Subgroups — #20252546

Order 6016.

364 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689651	C₂	1	1	yes	maximal, characteristic
2	#15689252	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689247	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689199	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689193	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689192	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689188	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×C₂²⋊D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×C₂²⋊D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689184	C₂	1	1	yes	maximal, characteristic, complemented
11	#15689183	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689175	C₂	1	1	yes	maximal, characteristic, complemented
13	#15689148	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄⋊C₈×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10625324	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₄₇	C₂²	1	1	yes	characteristic
20	#13094019	C₄	1	1	yes	characteristic
21	#13094017	C₄	1	1	yes	characteristic
22	#13094014	C₄	1	1	yes	characteristic
23	#13094010	C₄	1	1	yes	characteristic
24	#13094009	C₄	1	1	yes	characteristic
25	#13094006	C₄	1	1	yes	characteristic
26	#13094002	C₄	1	1	yes	characteristic
27	#13093997	C₄	1	1	yes	characteristic
28	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
29	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₄₇	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₄₇	C₂²	1	1	yes	characteristic
32	#13093981	C₂²	1	1	yes	characteristic
33	C₄²×D₄₇	C₂²	1	1	yes	characteristic
34	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
35	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
36	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
39	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
41	C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
42	(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
43	(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
46	#13093935	C₂²	1	1	yes	characteristic
47	#13093934	C₂²	1	1	yes	characteristic
48	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
49	C₈×Dic₄₇	C₂²	1	1	yes	characteristic
50	C₈×Dic₄₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₇	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
54	#117×C₄₇	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
57	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₄₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₄₇	C₂×C₄	1	1	yes	characteristic
62	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
67	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
68	M₁₆×C₄₇	—	4	1	no	—
69	C₂×C₈×C₄₇	—	4	1	no	—
70	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
76	C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
82	(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
83	(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
84	(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
89	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
90	Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
91	Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
92	Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
93	Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
94	Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
95	Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
99	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
100	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
104	#10625324	—	4	1	no	—
105	C₈×D₄₇	—	4	1	no	—
106	C₈×D₄₇	—	4	1	no	—
107	C₈×D₄₇	—	4	1	no	—
108	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₄₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₄₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₄₇	—	2	2	no	—
120	C₂²⋊D₄₇	—	2	2	no	—
121	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
127	D₁₈₈	—	2	2	no	—
128	C₄×D₄₇	—	2	2	no	—
129	C₄×D₄₇	—	2	1	no	—
130	C₄×D₄₇	—	2	1	no	—
131	C₄×D₄₇	—	2	1	no	—
132	C₄×D₄₇	—	2	1	no	—
133	C₄×D₄₇	—	2	2	no	—
134	Dic₉₄	—	2	2	no	—
135	C₈×C₄₇	—	2	1	no	—
136	C₈×C₄₇	—	2	1	no	—
137	C₈×C₄₇	—	2	1	no	—
138	C₈×C₄₇	—	2	1	no	—
139	C₄₇⋊C₈	—	2	1	no	—
140	C₄₇⋊C₈	—	2	1	no	—
141	C₄₇⋊C₈	—	2	1	no	—
142	C₄₇⋊C₈	—	2	1	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×D₄₇	—	2	1	no	—
148	C₂×D₄₇	—	2	1	no	—
149	C₂×D₄₇	—	2	1	no	—
150	C₂×D₄₇	—	2	1	no	—
151	C₂×D₄₇	—	2	1	no	—
152	C₂×D₄₇	#131	1	2	yes	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
159	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₄₇	—	2	1	no	—
161	Dic₄₇	—	2	1	no	—
162	Dic₄₇	—	2	1	no	—
163	Dic₄₇	—	2	1	no	—
164	Dic₄₇	—	2	1	no	—
165	Dic₄₇	#131	1	2	yes	—
166	#2994	—	47	1	no	maximal, Sylow
167	C₂×C₄₇	—	4	1	no	—
168	C₂×C₄₇	#520	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
171	D₄₇	—	2	2	no	—
172	D₄₇	—	4	1	no	—
173	#529	—	47	1	no	—
174	#446	—	47	1	no	—
175	C₈×Q₈	—	47	1	no	—
176	#436	—	47	1	no	—
177	#435	—	47	1	no	—
178	#435	—	47	1	no	—
179	C₈×D₄	—	47	1	no	—
180	C₈×D₄	—	47	1	no	—
181	C₈×D₄	—	47	1	no	—
182	#433	—	47	1	no	—
183	#433	—	47	1	no	—
184	#431	—	47	1	no	—
185	#424	—	47	1	no	—
186	#424	D₄₇	1	1	yes	characteristic, complemented
187	C₂×C₄⋊C₈	—	47	1	no	—
188	C₄₇	#2994	1	1	yes	characteristic, complemented, Sylow
189	C₂×M₁₆	—	47	1	no	—
190	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
191	C₂²×C₈	—	47	1	no	—
192	C₂²×C₈	—	47	1	no	—
193	C₂²×C₈	—	47	1	no	—
194	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
195	#126	—	47	1	no	—
196	#126	—	47	1	no	—
197	#125	—	47	1	no	—
198	#124	—	47	1	no	—
199	#123	—	47	1	no	—
200	#123	—	47	1	no	—
201	C₂²⋊Q₈	—	47	1	no	—
202	#121	—	47	1	no	—
203	C₄×Q₈	—	47	1	no	—
204	C₄×D₄	—	47	1	no	—
205	C₄×D₄	—	47	1	no	—
206	C₄×D₄	—	47	1	no	—
207	#117	—	47	1	no	—
208	#117	C₂×D₄₇	1	1	yes	characteristic
209	C₂×C₄²	—	47	1	no	—
210	C₄⋊C₈	—	47	1	no	—
211	C₄⋊C₈	—	47	1	no	—
212	C₄⋊C₈	—	47	1	no	—
213	C₄⋊C₈	—	47	1	no	—
214	C₄⋊C₈	—	47	1	no	—
215	C₄⋊C₈	—	47	1	no	—
216	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
217	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
218	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
219	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
220	C₂²⋊C₈	—	47	1	no	—
221	C₂²⋊C₈	—	47	1	no	—
222	C₂²⋊C₈	—	47	1	no	—
223	C₂²⋊C₈	—	47	1	no	—
224	C₂²⋊C₈	—	47	1	no	—
225	C₂²⋊C₈	—	47	1	no	—
226	#97	—	47	1	no	—
227	#97	—	47	1	no	—
228	C₄×C₈	—	47	1	no	—
229	C₄×C₈	—	47	1	no	—
230	C₄×C₈	—	47	1	no	—
231	C₄×C₈	—	47	1	no	—
232	C₂×Q₈	—	47	1	no	—
233	C₂×D₄	—	47	1	no	—
234	C₂×D₄	—	47	1	no	—
235	C₂×D₄	—	47	1	no	—
236	C₂²×C₄	—	47	1	no	—
237	C₂²×C₄	—	47	1	no	—
238	C₂²×C₄	—	47	1	no	—
239	C₂²×C₄	—	47	1	no	—
240	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
241	M₁₆	—	4	1	no	—
242	M₁₆	—	188	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	47	1	no	—
245	C₂×C₈	—	47	1	no	—
246	C₂×C₈	—	47	1	no	—
247	C₂×C₈	—	47	1	no	—
248	C₂×C₈	—	188	1	no	—
249	C₂×C₈	—	188	1	no	—
250	C₂×C₈	—	188	1	no	—
251	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
255	C₄⋊C₄	—	47	1	no	—
256	C₄⋊C₄	—	47	1	no	—
257	C₄⋊C₄	—	47	1	no	—
258	C₄⋊C₄	—	47	1	no	—
259	C₄⋊C₄	—	47	1	no	—
260	C₄⋊C₄	—	47	1	no	—
261	C₄⋊C₄	—	47	1	no	—
262	C₄⋊C₄	—	47	1	no	—
263	C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
264	C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	47	1	no	—
266	C₂²⋊C₄	—	47	1	no	—
267	C₂²⋊C₄	—	47	1	no	—
268	C₂²⋊C₄	—	47	1	no	—
269	C₂²⋊C₄	—	47	1	no	—
270	C₂²⋊C₄	—	47	1	no	—
271	C₂²⋊C₄	—	47	1	no	—
272	C₂²⋊C₄	—	47	1	no	—
273	C₂²⋊C₄	C₄×D₄₇	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₄₇	1	1	yes	characteristic
275	C₄²	—	47	1	no	—
276	C₄²	—	47	1	no	—
277	C₄²	—	47	1	no	—
278	C₄²	—	47	1	no	—
279	C₄²	C₂²×D₄₇	1	1	yes	characteristic
280	C₄²	C₂²×D₄₇	1	1	yes	characteristic
281	C₂³	—	47	1	no	—
282	C₂³	—	47	1	no	—
283	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
284	Q₈	—	94	2	no	—
285	D₄	—	94	2	no	—
286	D₄	—	94	2	no	—
287	D₄	—	94	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	47	1	no	—
291	C₂×C₄	—	47	1	no	—
292	C₂×C₄	—	47	1	no	—
293	C₂×C₄	—	47	1	no	—
294	C₂×C₄	—	47	1	no	—
295	C₂×C₄	—	47	1	no	—
296	C₂×C₄	—	94	2	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	1	no	—
299	C₂×C₄	—	94	1	no	—
300	C₂×C₄	—	94	1	no	—
301	C₂×C₄	—	94	1	no	—
302	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	94	1	no	—
313	C₈	—	94	1	no	—
314	C₈	—	94	1	no	—
315	C₈	—	94	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	47	2	no	—
320	C₂²	—	94	1	no	—
321	C₂²	—	94	1	no	—
322	C₂²	—	94	1	no	—
323	C₂²	—	94	1	no	—
324	C₂²	—	94	1	no	—
325	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	47	2	no	—
332	C₄	—	94	1	no	—
333	C₄	—	94	1	no	—
334	C₄	—	94	1	no	—
335	C₄	—	94	1	no	—
336	C₄	—	94	1	no	—
337	C₄	#13094069	1	1	yes	characteristic
338	C₄	#13094072	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	94	2	no	—
341	C₂	—	188	1	no	—
342	C₂	#15689604	1	1	yes	characteristic
343	C₂	#131×D₄₇	1	1	yes	characteristic
344	C₂	#131×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#424	C₂
hypercenter	#424	D₄₇
hypocenter	C₄₇	#2994
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #15689252	C₂	semidirect
3. #15689247	C₂	semidirect
4. #15689199	C₂	semidirect
5. #15689193	C₂	semidirect
6. #15689192	C₂	semidirect
7. #15689188	C₂	semidirect
8. C₈×C₂²⋊D₄₇	C₂	semidirect
9. C₈×C₂²⋊D₄₇	C₂	semidirect
10. #15689184	C₂	semidirect
11. #15689183	C₂	semidirect
12. #15689175	C₂	semidirect
13. #15689148	C₂	semidirect
14. C₄⋊C₈×D₄₇	C₂	semidirect
15. C₄₇×#424	C₂	semidirect
186. #424	D₄₇	semidirect
188. C₄₇	#2994	semidirect