Subgroups — #14284823

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 368 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	#11747011	C₂	1	1	yes	maximal, characteristic
3	#11746853	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	#10602566	—	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	C₂×#6650	—	2	1	no	—
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
23	Dic₃⋊#168	—	2	1	no	—
24	(C₂×D₃)⋊#168	—	2	1	no	—
25	Dic₃×#168	—	2	1	no	—
26	#68420	—	5	1	no	maximal, Hall
27	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
28	#65214	—	3	1	no	—
29	#65213	—	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	#6650	—	4	2	no	—
58	D₃×C₄⋊Q₈	D₅	1	1	yes	characteristic
59	#4837	—	5	1	no	—
60	#4833	—	5	1	no	—
61	C₃×#332	—	5	1	no	—
62	#4547	—	5	1	no	—
63	#4486	—	5	1	no	—
64	#4482	—	5	1	no	—
65	C₂²×C₅×Q₈	—	6	1	no	—
66	C₅×C₄⋊Q₈	—	3	2	no	—
67	C₅×C₄⋊Q₈	—	3	1	no	—
68	C₅×C₄⋊Q₈	—	6	1	no	—
69	C₅×C₄⋊Q₈	—	6	1	no	—
70	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
71	C₂×C₅×C₄⋊C₄	—	3	2	no	—
72	C₂×C₅×C₄⋊C₄	—	6	1	no	—
73	C₂×C₄²×C₅	—	3	1	no	—
74	C₂×#681	—	6	1	no	—
75	C₂²⋊#168	—	6	1	no	—
76	C₄⋊#168	—	3	1	no	—
77	C₄⋊#168	—	3	1	no	—
78	C₄⋊#168	—	6	1	no	—
79	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
80	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
81	C₄×#168	—	6	1	no	—
82	#1546	—	15	1	no	Sylow
83	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₅×Q₈	—	4	1	no	—
85	C₃×C₅×Q₈	—	4	1	no	—
86	C₂×C₃×C₄×C₅	—	2	2	no	—
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×Dic₃	—	2	2	no	—
91	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₅×D₃	—	2	1	no	—
95	C₄×C₅×D₃	—	2	1	no	—
96	C₄×C₅×D₃	—	2	2	no	—
97	C₄×C₅×D₃	—	4	4	no	—
98	C₄×C₅×D₃	D₈	1	2	yes	—
99	C₄×C₅×D₃	SD₁₆	1	2	yes	—
100	C₅×Dic₆	—	4	1	no	—
101	C₅×Dic₆	—	4	1	no	—
102	C₅×Dic₆	—	4	4	no	—
103	C₃⋊#168	—	4	1	no	—
104	C₃⋊#168	—	4	1	no	—
105	C₃×#168	—	4	1	no	—
106	C₃×#168	—	4	1	no	—
107	C₂×D₃×Q₈	—	2	1	no	—
108	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
109	#944	—	2	1	no	—
110	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
111	D₃×C₄⋊C₄	—	2	1	no	—
112	D₃×C₄⋊C₄	Dic₅	1	2	yes	—
113	#888	—	2	1	no	—
114	#888	Dic₅	1	2	yes	—
115	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
116	#869	C₂×D₅	1	1	yes	characteristic
117	C₃×C₄⋊C₈	—	5	1	no	—
118	C₃×C₄⋊C₈	—	5	1	no	—
119	(C₂×D₃)⋊C₈	—	10	1	no	—
120	Dic₃⋊C₈	—	10	1	no	—
121	C₈×Dic₃	—	10	1	no	—
122	C₄⋊(C₃⋊C₈)	—	5	1	no	—
123	C₄⋊(C₃⋊C₈)	—	5	1	no	—
124	C₂×C₅×Q₈	—	2	1	no	—
125	C₂×C₅×Q₈	—	6	2	no	—
126	C₂×C₅×Q₈	—	6	1	no	—
127	C₂×C₅×Q₈	—	12	2	no	—
128	C₂×C₅×Q₈	—	12	2	no	—
129	C₂²×C₄×C₅	—	3	1	no	—
130	C₂²×C₄×C₅	—	3	1	no	—
131	C₂²×C₄×C₅	—	3	1	no	—
132	C₂²×C₄×C₅	—	6	2	no	—
133	C₅×C₄⋊C₄	—	2	1	no	—
134	C₅×C₄⋊C₄	—	3	2	no	—
135	C₅×C₄⋊C₄	—	6	2	no	—
136	C₅×C₄⋊C₄	—	6	2	no	—
137	C₅×C₄⋊C₄	—	6	1	no	—
138	C₅×C₄⋊C₄	C₄×D₃	1	2	yes	—
139	C₄²×C₅	—	3	1	no	—
140	C₄²×C₅	—	3	1	no	—
141	C₄²×C₅	—	3	1	no	—
142	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
143	#681	—	12	2	no	—
144	C₂×#168	—	2	1	no	—
145	C₂×#168	—	2	1	no	—
146	C₂×#168	—	6	1	no	—
147	C₂×#168	—	6	1	no	—
148	C₂×C₄⋊Q₈	—	3	1	no	—
149	#431	—	15	1	no	—
150	#423	—	15	1	no	—
151	#332	—	5	1	no	—
152	#332	—	15	1	no	—
153	#332	—	15	1	no	—
154	#332	—	15	1	no	—
155	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₅×D₃	—	2	1	no	—
157	C₂×C₅×D₃	—	2	1	no	—
158	C₂×C₅×D₃	#102	1	2	yes	—
159	C₃×C₄×C₅	—	2	1	no	—
160	C₃×C₄×C₅	—	4	2	no	—
161	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
162	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
163	C₃×C₄×C₅	#137	1	1	yes	characteristic
164	C₃×C₄×C₅	#137	1	1	yes	characteristic
165	C₅×Dic₃	—	2	1	no	—
166	C₅×Dic₃	—	4	2	no	—
167	C₅×Dic₃	#101	1	2	yes	—
168	C₅×Dic₃	#102	1	2	yes	—
169	C₂×C₃×Q₈	—	2	1	no	—
170	D₃×Q₈	—	4	2	no	—
171	D₃×Q₈	—	4	2	no	—
172	C₂×C₄×D₃	—	2	2	no	—
173	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
174	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
175	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
176	C₂×Dic₆	—	2	1	no	—
177	C₂×Dic₆	—	2	2	no	—
178	C₂×C₃×C₈	—	10	1	no	—
179	C₂×C₃×C₈	—	10	1	no	—
180	C₃×C₄⋊C₄	—	2	1	no	—
181	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
182	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
183	C₄⋊Dic₃	—	2	1	no	—
184	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
185	Dic₃⋊C₄	—	2	2	no	—
186	Dic₃⋊C₄	—	2	2	no	—
187	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
188	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
189	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
190	C₂×C₃⋊C₈	—	10	1	no	—
191	C₂×C₃⋊C₈	—	10	1	no	—
192	ΓL(1,𝔽₂₅)	—	20	2	no	—
193	C₂³×C₅	—	3	1	no	—
194	C₅×Q₈	—	4	1	no	—
195	C₅×Q₈	—	4	1	no	—
196	C₅×Q₈	—	12	4	no	—
197	C₅×Q₈	—	12	1	no	—
198	C₅×Q₈	—	12	1	no	—
199	C₂×C₄×C₅	—	2	2	no	—
200	C₂×C₄×C₅	—	3	1	no	—
201	C₂×C₄×C₅	—	3	1	no	—
202	C₂×C₄×C₅	—	3	2	no	—
203	C₂×C₄×C₅	—	3	2	no	—
204	C₂×C₄×C₅	—	3	1	no	—
205	C₂×C₄×C₅	—	6	2	no	—
206	C₂×C₄×C₅	—	6	1	no	—
207	C₂×C₄×C₅	—	6	1	no	—
208	C₂×C₄×C₅	—	6	2	no	—
209	C₂×C₄×C₅	—	12	4	no	—
210	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
211	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
212	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
213	#168	—	4	1	no	—
214	#168	—	4	1	no	—
215	#168	—	12	1	no	—
216	#168	—	12	1	no	—
217	C₂²×Q₈	—	6	1	no	—
218	C₂×M₁₆	—	30	1	no	—
219	C₄⋊Q₈	—	3	2	no	—
220	C₄⋊Q₈	—	3	1	no	—
221	C₄⋊Q₈	—	6	1	no	—
222	C₄⋊Q₈	—	6	1	no	—
223	C₄⋊Q₈	D₃×D₅	1	1	yes	characteristic
224	C₂×C₄⋊C₄	—	3	2	no	—
225	C₂×C₄⋊C₄	—	6	1	no	—
226	C₂×C₄²	—	3	1	no	—
227	C₄⋊C₈	—	5	1	no	—
228	C₄⋊C₈	—	5	1	no	—
229	C₄⋊C₈	—	15	1	no	—
230	C₄⋊C₈	—	15	1	no	—
231	C₄⋊C₈	—	30	1	no	—
232	C₂²⋊C₈	—	30	1	no	—
233	C₄×C₈	—	30	1	no	—
234	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
235	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
236	C₂×C₃×C₅	#419	1	1	yes	characteristic
237	C₅×D₃	—	2	2	no	—
238	C₂²×D₃	C₂²⋊Dic₅	1	1	yes	characteristic
239	C₃×Q₈	—	4	1	no	—
240	C₃×Q₈	—	4	1	no	—
241	C₂×C₃×C₄	—	2	2	no	—
242	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
245	C₂×Dic₃	—	2	2	no	—
246	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
247	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
248	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
249	C₄×D₃	—	2	1	no	—
250	C₄×D₃	—	2	1	no	—
251	C₄×D₃	—	2	2	no	—
252	C₄×D₃	—	4	4	no	—
253	C₄×D₃	#686	1	2	yes	—
254	C₄×D₃	#687	1	2	yes	—
255	Dic₆	—	4	1	no	—
256	Dic₆	—	4	1	no	—
257	Dic₆	—	4	4	no	—
258	C₃×C₈	—	20	1	no	—
259	C₃×C₈	—	20	1	no	—
260	C₃⋊C₈	—	20	1	no	—
261	C₃⋊C₈	—	20	1	no	—
262	C₂²×C₅	—	3	2	no	—
263	C₂²×C₅	—	6	1	no	—
264	C₂²×C₅	—	6	1	no	—
265	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	—	3	2	no	—
268	C₄×C₅	—	3	2	no	—
269	C₄×C₅	—	4	2	no	—
270	C₄×C₅	—	6	1	no	—
271	C₄×C₅	—	12	2	no	—
272	C₄×C₅	D₃×D₈	1	1	yes	characteristic
273	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
274	C₄×C₅	#915	1	1	yes	characteristic
275	C₄×C₅	#918	1	1	yes	characteristic
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	6	2	no	—
278	C₂×Q₈	—	6	1	no	—
279	C₂×Q₈	—	12	2	no	—
280	C₂×Q₈	—	12	2	no	—
281	C₂²×C₄	—	3	1	no	—
282	C₂²×C₄	—	3	1	no	—
283	C₂²×C₄	—	3	1	no	—
284	C₂²×C₄	—	6	2	no	—
285	M₁₆	—	60	2	no	—
286	C₂×C₈	—	10	1	no	—
287	C₂×C₈	—	10	1	no	—
288	C₂×C₈	—	30	1	no	—
289	C₂×C₈	—	30	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	3	2	no	—
292	C₄⋊C₄	—	6	2	no	—
293	C₄⋊C₄	—	6	2	no	—
294	C₄⋊C₄	—	6	1	no	—
295	C₄⋊C₄	D₃×Dic₅	1	2	yes	—
296	C₄²	—	3	1	no	—
297	C₄²	—	3	1	no	—
298	C₄²	—	3	1	no	—
299	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
300	C₃×C₅	#1546	1	1	yes	characteristic, complemented, Hall
301	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
302	C₂×D₃	—	2	1	no	—
303	C₂×D₃	—	2	1	no	—
304	C₂×D₃	#3964	1	2	yes	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	4	2	no	—
307	C₃×C₄	C₂×#686	1	1	yes	characteristic
308	C₃×C₄	C₂×#687	1	1	yes	characteristic
309	C₃×C₄	#4088	1	1	yes	characteristic
310	C₃×C₄	#4088	1	1	yes	characteristic
311	Dic₃	—	2	1	no	—
312	Dic₃	—	4	2	no	—
313	Dic₃	#3964	1	2	yes	—
314	Dic₃	#3968	1	2	yes	—
315	C₂×C₅	—	6	2	no	—
316	C₂×C₅	D₃×#101	1	1	yes	characteristic
317	C₂×C₅	D₃×#102	1	1	yes	characteristic
318	C₂×C₅	#4802	1	1	yes	characteristic
319	C₂³	—	3	1	no	—
320	Q₈	—	4	1	no	—
321	Q₈	—	4	1	no	—
322	Q₈	—	12	4	no	—
323	Q₈	—	12	1	no	—
324	Q₈	—	12	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	3	1	no	—
327	C₂×C₄	—	3	1	no	—
328	C₂×C₄	—	3	2	no	—
329	C₂×C₄	—	3	2	no	—
330	C₂×C₄	—	3	1	no	—
331	C₂×C₄	—	6	2	no	—
332	C₂×C₄	—	6	1	no	—
333	C₂×C₄	—	6	1	no	—
334	C₂×C₄	—	6	2	no	—
335	C₂×C₄	—	12	4	no	—
336	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
337	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
338	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
339	C₈	—	20	1	no	—
340	C₈	—	20	1	no	—
341	C₈	—	60	1	no	—
342	C₈	—	60	1	no	—
343	C₂×C₃	C₂×#3964	1	1	yes	characteristic
344	C₂×C₃	#65514	1	1	yes	characteristic
345	C₂×C₃	C₂×#3968	1	1	yes	characteristic
346	D₃	—	2	2	no	—
347	C₅	#68420	1	1	yes	characteristic, complemented, Sylow
348	C₂²	—	3	2	no	—
349	C₂²	—	6	1	no	—
350	C₂²	—	6	1	no	—
351	C₂²	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	3	2	no	—
354	C₄	—	3	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	6	1	no	—
357	C₄	—	12	2	no	—
358	C₄	D₃×#686	1	1	yes	characteristic
359	C₄	D₃×#687	1	1	yes	characteristic
360	C₄	#91518	1	1	yes	characteristic
361	C₄	#91520	1	1	yes	characteristic
362	C₃	#10602566	1	1	yes	characteristic, complemented, Sylow
363	C₂	—	6	2	no	—
364	C₂	D₃×#3964	1	1	yes	characteristic
365	C₂	#11749213	1	1	yes	characteristic
366	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	#1546
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
300. C₃×C₅	#1546	semidirect
347. C₅	#68420	semidirect
362. C₃	#10602566	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
#11747011	2
#11746853	2
#11746368	2
C₃×#64767	2
#11745858	2
#11745857	2
#10602566	3
#68420	5