Subgroups — #26220606

Order 12032.

798 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689231	C₂	1	1	yes	maximal, characteristic
2	#20245865	C₂	1	2	yes	maximal
3	#20244259	C₂	1	1	yes	maximal, characteristic
4	#20244249	C₂	1	1	yes	maximal, characteristic
5	#20244131	C₂	1	1	yes	maximal, characteristic
6	#20244116	C₂	1	1	yes	maximal, characteristic
7	#20244078	C₂	1	2	yes	maximal
8	#20244064	C₂	1	2	yes	maximal
9	#20243285	C₂	1	1	yes	maximal, characteristic
10	#20243219	C₂	1	2	yes	maximal
11	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
14	#15689231	C₄	1	8	yes	complemented
15	C₂×#13093958	C₂²	1	1	yes	characteristic
16	C₂×#13093956	C₂²	1	1	yes	characteristic
17	C₂×#13093956	C₂²	1	1	yes	characteristic
18	C₂×#13093949	C₂²	1	2	yes	—
19	C₂×#13093949	C₂²	1	2	yes	—
20	C₂×#13093949	C₂²	1	2	yes	—
21	#15689029	C₂²	1	2	yes	—
22	#15689029	C₂²	1	2	yes	—
23	#15689021	C₂²	1	2	yes	—
24	#15689021	C₂²	1	2	yes	—
25	#15689007	—	2	2	no	—
26	#15689007	—	2	2	no	—
27	C₄×Dic₄₇⋊C₄	—	2	2	no	—
28	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
29	#15688969	—	2	2	no	—
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×#97×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	C₄³×C₄₇	C₂²	1	1	yes	characteristic
40	C₂²×Dic₉₄	C₂³	1	2	yes	—
41	C₂×Dic₁₈₈	—	2	2	no	—
42	C₂×Dic₁₈₈	—	2	2	no	—
43	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
44	C₄×Dic₉₄	—	2	4	no	—
45	C₄×Dic₉₄	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
49	C₂×Dic₄₇⋊C₄	—	2	2	no	—
50	C₂×Dic₄₇⋊C₄	—	2	2	no	—
51	C₂×Dic₄₇⋊C₄	—	2	2	no	—
52	C₂×C₄×Dic₄₇	—	2	2	no	—
53	C₂×C₄×Dic₄₇	—	2	2	no	—
54	#13093958	—	2	2	no	—
55	#13093956	—	2	2	no	—
56	#13093956	C₂×C₄	1	4	yes	—
57	#13093949	—	2	4	no	—
58	#13093949	—	2	4	no	—
59	#13093949	C₂×C₄	1	8	yes	—
60	C₄₇⋊#95	—	2	2	no	—
61	C₄₇⋊#95	—	2	2	no	—
62	C₄₇⋊#95	—	2	2	no	—
63	C₄₇⋊#95	—	2	2	no	—
64	C₄₇⋊#95	—	2	2	no	—
65	C₄₇⋊#95	—	2	2	no	—
66	C₄₇⋊#95	—	2	2	no	—
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	D₄	1	2	yes	—
72	C₂×C₄²×C₄₇	D₄	1	2	yes	—
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	#97×C₄₇	C₂×C₄	1	4	yes	—
80	C₄×C₈×C₄₇	—	2	2	no	—
81	C₄×C₈×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₇	—	2	2	no	—
88	C₂²×Dic₄₇	—	2	2	no	—
89	C₂²×Dic₄₇	—	2	2	no	—
90	C₂²×Dic₄₇	—	2	2	no	—
91	C₂×Dic₉₄	—	2	2	no	—
92	C₂×Dic₉₄	—	2	2	no	—
93	C₂×Dic₉₄	—	2	2	no	—
94	C₂×Dic₉₄	—	2	2	no	—
95	C₂×Dic₉₄	—	2	2	no	—
96	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
97	C₂×C₈×C₄₇	—	2	1	no	—
98	C₂×C₈×C₄₇	—	2	1	no	—
99	C₂×C₈×C₄₇	—	2	1	no	—
100	C₂×C₈×C₄₇	—	2	1	no	—
101	C₂×C₈×C₄₇	—	2	1	no	—
102	C₂×C₈×C₄₇	—	2	1	no	—
103	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₄₇	#41	1	2	yes	—
106	C₂×C₈×C₄₇	#41	1	2	yes	—
107	C₂×C₈×C₄₇	#41	1	2	yes	—
108	C₂×C₈×C₄₇	#41	1	2	yes	—
109	C₄²×C₄₇	—	2	4	no	—
110	C₄²×C₄₇	—	2	1	no	—
111	C₄²×C₄₇	—	2	1	no	—
112	C₄²×C₄₇	—	2	1	no	—
113	C₄²×C₄₇	—	2	1	no	—
114	C₄²×C₄₇	—	4	2	no	—
115	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
116	C₄²×C₄₇	#41	1	2	yes	—
117	C₄⋊Dic₄₇	—	2	2	no	—
118	C₄⋊Dic₄₇	—	2	2	no	—
119	C₄⋊Dic₄₇	—	2	2	no	—
120	C₄⋊Dic₄₇	—	2	2	no	—
121	C₄⋊Dic₄₇	—	2	2	no	—
122	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
123	Dic₄₇⋊C₄	—	2	8	no	—
124	Dic₄₇⋊C₄	—	4	4	no	—
125	Dic₄₇⋊C₄	—	4	4	no	—
126	C₄×Dic₄₇	—	2	8	no	—
127	C₄×Dic₄₇	—	4	4	no	—
128	Dic₁₈₈	—	2	8	no	—
129	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	—	2	2	no	—
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	1	no	—
135	C₂×C₄×C₄₇	—	2	1	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
139	C₂×C₄×C₄₇	#137	1	4	yes	—
140	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	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×Dic₄₇	—	2	4	no	—
147	C₂×Dic₄₇	—	2	4	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	4	2	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	Dic₉₄	—	2	4	no	—
159	Dic₉₄	—	2	4	no	—
160	Dic₉₄	—	4	2	no	—
161	Dic₉₄	—	4	2	no	—
162	C₈×C₄₇	—	2	2	no	—
163	C₈×C₄₇	—	2	2	no	—
164	C₈×C₄₇	—	2	2	no	—
165	C₈×C₄₇	#117	1	4	yes	—
166	#21287	—	47	1	no	maximal, Sylow
167	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₄₇	#532	1	1	yes	characteristic
169	C₂²×C₄₇	#532	1	1	yes	characteristic
170	C₂²×C₄₇	C₂×#137	1	2	yes	—
171	C₂²×C₄₇	#575	1	2	yes	—
172	C₄×C₄₇	—	2	4	no	—
173	C₄×C₄₇	—	2	4	no	—
174	C₄×C₄₇	—	4	1	no	—
175	C₄×C₄₇	—	4	1	no	—
176	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
177	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
178	C₄×C₄₇	#529	1	1	yes	characteristic
179	C₄×C₄₇	#529	1	1	yes	characteristic
180	Dic₄₇	—	4	4	no	—
181	Dic₄₇	—	4	4	no	—
182	Dic₄₇	—	8	2	no	—
183	Dic₄₇	—	8	2	no	—
184	C₂×#441	—	47	1	no	—
185	#2295	—	47	2	no	—
186	#1971	—	47	1	no	—
187	#1969	—	47	1	no	—
188	#1944	—	47	1	no	—
189	#1941	—	47	1	no	—
190	#1935	—	47	2	no	—
191	#1933	—	47	2	no	—
192	#1776	—	47	1	no	—
193	#1763	—	47	2	no	—
194	C₄×#97	D₄₇	1	1	yes	characteristic
195	C₂×C₄₇	#2293	1	1	yes	characteristic
196	C₂×C₄₇	C₄×#137	1	2	yes	—
197	C₂×C₄₇	#3181	1	2	yes	—
198	C₂×C₄₇	#3183	1	1	yes	characteristic
199	C₂×C₄₇	#3185	1	1	yes	characteristic
200	C₂²×Q₁₆	—	47	1	no	—
201	C₂×C₄×Q₈	—	47	2	no	—
202	#441	—	47	8	no	—
203	C₂×#107	—	47	1	no	—
204	C₂×#106	—	47	1	no	—
205	C₂×#106	—	47	1	no	—
206	C₂×#103	—	47	2	no	—
207	C₂×#103	—	47	2	no	—
208	C₂×#103	—	47	2	no	—
209	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
210	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
211	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
212	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
214	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
215	#391	—	47	2	no	—
216	#391	—	47	2	no	—
217	#389	—	47	2	no	—
218	#389	—	47	2	no	—
219	#387	—	94	2	no	—
220	#387	—	94	2	no	—
221	C₄×C₄⋊C₄	—	47	2	no	—
222	C₄×C₄⋊C₄	—	94	2	no	—
223	#376	—	94	2	no	—
224	C₄³	C₂×D₄₇	1	1	yes	characteristic
225	#340	—	47	2	no	—
226	#340	—	47	2	no	—
227	#340	—	47	2	no	—
228	C₄₇	#21287	1	1	yes	characteristic, complemented, Sylow
229	C₂²×Q₈	—	47	2	no	—
230	C₂×Q₁₆	—	47	4	no	—
231	C₂×Q₁₆	—	94	2	no	—
232	C₂×Q₁₆	—	94	2	no	—
233	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
236	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
237	C₄×Q₈	—	47	8	no	—
238	C₄×Q₈	—	94	4	no	—
239	C₂×C₄⋊C₄	—	47	2	no	—
240	C₂×C₄⋊C₄	—	47	2	no	—
241	C₂×C₄⋊C₄	—	47	2	no	—
242	C₂×C₄⋊C₄	—	94	2	no	—
243	C₂×C₄⋊C₄	—	94	2	no	—
244	C₂×C₄⋊C₄	—	94	2	no	—
245	C₂×C₄²	—	94	2	no	—
246	C₂×C₄²	—	94	2	no	—
247	C₂×C₄²	D₁₈₈	1	2	yes	—
248	C₂×C₄²	D₁₈₈	1	2	yes	—
249	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
250	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
251	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
252	#107	—	94	2	no	—
253	#106	—	47	4	no	—
254	#106	—	94	2	no	—
255	#103	—	47	8	no	—
256	#103	—	94	4	no	—
257	#103	—	94	4	no	—
258	#97	—	2	2	no	—
259	#97	—	2	2	no	—
260	#97	—	2	2	no	—
261	#97	C₄×D₄₇	1	4	yes	—
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	—	2	2	no	—
264	#95	—	94	2	no	—
265	#95	—	94	2	no	—
266	#95	—	94	2	no	—
267	#95	—	94	2	no	—
268	#95	—	94	2	no	—
269	#95	—	94	2	no	—
270	#95	—	94	2	no	—
271	C₂×Q₈	—	47	4	no	—
272	C₂×Q₈	—	94	2	no	—
273	C₂×Q₈	—	94	2	no	—
274	C₂×Q₈	—	94	2	no	—
275	C₂×Q₈	—	94	2	no	—
276	C₂×Q₈	—	94	2	no	—
277	C₂²×C₄	—	94	2	no	—
278	C₂²×C₄	—	94	2	no	—
279	C₂²×C₄	—	94	2	no	—
280	C₂²×C₄	—	94	2	no	—
281	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
284	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
285	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
286	Q₁₆	—	94	8	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
294	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
295	C₂×C₈	#10625350	1	2	yes	—
296	C₂×C₈	#10625350	1	2	yes	—
297	C₂×C₈	#10625350	1	2	yes	—
298	C₂×C₈	#10625350	1	2	yes	—
299	C₄⋊C₄	—	47	4	no	—
300	C₄⋊C₄	—	94	8	no	—
301	C₄⋊C₄	—	94	2	no	—
302	C₄⋊C₄	—	94	2	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₄⋊C₄	—	94	2	no	—
305	C₄⋊C₄	—	94	2	no	—
306	C₄⋊C₄	—	188	4	no	—
307	C₄⋊C₄	—	188	4	no	—
308	C₄²	—	2	4	no	—
309	C₄²	—	2	1	no	—
310	C₄²	—	2	1	no	—
311	C₄²	—	2	1	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	4	2	no	—
314	C₄²	—	94	8	no	—
315	C₄²	—	188	4	no	—
316	C₄²	C₂×C₄×D₄₇	1	2	yes	—
317	C₄²	#10625350	1	2	yes	—
318	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
319	Q₈	—	94	4	no	—
320	Q₈	—	94	4	no	—
321	Q₈	—	188	2	no	—
322	Q₈	—	188	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	94	4	no	—
332	C₂×C₄	—	94	4	no	—
333	C₂×C₄	—	188	2	no	—
334	C₂×C₄	—	188	2	no	—
335	C₂×C₄	—	188	2	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	—	188	2	no	—
338	C₂×C₄	—	188	2	no	—
339	C₂×C₄	—	188	2	no	—
340	C₂×C₄	—	188	2	no	—
341	C₂×C₄	—	188	2	no	—
342	C₂×C₄	—	188	2	no	—
343	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
344	C₂×C₄	#13094057	1	4	yes	—
345	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
346	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
347	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
348	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
349	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
350	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	2	2	no	—
353	C₈	—	2	2	no	—
354	C₈	#13093981	1	4	yes	—
355	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
356	C₂²	#15689673	1	1	yes	characteristic
357	C₂²	#15689673	1	1	yes	characteristic
358	C₂²	C₂×#13094057	1	2	yes	—
359	C₂²	#15689923	1	2	yes	—
360	C₄	—	2	4	no	—
361	C₄	—	2	4	no	—
362	C₄	—	4	1	no	—
363	C₄	—	4	1	no	—
364	C₄	—	376	2	no	—
365	C₄	—	376	2	no	—
366	C₄	—	188	4	no	—
367	C₄	—	188	4	no	—
368	C₄	C₂×#13093981	1	1	yes	characteristic
369	C₄	C₄×#10625350	1	1	yes	characteristic
370	C₄	#15689657	1	1	yes	characteristic
371	C₄	#15689657	1	1	yes	characteristic
372	C₂	#20245852	1	1	yes	characteristic
373	C₂	C₄×#13094057	1	2	yes	—
374	C₂	#20254097	1	2	yes	—
375	C₂	#20254117	1	1	yes	characteristic
376	C₂	#20254133	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	#21287
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
14. #15689231	C₄	semidirect
228. C₄₇	#21287	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689231	2
#20245865	2
#20244259	2
#20244249	2
#20244131	2
#20244116	2
#20244078	2
#20244064	2
#20243285	2
#20243219	2
C₄×#97×C₄₇	2
#21287	47