Subgroups — #26204256

Order 12032.

878 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254325	C₂	1	2	yes	maximal, complemented
2	#20253975	C₂	1	1	yes	maximal, characteristic, complemented
3	#493×D₄₇	C₂	1	1	yes	maximal, characteristic
4	#20253759	C₂	1	2	yes	maximal, complemented
5	#20243028	C₂	1	2	yes	maximal, complemented
6	#20243025	C₂	1	2	yes	maximal, complemented
7	#20242994	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242993	C₂	1	1	yes	maximal, characteristic, complemented
9	#20242981	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10625324	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₄₇	—	2	2	no	—
13	#15689678	C₂²	1	2	yes	—
14	#127×D₄₇	C₂²	1	2	yes	—
15	#15689503	C₂²	1	1	yes	characteristic, complemented
16	#15689500	C₂²	1	1	yes	characteristic
17	#15689499	C₂²	1	2	yes	—
18	#15689498	—	2	2	no	—
19	#15689497	C₂²	1	2	yes	complemented
20	#15689486	—	2	4	no	—
21	#15689475	C₂²	1	2	yes	complemented
22	#15689471	C₂²	1	2	yes	complemented
23	#15689470	C₂²	1	2	yes	complemented
24	#15689470	C₂²	1	2	yes	complemented
25	#15689469	C₂²	1	2	yes	complemented
26	#15689469	C₂²	1	2	yes	complemented
27	#15689365	—	2	2	no	—
28	#15689321	—	2	4	no	—
29	#15689272	—	2	4	no	—
30	#15689217	—	2	2	no	—
31	#15689174	C₂²	1	1	yes	characteristic
32	#15689126	—	2	4	no	—
33	C₄×#10625324	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
35	#15688845	C₂²	1	1	yes	characteristic, complemented
36	#15688844	C₂²	1	1	yes	characteristic, complemented
37	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	C₄₇×#500	C₂²	1	1	yes	characteristic, complemented
39	C₄₇×#493	C₂²	1	1	yes	characteristic
40	C₄₇×#492	C₂²	1	2	yes	complemented
41	C₄₇×#486	C₂²	1	2	yes	complemented
42	C₈²×C₄₇	C₂²	1	1	yes	characteristic, complemented
43	C₂×D₄×D₄₇	—	2	2	no	—
44	C₂×D₄×D₄₇	—	2	2	no	—
45	C₂×D₄×D₄₇	—	2	2	no	—
46	C₂×Dic₁₈₈	—	2	2	no	—
47	C₂×#10625336	—	2	4	no	—
48	C₂×D₃₇₆	—	2	2	no	—
49	C₂×#10625334	—	2	4	no	—
50	D₈×D₄₇	—	4	8	no	—
51	C₂×#10625324	D₄	1	2	yes	—
52	C₂×C₈×D₄₇	D₄	1	2	yes	—
53	#13094024	C₂³	1	2	yes	—
54	#13094021	C₂³	1	2	yes	—
55	#13094020	—	2	2	no	—
56	#13094020	—	2	2	no	—
57	#13094020	—	2	2	no	—
58	#13094015	—	2	2	no	—
59	#13094004	—	2	2	no	—
60	#13094000	—	2	2	no	—
61	Q₈×Dic₄₇	—	2	2	no	—
62	#13093989	—	2	2	no	—
63	#13093982	—	2	2	no	—
64	C₄²×D₄₇	C₂³	1	1	yes	characteristic
65	Dic₄₇⋊C₈	D₄	1	2	yes	—
66	#13093947	—	2	4	no	—
67	#13093946	—	2	4	no	—
68	#13093946	—	2	4	no	—
69	#13093945	—	2	4	no	—
70	#13093945	—	2	4	no	—
71	#13093944	—	2	4	no	—
72	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
73	#13093934	D₄	1	2	yes	—
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₄₇	D₄	1	2	yes	—
78	C₂×SD₁₆×C₄₇	—	2	4	no	—
79	C₂×D₈×C₄₇	—	2	2	no	—
80	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
81	#127×C₄₇	C₂³	1	2	yes	—
82	#107×C₄₇	—	2	2	no	—
83	#102×C₄₇	—	2	4	no	—
84	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
87	C₂³×D₄₇	—	4	2	no	—
88	C₂×Q₈×C₄₇	—	2	2	no	—
89	C₂×D₄×C₄₇	—	2	2	no	—
90	C₂×D₄×C₄₇	—	2	2	no	—
91	C₂×D₄×C₄₇	—	2	2	no	—
92	C₂×C₂²⋊D₄₇	—	4	2	no	—
93	C₂×C₂²⋊D₄₇	—	4	2	no	—
94	C₂×C₂²⋊D₄₇	—	4	2	no	—
95	D₄×D₄₇	—	4	8	no	—
96	D₄×D₄₇	—	4	8	no	—
97	D₄×D₄₇	—	8	4	no	—
98	C₂×D₁₈₈	—	2	2	no	—
99	C₂×D₁₈₈	—	2	2	no	—
100	C₂×D₁₈₈	—	2	2	no	—
101	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₉₄	—	2	2	no	—
105	SD₁₆×C₄₇	—	8	4	no	—
106	D₈×C₄₇	—	4	4	no	—
107	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₄⋊C₄×C₄₇	—	2	2	no	—
112	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
113	#10625336	—	8	4	no	—
114	#10625334	—	4	8	no	—
115	(C₂×D₄₇)⋊C₄	—	4	2	no	—
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	—	2	2	no	—
118	C₄⋊Dic₄₇	—	4	2	no	—
119	Dic₄₇⋊C₄	—	4	2	no	—
120	C₄×Dic₄₇	—	4	2	no	—
121	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
123	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
125	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
126	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
127	Dic₁₈₈	—	4	4	no	—
128	D₃₇₆	—	4	4	no	—
129	#10625324	—	2	4	no	—
130	C₈×D₄₇	—	2	4	no	—
131	C₂³×C₄₇	—	4	2	no	—
132	C₂²×D₄₇	—	4	2	no	—
133	C₂²×D₄₇	—	8	4	no	—
134	C₂²×D₄₇	—	8	4	no	—
135	C₂²×D₄₇	—	8	4	no	—
136	C₂²×D₄₇	#127	1	1	yes	characteristic
137	Q₈×C₄₇	—	4	4	no	—
138	D₄×C₄₇	—	4	4	no	—
139	D₄×C₄₇	—	4	4	no	—
140	D₄×C₄₇	—	4	4	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₄₇	—	8	4	no	—
146	C₂²⋊D₄₇	—	8	4	no	—
147	C₂²⋊D₄₇	—	8	4	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	#127	1	1	yes	characteristic
150	C₂×Dic₄₇	#127	1	1	yes	characteristic
151	C₂×Dic₄₇	#127	1	1	yes	characteristic
152	D₁₈₈	—	4	4	no	—
153	D₁₈₈	—	4	4	no	—
154	D₁₈₈	—	4	4	no	—
155	C₄×D₄₇	—	2	2	no	—
156	C₄×D₄₇	—	2	2	no	—
157	C₄×D₄₇	#135	1	4	yes	—
158	Dic₉₄	—	4	4	no	—
159	C₈×C₄₇	—	2	2	no	—
160	C₈×C₄₇	—	2	2	no	—
161	C₈×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
162	C₄₇⋊C₈	—	2	2	no	—
163	C₄₇⋊C₈	—	2	2	no	—
164	C₄₇⋊C₈	—	2	2	no	—
165	#19635	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	—	8	2	no	—
167	C₂²×C₄₇	—	8	2	no	—
168	C₂²×C₄₇	—	8	2	no	—
169	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
170	C₂×D₄₇	—	2	1	no	—
171	C₂×D₄₇	—	2	1	no	—
172	C₂×D₄₇	—	8	2	no	—
173	C₂×D₄₇	—	8	2	no	—
174	C₂×D₄₇	—	8	2	no	—
175	C₂×D₄₇	—	16	8	no	—
176	C₂×D₄₇	#495	1	2	yes	—
177	C₄×C₄₇	—	8	2	no	—
178	C₄×C₄₇	C₂×#135	1	2	yes	—
179	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
180	C₄×C₄₇	#576	1	2	yes	—
181	Dic₄₇	—	2	1	no	—
182	Dic₄₇	—	2	1	no	—
183	Dic₄₇	—	8	2	no	—
184	Dic₄₇	#495	1	2	yes	—
185	#3208	—	47	2	no	—
186	#3170	—	47	1	no	—
187	C₂×#493	—	47	1	no	—
188	#3138	—	47	2	no	—
189	#1717	—	47	2	no	—
190	#1717	—	47	2	no	—
191	#1709	—	47	1	no	—
192	#1709	—	47	1	no	—
193	#1706	—	47	1	no	—
194	#1706	D₄₇	1	1	yes	characteristic, complemented
195	C₈×M₁₆	—	47	1	no	—
196	C₂×C₄₇	—	16	2	no	—
197	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
198	C₂×C₄₇	#3151	1	1	yes	characteristic
199	C₂×C₄₇	#3155	1	1	yes	characteristic
200	D₄₇	—	2	2	no	—
201	D₄₇	—	16	2	no	—
202	C₂²×D₈	—	94	2	no	—
203	#533	—	47	2	no	—
204	C₂×#127	—	47	2	no	—
205	#500	C₂×D₄₇	1	1	yes	characteristic, complemented
206	#494	—	47	1	no	—
207	#493	—	47	2	no	—
208	#493	—	47	1	no	—
209	#493	—	94	2	no	—
210	#493	C₂×D₄₇	1	1	yes	characteristic
211	#492	—	47	2	no	—
212	#492	C₂×D₄₇	1	2	yes	complemented
213	#490	—	94	4	no	—
214	#487	—	47	2	no	—
215	#486	—	47	2	no	—
216	#486	—	47	2	no	—
217	#486	—	47	2	no	—
218	#486	—	47	2	no	—
219	#486	—	47	2	no	—
220	#486	C₂×D₄₇	1	2	yes	complemented
221	#467	—	94	2	no	—
222	#460	—	94	4	no	—
223	#450	—	94	4	no	—
224	C₄×Q₁₆	—	94	2	no	—
225	#431	—	47	1	no	—
226	#418	—	94	4	no	—
227	C₄×M₁₆	—	47	1	no	—
228	C₂×C₄×C₈	—	47	1	no	—
229	#322	—	47	1	no	—
230	#322	—	47	1	no	—
231	C₈²	—	47	1	no	—
232	C₈²	C₂×D₄₇	1	1	yes	characteristic, complemented
233	C₄₇	#19635	1	1	yes	characteristic, complemented, Sylow
234	C₂²×D₄	—	94	2	no	—
235	C₂²×D₄	—	94	2	no	—
236	C₂²×D₄	—	94	2	no	—
237	C₂×Q₁₆	—	94	2	no	—
238	C₂×SD₁₆	—	2	4	no	—
239	C₂×SD₁₆	—	94	4	no	—
240	C₂×D₈	—	2	2	no	—
241	C₂×D₈	—	94	4	no	—
242	C₂×D₈	—	94	2	no	—
243	C₂×D₈	—	188	8	no	—
244	C₂×M₁₆	—	47	2	no	—
245	C₂²×C₈	—	47	2	no	—
246	C₄⋊Q₈	—	47	2	no	—
247	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
248	#127	—	47	2	no	—
249	#127	—	94	2	no	—
250	#127	—	94	2	no	—
251	#127	—	94	2	no	—
252	#127	C₂²×D₄₇	1	2	yes	—
253	#125	—	94	2	no	—
254	C₂²⋊Q₈	—	94	2	no	—
255	C₂²⋊Q₈	—	94	2	no	—
256	C₄×Q₈	—	94	2	no	—
257	C₄×Q₈	—	94	2	no	—
258	#117	—	94	2	no	—
259	C₂×C₄²	—	47	1	no	—
260	#107	—	2	2	no	—
261	C₄⋊C₈	—	47	2	no	—
262	#103	—	94	4	no	—
263	#102	—	2	4	no	—
264	#102	—	94	4	no	—
265	#102	—	94	4	no	—
266	#102	—	94	4	no	—
267	#102	—	94	4	no	—
268	#102	—	94	4	no	—
269	C₂²⋊C₈	—	47	2	no	—
270	#97	—	47	2	no	—
271	C₄×C₈	—	47	2	no	—
272	C₄×C₈	—	47	1	no	—
273	C₄×C₈	—	47	1	no	—
274	C₄×C₈	—	47	1	no	—
275	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
278	C₂⁴	—	188	2	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	94	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	94	2	no	—
285	C₂×D₄	—	94	2	no	—
286	C₂×D₄	—	94	2	no	—
287	C₂×D₄	—	376	4	no	—
288	C₂×D₄	—	188	8	no	—
289	C₂×D₄	—	188	2	no	—
290	C₂×D₄	—	188	2	no	—
291	C₂×D₄	—	188	2	no	—
292	C₂×D₄	—	188	8	no	—
293	C₂²×C₄	—	47	1	no	—
294	C₂²×C₄	—	47	1	no	—
295	C₂²×C₄	—	47	1	no	—
296	Q₁₆	—	188	4	no	—
297	SD₁₆	—	8	4	no	—
298	SD₁₆	—	376	4	no	—
299	D₈	—	4	4	no	—
300	D₈	—	188	8	no	—
301	D₈	—	188	4	no	—
302	M₁₆	—	94	4	no	—
303	C₂×C₈	—	47	2	no	—
304	C₂×C₈	—	47	2	no	—
305	C₂×C₈	—	47	2	no	—
306	C₂×C₈	—	94	4	no	—
307	C₂×C₈	D₄×D₄₇	1	2	yes	—
308	C₂×C₈	D₄×D₄₇	1	2	yes	—
309	C₂×C₈	D₄×D₄₇	1	2	yes	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	94	2	no	—
313	C₄⋊C₄	—	94	2	no	—
314	C₄⋊C₄	—	188	2	no	—
315	C₄⋊C₄	—	188	2	no	—
316	C₂²⋊C₄	—	188	2	no	—
317	C₄²	—	47	1	no	—
318	C₄²	—	47	1	no	—
319	C₄²	—	47	1	no	—
320	C₄²	—	188	2	no	—
321	C₄²	C₂³×D₄₇	1	1	yes	characteristic
322	C₂³	—	4	2	no	—
323	C₂³	—	47	1	no	—
324	C₂³	—	376	4	no	—
325	C₂³	—	376	4	no	—
326	C₂³	—	376	4	no	—
327	C₂³	—	188	2	no	—
328	Q₈	—	4	4	no	—
329	Q₈	—	188	4	no	—
330	D₄	—	4	4	no	—
331	D₄	—	4	4	no	—
332	D₄	—	4	4	no	—
333	D₄	—	376	4	no	—
334	D₄	—	376	4	no	—
335	D₄	—	376	4	no	—
336	D₄	—	188	4	no	—
337	D₄	—	188	4	no	—
338	D₄	—	188	4	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	47	1	no	—
341	C₂×C₄	—	47	4	no	—
342	C₂×C₄	—	47	1	no	—
343	C₂×C₄	—	47	1	no	—
344	C₂×C₄	—	94	2	no	—
345	C₂×C₄	—	94	2	no	—
346	C₂×C₄	—	188	2	no	—
347	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
349	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
350	C₈	—	2	2	no	—
351	C₈	—	2	2	no	—
352	C₈	—	94	2	no	—
353	C₈	—	94	2	no	—
354	C₈	—	94	2	no	—
355	C₈	#13094051	1	4	yes	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	47	2	no	—
360	C₂²	—	94	1	no	—
361	C₂²	—	94	1	no	—
362	C₂²	—	376	2	no	—
363	C₂²	—	376	2	no	—
364	C₂²	—	376	2	no	—
365	C₂²	—	752	8	no	—
366	C₂²	#127×D₄₇	1	1	yes	characteristic
367	C₄	—	8	2	no	—
368	C₄	—	47	2	no	—
369	C₄	—	94	1	no	—
370	C₄	—	94	1	no	—
371	C₄	—	376	2	no	—
372	C₄	#135×D₄₇	1	2	yes	—
373	C₄	C₂×#13094051	1	2	yes	—
374	C₄	#15689924	1	2	yes	—
375	C₂	—	16	2	no	—
376	C₂	—	94	2	no	—
377	C₂	—	752	2	no	—
378	C₂	#495×D₄₇	1	1	yes	characteristic
379	C₂	#20253835	1	1	yes	characteristic
380	C₂	#20253868	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₄₇
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1706	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	#493×D₄₇	C₂
hypercenter	#1706	D₄₇
hypocenter	C₄₇	#19635
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1706

Complements

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

N	H ≅ G/N	type
1. #20254325	C₂	semidirect
2. #20253975	C₂	semidirect
4. #20253759	C₂	semidirect
5. #20243028	C₂	semidirect
6. #20243025	C₂	semidirect
7. #20242994	C₂	semidirect
8. #20242993	C₂	semidirect
9. #20242981	C₂	semidirect
10. C₈×#10625324	C₂	semidirect
11. C₄₇×#1706	C₂	semidirect
15. #15689503	C₂²	semidirect
19. #15689497	C₂²	semidirect
21. #15689475	C₂²	semidirect
22. #15689471	C₂²	semidirect
23. #15689470	C₂²	semidirect
24. #15689470	C₂²	semidirect
25. #15689469	C₂²	semidirect
26. #15689469	C₂²	semidirect
35. #15688845	C₂²	semidirect
36. #15688844	C₂²	semidirect
37. C₈×C₄₇⋊C₈	C₂²	semidirect
38. C₄₇×#500	C₂²	semidirect
40. C₄₇×#492	C₂²	semidirect
41. C₄₇×#486	C₂²	semidirect
42. C₈²×C₄₇	C₂²	semidirect
194. #1706	D₄₇	semidirect
205. #500	C₂×D₄₇	semidirect
212. #492	C₂×D₄₇	semidirect
220. #486	C₂×D₄₇	semidirect
232. C₈²	C₂×D₄₇	semidirect
233. C₄₇	#19635	semidirect

Maximal subgroups

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

subgroup	index
#20254325	2
#20253975	2
#493×D₄₇	2
#20253759	2
#20243028	2
#20243025	2
#20242994	2
#20242993	2
#20242981	2
C₈×#10625324	2
C₄₇×#1706	2
#19635	47