Subgroups — #14304775

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	#11751230	C₂	1	1	yes	maximal, characteristic
2	#11751103	C₂	1	1	yes	maximal, characteristic
3	#11751100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#91089	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#339	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746379	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599538	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₃×C₅×#117	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×#6710	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
16	#91104	—	2	2	no	—
17	#91094	C₄	1	2	yes	complemented
18	#91093	C₄	1	2	yes	complemented
19	#91089	—	2	2	no	—
20	C₄⋊(C₃⋊#168)	—	2	2	no	—
21	C₄×C₃⋊#168	—	2	2	no	—
22	#68745	—	5	1	no	maximal, Hall
23	#65418	—	3	1	no	—
24	#65291	—	3	1	no	—
25	#65288	—	3	1	no	—
26	C₂×#3937	—	3	1	no	—
27	C₂×C₄×#168	—	3	1	no	—
28	C₅×#339	D₃	1	1	yes	characteristic, complemented
29	#64778	—	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	C₃×C₅×M₁₆	—	2	1	no	—
41	C₃×C₅×M₁₆	—	2	1	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
45	C₃×C₄²×C₅	—	2	1	no	—
46	C₃×C₄²×C₅	—	2	1	no	—
47	C₃×C₄²×C₅	D₄	1	2	yes	—
48	#6710	—	2	1	no	—
49	#6710	—	2	1	no	—
50	C₂×C₃⋊#168	—	2	1	no	—
51	C₂×C₃⋊#168	—	2	1	no	—
52	C₂×C₃⋊#168	—	2	1	no	—
53	C₂×C₃⋊#168	—	2	1	no	—
54	C₂×C₃⋊#168	—	2	1	no	—
55	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
56	C₂×C₃⋊#168	D₄	1	2	yes	—
57	#5129	—	5	1	no	—
58	#5002	—	5	1	no	—
59	#4999	—	5	1	no	—
60	C₂×#805	—	5	1	no	—
61	C₂×C₄×C₃⋊C₈	—	5	1	no	—
62	C₃×#339	D₅	1	1	yes	characteristic, complemented
63	#4558	—	5	1	no	—
64	C₂×#709	—	3	1	no	—
65	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
66	C₅×#117	C₂×D₃	1	1	yes	characteristic
67	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
68	C₂×#681	—	3	1	no	—
69	C₂²×#168	—	3	1	no	—
70	C₂²×#168	—	3	1	no	—
71	#3952	—	6	2	no	—
72	#3942	—	3	2	no	—
73	#3941	—	3	2	no	—
74	#3937	—	6	2	no	—
75	C₄⋊#168	—	6	2	no	—
76	C₄×#168	—	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	—	2	2	no	—
83	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	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	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
89	#1220	—	4	1	no	—
90	#1220	—	4	1	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	—	4	1	no	—
99	C₃⋊#168	—	2	2	no	—
100	C₃⋊#168	—	2	2	no	—
101	C₃⋊#168	—	4	1	no	—
102	C₂×#235	—	5	1	no	—
103	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
104	C₃×#117	C₂×D₅	1	1	yes	characteristic
105	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
106	C₂×#208	—	5	1	no	—
107	C₂²×C₃⋊C₈	—	5	1	no	—
108	C₂²×C₃⋊C₈	—	5	1	no	—
109	(C₂×D₃)⋊C₈	—	10	2	no	—
110	#810	—	5	2	no	—
111	#809	—	5	2	no	—
112	#805	—	10	2	no	—
113	C₄⋊(C₃⋊C₈)	—	10	2	no	—
114	C₄×C₃⋊C₈	—	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	C₅×M₁₆	—	2	1	no	—
126	C₅×M₁₆	—	2	1	no	—
127	C₂×C₅×C₈	—	2	1	no	—
128	C₅×C₄⋊C₄	C₄×D₃	1	2	yes	—
129	C₅×C₂²⋊C₄	—	2	1	no	—
130	C₄²×C₅	—	2	1	no	—
131	C₄²×C₅	—	2	1	no	—
132	C₄²×C₅	D₁₂	1	2	yes	—
133	#681	—	6	1	no	—
134	#681	—	6	1	no	—
135	C₂×#168	—	3	2	no	—
136	C₂×#168	—	3	2	no	—
137	C₂×#168	—	6	1	no	—
138	C₂×#168	—	6	1	no	—
139	C₂×#168	—	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	—	15	1	no	—
147	#341	—	15	1	no	—
148	#339	D₁₅	1	1	yes	characteristic, complemented
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	—	4	2	no	—
157	C₃×C₄×C₅	#117	1	1	yes	characteristic
158	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
159	C₃×C₄×C₅	#121	1	1	yes	characteristic
160	C₃×C₄×C₅	#124	1	1	yes	characteristic
161	Dic₁₅	—	4	1	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	C₃×M₁₆	—	2	1	no	—
173	C₃×M₁₆	—	2	1	no	—
174	C₂×C₃×C₈	—	2	1	no	—
175	C₃×C₄⋊C₄	C₄×D₅	1	2	yes	—
176	C₃×C₂²⋊C₄	—	2	1	no	—
177	C₃×C₄²	—	2	1	no	—
178	C₃×C₄²	—	2	1	no	—
179	C₃×C₄²	D₂₀	1	2	yes	—
180	#208	—	10	1	no	—
181	#208	—	10	1	no	—
182	C₂×C₃⋊C₈	—	5	2	no	—
183	C₂×C₃⋊C₈	—	5	2	no	—
184	C₂×C₃⋊C₈	—	10	1	no	—
185	C₂×C₃⋊C₈	—	10	1	no	—
186	C₂×C₃⋊C₈	—	10	1	no	—
187	C₂×C₃⋊C₈	—	10	1	no	—
188	C₂×C₃⋊C₈	—	10	1	no	—
189	C₂³×C₅	#237	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	—	2	2	no	—
194	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
196	C₂×C₄×C₅	#235	1	1	yes	characteristic
197	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
198	C₂×C₄×C₅	#237	1	1	yes	characteristic
199	C₂×C₄×C₅	#239	1	1	yes	characteristic
200	C₂²⋊D₅	—	12	1	no	—
201	C₂²⋊D₅	—	12	1	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	—	4	1	no	—
210	#168	—	6	2	no	—
211	#168	—	6	2	no	—
212	#168	—	12	1	no	—
213	C₂×#41	—	15	1	no	—
214	C₂×M₁₆	—	15	1	no	—
215	C₂×M₁₆	C₂×D₁₅	1	1	yes	characteristic
216	C₂²×C₈	—	15	1	no	—
217	C₂²×C₈	—	15	1	no	—
218	#117	C₂×D₁₅	1	1	yes	characteristic
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	—	30	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	#711	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	—	2	2	no	—
236	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
238	C₂×C₃×C₄	#709	1	1	yes	characteristic
239	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
240	C₂×C₃×C₄	#711	1	1	yes	characteristic
241	C₂×C₃×C₄	#713	1	1	yes	characteristic
242	#67	—	20	1	no	—
243	#67	—	20	1	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	—	4	1	no	—
252	C₃⋊C₈	—	10	2	no	—
253	C₃⋊C₈	—	10	2	no	—
254	C₃⋊C₈	—	20	1	no	—
255	C₂²×C₅	—	2	2	no	—
256	C₂²×C₅	#67⋊C₄	1	1	yes	characteristic
257	C₂²×C₅	#886	1	1	yes	characteristic
258	C₂²×C₅	#898	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	—	4	2	no	—
263	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
264	C₄×C₅	#885	1	1	yes	characteristic
265	C₄×C₅	#892	1	1	yes	characteristic
266	C₄×C₅	#895	1	1	yes	characteristic
267	Dic₅	—	12	1	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	—	2	1	no	—
279	M₁₆	—	2	1	no	—
280	M₁₆	—	30	1	no	—
281	M₁₆	—	30	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	15	2	no	—
284	C₂×C₈	—	15	2	no	—
285	C₂×C₈	—	30	1	no	—
286	C₂×C₈	—	30	1	no	—
287	C₂×C₈	—	30	1	no	—
288	C₂×C₈	—	30	1	no	—
289	C₂×C₈	—	30	1	no	—
290	C₄⋊C₄	C₄×D₁₅	1	2	yes	—
291	C₂²⋊C₄	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	D₆₀	1	2	yes	—
295	C₃×C₅	#1937	1	1	yes	characteristic, complemented, Hall
296	C₂²×C₃	—	2	2	no	—
297	C₂²×C₃	#4027	1	1	yes	characteristic
298	C₂²×C₃	#4032	1	1	yes	characteristic
299	C₂²×C₃	#4044	1	1	yes	characteristic
300	C₂×D₃	—	20	2	no	—
301	C₂×D₃	—	20	1	no	—
302	C₃×C₄	—	2	4	no	—
303	C₃×C₄	—	4	2	no	—
304	C₃×C₄	C₄×D₂₀	1	1	yes	characteristic
305	C₃×C₄	#4031	1	1	yes	characteristic
306	C₃×C₄	#4038	1	1	yes	characteristic
307	C₃×C₄	#4041	1	1	yes	characteristic
308	Dic₃	—	20	1	no	—
309	C₂×C₅	—	2	1	no	—
310	C₂×C₅	—	2	1	no	—
311	C₂×C₅	#4670	1	1	yes	characteristic
312	C₂×C₅	#4824	1	2	yes	—
313	D₅	—	24	1	no	—
314	C₂³	—	30	1	no	—
315	C₂³	#6819	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	—	2	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	D₄×D₁₅	1	1	yes	characteristic
332	C₂×C₄	#6819	1	1	yes	characteristic
333	C₂×C₄	#6821	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	30	2	no	—
336	C₈	—	30	2	no	—
337	C₈	—	60	1	no	—
338	C₂×C₃	—	2	1	no	—
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	#64959	1	1	yes	characteristic
341	C₂×C₃	#65113	1	2	yes	—
342	D₃	—	40	1	no	—
343	C₅	#68745	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	2	2	no	—
345	C₂²	—	60	2	no	—
346	C₂²	—	60	1	no	—
347	C₂²	#91770	1	1	yes	characteristic
348	C₂²	#91775	1	1	yes	characteristic
349	C₂²	#91787	1	1	yes	characteristic
350	C₄	—	2	4	no	—
351	C₄	—	4	2	no	—
352	C₄	—	60	1	no	—
353	C₄	C₄×D₆₀	1	1	yes	characteristic
354	C₄	#91774	1	1	yes	characteristic
355	C₄	#91781	1	1	yes	characteristic
356	C₄	#91784	1	1	yes	characteristic
357	C₃	#10599538	1	1	yes	characteristic, complemented, Sylow
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	120	1	no	—
361	C₂	#11750771	1	1	yes	characteristic
362	C₂	#11750925	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#339	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×#6817	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#339	D₁₅
hypocenter	C₃×C₅	#1937
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

Complements

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

N	H ≅ G/N	type
3. #11751100	C₂	semidirect
5. C₂×C₄×C₃⋊#168	C₂	semidirect
6. C₃×C₅×#339	C₂	semidirect
7. #11746379	C₂	semidirect
17. #91094	C₄	semidirect
18. #91093	C₄	semidirect
28. C₅×#339	D₃	semidirect
62. C₃×#339	D₅	semidirect
148. #339	D₁₅	semidirect
295. C₃×C₅	#1937	semidirect
343. C₅	#68745	semidirect
357. C₃	#10599538	semidirect

Maximal subgroups

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

subgroup	index
#11751230	2
#11751103	2
#11751100	2
C₂×#91089	2
C₂×C₄×C₃⋊#168	2
C₃×C₅×#339	2
#11746379	2
#10599538	3
#68745	5