Subgroups — #14299682

Order 1920.

504 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756504	C₂	1	1	yes	maximal, characteristic
2	C₃×#65413	C₂	1	1	yes	maximal, characteristic, complemented
3	#11748480	C₂	1	1	yes	maximal, characteristic, complemented
4	#11748357	C₂	1	1	yes	maximal, characteristic
5	#11748350	C₂	1	1	yes	maximal, characteristic
6	C₅×#4558	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745983	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599738	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×#6710	C₂²	1	1	yes	characteristic
12	#91796	—	2	2	no	—
13	M₁₆×D₁₅	—	2	2	no	—
14	#91790	—	2	2	no	—
15	C₂×C₅×#208	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×#681	C₂²	1	1	yes	characteristic
18	C₂×C₃×#681	C₂²	1	1	yes	characteristic
19	C₃×#3954	—	2	2	no	—
20	#90993	—	2	2	no	—
21	#90963	C₄	1	2	yes	—
22	#90960	C₄	1	2	yes	—
23	#90957	—	2	2	no	—
24	#70734	—	5	1	no	maximal, Hall
25	#66079	—	3	1	no	—
26	#65421	—	3	1	no	—
27	#65414	—	3	1	no	—
28	#65413	D₃	1	1	yes	characteristic, complemented
29	#65399	—	3	1	no	—
30	C₅×#341	—	3	1	no	—
31	#64772	—	3	1	no	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×#1220	Q₈	1	2	yes	—
34	#6817	—	2	1	no	—
35	#6817	—	2	1	no	—
36	#6817	—	2	1	no	—
37	#6817	—	2	1	no	—
38	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₅	D₄	1	2	yes	—
40	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
41	C₃×C₅×M₁₆	—	2	1	no	—
42	C₃×C₅×M₁₆	D₄	1	2	yes	—
43	C₂×C₃×C₅×C₈	—	2	1	no	—
44	#6710	—	2	1	no	—
45	#6710	D₄	1	2	yes	—
46	C₂×C₃⋊#168	—	2	1	no	—
47	#6705	—	2	4	no	—
48	C₈×D₁₅	—	2	4	no	—
49	C₅×#208	—	2	1	no	—
50	C₅×#208	C₂×C₄	1	2	yes	—
51	C₂×C₅×C₃⋊C₈	—	2	1	no	—
52	C₂×C₅×C₃⋊C₈	—	2	1	no	—
53	C₂×C₅×C₃⋊C₈	—	2	1	no	—
54	C₂×C₅×C₃⋊C₈	—	2	1	no	—
55	C₃×#681	—	2	1	no	—
56	C₃×#681	—	2	1	no	—
57	C₃×#681	—	2	1	no	—
58	C₃×#681	C₂×C₄	1	2	yes	—
59	C₂×C₃×#168	—	2	1	no	—
60	C₂×C₃×#168	—	2	1	no	—
61	#5745	—	5	1	no	—
62	C₃×#430	—	5	1	no	—
63	#5132	—	5	1	no	—
64	#5129	—	5	1	no	—
65	#5124	—	5	1	no	—
66	#4558	—	5	1	no	—
67	#4558	D₅	1	1	yes	characteristic, complemented
68	C₂×#709	—	3	1	no	—
69	C₂×C₅×M₁₆	—	3	1	no	—
70	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
71	C₂²×C₅×C₈	—	3	1	no	—
72	C₂×#681	—	3	1	no	—
73	C₂×#681	C₂×D₃	1	1	yes	characteristic
74	C₂×#681	C₂×D₃	1	1	yes	characteristic
75	#4053	—	6	2	no	—
76	D₅×M₁₆	—	6	2	no	—
77	#4047	—	6	2	no	—
78	#3956	—	3	2	no	—
79	#3955	—	3	2	no	—
80	#3954	—	2	2	no	—
81	#3952	—	6	2	no	—
82	#3951	—	6	2	no	—
83	#1858	—	15	1	no	Sylow
84	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₂²×D₁₅	C₄⋊C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
92	#1220	—	2	2	no	—
93	#1220	—	2	2	no	—
94	C₂×Dic₁₅	C₄⋊C₄	1	2	yes	—
95	D₆₀	—	2	1	no	—
96	D₆₀	—	2	1	no	—
97	C₄×D₁₅	—	2	2	no	—
98	C₄×D₁₅	—	2	2	no	—
99	Dic₃₀	—	2	1	no	—
100	Dic₃₀	—	2	1	no	—
101	C₃×C₅×C₈	—	2	2	no	—
102	C₃⋊#168	—	2	2	no	—
103	C₃×#168	—	2	2	no	—
104	C₃×#168	—	4	1	no	—
105	C₅×C₃⋊C₈	—	2	2	no	—
106	C₅×C₃⋊C₈	—	4	1	no	—
107	C₂×#235	—	5	1	no	—
108	C₂×C₃×M₁₆	—	5	1	no	—
109	C₂×C₃×M₁₆	—	5	1	no	—
110	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
111	C₂×#208	—	5	1	no	—
112	C₂×#208	C₂×D₅	1	1	yes	characteristic
113	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
114	#907	—	10	2	no	—
115	D₃×M₁₆	—	10	2	no	—
116	#901	—	10	2	no	—
117	C₃×#108	—	10	2	no	—
118	#824	—	5	2	no	—
119	#823	—	5	2	no	—
120	#822	—	10	2	no	—
121	(C₂×D₃)⋊C₈	—	10	2	no	—
122	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
123	C₂×C₂²⋊D₅	—	3	2	no	—
124	#709	—	6	1	no	—
125	#709	—	6	1	no	—
126	#709	—	6	1	no	—
127	#709	—	6	1	no	—
128	C₂×D₂₀	—	3	1	no	—
129	C₂×C₄×D₅	—	3	2	no	—
130	C₂×Dic₁₀	—	3	1	no	—
131	C₅×M₁₆	—	2	1	no	—
132	C₅×M₁₆	—	3	2	no	—
133	C₅×M₁₆	—	6	1	no	—
134	C₅×M₁₆	#67	1	2	yes	—
135	C₂×C₅×C₈	—	2	1	no	—
136	C₂×C₅×C₈	—	6	1	no	—
137	C₂×C₅×C₈	—	6	1	no	—
138	C₂×C₅×C₈	—	6	1	no	—
139	C₂×C₅×C₈	—	6	1	no	—
140	#681	—	2	1	no	—
141	#681	—	2	1	no	—
142	#681	—	2	1	no	—
143	#681	—	3	2	no	—
144	#681	—	6	1	no	—
145	#681	C₄×D₃	1	2	yes	—
146	C₂×#168	—	2	1	no	—
147	C₂×#168	—	2	1	no	—
148	C₂×#168	—	6	1	no	—
149	#676	—	6	4	no	—
150	C₈×D₅	—	6	4	no	—
151	#568	—	15	1	no	—
152	#430	—	5	1	no	—
153	#430	—	15	1	no	—
154	#413	—	15	1	no	—
155	#408	—	15	1	no	—
156	#341	—	3	1	no	—
157	#341	—	15	1	no	—
158	C₂²×C₃×C₅	—	4	1	no	—
159	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×D₁₅	—	2	2	no	—
163	C₂×D₁₅	—	4	2	no	—
164	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
165	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
166	C₃×C₄×C₅	#120	1	1	yes	characteristic
167	C₃×C₄×C₅	#123	1	1	yes	characteristic
168	Dic₁₅	—	2	2	no	—
169	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
170	C₂×#67	—	5	2	no	—
171	#235	—	10	1	no	—
172	#235	—	10	1	no	—
173	#235	—	10	1	no	—
174	#235	—	10	1	no	—
175	C₂×D₁₂	—	5	1	no	—
176	C₂×C₄×D₃	—	5	2	no	—
177	C₂×Dic₆	—	5	1	no	—
178	C₃×M₁₆	—	2	1	no	—
179	C₃×M₁₆	—	5	2	no	—
180	C₃×M₁₆	—	10	1	no	—
181	C₃×M₁₆	—	10	1	no	—
182	C₃×M₁₆	—	10	1	no	—
183	C₃×M₁₆	D₂₀	1	2	yes	—
184	C₂×C₃×C₈	—	2	1	no	—
185	C₂×C₃×C₈	—	10	1	no	—
186	C₂×C₃×C₈	—	10	1	no	—
187	#208	—	2	1	no	—
188	#208	—	5	2	no	—
189	#208	—	10	1	no	—
190	#208	C₄×D₅	1	2	yes	—
191	C₂×C₃⋊C₈	—	2	1	no	—
192	C₂×C₃⋊C₈	—	2	1	no	—
193	C₂×C₃⋊C₈	—	2	1	no	—
194	C₂×C₃⋊C₈	—	2	1	no	—
195	C₂×C₃⋊C₈	—	10	1	no	—
196	ΓL(1,𝔽₂₅)	—	10	4	no	—
197	D₃×C₈	—	10	4	no	—
198	C₂³×C₅	D₃×Q₈	1	1	yes	characteristic
199	C₂²×D₅	—	3	2	no	—
200	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
201	C₂×C₄×C₅	#235	1	1	yes	characteristic
202	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
203	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅	#239	1	1	yes	characteristic
205	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
206	C₂²⋊D₅	—	6	2	no	—
207	C₂²⋊D₅	—	6	2	no	—
208	C₂×Dic₅	—	3	2	no	—
209	D₂₀	—	6	1	no	—
210	D₂₀	—	6	1	no	—
211	C₄×D₅	—	6	2	no	—
212	C₄×D₅	—	6	2	no	—
213	Dic₁₀	—	6	1	no	—
214	Dic₁₀	—	6	1	no	—
215	C₅×C₈	—	2	2	no	—
216	C₅×C₈	—	6	2	no	—
217	C₅×C₈	—	12	1	no	—
218	#168	—	2	2	no	—
219	#168	—	4	1	no	—
220	#168	—	6	2	no	—
221	C₂×#41	—	15	1	no	—
222	#131	—	30	2	no	—
223	#131	—	30	2	no	—
224	C₂×M₁₆	—	3	1	no	—
225	C₂×M₁₆	—	5	1	no	—
226	C₂×M₁₆	—	5	1	no	—
227	C₂×M₁₆	—	15	1	no	—
228	C₂×M₁₆	—	30	2	no	—
229	C₂×M₁₆	D₃×D₅	1	1	yes	characteristic
230	C₂²×C₈	—	3	1	no	—
231	#108	—	10	2	no	—
232	#108	—	30	2	no	—
233	#101	—	15	2	no	—
234	#100	—	15	2	no	—
235	C₂²⋊C₈	—	30	2	no	—
236	C₂×C₃×C₅	—	2	1	no	—
237	C₂×C₃×C₅	—	2	1	no	—
238	C₂×C₃×C₅	—	2	1	no	—
239	C₂×C₃×C₅	#385	1	1	yes	characteristic
240	D₁₅	—	4	2	no	—
241	C₂³×C₃	Q₈×D₅	1	1	yes	characteristic
242	C₂²×D₃	—	5	2	no	—
243	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
245	C₂×C₃×C₄	#709	1	1	yes	characteristic
246	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
247	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
248	C₂×C₃×C₄	#711	1	1	yes	characteristic
249	#67	—	10	2	no	—
250	#67	—	10	2	no	—
251	C₂×Dic₃	—	5	2	no	—
252	D₁₂	—	10	1	no	—
253	D₁₂	—	10	1	no	—
254	C₄×D₃	—	10	2	no	—
255	C₄×D₃	—	10	2	no	—
256	Dic₆	—	10	1	no	—
257	Dic₆	—	10	1	no	—
258	C₃×C₈	—	2	2	no	—
259	C₃×C₈	—	10	2	no	—
260	C₃×C₈	—	20	1	no	—
261	C₃⋊C₈	—	2	2	no	—
262	C₃⋊C₈	—	4	1	no	—
263	C₃⋊C₈	—	10	2	no	—
264	C₂²×C₅	—	4	1	no	—
265	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
266	C₂²×C₅	#896	1	1	yes	characteristic
267	C₂²×C₅	#946	1	1	yes	characteristic
268	C₂×D₅	—	6	2	no	—
269	C₂×D₅	—	12	2	no	—
270	C₄×C₅	#893	1	1	yes	characteristic
271	C₄×C₅	#894	1	1	yes	characteristic
272	C₄×C₅	C₄×#67	1	1	yes	characteristic
273	C₄×C₅	#937	1	1	yes	characteristic
274	Dic₅	—	6	2	no	—
275	#41	—	30	1	no	—
276	#41	—	30	1	no	—
277	#41	—	30	1	no	—
278	#41	—	30	1	no	—
279	C₂×Q₈	—	15	1	no	—
280	C₂×D₄	—	15	2	no	—
281	C₂×D₄	—	15	1	no	—
282	C₂²×C₄	—	15	2	no	—
283	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
284	M₁₆	—	2	1	no	—
285	M₁₆	—	3	2	no	—
286	M₁₆	—	5	2	no	—
287	M₁₆	—	6	1	no	—
288	M₁₆	—	10	1	no	—
289	M₁₆	—	10	1	no	—
290	M₁₆	—	10	1	no	—
291	M₁₆	—	15	2	no	—
292	M₁₆	—	30	4	no	—
293	M₁₆	—	30	1	no	—
294	M₁₆	#1202	1	2	yes	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	6	1	no	—
297	C₂×C₈	—	6	1	no	—
298	C₂×C₈	—	6	1	no	—
299	C₂×C₈	—	6	1	no	—
300	C₂×C₈	—	10	1	no	—
301	C₂×C₈	—	10	1	no	—
302	C₂×C₈	—	30	4	no	—
303	C₂×C₈	—	30	1	no	—
304	C₃×C₅	#1858	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	4	1	no	—
306	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
307	C₂²×C₃	#4042	1	1	yes	characteristic
308	C₂²×C₃	#4043	1	1	yes	characteristic
309	C₂×D₃	—	10	2	no	—
310	C₂×D₃	—	20	2	no	—
311	C₃×C₄	C₄×D₂₀	1	1	yes	characteristic
312	C₃×C₄	#4027	1	1	yes	characteristic
313	C₃×C₄	#4028	1	1	yes	characteristic
314	C₃×C₄	#4029	1	1	yes	characteristic
315	Dic₃	—	10	2	no	—
316	C₂×C₅	—	2	1	no	—
317	C₂×C₅	—	2	1	no	—
318	C₂×C₅	—	2	1	no	—
319	C₂×C₅	#4989	1	1	yes	characteristic
320	D₅	—	12	2	no	—
321	C₂³	—	15	2	no	—
322	C₂³	D₁₅⋊Q₈	1	1	yes	characteristic
323	Q₈	—	30	1	no	—
324	Q₈	—	30	1	no	—
325	D₄	—	30	2	no	—
326	D₄	—	30	2	no	—
327	D₄	—	30	1	no	—
328	D₄	—	30	1	no	—
329	C₂×C₄	—	15	2	no	—
330	C₂×C₄	—	30	2	no	—
331	C₂×C₄	—	30	2	no	—
332	C₂×C₄	#6768	1	1	yes	characteristic
333	C₂×C₄	#6771	1	1	yes	characteristic
334	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
335	C₂×C₄	C₂×#1202	1	1	yes	characteristic
336	C₂×C₄	#6790	1	1	yes	characteristic
337	C₂×C₄	#6790	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	—	6	2	no	—
340	C₈	—	10	2	no	—
341	C₈	—	12	1	no	—
342	C₈	—	20	1	no	—
343	C₈	—	30	2	no	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	2	1	no	—
346	C₂×C₃	—	2	1	no	—
347	C₂×C₃	#64957	1	1	yes	characteristic
348	D₃	—	20	2	no	—
349	C₅	#70734	1	1	yes	characteristic, complemented, Sylow
350	C₂²	—	4	1	no	—
351	C₂²	—	30	2	no	—
352	C₂²	—	60	2	no	—
353	C₂²	#91377	1	1	yes	characteristic
354	C₂²	#91390	1	1	yes	characteristic
355	C₂²	#91404	1	1	yes	characteristic
356	C₄	—	30	2	no	—
357	C₄	#91355	1	1	yes	characteristic
358	C₄	#91406	1	1	yes	characteristic
359	C₄	C₄×#1202	1	1	yes	characteristic
360	C₄	#91572	1	1	yes	characteristic
361	C₃	#10599738	1	1	yes	characteristic, complemented, Sylow
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	60	2	no	—
366	C₂	#11748729	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#91572
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂²
socle	C₂×C₃×C₅	#385
prime radical	C₂×#6817	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×M₁₆	D₃×D₅
hypocenter	C₃×C₅	#1858
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×#65413	C₂	semidirect
3. #11748480	C₂	semidirect
6. C₅×#4558	C₂	semidirect
7. #11745983	C₂	semidirect
28. #65413	D₃	semidirect
67. #4558	D₅	semidirect
304. C₃×C₅	#1858	semidirect
349. C₅	#70734	semidirect
361. C₃	#10599738	semidirect

Maximal subgroups

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

subgroup	index
#11756504	2
C₃×#65413	2
#11748480	2
#11748357	2
#11748350	2
C₅×#4558	2
#11745983	2
#10599738	3
#70734	5