Subgroups — #26216808

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245792	C₂	1	2	yes	maximal
3	#20243543	C₂	1	2	yes	maximal, complemented
4	#20243514	C₂	1	2	yes	maximal, complemented
5	#20243412	C₂	1	2	yes	maximal
6	#20243410	C₂	1	2	yes	maximal
7	C₄×#13093955	C₂	1	2	yes	maximal, complemented
8	C₄×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#106×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
12	C₂×#13093981	—	2	2	no	—
13	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
14	C₂×#13093956	C₂²	1	2	yes	—
15	C₂×#13093955	C₂²	1	4	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₄₇	C₄	1	8	yes	complemented
20	#15688992	C₂²	1	4	yes	—
21	#15688987	—	2	2	no	—
22	#15688982	—	2	2	no	—
23	#15688980	C₂²	1	4	yes	—
24	C₄×Dic₄₇⋊C₄	—	2	2	no	—
25	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
26	#15688886	C₂²	1	2	yes	—
27	#15688885	C₂²	1	4	yes	—
28	C₂×#106×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
31	C₄₇×#340	C₂²	1	2	yes	—
32	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
37	C₂×C₈×D₄₇	—	2	4	no	—
38	C₂×C₈×D₄₇	—	2	4	no	—
39	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
41	#13093981	—	4	8	no	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
43	C₂×Dic₄₇⋊C₄	—	2	4	no	—
44	C₂×Dic₄₇⋊C₄	—	2	4	no	—
45	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
46	C₂×C₄×Dic₄₇	—	2	2	no	—
47	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
50	C₄²×D₄₇	C₂×C₄	1	4	yes	—
51	C₄²×D₄₇	Q₈	1	4	yes	—
52	#13093956	—	2	4	no	—
53	#13093955	—	2	8	no	—
54	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
55	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
56	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
57	C₄₇⋊#95	—	2	2	no	—
58	C₄₇⋊#95	—	2	4	no	—
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
62	C₂×C₄²×C₄₇	—	2	2	no	—
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#106×C₄₇	—	2	4	no	—
65	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
66	#95×C₄₇	—	2	2	no	—
67	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×D₄₇	—	2	2	no	—
77	C₂×C₄×D₄₇	—	2	2	no	—
78	C₂×C₄×D₄₇	—	2	4	no	—
79	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
80	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
81	C₂×C₄×D₄₇	SD₁₆	1	8	yes	—
82	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	#41	1	4	yes	—
88	C₄⋊C₄×C₄₇	—	2	4	no	—
89	C₄⋊C₄×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	4	4	no	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
93	(C₂×D₄₇)⋊C₄	—	4	8	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	Dic₄₇⋊C₄	—	4	8	no	—
97	Dic₄₇⋊C₄	—	4	8	no	—
98	C₄×Dic₄₇	—	4	4	no	—
99	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
100	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
101	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
102	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
103	C₂×C₄₇⋊C₈	—	2	2	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	C₈×D₄₇	—	2	8	no	—
108	C₈×D₄₇	—	2	8	no	—
109	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	C₂²×D₄₇	—	2	2	no	—
111	C₂²×D₄₇	—	2	2	no	—
112	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂²×D₄₇	C₂×SD₁₆	1	4	yes	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	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	—	4	4	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×Dic₄₇	C₂×SD₁₆	1	4	yes	—
133	C₂×Dic₄₇	#135	1	4	yes	—
134	C₄×D₄₇	—	2	2	no	—
135	C₄×D₄₇	—	2	2	no	—
136	C₄×D₄₇	—	2	4	no	—
137	C₄×D₄₇	—	2	8	no	—
138	C₄×D₄₇	#106	1	16	yes	complemented
139	C₈×C₄₇	—	2	2	no	—
140	C₈×C₄₇	#117	1	4	yes	complemented
141	C₄₇⋊C₈	—	2	2	no	—
142	C₄₇⋊C₈	#117	1	4	yes	complemented
143	#20816	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₄₇	#532	1	2	yes	—
146	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
147	C₂²×C₄₇	C₂×#135	1	2	yes	—
148	C₂×D₄₇	—	2	2	no	—
149	C₂×D₄₇	—	2	8	no	—
150	C₂×D₄₇	—	2	2	no	—
151	C₂×D₄₇	C₂×#106	1	4	yes	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	8	4	no	—
154	C₄×C₄₇	C₂×#106	1	4	yes	—
155	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
156	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
157	C₄×C₄₇	#533	1	2	yes	—
158	Dic₄₇	—	2	2	no	—
159	Dic₄₇	—	8	4	no	—
160	Dic₄₇	C₂×#106	1	4	yes	—
161	Dic₄₇	#427	1	4	yes	—
162	C₂²×C₄×C₈	—	47	1	no	—
163	#2285	—	47	2	no	—
164	#1844	—	47	2	no	—
165	#1834	—	47	2	no	—
166	#1807	—	47	2	no	—
167	#1807	—	47	2	no	—
168	C₄×#106	—	47	1	no	—
169	C₄×#106	—	47	2	no	—
170	C₄×#106	D₄₇	1	1	yes	characteristic, complemented
171	C₂×C₄₇	#2285	1	1	yes	characteristic
172	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
173	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
174	C₂×C₄₇	#3167	1	4	yes	—
175	D₄₇	—	2	4	no	—
176	C₂³×C₈	—	47	1	no	—
177	C₂³×C₈	—	47	1	no	—
178	C₂×#117	—	94	2	no	—
179	C₂²×C₄²	—	47	1	no	—
180	C₂×#106	—	47	2	no	—
181	C₂×#106	—	47	4	no	—
182	C₂×#106	C₂×D₄₇	1	2	yes	—
183	C₂×C₄×C₈	—	47	1	no	—
184	C₂×C₄×C₈	—	47	1	no	—
185	C₂×C₄×C₈	—	47	1	no	—
186	C₂×C₄×C₈	—	47	8	no	—
187	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
188	#384	—	47	4	no	—
189	#382	—	94	2	no	—
190	#380	—	47	4	no	—
191	#380	—	94	2	no	—
192	C₄×C₄⋊C₄	—	47	2	no	—
193	C₄×C₄⋊C₄	—	94	2	no	—
194	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
195	#340	—	47	2	no	—
196	#340	—	47	4	no	—
197	#340	C₂×D₄₇	1	2	yes	—
198	C₄₇	#20816	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₄	—	47	1	no	—
200	C₂³×C₄	—	47	1	no	—
201	C₂³×C₄	—	47	1	no	—
202	C₂²×C₈	—	47	1	no	—
203	C₂²×C₈	—	47	1	no	—
204	C₂²×C₈	—	47	4	no	—
205	C₂²×C₈	—	47	4	no	—
206	C₂²×C₈	—	94	4	no	—
207	C₂²×C₈	—	94	4	no	—
208	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
210	#117	—	188	8	no	—
211	C₂×C₄⋊C₄	—	47	4	no	—
212	C₂×C₄⋊C₄	—	94	4	no	—
213	C₂×C₄⋊C₄	—	94	4	no	—
214	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
215	C₂×C₂²⋊C₄	—	94	4	no	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	—	47	1	no	—
218	C₂×C₄²	—	47	1	no	—
219	C₂×C₄²	—	47	1	no	—
220	C₂×C₄²	—	47	4	no	—
221	C₂×C₄²	—	47	4	no	—
222	C₂×C₄²	—	94	2	no	—
223	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
224	#106	—	2	4	no	—
225	#106	—	94	8	no	—
226	#106	—	94	4	no	—
227	C₄×C₈	—	47	4	no	—
228	C₄×C₈	—	47	4	no	—
229	C₄×C₈	—	47	4	no	—
230	C₄×C₈	C₄×D₄₇	1	4	yes	—
231	#95	—	2	2	no	—
232	#95	—	94	4	no	—
233	#95	—	94	2	no	—
234	C₂⁴	—	47	1	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	47	1	no	—
237	C₂²×C₄	—	47	1	no	—
238	C₂²×C₄	—	47	4	no	—
239	C₂²×C₄	—	47	8	no	—
240	C₂²×C₄	—	47	1	no	—
241	C₂²×C₄	—	47	4	no	—
242	C₂²×C₄	—	47	2	no	—
243	C₂²×C₄	—	47	2	no	—
244	C₂²×C₄	—	94	4	no	—
245	C₂²×C₄	—	94	2	no	—
246	C₂²×C₄	—	94	2	no	—
247	C₂²×C₄	—	94	4	no	—
248	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	47	2	no	—
253	C₂×C₈	—	47	4	no	—
254	C₂×C₈	—	47	2	no	—
255	C₂×C₈	—	94	8	no	—
256	C₂×C₈	—	94	2	no	—
257	C₂×C₈	—	94	8	no	—
258	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
259	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
260	C₂×C₈	#10625350	1	4	yes	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	94	4	no	—
264	C₄⋊C₄	—	94	4	no	—
265	C₄⋊C₄	—	188	8	no	—
266	C₄⋊C₄	—	188	8	no	—
267	C₂²⋊C₄	—	188	8	no	—
268	C₄²	—	4	4	no	—
269	C₄²	—	47	4	no	—
270	C₄²	—	47	4	no	—
271	C₄²	—	47	2	no	—
272	C₄²	—	47	2	no	—
273	C₄²	—	188	4	no	—
274	C₄²	C₂×C₄×D₄₇	1	2	yes	—
275	C₄²	Q₈×D₄₇	1	2	yes	—
276	C₂³	—	47	2	no	—
277	C₂³	—	47	4	no	—
278	C₂³	—	94	2	no	—
279	C₂³	—	94	2	no	—
280	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	47	2	no	—
286	C₂×C₄	—	47	2	no	—
287	C₂×C₄	—	47	4	no	—
288	C₂×C₄	—	47	4	no	—
289	C₂×C₄	—	47	16	no	—
290	C₂×C₄	—	47	2	no	—
291	C₂×C₄	—	94	8	no	—
292	C₂×C₄	—	94	2	no	—
293	C₂×C₄	—	94	2	no	—
294	C₂×C₄	—	94	2	no	—
295	C₂×C₄	—	94	4	no	—
296	C₂×C₄	—	188	4	no	—
297	C₂×C₄	—	188	4	no	—
298	C₂×C₄	—	188	4	no	—
299	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
300	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
301	C₂×C₄	SD₁₆×D₄₇	1	4	yes	—
302	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
303	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
304	C₂×C₄	#41×D₄₇	1	2	yes	—
305	C₂×C₄	C₂×#10625350	1	2	yes	—
306	C₈	—	2	2	no	—
307	C₈	—	47	4	no	—
308	C₈	—	94	2	no	—
309	C₈	#13093981	1	4	yes	complemented
310	C₂²	—	47	4	no	—
311	C₂²	—	94	2	no	—
312	C₂²	—	94	2	no	—
313	C₂²	—	94	8	no	—
314	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
315	C₂²	#15689670	1	2	yes	—
316	C₂²	C₂×SD₁₆×D₄₇	1	2	yes	—
317	C₂²	#135×D₄₇	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	8	4	no	—
320	C₄	—	47	4	no	—
321	C₄	—	47	4	no	—
322	C₄	—	94	2	no	—
323	C₄	—	376	4	no	—
324	C₄	#106×D₄₇	1	4	yes	—
325	C₄	#117×D₄₇	1	1	yes	characteristic
326	C₄	C₂×#13093981	1	1	yes	characteristic
327	C₄	#15689677	1	2	yes	—
328	C₂	—	94	4	no	—
329	C₂	#20245792	1	1	yes	characteristic
330	C₂	C₂×#106×D₄₇	1	1	yes	characteristic
331	C₂	#427×D₄₇	1	1	yes	characteristic
332	C₂	#20253944	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#106×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#106	D₄₇
hypocenter	C₄₇	#20816
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
3. #20243543	C₂	semidirect
4. #20243514	C₂	semidirect
7. C₄×#13093955	C₂	semidirect
8. C₄×#13093956	C₂	semidirect
9. C₄×#106×C₄₇	C₂	semidirect
19. C₄×C₈×D₄₇	C₄	semidirect
138. C₄×D₄₇	#106	semidirect
140. C₈×C₄₇	#117	semidirect
142. C₄₇⋊C₈	#117	semidirect
170. C₄×#106	D₄₇	semidirect
198. C₄₇	#20816	semidirect
309. C₈	#13093981	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₄₇	2
#20245792	2
#20243543	2
#20243514	2
#20243412	2
#20243410	2
C₄×#13093955	2
C₄×#13093956	2
C₄×#106×C₄₇	2
#20816	47