Subgroups — #14418068

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	C₃×#66082	C₂	1	1	yes	maximal, characteristic
2	C₂×#91283	C₂	1	2	yes	maximal, complemented
3	#11751226	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5122	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748421	C₂	1	2	yes	maximal, complemented
6	#11748067	C₂	1	4	yes	maximal, complemented
7	#11748066	C₂	1	4	yes	maximal, complemented
8	#10610310	—	3	1	no	maximal, Hall
9	C₂×C₃×#709	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×#6710	C₂²	1	2	yes	—
12	C₅×#926	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×#4055	C₄	1	4	yes	complemented
15	C₃×#4054	C₄	1	4	yes	complemented
16	C₃×#4050	C₂²	1	4	yes	—
17	C₂×#6649	—	2	2	no	—
18	#91283	—	2	2	no	—
19	#91283	—	2	2	no	—
20	#91283	C₂²	1	4	yes	complemented
21	#91283	C₂²	1	4	yes	complemented
22	C₂×D₅×C₃⋊C₈	—	2	2	no	—
23	#91103	C₂²	1	4	yes	complemented
24	C₅×#815	C₂²	1	2	yes	complemented
25	C₅×C₈×Dic₃	C₂²	1	2	yes	complemented
26	#90977	C₂²	1	8	yes	complemented
27	#83999	—	5	1	no	maximal, Hall
28	#66082	D₃	1	1	yes	characteristic
29	C₂×#4047	—	3	2	no	—
30	C₅×#405	—	3	1	no	—
31	#65414	—	3	1	no	—
32	#65294	—	3	2	no	—
33	#65006	—	3	4	no	—
34	#64989	—	3	4	no	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
37	C₃×#709	—	2	2	no	—
38	C₃×#709	—	2	2	no	—
39	C₃×#709	C₂×C₄	1	4	yes	—
40	C₃×#709	C₂³	1	4	yes	—
41	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
42	C₂×C₃×C₄×D₅	—	2	2	no	—
43	C₂×C₃×Dic₁₀	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×#217	—	2	1	no	—
50	C₅×C₄⋊Dic₃	—	2	1	no	—
51	C₄×C₅×Dic₃	C₂³	1	2	yes	—
52	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
53	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
54	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
56	C₃×Dic₂₀	C₂×C₄	1	4	yes	—
57	C₃×D₄₀	C₂×C₄	1	4	yes	—
58	C₃×#677	C₂×C₄	1	8	yes	—
59	#6649	—	2	4	no	—
60	#6649	—	2	4	no	—
61	D₅×C₃⋊C₈	—	2	4	no	—
62	D₅×C₃⋊C₈	—	2	4	no	—
63	C₃×#575	—	5	1	no	—
64	C₂×#948	—	5	2	no	—
65	#5236	—	5	2	no	—
66	#5197	—	5	4	no	—
67	#5196	—	5	4	no	—
68	#5125	—	5	1	no	—
69	#5122	D₅	1	1	yes	characteristic, complemented
70	C₂×#709	C₂×D₃	1	2	yes	—
71	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
72	C₂²×C₅×C₈	—	3	1	no	—
73	C₅×#117	—	3	1	no	—
74	C₂×#681	—	3	2	no	—
75	#4055	Dic₃	1	4	yes	complemented
76	#4054	Dic₃	1	4	yes	complemented
77	#4050	C₂×D₃	1	4	yes	—
78	#4047	—	3	4	no	—
79	#4047	—	3	4	no	—
80	#4047	—	6	2	no	—
81	#4047	—	6	2	no	—
82	C₂×#676	—	6	2	no	—
83	C₂×C₈×D₅	—	6	2	no	—
84	C₅×#97	—	3	2	no	—
85	C₄×C₅×C₈	—	3	2	no	—
86	#3951	—	3	4	no	—
87	#3937	—	3	8	no	—
88	#2955	—	15	1	no	Sylow
89	C₂³×C₃×C₅	#41	1	1	yes	characteristic
90	C₂²×C₃×D₅	—	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	C₂×C₅×Dic₃	—	2	2	no	—
96	C₃×C₂²⋊D₅	—	2	4	no	—
97	C₃×C₂²⋊D₅	—	2	4	no	—
98	C₂×C₃×Dic₅	—	2	2	no	—
99	C₃×D₂₀	—	2	2	no	—
100	C₃×D₂₀	C₂²×C₄	1	4	yes	—
101	C₃×C₄×D₅	—	2	4	no	—
102	C₃×C₄×D₅	—	2	4	no	—
103	C₃×Dic₁₀	—	2	2	no	—
104	C₃×Dic₁₀	C₂²×C₄	1	4	yes	—
105	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
106	C₃⋊#168	—	2	4	no	—
107	C₅×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
109	C₅×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₃×#41	—	5	2	no	—
111	C₃×#137	—	5	4	no	—
112	C₃×AGL(1,ℤ/8)	—	5	4	no	—
113	C₃×#135	—	5	4	no	—
114	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
115	#948	—	5	4	no	—
116	#948	—	5	4	no	—
117	#948	—	10	2	no	—
118	#948	—	10	2	no	—
119	#926	C₂×D₅	1	1	yes	characteristic
120	C₂×#208	—	5	2	no	—
121	C₂×#208	—	10	2	no	—
122	C₂²×C₃⋊C₈	—	10	2	no	—
123	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
124	#837	—	5	8	no	—
125	#819	—	5	4	no	—
126	#815	C₂×D₅	1	2	yes	complemented
127	C₈×Dic₃	C₂×D₅	1	2	yes	complemented
128	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
129	C₂×C₂²⋊D₅	—	2	2	no	—
130	#709	—	2	2	no	—
131	#709	—	2	2	no	—
132	#709	C₂×Dic₃	1	4	yes	—
133	#709	C₂²×D₃	1	4	yes	—
134	C₂×D₂₀	C₂×Dic₃	1	2	yes	—
135	C₂×C₄×D₅	—	2	2	no	—
136	C₂×Dic₁₀	C₂×Dic₃	1	2	yes	—
137	C₅×M₁₆	C₂×Dic₃	1	4	yes	—
138	C₂×C₅×C₈	—	3	2	no	—
139	C₂×C₅×C₈	—	3	2	no	—
140	C₂×C₅×C₈	—	3	1	no	—
141	C₂×C₅×C₈	—	3	1	no	—
142	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
143	C₅×C₄⋊C₄	—	6	1	no	—
144	C₅×C₂²⋊C₄	—	6	1	no	—
145	C₄²×C₅	—	3	2	no	—
146	#681	—	3	4	no	—
147	#681	—	6	2	no	—
148	C₂×#168	—	6	2	no	—
149	Dic₂₀	C₂×Dic₃	1	4	yes	—
150	D₄₀	C₂×Dic₃	1	4	yes	—
151	#677	C₂×Dic₃	1	8	yes	—
152	#676	—	6	4	no	—
153	#676	—	6	4	no	—
154	C₈×D₅	—	6	4	no	—
155	C₈×D₅	—	6	4	no	—
156	#575	—	5	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	—	3	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×C₃×D₅	—	2	4	no	—
167	C₂×C₃×D₅	—	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	—	2	4	no	—
172	C₅×Dic₃	—	4	2	no	—
173	C₃×#41	—	5	4	no	—
174	C₃×#41	—	5	4	no	—
175	C₃×#41	—	10	2	no	—
176	C₃×#41	—	10	2	no	—
177	C₂×C₃×Q₈	—	5	2	no	—
178	C₂×C₃×D₄	—	5	2	no	—
179	C₂×C₃×D₄	—	10	2	no	—
180	C₂²×C₃×C₄	—	10	2	no	—
181	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₃×Q₁₆	—	5	4	no	—
183	C₃×SD₁₆	—	5	8	no	—
184	C₃×D₈	—	5	4	no	—
185	C₃×M₁₆	C₄×D₅	1	4	yes	—
186	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
187	#217	—	2	1	no	—
188	C₄⋊Dic₃	—	2	1	no	—
189	C₄×Dic₃	C₂²×D₅	1	2	yes	—
190	#208	—	5	4	no	—
191	#208	—	10	2	no	—
192	#208	—	10	4	no	—
193	#208	—	10	4	no	—
194	C₂×C₃⋊C₈	—	10	2	no	—
195	C₂×C₃⋊C₈	—	10	4	no	—
196	C₂×C₃⋊C₈	—	10	4	no	—
197	C₂×C₃⋊C₈	D₂₀	1	2	yes	—
198	C₂×C₃⋊C₈	D₂₀	1	2	yes	—
199	C₂×C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
200	C₂×C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
201	C₂³×C₅	#237	1	1	yes	characteristic
202	C₂²×D₅	—	2	2	no	—
203	C₂×C₄×C₅	—	6	2	no	—
204	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
205	C₂×C₄×C₅	#237	1	1	yes	characteristic
206	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
207	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
208	C₂²⋊D₅	—	2	4	no	—
209	C₂²⋊D₅	—	2	4	no	—
210	C₂×Dic₅	—	2	2	no	—
211	D₂₀	—	2	2	no	—
212	D₂₀	C₂²×Dic₃	1	4	yes	—
213	C₄×D₅	—	2	4	no	—
214	C₄×D₅	—	2	4	no	—
215	Dic₁₀	—	2	2	no	—
216	Dic₁₀	C₂²×Dic₃	1	4	yes	—
217	C₅×C₈	—	3	1	no	—
218	C₅×C₈	—	3	2	no	—
219	C₅×C₈	—	3	1	no	—
220	C₅×C₈	C₂²×Dic₃	1	4	yes	—
221	#168	—	6	4	no	—
222	C₂×#41	—	5	2	no	—
223	#137	—	5	4	no	—
224	AGL(1,ℤ/8)	—	5	4	no	—
225	#135	—	5	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	—	3	1	no	—
234	C₂²×C₈	—	30	2	no	—
235	#117	—	3	1	no	—
236	#108	—	15	4	no	—
237	#104	—	15	8	no	—
238	#97	—	3	2	no	—
239	C₄×C₈	—	3	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	C₃×D₅	—	4	4	no	—
244	C₂³×C₃	—	10	2	no	—
245	C₂³×C₃	#709	1	1	yes	characteristic
246	C₃×Q₈	—	5	4	no	—
247	C₃×Q₈	—	10	2	no	—
248	C₃×D₄	—	5	4	no	—
249	C₃×D₄	—	10	4	no	—
250	C₃×D₄	—	10	4	no	—
251	C₃×D₄	—	10	2	no	—
252	C₂×C₃×C₄	—	10	2	no	—
253	C₂×C₃×C₄	—	10	4	no	—
254	C₂×C₃×C₄	—	10	4	no	—
255	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
256	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
257	C₂×C₃×C₄	#709	1	1	yes	characteristic
258	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
259	C₂×Dic₃	—	2	2	no	—
260	C₃×C₈	C₂×C₄×D₅	1	4	yes	—
261	C₃⋊C₈	—	10	4	no	—
262	C₃⋊C₈	C₂×D₂₀	1	1	yes	characteristic
263	C₃⋊C₈	C₂×D₂₀	1	2	yes	—
264	C₃⋊C₈	C₂×D₂₀	1	1	yes	characteristic
265	C₂²×C₅	—	4	1	no	—
266	C₂²×C₅	D₄×Dic₃	1	2	yes	—
267	C₂²×C₅	C₂×#237	1	1	yes	characteristic
268	C₂×D₅	—	2	4	no	—
269	C₂×D₅	—	4	4	no	—
270	C₄×C₅	—	12	2	no	—
271	C₄×C₅	D₄×Dic₃	1	2	yes	—
272	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
273	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
274	Dic₅	—	2	4	no	—
275	#41	—	5	4	no	—
276	#41	—	5	4	no	—
277	#41	—	10	2	no	—
278	#41	—	10	2	no	—
279	C₂×Q₈	—	5	2	no	—
280	C₂×D₄	—	5	2	no	—
281	C₂×D₄	—	10	2	no	—
282	C₂²×C₄	—	10	2	no	—
283	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
284	Q₁₆	—	5	4	no	—
285	SD₁₆	—	5	8	no	—
286	D₈	—	5	4	no	—
287	M₁₆	—	15	4	no	—
288	M₁₆	—	30	4	no	—
289	M₁₆	—	30	4	no	—
290	M₁₆	—	30	2	no	—
291	M₁₆	D₅×Dic₃	1	4	yes	—
292	C₂×C₈	—	3	2	no	—
293	C₂×C₈	—	3	2	no	—
294	C₂×C₈	—	3	1	no	—
295	C₂×C₈	—	3	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	—	6	1	no	—
301	C₂²⋊C₄	—	6	1	no	—
302	C₄²	—	3	2	no	—
303	C₃×C₅	#2955	1	1	yes	characteristic, complemented, Hall
304	C₂²×C₃	—	4	1	no	—
305	C₂²×C₃	—	10	4	no	—
306	C₂²×C₃	—	20	4	no	—
307	C₂²×C₃	C₄×D₂₀	1	2	yes	—
308	C₂²×C₃	C₂×#709	1	1	yes	characteristic
309	C₃×C₄	—	10	4	no	—
310	C₃×C₄	C₄×D₂₀	1	2	yes	—
311	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
312	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
313	Dic₃	—	4	2	no	—
314	C₂×C₅	—	2	1	no	—
315	C₂×C₅	—	2	2	no	—
316	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
317	D₅	—	4	4	no	—
318	C₂³	—	10	2	no	—
319	C₂³	#6765	1	1	yes	characteristic
320	Q₈	—	5	4	no	—
321	Q₈	—	10	2	no	—
322	D₄	—	5	4	no	—
323	D₄	—	10	4	no	—
324	D₄	—	10	4	no	—
325	D₄	—	10	2	no	—
326	C₂×C₄	—	6	2	no	—
327	C₂×C₄	—	10	2	no	—
328	C₂×C₄	—	10	4	no	—
329	C₂×C₄	—	10	4	no	—
330	C₂×C₄	#6765	1	1	yes	characteristic
331	C₂×C₄	D₃×D₂₀	1	2	yes	—
332	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
333	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
334	C₈	—	3	1	no	—
335	C₈	—	3	2	no	—
336	C₈	—	3	1	no	—
337	C₈	—	30	4	no	—
338	C₈	C₂×D₅×Dic₃	1	4	yes	—
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	—	2	2	no	—
341	C₂×C₃	—	20	4	no	—
342	C₂×C₃	C₂×C₄×D₂₀	1	1	yes	characteristic
343	C₅	#83999	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	4	1	no	—
345	C₂²	—	10	4	no	—
346	C₂²	—	20	4	no	—
347	C₂²	Dic₃×D₂₀	1	2	yes	—
348	C₂²	C₂×#6765	1	1	yes	characteristic
349	C₄	—	10	4	no	—
350	C₄	—	12	2	no	—
351	C₄	Dic₃×D₂₀	1	2	yes	—
352	C₄	C₂×D₃×D₂₀	1	1	yes	characteristic
353	C₄	C₂²×D₅×Dic₃	1	1	yes	characteristic
354	C₃	#10610310	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	20	4	no	—
358	C₂	C₂×Dic₃×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₃×D₂₀
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	C₃×#66082	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×#91283	C₂	semidirect
3. #11751226	C₂	semidirect
4. C₅×#5122	C₂	semidirect
5. #11748421	C₂	semidirect
6. #11748067	C₂	semidirect
7. #11748066	C₂	semidirect
14. C₃×#4055	C₄	semidirect
15. C₃×#4054	C₄	semidirect
20. #91283	C₂²	semidirect
21. #91283	C₂²	semidirect
23. #91103	C₂²	semidirect
24. C₅×#815	C₂²	semidirect
25. C₅×C₈×Dic₃	C₂²	semidirect
26. #90977	C₂²	semidirect
69. #5122	D₅	semidirect
75. #4055	Dic₃	semidirect
76. #4054	Dic₃	semidirect
126. #815	C₂×D₅	semidirect
127. C₈×Dic₃	C₂×D₅	semidirect
303. C₃×C₅	#2955	semidirect
343. C₅	#83999	semidirect
354. C₃	#10610310	semidirect

Maximal subgroups

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

subgroup	index
C₃×#66082	2
C₂×#91283	2
#11751226	2
C₅×#5122	2
#11748421	2
#11748067	2
#11748066	2
#10610310	3
#83999	5