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#432	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#91819	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#432	D₁₅
hypocenter	C₃×C₅	#1509
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#432	C₂	semidirect
3. C₄×C₄⋊Dic₁₅	C₂	semidirect
5. #11746288	C₂	semidirect
15. #91764	C₄	semidirect
16. C₄×D₆₀	C₄	semidirect
18. #91759	C₄	semidirect
19. C₄×Dic₃₀	C₄	semidirect
26. C₅×#432	D₃	semidirect
58. C₃×#432	D₅	semidirect
149. #432	D₁₅	semidirect
314. C₃×C₅	#1509	semidirect
369. C₅	#70833	semidirect
388. C₃	#10601626	semidirect

Maximal subgroups

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

subgroup	index
#11756238	2
C₃×C₅×#432	2
C₄×C₄⋊Dic₁₅	2
#11746291	2
#11746288	2
#10601626	3
#70833	5