Subgroups — D₄₇×#1336 #26126067

Order 12032.

398 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	#443×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#364×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#322×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241365	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241364	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241363	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1336	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₄₇	C₄	1	1	yes	characteristic
9	#131×D₄₇	—	2	1	no	—
10	M₃₂×D₄₇	—	2	1	no	—
11	#15689241	C₂²	1	1	yes	characteristic
12	#15689240	C₄	1	1	yes	characteristic
13	#15689239	C₄	1	1	yes	characteristic
14	#15689238	—	2	1	no	—
15	#15689237	—	2	1	no	—
16	#108×D₄₇	C₄	1	1	yes	characteristic
17	#104×D₄₇	—	2	1	no	—
18	C₄×C₈×D₄₇	—	2	1	no	—
19	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
20	#15688944	C₂²	1	1	yes	characteristic, complemented
21	#15688943	—	2	1	no	—
22	#15688845	—	2	1	no	—
23	#15688844	C₂²	1	1	yes	characteristic, complemented
24	C₄₇×#443	C₂²	1	1	yes	characteristic
25	C₄₇×#364	C₂²	1	1	yes	characteristic, complemented
26	C₄₇×#322	C₂²	1	1	yes	characteristic, complemented
27	#41×D₄₇	—	2	1	no	—
28	#13094047	C₈	1	1	yes	characteristic, complemented
29	#13094046	C₂×C₄	1	1	yes	characteristic
30	#13094045	—	2	1	no	—
31	#13094044	—	2	1	no	—
32	#13094043	C₈	1	1	yes	characteristic, complemented
33	Q₁₆×D₄₇	C₈	1	1	yes	characteristic, complemented
34	SD₁₆×D₄₇	—	2	1	no	—
35	D₈×D₄₇	C₈	1	1	yes	characteristic, complemented
36	#13094032	—	2	1	no	—
37	#13094032	—	4	1	no	—
38	#13094031	—	2	1	no	—
39	#13094030	—	2	1	no	—
40	#13094030	—	4	1	no	—
41	M₁₆×D₄₇	—	2	1	no	—
42	M₁₆×D₄₇	—	4	1	no	—
43	C₂×C₈×D₄₇	—	2	2	no	—
44	C₂×C₈×D₄₇	—	4	1	no	—
45	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
47	C₄²×D₄₇	D₄	1	1	yes	characteristic
48	#13093964	—	2	1	no	—
49	#13093963	—	4	1	no	—
50	C₁₆×D₄₇	—	4	1	no	—
51	#13093960	C₂×C₄	1	1	yes	characteristic
52	#13093959	—	2	1	no	—
53	#13093952	—	2	1	no	—
54	#13093951	—	2	1	no	—
55	#13093950	—	2	1	no	—
56	C₄×C₄₇⋊C₈	—	2	1	no	—
57	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
58	C₈×Dic₄₇	—	2	2	no	—
59	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
60	C₈×Dic₄₇	D₄	1	1	yes	characteristic
61	#135×C₄₇	C₂×C₄	1	1	yes	characteristic
62	#131×C₄₇	—	2	1	no	—
63	M₃₂×C₄₇	—	2	1	no	—
64	#108×C₄₇	C₂×C₄	1	1	yes	characteristic
65	#104×C₄₇	—	2	1	no	—
66	C₄×C₈×C₄₇	—	2	1	no	—
67	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	#41×C₄₇	—	2	1	no	—
69	#10625354	—	2	1	no	—
70	Q₈×D₄₇	—	2	1	no	—
71	#10625352	—	2	1	no	—
72	#10625352	—	4	1	no	—
73	D₄×D₄₇	—	2	1	no	—
74	D₄×D₄₇	—	4	1	no	—
75	#10625350	—	2	1	no	—
76	#10625350	—	4	1	no	—
77	C₂×C₄×D₄₇	—	2	1	no	—
78	C₂×C₄×D₄₇	—	4	1	no	—
79	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
80	Q₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic, complemented
81	SD₁₆×C₄₇	—	2	1	no	—
82	D₈×C₄₇	C₂×C₈	1	1	yes	characteristic, complemented
83	M₁₆×C₄₇	—	2	1	no	—
84	M₁₆×C₄₇	—	4	1	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	4	1	no	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	#10625337	—	2	1	no	—
91	#10625336	—	2	1	no	—
92	#10625335	—	2	1	no	—
93	#10625334	—	2	1	no	—
94	C₄×Dic₄₇	—	2	1	no	—
95	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
96	#10625329	—	2	1	no	—
97	#10625329	—	4	1	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	4	1	no	—
100	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
102	Dic₁₈₈	C₂×C₈	1	1	yes	characteristic, complemented
103	D₃₇₆	C₂×C₈	1	1	yes	characteristic, complemented
104	#10625325	—	2	1	no	—
105	#10625324	—	4	1	no	—
106	#10625324	—	4	1	no	—
107	#10625324	—	4	1	no	—
108	C₈×D₄₇	—	4	4	no	—
109	C₈×D₄₇	—	4	1	no	—
110	C₈×D₄₇	—	4	1	no	—
111	C₈×D₄₇	—	4	1	no	—
112	C₈×D₄₇	C₂×C₈	1	1	yes	characteristic
113	C₈×D₄₇	C₂×C₈	1	1	yes	characteristic
114	C₈×D₄₇	M₁₆	1	1	yes	characteristic
115	C₈×D₄₇	M₁₆	1	1	yes	characteristic
116	C₈×D₄₇	D₈	1	1	yes	characteristic
117	C₈×D₄₇	D₈	1	1	yes	characteristic
118	C₈×D₄₇	SD₁₆	1	1	yes	characteristic
119	C₈×D₄₇	SD₁₆	1	1	yes	characteristic
120	C₁₆×C₄₇	—	4	1	no	—
121	C₄₇⋊C₁₆	—	4	1	no	—
122	C₂²×D₄₇	—	4	1	no	—
123	C₂²×D₄₇	#104	1	1	yes	characteristic
124	Q₈×C₄₇	—	2	1	no	—
125	D₄×C₄₇	—	2	1	no	—
126	D₄×C₄₇	—	4	1	no	—
127	C₂×C₄×C₄₇	—	2	1	no	—
128	C₂×C₄×C₄₇	—	4	1	no	—
129	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²⋊D₄₇	—	4	1	no	—
131	C₂²⋊D₄₇	—	4	1	no	—
132	C₂²⋊D₄₇	—	4	1	no	—
133	C₂×Dic₄₇	—	2	1	no	—
134	C₂×Dic₄₇	—	4	1	no	—
135	C₂×Dic₄₇	#104	1	1	yes	characteristic
136	D₁₈₈	—	2	1	no	—
137	D₁₈₈	—	4	1	no	—
138	C₄×D₄₇	—	2	4	no	—
139	C₄×D₄₇	—	4	1	no	—
140	C₄×D₄₇	—	4	1	no	—
141	C₄×D₄₇	—	4	1	no	—
142	C₄×D₄₇	C₂²⋊C₈	1	1	yes	characteristic
143	C₄×D₄₇	C₂²⋊C₈	1	1	yes	characteristic
144	C₄×D₄₇	#102	1	1	yes	characteristic
145	C₄×D₄₇	#102	1	1	yes	characteristic
146	Dic₉₄	—	2	1	no	—
147	Dic₉₄	—	4	1	no	—
148	C₈×C₄₇	—	4	2	no	—
149	C₈×C₄₇	—	4	1	no	—
150	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
151	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
152	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
153	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
154	C₄₇⋊C₈	—	4	2	no	—
155	C₄₇⋊C₈	—	4	1	no	—
156	C₄₇⋊C₈	C₂²⋊C₈	1	1	yes	characteristic
157	C₄₇⋊C₈	C₂²⋊C₈	1	1	yes	characteristic
158	C₄₇⋊C₈	#102	1	1	yes	characteristic
159	C₄₇⋊C₈	#102	1	1	yes	characteristic
160	C₂×#1336	—	47	1	no	maximal, Sylow
161	C₂²×C₄₇	—	4	1	no	—
162	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
163	C₂×D₄₇	—	2	2	no	—
164	C₂×D₄₇	—	4	1	no	—
165	C₂×D₄₇	—	8	2	no	—
166	C₂×D₄₇	D₄⋊C₈	1	1	yes	characteristic
167	C₂×D₄₇	D₄⋊C₈	1	1	yes	characteristic
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	4	1	no	—
170	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
172	Dic₄₇	—	2	2	no	—
173	Dic₄₇	—	4	1	no	—
174	Dic₄₇	D₄⋊C₈	1	1	yes	characteristic
175	Dic₄₇	D₄⋊C₈	1	1	yes	characteristic
176	C₂×#443	—	47	1	no	—
177	C₂×#364	—	47	1	no	—
178	C₂×#322	—	47	1	no	—
179	#1336	—	47	1	no	—
180	#1336	—	47	1	no	—
181	#1336	—	47	1	no	—
182	#1336	D₄₇	1	1	yes	characteristic, direct product factor
183	C₂×C₄₇	—	2	1	no	—
184	C₂×C₄₇	—	8	1	no	—
185	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
186	D₄₇	—	2	1	no	—
187	D₄₇	—	8	1	no	—
188	D₄₇	#1336	1	2	yes	direct product factor
189	C₂×#135	—	47	1	no	—
190	C₂×#131	—	94	1	no	—
191	C₂×M₃₂	—	94	1	no	—
192	#443	—	47	1	no	—
193	#443	—	47	1	no	—
194	#443	—	47	1	no	—
195	#443	—	94	1	no	—
196	#443	—	94	1	no	—
197	#443	C₂×D₄₇	1	1	yes	characteristic
198	C₂×#108	—	47	1	no	—
199	C₂×#104	—	94	1	no	—
200	C₂×C₄×C₈	—	47	1	no	—
201	C₂×C₄×C₈	—	94	1	no	—
202	#364	—	47	1	no	—
203	#364	—	94	1	no	—
204	#364	C₂×D₄₇	1	1	yes	characteristic, complemented
205	#322	—	47	1	no	—
206	#322	—	94	1	no	—
207	#322	C₂×D₄₇	1	1	yes	characteristic, complemented
208	C₄₇	C₂×#1336	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	94	1	no	—
210	#135	—	47	1	no	—
211	#135	—	47	1	no	—
212	#135	—	47	1	no	—
213	#135	—	94	1	no	—
214	#135	—	94	1	no	—
215	#135	C₄×D₄₇	1	1	yes	characteristic
216	C₂×Q₁₆	—	47	1	no	—
217	C₂×SD₁₆	—	94	1	no	—
218	C₂×D₈	—	47	1	no	—
219	#131	—	2	1	no	—
220	#131	—	94	1	no	—
221	#131	—	94	1	no	—
222	#131	—	94	1	no	—
223	#131	—	188	1	no	—
224	#131	—	188	1	no	—
225	C₂×M₁₆	—	94	1	no	—
226	C₂×M₁₆	—	188	1	no	—
227	C₂²×C₈	—	47	1	no	—
228	C₂²×C₈	—	47	1	no	—
229	C₂²×C₈	—	94	2	no	—
230	C₂²×C₈	—	188	1	no	—
231	C₂×C₄²	—	47	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	94	1	no	—
234	M₃₂	—	188	1	no	—
235	C₂×C₁₆	—	188	1	no	—
236	#108	—	47	1	no	—
237	#108	—	94	1	no	—
238	#108	C₄×D₄₇	1	1	yes	characteristic
239	#104	—	2	1	no	—
240	#104	—	94	1	no	—
241	#104	—	94	1	no	—
242	#104	—	94	1	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	—	47	1	no	—
245	C₄×C₈	—	47	1	no	—
246	C₄×C₈	—	47	1	no	—
247	C₄×C₈	—	94	2	no	—
248	C₄×C₈	—	94	1	no	—
249	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
250	#41	—	2	1	no	—
251	#41	—	94	1	no	—
252	#41	—	94	1	no	—
253	#41	—	94	1	no	—
254	#41	—	188	1	no	—
255	#41	—	188	1	no	—
256	C₂×Q₈	—	94	1	no	—
257	C₂×D₄	—	94	1	no	—
258	C₂×D₄	—	188	1	no	—
259	C₂²×C₄	—	47	1	no	—
260	C₂²×C₄	—	94	1	no	—
261	C₂²×C₄	—	188	1	no	—
262	Q₁₆	—	47	1	no	—
263	Q₁₆	—	94	1	no	—
264	Q₁₆	C₈×D₄₇	1	1	yes	characteristic, complemented
265	SD₁₆	—	2	1	no	—
266	SD₁₆	—	94	1	no	—
267	SD₁₆	—	94	1	no	—
268	SD₁₆	—	94	1	no	—
269	D₈	—	47	1	no	—
270	D₈	—	94	1	no	—
271	D₈	C₈×D₄₇	1	1	yes	characteristic, complemented
272	M₁₆	—	2	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	94	1	no	—
275	M₁₆	—	188	1	no	—
276	M₁₆	—	188	1	no	—
277	M₁₆	—	188	1	no	—
278	M₁₆	—	188	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	47	1	no	—
282	C₂×C₈	—	47	1	no	—
283	C₂×C₈	—	47	1	no	—
284	C₂×C₈	—	47	1	no	—
285	C₂×C₈	—	47	1	no	—
286	C₂×C₈	—	47	1	no	—
287	C₂×C₈	—	47	1	no	—
288	C₂×C₈	—	47	1	no	—
289	C₂×C₈	—	47	1	no	—
290	C₂×C₈	—	47	1	no	—
291	C₂×C₈	—	94	2	no	—
292	C₂×C₈	—	188	1	no	—
293	C₂×C₈	—	188	1	no	—
294	C₂×C₈	—	188	4	no	—
295	C₂×C₈	—	188	1	no	—
296	C₂×C₈	—	188	1	no	—
297	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
299	C₄²	—	47	1	no	—
300	C₄²	—	94	1	no	—
301	C₄²	D₄×D₄₇	1	1	yes	characteristic
302	C₁₆	—	4	1	no	—
303	C₁₆	—	188	1	no	—
304	C₂³	—	47	1	no	—
305	C₂³	—	188	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	94	1	no	—
308	Q₈	—	188	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	1	no	—
311	D₄	—	94	1	no	—
312	D₄	—	188	1	no	—
313	D₄	—	188	1	no	—
314	D₄	—	188	1	no	—
315	D₄	—	188	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	47	1	no	—
319	C₂×C₄	—	47	1	no	—
320	C₂×C₄	—	47	1	no	—
321	C₂×C₄	—	47	1	no	—
322	C₂×C₄	—	47	1	no	—
323	C₂×C₄	—	94	4	no	—
324	C₂×C₄	—	94	1	no	—
325	C₂×C₄	—	188	1	no	—
326	C₂×C₄	—	188	1	no	—
327	C₂×C₄	—	188	1	no	—
328	C₂×C₄	—	188	1	no	—
329	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
330	C₈	—	4	2	no	—
331	C₈	—	4	1	no	—
332	C₈	—	47	1	no	—
333	C₈	—	47	1	no	—
334	C₈	—	47	1	no	—
335	C₈	—	47	1	no	—
336	C₈	—	188	2	no	—
337	C₈	—	188	1	no	—
338	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
339	C₈	M₁₆×D₄₇	1	1	yes	characteristic
340	C₈	D₈×D₄₇	1	1	yes	characteristic
341	C₈	SD₁₆×D₄₇	1	1	yes	characteristic
342	C₂²	—	4	1	no	—
343	C₂²	—	47	1	no	—
344	C₂²	—	47	1	no	—
345	C₂²	—	94	2	no	—
346	C₂²	—	376	2	no	—
347	C₂²	—	188	1	no	—
348	C₂²	#104×D₄₇	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	47	1	no	—
352	C₄	—	47	1	no	—
353	C₄	—	94	2	no	—
354	C₄	—	188	1	no	—
355	C₄	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
356	C₄	#102×D₄₇	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	8	1	no	—
359	C₂	—	47	2	no	—
360	C₂	—	94	1	no	—
361	C₂	—	376	1	no	—
362	C₂	D₄⋊C₈×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#102×D₄₇
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1336	C₂
socle	C₂×C₄₇	C₂×D₄⋊C₈
prime radical	#135×D₄₇	C₄
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1336	D₄₇
hypocenter	C₄₇	C₂×#1336
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1336

Complements

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

N	H ≅ G/N	type
2. #364×D₄₇	C₂	semidirect
3. #322×D₄₇	C₂	semidirect
4. #20241365	C₂	semidirect
5. #20241364	C₂	semidirect
6. #20241363	C₂	semidirect
7. C₄₇×#1336	C₂	semidirect
20. #15688944	C₂²	semidirect
23. #15688844	C₂²	semidirect
25. C₄₇×#364	C₂²	semidirect
26. C₄₇×#322	C₂²	semidirect
28. #13094047	C₈	semidirect
32. #13094043	C₈	semidirect
33. Q₁₆×D₄₇	C₈	semidirect
35. D₈×D₄₇	C₈	semidirect
80. Q₁₆×C₄₇	C₂×C₈	semidirect
82. D₈×C₄₇	C₂×C₈	semidirect
102. Dic₁₈₈	C₂×C₈	semidirect
103. D₃₇₆	C₂×C₈	semidirect
182. #1336	D₄₇	direct (N × H)
188. D₄₇	#1336	direct (N × H)
204. #364	C₂×D₄₇	semidirect
207. #322	C₂×D₄₇	semidirect
208. C₄₇	C₂×#1336	semidirect
264. Q₁₆	C₈×D₄₇	semidirect
271. D₈	C₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#443×D₄₇	2
#364×D₄₇	2
#322×D₄₇	2
#20241365	2
#20241364	2
#20241363	2
C₄₇×#1336	2
C₂×#1336	47

Subgroups — D47×#1336 #26126067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₇×#1336 #26126067