Subgroups — #26224302

Order 12032.

778 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689474	C₂	1	1	yes	maximal, characteristic
2	#20246136	C₂	1	1	yes	maximal, characteristic
3	#20246107	C₂	1	1	yes	maximal, characteristic
4	#20244361	C₂	1	1	yes	maximal, characteristic
5	#20244208	C₂	1	1	yes	maximal, characteristic
6	#20244078	C₂	1	2	yes	maximal
7	#20244064	C₂	1	2	yes	maximal
8	#20243519	C₂	1	1	yes	maximal, characteristic
9	#20243219	C₂	1	2	yes	maximal
10	C₄×#13093949	C₂	1	2	yes	maximal
11	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
12	C₂×#13094024	C₂²	1	1	yes	characteristic
13	C₂×#13094024	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
15	#15689474	C₄	1	8	yes	complemented
16	C₂×#13093949	C₂²	1	2	yes	—
17	C₂×#13093949	C₂²	1	2	yes	—
18	C₂×#13093949	C₂²	1	2	yes	—
19	C₂×#13093949	C₂²	1	2	yes	—
20	#15689029	C₂²	1	2	yes	—
21	#15689027	—	2	2	no	—
22	#15689021	C₂²	1	2	yes	—
23	#15689008	—	2	2	no	—
24	#15689008	—	2	2	no	—
25	#15689006	—	2	2	no	—
26	#15688989	C₂²	1	1	yes	characteristic
27	#15688989	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
29	#15688886	C₂²	1	2	yes	—
30	#15688886	C₂²	1	2	yes	—
31	#15688886	C₂²	1	2	yes	—
32	#15688886	C₂²	1	2	yes	—
33	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#376	C₂²	1	1	yes	characteristic
40	C₂²×Dic₉₄	—	2	1	no	—
41	C₂²×Dic₉₄	C₂³	1	2	yes	—
42	#13094024	—	2	2	no	—
43	#13094024	—	2	2	no	—
44	#13094024	C₂×C₄	1	4	yes	—
45	#13094024	C₂×C₄	1	4	yes	—
46	C₄×Dic₉₄	—	2	4	no	—
47	C₄×Dic₉₄	—	2	4	no	—
48	C₂×C₄⋊Dic₄₇	—	2	1	no	—
49	C₂×C₄⋊Dic₄₇	—	2	1	no	—
50	C₂×C₄⋊Dic₄₇	—	2	1	no	—
51	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
52	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
53	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
54	C₂×Dic₄₇⋊C₄	—	2	2	no	—
55	C₂×Dic₄₇⋊C₄	—	2	2	no	—
56	C₂×C₄×Dic₄₇	—	2	2	no	—
57	C₂×C₄×Dic₄₇	—	2	2	no	—
58	#13093949	—	2	4	no	—
59	#13093949	—	2	4	no	—
60	#13093949	C₂×C₄	1	8	yes	—
61	#13093949	C₂×C₄	1	8	yes	—
62	C₄₇⋊#95	—	2	2	no	—
63	C₄₇⋊#95	—	2	1	no	—
64	C₄₇⋊#95	—	2	1	no	—
65	C₄₇⋊#95	—	2	1	no	—
66	C₄₇⋊#95	—	2	1	no	—
67	C₄₇⋊#95	—	2	2	no	—
68	C₄₇⋊#95	—	2	4	no	—
69	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	#97×C₄₇	—	2	2	no	—
77	#97×C₄₇	—	2	2	no	—
78	#97×C₄₇	—	2	2	no	—
79	C₄×C₈×C₄₇	—	2	2	no	—
80	C₄×C₈×C₄₇	—	2	2	no	—
81	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
82	#95×C₄₇	D₄	1	2	yes	—
83	#95×C₄₇	D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₄₇	—	2	2	no	—
90	C₂²×Dic₄₇	—	2	2	no	—
91	C₂²×Dic₄₇	—	2	2	no	—
92	C₂²×Dic₄₇	—	2	2	no	—
93	C₂×Dic₉₄	—	2	2	no	—
94	C₂×Dic₉₄	—	2	2	no	—
95	C₂×Dic₉₄	—	2	2	no	—
96	C₂×Dic₉₄	—	2	2	no	—
97	C₂×Dic₉₄	—	2	2	no	—
98	C₂×Dic₉₄	—	2	4	no	—
99	C₂×Dic₉₄	—	8	2	no	—
100	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₄₇	—	2	1	no	—
102	C₂×C₈×C₄₇	—	2	1	no	—
103	C₂×C₈×C₄₇	—	2	1	no	—
104	C₂×C₈×C₄₇	—	2	1	no	—
105	C₂×C₈×C₄₇	—	2	1	no	—
106	C₂×C₈×C₄₇	—	2	1	no	—
107	C₂×C₈×C₄₇	—	2	1	no	—
108	C₂×C₈×C₄₇	—	2	1	no	—
109	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₄₇	#41	1	2	yes	—
112	C₂×C₈×C₄₇	#41	1	2	yes	—
113	C₄²×C₄₇	—	2	1	no	—
114	C₄²×C₄₇	—	2	1	no	—
115	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
116	C₄²×C₄₇	#41	1	2	yes	—
117	C₄²×C₄₇	#41	1	2	yes	—
118	C₄²×C₄₇	#41	1	2	yes	—
119	C₄⋊Dic₄₇	—	2	4	no	—
120	C₄⋊Dic₄₇	—	2	4	no	—
121	C₄⋊Dic₄₇	—	2	4	no	—
122	C₄⋊Dic₄₇	—	2	2	no	—
123	C₄⋊Dic₄₇	—	2	2	no	—
124	C₄⋊Dic₄₇	—	2	2	no	—
125	C₄⋊Dic₄₇	—	2	2	no	—
126	C₄⋊Dic₄₇	—	2	2	no	—
127	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
128	Dic₄₇⋊C₄	—	4	4	no	—
129	Dic₄₇⋊C₄	—	8	2	no	—
130	C₄×Dic₄₇	—	4	4	no	—
131	C₄×Dic₄₇	—	8	2	no	—
132	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₄₇	#124	1	4	yes	—
143	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
149	C₂×Dic₄₇	—	2	4	no	—
150	C₂×Dic₄₇	—	2	4	no	—
151	C₂×Dic₄₇	—	4	2	no	—
152	C₂×Dic₄₇	—	4	2	no	—
153	C₂×Dic₄₇	—	4	2	no	—
154	C₂×Dic₄₇	—	4	2	no	—
155	C₂×Dic₄₇	—	4	2	no	—
156	C₂×Dic₄₇	—	4	2	no	—
157	C₂×Dic₄₇	—	4	2	no	—
158	C₂×Dic₄₇	—	4	2	no	—
159	C₂×Dic₄₇	—	4	2	no	—
160	C₂×Dic₄₇	—	4	2	no	—
161	Dic₉₄	—	2	4	no	—
162	Dic₉₄	—	2	4	no	—
163	Dic₉₄	—	4	2	no	—
164	Dic₉₄	—	4	2	no	—
165	Dic₉₄	—	8	4	no	—
166	C₈×C₄₇	—	2	2	no	—
167	C₈×C₄₇	—	2	2	no	—
168	C₈×C₄₇	—	2	2	no	—
169	C₈×C₄₇	—	2	2	no	—
170	#21645	—	47	1	no	maximal, Sylow
171	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
173	C₂²×C₄₇	#532	1	1	yes	characteristic
174	C₂²×C₄₇	#577	1	2	yes	—
175	C₂²×C₄₇	#578	1	2	yes	—
176	C₄×C₄₇	—	2	4	no	—
177	C₄×C₄₇	—	2	2	no	—
178	C₄×C₄₇	—	2	2	no	—
179	C₄×C₄₇	—	4	2	no	—
180	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
181	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
182	C₄×C₄₇	#529	1	1	yes	characteristic
183	C₄×C₄₇	#529	1	1	yes	characteristic
184	Dic₄₇	—	4	4	no	—
185	Dic₄₇	—	4	4	no	—
186	Dic₄₇	—	8	2	no	—
187	Dic₄₇	—	8	2	no	—
188	C₂×#487	—	47	1	no	—
189	#2331	—	47	1	no	—
190	#2327	—	47	1	no	—
191	#1993	—	47	1	no	—
192	#1960	—	47	1	no	—
193	#1935	—	47	2	no	—
194	#1933	—	47	2	no	—
195	#1835	—	47	1	no	—
196	#1763	—	47	2	no	—
197	C₄×#103	—	47	2	no	—
198	#1726	D₄₇	1	1	yes	characteristic
199	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
200	C₂×C₄₇	#2949	1	1	yes	characteristic
201	C₂×C₄₇	#2950	1	1	yes	characteristic
202	C₂×C₄₇	#3137	1	2	yes	—
203	C₂×C₄₇	#3178	1	2	yes	—
204	C₂×C₄⋊Q₈	—	47	1	no	—
205	C₂×C₄⋊Q₈	—	47	1	no	—
206	C₂×C₄×Q₈	—	47	2	no	—
207	#487	—	47	8	no	—
208	C₂×#103	—	47	2	no	—
209	C₂×#103	—	47	2	no	—
210	C₂×#103	—	47	2	no	—
211	C₂×#103	—	47	2	no	—
212	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
213	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
214	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
215	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
216	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
217	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
218	#391	—	47	2	no	—
219	#391	—	94	2	no	—
220	#389	—	47	2	no	—
221	#387	—	94	2	no	—
222	#387	—	94	2	no	—
223	#387	—	94	2	no	—
224	#382	—	47	1	no	—
225	#382	—	47	1	no	—
226	C₄×C₄⋊C₄	—	47	2	no	—
227	#376	C₂×D₄₇	1	1	yes	characteristic
228	#340	—	47	2	no	—
229	#340	—	47	2	no	—
230	#340	—	47	2	no	—
231	#340	—	47	2	no	—
232	C₄₇	#21645	1	1	yes	characteristic, complemented, Sylow
233	C₂²×Q₈	—	47	2	no	—
234	C₂²×Q₈	—	94	1	no	—
235	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
236	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
239	C₄⋊Q₈	—	47	4	no	—
240	C₄⋊Q₈	—	47	4	no	—
241	C₄⋊Q₈	—	94	2	no	—
242	C₄⋊Q₈	—	94	2	no	—
243	C₄×Q₈	—	94	4	no	—
244	C₄×Q₈	—	94	4	no	—
245	C₂×C₄⋊C₄	—	47	2	no	—
246	C₂×C₄⋊C₄	—	47	2	no	—
247	C₂×C₄⋊C₄	—	47	2	no	—
248	C₂×C₄⋊C₄	—	94	1	no	—
249	C₂×C₄⋊C₄	—	94	1	no	—
250	C₂×C₄⋊C₄	—	94	1	no	—
251	C₂×C₄⋊C₄	—	94	2	no	—
252	C₂×C₄⋊C₄	—	94	2	no	—
253	C₂×C₄²	—	94	2	no	—
254	C₂×C₄²	—	94	2	no	—
255	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
256	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
257	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
258	#103	—	47	8	no	—
259	#103	—	47	8	no	—
260	#103	—	94	4	no	—
261	#103	—	94	4	no	—
262	#97	—	2	2	no	—
263	#97	—	2	2	no	—
264	#97	—	2	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	2	2	no	—
267	C₄×C₈	C₄×D₄₇	1	4	yes	—
268	#95	—	94	2	no	—
269	#95	—	94	1	no	—
270	#95	—	94	1	no	—
271	#95	—	94	1	no	—
272	#95	—	94	1	no	—
273	#95	—	94	4	no	—
274	#95	—	94	2	no	—
275	#95	D₁₈₈	1	2	yes	—
276	#95	D₁₈₈	1	2	yes	—
277	C₂×Q₈	—	47	4	no	—
278	C₂×Q₈	—	94	4	no	—
279	C₂×Q₈	—	94	2	no	—
280	C₂×Q₈	—	94	2	no	—
281	C₂×Q₈	—	94	2	no	—
282	C₂×Q₈	—	94	2	no	—
283	C₂×Q₈	—	94	2	no	—
284	C₂×Q₈	—	376	2	no	—
285	C₂²×C₄	—	94	2	no	—
286	C₂²×C₄	—	94	2	no	—
287	C₂²×C₄	—	94	2	no	—
288	C₂²×C₄	—	94	2	no	—
289	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
290	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
293	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
303	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
304	C₂×C₈	#10625350	1	2	yes	—
305	C₂×C₈	#10625350	1	2	yes	—
306	C₄⋊C₄	—	47	4	no	—
307	C₄⋊C₄	—	94	4	no	—
308	C₄⋊C₄	—	94	4	no	—
309	C₄⋊C₄	—	94	4	no	—
310	C₄⋊C₄	—	94	2	no	—
311	C₄⋊C₄	—	94	2	no	—
312	C₄⋊C₄	—	94	2	no	—
313	C₄⋊C₄	—	94	2	no	—
314	C₄⋊C₄	—	94	2	no	—
315	C₄⋊C₄	—	376	2	no	—
316	C₄⋊C₄	—	188	4	no	—
317	C₄²	—	2	1	no	—
318	C₄²	—	2	1	no	—
319	C₄²	—	376	2	no	—
320	C₄²	—	188	4	no	—
321	C₄²	C₂×C₄×D₄₇	1	2	yes	—
322	C₄²	#10625350	1	2	yes	—
323	C₄²	#10625350	1	2	yes	—
324	C₄²	#10625350	1	2	yes	—
325	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
326	Q₈	—	94	4	no	—
327	Q₈	—	94	4	no	—
328	Q₈	—	376	4	no	—
329	Q₈	—	188	2	no	—
330	Q₈	—	188	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	94	4	no	—
339	C₂×C₄	—	94	4	no	—
340	C₂×C₄	—	188	2	no	—
341	C₂×C₄	—	188	2	no	—
342	C₂×C₄	—	188	2	no	—
343	C₂×C₄	—	188	2	no	—
344	C₂×C₄	—	188	2	no	—
345	C₂×C₄	—	188	2	no	—
346	C₂×C₄	—	188	2	no	—
347	C₂×C₄	—	188	2	no	—
348	C₂×C₄	—	188	2	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
351	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
352	C₂×C₄	#13094012	1	4	yes	—
353	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
354	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
355	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
356	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
357	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
358	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	2	2	no	—
362	C₈	—	2	2	no	—
363	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
364	C₂²	C₂×#13094012	1	1	yes	characteristic
365	C₂²	#15689673	1	1	yes	characteristic
366	C₂²	#15689935	1	2	yes	—
367	C₂²	#15689940	1	2	yes	—
368	C₄	—	2	4	no	—
369	C₄	—	2	2	no	—
370	C₄	—	2	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	376	2	no	—
373	C₄	—	376	2	no	—
374	C₄	—	188	4	no	—
375	C₄	—	188	4	no	—
376	C₄	C₄×#10625350	1	1	yes	characteristic
377	C₄	C₄×#10625350	1	1	yes	characteristic
378	C₄	#15689657	1	1	yes	characteristic
379	C₄	#15689657	1	1	yes	characteristic
380	C₂	C₄×#13094012	1	1	yes	characteristic
381	C₂	#20252249	1	1	yes	characteristic
382	C₂	#20252257	1	1	yes	characteristic
383	C₂	#20253755	1	2	yes	—
384	C₂	#20254056	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1726	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#21645
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
15. #15689474	C₄	semidirect
232. C₄₇	#21645	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689474	2
#20246136	2
#20246107	2
#20244361	2
#20244208	2
#20244078	2
#20244064	2
#20243519	2
#20243219	2
C₄×#13093949	2
C₄₇×#1726	2
#21645	47