Subgroups — #14284656

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756487	C₂	1	1	yes	maximal, characteristic
2	#11750934	C₂	1	1	yes	maximal, characteristic
3	D₁₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746368	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746302	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746302	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599211	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#91835	—	2	1	no	—
11	Q₈×Dic₁₅	—	2	1	no	—
12	C₂×C₈×D₁₅	—	2	1	no	—
13	#91786	C₄	1	2	yes	—
14	#91781	—	2	1	no	—
15	#91779	—	2	1	no	—
16	#91776	C₄	1	2	yes	—
17	C₄²×D₁₅	C₂²	1	1	yes	characteristic
18	#91758	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	#91104	—	2	1	no	—
22	#91098	—	2	1	no	—
23	C₈×Dic₁₅	—	2	1	no	—
24	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
26	#68418	—	5	1	no	maximal, Hall
27	#66062	—	3	1	no	—
28	#65122	—	3	1	no	—
29	D₅×C₄⋊C₈	—	3	1	no	—
30	C₅×#332	D₃	1	1	yes	characteristic, complemented
31	#64767	—	3	1	no	—
32	#64701	—	3	1	no	—
33	#64701	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
37	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
38	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	#6714	—	2	2	no	—
45	C₄⋊Dic₁₅	—	2	1	no	—
46	C₄⋊Dic₁₅	—	2	1	no	—
47	C₄⋊Dic₁₅	—	2	2	no	—
48	#6712	—	2	2	no	—
49	C₄×Dic₁₅	—	2	2	no	—
50	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₅	D₄	1	1	yes	characteristic
52	C₄×Dic₁₅	D₄	1	1	yes	characteristic
53	C₂×C₃⋊#168	—	2	1	no	—
54	C₂×C₃⋊#168	—	2	1	no	—
55	C₈×D₁₅	—	4	2	no	—
56	#5728	—	5	1	no	—
57	#4833	—	5	1	no	—
58	D₃×C₄⋊C₈	—	5	1	no	—
59	C₃×#332	D₅	1	1	yes	characteristic, complemented
60	#4547	—	5	1	no	—
61	#4481	—	5	1	no	—
62	#4481	—	5	1	no	—
63	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
64	#4092	—	6	1	no	—
65	Q₈×Dic₅	—	6	1	no	—
66	C₂×C₈×D₅	—	6	1	no	—
67	#4043	—	3	2	no	—
68	#4038	—	6	1	no	—
69	#4036	—	6	1	no	—
70	#4033	—	3	2	no	—
71	C₄²×D₅	—	3	1	no	—
72	#4015	—	3	1	no	—
73	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
74	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
75	#3952	—	6	1	no	—
76	#3946	—	6	1	no	—
77	C₈×Dic₅	—	6	1	no	—
78	C₄⋊#168	—	3	1	no	—
79	C₄⋊#168	—	3	1	no	—
80	#1542	—	15	1	no	Sylow
81	C₂²×D₁₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₃×C₅×Q₈	—	4	1	no	—
83	C₃×C₅×Q₈	—	4	1	no	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×Dic₁₅	—	2	2	no	—
89	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×D₁₅	—	2	1	no	—
93	C₄×D₁₅	—	2	1	no	—
94	C₄×D₁₅	—	4	1	no	—
95	C₄×D₁₅	SD₁₆	1	2	yes	—
96	C₄×D₁₅	Q₁₆	1	2	yes	—
97	Dic₃₀	—	4	2	no	—
98	C₃×C₅×C₈	—	4	1	no	—
99	C₃×C₅×C₈	—	4	1	no	—
100	C₃⋊#168	—	4	1	no	—
101	C₃⋊#168	—	4	1	no	—
102	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
103	#946	—	10	1	no	—
104	Q₈×Dic₃	—	10	1	no	—
105	C₂×D₃×C₈	—	10	1	no	—
106	#897	—	5	2	no	—
107	#892	—	10	1	no	—
108	#890	—	10	1	no	—
109	#887	—	5	2	no	—
110	C₄²×D₃	—	5	1	no	—
111	#869	—	5	1	no	—
112	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
113	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
114	(C₂×D₃)⋊C₈	—	10	1	no	—
115	Dic₃⋊C₈	—	10	1	no	—
116	C₈×Dic₃	—	10	1	no	—
117	C₄⋊(C₃⋊C₈)	—	5	1	no	—
118	C₄⋊(C₃⋊C₈)	—	5	1	no	—
119	C₂×C₅×Q₈	—	2	1	no	—
120	C₂×C₄×D₅	—	3	1	no	—
121	C₂×C₄×D₅	—	3	1	no	—
122	C₂×C₄×D₅	—	3	1	no	—
123	C₂×Dic₁₀	—	3	2	no	—
124	C₂×C₅×C₈	—	2	1	no	—
125	C₂×C₅×C₈	—	2	1	no	—
126	C₅×C₄⋊C₄	—	2	1	no	—
127	C₅×C₄⋊C₄	C₄×D₃	1	2	yes	—
128	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
129	#685	—	6	2	no	—
130	C₄⋊Dic₅	—	6	2	no	—
131	C₄⋊Dic₅	—	6	1	no	—
132	C₄⋊Dic₅	—	6	1	no	—
133	#683	—	6	2	no	—
134	C₄×Dic₅	—	3	1	no	—
135	C₄×Dic₅	—	3	1	no	—
136	C₄×Dic₅	—	3	1	no	—
137	C₄×Dic₅	—	6	2	no	—
138	C₂×#168	—	6	1	no	—
139	C₂×#168	—	6	1	no	—
140	C₈×D₅	—	12	2	no	—
141	#533	—	15	1	no	—
142	#431	—	15	1	no	—
143	C₂×C₄⋊C₈	—	15	1	no	—
144	#333	—	15	1	no	—
145	#333	—	15	1	no	—
146	#332	—	15	1	no	—
147	#332	D₁₅	1	1	yes	characteristic, complemented
148	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×D₁₅	—	2	1	no	—
150	C₂×D₁₅	—	2	1	no	—
151	C₂×D₁₅	#103	1	2	yes	—
152	C₃×C₄×C₅	—	2	1	no	—
153	C₃×C₄×C₅	—	4	2	no	—
154	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
155	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
156	C₃×C₄×C₅	#135	1	1	yes	characteristic
157	C₃×C₄×C₅	#135	1	1	yes	characteristic
158	Dic₁₅	—	2	1	no	—
159	Dic₁₅	—	2	1	no	—
160	Dic₁₅	—	4	2	no	—
161	Dic₁₅	#103	1	2	yes	—
162	C₂×C₃×Q₈	—	2	1	no	—
163	C₂×C₄×D₃	—	5	1	no	—
164	C₂×C₄×D₃	—	5	1	no	—
165	C₂×C₄×D₃	—	5	1	no	—
166	C₂×Dic₆	—	5	2	no	—
167	C₂×C₃×C₈	—	2	1	no	—
168	C₂×C₃×C₈	—	2	1	no	—
169	C₃×C₄⋊C₄	—	2	1	no	—
170	C₃×C₄⋊C₄	C₄×D₅	1	2	yes	—
171	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
172	(C₂×D₃)⋊C₄	—	10	2	no	—
173	C₄⋊Dic₃	—	10	2	no	—
174	C₄⋊Dic₃	—	10	1	no	—
175	C₄⋊Dic₃	—	10	1	no	—
176	Dic₃⋊C₄	—	10	2	no	—
177	C₄×Dic₃	—	5	1	no	—
178	C₄×Dic₃	—	5	1	no	—
179	C₄×Dic₃	—	5	1	no	—
180	C₄×Dic₃	—	10	2	no	—
181	C₂×C₃⋊C₈	—	10	1	no	—
182	C₂×C₃⋊C₈	—	10	1	no	—
183	D₃×C₈	—	20	2	no	—
184	C₂²×D₅	—	3	1	no	—
185	C₅×Q₈	—	4	1	no	—
186	C₅×Q₈	—	4	1	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
189	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
191	C₂×Dic₅	—	3	1	no	—
192	C₂×Dic₅	—	3	1	no	—
193	C₂×Dic₅	—	3	1	no	—
194	C₂×Dic₅	—	6	2	no	—
195	C₄×D₅	—	3	2	no	—
196	C₄×D₅	—	3	2	no	—
197	C₄×D₅	—	6	1	no	—
198	C₄×D₅	—	6	1	no	—
199	C₄×D₅	—	12	1	no	—
200	Dic₁₀	—	12	2	no	—
201	C₅×C₈	—	4	1	no	—
202	C₅×C₈	—	4	1	no	—
203	#168	—	12	1	no	—
204	#168	—	12	1	no	—
205	C₂²×C₈	—	30	1	no	—
206	C₄⋊Q₈	—	15	1	no	—
207	C₄⋊Q₈	C₂×D₁₅	1	1	yes	characteristic
208	#125	—	30	1	no	—
209	C₂²⋊Q₈	—	15	2	no	—
210	C₂²⋊Q₈	—	30	1	no	—
211	C₄×Q₈	—	15	2	no	—
212	C₄×Q₈	—	30	1	no	—
213	#117	—	30	1	no	—
214	C₂×C₄²	—	15	1	no	—
215	C₄⋊C₈	—	15	1	no	—
216	C₄⋊C₈	—	15	1	no	—
217	C₄⋊C₈	—	30	1	no	—
218	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
219	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
220	C₂²⋊C₈	—	30	1	no	—
221	C₄×C₈	—	30	1	no	—
222	C₂×C₃×C₅	#413	1	1	yes	characteristic
223	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
224	C₂×C₃×C₅	#416	1	1	yes	characteristic
225	D₁₅	—	2	2	no	—
226	C₂²×D₃	—	5	1	no	—
227	C₃×Q₈	—	4	1	no	—
228	C₃×Q₈	—	4	1	no	—
229	C₂×C₃×C₄	—	2	2	no	—
230	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
232	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
233	C₂×Dic₃	—	5	1	no	—
234	C₂×Dic₃	—	5	1	no	—
235	C₂×Dic₃	—	5	1	no	—
236	C₂×Dic₃	—	10	2	no	—
237	C₄×D₃	—	5	2	no	—
238	C₄×D₃	—	5	2	no	—
239	C₄×D₃	—	10	1	no	—
240	C₄×D₃	—	10	1	no	—
241	C₄×D₃	—	20	1	no	—
242	Dic₆	—	20	2	no	—
243	C₃×C₈	—	4	1	no	—
244	C₃×C₈	—	4	1	no	—
245	C₃⋊C₈	—	20	1	no	—
246	C₃⋊C₈	—	20	1	no	—
247	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×D₅	—	3	2	no	—
249	C₂×D₅	—	6	1	no	—
250	C₂×D₅	—	6	1	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	4	2	no	—
253	C₄×C₅	#912	1	1	yes	characteristic
254	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
255	C₄×C₅	#916	1	1	yes	characteristic
256	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
257	Dic₅	—	3	2	no	—
258	Dic₅	—	6	1	no	—
259	Dic₅	—	6	1	no	—
260	Dic₅	—	12	2	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	15	2	no	—
263	C₂²×C₄	—	15	1	no	—
264	C₂²×C₄	—	15	1	no	—
265	C₂²×C₄	—	15	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	30	1	no	—
269	C₂×C₈	—	30	1	no	—
270	C₂×C₈	—	60	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	30	2	no	—
273	C₄⋊C₄	—	30	2	no	—
274	C₄⋊C₄	—	30	1	no	—
275	C₄⋊C₄	—	30	1	no	—
276	C₄⋊C₄	C₄×D₁₅	1	2	yes	—
277	C₂²⋊C₄	—	30	2	no	—
278	C₄²	—	15	1	no	—
279	C₄²	—	15	1	no	—
280	C₄²	—	15	1	no	—
281	C₄²	—	30	2	no	—
282	C₄²	C₂²×D₁₅	1	1	yes	characteristic
283	C₃×C₅	#1542	1	1	yes	characteristic, complemented, Hall
284	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
285	C₂×D₃	—	5	2	no	—
286	C₂×D₃	—	10	1	no	—
287	C₂×D₃	—	10	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	#4058	1	1	yes	characteristic
291	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
292	C₃×C₄	#4062	1	1	yes	characteristic
293	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
294	Dic₃	—	5	2	no	—
295	Dic₃	—	10	1	no	—
296	Dic₃	—	10	1	no	—
297	Dic₃	—	20	2	no	—
298	C₂×C₅	#4751	1	1	yes	characteristic
299	C₂×C₅	#4772	1	1	yes	characteristic
300	C₂×C₅	D₃×#103	1	1	yes	characteristic
301	D₅	—	6	2	no	—
302	C₂³	—	15	1	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	60	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	15	1	no	—
308	C₂×C₄	—	15	2	no	—
309	C₂×C₄	—	15	2	no	—
310	C₂×C₄	—	15	1	no	—
311	C₂×C₄	—	15	1	no	—
312	C₂×C₄	—	30	1	no	—
313	C₂×C₄	—	30	1	no	—
314	C₂×C₄	—	30	2	no	—
315	C₂×C₄	—	60	1	no	—
316	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
317	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
318	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
319	C₈	—	4	1	no	—
320	C₈	—	4	1	no	—
321	C₈	—	60	1	no	—
322	C₈	—	60	1	no	—
323	C₂×C₃	#65040	1	1	yes	characteristic
324	C₂×C₃	#65061	1	1	yes	characteristic
325	C₂×C₃	D₅×#103	1	1	yes	characteristic
326	D₃	—	10	2	no	—
327	C₅	#68418	1	1	yes	characteristic, complemented, Sylow
328	C₂²	—	15	2	no	—
329	C₂²	—	30	1	no	—
330	C₂²	—	30	1	no	—
331	C₂²	C₂²⋊C₄×D₁₅	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	15	2	no	—
335	C₄	—	30	1	no	—
336	C₄	—	30	1	no	—
337	C₄	—	60	2	no	—
338	C₄	#91801	1	1	yes	characteristic
339	C₄	SD₁₆×D₁₅	1	1	yes	characteristic
340	C₄	#91805	1	1	yes	characteristic
341	C₄	Q₁₆×D₁₅	1	1	yes	characteristic
342	C₃	#10599211	1	1	yes	characteristic, complemented, Sylow
343	C₂	—	30	2	no	—
344	C₂	#11750852	1	1	yes	characteristic
345	C₂	#11750873	1	1	yes	characteristic
346	C₂	D₁₅×#103	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#332	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₅	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#332	D₁₅
hypocenter	C₃×C₅	#1542
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#332	C₂	semidirect
5. #11746368	C₂	semidirect
6. #11746302	C₂	semidirect
7. #11746302	C₂	semidirect
30. C₅×#332	D₃	semidirect
59. C₃×#332	D₅	semidirect
147. #332	D₁₅	semidirect
283. C₃×C₅	#1542	semidirect
327. C₅	#68418	semidirect
342. C₃	#10599211	semidirect

Maximal subgroups

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

subgroup	index
#11756487	2
#11750934	2
D₁₅×C₄⋊C₈	2
C₃×C₅×#332	2
#11746368	2
#11746302	2
#10599211	3
#68418	5