Subgroups — #26233875

Order 12032.

704 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252048	C₂	1	2	yes	maximal
2	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
3	#20244134	C₂	1	2	yes	maximal, complemented
4	#20243986	C₂	1	4	yes	maximal, complemented
5	#20243485	C₂	1	2	yes	maximal
6	#20243270	C₂	1	2	yes	maximal, complemented
7	C₈×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#1810	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	C₂²	1	2	yes	—
10	#15689181	—	2	4	no	—
11	#15689177	—	2	2	no	—
12	#15689173	—	2	2	no	—
13	#15689161	C₄	1	1	yes	characteristic, complemented
14	#15689161	C₄	1	1	yes	characteristic, complemented
15	#15689161	C₄	1	2	yes	complemented
16	#15689161	C₂²	1	2	yes	—
17	#15689161	C₂²	1	1	yes	characteristic
18	#15689161	C₂²	1	1	yes	characteristic
19	C₂×#13093958	C₄	1	1	yes	characteristic, complemented
20	C₂×#13093958	C₄	1	1	yes	characteristic, complemented
21	C₂×#13093956	C₄	1	2	yes	complemented
22	#15689151	C₂²	1	4	yes	—
23	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
24	#15689081	C₂²	1	4	yes	—
25	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
26	C₄×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
27	#15688884	C₂²	1	4	yes	—
28	#15688884	C₂²	1	4	yes	—
29	#15688877	C₂²	1	2	yes	—
30	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#108×C₄₇	C₂²	1	2	yes	—
32	C₄₇×#407	C₂²	1	2	yes	—
33	C₄₇×#341	C₂²	1	2	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	#13093983	—	2	2	no	—
36	#13093983	—	2	2	no	—
37	#13093983	C₂×C₄	1	2	yes	—
38	#13093983	C₂×C₄	1	2	yes	—
39	#13093983	C₂³	1	2	yes	—
40	#13093983	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
44	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
45	C₂×C₄×Dic₄₇	—	2	2	no	—
46	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
47	#13093958	C₂×C₄	1	2	yes	—
48	#13093958	C₂×C₄	1	2	yes	—
49	#13093958	C₂×C₄	1	1	yes	characteristic
50	#13093958	C₂×C₄	1	1	yes	characteristic
51	#13093958	C₂×C₄	1	1	yes	characteristic
52	#13093958	C₂×C₄	1	1	yes	characteristic
53	#13093956	C₂×C₄	1	2	yes	—
54	#13093956	C₂×C₄	1	2	yes	—
55	#13093956	C₂×C₄	1	2	yes	—
56	#13093956	C₂×C₄	1	2	yes	—
57	Dic₄₇⋊C₈	—	2	4	no	—
58	Dic₄₇⋊C₈	—	2	4	no	—
59	C₂²⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
60	C₂²⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
61	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
62	#13093934	—	2	4	no	—
63	C₈×Dic₄₇	—	2	4	no	—
64	C₈×Dic₄₇	—	2	4	no	—
65	C₄₇⋊#95	—	2	1	no	—
66	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
68	C₂²×C₈×C₄₇	—	2	4	no	—
69	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	#108×C₄₇	—	2	4	no	—
76	C₂²⋊C₈×C₄₇	—	2	2	no	—
77	C₂²⋊C₈×C₄₇	—	2	2	no	—
78	C₂⁴×C₄₇	#41	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	#41	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	—	2	2	no	—
90	M₁₆×C₄₇	—	2	4	no	—
91	M₁₆×C₄₇	—	2	4	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	4	no	—
98	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	#41	1	4	yes	—
103	C₂²⋊Dic₄₇	—	2	4	no	—
104	C₂²⋊Dic₄₇	—	2	2	no	—
105	C₂²⋊Dic₄₇	—	2	2	no	—
106	C₂²⋊Dic₄₇	—	2	2	no	—
107	C₂²⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
111	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
112	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
113	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
114	C₄×Dic₄₇	—	2	2	no	—
115	C₄×Dic₄₇	—	2	2	no	—
116	C₄×Dic₄₇	—	2	2	no	—
117	C₄×Dic₄₇	—	2	2	no	—
118	C₄×Dic₄₇	—	2	2	no	—
119	C₄×Dic₄₇	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
123	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
124	C₂³×C₄₇	—	2	2	no	—
125	C₂³×C₄₇	—	2	2	no	—
126	C₂³×C₄₇	—	4	1	no	—
127	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
129	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₄₇	—	2	2	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	—	2	1	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
142	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
143	C₂×Dic₄₇	—	2	4	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	—	4	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	4	4	no	—
151	C₂×Dic₄₇	—	8	2	no	—
152	C₈×C₄₇	—	2	1	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₈×C₄₇	—	4	4	no	—
155	C₈×C₄₇	#117	1	2	yes	—
156	C₈×C₄₇	#117	1	2	yes	—
157	C₄₇⋊C₈	—	2	2	no	—
158	C₄₇⋊C₈	—	2	2	no	—
159	#22606	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	—	2	1	no	—
165	C₂²×C₄₇	—	2	1	no	—
166	C₂²×C₄₇	—	2	1	no	—
167	C₂²×C₄₇	—	4	4	no	—
168	C₂²×C₄₇	#443	1	2	yes	—
169	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₄₇	#552	1	2	yes	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
174	C₄×C₄₇	#519	1	1	yes	characteristic
175	C₄×C₄₇	#547	1	2	yes	—
176	Dic₄₇	—	4	4	no	—
177	Dic₄₇	—	4	2	no	—
178	Dic₄₇	—	4	2	no	—
179	Dic₄₇	—	4	2	no	—
180	Dic₄₇	—	4	2	no	—
181	#2924	—	47	2	no	—
182	C₂×#427	—	47	1	no	—
183	#1945	—	47	2	no	—
184	#1922	—	47	4	no	—
185	#1826	—	47	2	no	—
186	#1810	D₄₇	1	1	yes	characteristic
187	#1772	—	47	2	no	—
188	C₈×C₂²⋊C₄	—	47	1	no	—
189	C₂×C₄₇	—	2	2	no	—
190	C₂×C₄₇	—	2	2	no	—
191	C₂×C₄₇	—	4	1	no	—
192	C₂×C₄₇	#2344	1	1	yes	characteristic
193	C₂×C₄₇	C₂×#443	1	1	yes	characteristic
194	C₂×C₄₇	#2957	1	1	yes	characteristic
195	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
196	C₂×#117	—	47	2	no	—
197	#433	—	94	4	no	—
198	#432	—	94	2	no	—
199	#431	—	94	2	no	—
200	C₂×#108	C₂×D₄₇	1	2	yes	—
201	#427	—	47	1	no	—
202	#427	—	47	1	no	—
203	#427	—	47	2	no	—
204	#427	—	47	2	no	—
205	#427	—	47	1	no	—
206	#427	—	47	1	no	—
207	C₂×#107	—	47	1	no	—
208	C₂×#107	—	47	1	no	—
209	C₂×#106	—	47	2	no	—
210	#424	—	47	4	no	—
211	#407	C₂×D₄₇	1	2	yes	—
212	C₂×C₂²⋊C₈	—	47	1	no	—
213	#405	—	47	4	no	—
214	C₂×C₄×C₈	—	47	2	no	—
215	C₄×C₂²⋊C₄	—	47	1	no	—
216	#341	C₂×D₄₇	1	2	yes	—
217	#339	—	47	4	no	—
218	#339	—	47	4	no	—
219	#336	—	47	2	no	—
220	C₄₇	#22606	1	1	yes	characteristic, complemented, Sylow
221	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
222	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
223	C₂²×C₈	—	2	4	no	—
224	C₂²×C₈	—	47	2	no	—
225	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
226	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
230	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
231	#117	—	47	2	no	—
232	#117	—	47	2	no	—
233	#117	—	47	2	no	—
234	#117	—	47	2	no	—
235	#117	—	94	2	no	—
236	#117	—	94	2	no	—
237	C₂×C₄⋊C₄	—	47	2	no	—
238	C₂×C₄⋊C₄	—	47	2	no	—
239	C₂×C₂²⋊C₄	—	94	2	no	—
240	C₂×C₂²⋊C₄	—	94	1	no	—
241	C₂×C₄²	—	47	2	no	—
242	C₂×C₄²	—	94	2	no	—
243	#108	—	2	4	no	—
244	#107	—	47	2	no	—
245	#107	—	47	2	no	—
246	#107	—	47	1	no	—
247	#107	—	47	1	no	—
248	#107	—	47	1	no	—
249	#107	—	47	1	no	—
250	#106	—	47	2	no	—
251	#106	—	47	2	no	—
252	#106	—	47	2	no	—
253	#106	—	47	2	no	—
254	C₄⋊C₈	—	94	4	no	—
255	C₄⋊C₈	—	94	4	no	—
256	C₂²⋊C₈	—	2	2	no	—
257	C₂²⋊C₈	—	2	2	no	—
258	C₂²⋊C₈	—	47	1	no	—
259	C₂²⋊C₈	—	47	1	no	—
260	C₂²⋊C₈	—	47	2	no	—
261	#97	—	94	4	no	—
262	C₄×C₈	—	94	4	no	—
263	C₄×C₈	—	94	4	no	—
264	#95	—	94	1	no	—
265	C₂⁴	#10625354	1	1	yes	characteristic
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	94	2	no	—
269	C₂²×C₄	—	94	2	no	—
270	C₂²×C₄	—	94	2	no	—
271	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
274	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
275	C₂²×C₄	#10625354	1	1	yes	characteristic
276	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	47	2	no	—
286	C₂×C₈	—	47	2	no	—
287	C₂×C₈	—	94	2	no	—
288	C₂×C₈	—	94	2	no	—
289	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
293	C₂×C₈	#10625352	1	4	yes	—
294	C₄⋊C₄	—	47	2	no	—
295	C₄⋊C₄	—	47	2	no	—
296	C₄⋊C₄	—	47	2	no	—
297	C₄⋊C₄	—	47	2	no	—
298	C₄⋊C₄	—	94	2	no	—
299	C₄⋊C₄	—	94	2	no	—
300	C₂²⋊C₄	—	94	4	no	—
301	C₂²⋊C₄	—	94	2	no	—
302	C₂²⋊C₄	—	94	2	no	—
303	C₂²⋊C₄	—	94	2	no	—
304	C₂²⋊C₄	—	94	2	no	—
305	C₄²	—	94	2	no	—
306	C₄²	—	94	2	no	—
307	C₄²	—	94	2	no	—
308	C₄²	—	94	2	no	—
309	C₄²	—	94	2	no	—
310	C₄²	—	94	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	—	2	2	no	—
313	C₂³	—	4	1	no	—
314	C₂³	#13093986	1	1	yes	characteristic
315	C₂³	#13093986	1	1	yes	characteristic
316	C₂³	C₂×#10625354	1	1	yes	characteristic
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	94	4	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	—	376	2	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	—	188	4	no	—
332	C₂×C₄	—	188	2	no	—
333	C₂×C₄	#13093986	1	1	yes	characteristic
334	C₂×C₄	#13093986	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
336	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
337	C₂×C₄	C₂×#10625352	1	2	yes	—
338	C₂×C₄	#13094071	1	2	yes	—
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	4	4	no	—
342	C₈	—	94	2	no	—
343	C₈	—	94	2	no	—
344	C₈	#13093980	1	2	yes	—
345	C₈	#13093980	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	4	4	no	—
354	C₂²	#15689239	1	2	yes	—
355	C₂²	C₂×#13093986	1	1	yes	characteristic
356	C₂²	#15689815	1	2	yes	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	188	4	no	—
360	C₄	—	188	2	no	—
361	C₄	—	188	2	no	—
362	C₄	—	188	2	no	—
363	C₄	—	188	2	no	—
364	C₄	C₂×#13093980	1	1	yes	characteristic
365	C₄	#15689600	1	1	yes	characteristic
366	C₄	#15689773	1	2	yes	—
367	C₂	—	2	2	no	—
368	C₂	—	2	2	no	—
369	C₂	—	4	1	no	—
370	C₂	#20246259	1	1	yes	characteristic
371	C₂	C₂×#15689239	1	1	yes	characteristic
372	C₂	#20252304	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1810	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1810	D₄₇
hypocenter	C₄₇	#22606
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1810

Complements

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

N	H ≅ G/N	type
3. #20244134	C₂	semidirect
4. #20243986	C₂	semidirect
6. #20243270	C₂	semidirect
13. #15689161	C₄	semidirect
14. #15689161	C₄	semidirect
15. #15689161	C₄	semidirect
19. C₂×#13093958	C₄	semidirect
20. C₂×#13093958	C₄	semidirect
21. C₂×#13093956	C₄	semidirect
220. C₄₇	#22606	semidirect

Maximal subgroups

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

subgroup	index
#20252048	2
C₂×#15689161	2
#20244134	2
#20243986	2
#20243485	2
#20243270	2
C₈×C₂²⋊Dic₄₇	2
C₄₇×#1810	2
#22606	47