Subgroups — #26116342

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251740	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20241267	C₂	1	4	yes	maximal
4	C₄₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic, complemented
6	C₂×#13094003	—	2	1	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15689029	—	2	1	no	—
10	#15689021	—	2	1	no	—
11	#15689021	C₄	1	2	yes	complemented
12	#15688993	C₄	1	1	yes	characteristic, complemented
13	#15688983	—	2	1	no	—
14	#15688983	C₂²	1	2	yes	—
15	#15688878	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#95×C₄₇	—	2	1	no	—
19	C₂×#95×C₄₇	C₂²	1	2	yes	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	#13094024	—	4	2	no	—
23	#13094003	—	4	2	no	—
24	#13094003	—	4	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₄₇⋊C₄	—	2	2	no	—
31	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
32	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
33	C₂²⋊(C₄₇⋊C₈)	Q₈	1	4	yes	complemented
34	C₄×C₄₇⋊C₈	—	4	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₂³×C₄×C₄₇	—	2	2	no	—
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
44	C₂×C₄²×C₄₇	—	4	2	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	#95×C₄₇	—	2	4	no	—
50	#95×C₄₇	—	4	2	no	—
51	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	4	1	no	—
56	C₂²×C₄×C₄₇	—	4	1	no	—
57	C₂²×C₄×C₄₇	—	4	1	no	—
58	C₂²×C₄×C₄₇	—	4	1	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	D₈	1	2	yes	—
70	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₄₇	#41	1	2	yes	—
77	C₂²×Dic₄₇	—	2	2	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂×Dic₉₄	—	4	2	no	—
80	C₂×Dic₉₄	—	4	2	no	—
81	C₂×Dic₉₄	—	4	2	no	—
82	C₄⋊C₄×C₄₇	—	4	8	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	4	1	no	—
86	C₄²×C₄₇	—	4	1	no	—
87	C₂²⋊Dic₄₇	—	4	2	no	—
88	C₂²⋊Dic₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	4	2	no	—
90	C₄⋊Dic₄₇	—	4	2	no	—
91	C₄⋊Dic₄₇	—	4	2	no	—
92	C₄⋊Dic₄₇	—	4	2	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	Dic₄₇⋊C₄	—	4	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	—	2	1	no	—
101	C₂³×C₄₇	—	2	1	no	—
102	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₄₇	#124	1	1	yes	characteristic
104	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
105	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
106	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	—	4	4	no	—
127	C₂×C₄×C₄₇	—	4	4	no	—
128	C₂×C₄×C₄₇	—	8	2	no	—
129	C₂×C₄×C₄₇	—	8	2	no	—
130	C₂×C₄×C₄₇	—	8	2	no	—
131	C₂×C₄×C₄₇	—	8	2	no	—
132	C₂×C₄×C₄₇	#99	1	2	yes	—
133	C₂×C₄×C₄₇	#102	1	2	yes	—
134	C₂×C₄×C₄₇	#102	1	2	yes	—
135	C₂×C₄×C₄₇	#103	1	2	yes	—
136	C₂×C₄×C₄₇	#103	1	2	yes	—
137	C₂×C₄×C₄₇	#104	1	2	yes	—
138	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	Dic₉₄	—	8	4	no	—
147	C₄₇⋊C₈	—	4	4	no	—
148	#9884	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	—	2	1	no	—
153	C₂²×C₄₇	—	2	1	no	—
154	C₂²×C₄₇	—	2	1	no	—
155	C₂²×C₄₇	—	2	1	no	—
156	C₂²×C₄₇	—	2	1	no	—
157	C₂²×C₄₇	—	2	1	no	—
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	#391	1	1	yes	characteristic
162	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
163	C₂²×C₄₇	#410	1	1	yes	characteristic
164	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
165	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
166	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
167	C₂²×C₄₇	#421	1	1	yes	characteristic
168	C₂²×C₄₇	#486	1	1	yes	characteristic
169	C₂²×C₄₇	#487	1	1	yes	characteristic
170	C₂²×C₄₇	#498	1	1	yes	characteristic
171	C₂²×C₄₇	#500	1	1	yes	characteristic
172	C₄×C₄₇	—	2	2	no	—
173	C₄×C₄₇	—	2	2	no	—
174	C₄×C₄₇	—	2	2	no	—
175	C₄×C₄₇	—	2	2	no	—
176	C₄×C₄₇	—	2	2	no	—
177	C₄×C₄₇	—	8	4	no	—
178	C₄×C₄₇	#328	1	4	yes	—
179	Dic₄₇	—	8	2	no	—
180	Dic₄₇	—	8	2	no	—
181	#2857	—	47	1	no	—
182	C₄×C₂²⋊C₈	—	47	1	no	—
183	C₄⋊#95	D₄₇	1	1	yes	characteristic
184	#1294	—	47	4	no	—
185	C₂×C₄₇	—	2	1	no	—
186	C₂×C₄₇	—	2	1	no	—
187	C₂×C₄₇	—	2	1	no	—
188	C₂×C₄₇	—	2	1	no	—
189	C₂×C₄₇	C₂×#328	1	1	yes	characteristic
190	C₂×C₄₇	#1502	1	1	yes	characteristic
191	C₂×C₄₇	#1518	1	1	yes	characteristic
192	C₂×C₄₇	#1519	1	1	yes	characteristic
193	C₂×C₄₇	#1986	1	1	yes	characteristic
194	C₂×C₄₇	#1990	1	1	yes	characteristic
195	C₂×C₄₇	#1996	1	1	yes	characteristic
196	C₂×C₄⋊Q₈	—	47	1	no	—
197	C₂×C₂²⋊Q₈	—	94	1	no	—
198	C₂²×C₄⋊C₄	—	2	1	no	—
199	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
200	C₂×C₂²⋊C₈	—	47	2	no	—
201	C₂×C₄×C₈	—	94	1	no	—
202	#391	—	94	1	no	—
203	#389	—	47	2	no	—
204	#389	—	94	1	no	—
205	#384	—	47	1	no	—
206	#380	—	47	2	no	—
207	#380	—	94	1	no	—
208	C₂×#95	—	2	1	no	—
209	C₂×#95	C₂×D₄₇	1	2	yes	—
210	#336	—	94	1	no	—
211	C₄₇	#9884	1	1	yes	characteristic, complemented, Sylow
212	C₂²×Q₈	—	94	1	no	—
213	C₂³×C₄	—	2	2	no	—
214	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
215	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
217	C₂²×C₈	—	94	2	no	—
218	C₄⋊Q₈	—	188	2	no	—
219	C₂²⋊Q₈	—	188	2	no	—
220	C₂²⋊Q₈	—	188	2	no	—
221	C₂×C₄⋊C₄	—	2	4	no	—
222	C₂×C₄⋊C₄	—	4	2	no	—
223	C₂×C₄⋊C₄	—	4	2	no	—
224	C₂×C₄⋊C₄	—	47	2	no	—
225	C₂×C₄⋊C₄	—	47	2	no	—
226	C₂×C₄⋊C₄	—	94	2	no	—
227	C₂×C₄⋊C₄	—	94	1	no	—
228	C₂×C₄⋊C₄	—	94	1	no	—
229	C₂×C₄⋊C₄	—	94	1	no	—
230	C₂×C₂²⋊C₄	—	94	2	no	—
231	C₂×C₄²	—	4	2	no	—
232	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
234	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
235	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
236	C₂²⋊C₈	—	47	4	no	—
237	C₂²⋊C₈	—	94	2	no	—
238	C₄×C₈	—	188	2	no	—
239	#95	—	2	4	no	—
240	#95	—	4	2	no	—
241	#95	—	94	2	no	—
242	#95	—	94	2	no	—
243	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
244	C₂×Q₈	—	188	2	no	—
245	C₂×Q₈	—	188	2	no	—
246	C₂×Q₈	—	188	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	94	2	no	—
261	C₂²×C₄	—	94	2	no	—
262	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
263	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
264	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
265	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
266	C₂²×C₄	#10625334	1	2	yes	—
267	C₂²×C₄	#10625335	1	2	yes	—
268	C₂²×C₄	#10625336	1	2	yes	—
269	C₂²×C₄	#10625337	1	2	yes	—
270	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
272	C₂²×C₄	#10625352	1	2	yes	—
273	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
274	C₂×C₈	—	94	4	no	—
275	C₂×C₈	—	188	2	no	—
276	C₂×C₈	—	188	2	no	—
277	C₄⋊C₄	—	4	8	no	—
278	C₄⋊C₄	—	188	2	no	—
279	C₄⋊C₄	—	188	2	no	—
280	C₄⋊C₄	—	188	2	no	—
281	C₄⋊C₄	—	188	4	no	—
282	C₄⋊C₄	—	188	2	no	—
283	C₄⋊C₄	—	188	2	no	—
284	C₂²⋊C₄	—	188	2	no	—
285	C₂²⋊C₄	—	188	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	4	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
295	C₂³	#13094011	1	1	yes	characteristic
296	C₂³	#13094022	1	1	yes	characteristic
297	C₂³	C₂×#10625334	1	1	yes	characteristic
298	C₂³	C₂×#10625336	1	1	yes	characteristic
299	C₂³	C₂×#10625335	1	1	yes	characteristic
300	C₂³	C₂×#10625337	1	1	yes	characteristic
301	Q₈	—	376	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	—	188	2	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	#13093938	1	2	yes	—
332	C₂×C₄	#13093944	1	2	yes	—
333	C₂×C₄	#13093944	1	2	yes	—
334	C₂×C₄	#13093947	1	2	yes	—
335	C₂×C₄	#13093947	1	2	yes	—
336	C₂×C₄	#13093952	1	2	yes	—
337	C₂×C₄	#13093989	1	2	yes	—
338	C₂×C₄	#13094004	1	2	yes	—
339	C₈	—	188	4	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	#15689028	1	1	yes	characteristic
353	C₂²	C₂×#13093938	1	1	yes	characteristic
354	C₂²	#15689096	1	1	yes	characteristic
355	C₂²	C₂×#13093944	1	1	yes	characteristic
356	C₂²	C₂×#13093947	1	1	yes	characteristic
357	C₂²	C₂×#13093952	1	1	yes	characteristic
358	C₂²	#15689141	1	1	yes	characteristic
359	C₂²	#15689469	1	1	yes	characteristic
360	C₂²	#15689473	1	1	yes	characteristic
361	C₂²	#15689519	1	1	yes	characteristic
362	C₂²	#15689526	1	1	yes	characteristic
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	2	no	—
368	C₄	—	8	4	no	—
369	C₄	—	376	2	no	—
370	C₄	—	376	2	no	—
371	C₄	#15688859	1	4	yes	—
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	C₂×#15688859	1	1	yes	characteristic
377	C₂	#20241882	1	1	yes	characteristic
378	C₂	#20241951	1	1	yes	characteristic
379	C₂	#20241955	1	1	yes	characteristic
380	C₂	#20244331	1	1	yes	characteristic
381	C₂	#20244348	1	1	yes	characteristic
382	C₂	#20244372	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×C₄⋊#95	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₄⋊#95	D₄₇
hypocenter	C₄₇	#9884
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄⋊#95

Complements

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

N	H ≅ G/N	type
5. C₂×#13094024	C₄	semidirect
11. #15689021	C₄	semidirect
12. #15688993	C₄	semidirect
33. C₂²⋊(C₄₇⋊C₈)	Q₈	semidirect
211. C₄₇	#9884	semidirect

Maximal subgroups

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

subgroup	index
#20251740	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
#20241267	2
C₄₇×C₄⋊#95	2
#9884	47