Subgroups — #14285266

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11746875	C₂	1	1	yes	maximal, characteristic
3	#11746875	C₂	1	1	yes	maximal, characteristic
4	#11746368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64767	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745858	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745857	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602569	—	3	1	no	maximal, Hall
9	C₂×C₅×D₃×Q₈	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₅×#944	—	2	1	no	—
12	C₅×D₃×C₄⋊C₄	—	2	1	no	—
13	C₅×D₃×C₄⋊C₄	C₄	1	2	yes	—
14	C₅×#888	—	2	1	no	—
15	C₅×#888	C₄	1	2	yes	—
16	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₅×#869	C₂²	1	1	yes	characteristic
18	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
22	(C₂×D₃)⋊#168	—	2	1	no	—
23	(C₂×D₃)⋊#168	—	2	1	no	—
24	#90953	—	2	1	no	—
25	#90953	—	2	1	no	—
26	#68428	—	5	1	no	maximal, Hall
27	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
28	#65215	—	3	1	no	—
29	#65215	—	3	1	no	—
30	#64767	—	3	1	no	—
31	#64767	—	3	1	no	—
32	#64767	—	3	1	no	—
33	#64767	D₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₅×D₃×Q₈	—	4	2	no	—
36	C₅×D₃×Q₈	—	4	2	no	—
37	C₂×C₄×C₅×D₃	—	2	2	no	—
38	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
40	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
41	C₂×C₅×Dic₆	—	2	1	no	—
42	C₂×C₅×Dic₆	—	2	2	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₅×C₄⋊Dic₃	—	2	1	no	—
49	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
50	C₅×Dic₃⋊C₄	—	2	2	no	—
51	C₅×Dic₃⋊C₄	—	2	2	no	—
52	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
54	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
55	C₂×C₃×#168	—	2	1	no	—
56	C₂×C₃×#168	—	2	1	no	—
57	D₃×C₄⋊Q₈	D₅	1	1	yes	characteristic
58	#4835	—	5	1	no	—
59	#4835	—	5	1	no	—
60	C₃×#332	—	5	1	no	—
61	#4547	—	5	1	no	—
62	#4486	—	5	1	no	—
63	#4482	—	5	1	no	—
64	C₂²×C₅×Q₈	—	6	1	no	—
65	C₅×C₄⋊Q₈	—	3	2	no	—
66	C₅×C₄⋊Q₈	—	3	1	no	—
67	C₅×C₄⋊Q₈	—	6	1	no	—
68	C₅×C₄⋊Q₈	—	6	1	no	—
69	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
70	C₂×C₅×C₄⋊C₄	—	3	2	no	—
71	C₂×C₅×C₄⋊C₄	—	6	1	no	—
72	C₂×C₄²×C₅	—	3	1	no	—
73	C₂²⋊#168	—	6	1	no	—
74	C₂²⋊#168	—	6	1	no	—
75	C₄⋊#168	—	3	1	no	—
76	C₄⋊#168	—	3	1	no	—
77	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
78	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
79	#3935	—	6	1	no	—
80	#3935	—	6	1	no	—
81	#1556	—	15	1	no	Sylow
82	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₃×C₅×Q₈	—	4	1	no	—
84	C₃×C₅×Q₈	—	4	1	no	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×Dic₃	—	2	2	no	—
90	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₅×D₃	—	2	2	no	—
94	C₄×C₅×D₃	—	2	1	no	—
95	C₄×C₅×D₃	—	2	1	no	—
96	C₄×C₅×D₃	—	2	1	no	—
97	C₄×C₅×D₃	—	2	1	no	—
98	C₄×C₅×D₃	—	4	4	no	—
99	C₅×Dic₆	—	4	4	no	—
100	C₅×Dic₆	—	4	1	no	—
101	C₅×Dic₆	—	4	1	no	—
102	C₃⋊#168	—	4	1	no	—
103	C₃⋊#168	—	4	1	no	—
104	C₃×#168	—	4	1	no	—
105	C₃×#168	—	4	1	no	—
106	C₂×D₃×Q₈	—	2	1	no	—
107	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
108	#944	—	2	1	no	—
109	D₃×C₄⋊C₄	—	2	1	no	—
110	D₃×C₄⋊C₄	Dic₅	1	2	yes	—
111	#888	—	2	1	no	—
112	#888	Dic₅	1	2	yes	—
113	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
114	#869	C₂×D₅	1	1	yes	characteristic
115	C₃×C₄⋊C₈	—	5	1	no	—
116	C₃×C₄⋊C₈	—	5	1	no	—
117	(C₂×D₃)⋊C₈	—	10	1	no	—
118	(C₂×D₃)⋊C₈	—	10	1	no	—
119	#815	—	10	1	no	—
120	#815	—	10	1	no	—
121	C₄⋊(C₃⋊C₈)	—	5	1	no	—
122	C₄⋊(C₃⋊C₈)	—	5	1	no	—
123	C₂×C₅×Q₈	—	2	1	no	—
124	C₂×C₅×Q₈	—	6	2	no	—
125	C₂×C₅×Q₈	—	6	1	no	—
126	C₂×C₅×Q₈	—	12	2	no	—
127	C₂×C₅×Q₈	—	12	2	no	—
128	C₂²×C₄×C₅	—	3	1	no	—
129	C₂²×C₄×C₅	—	3	1	no	—
130	C₂²×C₄×C₅	—	3	1	no	—
131	C₂²×C₄×C₅	—	6	2	no	—
132	C₅×C₄⋊C₄	—	2	1	no	—
133	C₅×C₄⋊C₄	—	3	2	no	—
134	C₅×C₄⋊C₄	—	6	2	no	—
135	C₅×C₄⋊C₄	—	6	2	no	—
136	C₅×C₄⋊C₄	—	6	1	no	—
137	C₅×C₄⋊C₄	C₄×D₃	1	2	yes	—
138	C₄²×C₅	—	3	1	no	—
139	C₄²×C₅	—	3	1	no	—
140	C₄²×C₅	—	3	1	no	—
141	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂×#168	—	2	1	no	—
143	C₂×#168	—	2	1	no	—
144	C₂×#168	—	6	1	no	—
145	C₂×#168	—	6	1	no	—
146	C₂×C₄⋊Q₈	—	3	1	no	—
147	#432	—	15	1	no	—
148	#432	—	15	1	no	—
149	#332	—	5	1	no	—
150	#332	—	15	1	no	—
151	#332	—	15	1	no	—
152	#332	—	15	1	no	—
153	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₅×D₃	—	2	1	no	—
155	C₂×C₅×D₃	—	2	1	no	—
156	C₂×C₅×D₃	#101	1	2	yes	—
157	C₃×C₄×C₅	—	2	1	no	—
158	C₃×C₄×C₅	—	4	2	no	—
159	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₃×C₄×C₅	#137	1	1	yes	characteristic
162	C₃×C₄×C₅	#137	1	1	yes	characteristic
163	C₅×Dic₃	—	2	1	no	—
164	C₅×Dic₃	—	2	1	no	—
165	C₅×Dic₃	—	2	1	no	—
166	C₅×Dic₃	—	4	2	no	—
167	C₂×C₃×Q₈	—	2	1	no	—
168	D₃×Q₈	—	4	2	no	—
169	D₃×Q₈	—	4	2	no	—
170	C₂×C₄×D₃	—	2	2	no	—
171	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
172	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
173	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
174	C₂×Dic₆	—	2	1	no	—
175	C₂×Dic₆	—	2	2	no	—
176	C₂×C₃×C₈	—	10	1	no	—
177	C₂×C₃×C₈	—	10	1	no	—
178	C₃×C₄⋊C₄	—	2	1	no	—
179	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
180	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
181	C₄⋊Dic₃	—	2	1	no	—
182	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
183	Dic₃⋊C₄	—	2	2	no	—
184	Dic₃⋊C₄	—	2	2	no	—
185	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
186	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
187	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
188	C₂×C₃⋊C₈	—	10	1	no	—
189	C₂×C₃⋊C₈	—	10	1	no	—
190	C₂³×C₅	—	3	1	no	—
191	C₅×Q₈	—	4	1	no	—
192	C₅×Q₈	—	4	1	no	—
193	C₅×Q₈	—	12	4	no	—
194	C₅×Q₈	—	12	1	no	—
195	C₅×Q₈	—	12	1	no	—
196	C₂×C₄×C₅	—	2	2	no	—
197	C₂×C₄×C₅	—	3	1	no	—
198	C₂×C₄×C₅	—	3	1	no	—
199	C₂×C₄×C₅	—	3	1	no	—
200	C₂×C₄×C₅	—	6	2	no	—
201	C₂×C₄×C₅	—	6	1	no	—
202	C₂×C₄×C₅	—	6	1	no	—
203	C₂×C₄×C₅	—	6	1	no	—
204	C₂×C₄×C₅	—	6	1	no	—
205	C₂×C₄×C₅	—	6	2	no	—
206	C₂×C₄×C₅	—	12	4	no	—
207	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
208	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
209	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
210	#168	—	4	1	no	—
211	#168	—	4	1	no	—
212	#168	—	12	1	no	—
213	#168	—	12	1	no	—
214	C₂²×Q₈	—	6	1	no	—
215	C₄⋊Q₈	—	3	2	no	—
216	C₄⋊Q₈	—	3	1	no	—
217	C₄⋊Q₈	—	6	1	no	—
218	C₄⋊Q₈	—	6	1	no	—
219	C₄⋊Q₈	D₃×D₅	1	1	yes	characteristic
220	C₂×C₄⋊C₄	—	3	2	no	—
221	C₂×C₄⋊C₄	—	6	1	no	—
222	C₂×C₄²	—	3	1	no	—
223	C₄⋊C₈	—	5	1	no	—
224	C₄⋊C₈	—	5	1	no	—
225	C₄⋊C₈	—	15	1	no	—
226	C₄⋊C₈	—	15	1	no	—
227	C₂²⋊C₈	—	30	1	no	—
228	C₂²⋊C₈	—	30	1	no	—
229	#97	—	30	1	no	—
230	#97	—	30	1	no	—
231	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
232	C₂×C₃×C₅	#418	1	1	yes	characteristic
233	C₂×C₃×C₅	#419	1	1	yes	characteristic
234	C₅×D₃	—	2	2	no	—
235	C₂²×D₃	C₂²⋊Dic₅	1	1	yes	characteristic
236	C₃×Q₈	—	4	1	no	—
237	C₃×Q₈	—	4	1	no	—
238	C₂×C₃×C₄	—	2	2	no	—
239	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
241	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
242	C₂×Dic₃	—	2	2	no	—
243	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
244	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
245	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
246	C₄×D₃	—	2	2	no	—
247	C₄×D₃	—	2	1	no	—
248	C₄×D₃	—	2	1	no	—
249	C₄×D₃	—	2	1	no	—
250	C₄×D₃	—	2	1	no	—
251	C₄×D₃	—	4	4	no	—
252	Dic₆	—	4	4	no	—
253	Dic₆	—	4	1	no	—
254	Dic₆	—	4	1	no	—
255	C₃×C₈	—	20	1	no	—
256	C₃×C₈	—	20	1	no	—
257	C₃⋊C₈	—	20	1	no	—
258	C₃⋊C₈	—	20	1	no	—
259	C₂²×C₅	—	3	2	no	—
260	C₂²×C₅	—	6	1	no	—
261	C₂²×C₅	—	6	1	no	—
262	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	4	2	no	—
265	C₄×C₅	—	6	1	no	—
266	C₄×C₅	—	6	1	no	—
267	C₄×C₅	—	6	1	no	—
268	C₄×C₅	—	12	2	no	—
269	C₄×C₅	#911	1	1	yes	characteristic
270	C₄×C₅	#914	1	1	yes	characteristic
271	C₄×C₅	#915	1	1	yes	characteristic
272	C₄×C₅	#918	1	1	yes	characteristic
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	6	2	no	—
275	C₂×Q₈	—	6	1	no	—
276	C₂×Q₈	—	12	2	no	—
277	C₂×Q₈	—	12	2	no	—
278	C₂²×C₄	—	3	1	no	—
279	C₂²×C₄	—	3	1	no	—
280	C₂²×C₄	—	3	1	no	—
281	C₂²×C₄	—	6	2	no	—
282	C₂×C₈	—	10	1	no	—
283	C₂×C₈	—	10	1	no	—
284	C₂×C₈	—	30	1	no	—
285	C₂×C₈	—	30	1	no	—
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	3	2	no	—
288	C₄⋊C₄	—	6	2	no	—
289	C₄⋊C₄	—	6	2	no	—
290	C₄⋊C₄	—	6	1	no	—
291	C₄⋊C₄	D₃×Dic₅	1	2	yes	—
292	C₄²	—	3	1	no	—
293	C₄²	—	3	1	no	—
294	C₄²	—	3	1	no	—
295	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
296	C₃×C₅	#1556	1	1	yes	characteristic, complemented, Hall
297	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
298	C₂×D₃	—	2	1	no	—
299	C₂×D₃	—	2	1	no	—
300	C₂×D₃	#3968	1	2	yes	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	4	2	no	—
303	C₃×C₄	#4078	1	1	yes	characteristic
304	C₃×C₄	#4078	1	1	yes	characteristic
305	C₃×C₄	#4088	1	1	yes	characteristic
306	C₃×C₄	#4088	1	1	yes	characteristic
307	Dic₃	—	2	1	no	—
308	Dic₃	—	2	1	no	—
309	Dic₃	—	2	1	no	—
310	Dic₃	—	4	2	no	—
311	C₂×C₅	—	6	2	no	—
312	C₂×C₅	D₃×#101	1	1	yes	characteristic
313	C₂×C₅	#4770	1	1	yes	characteristic
314	C₂×C₅	#4802	1	1	yes	characteristic
315	C₂³	—	3	1	no	—
316	Q₈	—	4	1	no	—
317	Q₈	—	4	1	no	—
318	Q₈	—	12	4	no	—
319	Q₈	—	12	1	no	—
320	Q₈	—	12	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	3	1	no	—
323	C₂×C₄	—	3	1	no	—
324	C₂×C₄	—	3	1	no	—
325	C₂×C₄	—	6	2	no	—
326	C₂×C₄	—	6	1	no	—
327	C₂×C₄	—	6	1	no	—
328	C₂×C₄	—	6	1	no	—
329	C₂×C₄	—	6	1	no	—
330	C₂×C₄	—	6	2	no	—
331	C₂×C₄	—	12	4	no	—
332	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
333	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
334	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
335	C₈	—	20	1	no	—
336	C₈	—	20	1	no	—
337	C₈	—	60	1	no	—
338	C₈	—	60	1	no	—
339	C₂×C₃	#65504	1	1	yes	characteristic
340	C₂×C₃	#65514	1	1	yes	characteristic
341	C₂×C₃	C₂×#3968	1	1	yes	characteristic
342	D₃	—	2	2	no	—
343	C₅	#68428	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	3	2	no	—
345	C₂²	—	6	1	no	—
346	C₂²	—	6	1	no	—
347	C₂²	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	6	1	no	—
351	C₄	—	6	1	no	—
352	C₄	—	6	1	no	—
353	C₄	—	12	2	no	—
354	C₄	#91494	1	1	yes	characteristic
355	C₄	#91496	1	1	yes	characteristic
356	C₄	#91518	1	1	yes	characteristic
357	C₄	#91520	1	1	yes	characteristic
358	C₃	#10602569	1	1	yes	characteristic, complemented, Sylow
359	C₂	—	6	2	no	—
360	C₂	#11748967	1	1	yes	characteristic
361	C₂	#11749213	1	1	yes	characteristic
362	C₂	D₃×#3968	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×C₂²⋊Dic₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊Q₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₃	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊Q₈	D₃×D₅
hypocenter	C₃×C₅	#1556
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. #11746368	C₂	semidirect
5. C₃×#64767	C₂	semidirect
6. #11745858	C₂	semidirect
7. #11745857	C₂	semidirect
33. #64767	D₃	semidirect
296. C₃×C₅	#1556	semidirect
343. C₅	#68428	semidirect
358. C₃	#10602569	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₃×C₄⋊Q₈	2
#11746875	2
#11746368	2
C₃×#64767	2
#11745858	2
#11745857	2
#10602569	3
#68428	5