Subgroups — #14418066

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5748	C₂	1	1	yes	maximal, characteristic
2	C₂×#91286	C₂	1	2	yes	maximal, complemented
3	#11751226	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65411	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748425	C₂	1	2	yes	maximal, complemented
6	#11748063	C₂	1	4	yes	maximal, complemented
7	#11748062	C₂	1	4	yes	maximal, complemented
8	#10615730	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×#6710	C₂²	1	2	yes	—
12	C₅×#909	C₄	1	4	yes	complemented
13	C₅×#908	C₄	1	4	yes	complemented
14	C₅×#904	C₂²	1	4	yes	—
15	C₃×#4072	C₂²	1	1	yes	characteristic
16	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
17	C₂×#6650	—	2	2	no	—
18	#91286	—	2	2	no	—
19	#91286	—	2	2	no	—
20	#91286	C₂²	1	4	yes	complemented
21	#91286	C₂²	1	4	yes	complemented
22	C₂×D₃×#168	—	2	2	no	—
23	#91103	C₂²	1	4	yes	complemented
24	C₃×#3947	C₂²	1	2	yes	complemented
25	C₃×C₈×Dic₅	C₂²	1	2	yes	complemented
26	#90979	C₂²	1	8	yes	complemented
27	#78579	—	5	1	no	maximal, Hall
28	C₅×#575	—	3	1	no	—
29	C₂×#4094	—	3	2	no	—
30	#65525	—	3	2	no	—
31	#65486	—	3	4	no	—
32	#65485	—	3	4	no	—
33	#65414	—	3	1	no	—
34	#65411	D₃	1	1	yes	characteristic, complemented
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₅×#67	—	2	2	no	—
37	C₅×#235	—	2	2	no	—
38	C₅×#235	—	2	2	no	—
39	C₅×#235	C₂×C₄	1	4	yes	—
40	C₅×#235	C₂³	1	4	yes	—
41	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
42	C₂×C₄×C₅×D₃	—	2	2	no	—
43	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
44	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
45	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
46	#6710	—	2	2	no	—
47	#6710	C₂³	1	4	yes	—
48	C₂×C₃⋊#168	—	2	2	no	—
49	C₅×Dic₁₂	C₂×C₄	1	4	yes	—
50	C₅×D₂₄	C₂×C₄	1	4	yes	—
51	C₅×#204	C₂×C₄	1	8	yes	—
52	C₃×C₂²⋊Dic₅	—	2	1	no	—
53	C₃×C₄⋊Dic₅	—	2	1	no	—
54	C₃×C₄×Dic₅	C₂³	1	2	yes	—
55	C₂×C₃×#168	D₄	1	2	yes	—
56	C₂×C₃×#168	D₄	1	2	yes	—
57	C₂×C₃×#168	C₂³	1	1	yes	characteristic
58	C₂×C₃×#168	C₂³	1	1	yes	characteristic
59	#6650	—	2	4	no	—
60	#6650	—	2	4	no	—
61	D₃×#168	—	2	4	no	—
62	D₃×#168	—	2	4	no	—
63	#5748	D₅	1	1	yes	characteristic
64	C₂×#901	—	5	2	no	—
65	C₃×#405	—	5	1	no	—
66	#5125	—	5	1	no	—
67	#5005	—	5	2	no	—
68	#4717	—	5	4	no	—
69	#4700	—	5	4	no	—
70	C₂×C₅×#41	—	3	2	no	—
71	C₅×#137	—	3	4	no	—
72	C₅×AGL(1,ℤ/8)	—	3	4	no	—
73	C₅×#135	—	3	4	no	—
74	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
75	#4094	—	3	4	no	—
76	#4094	—	3	4	no	—
77	#4094	—	6	2	no	—
78	#4094	—	6	2	no	—
79	#4072	C₂×D₃	1	1	yes	characteristic
80	C₂×#681	—	3	2	no	—
81	C₂×#681	—	6	2	no	—
82	C₂²×#168	—	6	2	no	—
83	C₂²×#168	C₂×D₃	1	1	yes	characteristic
84	#3969	—	3	8	no	—
85	#3951	—	3	4	no	—
86	#3947	C₂×D₃	1	2	yes	complemented
87	C₈×Dic₅	C₂×D₃	1	2	yes	complemented
88	#2955	—	15	1	no	Sylow
89	C₂³×C₃×C₅	#41	1	1	yes	characteristic
90	C₂²×C₅×D₃	—	2	2	no	—
91	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
93	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
95	C₅×#67	—	2	4	no	—
96	C₅×#67	—	2	4	no	—
97	C₂×C₅×Dic₃	—	2	2	no	—
98	C₅×D₁₂	—	2	2	no	—
99	C₅×D₁₂	C₂²×C₄	1	4	yes	—
100	C₄×C₅×D₃	—	2	4	no	—
101	C₄×C₅×D₃	—	2	4	no	—
102	C₅×Dic₆	—	2	2	no	—
103	C₅×Dic₆	C₂²×C₄	1	4	yes	—
104	C₂×C₃×Dic₅	—	2	2	no	—
105	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
106	C₃⋊#168	—	2	4	no	—
107	C₃×#168	C₂×D₄	1	1	yes	characteristic
108	C₃×#168	C₂×D₄	1	2	yes	—
109	C₃×#168	C₂×D₄	1	1	yes	characteristic
110	C₂×#235	C₂×D₅	1	2	yes	—
111	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
112	C₂²×C₃×C₈	—	5	1	no	—
113	C₃×#117	—	5	1	no	—
114	C₂×#208	—	5	2	no	—
115	#909	Dic₅	1	4	yes	complemented
116	#908	Dic₅	1	4	yes	complemented
117	#904	C₂×D₅	1	4	yes	—
118	#901	—	5	4	no	—
119	#901	—	5	4	no	—
120	#901	—	10	2	no	—
121	#901	—	10	2	no	—
122	C₂×ΓL(1,𝔽₂₅)	—	10	2	no	—
123	C₂×D₃×C₈	—	10	2	no	—
124	C₃×#97	—	5	2	no	—
125	C₃×C₄×C₈	—	5	2	no	—
126	#819	—	5	4	no	—
127	#805	—	5	8	no	—
128	C₅×#41	—	3	4	no	—
129	C₅×#41	—	3	4	no	—
130	C₅×#41	—	6	2	no	—
131	C₅×#41	—	6	2	no	—
132	C₂×C₅×Q₈	—	3	2	no	—
133	C₂×C₅×D₄	—	3	2	no	—
134	C₂×C₅×D₄	—	6	2	no	—
135	C₂²×C₄×C₅	—	6	2	no	—
136	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
137	C₅×Q₁₆	—	3	4	no	—
138	C₅×SD₁₆	—	3	8	no	—
139	C₅×D₈	—	3	4	no	—
140	C₅×M₁₆	C₄×D₃	1	4	yes	—
141	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
142	C₂²⋊Dic₅	—	2	1	no	—
143	C₄⋊Dic₅	—	2	1	no	—
144	C₄×Dic₅	C₂²×D₃	1	2	yes	—
145	#681	—	3	4	no	—
146	#681	—	6	2	no	—
147	#681	—	6	4	no	—
148	#681	—	6	4	no	—
149	C₂×#168	—	6	2	no	—
150	C₂×#168	—	6	4	no	—
151	C₂×#168	—	6	4	no	—
152	C₂×#168	D₁₂	1	2	yes	—
153	C₂×#168	D₁₂	1	2	yes	—
154	C₂×#168	C₂²×D₃	1	1	yes	characteristic
155	C₂×#168	C₂²×D₃	1	1	yes	characteristic
156	#575	—	3	1	no	—
157	C₂×#131	—	15	2	no	—
158	#444	—	15	4	no	—
159	#443	—	15	4	no	—
160	#430	—	15	1	no	—
161	#421	—	15	2	no	—
162	#405	—	5	1	no	—
163	C₂²×C₃×C₅	—	4	1	no	—
164	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
165	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
166	C₂×C₅×D₃	—	2	4	no	—
167	C₂×C₅×D₃	—	4	4	no	—
168	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
169	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
170	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₃×Dic₅	—	4	2	no	—
172	C₅×Dic₃	—	2	4	no	—
173	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×#67	—	2	2	no	—
175	#235	—	2	2	no	—
176	#235	—	2	2	no	—
177	#235	C₂×Dic₅	1	4	yes	—
178	#235	C₂²×D₅	1	4	yes	—
179	C₂×D₁₂	C₂×Dic₅	1	2	yes	—
180	C₂×C₄×D₃	—	2	2	no	—
181	C₂×Dic₆	C₂×Dic₅	1	2	yes	—
182	C₃×M₁₆	C₂×Dic₅	1	4	yes	—
183	C₂×C₃×C₈	—	5	2	no	—
184	C₂×C₃×C₈	—	5	2	no	—
185	C₂×C₃×C₈	—	5	1	no	—
186	C₂×C₃×C₈	—	5	1	no	—
187	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
188	C₃×C₄⋊C₄	—	10	1	no	—
189	C₃×C₂²⋊C₄	—	10	1	no	—
190	C₃×C₄²	—	5	2	no	—
191	#208	—	5	4	no	—
192	#208	—	10	2	no	—
193	C₂×C₃⋊C₈	—	10	2	no	—
194	Dic₁₂	C₂×Dic₅	1	4	yes	—
195	D₂₄	C₂×Dic₅	1	4	yes	—
196	#204	C₂×Dic₅	1	8	yes	—
197	ΓL(1,𝔽₂₅)	—	10	4	no	—
198	ΓL(1,𝔽₂₅)	—	10	4	no	—
199	D₃×C₈	—	10	4	no	—
200	D₃×C₈	—	10	4	no	—
201	C₂³×C₅	—	6	2	no	—
202	C₂³×C₅	#235	1	1	yes	characteristic
203	C₅×Q₈	—	3	4	no	—
204	C₅×Q₈	—	6	2	no	—
205	C₅×D₄	—	3	4	no	—
206	C₅×D₄	—	6	4	no	—
207	C₅×D₄	—	6	4	no	—
208	C₅×D₄	—	6	2	no	—
209	C₂×C₄×C₅	—	6	2	no	—
210	C₂×C₄×C₅	—	6	4	no	—
211	C₂×C₄×C₅	—	6	4	no	—
212	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
213	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
214	C₂×C₄×C₅	#235	1	1	yes	characteristic
215	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
216	C₂×Dic₅	—	2	2	no	—
217	C₅×C₈	C₂×C₄×D₃	1	4	yes	—
218	#168	—	6	4	no	—
219	#168	C₂×D₁₂	1	1	yes	characteristic
220	#168	C₂×D₁₂	1	2	yes	—
221	#168	C₂×D₁₂	1	1	yes	characteristic
222	C₂×#41	—	3	2	no	—
223	#137	—	3	4	no	—
224	AGL(1,ℤ/8)	—	3	4	no	—
225	#135	—	3	4	no	—
226	#131	—	15	4	no	—
227	#131	—	15	4	no	—
228	#131	—	30	2	no	—
229	#131	—	30	2	no	—
230	C₂×M₁₆	—	15	2	no	—
231	C₂×M₁₆	—	30	2	no	—
232	C₂×M₁₆	D₃×D₅	1	1	yes	characteristic
233	C₂²×C₈	—	5	1	no	—
234	C₂²×C₈	—	30	2	no	—
235	#117	—	5	1	no	—
236	#108	—	15	4	no	—
237	#104	—	15	8	no	—
238	#97	—	5	2	no	—
239	C₄×C₈	—	5	2	no	—
240	C₂×C₃×C₅	—	2	1	no	—
241	C₂×C₃×C₅	—	2	2	no	—
242	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
243	C₅×D₃	—	4	4	no	—
244	C₂³×C₃	#711	1	1	yes	characteristic
245	C₂²×D₃	—	2	2	no	—
246	C₂×C₃×C₄	—	10	2	no	—
247	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
248	C₂×C₃×C₄	#711	1	1	yes	characteristic
249	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
250	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
251	#67	—	2	4	no	—
252	#67	—	2	4	no	—
253	C₂×Dic₃	—	2	2	no	—
254	D₁₂	—	2	2	no	—
255	D₁₂	C₂²×Dic₅	1	4	yes	—
256	C₄×D₃	—	2	4	no	—
257	C₄×D₃	—	2	4	no	—
258	Dic₆	—	2	2	no	—
259	Dic₆	C₂²×Dic₅	1	4	yes	—
260	C₃×C₈	—	5	1	no	—
261	C₃×C₈	—	5	2	no	—
262	C₃×C₈	—	5	1	no	—
263	C₃×C₈	C₂²×Dic₅	1	4	yes	—
264	C₃⋊C₈	—	10	4	no	—
265	C₂²×C₅	—	4	1	no	—
266	C₂²×C₅	—	6	4	no	—
267	C₂²×C₅	—	12	4	no	—
268	C₂²×C₅	C₄×D₁₂	1	2	yes	—
269	C₂²×C₅	C₂×#235	1	1	yes	characteristic
270	C₄×C₅	—	6	4	no	—
271	C₄×C₅	C₄×D₁₂	1	2	yes	—
272	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
273	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
274	Dic₅	—	4	2	no	—
275	#41	—	3	4	no	—
276	#41	—	3	4	no	—
277	#41	—	6	2	no	—
278	#41	—	6	2	no	—
279	C₂×Q₈	—	3	2	no	—
280	C₂×D₄	—	3	2	no	—
281	C₂×D₄	—	6	2	no	—
282	C₂²×C₄	—	6	2	no	—
283	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
284	Q₁₆	—	3	4	no	—
285	SD₁₆	—	3	8	no	—
286	D₈	—	3	4	no	—
287	M₁₆	—	15	4	no	—
288	M₁₆	—	30	4	no	—
289	M₁₆	—	30	4	no	—
290	M₁₆	—	30	2	no	—
291	M₁₆	D₃×Dic₅	1	4	yes	—
292	C₂×C₈	—	5	2	no	—
293	C₂×C₈	—	5	2	no	—
294	C₂×C₈	—	5	1	no	—
295	C₂×C₈	—	5	1	no	—
296	C₂×C₈	—	30	4	no	—
297	C₂×C₈	—	30	4	no	—
298	C₂×C₈	—	30	2	no	—
299	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
300	C₄⋊C₄	—	10	1	no	—
301	C₂²⋊C₄	—	10	1	no	—
302	C₄²	—	5	2	no	—
303	C₃×C₅	#2955	1	1	yes	characteristic, complemented, Hall
304	C₂²×C₃	—	4	1	no	—
305	C₂²×C₃	D₄×Dic₅	1	2	yes	—
306	C₂²×C₃	C₂×#711	1	1	yes	characteristic
307	C₂×D₃	—	2	4	no	—
308	C₂×D₃	—	4	4	no	—
309	C₃×C₄	—	20	2	no	—
310	C₃×C₄	D₄×Dic₅	1	2	yes	—
311	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
312	C₃×C₄	C₂³×Dic₅	1	1	yes	characteristic
313	Dic₃	—	2	4	no	—
314	C₂×C₅	—	2	1	no	—
315	C₂×C₅	—	2	2	no	—
316	C₂×C₅	—	12	4	no	—
317	C₂×C₅	C₂×C₄×D₁₂	1	1	yes	characteristic
318	C₂³	—	6	2	no	—
319	C₂³	#6772	1	1	yes	characteristic
320	Q₈	—	3	4	no	—
321	Q₈	—	6	2	no	—
322	D₄	—	3	4	no	—
323	D₄	—	6	4	no	—
324	D₄	—	6	4	no	—
325	D₄	—	6	2	no	—
326	C₂×C₄	—	6	2	no	—
327	C₂×C₄	—	6	4	no	—
328	C₂×C₄	—	6	4	no	—
329	C₂×C₄	—	10	2	no	—
330	C₂×C₄	#6772	1	1	yes	characteristic
331	C₂×C₄	D₅×D₁₂	1	2	yes	—
332	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
333	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
334	C₈	—	5	1	no	—
335	C₈	—	5	2	no	—
336	C₈	—	5	1	no	—
337	C₈	—	30	4	no	—
338	C₈	C₂×D₃×Dic₅	1	4	yes	—
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	—	2	2	no	—
341	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
342	D₃	—	4	4	no	—
343	C₅	#78579	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	4	1	no	—
345	C₂²	—	6	4	no	—
346	C₂²	—	12	4	no	—
347	C₂²	Dic₅×D₁₂	1	2	yes	—
348	C₂²	C₂×#6772	1	1	yes	characteristic
349	C₄	—	6	4	no	—
350	C₄	—	20	2	no	—
351	C₄	Dic₅×D₁₂	1	2	yes	—
352	C₄	C₂×D₅×D₁₂	1	1	yes	characteristic
353	C₄	C₂²×D₃×Dic₅	1	1	yes	characteristic
354	C₃	#10615730	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	12	4	no	—
358	C₂	C₂×Dic₅×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₅×D₁₂
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂²
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	C₅×#5748	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×M₁₆	D₃×D₅
hypocenter	C₃×C₅	#2955
LCS	G ▷ 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. C₂×#91286	C₂	semidirect
3. #11751226	C₂	semidirect
4. C₃×#65411	C₂	semidirect
5. #11748425	C₂	semidirect
6. #11748063	C₂	semidirect
7. #11748062	C₂	semidirect
12. C₅×#909	C₄	semidirect
13. C₅×#908	C₄	semidirect
20. #91286	C₂²	semidirect
21. #91286	C₂²	semidirect
23. #91103	C₂²	semidirect
24. C₃×#3947	C₂²	semidirect
25. C₃×C₈×Dic₅	C₂²	semidirect
26. #90979	C₂²	semidirect
34. #65411	D₃	semidirect
86. #3947	C₂×D₃	semidirect
87. C₈×Dic₅	C₂×D₃	semidirect
115. #909	Dic₅	semidirect
116. #908	Dic₅	semidirect
303. C₃×C₅	#2955	semidirect
343. C₅	#78579	semidirect
354. C₃	#10615730	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5748	2
C₂×#91286	2
#11751226	2
C₃×#65411	2
#11748425	2
#11748063	2
#11748062	2
#10615730	3
#78579	5