Subgroups — #14282427

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756337	C₂	1	1	yes	maximal, characteristic
2	#11747015	C₂	1	1	yes	maximal, characteristic
3	C₄×#6648	C₂	1	1	yes	maximal, characteristic
4	C₅×#4535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Q₈⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745845	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745844	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599033	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	#91835	C₄	1	1	yes	characteristic
11	#91833	C₄	1	1	yes	characteristic
12	#91786	C₄	1	1	yes	characteristic
13	#91785	—	2	1	no	—
14	#91778	—	2	1	no	—
15	#91777	C₄	1	1	yes	characteristic
16	#91761	—	2	1	no	—
17	C₄²×D₁₅	C₂²	1	1	yes	characteristic
18	C₄×Dic₃₀	—	2	1	no	—
19	C₄×Dic₃₀	C₂²	1	1	yes	characteristic
20	C₂×#6651	—	2	1	no	—
21	C₂×#6648	—	2	1	no	—
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	C₃×C₄×#168	C₂²	1	1	yes	characteristic
26	#90983	—	2	1	no	—
27	#90951	—	2	1	no	—
28	#68240	—	5	1	no	maximal, Hall
29	#65912	—	3	1	no	—
30	#65126	—	3	1	no	—
31	C₄×C₈×D₅	—	3	1	no	—
32	C₅×Q₈⋊C₈	—	3	1	no	—
33	Q₈⋊#168	D₃	1	1	yes	characteristic, complemented
34	#64704	—	3	1	no	—
35	#64676	—	3	1	no	—
36	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
39	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
40	C₂×Dic₃₀	—	2	1	no	—
41	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	—	2	1	no	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	#6714	—	2	1	no	—
47	#6714	—	2	1	no	—
48	C₄⋊Dic₁₅	—	2	1	no	—
49	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
50	#6712	—	2	1	no	—
51	#6712	—	2	1	no	—
52	#6712	—	2	1	no	—
53	#6712	—	2	1	no	—
54	#6712	—	2	1	no	—
55	C₄×Dic₁₅	—	2	1	no	—
56	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₅	D₄	1	1	yes	characteristic
58	C₄×Dic₁₅	D₄	1	1	yes	characteristic
59	C₂×C₅×C₃⋊C₈	—	2	1	no	—
60	C₂×C₅×C₃⋊C₈	—	2	1	no	—
61	C₂×C₃×#168	—	2	1	no	—
62	C₂×C₃×#168	—	2	1	no	—
63	#6651	—	4	2	no	—
64	#6648	—	2	4	no	—
65	#5578	—	5	1	no	—
66	#4837	—	5	1	no	—
67	C₄×D₃×C₈	—	5	1	no	—
68	C₃×Q₈⋊C₈	—	5	1	no	—
69	#4535	D₅	1	1	yes	characteristic, complemented
70	#4484	—	5	1	no	—
71	#4456	—	5	1	no	—
72	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
73	#4092	—	3	1	no	—
74	#4090	—	3	1	no	—
75	C₂×#676	—	6	1	no	—
76	C₂×C₈×D₅	—	6	1	no	—
77	#4043	—	3	1	no	—
78	#4042	—	6	1	no	—
79	#4035	—	6	1	no	—
80	#4034	—	3	1	no	—
81	#4018	—	6	1	no	—
82	C₄²×D₅	—	3	1	no	—
83	C₄×Dic₁₀	—	3	1	no	—
84	C₄×Dic₁₀	—	6	1	no	—
85	C₅×C₄⋊C₈	—	3	1	no	—
86	C₄×C₅×C₈	—	3	1	no	—
87	#3946	—	6	1	no	—
88	C₈×Dic₅	—	6	1	no	—
89	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
90	C₄×#168	C₂×D₃	1	1	yes	characteristic
91	#1493	—	15	1	no	Sylow
92	C₂²×D₁₅	C₂²⋊C₄	1	1	yes	characteristic
93	C₃×C₅×Q₈	—	2	1	no	—
94	C₃×C₅×Q₈	—	2	1	no	—
95	C₂×C₃×C₄×C₅	—	2	1	no	—
96	C₂×C₃×C₄×C₅	—	2	1	no	—
97	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂×Dic₁₅	—	2	1	no	—
101	C₂×Dic₁₅	—	2	1	no	—
102	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
105	C₄×D₁₅	—	2	1	no	—
106	C₄×D₁₅	—	2	1	no	—
107	C₄×D₁₅	—	2	2	no	—
108	C₄×D₁₅	SD₁₆	1	2	yes	—
109	C₄×D₁₅	Q₁₆	1	2	yes	—
110	Dic₃₀	—	2	1	no	—
111	Dic₃₀	—	2	1	no	—
112	Dic₃₀	—	4	2	no	—
113	C₃×#168	—	2	2	no	—
114	C₃×#168	—	4	1	no	—
115	C₅×C₃⋊C₈	—	2	2	no	—
116	C₅×C₃⋊C₈	—	4	1	no	—
117	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
118	#946	—	5	1	no	—
119	#944	—	5	1	no	—
120	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
121	C₂×D₃×C₈	—	10	1	no	—
122	#897	—	5	1	no	—
123	#896	—	10	1	no	—
124	#889	—	10	1	no	—
125	#888	—	5	1	no	—
126	#872	—	10	1	no	—
127	C₄²×D₃	—	5	1	no	—
128	C₄×Dic₆	—	5	1	no	—
129	C₄×Dic₆	—	10	1	no	—
130	C₃×C₄⋊C₈	—	5	1	no	—
131	C₃×C₄×C₈	—	5	1	no	—
132	Dic₃⋊C₈	—	10	1	no	—
133	C₈×Dic₃	—	10	1	no	—
134	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
135	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
136	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
137	C₂×C₄×D₅	—	3	1	no	—
138	C₂×C₄×D₅	—	3	1	no	—
139	C₂×C₄×D₅	—	3	1	no	—
140	C₂×Dic₁₀	—	3	1	no	—
141	C₂×Dic₁₀	—	6	1	no	—
142	C₂×C₅×C₈	—	6	1	no	—
143	C₂×C₅×C₈	—	6	1	no	—
144	C₅×C₄⋊C₄	—	2	1	no	—
145	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
146	C₄²×C₅	—	2	1	no	—
147	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
148	#685	—	6	1	no	—
149	#685	—	6	1	no	—
150	C₄⋊Dic₅	—	3	1	no	—
151	C₄⋊Dic₅	—	6	1	no	—
152	#683	—	6	1	no	—
153	#683	—	6	1	no	—
154	#683	—	6	1	no	—
155	#683	—	6	1	no	—
156	#683	—	6	1	no	—
157	C₄×Dic₅	—	3	1	no	—
158	C₄×Dic₅	—	3	1	no	—
159	C₄×Dic₅	—	3	1	no	—
160	C₄×Dic₅	—	6	1	no	—
161	C₂×#168	—	2	1	no	—
162	C₂×#168	—	2	1	no	—
163	#676	—	12	2	no	—
164	C₈×D₅	—	6	4	no	—
165	#533	—	15	1	no	—
166	#423	—	15	1	no	—
167	C₂×C₄×C₈	—	15	1	no	—
168	Q₈⋊C₈	—	3	1	no	—
169	Q₈⋊C₈	—	5	1	no	—
170	Q₈⋊C₈	—	15	1	no	—
171	Q₈⋊C₈	—	15	1	no	—
172	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×D₁₅	—	2	1	no	—
174	C₂×D₁₅	—	2	1	no	—
175	C₂×D₁₅	#103	1	2	yes	—
176	C₃×C₄×C₅	—	2	1	no	—
177	C₃×C₄×C₅	—	4	1	no	—
178	C₃×C₄×C₅	—	4	1	no	—
179	C₃×C₄×C₅	#131	1	1	yes	characteristic
180	C₃×C₄×C₅	#131	1	1	yes	characteristic
181	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
182	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
183	Dic₁₅	—	2	1	no	—
184	Dic₁₅	—	4	1	no	—
185	Dic₁₅	—	4	1	no	—
186	Dic₁₅	#103	1	2	yes	—
187	Dic₁₅	#104	1	2	yes	—
188	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
189	C₂×C₄×D₃	—	5	1	no	—
190	C₂×C₄×D₃	—	5	1	no	—
191	C₂×C₄×D₃	—	5	1	no	—
192	C₂×Dic₆	—	5	1	no	—
193	C₂×Dic₆	—	10	1	no	—
194	C₂×C₃×C₈	—	10	1	no	—
195	C₂×C₃×C₈	—	10	1	no	—
196	C₃×C₄⋊C₄	—	2	1	no	—
197	C₃×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
198	C₃×C₄²	—	2	1	no	—
199	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
200	(C₂×D₃)⋊C₄	—	10	1	no	—
201	(C₂×D₃)⋊C₄	—	10	1	no	—
202	C₄⋊Dic₃	—	5	1	no	—
203	C₄⋊Dic₃	—	10	1	no	—
204	Dic₃⋊C₄	—	10	1	no	—
205	Dic₃⋊C₄	—	10	1	no	—
206	Dic₃⋊C₄	—	10	1	no	—
207	Dic₃⋊C₄	—	10	1	no	—
208	Dic₃⋊C₄	—	10	1	no	—
209	C₄×Dic₃	—	5	1	no	—
210	C₄×Dic₃	—	5	1	no	—
211	C₄×Dic₃	—	5	1	no	—
212	C₄×Dic₃	—	10	1	no	—
213	C₂×C₃⋊C₈	—	2	1	no	—
214	C₂×C₃⋊C₈	—	2	1	no	—
215	ΓL(1,𝔽₂₅)	—	20	2	no	—
216	D₃×C₈	—	10	4	no	—
217	C₂²×D₅	—	3	1	no	—
218	C₅×Q₈	—	2	1	no	—
219	C₅×Q₈	—	2	1	no	—
220	C₂×C₄×C₅	—	2	1	no	—
221	C₂×C₄×C₅	—	2	1	no	—
222	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
223	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
224	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
225	C₂×Dic₅	—	3	1	no	—
226	C₂×Dic₅	—	3	1	no	—
227	C₂×Dic₅	—	3	1	no	—
228	C₂×Dic₅	—	6	1	no	—
229	C₂×Dic₅	—	6	1	no	—
230	C₄×D₅	—	3	2	no	—
231	C₄×D₅	—	3	2	no	—
232	C₄×D₅	—	6	2	no	—
233	C₄×D₅	—	6	1	no	—
234	C₄×D₅	—	6	1	no	—
235	Dic₁₀	—	6	1	no	—
236	Dic₁₀	—	6	1	no	—
237	Dic₁₀	—	12	2	no	—
238	C₅×C₈	—	6	2	no	—
239	C₅×C₈	—	12	1	no	—
240	#168	—	2	2	no	—
241	#168	—	4	1	no	—
242	C₂×M₁₆	—	30	1	no	—
243	C₂²×C₈	—	30	1	no	—
244	C₄⋊Q₈	—	15	1	no	—
245	C₄⋊Q₈	—	15	1	no	—
246	#125	—	30	1	no	—
247	C₂²⋊Q₈	—	15	1	no	—
248	C₂²⋊Q₈	—	15	1	no	—
249	C₂²⋊Q₈	—	30	1	no	—
250	C₄×Q₈	—	15	1	no	—
251	C₄×Q₈	—	30	1	no	—
252	C₄×Q₈	D₃×D₅	1	1	yes	characteristic
253	#117	—	30	1	no	—
254	C₂×C₄²	—	15	1	no	—
255	C₄⋊C₈	—	3	1	no	—
256	C₄⋊C₈	—	5	1	no	—
257	C₄⋊C₈	—	30	1	no	—
258	C₄×C₈	—	3	1	no	—
259	C₄×C₈	—	5	1	no	—
260	C₄×C₈	—	30	1	no	—
261	C₂×C₃×C₅	#408	1	1	yes	characteristic
262	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
263	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
264	D₁₅	—	2	2	no	—
265	C₂²×D₃	—	5	1	no	—
266	C₃×Q₈	—	2	1	no	—
267	C₃×Q₈	—	2	1	no	—
268	C₂×C₃×C₄	—	2	1	no	—
269	C₂×C₃×C₄	—	2	1	no	—
270	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
271	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
272	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
273	C₂×Dic₃	—	5	1	no	—
274	C₂×Dic₃	—	5	1	no	—
275	C₂×Dic₃	—	5	1	no	—
276	C₂×Dic₃	—	10	1	no	—
277	C₂×Dic₃	—	10	1	no	—
278	C₄×D₃	—	5	2	no	—
279	C₄×D₃	—	5	2	no	—
280	C₄×D₃	—	10	2	no	—
281	C₄×D₃	—	10	1	no	—
282	C₄×D₃	—	10	1	no	—
283	Dic₆	—	10	1	no	—
284	Dic₆	—	10	1	no	—
285	Dic₆	—	20	2	no	—
286	C₃×C₈	—	10	2	no	—
287	C₃×C₈	—	20	1	no	—
288	C₃⋊C₈	—	2	2	no	—
289	C₃⋊C₈	—	4	1	no	—
290	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
291	C₂×D₅	—	3	2	no	—
292	C₂×D₅	—	6	1	no	—
293	C₂×D₅	—	6	1	no	—
294	C₄×C₅	—	2	1	no	—
295	C₄×C₅	—	4	1	no	—
296	C₄×C₅	—	4	1	no	—
297	C₄×C₅	#901	1	1	yes	characteristic
298	C₄×C₅	#907	1	1	yes	characteristic
299	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
300	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
301	Dic₅	—	3	2	no	—
302	Dic₅	—	3	2	no	—
303	Dic₅	—	6	1	no	—
304	Dic₅	—	12	1	no	—
305	Dic₅	—	12	1	no	—
306	C₂×Q₈	—	15	1	no	—
307	C₂×Q₈	—	30	1	no	—
308	C₂×Q₈	#1200	1	1	yes	characteristic
309	C₂²×C₄	—	15	1	no	—
310	C₂²×C₄	—	15	1	no	—
311	C₂²×C₄	—	15	1	no	—
312	M₁₆	—	60	2	no	—
313	C₂×C₈	—	6	1	no	—
314	C₂×C₈	—	6	1	no	—
315	C₂×C₈	—	10	1	no	—
316	C₂×C₈	—	10	1	no	—
317	C₂×C₈	—	30	4	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	15	1	no	—
320	C₄⋊C₄	—	30	1	no	—
321	C₄⋊C₄	—	30	1	no	—
322	C₄⋊C₄	—	30	1	no	—
323	C₄⋊C₄	—	30	1	no	—
324	C₄⋊C₄	—	30	1	no	—
325	C₄⋊C₄	—	30	1	no	—
326	C₄⋊C₄	#1200	1	1	yes	characteristic
327	C₂²⋊C₄	—	30	1	no	—
328	C₂²⋊C₄	—	30	1	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	15	1	no	—
331	C₄²	—	15	1	no	—
332	C₄²	—	15	1	no	—
333	C₄²	—	30	1	no	—
334	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
335	C₃×C₅	#1493	1	1	yes	characteristic, complemented, Hall
336	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
337	C₂×D₃	—	5	2	no	—
338	C₂×D₃	—	10	1	no	—
339	C₂×D₃	—	10	1	no	—
340	C₃×C₄	—	2	1	no	—
341	C₃×C₄	—	4	1	no	—
342	C₃×C₄	—	4	1	no	—
343	C₃×C₄	#4047	1	1	yes	characteristic
344	C₃×C₄	#4053	1	1	yes	characteristic
345	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
346	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
347	Dic₃	—	5	2	no	—
348	Dic₃	—	5	2	no	—
349	Dic₃	—	10	1	no	—
350	Dic₃	—	20	1	no	—
351	Dic₃	—	20	1	no	—
352	C₂×C₅	#4727	1	1	yes	characteristic
353	C₂×C₅	D₃×#103	1	1	yes	characteristic
354	C₂×C₅	D₃×#104	1	1	yes	characteristic
355	D₅	—	6	2	no	—
356	C₂³	—	15	1	no	—
357	Q₈	—	2	1	no	—
358	Q₈	—	2	1	no	—
359	Q₈	—	30	1	no	—
360	Q₈	—	30	1	no	—
361	Q₈	—	60	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	15	1	no	—
365	C₂×C₄	—	15	1	no	—
366	C₂×C₄	—	15	2	no	—
367	C₂×C₄	—	15	2	no	—
368	C₂×C₄	—	15	1	no	—
369	C₂×C₄	—	30	2	no	—
370	C₂×C₄	—	30	1	no	—
371	C₂×C₄	—	30	1	no	—
372	C₂×C₄	—	30	1	no	—
373	C₂×C₄	—	30	1	no	—
374	C₂×C₄	C₂×#1200	1	1	yes	characteristic
375	C₂×C₄	#6790	1	1	yes	characteristic
376	C₂×C₄	#6790	1	1	yes	characteristic
377	C₈	—	6	2	no	—
378	C₈	—	10	2	no	—
379	C₈	—	12	1	no	—
380	C₈	—	20	1	no	—
381	C₂×C₃	#65016	1	1	yes	characteristic
382	C₂×C₃	D₅×#103	1	1	yes	characteristic
383	C₂×C₃	D₅×#104	1	1	yes	characteristic
384	D₃	—	10	2	no	—
385	C₅	#68240	1	1	yes	characteristic, complemented, Sylow
386	C₂²	—	15	2	no	—
387	C₂²	—	30	1	no	—
388	C₂²	—	30	1	no	—
389	C₂²	#91561	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	15	2	no	—
394	C₄	—	15	2	no	—
395	C₄	—	30	1	no	—
396	C₄	—	60	1	no	—
397	C₄	—	60	1	no	—
398	C₄	#91290	1	1	yes	characteristic
399	C₄	#91291	1	1	yes	characteristic
400	C₄	#91505	1	1	yes	characteristic
401	C₄	#91511	1	1	yes	characteristic
402	C₃	#10599033	1	1	yes	characteristic, complemented, Sylow
403	C₂	—	30	2	no	—
404	C₂	#11748498	1	1	yes	characteristic
405	C₂	#11749203	1	1	yes	characteristic
406	C₂	#11749430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6790
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×Q₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₅	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×Q₈	D₃×D₅
hypocenter	C₃×C₅	#1493
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
4. C₅×#4535	C₂	semidirect
5. C₃×Q₈⋊#168	C₂	semidirect
6. #11745845	C₂	semidirect
7. #11745844	C₂	semidirect
33. Q₈⋊#168	D₃	semidirect
69. #4535	D₅	semidirect
335. C₃×C₅	#1493	semidirect
385. C₅	#68240	semidirect
402. C₃	#10599033	semidirect

Maximal subgroups

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

subgroup	index
#11756337	2
#11747015	2
C₄×#6648	2
C₅×#4535	2
C₃×Q₈⋊#168	2
#11745845	2
#11745844	2
#10599033	3
#68240	5