Subgroups — #26242872

Order 12032.

636 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094024	C₂	1	1	yes	maximal, characteristic
2	#20244326	C₂	1	2	yes	maximal
3	#20244322	C₂	1	2	yes	maximal
4	#20243953	C₂	1	2	yes	maximal
5	#20243951	C₂	1	2	yes	maximal
6	#20243946	C₂	1	2	yes	maximal
7	#20243944	C₂	1	2	yes	maximal
8	#20243531	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1839	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094024	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
14	C₂×#13093934	C₂²	1	2	yes	—
15	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
16	#15689029	C₄	1	2	yes	—
17	#15689021	C₄	1	2	yes	—
18	#15689021	C₄	1	1	yes	characteristic
19	#15689021	C₄	1	1	yes	characteristic
20	#15688993	C₄	1	1	yes	characteristic
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
24	#15688878	C₂²	1	2	yes	—
25	#15688878	C₂²	1	1	yes	characteristic
26	#15688878	C₂²	1	2	yes	—
27	#15688878	C₂²	1	1	yes	characteristic
28	#15688877	C₂²	1	2	yes	—
29	#15688877	C₂²	1	2	yes	—
30	#15688877	C₂²	1	2	yes	—
31	#15688877	C₂²	1	2	yes	—
32	C₄³×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#336	C₂²	1	1	yes	characteristic
34	C₄₇×#336	C₂²	1	1	yes	characteristic
35	C₄₇×#336	C₂²	1	2	yes	—
36	C₄₇×#336	C₂²	1	2	yes	—
37	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	#13094024	—	2	4	no	—
41	C₄×Dic₉₄	—	4	2	no	—
42	C₄×Dic₉₄	—	4	2	no	—
43	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
53	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
54	Dic₄₇⋊C₈	—	4	2	no	—
55	Dic₄₇⋊C₈	—	4	2	no	—
56	#13093934	—	4	2	no	—
57	C₈×Dic₄₇	—	4	2	no	—
58	C₄₇⋊#95	C₂×C₄	1	2	yes	—
59	C₄₇⋊#95	C₂×C₄	1	2	yes	—
60	C₄₇⋊#95	C₂×C₄	1	2	yes	—
61	C₄₇⋊#95	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
82	C₂×Dic₉₄	—	2	4	no	—
83	C₂×Dic₉₄	—	2	4	no	—
84	C₂×Dic₉₄	—	4	2	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	—	2	1	no	—
96	C₄²×C₄₇	—	2	1	no	—
97	C₄²×C₄₇	—	2	2	no	—
98	C₄²×C₄₇	—	4	1	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄⋊Dic₄₇	—	2	4	no	—
102	C₄⋊Dic₄₇	—	4	1	no	—
103	C₄⋊Dic₄₇	—	4	1	no	—
104	C₄⋊Dic₄₇	—	4	2	no	—
105	C₄⋊Dic₄₇	M₁₆	1	4	yes	—
106	Dic₄₇⋊C₄	—	4	2	no	—
107	Dic₄₇⋊C₄	—	4	2	no	—
108	C₄×Dic₄₇	—	2	2	no	—
109	C₄×Dic₄₇	—	2	2	no	—
110	C₄×Dic₄₇	—	2	2	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₂×C₄₇⋊C₈	—	2	2	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	2	1	no	—
126	C₂×C₄×C₄₇	—	2	1	no	—
127	C₂×C₄×C₄₇	—	2	1	no	—
128	C₂×C₄×C₄₇	—	2	1	no	—
129	C₂×C₄×C₄₇	—	2	1	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	1	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	1	no	—
134	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₄₇	#131	1	2	yes	—
136	C₂×C₄×C₄₇	#131	1	2	yes	—
137	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
138	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
139	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	—	2	2	no	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	C₂×M₁₆	1	4	yes	—
153	Dic₉₄	—	4	8	no	—
154	C₈×C₄₇	—	4	2	no	—
155	C₈×C₄₇	—	4	2	no	—
156	C₄₇⋊C₈	—	4	2	no	—
157	C₄₇⋊C₈	—	4	2	no	—
158	#23364	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	#518	1	1	yes	characteristic
160	C₂²×C₄₇	#519	1	1	yes	characteristic
161	C₂²×C₄₇	#519	1	1	yes	characteristic
162	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
163	C₂²×C₄₇	C₂×#131	1	1	yes	characteristic
164	C₂²×C₄₇	C₂×#131	1	1	yes	characteristic
165	C₂²×C₄₇	#568	1	1	yes	characteristic
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	2	2	no	—
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	4	1	no	—
170	C₄×C₄₇	—	4	1	no	—
171	C₄×C₄₇	—	4	1	no	—
172	C₄×C₄₇	#563	1	2	yes	—
173	C₄×C₄₇	#563	1	2	yes	—
174	Dic₄₇	—	2	4	no	—
175	Dic₄₇	—	4	2	no	—
176	Dic₄₇	—	4	2	no	—
177	Dic₄₇	—	4	2	no	—
178	C₄×C₄⋊Q₈	—	47	1	no	—
179	#1985	—	47	2	no	—
180	#1984	—	47	2	no	—
181	#1917	—	47	2	no	—
182	#1917	—	47	2	no	—
183	#1916	—	47	2	no	—
184	#1916	—	47	2	no	—
185	#1839	—	47	1	no	—
186	#1839	D₄₇	1	1	yes	characteristic
187	C₂×C₄₇	#2381	1	1	yes	characteristic
188	C₂×C₄₇	#2974	1	1	yes	characteristic
189	C₂×C₄₇	#2981	1	1	yes	characteristic
190	C₂×C₄₇	#2983	1	1	yes	characteristic
191	C₂×C₄₇	#2984	1	1	yes	characteristic
192	C₂×C₄₇	#2985	1	1	yes	characteristic
193	C₂×C₄₇	#2986	1	1	yes	characteristic
194	C₂×C₄⋊Q₈	—	47	1	no	—
195	C₂×C₄×Q₈	—	47	2	no	—
196	C₂×C₄⋊C₈	—	47	2	no	—
197	C₂×C₄⋊C₈	—	47	2	no	—
198	C₂×#97	—	47	2	no	—
199	C₂×C₄×C₈	—	47	2	no	—
200	#391	—	47	2	no	—
201	#389	—	47	2	no	—
202	#389	—	47	1	no	—
203	#389	—	47	1	no	—
204	#384	—	47	1	no	—
205	C₄×C₄⋊C₄	—	47	1	no	—
206	C₄×C₄⋊C₄	—	47	2	no	—
207	C₄×C₄⋊C₄	—	47	1	no	—
208	C₄³	C₂×D₄₇	1	1	yes	characteristic
209	#336	—	47	2	no	—
210	#336	—	47	2	no	—
211	#336	—	47	2	no	—
212	#336	—	47	2	no	—
213	#336	—	47	1	no	—
214	#336	—	47	2	no	—
215	#336	—	47	1	no	—
216	#336	—	47	2	no	—
217	#336	C₂×D₄₇	1	2	yes	—
218	#336	C₂×D₄₇	1	2	yes	—
219	#336	C₂×D₄₇	1	1	yes	characteristic
220	#336	C₂×D₄₇	1	1	yes	characteristic
221	C₄₇	#23364	1	1	yes	characteristic, complemented, Sylow
222	C₂²×Q₈	—	47	2	no	—
223	C₂²×C₈	—	47	2	no	—
224	C₂²×C₈	—	47	2	no	—
225	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
226	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
227	C₄⋊Q₈	—	94	4	no	—
228	C₄×Q₈	—	188	2	no	—
229	C₄×Q₈	—	188	2	no	—
230	C₂×C₄⋊C₄	—	47	1	no	—
231	C₂×C₄⋊C₄	—	47	2	no	—
232	C₂×C₄⋊C₄	—	47	1	no	—
233	C₂×C₄⋊C₄	—	47	2	no	—
234	C₂×C₄⋊C₄	—	47	2	no	—
235	C₂×C₄⋊C₄	—	47	2	no	—
236	C₂×C₄⋊C₄	—	47	2	no	—
237	C₂×C₄⋊C₄	—	47	1	no	—
238	C₂×C₄⋊C₄	—	47	1	no	—
239	C₂×C₄²	—	47	2	no	—
240	C₂×C₄²	—	47	2	no	—
241	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
242	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
243	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
244	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
245	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
246	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
247	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
248	C₄⋊C₈	—	188	2	no	—
249	C₄⋊C₈	—	188	2	no	—
250	#97	—	188	2	no	—
251	C₄×C₈	—	188	2	no	—
252	#95	—	47	2	no	—
253	#95	—	47	2	no	—
254	#95	—	47	2	no	—
255	#95	—	47	2	no	—
256	C₂×Q₈	—	94	4	no	—
257	C₂×Q₈	—	94	4	no	—
258	C₂×Q₈	—	188	2	no	—
259	C₂²×C₄	—	47	2	no	—
260	C₂²×C₄	—	47	2	no	—
261	C₂²×C₄	—	47	2	no	—
262	C₂²×C₄	—	47	2	no	—
263	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
269	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	94	2	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	—	94	2	no	—
279	C₂×C₈	—	94	2	no	—
280	C₂×C₈	—	94	2	no	—
281	C₂×C₈	—	94	2	no	—
282	C₄⋊C₄	—	47	4	no	—
283	C₄⋊C₄	—	94	4	no	—
284	C₄⋊C₄	—	94	2	no	—
285	C₄⋊C₄	—	94	4	no	—
286	C₄⋊C₄	—	188	2	no	—
287	C₄⋊C₄	—	188	2	no	—
288	C₄⋊C₄	—	188	1	no	—
289	C₄⋊C₄	—	188	1	no	—
290	C₄⋊C₄	—	188	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	4	1	no	—
299	C₄²	—	94	2	no	—
300	C₄²	—	94	2	no	—
301	C₄²	—	94	2	no	—
302	C₄²	—	94	2	no	—
303	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
304	Q₈	—	188	8	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	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	47	4	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	94	2	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	94	2	no	—
330	C₂×C₄	—	94	2	no	—
331	C₂×C₄	—	94	2	no	—
332	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
333	C₂×C₄	#13094030	1	2	yes	—
334	C₂×C₄	#13094032	1	2	yes	—
335	C₂×C₄	#13094070	1	2	yes	—
336	C₂×C₄	#13094071	1	1	yes	characteristic
337	C₂×C₄	#13094071	1	1	yes	characteristic
338	C₂×C₄	#13094071	1	2	yes	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	188	2	no	—
342	C₈	—	188	2	no	—
343	C₂²	#15689597	1	1	yes	characteristic
344	C₂²	#15689600	1	1	yes	characteristic
345	C₂²	#15689600	1	1	yes	characteristic
346	C₂²	C₂×M₁₆×D₄₇	1	1	yes	characteristic
347	C₂²	C₂×#13094030	1	1	yes	characteristic
348	C₂²	C₂×#13094032	1	1	yes	characteristic
349	C₂²	#15689889	1	1	yes	characteristic
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	94	4	no	—
357	C₄	—	188	2	no	—
358	C₄	—	188	2	no	—
359	C₄	—	188	2	no	—
360	C₄	#15689875	1	2	yes	—
361	C₄	#15689875	1	2	yes	—
362	C₂	#20246568	1	1	yes	characteristic
363	C₂	#20252422	1	1	yes	characteristic
364	C₂	#20252466	1	1	yes	characteristic
365	C₂	#20252472	1	1	yes	characteristic
366	C₂	#20252478	1	1	yes	characteristic
367	C₂	#20252482	1	1	yes	characteristic
368	C₂	#20252489	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1839	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1839	D₄₇
hypocenter	C₄₇	#23364
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

Complements

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

N	H ≅ G/N	type
221. C₄₇	#23364	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094024	2
#20244326	2
#20244322	2
#20243953	2
#20243951	2
#20243946	2
#20243944	2
#20243531	2
C₄₇×#1839	2
#23364	47