Subgroups — #26106092

Order 12032.

542 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252018	C₂	1	1	yes	maximal, characteristic
2	#20252018	C₂	1	1	yes	maximal, characteristic
3	#431×D₄₇	C₂	1	1	yes	maximal, characteristic
4	#20241251	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241250	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241249	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1287	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
9	#15689180	C₂²	1	1	yes	characteristic
10	#15689180	C₂²	1	1	yes	characteristic
11	#15689178	—	2	1	no	—
12	#15689178	C₄	1	2	yes	—
13	#15689176	C₂²	1	1	yes	characteristic
14	#15689175	—	2	1	no	—
15	#15689175	—	2	1	no	—
16	#15689175	—	2	1	no	—
17	#15689174	C₄	1	2	yes	—
18	#15689173	—	2	1	no	—
19	#15689173	—	2	1	no	—
20	#15689173	C₄	1	2	yes	—
21	#15689145	—	2	1	no	—
22	#15689145	—	2	1	no	—
23	C₄⋊C₈×D₄₇	—	2	1	no	—
24	#15689085	—	2	1	no	—
25	#15689085	C₄	1	2	yes	—
26	C₂²⋊C₈×D₄₇	—	2	1	no	—
27	#15689075	—	2	1	no	—
28	#15689075	C₄	1	2	yes	—
29	C₄×C₈×D₄₇	C₄	1	2	yes	—
30	C₄₇×#432	C₂²	1	1	yes	characteristic
31	C₄₇×#432	C₂²	1	1	yes	characteristic
32	C₄₇×#431	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×D₄₇	Q₈	1	1	yes	characteristic
36	C₂×#10625324	—	2	2	no	—
37	C₂×#10625324	—	2	2	no	—
38	C₂×C₈×D₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₄₇	Q₈	1	1	yes	characteristic
42	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
45	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₄₇	D₄	1	1	yes	characteristic
47	C₄²×D₄₇	D₄	1	1	yes	characteristic
48	C₄²×D₄₇	D₄	1	1	yes	characteristic
49	C₄²×D₄₇	D₄	1	1	yes	characteristic
50	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
51	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
52	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
53	Dic₄₇⋊C₈	—	2	2	no	—
54	Dic₄₇⋊C₈	—	2	2	no	—
55	Dic₄₇⋊C₈	—	2	2	no	—
56	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
57	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
58	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
59	(C₂×D₄₇)⋊C₈	—	2	2	no	—
60	(C₂×D₄₇)⋊C₈	—	2	2	no	—
61	(C₂×D₄₇)⋊C₈	—	2	2	no	—
62	#13093935	—	2	1	no	—
63	#13093935	C₂×C₄	1	2	yes	—
64	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
65	C₈×Dic₄₇	—	2	2	no	—
66	C₈×Dic₄₇	—	2	2	no	—
67	C₈×Dic₄₇	—	2	2	no	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₇	—	2	1	no	—
70	C₄⋊C₈×C₄₇	—	2	1	no	—
71	C₄⋊C₈×C₄₇	—	2	1	no	—
72	C₂²⋊C₈×C₄₇	—	2	1	no	—
73	C₂²⋊C₈×C₄₇	—	2	1	no	—
74	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
75	#97×C₄₇	—	2	1	no	—
76	#97×C₄₇	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
78	C₂³×D₄₇	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₄₇	—	2	4	no	—
86	C₂×C₄×D₄₇	—	4	2	no	—
87	C₂×C₄×D₄₇	—	4	2	no	—
88	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
89	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
90	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×D₄₇	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
108	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
109	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
110	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
111	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
112	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
113	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
114	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
115	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
116	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
117	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
118	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	#10625324	—	4	4	no	—
123	#10625324	—	4	4	no	—
124	C₈×D₄₇	—	4	4	no	—
125	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×D₄₇	—	2	2	no	—
127	C₂²×D₄₇	—	4	1	no	—
128	C₂²×D₄₇	—	4	1	no	—
129	C₂²×D₄₇	#95	1	1	yes	characteristic
130	C₂²×D₄₇	#95	1	1	yes	characteristic
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	4	1	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	#95	1	1	yes	characteristic
144	C₂×Dic₄₇	#95	1	1	yes	characteristic
145	C₂×Dic₄₇	#95	1	1	yes	characteristic
146	C₂×Dic₄₇	#95	1	1	yes	characteristic
147	C₂×Dic₄₇	#95	1	1	yes	characteristic
148	C₂×Dic₄₇	#95	1	1	yes	characteristic
149	C₄×D₄₇	—	2	2	no	—
150	C₄×D₄₇	—	2	2	no	—
151	C₄×D₄₇	—	2	2	no	—
152	C₄×D₄₇	—	2	2	no	—
153	C₄×D₄₇	—	2	2	no	—
154	C₄×D₄₇	—	2	2	no	—
155	C₄×D₄₇	—	2	2	no	—
156	C₄×D₄₇	—	4	1	no	—
157	C₄×D₄₇	—	4	1	no	—
158	C₄×D₄₇	—	4	1	no	—
159	C₄×D₄₇	—	4	1	no	—
160	C₄×D₄₇	#108	1	4	yes	—
161	C₈×C₄₇	—	4	2	no	—
162	C₈×C₄₇	—	4	2	no	—
163	C₈×C₄₇	—	4	2	no	—
164	C₄₇⋊C₈	—	4	2	no	—
165	C₄₇⋊C₈	—	4	2	no	—
166	C₄₇⋊C₈	—	4	2	no	—
167	#7965	—	47	1	no	maximal, Sylow
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	—	2	1	no	—
170	C₂²×C₄₇	—	2	1	no	—
171	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
172	C₂×D₄₇	—	2	1	no	—
173	C₂×D₄₇	—	2	1	no	—
174	C₂×D₄₇	—	2	1	no	—
175	C₂×D₄₇	—	2	1	no	—
176	C₂×D₄₇	—	2	1	no	—
177	C₂×D₄₇	—	4	4	no	—
178	C₂×D₄₇	#341	1	2	yes	—
179	C₄×C₄₇	—	2	1	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	2	1	no	—
182	C₄×C₄₇	—	2	1	no	—
183	C₄×C₄₇	—	2	1	no	—
184	C₄×C₄₇	C₂×#108	1	2	yes	—
185	Dic₄₇	—	2	1	no	—
186	Dic₄₇	—	2	1	no	—
187	Dic₄₇	—	2	1	no	—
188	Dic₄₇	H(3,ℤ/4)	1	2	yes	—
189	Dic₄₇	#341	1	2	yes	—
190	Dic₄₇	#344	1	2	yes	—
191	#2920	—	47	1	no	—
192	#2920	—	47	1	no	—
193	C₂×#431	—	47	1	no	—
194	#1287	—	47	1	no	—
195	#1287	—	47	1	no	—
196	#1287	—	47	1	no	—
197	#1287	D₄₇	1	1	yes	characteristic, complemented
198	C₂×C₄₇	—	4	1	no	—
199	C₂×C₄₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
200	C₂×C₄₇	C₂×#341	1	1	yes	characteristic
201	C₂×C₄₇	C₂×#344	1	1	yes	characteristic
202	D₄₇	—	2	2	no	—
203	D₄₇	—	4	1	no	—
204	C₂²×C₄²	—	47	1	no	—
205	#432	—	47	1	no	—
206	#432	—	47	2	no	—
207	#432	—	47	1	no	—
208	#432	—	94	1	no	—
209	#432	C₂×D₄₇	1	1	yes	characteristic
210	#432	C₂×D₄₇	1	1	yes	characteristic
211	#431	—	47	1	no	—
212	#431	—	47	2	no	—
213	#431	—	47	2	no	—
214	#431	—	94	1	no	—
215	#431	—	94	1	no	—
216	#431	—	94	1	no	—
217	#431	—	94	1	no	—
218	#431	—	94	1	no	—
219	#431	C₂×D₄₇	1	1	yes	characteristic
220	#423	—	94	1	no	—
221	#423	—	94	1	no	—
222	C₂×C₄⋊C₈	—	94	1	no	—
223	#407	—	47	2	no	—
224	#407	—	94	1	no	—
225	C₂×C₂²⋊C₈	—	94	1	no	—
226	C₄×M₁₆	—	47	2	no	—
227	C₄×M₁₆	—	94	1	no	—
228	C₂×C₄×C₈	—	47	2	no	—
229	C₄₇	#7965	1	1	yes	characteristic, complemented, Sylow
230	C₂³×C₄	—	47	1	no	—
231	C₂³×C₄	—	47	1	no	—
232	C₂³×C₄	—	47	1	no	—
233	C₂×M₁₆	—	94	2	no	—
234	C₂×M₁₆	—	94	2	no	—
235	C₂²×C₈	—	94	2	no	—
236	C₂×C₄²	—	47	1	no	—
237	C₂×C₄²	—	47	1	no	—
238	C₂×C₄²	—	47	1	no	—
239	C₂×C₄²	—	47	1	no	—
240	C₂×C₄²	—	47	1	no	—
241	C₂×C₄²	—	47	1	no	—
242	C₂×C₄²	—	47	1	no	—
243	C₂×C₄²	—	47	1	no	—
244	C₂×C₄²	—	47	1	no	—
245	C₂×C₄²	—	47	1	no	—
246	C₂×C₄²	—	47	1	no	—
247	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	94	2	no	—
252	C₄⋊C₈	—	94	2	no	—
253	C₄⋊C₈	—	94	2	no	—
254	C₄⋊C₈	—	94	1	no	—
255	C₄⋊C₈	—	94	1	no	—
256	C₄⋊C₈	—	94	1	no	—
257	C₂²⋊C₈	—	2	1	no	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	47	2	no	—
260	C₂²⋊C₈	—	94	2	no	—
261	C₂²⋊C₈	—	94	2	no	—
262	C₂²⋊C₈	—	94	2	no	—
263	C₂²⋊C₈	—	94	1	no	—
264	C₂²⋊C₈	—	94	1	no	—
265	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
266	#97	—	2	1	no	—
267	#97	—	47	2	no	—
268	#97	—	94	1	no	—
269	#97	C₄×D₄₇	1	2	yes	—
270	C₄×C₈	—	47	2	no	—
271	C₄×C₈	—	94	2	no	—
272	C₄×C₈	—	94	2	no	—
273	C₄×C₈	—	94	2	no	—
274	C₄×C₈	C₄×D₄₇	1	2	yes	—
275	C₂⁴	—	47	1	no	—
276	C₂²×C₄	—	47	1	no	—
277	C₂²×C₄	—	47	1	no	—
278	C₂²×C₄	—	47	1	no	—
279	C₂²×C₄	—	47	1	no	—
280	C₂²×C₄	—	47	1	no	—
281	C₂²×C₄	—	47	1	no	—
282	C₂²×C₄	—	47	1	no	—
283	C₂²×C₄	—	47	1	no	—
284	C₂²×C₄	—	47	1	no	—
285	C₂²×C₄	—	47	1	no	—
286	C₂²×C₄	—	47	1	no	—
287	C₂²×C₄	—	47	1	no	—
288	C₂²×C₄	—	47	1	no	—
289	C₂²×C₄	—	47	1	no	—
290	C₂²×C₄	—	47	1	no	—
291	C₂²×C₄	—	94	4	no	—
292	C₂²×C₄	—	188	2	no	—
293	C₂²×C₄	—	188	2	no	—
294	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
295	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
296	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
297	M₁₆	—	188	4	no	—
298	M₁₆	—	188	4	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	94	2	no	—
303	C₂×C₈	—	94	2	no	—
304	C₂×C₈	—	94	2	no	—
305	C₂×C₈	—	188	4	no	—
306	C₄²	—	47	1	no	—
307	C₄²	—	47	1	no	—
308	C₄²	—	47	1	no	—
309	C₄²	—	47	1	no	—
310	C₄²	—	47	1	no	—
311	C₄²	—	47	1	no	—
312	C₄²	—	47	1	no	—
313	C₄²	—	47	1	no	—
314	C₄²	—	47	1	no	—
315	C₄²	—	47	1	no	—
316	C₄²	—	47	1	no	—
317	C₄²	—	47	1	no	—
318	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
319	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
320	C₄²	D₄×D₄₇	1	1	yes	characteristic
321	C₄²	D₄×D₄₇	1	1	yes	characteristic
322	C₂³	—	47	1	no	—
323	C₂³	—	47	1	no	—
324	C₂³	—	94	2	no	—
325	C₂³	—	188	1	no	—
326	C₂³	—	188	1	no	—
327	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	47	1	no	—
332	C₂×C₄	—	47	1	no	—
333	C₂×C₄	—	47	1	no	—
334	C₂×C₄	—	47	4	no	—
335	C₂×C₄	—	47	1	no	—
336	C₂×C₄	—	47	1	no	—
337	C₂×C₄	—	47	1	no	—
338	C₂×C₄	—	94	2	no	—
339	C₂×C₄	—	94	2	no	—
340	C₂×C₄	—	94	2	no	—
341	C₂×C₄	—	94	2	no	—
342	C₂×C₄	—	94	2	no	—
343	C₂×C₄	—	94	2	no	—
344	C₂×C₄	—	94	2	no	—
345	C₂×C₄	—	94	2	no	—
346	C₂×C₄	—	94	2	no	—
347	C₂×C₄	—	94	2	no	—
348	C₂×C₄	—	188	1	no	—
349	C₂×C₄	—	188	1	no	—
350	C₂×C₄	—	188	1	no	—
351	C₂×C₄	—	188	1	no	—
352	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
353	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
354	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
355	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
356	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
357	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
358	C₈	—	4	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	4	2	no	—
361	C₈	—	188	2	no	—
362	C₈	—	188	2	no	—
363	C₈	—	188	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	47	2	no	—
368	C₂²	—	94	1	no	—
369	C₂²	—	94	1	no	—
370	C₂²	—	94	1	no	—
371	C₂²	—	94	1	no	—
372	C₂²	—	94	1	no	—
373	C₂²	—	188	4	no	—
374	C₂²	#95×D₄₇	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	47	2	no	—
381	C₄	—	47	2	no	—
382	C₄	—	47	2	no	—
383	C₄	—	94	1	no	—
384	C₄	—	94	1	no	—
385	C₄	—	94	1	no	—
386	C₄	#108×D₄₇	1	2	yes	—
387	C₂	—	4	1	no	—
388	C₂	—	94	2	no	—
389	C₂	—	188	1	no	—
390	C₂	H(3,ℤ/4)×D₄₇	1	1	yes	characteristic
391	C₂	#341×D₄₇	1	1	yes	characteristic
392	C₂	#344×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#95×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1287	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂³×D₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1287	D₄₇
hypocenter	C₄₇	#7965
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1287

Complements

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

N	H ≅ G/N	type
4. #20241251	C₂	semidirect
5. #20241250	C₂	semidirect
6. #20241249	C₂	semidirect
7. C₄₇×#1287	C₂	semidirect
197. #1287	D₄₇	semidirect
229. C₄₇	#7965	semidirect

Maximal subgroups

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

subgroup	index
#20252018	2
#431×D₄₇	2
#20241251	2
#20241250	2
#20241249	2
C₄₇×#1287	2
#7965	47