Subgroups — #14416613

Order 1920.

984 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	C₂×C₃×#4050	C₂	1	1	yes	maximal, characteristic
2	#11754444	C₂	1	2	yes	maximal, complemented
3	C₂×#91099	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5106	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748364	C₂	1	2	yes	maximal, complemented
6	#11747884	C₂	1	2	yes	maximal, complemented
7	#11747815	C₂	1	2	yes	maximal, complemented
8	#11747812	C₂	1	4	yes	maximal, complemented
9	#10610897	—	3	1	no	maximal, Hall
10	C₂×C₃×#709	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
14	C₅×#926	C₂²	1	2	yes	—
15	C₂×C₃×Dic₂₀	C₂²	1	1	yes	characteristic
16	C₃×#4050	C₄	1	8	yes	complemented
17	C₂×C₃×D₄₀	C₂²	1	1	yes	characteristic
18	C₂×C₃×#677	C₂²	1	2	yes	—
19	#91559	—	2	4	no	—
20	#91434	C₂²	1	4	yes	—
21	C₄×D₅×Dic₃	—	2	2	no	—
22	#91357	—	2	2	no	—
23	#91331	C₂²	1	4	yes	—
24	#91099	C₂²	1	2	yes	—
25	#91099	C₂²	1	2	yes	—
26	C₅×#818	C₂²	1	2	yes	—
27	C₅×#817	C₂²	1	2	yes	—
28	#90970	C₂²	1	4	yes	—
29	#90964	C₂²	1	4	yes	—
30	#83897	—	5	1	no	maximal, Hall
31	C₂×#4050	D₃	1	1	yes	characteristic
32	#65856	—	3	2	no	—
33	C₅×#427	—	3	1	no	—
34	C₂×#3947	—	3	1	no	—
35	#65291	—	3	2	no	—
36	#65178	—	3	4	no	—
37	#65158	—	3	2	no	—
38	#65142	—	3	2	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₅	D₄	1	2	yes	—
41	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
42	C₃×#709	—	2	4	no	—
43	C₃×#709	C₂×C₄	1	8	yes	complemented
44	C₂×C₃×D₂₀	C₂³	1	2	yes	—
45	C₂×C₃×C₄×D₅	—	2	2	no	—
46	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
47	C₂×D₅×Dic₃	—	2	4	no	—
48	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
49	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₄×Dic₁₅	D₄	1	2	yes	—
52	C₄×Dic₁₅	C₂³	1	2	yes	—
53	C₂×C₃⋊#168	—	2	2	no	—
54	C₂×C₃⋊#168	C₂³	1	2	yes	—
55	C₅×#217	—	2	2	no	—
56	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
57	C₄×C₅×Dic₃	—	2	2	no	—
58	C₃×Dic₂₀	C₂×C₄	1	4	yes	complemented
59	C₃×D₄₀	C₂×C₄	1	4	yes	complemented
60	C₃×#677	C₂×C₄	1	8	yes	complemented
61	#6668	—	2	4	no	—
62	#6665	—	2	4	no	—
63	Dic₃×Dic₅	—	2	4	no	—
64	C₂×C₃×#135	—	5	1	no	—
65	Dic₃×#41	—	5	2	no	—
66	#5233	—	5	2	no	—
67	#5184	—	5	2	no	—
68	#5164	—	5	4	no	—
69	#5152	—	5	2	no	—
70	#5106	D₅	1	1	yes	characteristic, complemented
71	C₂×#815	—	5	1	no	—
72	C₂×#709	C₂×D₃	1	2	yes	—
73	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
74	C₅×#117	—	3	2	no	—
75	C₂×C₄×Dic₅	—	3	1	no	—
76	C₂²×#168	—	3	1	no	—
77	C₂×Dic₂₀	C₂×D₃	1	1	yes	characteristic
78	#4050	Dic₃	1	8	yes	complemented
79	C₂×D₄₀	C₂×D₃	1	1	yes	characteristic
80	C₂×#677	C₂×D₃	1	2	yes	—
81	#4039	—	3	4	no	—
82	#4033	—	3	4	no	—
83	#4027	—	6	4	no	—
84	#4018	—	6	2	no	—
85	C₄²×D₅	—	6	2	no	—
86	C₅×#107	—	3	2	no	—
87	C₅×#106	—	3	2	no	—
88	#3947	—	3	2	no	—
89	#3947	—	3	2	no	—
90	#3942	—	3	4	no	—
91	#3941	—	3	4	no	—
92	#2943	—	15	1	no	Sylow
93	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₃×D₅	—	2	2	no	—
95	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	#41	1	2	yes	—
98	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₁₅	—	4	2	no	—
100	C₂×Dic₁₅	C₂×D₄	1	4	yes	—
101	C₂×C₅×Dic₃	—	2	4	no	—
102	C₃×C₂²⋊D₅	—	2	8	no	—
103	C₂×C₃×Dic₅	—	2	2	no	—
104	C₃×D₂₀	—	2	2	no	—
105	C₃×D₂₀	C₂²×C₄	1	4	yes	—
106	C₃×C₄×D₅	—	2	8	no	—
107	C₃×Dic₁₀	—	2	2	no	—
108	C₃×Dic₁₀	C₂²×C₄	1	4	yes	—
109	D₅×Dic₃	—	4	8	no	—
110	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
111	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
112	C₃⋊#168	—	2	2	no	—
113	C₂×C₃×#41	—	5	2	no	—
114	C₃×#135	—	5	8	no	—
115	C₂×C₃×Q₁₆	—	5	1	no	—
116	C₂×C₃×SD₁₆	—	5	2	no	—
117	C₂×C₃×D₈	—	5	1	no	—
118	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
119	Q₈×Dic₃	—	5	4	no	—
120	D₄×Dic₃	—	5	4	no	—
121	D₄×Dic₃	—	10	4	no	—
122	#926	—	10	2	no	—
123	#926	C₂×D₅	1	2	yes	—
124	C₂×C₄×Dic₃	—	5	1	no	—
125	C₂×C₄×Dic₃	—	10	2	no	—
126	C₂²×C₃⋊C₈	—	5	1	no	—
127	#835	—	5	4	no	—
128	#832	—	5	4	no	—
129	#818	C₂×D₅	1	2	yes	—
130	#817	C₂×D₅	1	2	yes	—
131	#815	—	5	2	no	—
132	#815	—	5	2	no	—
133	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₂²⋊D₅	—	2	2	no	—
135	C₂²×Dic₅	—	3	2	no	—
136	#709	—	2	4	no	—
137	#709	C₂×Dic₃	1	8	yes	complemented
138	C₂×D₂₀	C₂²×D₃	1	2	yes	—
139	C₂×C₄×D₅	—	2	2	no	—
140	C₂×C₄×D₅	—	6	4	no	—
141	C₂×Dic₁₀	C₂²×D₃	1	2	yes	—
142	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	complemented
143	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
145	C₅×C₄⋊C₄	—	3	4	no	—
146	C₅×C₂²⋊C₄	—	6	2	no	—
147	C₄²×C₅	—	6	2	no	—
148	#685	—	6	4	no	—
149	#683	—	6	4	no	—
150	C₄×Dic₅	—	3	2	no	—
151	C₄×Dic₅	—	3	2	no	—
152	C₄×Dic₅	—	6	4	no	—
153	C₂×#168	—	3	2	no	—
154	C₂×#168	—	6	2	no	—
155	Dic₂₀	C₂×Dic₃	1	4	yes	complemented
156	D₄₀	C₂×Dic₃	1	4	yes	complemented
157	#677	C₂×Dic₃	1	8	yes	complemented
158	C₂×#135	—	5	1	no	—
159	C₄×#41	—	15	2	no	—
160	#442	—	15	2	no	—
161	#441	—	15	2	no	—
162	#440	—	15	4	no	—
163	#427	—	3	1	no	—
164	#416	—	15	2	no	—
165	C₂×#97	—	15	1	no	—
166	C₂²×C₃×C₅	—	2	2	no	—
167	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
168	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₃×D₅	—	2	4	no	—
170	C₂×C₃×D₅	—	4	4	no	—
171	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
172	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
173	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
174	Dic₁₅	—	2	4	no	—
175	C₃×Dic₅	—	2	4	no	—
176	C₅×Dic₃	—	4	4	no	—
177	C₃×#41	—	5	8	no	—
178	C₃×#41	—	10	4	no	—
179	C₂×C₃×Q₈	—	5	2	no	—
180	C₂×C₃×D₄	—	5	2	no	—
181	C₂×C₃×D₄	—	10	2	no	—
182	C₂²×C₃×C₄	—	10	2	no	—
183	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
184	C₂²×Dic₃	—	5	2	no	—
185	C₂²×Dic₃	—	10	4	no	—
186	C₃×Q₁₆	—	5	4	no	—
187	C₃×SD₁₆	—	5	8	no	—
188	C₃×D₈	—	5	4	no	—
189	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
190	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
191	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
192	#217	—	2	2	no	—
193	#217	—	10	4	no	—
194	C₄⋊Dic₃	—	10	4	no	—
195	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
196	C₄×Dic₃	—	2	2	no	—
197	C₄×Dic₃	—	5	2	no	—
198	C₄×Dic₃	—	5	2	no	—
199	C₄×Dic₃	—	10	4	no	—
200	C₂×C₃⋊C₈	—	5	2	no	—
201	C₂×C₃⋊C₈	—	10	2	no	—
202	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
203	C₂²×D₅	—	2	2	no	—
204	C₂×C₄×C₅	—	6	4	no	—
205	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅	#237	1	2	yes	—
207	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
208	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
209	C₂²⋊D₅	—	2	8	no	—
210	C₂×Dic₅	—	2	2	no	—
211	C₂×Dic₅	—	3	4	no	—
212	C₂×Dic₅	—	12	2	no	—
213	D₂₀	—	2	2	no	—
214	D₂₀	C₂²×Dic₃	1	4	yes	—
215	C₄×D₅	—	2	8	no	—
216	C₄×D₅	—	12	8	no	—
217	Dic₁₀	—	2	2	no	—
218	Dic₁₀	C₂²×Dic₃	1	4	yes	—
219	C₅×C₈	C₂²×Dic₃	1	2	yes	—
220	C₅×C₈	C₂²×Dic₃	1	2	yes	—
221	#168	—	6	2	no	—
222	C₂×#41	—	5	2	no	—
223	#135	—	5	8	no	—
224	C₂×Q₁₆	—	5	1	no	—
225	C₂×SD₁₆	—	5	2	no	—
226	C₂×D₈	—	5	1	no	—
227	C₂²×C₈	—	15	1	no	—
228	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
229	C₄×Q₈	—	15	4	no	—
230	C₄×D₄	—	15	4	no	—
231	C₄×D₄	—	30	4	no	—
232	#117	—	3	2	no	—
233	#117	—	30	2	no	—
234	C₂×C₄²	—	15	1	no	—
235	C₂×C₄²	—	30	2	no	—
236	#107	—	3	2	no	—
237	#106	—	3	2	no	—
238	#103	—	15	4	no	—
239	#102	—	15	4	no	—
240	#97	—	15	2	no	—
241	#97	—	15	2	no	—
242	C₂×C₃×C₅	—	2	2	no	—
243	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂×C₃×C₅	#575	1	2	yes	—
245	C₃×D₅	—	4	4	no	—
246	C₂³×C₃	—	10	2	no	—
247	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
248	C₃×Q₈	—	5	4	no	—
249	C₃×Q₈	—	10	2	no	—
250	C₃×D₄	—	5	4	no	—
251	C₃×D₄	—	10	8	no	—
252	C₃×D₄	—	10	2	no	—
253	C₂×C₃×C₄	—	10	2	no	—
254	C₂×C₃×C₄	—	10	8	no	—
255	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
256	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
257	C₂×C₃×C₄	#713	1	2	yes	—
258	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
259	C₂×Dic₃	—	2	4	no	—
260	C₂×Dic₃	—	5	4	no	—
261	C₂×Dic₃	—	20	8	no	—
262	C₂×Dic₃	—	20	2	no	—
263	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
264	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
265	C₃⋊C₈	—	10	2	no	—
266	C₂²×C₅	—	2	2	no	—
267	C₂²×C₅	D₄×Dic₃	1	2	yes	—
268	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
269	C₂×D₅	—	2	4	no	—
270	C₂×D₅	—	4	4	no	—
271	C₄×C₅	—	12	4	no	—
272	C₄×C₅	D₄×Dic₃	1	2	yes	—
273	C₄×C₅	C₂×#237	1	1	yes	characteristic
274	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
275	Dic₅	—	2	4	no	—
276	Dic₅	—	6	4	no	—
277	#41	—	5	8	no	—
278	#41	—	10	4	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	—	15	2	no	—
284	C₂²×C₄	—	30	4	no	—
285	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
286	Q₁₆	—	5	4	no	—
287	SD₁₆	—	5	8	no	—
288	D₈	—	5	4	no	—
289	C₂×C₈	—	15	2	no	—
290	C₂×C₈	—	30	2	no	—
291	C₂×C₈	D₅×Dic₃	1	4	yes	complemented
292	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
293	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
294	C₄⋊C₄	—	3	4	no	—
295	C₄⋊C₄	—	30	4	no	—
296	C₂²⋊C₄	—	6	2	no	—
297	C₂²⋊C₄	—	30	4	no	—
298	C₄²	—	6	2	no	—
299	C₄²	—	15	2	no	—
300	C₄²	—	15	2	no	—
301	C₄²	—	30	4	no	—
302	C₃×C₅	#2943	1	1	yes	characteristic, complemented, Hall
303	C₂²×C₃	—	2	2	no	—
304	C₂²×C₃	—	10	4	no	—
305	C₂²×C₃	—	20	4	no	—
306	C₂²×C₃	#4039	1	2	yes	—
307	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
308	C₃×C₄	—	10	4	no	—
309	C₃×C₄	#4039	1	2	yes	—
310	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
311	C₃×C₄	C₂×#713	1	1	yes	characteristic
312	Dic₃	—	4	4	no	—
313	Dic₃	—	10	4	no	—
314	C₂×C₅	—	2	2	no	—
315	C₂×C₅	#5768	1	2	yes	—
316	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
317	D₅	—	4	4	no	—
318	C₂³	—	10	2	no	—
319	C₂³	#6777	1	1	yes	characteristic
320	Q₈	—	5	4	no	—
321	Q₈	—	10	2	no	—
322	D₄	—	5	4	no	—
323	D₄	—	10	8	no	—
324	D₄	—	10	2	no	—
325	C₂×C₄	—	6	4	no	—
326	C₂×C₄	—	10	2	no	—
327	C₂×C₄	—	10	8	no	—
328	C₂×C₄	—	15	4	no	—
329	C₂×C₄	—	60	8	no	—
330	C₂×C₄	—	60	2	no	—
331	C₂×C₄	#6766	1	2	yes	—
332	C₂×C₄	#6777	1	1	yes	characteristic
333	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
334	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
335	C₈	—	30	2	no	—
336	C₈	C₂×D₅×Dic₃	1	2	yes	—
337	C₈	C₂×D₅×Dic₃	1	2	yes	—
338	C₂×C₃	—	2	2	no	—
339	C₂×C₃	—	20	4	no	—
340	C₂×C₃	C₂×#4039	1	1	yes	characteristic
341	C₂×C₃	#66102	1	2	yes	—
342	C₅	#83897	1	1	yes	characteristic, complemented, Sylow
343	C₂²	—	2	2	no	—
344	C₂²	—	10	4	no	—
345	C₂²	—	20	4	no	—
346	C₂²	#91434	1	2	yes	—
347	C₂²	C₂×#6777	1	1	yes	characteristic
348	C₄	—	10	4	no	—
349	C₄	—	12	4	no	—
350	C₄	—	30	4	no	—
351	C₄	#91434	1	2	yes	—
352	C₄	C₂×#6766	1	1	yes	characteristic
353	C₄	C₂²×D₅×Dic₃	1	1	yes	characteristic
354	C₃	#10610897	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	2	2	no	—
356	C₂	—	20	4	no	—
357	C₂	#11753979	1	2	yes	—
358	C₂	C₂×#91434	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6777
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₃×#4050	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2943
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #11754444	C₂	semidirect
3. C₂×#91099	C₂	semidirect
4. C₅×#5106	C₂	semidirect
5. #11748364	C₂	semidirect
6. #11747884	C₂	semidirect
7. #11747815	C₂	semidirect
8. #11747812	C₂	semidirect
16. C₃×#4050	C₄	semidirect
43. C₃×#709	C₂×C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
58. C₃×Dic₂₀	C₂×C₄	semidirect
59. C₃×D₄₀	C₂×C₄	semidirect
60. C₃×#677	C₂×C₄	semidirect
70. #5106	D₅	semidirect
78. #4050	Dic₃	semidirect
137. #709	C₂×Dic₃	semidirect
142. C₂×C₅×C₈	C₂×Dic₃	semidirect
155. Dic₂₀	C₂×Dic₃	semidirect
156. D₄₀	C₂×Dic₃	semidirect
157. #677	C₂×Dic₃	semidirect
189. C₂×C₃×C₈	C₄×D₅	semidirect
291. C₂×C₈	D₅×Dic₃	semidirect
302. C₃×C₅	#2943	semidirect
342. C₅	#83897	semidirect
354. C₃	#10610897	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4050	2
#11754444	2
C₂×#91099	2
C₅×#5106	2
#11748364	2
#11747884	2
#11747815	2
#11747812	2
#10610897	3
#83897	5