Subgroups — #26216820

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245783	C₂	1	2	yes	maximal
3	#20243551	C₂	1	2	yes	maximal, complemented
4	#20243517	C₂	1	2	yes	maximal, complemented
5	#20243417	C₂	1	1	yes	maximal, characteristic
6	#20243411	C₂	1	2	yes	maximal
7	#107×Dic₄₇	C₂	1	2	yes	maximal, complemented
8	#20243276	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1808	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093955	C₂²	1	4	yes	—
14	C₂×C₈×Dic₄₇	C₄	1	8	yes	complemented
15	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	#15688990	—	2	2	no	—
20	#15688988	C₂²	1	4	yes	—
21	#15688984	—	2	2	no	—
22	#15688983	C₂²	1	2	yes	—
23	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
24	C₂²⋊C₄×Dic₄₇	—	2	2	no	—
25	#15688968	—	2	2	no	—
26	#15688886	C₂²	1	2	yes	—
27	#15688885	C₂²	1	4	yes	—
28	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#107×C₄₇	C₂²	1	2	yes	—
30	C₄₇×#380	C₂²	1	2	yes	—
31	C₄₇×#340	C₂²	1	2	yes	—
32	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
33	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₄₇⋊C₈	—	2	4	no	—
35	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
39	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	—	2	4	no	—
41	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
47	#13093958	—	2	4	no	—
48	#13093955	—	2	8	no	—
49	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
50	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
51	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
52	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
53	C₄₇⋊#95	—	2	2	no	—
54	C₄₇⋊#95	—	2	4	no	—
55	C₄₇⋊#95	—	2	4	no	—
56	C₄₇⋊#95	—	2	4	no	—
57	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	—	2	4	no	—
59	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
63	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
64	#107×C₄₇	—	2	4	no	—
65	#95×C₄₇	—	2	2	no	—
66	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	2	no	—
75	C₂²×Dic₄₇	—	2	2	no	—
76	C₂²×Dic₄₇	—	2	4	no	—
77	C₂²×Dic₄₇	—	2	4	no	—
78	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
79	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
80	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	8	no	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	#41	1	4	yes	—
89	C₄⋊C₄×C₄₇	—	2	4	no	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₂²⋊C₄×C₄₇	—	4	4	no	—
92	C₂²⋊Dic₄₇	—	4	4	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄×Dic₄₇	—	4	8	no	—
96	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
97	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
98	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
99	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
100	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
101	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	—	2	8	no	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₄₇⋊C₈	#41	1	4	yes	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₄₇	#135	1	2	yes	—
111	C₂³×C₄₇	#135	1	2	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	1	no	—
127	C₂×Dic₄₇	—	2	1	no	—
128	C₂×Dic₄₇	—	2	1	no	—
129	C₂×Dic₄₇	—	2	1	no	—
130	C₂×Dic₄₇	—	2	8	no	—
131	C₂×Dic₄₇	—	2	8	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×Dic₄₇	#135	1	4	yes	—
140	C₈×C₄₇	—	2	2	no	—
141	C₈×C₄₇	#117	1	4	yes	—
142	C₄₇⋊C₈	—	2	2	no	—
143	C₄₇⋊C₈	#117	1	4	yes	—
144	#20818	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	4	no	—
147	C₂²×C₄₇	—	2	4	no	—
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	#427	1	4	yes	—
150	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₄₇	#532	1	2	yes	—
152	C₂²×C₄₇	C₂×#135	1	2	yes	—
153	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	8	4	no	—
157	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
158	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
159	C₄×C₄₇	#533	1	2	yes	—
160	Dic₄₇	—	2	2	no	—
161	Dic₄₇	—	2	2	no	—
162	Dic₄₇	—	2	2	no	—
163	Dic₄₇	—	8	4	no	—
164	Dic₄₇	#427	1	4	yes	—
165	C₂²×C₄×C₈	—	47	1	no	—
166	#2283	—	47	2	no	—
167	#1846	—	47	2	no	—
168	#1835	—	47	2	no	—
169	#1808	—	47	1	no	—
170	#1808	D₄₇	1	1	yes	characteristic
171	#1807	—	47	2	no	—
172	C₄×#107	—	47	2	no	—
173	C₄×#106	—	47	2	no	—
174	C₂×C₄₇	—	2	4	no	—
175	C₂×C₄₇	#2285	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
177	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
178	C₂×C₄₇	#3168	1	2	yes	—
179	C₂×C₄₇	#3168	1	2	yes	—
180	C₂³×C₈	—	47	1	no	—
181	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
182	C₂²×C₄²	—	47	1	no	—
183	C₂×#107	—	47	2	no	—
184	C₂×#107	C₂×D₄₇	1	2	yes	—
185	C₂×#106	—	47	4	no	—
186	C₂×C₄×C₈	—	47	8	no	—
187	C₂×C₄×C₈	—	47	1	no	—
188	C₂×C₄×C₈	—	47	1	no	—
189	C₂×C₄×C₈	—	47	1	no	—
190	C₂×C₄×C₈	—	47	1	no	—
191	#383	—	94	2	no	—
192	#382	—	47	4	no	—
193	#381	—	94	2	no	—
194	#380	—	47	2	no	—
195	#380	C₂×D₄₇	1	2	yes	—
196	C₄×C₄⋊C₄	—	47	4	no	—
197	C₄×C₂²⋊C₄	—	94	2	no	—
198	#376	—	94	2	no	—
199	#340	—	47	2	no	—
200	#340	—	47	4	no	—
201	#340	C₂×D₄₇	1	2	yes	—
202	C₄₇	#20818	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	—	47	1	no	—
204	C₂³×C₄	—	47	1	no	—
205	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	—	47	1	no	—
208	C₂²×C₈	—	47	4	no	—
209	C₂²×C₈	—	47	1	no	—
210	C₂²×C₈	—	94	4	no	—
211	C₂²×C₈	C₄×D₄₇	1	4	yes	—
212	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×C₄⋊C₄	—	47	4	no	—
215	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₂²⋊C₄	—	94	2	no	—
218	C₂×C₄²	—	47	4	no	—
219	C₂×C₄²	—	47	1	no	—
220	C₂×C₄²	—	47	1	no	—
221	C₂×C₄²	—	47	4	no	—
222	C₂×C₄²	—	47	1	no	—
223	C₂×C₄²	—	47	1	no	—
224	C₂×C₄²	—	94	4	no	—
225	#107	—	2	4	no	—
226	#107	—	94	4	no	—
227	#106	—	94	8	no	—
228	C₄×C₈	—	47	4	no	—
229	C₄×C₈	—	47	4	no	—
230	C₄×C₈	—	47	4	no	—
231	C₄×C₈	—	47	4	no	—
232	#95	—	2	2	no	—
233	#95	—	94	4	no	—
234	#95	—	94	4	no	—
235	#95	—	94	4	no	—
236	#95	—	94	2	no	—
237	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	47	1	no	—
242	C₂²×C₄	—	47	4	no	—
243	C₂²×C₄	—	47	4	no	—
244	C₂²×C₄	—	47	1	no	—
245	C₂²×C₄	—	47	1	no	—
246	C₂²×C₄	—	47	1	no	—
247	C₂²×C₄	—	94	4	no	—
248	C₂²×C₄	—	94	2	no	—
249	C₂²×C₄	—	94	2	no	—
250	C₂²×C₄	—	94	4	no	—
251	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
252	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
254	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	8	no	—
257	C₂×C₈	—	47	2	no	—
258	C₂×C₈	—	47	4	no	—
259	C₂×C₈	—	47	2	no	—
260	C₂×C₈	—	94	8	no	—
261	C₂×C₈	—	94	2	no	—
262	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
263	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
264	C₂×C₈	#10625352	1	4	yes	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	94	4	no	—
268	C₄⋊C₄	—	94	4	no	—
269	C₂²⋊C₄	—	4	4	no	—
270	C₂²⋊C₄	—	188	4	no	—
271	C₄²	—	47	4	no	—
272	C₄²	—	47	4	no	—
273	C₄²	—	47	2	no	—
274	C₄²	—	47	2	no	—
275	C₄²	—	47	2	no	—
276	C₄²	—	47	2	no	—
277	C₄²	—	188	8	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
281	C₂³	#13094043	1	2	yes	—
282	C₂³	#13094047	1	2	yes	—
283	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	47	2	no	—
292	C₂×C₄	—	47	4	no	—
293	C₂×C₄	—	47	2	no	—
294	C₂×C₄	—	47	2	no	—
295	C₂×C₄	—	47	2	no	—
296	C₂×C₄	—	94	8	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	1	no	—
299	C₂×C₄	—	94	1	no	—
300	C₂×C₄	—	94	1	no	—
301	C₂×C₄	—	94	1	no	—
302	C₂×C₄	—	94	8	no	—
303	C₂×C₄	—	188	4	no	—
304	C₂×C₄	—	188	4	no	—
305	C₂×C₄	—	188	4	no	—
306	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
307	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
308	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
309	C₂×C₄	#41×D₄₇	1	2	yes	—
310	C₂×C₄	C₂×#10625352	1	2	yes	—
311	C₈	—	2	2	no	—
312	C₈	—	47	4	no	—
313	C₈	—	94	2	no	—
314	C₈	#13093980	1	4	yes	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	#15689160	1	4	yes	—
320	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
321	C₂²	#15689668	1	2	yes	—
322	C₂²	#135×D₄₇	1	2	yes	—
323	C₂²	C₂×#13094043	1	1	yes	characteristic
324	C₂²	C₂×#13094047	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	8	4	no	—
327	C₄	—	47	4	no	—
328	C₄	—	94	2	no	—
329	C₄	—	94	2	no	—
330	C₄	—	94	2	no	—
331	C₄	—	376	4	no	—
332	C₄	#117×D₄₇	1	1	yes	characteristic
333	C₄	C₂×#13093980	1	1	yes	characteristic
334	C₄	#15689675	1	2	yes	—
335	C₂	—	2	4	no	—
336	C₂	#20245790	1	1	yes	characteristic
337	C₂	#427×D₄₇	1	1	yes	characteristic
338	C₂	C₂×#15689160	1	1	yes	characteristic
339	C₂	#20253953	1	2	yes	—
340	C₂	#20253959	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1808	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1808	D₄₇
hypocenter	C₄₇	#20818
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1808

Complements

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

N	H ≅ G/N	type
3. #20243551	C₂	semidirect
4. #20243517	C₂	semidirect
7. #107×Dic₄₇	C₂	semidirect
8. #20243276	C₂	semidirect
14. C₂×C₈×Dic₄₇	C₄	semidirect
202. C₄₇	#20818	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₄₇	2
#20245783	2
#20243551	2
#20243517	2
#20243417	2
#20243411	2
#107×Dic₄₇	2
#20243276	2
C₄₇×#1808	2
#20818	47