Subgroups — #20258161

Order 6016.

1424 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689959	C₂	1	2	yes	maximal, complemented
2	#15689939	C₂	1	4	yes	maximal, complemented
3	#15689933	C₂	1	4	yes	maximal, complemented
4	#15689932	C₂	1	4	yes	maximal, complemented
5	#15689927	C₂	1	4	yes	maximal, complemented
6	#15689923	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689920	C₂	1	4	yes	maximal, complemented
8	C₂×#13094044	C₂	1	2	yes	maximal, complemented
9	#135×D₄₇	C₂	1	4	yes	maximal, complemented
10	C₂×#13094032	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#575	C₂	1	1	yes	maximal, characteristic, complemented
12	#13094072	C₂²	1	4	yes	complemented
13	#13094071	—	2	2	no	—
14	#13094071	C₂²	1	4	yes	complemented
15	#13094070	—	2	2	no	—
16	#13094070	C₂²	1	4	yes	complemented
17	#13094069	—	2	4	no	—
18	#13094069	C₂²	1	4	yes	complemented
19	C₂×#10625350	—	2	4	no	—
20	C₂×#10625350	C₂²	1	1	yes	characteristic
21	C₂×#10625350	C₂²	1	2	yes	complemented
22	#41×D₄₇	—	2	4	no	—
23	#41×D₄₇	C₂²	1	8	yes	complemented
24	#13094057	C₂²	1	4	yes	complemented
25	#13094056	C₂²	1	8	yes	complemented
26	#13094053	C₂²	1	8	yes	complemented
27	#13094052	C₂²	1	4	yes	complemented
28	#13094051	C₂²	1	8	yes	complemented
29	#13094048	C₂²	1	8	yes	complemented
30	#13094047	C₂²	1	4	yes	complemented
31	#13094046	C₂²	1	4	yes	complemented
32	#13094045	C₂²	1	8	yes	complemented
33	#13094044	C₂²	1	4	yes	complemented
34	#13094044	C₂²	1	4	yes	complemented
35	#13094044	C₂²	1	4	yes	complemented
36	#13094044	C₂²	1	4	yes	complemented
37	#13094043	C₂²	1	4	yes	complemented
38	C₂×#10625337	C₂²	1	2	yes	complemented
39	Q₁₆×D₄₇	C₂²	1	4	yes	complemented
40	C₂×#10625335	C₂²	1	2	yes	complemented
41	C₂×#10625336	C₂²	1	2	yes	complemented
42	SD₁₆×D₄₇	C₂²	1	8	yes	complemented
43	C₂×#10625334	C₂²	1	2	yes	complemented
44	D₈×D₄₇	C₂²	1	4	yes	complemented
45	#13094032	C₂²	1	4	yes	complemented
46	#13094032	C₂²	1	4	yes	complemented
47	C₂×#10625324	C₂²	1	2	yes	complemented
48	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₈×D₄₇	C₂²	1	2	yes	complemented
50	C₂×#41×C₄₇	C₂²	1	2	yes	complemented
51	#137×C₄₇	C₂²	1	4	yes	complemented
52	AGL(1,ℤ/8)×C₄₇	C₂²	1	4	yes	complemented
53	#135×C₄₇	C₂²	1	4	yes	complemented
54	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
55	#41×C₄₇	—	2	2	no	—
56	#41×C₄₇	—	2	2	no	—
57	#41×C₄₇	C₂³	1	4	yes	—
58	#41×C₄₇	C₂³	1	4	yes	—
59	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
60	C₂×D₄×C₄₇	—	2	2	no	—
61	C₂×D₄×C₄₇	C₂³	1	2	yes	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₇	—	2	2	no	—
65	C₂×C₂²⋊D₄₇	—	2	8	no	—
66	C₂×C₂²⋊D₄₇	D₄	1	2	yes	—
67	#10625354	—	2	4	no	—
68	#10625354	C₂³	1	8	yes	—
69	Q₈×D₄₇	—	2	4	no	—
70	Q₈×D₄₇	C₂³	1	8	yes	—
71	#10625352	—	2	8	no	—
72	#10625352	—	2	8	no	—
73	#10625352	—	2	4	no	—
74	#10625352	C₂³	1	8	yes	—
75	D₄×D₄₇	—	2	8	no	—
76	D₄×D₄₇	—	2	8	no	—
77	D₄×D₄₇	—	2	4	no	—
78	D₄×D₄₇	C₂³	1	8	yes	—
79	#10625350	—	2	2	no	—
80	#10625350	—	2	2	no	—
81	#10625350	—	2	8	no	—
82	#10625350	—	2	4	no	—
83	#10625350	—	2	8	no	—
84	#10625350	—	2	4	no	—
85	#10625350	D₄	1	4	yes	—
86	#10625350	C₂³	1	4	yes	—
87	#10625350	C₂³	1	4	yes	—
88	#10625350	C₂³	1	4	yes	—
89	C₂×D₁₈₈	—	2	4	no	—
90	C₂×D₁₈₈	D₄	1	1	yes	characteristic
91	C₂×D₁₈₈	C₂³	1	2	yes	—
92	C₂×C₄×D₄₇	—	2	2	no	—
93	C₂×C₄×D₄₇	—	2	8	no	—
94	C₂×C₄×D₄₇	C₂³	1	2	yes	—
95	C₂×Dic₉₄	—	2	4	no	—
96	C₂×Dic₉₄	D₄	1	1	yes	characteristic
97	C₂×Dic₉₄	C₂³	1	2	yes	—
98	Q₁₆×C₄₇	C₂³	1	4	yes	—
99	SD₁₆×C₄₇	C₂³	1	8	yes	—
100	D₈×C₄₇	C₂³	1	4	yes	—
101	M₁₆×C₄₇	C₂³	1	4	yes	—
102	C₂×C₈×C₄₇	C₂³	1	2	yes	—
103	#10625337	C₂³	1	4	yes	—
104	#10625337	C₂³	1	4	yes	—
105	#10625336	C₂³	1	4	yes	—
106	#10625336	C₂³	1	4	yes	—
107	#10625335	C₂³	1	4	yes	—
108	#10625335	C₂³	1	4	yes	—
109	#10625334	C₂³	1	4	yes	—
110	#10625334	C₂³	1	4	yes	—
111	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
112	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
113	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
114	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
115	Dic₁₈₈	C₂³	1	4	yes	—
116	D₃₇₆	C₂³	1	4	yes	—
117	#10625325	C₂³	1	8	yes	—
118	#10625324	C₂³	1	8	yes	—
119	C₈×D₄₇	C₂³	1	4	yes	—
120	C₈×D₄₇	C₂³	1	4	yes	—
121	C₂³×C₄₇	—	2	2	no	—
122	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×D₄₇	—	2	2	no	—
124	C₂²×D₄₇	—	2	8	no	—
125	C₂²×D₄₇	C₂×D₄	1	2	yes	—
126	Q₈×C₄₇	—	2	2	no	—
127	Q₈×C₄₇	C₂⁴	1	4	yes	—
128	D₄×C₄₇	—	2	4	no	—
129	D₄×C₄₇	—	2	4	no	—
130	D₄×C₄₇	—	2	2	no	—
131	D₄×C₄₇	C₂⁴	1	4	yes	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	4	no	—
134	C₂×C₄×C₄₇	—	2	4	no	—
135	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
137	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
139	C₂²⋊D₄₇	—	2	4	no	—
140	C₂²⋊D₄₇	—	2	4	no	—
141	C₂²⋊D₄₇	—	2	8	no	—
142	C₂²⋊D₄₇	—	2	8	no	—
143	C₂²⋊D₄₇	C₂×D₄	1	8	yes	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	2	8	no	—
146	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
147	D₁₈₈	—	2	8	no	—
148	D₁₈₈	—	2	2	no	—
149	D₁₈₈	C₂×D₄	1	4	yes	—
150	D₁₈₈	C₂⁴	1	4	yes	—
151	C₄×D₄₇	—	2	8	no	—
152	C₄×D₄₇	—	2	4	no	—
153	C₄×D₄₇	—	2	4	no	—
154	C₄×D₄₇	—	2	8	no	—
155	C₄×D₄₇	C₂×D₄	1	4	yes	—
156	C₄×D₄₇	C₂⁴	1	4	yes	—
157	Dic₉₄	—	2	2	no	—
158	Dic₉₄	—	2	8	no	—
159	Dic₉₄	C₂×D₄	1	4	yes	—
160	Dic₉₄	C₂⁴	1	4	yes	—
161	C₈×C₄₇	C₂⁴	1	4	yes	—
162	C₄₇⋊C₈	C₂⁴	1	1	yes	characteristic
163	C₄₇⋊C₈	C₂⁴	1	1	yes	characteristic
164	C₄₇⋊C₈	C₂⁴	1	2	yes	—
165	C₂²×C₄₇	—	2	4	no	—
166	C₂²×C₄₇	—	4	1	no	—
167	C₂²×C₄₇	—	8	2	no	—
168	C₂²×C₄₇	C₂²×D₄	1	2	yes	—
169	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×D₄₇	—	2	4	no	—
171	C₂×D₄₇	—	4	2	no	—
172	C₂×D₄₇	—	8	8	no	—
173	C₂×D₄₇	—	8	2	no	—
174	C₂×D₄₇	C₂²×D₄	1	4	yes	—
175	C₄×C₄₇	—	2	4	no	—
176	C₄×C₄₇	C₂²×D₄	1	2	yes	—
177	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
178	C₄×C₄₇	C₂⁵	1	1	yes	characteristic
179	Dic₄₇	—	2	4	no	—
180	Dic₄₇	C₂²×D₄	1	4	yes	—
181	#3584	—	47	1	no	maximal, Sylow
182	C₂×C₄₇	—	2	1	no	—
183	C₂×C₄₇	—	2	2	no	—
184	C₂×C₄₇	—	4	4	no	—
185	C₂×C₄₇	C₂³×D₄	1	1	yes	characteristic
186	D₄₇	—	2	4	no	—
187	D₄₇	—	4	4	no	—
188	#585	—	47	2	no	—
189	#578	—	47	4	no	—
190	#577	—	47	4	no	—
191	#577	—	47	4	no	—
192	#576	—	47	4	no	—
193	#575	—	47	4	no	—
194	#575	—	47	1	no	—
195	#575	D₄₇	1	1	yes	characteristic, complemented
196	C₂×#135	—	47	2	no	—
197	C₂×#135	—	47	4	no	—
198	C₂×#131	—	47	1	no	—
199	C₄₇	#3584	1	1	yes	characteristic, complemented, Sylow
200	2₋¹⁺⁴	—	47	4	no	—
201	2₋¹⁺⁴	—	47	4	no	—
202	2₋¹⁺⁴	—	94	2	no	—
203	H(5,𝔽₂)	—	47	4	no	—
204	H(5,𝔽₂)	—	47	4	no	—
205	H(5,𝔽₂)	—	94	4	no	—
206	H(5,𝔽₂)	—	94	2	no	—
207	C₂×#41	—	47	1	no	—
208	C₂×#41	—	47	8	no	—
209	C₂×#41	—	47	2	no	—
210	C₂×#41	—	94	4	no	—
211	C₂×#41	—	94	4	no	—
212	C₂×#41	C₂×D₄₇	1	2	yes	complemented
213	#137	—	47	8	no	—
214	#137	—	47	8	no	—
215	#137	—	47	4	no	—
216	#137	C₂×D₄₇	1	4	yes	complemented
217	AGL(1,ℤ/8)	—	47	8	no	—
218	AGL(1,ℤ/8)	—	47	8	no	—
219	AGL(1,ℤ/8)	—	47	4	no	—
220	AGL(1,ℤ/8)	C₂×D₄₇	1	4	yes	complemented
221	#135	—	47	4	no	—
222	#135	—	47	4	no	—
223	#135	—	47	4	no	—
224	#135	—	47	4	no	—
225	#135	—	47	4	no	—
226	#135	—	47	8	no	—
227	#135	—	47	4	no	—
228	#135	—	47	4	no	—
229	#135	C₂×D₄₇	1	4	yes	complemented
230	C₂×Q₁₆	—	47	2	no	—
231	C₂×Q₁₆	—	47	4	no	—
232	C₂×SD₁₆	—	47	2	no	—
233	C₂×SD₁₆	—	47	2	no	—
234	C₂×SD₁₆	—	47	8	no	—
235	C₂×D₈	—	47	2	no	—
236	C₂×D₈	—	47	4	no	—
237	#131	—	47	4	no	—
238	#131	—	47	4	no	—
239	C₂×M₁₆	—	47	2	no	—
240	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
241	C₂²×C₈	—	47	1	no	—
242	C₂²×C₈	—	47	2	no	—
243	#41	—	2	2	no	—
244	#41	—	2	2	no	—
245	#41	—	47	4	no	—
246	#41	—	47	4	no	—
247	#41	—	47	8	no	—
248	#41	—	47	8	no	—
249	#41	—	47	4	no	—
250	#41	—	47	4	no	—
251	#41	—	94	8	no	—
252	#41	—	94	8	no	—
253	#41	—	94	4	no	—
254	#41	—	94	4	no	—
255	#41	—	94	8	no	—
256	#41	—	94	8	no	—
257	#41	—	94	4	no	—
258	#41	—	94	4	no	—
259	#41	—	94	2	no	—
260	#41	—	94	2	no	—
261	#41	C₂²×D₄₇	1	4	yes	—
262	#41	C₂²×D₄₇	1	4	yes	—
263	C₂×Q₈	—	47	1	no	—
264	C₂×Q₈	—	47	8	no	—
265	C₂×Q₈	—	47	2	no	—
266	C₂×Q₈	—	94	4	no	—
267	C₂×Q₈	—	94	4	no	—
268	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	47	2	no	—
271	C₂×D₄	—	47	8	no	—
272	C₂×D₄	—	47	1	no	—
273	C₂×D₄	—	47	2	no	—
274	C₂×D₄	—	94	4	no	—
275	C₂×D₄	—	94	8	no	—
276	C₂×D₄	—	94	4	no	—
277	C₂×D₄	—	94	8	no	—
278	C₂×D₄	—	94	8	no	—
279	C₂×D₄	—	94	2	no	—
280	C₂×D₄	C₂²×D₄₇	1	2	yes	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	47	2	no	—
283	C₂²×C₄	—	94	8	no	—
284	C₂²×C₄	—	94	2	no	—
285	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
286	Q₁₆	—	47	4	no	—
287	Q₁₆	—	47	4	no	—
288	Q₁₆	—	47	4	no	—
289	Q₁₆	C₂²×D₄₇	1	4	yes	—
290	SD₁₆	—	47	4	no	—
291	SD₁₆	—	47	4	no	—
292	SD₁₆	—	47	4	no	—
293	SD₁₆	—	47	4	no	—
294	SD₁₆	—	47	8	no	—
295	SD₁₆	C₂²×D₄₇	1	8	yes	—
296	D₈	—	47	4	no	—
297	D₈	—	47	4	no	—
298	D₈	—	47	4	no	—
299	D₈	C₂²×D₄₇	1	4	yes	—
300	M₁₆	—	47	8	no	—
301	M₁₆	C₂²×D₄₇	1	4	yes	—
302	C₂×C₈	—	47	1	no	—
303	C₂×C₈	—	47	1	no	—
304	C₂×C₈	—	47	2	no	—
305	C₂×C₈	—	47	2	no	—
306	C₂×C₈	—	47	4	no	—
307	C₂×C₈	—	47	4	no	—
308	C₂×C₈	C₂²×D₄₇	1	2	yes	—
309	C₂³	—	2	2	no	—
310	C₂³	—	47	2	no	—
311	C₂³	—	94	8	no	—
312	C₂³	—	94	2	no	—
313	C₂³	D₄×D₄₇	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	Q₈	—	47	4	no	—
316	Q₈	—	47	4	no	—
317	Q₈	—	94	8	no	—
318	Q₈	—	94	2	no	—
319	Q₈	C₂³×D₄₇	1	4	yes	—
320	D₄	—	2	4	no	—
321	D₄	—	2	4	no	—
322	D₄	—	2	2	no	—
323	D₄	—	47	8	no	—
324	D₄	—	47	4	no	—
325	D₄	—	47	4	no	—
326	D₄	—	94	8	no	—
327	D₄	—	94	8	no	—
328	D₄	—	94	8	no	—
329	D₄	—	94	4	no	—
330	D₄	—	94	4	no	—
331	D₄	—	94	2	no	—
332	D₄	C₂³×D₄₇	1	4	yes	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	47	2	no	—
337	C₂×C₄	—	47	4	no	—
338	C₂×C₄	—	47	4	no	—
339	C₂×C₄	—	94	8	no	—
340	C₂×C₄	—	94	8	no	—
341	C₂×C₄	—	94	8	no	—
342	C₂×C₄	—	94	2	no	—
343	C₂×C₄	—	94	4	no	—
344	C₂×C₄	—	94	4	no	—
345	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
346	C₂×C₄	D₄×D₄₇	1	2	yes	—
347	C₂×C₄	C₂³×D₄₇	1	2	yes	—
348	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
349	C₈	—	47	1	no	—
350	C₈	—	47	1	no	—
351	C₈	—	47	2	no	—
352	C₈	C₂³×D₄₇	1	4	yes	—
353	C₂²	—	2	4	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	47	4	no	—
357	C₂²	—	94	4	no	—
358	C₂²	—	376	2	no	—
359	C₂²	—	376	8	no	—
360	C₂²	—	188	2	no	—
361	C₂²	C₂×D₄×D₄₇	1	2	yes	—
362	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
363	C₄	—	2	4	no	—
364	C₄	—	47	4	no	—
365	C₄	—	94	4	no	—
366	C₄	C₂×D₄×D₄₇	1	2	yes	—
367	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
368	C₄	C₂⁴×D₄₇	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	2	1	no	—
371	C₂	—	4	4	no	—
372	C₂	—	94	4	no	—
373	C₂	—	188	4	no	—
374	C₂	C₂²×D₄×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂⁵
Frattini	C₄	C₂⁴×D₄₇
Fitting	C₄₇×#575	C₂
socle	C₂×C₄₇	C₂³×D₄
hypercenter	#575	D₄₇
hypocenter	C₄₇	#3584
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

Complements

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

N	H ≅ G/N	type
1. #15689959	C₂	semidirect
2. #15689939	C₂	semidirect
3. #15689933	C₂	semidirect
4. #15689932	C₂	semidirect
5. #15689927	C₂	semidirect
6. #15689923	C₂	semidirect
7. #15689920	C₂	semidirect
8. C₂×#13094044	C₂	semidirect
9. #135×D₄₇	C₂	semidirect
10. C₂×#13094032	C₂	semidirect
11. C₄₇×#575	C₂	semidirect
12. #13094072	C₂²	semidirect
14. #13094071	C₂²	semidirect
16. #13094070	C₂²	semidirect
18. #13094069	C₂²	semidirect
21. C₂×#10625350	C₂²	semidirect
23. #41×D₄₇	C₂²	semidirect
24. #13094057	C₂²	semidirect
25. #13094056	C₂²	semidirect
26. #13094053	C₂²	semidirect
27. #13094052	C₂²	semidirect
28. #13094051	C₂²	semidirect
29. #13094048	C₂²	semidirect
30. #13094047	C₂²	semidirect
31. #13094046	C₂²	semidirect
32. #13094045	C₂²	semidirect
33. #13094044	C₂²	semidirect
34. #13094044	C₂²	semidirect
35. #13094044	C₂²	semidirect
36. #13094044	C₂²	semidirect
37. #13094043	C₂²	semidirect
38. C₂×#10625337	C₂²	semidirect
39. Q₁₆×D₄₇	C₂²	semidirect
40. C₂×#10625335	C₂²	semidirect
41. C₂×#10625336	C₂²	semidirect
42. SD₁₆×D₄₇	C₂²	semidirect
43. C₂×#10625334	C₂²	semidirect
44. D₈×D₄₇	C₂²	semidirect
45. #13094032	C₂²	semidirect
46. #13094032	C₂²	semidirect
47. C₂×#10625324	C₂²	semidirect
48. C₂²×C₄₇⋊C₈	C₂²	semidirect
49. C₂×C₈×D₄₇	C₂²	semidirect
50. C₂×#41×C₄₇	C₂²	semidirect
51. #137×C₄₇	C₂²	semidirect
52. AGL(1,ℤ/8)×C₄₇	C₂²	semidirect
53. #135×C₄₇	C₂²	semidirect
54. C₂×M₁₆×C₄₇	C₂²	semidirect
195. #575	D₄₇	semidirect
199. C₄₇	#3584	semidirect
212. C₂×#41	C₂×D₄₇	semidirect
216. #137	C₂×D₄₇	semidirect
220. AGL(1,ℤ/8)	C₂×D₄₇	semidirect
229. #135	C₂×D₄₇	semidirect
240. C₂×M₁₆	C₂×D₄₇	semidirect