Subgroups — #26235106

Order 12032.

750 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	#20246340	C₂	1	2	yes	maximal
3	#20244154	C₂	1	2	yes	maximal
4	#20244153	C₂	1	2	yes	maximal
5	#20244143	C₂	1	2	yes	maximal
6	#20243549	C₂	1	1	yes	maximal, characteristic
7	#20243543	C₂	1	1	yes	maximal, characteristic
8	#106×Dic₄₇	C₂	1	1	yes	maximal, characteristic
9	#106×Dic₄₇	C₂	1	1	yes	maximal, characteristic
10	C₄×#13093956	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1846	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
13	C₈×Dic₉₄	C₄	1	8	yes	complemented
14	C₂×#13093957	C₂²	1	2	yes	—
15	C₂×#13093956	C₂²	1	1	yes	characteristic
16	C₂×#13093956	C₂²	1	1	yes	characteristic
17	C₂×#13093955	C₂²	1	2	yes	—
18	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
23	#15689027	—	2	2	no	—
24	#15689008	—	2	2	no	—
25	#15689005	—	2	2	no	—
26	#15688992	C₂²	1	2	yes	—
27	#15688988	C₂²	1	2	yes	—
28	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
29	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
30	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
31	#15688974	—	2	2	no	—
32	#15688886	C₂²	1	2	yes	—
33	#15688885	C₂²	1	2	yes	—
34	#15688885	C₂²	1	2	yes	—
35	C₂×#107×C₄₇	C₂²	1	2	yes	—
36	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#382	C₂²	1	2	yes	—
40	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
44	C₄×Dic₉₄	Q₈	1	4	yes	—
45	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
48	C₂×Dic₄₇⋊C₄	—	2	2	no	—
49	C₂×Dic₄₇⋊C₄	—	2	2	no	—
50	C₂×Dic₄₇⋊C₄	D₄	1	1	yes	characteristic
51	C₂×Dic₄₇⋊C₄	D₄	1	1	yes	characteristic
52	C₂×C₄×Dic₄₇	—	2	2	no	—
53	C₂×C₄×Dic₄₇	—	2	2	no	—
54	C₂×C₄×Dic₄₇	—	2	2	no	—
55	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
57	#13093957	—	4	2	no	—
58	#13093956	—	2	2	no	—
59	#13093956	Q₈	1	4	yes	complemented
60	#13093955	—	4	2	no	—
61	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
62	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
63	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
64	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
65	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
66	C₄₇⋊#95	—	2	2	no	—
67	C₄₇⋊#95	—	2	2	no	—
68	C₄₇⋊#95	—	2	2	no	—
69	C₄₇⋊#95	—	2	2	no	—
70	C₄₇⋊#95	—	2	2	no	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
75	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	#107×C₄₇	—	4	2	no	—
77	#106×C₄₇	—	2	2	no	—
78	#106×C₄₇	—	2	2	no	—
79	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
80	#95×C₄₇	—	2	2	no	—
81	#95×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	—	2	2	no	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₇	—	2	2	no	—
88	C₂²×Dic₄₇	—	2	2	no	—
89	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₉₄	—	2	4	no	—
94	C₂×Dic₉₄	C₄⋊C₄	1	4	yes	—
95	C₂×C₈×C₄₇	—	2	1	no	—
96	C₂×C₈×C₄₇	—	2	1	no	—
97	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
100	C₂×C₈×C₄₇	#41	1	2	yes	—
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₄⋊C₄×C₄₇	—	2	2	no	—
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
111	C₄⋊Dic₄₇	C₂×Q₈	1	4	yes	—
112	Dic₄₇⋊C₄	—	4	4	no	—
113	Dic₄₇⋊C₄	—	8	2	no	—
114	Dic₄₇⋊C₄	C₄⋊C₄	1	4	yes	—
115	Dic₄₇⋊C₄	C₄⋊C₄	1	4	yes	—
116	C₄×Dic₄₇	—	4	4	no	—
117	C₄×Dic₄₇	—	8	2	no	—
118	C₄×Dic₄₇	—	8	2	no	—
119	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
120	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
121	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
122	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
123	C₂×C₄₇⋊C₈	—	2	1	no	—
124	C₂×C₄₇⋊C₈	—	2	1	no	—
125	C₂×C₄₇⋊C₈	—	2	1	no	—
126	C₂×C₄₇⋊C₈	—	2	1	no	—
127	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
128	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
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	—	4	2	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	2	4	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	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
157	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
158	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×Dic₄₇	#135	1	4	yes	—
160	Dic₉₄	—	2	8	no	—
161	C₈×C₄₇	—	2	2	no	—
162	C₈×C₄₇	C₄×Q₈	1	4	yes	—
163	C₄₇⋊C₈	—	2	2	no	—
164	C₄₇⋊C₈	—	2	2	no	—
165	#22725	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
167	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
168	C₂²×C₄₇	#550	1	1	yes	characteristic
169	C₂²×C₄₇	C₂×#135	1	2	yes	—
170	C₂²×C₄₇	#577	1	2	yes	—
171	C₄×C₄₇	—	2	2	no	—
172	C₄×C₄₇	—	2	2	no	—
173	C₄×C₄₇	—	8	2	no	—
174	C₄×C₄₇	—	8	2	no	—
175	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
176	C₄×C₄₇	#520	1	1	yes	characteristic
177	C₄×C₄₇	#557	1	1	yes	characteristic
178	C₄×C₄₇	Q₈²	1	1	yes	characteristic
179	Dic₄₇	—	2	2	no	—
180	Dic₄₇	—	2	2	no	—
181	Dic₄₇	—	2	2	no	—
182	Dic₄₇	—	4	4	no	—
183	Dic₄₇	—	8	2	no	—
184	Dic₄₇	#427	1	4	yes	—
185	C₂×C₈×Q₈	—	47	1	no	—
186	#2355	—	47	2	no	—
187	#1949	—	47	2	no	—
188	#1949	—	47	2	no	—
189	#1947	—	47	2	no	—
190	#1846	—	47	1	no	—
191	#1846	D₄₇	1	1	yes	characteristic
192	#1844	—	47	1	no	—
193	C₄×#106	—	47	1	no	—
194	C₄×#106	—	47	1	no	—
195	C₄×#106	—	47	1	no	—
196	C₂×C₄₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
198	C₂×C₄₇	#2912	1	1	yes	characteristic
199	C₂×C₄₇	#3381	1	2	yes	—
200	C₂×C₄₇	#3394	1	2	yes	—
201	C₂×C₄×Q₈	—	47	1	no	—
202	C₈×Q₈	—	47	8	no	—
203	C₂×#107	—	47	2	no	—
204	C₂×#107	C₂×D₄₇	1	2	yes	—
205	C₂×#106	—	47	1	no	—
206	C₂×#106	—	47	1	no	—
207	C₂×#106	—	47	2	no	—
208	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
209	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
210	C₂×C₄⋊C₈	—	47	1	no	—
211	C₂×C₄⋊C₈	—	47	1	no	—
212	C₂×C₄⋊C₈	—	47	1	no	—
213	C₂×C₄×C₈	—	47	1	no	—
214	C₂×C₄×C₈	—	47	1	no	—
215	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
216	#391	—	94	2	no	—
217	#387	—	94	2	no	—
218	#387	—	94	2	no	—
219	#384	—	47	2	no	—
220	#382	—	47	2	no	—
221	#382	C₂×D₄₇	1	2	yes	—
222	C₄×C₄⋊C₄	—	47	2	no	—
223	C₄×C₄⋊C₄	—	47	2	no	—
224	C₄×C₄⋊C₄	—	47	2	no	—
225	C₄×C₄⋊C₄	—	94	2	no	—
226	#340	—	47	2	no	—
227	#340	—	47	2	no	—
228	#340	—	47	2	no	—
229	C₄₇	#22725	1	1	yes	characteristic, complemented, Sylow
230	C₂²×Q₈	—	47	1	no	—
231	C₂²×C₈	—	47	1	no	—
232	C₂²×C₈	—	47	1	no	—
233	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
235	C₄×Q₈	—	47	4	no	—
236	C₄×Q₈	—	47	4	no	—
237	C₂×C₄⋊C₄	—	47	1	no	—
238	C₂×C₄⋊C₄	—	47	1	no	—
239	C₂×C₄⋊C₄	—	47	1	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	C₂²×D₄₇	1	2	yes	—
245	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
246	C₂×C₄²	—	47	1	no	—
247	C₂×C₄²	—	47	1	no	—
248	C₂×C₄²	—	94	2	no	—
249	C₂×C₄²	—	94	2	no	—
250	C₂×C₄²	—	94	2	no	—
251	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
252	#107	—	4	2	no	—
253	#107	—	188	2	no	—
254	#106	—	2	2	no	—
255	#106	—	2	2	no	—
256	#106	—	47	4	no	—
257	#106	—	94	2	no	—
258	#106	—	188	2	no	—
259	C₄⋊C₈	—	47	4	no	—
260	C₄⋊C₈	—	47	4	no	—
261	C₄⋊C₈	—	47	4	no	—
262	C₄×C₈	—	47	4	no	—
263	C₄×C₈	—	47	4	no	—
264	C₄×C₈	C₄×D₄₇	1	4	yes	—
265	#95	—	2	2	no	—
266	#95	—	2	2	no	—
267	#95	—	94	2	no	—
268	#95	—	94	2	no	—
269	#95	—	94	2	no	—
270	#95	—	94	2	no	—
271	#95	—	94	2	no	—
272	C₂×Q₈	—	47	4	no	—
273	C₂×Q₈	—	94	4	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	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	—	94	2	no	—
281	C₂²×C₄	—	94	2	no	—
282	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
283	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
284	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	47	2	no	—
288	C₂×C₈	—	47	2	no	—
289	C₂×C₈	—	94	1	no	—
290	C₂×C₈	—	94	1	no	—
291	C₂×C₈	—	94	1	no	—
292	C₂×C₈	—	94	1	no	—
293	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
294	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
295	C₂×C₈	#10625352	1	2	yes	—
296	C₂×C₈	Q₈×D₄₇	1	2	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	47	4	no	—
302	C₄⋊C₄	—	47	4	no	—
303	C₄⋊C₄	—	47	4	no	—
304	C₄⋊C₄	—	47	4	no	—
305	C₄⋊C₄	—	94	2	no	—
306	C₄⋊C₄	—	94	2	no	—
307	C₄⋊C₄	—	94	2	no	—
308	C₄⋊C₄	—	376	2	no	—
309	C₄⋊C₄	—	188	4	no	—
310	C₄²	—	47	2	no	—
311	C₄²	—	47	2	no	—
312	C₄²	—	47	2	no	—
313	C₄²	—	47	2	no	—
314	C₄²	—	376	2	no	—
315	C₄²	—	376	2	no	—
316	C₄²	—	188	4	no	—
317	C₄²	C₂×C₄×D₄₇	1	2	yes	—
318	C₄²	Q₈×D₄₇	1	2	yes	—
319	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
320	Q₈	—	94	8	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	47	2	no	—
330	C₂×C₄	—	47	4	no	—
331	C₂×C₄	—	47	2	no	—
332	C₂×C₄	—	47	2	no	—
333	C₂×C₄	—	47	2	no	—
334	C₂×C₄	—	94	2	no	—
335	C₂×C₄	—	94	2	no	—
336	C₂×C₄	—	94	2	no	—
337	C₂×C₄	—	94	4	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⋊C₄×D₄₇	1	2	yes	—
344	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
345	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
346	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
347	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
348	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
349	C₂×C₄	#13094069	1	1	yes	characteristic
350	C₂×C₄	#13094072	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	94	2	no	—
353	C₈	—	94	2	no	—
354	C₈	#13093989	1	4	yes	—
355	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
356	C₂²	#15689788	1	1	yes	characteristic
357	C₂²	#15689793	1	1	yes	characteristic
358	C₂²	#135×D₄₇	1	2	yes	—
359	C₂²	#15689930	1	2	yes	—
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	2	no	—
364	C₄	—	47	4	no	—
365	C₄	—	94	2	no	—
366	C₄	—	94	2	no	—
367	C₄	—	94	2	no	—
368	C₄	—	376	2	no	—
369	C₄	—	188	4	no	—
370	C₄	C₂×#13093989	1	1	yes	characteristic
371	C₄	#15689604	1	1	yes	characteristic
372	C₄	#15689848	1	1	yes	characteristic
373	C₄	#15689853	1	1	yes	characteristic
374	C₂	#20246300	1	1	yes	characteristic
375	C₂	#427×D₄₇	1	1	yes	characteristic
376	C₂	#20251980	1	1	yes	characteristic
377	C₂	#20256260	1	2	yes	—
378	C₂	#20256381	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1846	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1846	D₄₇
hypocenter	C₄₇	#22725
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
13. C₈×Dic₉₄	C₄	semidirect
59. #13093956	Q₈	semidirect
229. C₄₇	#22725	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₉₄	2
#20246340	2
#20244154	2
#20244153	2
#20244143	2
#20243549	2
#20243543	2
#106×Dic₄₇	2
C₄×#13093956	2
C₄₇×#1846	2
#22725	47