Subgroups — #14416281

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 372 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×C₈×Dic₁₅	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	#11747883	C₂	1	2	yes	maximal, complemented
7	#11747809	C₂	1	2	yes	maximal, complemented
8	#11747806	C₂	1	4	yes	maximal, complemented
9	#10610896	—	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	C₈×Dic₁₅	C₂²	1	2	yes	—
25	C₈×Dic₁₅	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	Dic₃×#135	—	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×C₈×Dic₅	—	3	1	no	—
35	#65291	—	3	2	no	—
36	#65162	—	3	2	no	—
37	#65161	—	3	2	no	—
38	#65141	—	3	4	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	1	yes	characteristic
52	C₄×Dic₁₅	D₄	1	1	yes	characteristic
53	C₄×Dic₁₅	C₂³	1	2	yes	—
54	C₂×C₃⋊#168	—	2	2	no	—
55	C₂×C₃⋊#168	C₂³	1	2	yes	—
56	C₅×#217	—	2	2	no	—
57	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
58	C₄×C₅×Dic₃	—	2	2	no	—
59	C₃×Dic₂₀	C₂×C₄	1	4	yes	complemented
60	C₃×D₄₀	C₂×C₄	1	4	yes	complemented
61	C₃×#677	C₂×C₄	1	8	yes	complemented
62	#6668	—	2	4	no	—
63	#6665	—	2	4	no	—
64	Dic₃×Dic₅	—	2	4	no	—
65	C₂×C₃×#135	—	5	1	no	—
66	Dic₃×#41	—	5	2	no	—
67	#5233	—	5	2	no	—
68	Dic₃×Q₁₆	—	5	2	no	—
69	Dic₃×SD₁₆	—	5	4	no	—
70	Dic₃×D₈	—	5	2	no	—
71	#5106	D₅	1	1	yes	characteristic, complemented
72	C₂×C₈×Dic₃	—	5	1	no	—
73	C₂×#709	C₂×D₃	1	2	yes	—
74	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
75	C₅×#117	—	3	2	no	—
76	C₂×C₄×Dic₅	—	3	1	no	—
77	C₂²×#168	—	3	1	no	—
78	C₂×Dic₂₀	C₂×D₃	1	1	yes	characteristic
79	#4050	Dic₃	1	8	yes	complemented
80	C₂×D₄₀	C₂×D₃	1	1	yes	characteristic
81	C₂×#677	C₂×D₃	1	2	yes	—
82	#4039	—	3	4	no	—
83	#4033	—	3	4	no	—
84	#4027	—	6	4	no	—
85	#4018	—	6	2	no	—
86	C₄²×D₅	—	6	2	no	—
87	C₅×#107	—	3	2	no	—
88	C₅×#106	—	3	2	no	—
89	C₈×Dic₅	—	3	2	no	—
90	C₈×Dic₅	—	3	2	no	—
91	#3942	—	3	4	no	—
92	#3941	—	3	4	no	—
93	C₄×#135	—	15	1	no	Sylow
94	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₃×D₅	—	2	2	no	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
97	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×C₄×C₅	#41	1	2	yes	—
99	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₁₅	—	2	4	no	—
101	C₂×Dic₁₅	C₂×D₄	1	2	yes	—
102	C₂×Dic₁₅	C₂×D₄	1	2	yes	—
103	C₂×C₅×Dic₃	—	2	4	no	—
104	C₃×C₂²⋊D₅	—	2	8	no	—
105	C₂×C₃×Dic₅	—	2	2	no	—
106	C₃×D₂₀	—	2	2	no	—
107	C₃×D₂₀	C₂²×C₄	1	4	yes	—
108	C₃×C₄×D₅	—	2	8	no	—
109	C₃×Dic₁₀	—	2	2	no	—
110	C₃×Dic₁₀	C₂²×C₄	1	4	yes	—
111	D₅×Dic₃	—	4	8	no	—
112	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
113	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
114	C₃⋊#168	—	2	2	no	—
115	C₂×C₃×#41	—	5	2	no	—
116	C₃×#135	—	5	8	no	—
117	C₂×C₃×Q₁₆	—	5	1	no	—
118	C₂×C₃×SD₁₆	—	5	2	no	—
119	C₂×C₃×D₈	—	5	1	no	—
120	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
121	Q₈×Dic₃	—	5	4	no	—
122	D₄×Dic₃	—	5	4	no	—
123	D₄×Dic₃	—	10	4	no	—
124	#926	—	10	2	no	—
125	#926	C₂×D₅	1	2	yes	—
126	C₂×C₄×Dic₃	—	5	1	no	—
127	C₂×C₄×Dic₃	—	10	2	no	—
128	C₂²×C₃⋊C₈	—	5	1	no	—
129	#835	—	5	4	no	—
130	#832	—	5	4	no	—
131	#818	C₂×D₅	1	2	yes	—
132	#817	C₂×D₅	1	2	yes	—
133	C₈×Dic₃	—	5	2	no	—
134	C₈×Dic₃	—	5	2	no	—
135	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₂²⋊D₅	—	2	2	no	—
137	C₂²×Dic₅	—	3	2	no	—
138	#709	—	2	4	no	—
139	#709	C₂×Dic₃	1	8	yes	complemented
140	C₂×D₂₀	C₂²×D₃	1	2	yes	—
141	C₂×C₄×D₅	—	2	2	no	—
142	C₂×C₄×D₅	—	6	4	no	—
143	C₂×Dic₁₀	C₂²×D₃	1	2	yes	—
144	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	complemented
145	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
146	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
147	C₅×C₄⋊C₄	—	3	4	no	—
148	C₅×C₂²⋊C₄	—	6	2	no	—
149	C₄²×C₅	—	6	2	no	—
150	#685	—	6	4	no	—
151	#683	—	6	4	no	—
152	C₄×Dic₅	—	3	1	no	—
153	C₄×Dic₅	—	3	1	no	—
154	C₄×Dic₅	—	3	2	no	—
155	C₄×Dic₅	—	6	4	no	—
156	C₂×#168	—	3	2	no	—
157	C₂×#168	—	6	2	no	—
158	Dic₂₀	C₂×Dic₃	1	4	yes	complemented
159	D₄₀	C₂×Dic₃	1	4	yes	complemented
160	#677	C₂×Dic₃	1	8	yes	complemented
161	C₂×#135	—	5	1	no	—
162	C₄×#41	—	15	2	no	—
163	C₄×Q₁₆	—	15	2	no	—
164	C₄×SD₁₆	—	15	4	no	—
165	C₄×D₈	—	15	2	no	—
166	#427	—	3	1	no	—
167	#416	—	15	2	no	—
168	C₂×C₄×C₈	—	15	1	no	—
169	C₂²×C₃×C₅	—	2	2	no	—
170	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
171	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₃×D₅	—	2	4	no	—
173	C₂×C₃×D₅	—	4	4	no	—
174	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
175	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
176	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
177	Dic₁₅	—	2	2	no	—
178	Dic₁₅	#135	1	4	yes	complemented
179	C₃×Dic₅	—	2	4	no	—
180	C₅×Dic₃	—	4	4	no	—
181	C₃×#41	—	5	8	no	—
182	C₃×#41	—	10	4	no	—
183	C₂×C₃×Q₈	—	5	2	no	—
184	C₂×C₃×D₄	—	5	2	no	—
185	C₂×C₃×D₄	—	10	2	no	—
186	C₂²×C₃×C₄	—	10	2	no	—
187	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
188	C₂²×Dic₃	—	5	2	no	—
189	C₂²×Dic₃	—	10	4	no	—
190	C₃×Q₁₆	—	5	4	no	—
191	C₃×SD₁₆	—	5	8	no	—
192	C₃×D₈	—	5	4	no	—
193	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
194	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
195	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
196	#217	—	2	2	no	—
197	#217	—	10	4	no	—
198	C₄⋊Dic₃	—	10	4	no	—
199	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
200	C₄×Dic₃	—	2	2	no	—
201	C₄×Dic₃	—	5	1	no	—
202	C₄×Dic₃	—	5	1	no	—
203	C₄×Dic₃	—	5	2	no	—
204	C₄×Dic₃	—	10	4	no	—
205	C₂×C₃⋊C₈	—	5	2	no	—
206	C₂×C₃⋊C₈	—	10	2	no	—
207	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
208	C₂²×D₅	—	2	2	no	—
209	C₂×C₄×C₅	—	6	4	no	—
210	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅	#237	1	2	yes	—
212	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
213	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
214	C₂²⋊D₅	—	2	8	no	—
215	C₂×Dic₅	—	2	2	no	—
216	C₂×Dic₅	—	3	2	no	—
217	C₂×Dic₅	—	3	2	no	—
218	C₂×Dic₅	—	6	4	no	—
219	D₂₀	—	2	2	no	—
220	D₂₀	C₂²×Dic₃	1	4	yes	—
221	C₄×D₅	—	2	8	no	—
222	C₄×D₅	—	12	8	no	—
223	Dic₁₀	—	2	2	no	—
224	Dic₁₀	C₂²×Dic₃	1	4	yes	—
225	C₅×C₈	C₂²×Dic₃	1	2	yes	—
226	C₅×C₈	C₂²×Dic₃	1	2	yes	—
227	#168	—	6	2	no	—
228	C₂×#41	—	5	2	no	—
229	#135	—	5	8	no	—
230	C₂×Q₁₆	—	5	1	no	—
231	C₂×SD₁₆	—	5	2	no	—
232	C₂×D₈	—	5	1	no	—
233	C₂²×C₈	—	15	1	no	—
234	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
235	C₄×Q₈	—	15	4	no	—
236	C₄×D₄	—	15	4	no	—
237	C₄×D₄	—	30	4	no	—
238	#117	—	3	2	no	—
239	#117	—	30	2	no	—
240	C₂×C₄²	—	15	1	no	—
241	C₂×C₄²	—	30	2	no	—
242	#107	—	3	2	no	—
243	#106	—	3	2	no	—
244	#103	—	15	4	no	—
245	#102	—	15	4	no	—
246	C₄×C₈	—	15	2	no	—
247	C₄×C₈	—	15	2	no	—
248	C₂×C₃×C₅	—	2	2	no	—
249	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
250	C₂×C₃×C₅	C₂×#135	1	2	yes	—
251	C₃×D₅	—	4	4	no	—
252	C₂³×C₃	—	10	2	no	—
253	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
254	C₃×Q₈	—	5	4	no	—
255	C₃×Q₈	—	10	2	no	—
256	C₃×D₄	—	5	4	no	—
257	C₃×D₄	—	10	8	no	—
258	C₃×D₄	—	10	2	no	—
259	C₂×C₃×C₄	—	10	2	no	—
260	C₂×C₃×C₄	—	10	8	no	—
261	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
262	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
263	C₂×C₃×C₄	#713	1	2	yes	—
264	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
265	C₂×Dic₃	—	2	4	no	—
266	C₂×Dic₃	—	5	2	no	—
267	C₂×Dic₃	—	5	2	no	—
268	C₂×Dic₃	—	10	4	no	—
269	C₂×Dic₃	—	20	8	no	—
270	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
271	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
272	C₃⋊C₈	—	10	2	no	—
273	C₂²×C₅	—	2	2	no	—
274	C₂²×C₅	D₄×Dic₃	1	2	yes	—
275	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
276	C₂×D₅	—	2	4	no	—
277	C₂×D₅	—	4	4	no	—
278	C₄×C₅	—	12	4	no	—
279	C₄×C₅	D₄×Dic₃	1	2	yes	—
280	C₄×C₅	C₂×#237	1	1	yes	characteristic
281	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
282	Dic₅	—	2	4	no	—
283	Dic₅	—	3	4	no	—
284	Dic₅	—	6	2	no	—
285	#41	—	5	8	no	—
286	#41	—	10	4	no	—
287	C₂×Q₈	—	5	2	no	—
288	C₂×D₄	—	5	2	no	—
289	C₂×D₄	—	10	2	no	—
290	C₂²×C₄	—	10	2	no	—
291	C₂²×C₄	—	15	2	no	—
292	C₂²×C₄	—	30	4	no	—
293	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
294	Q₁₆	—	5	4	no	—
295	SD₁₆	—	5	8	no	—
296	D₈	—	5	4	no	—
297	C₂×C₈	—	15	2	no	—
298	C₂×C₈	—	30	2	no	—
299	C₂×C₈	D₅×Dic₃	1	4	yes	complemented
300	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
301	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
302	C₄⋊C₄	—	3	4	no	—
303	C₄⋊C₄	—	30	4	no	—
304	C₂²⋊C₄	—	6	2	no	—
305	C₂²⋊C₄	—	30	4	no	—
306	C₄²	—	6	2	no	—
307	C₄²	—	15	1	no	—
308	C₄²	—	15	1	no	—
309	C₄²	—	15	2	no	—
310	C₄²	—	30	4	no	—
311	C₃×C₅	C₄×#135	1	1	yes	characteristic, complemented, Hall
312	C₂²×C₃	—	2	2	no	—
313	C₂²×C₃	—	10	4	no	—
314	C₂²×C₃	—	20	4	no	—
315	C₂²×C₃	#4039	1	2	yes	—
316	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
317	C₃×C₄	—	10	4	no	—
318	C₃×C₄	#4039	1	2	yes	—
319	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
320	C₃×C₄	C₂×#713	1	1	yes	characteristic
321	Dic₃	—	4	4	no	—
322	Dic₃	—	5	4	no	—
323	Dic₃	—	10	2	no	—
324	C₂×C₅	—	2	2	no	—
325	C₂×C₅	D₃×#135	1	2	yes	—
326	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
327	D₅	—	4	4	no	—
328	C₂³	—	10	2	no	—
329	C₂³	#6777	1	1	yes	characteristic
330	Q₈	—	5	4	no	—
331	Q₈	—	10	2	no	—
332	D₄	—	5	4	no	—
333	D₄	—	10	8	no	—
334	D₄	—	10	2	no	—
335	C₂×C₄	—	6	4	no	—
336	C₂×C₄	—	10	2	no	—
337	C₂×C₄	—	10	8	no	—
338	C₂×C₄	—	15	2	no	—
339	C₂×C₄	—	15	2	no	—
340	C₂×C₄	—	30	4	no	—
341	C₂×C₄	—	60	8	no	—
342	C₂×C₄	#6766	1	2	yes	—
343	C₂×C₄	#6777	1	1	yes	characteristic
344	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
345	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
346	C₈	—	30	2	no	—
347	C₈	C₂×D₅×Dic₃	1	2	yes	—
348	C₈	C₂×D₅×Dic₃	1	2	yes	—
349	C₂×C₃	—	2	2	no	—
350	C₂×C₃	—	20	4	no	—
351	C₂×C₃	C₂×#4039	1	1	yes	characteristic
352	C₂×C₃	D₅×#135	1	2	yes	—
353	C₅	Dic₃×#135	1	1	yes	characteristic, complemented, Sylow
354	C₂²	—	2	2	no	—
355	C₂²	—	10	4	no	—
356	C₂²	—	20	4	no	—
357	C₂²	#91434	1	2	yes	—
358	C₂²	C₂×#6777	1	1	yes	characteristic
359	C₄	—	10	4	no	—
360	C₄	—	12	4	no	—
361	C₄	—	15	4	no	—
362	C₄	—	30	2	no	—
363	C₄	#91434	1	2	yes	—
364	C₄	C₂×#6766	1	1	yes	characteristic
365	C₄	C₂²×D₅×Dic₃	1	1	yes	characteristic
366	C₃	#10610896	1	1	yes	characteristic, complemented, Sylow
367	C₂	—	2	2	no	—
368	C₂	—	20	4	no	—
369	C₂	#11753978	1	2	yes	—
370	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	C₄×#135
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×C₈×Dic₁₅	C₂	semidirect
4. C₅×#5106	C₂	semidirect
5. #11748364	C₂	semidirect
6. #11747883	C₂	semidirect
7. #11747809	C₂	semidirect
8. #11747806	C₂	semidirect
16. C₃×#4050	C₄	semidirect
43. C₃×#709	C₂×C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
59. C₃×Dic₂₀	C₂×C₄	semidirect
60. C₃×D₄₀	C₂×C₄	semidirect
61. C₃×#677	C₂×C₄	semidirect
71. #5106	D₅	semidirect
79. #4050	Dic₃	semidirect
139. #709	C₂×Dic₃	semidirect
144. C₂×C₅×C₈	C₂×Dic₃	semidirect
158. Dic₂₀	C₂×Dic₃	semidirect
159. D₄₀	C₂×Dic₃	semidirect
160. #677	C₂×Dic₃	semidirect
178. Dic₁₅	#135	semidirect
193. C₂×C₃×C₈	C₄×D₅	semidirect
299. C₂×C₈	D₅×Dic₃	semidirect
311. C₃×C₅	C₄×#135	semidirect
353. C₅	Dic₃×#135	semidirect
366. C₃	#10610896	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×C₈×Dic₁₅	2
C₅×#5106	2
#11748364	2
#11747883	2
#11747809	2
#11747806	2
#10610896	3
Dic₃×#135	5