Subgroups — #433×Dic₄₇ #26214425

Order 12032.

970 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689184	C₂	1	1	yes	maximal, characteristic
2	#117×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243247	C₂	1	2	yes	maximal, complemented
4	C₄⋊C₈×Dic₄₇	C₂	1	2	yes	maximal, complemented
5	#20243156	C₂	1	2	yes	maximal
6	C₂²⋊C₈×Dic₄₇	C₂	1	2	yes	maximal
7	#20243141	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243139	C₂	1	1	yes	maximal, characteristic, complemented
9	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₇×#433	C₂	1	1	yes	maximal, characteristic
12	C₂×#13093983	C₂²	1	1	yes	characteristic
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	#15689184	C₄	1	8	yes	complemented
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×#13093935	C₂²	1	1	yes	characteristic
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
21	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
22	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
23	C₄⋊C₄×Dic₄₇	C₄	1	4	yes	—
24	C₂²⋊C₄×Dic₄₇	C₄	1	4	yes	—
25	#15688969	C₂²	1	1	yes	characteristic
26	#15688969	C₂²	1	1	yes	characteristic
27	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
28	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
29	#15688877	C₂²	1	2	yes	—
30	#15688877	C₂²	1	2	yes	—
31	#15688877	C₂²	1	2	yes	—
32	#15688877	C₂²	1	2	yes	—
33	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#433	C₄	1	8	yes	complemented
35	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
37	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
41	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
42	#13093983	—	2	4	no	—
43	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₄₇	—	2	4	no	—
46	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
54	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
55	#13093935	C₂×C₄	1	4	yes	complemented
56	#13093934	—	2	4	no	—
57	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
58	C₈×Dic₄₇	—	2	4	no	—
59	C₈×Dic₄₇	—	2	4	no	—
60	C₈×Dic₄₇	—	2	4	no	—
61	C₄₇⋊#95	C₂×C₄	1	4	yes	—
62	C₄₇⋊#95	C₂×C₄	1	4	yes	—
63	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
65	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
66	#117×C₄₇	C₂×C₄	1	4	yes	—
67	#117×C₄₇	C₂×C₄	1	4	yes	—
68	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	complemented
73	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
74	#97×C₄₇	C₂×C₄	1	4	yes	complemented
75	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
76	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	—	2	2	no	—
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₇	—	2	4	no	—
86	C₂²×Dic₄₇	—	2	8	no	—
87	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
88	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
89	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
90	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₇	—	2	2	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	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
97	C₄⋊C₄×C₄₇	C₄²	1	8	yes	—
98	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
99	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
102	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
103	C₂²⋊Dic₄₇	—	2	4	no	—
104	C₄⋊Dic₄₇	—	2	4	no	—
105	C₄×Dic₄₇	—	2	2	no	—
106	C₄×Dic₄₇	—	2	2	no	—
107	C₄×Dic₄₇	—	2	4	no	—
108	C₄×Dic₄₇	—	2	8	no	—
109	C₄×Dic₄₇	—	2	8	no	—
110	C₄×Dic₄₇	#41	1	4	yes	—
111	C₄×Dic₄₇	#41	1	4	yes	—
112	C₂×C₄₇⋊C₈	—	2	2	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
117	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
118	C₂³×C₄₇	—	2	1	no	—
119	C₂³×C₄₇	—	2	1	no	—
120	C₂³×C₄₇	—	2	4	no	—
121	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	2	4	no	—
128	C₂×C₄×C₄₇	—	2	4	no	—
129	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
130	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
131	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
136	C₂×Dic₄₇	—	2	4	no	—
137	C₂×Dic₄₇	—	2	2	no	—
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	2	4	no	—
141	C₂×Dic₄₇	—	4	8	no	—
142	C₂×Dic₄₇	#117	1	4	yes	—
143	C₂×Dic₄₇	#117	1	4	yes	—
144	C₂×Dic₄₇	#131	1	8	yes	—
145	C₈×C₄₇	—	2	4	no	—
146	C₈×C₄₇	—	2	4	no	—
147	C₄₇⋊C₈	—	2	4	no	—
148	C₄₇⋊C₈	—	2	4	no	—
149	C₄×#433	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	4	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	4	2	no	—
153	C₂²×C₄₇	—	4	2	no	—
154	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#131	1	4	yes	—
158	C₄×C₄₇	—	2	4	no	—
159	C₄×C₄₇	—	2	4	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	C₄×#41	1	2	yes	—
162	C₄×C₄₇	C₄×#41	1	2	yes	—
163	Dic₄₇	—	2	4	no	—
164	Dic₄₇	—	4	2	no	—
165	Dic₄₇	—	4	2	no	—
166	Dic₄₇	#433	1	8	yes	direct product factor
167	C₂×#433	—	47	1	no	—
168	C₂×#433	D₄₇	1	1	yes	characteristic
169	C₄×#117	—	47	1	no	—
170	#1768	—	47	2	no	—
171	C₄×C₄⋊C₈	—	47	2	no	—
172	#1750	—	47	2	no	—
173	C₄×C₂²⋊C₈	—	47	2	no	—
174	#1746	—	47	1	no	—
175	#1746	—	47	1	no	—
176	C₄×#97	—	47	1	no	—
177	C₄²×C₈	—	47	1	no	—
178	C₂×C₄₇	—	4	2	no	—
179	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
180	C₂×C₄₇	C₄×#131	1	2	yes	—
181	C₂×C₄₇	C₂×#433	1	2	yes	—
182	C₂×C₄₇	#2922	1	2	yes	—
183	C₂×#117	—	47	1	no	—
184	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
185	C₂²×C₄²	—	47	1	no	—
186	#433	—	47	8	no	—
187	#433	Dic₄₇	1	8	yes	direct product factor
188	C₂×C₄⋊C₈	—	47	2	no	—
189	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
190	C₂×C₂²⋊C₈	—	47	2	no	—
191	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
192	C₂×#97	—	47	1	no	—
193	C₂×#97	—	47	2	no	—
194	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
195	C₂×C₄×C₈	—	47	1	no	—
196	C₂×C₄×C₈	—	47	2	no	—
197	C₂×C₄×C₈	—	47	2	no	—
198	C₂×C₄×C₈	—	47	2	no	—
199	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
200	C₄×C₄⋊C₄	—	47	4	no	—
201	C₄×C₂²⋊C₄	—	47	4	no	—
202	#376	—	47	1	no	—
203	#376	—	47	1	no	—
204	C₄³	—	47	1	no	—
205	C₄³	—	47	1	no	—
206	#336	—	47	2	no	—
207	#336	—	47	2	no	—
208	#336	—	47	2	no	—
209	#336	—	47	2	no	—
210	C₄₇	C₄×#433	1	1	yes	characteristic, complemented, Sylow
211	C₂³×C₄	—	47	2	no	—
212	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
213	C₂²×C₈	—	47	2	no	—
214	C₂²×C₈	—	47	2	no	—
215	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
216	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
217	#117	—	94	4	no	—
218	#117	C₄×D₄₇	1	4	yes	—
219	#117	C₂×Dic₄₇	1	4	yes	—
220	C₂×C₄⋊C₄	—	47	2	no	—
221	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
222	C₂×C₂²⋊C₄	—	47	2	no	—
223	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
224	C₂×C₄²	—	47	1	no	—
225	C₂×C₄²	—	47	1	no	—
226	C₂×C₄²	—	47	4	no	—
227	C₂×C₄²	—	47	4	no	—
228	C₂×C₄²	—	47	2	no	—
229	C₂×C₄²	—	47	1	no	—
230	C₂×C₄²	—	47	1	no	—
231	C₂×C₄²	—	94	4	no	—
232	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
234	C₄⋊C₈	—	47	8	no	—
235	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	complemented
236	C₂²⋊C₈	—	47	8	no	—
237	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
238	#97	—	47	4	no	—
239	#97	—	94	4	no	—
240	#97	C₂×Dic₄₇	1	4	yes	complemented
241	C₄×C₈	—	47	4	no	—
242	C₄×C₈	—	94	4	no	—
243	C₄×C₈	—	94	4	no	—
244	C₄×C₈	—	94	4	no	—
245	C₄×C₈	C₂×Dic₄₇	1	4	yes	complemented
246	#95	—	47	4	no	—
247	#95	—	47	4	no	—
248	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	—	47	2	no	—
253	C₂²×C₄	—	47	2	no	—
254	C₂²×C₄	—	47	2	no	—
255	C₂²×C₄	—	94	4	no	—
256	C₂²×C₄	—	94	8	no	—
257	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
258	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
259	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
262	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	47	4	no	—
268	C₂×C₈	—	47	4	no	—
269	C₂×C₈	—	94	2	no	—
270	C₂×C₈	—	94	2	no	—
271	C₂×C₈	—	94	2	no	—
272	C₂×C₈	—	94	2	no	—
273	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
274	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
275	C₄⋊C₄	—	94	4	no	—
276	C₄⋊C₄	C₄×Dic₄₇	1	8	yes	—
277	C₂²⋊C₄	—	94	4	no	—
278	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
279	C₄²	—	47	4	no	—
280	C₄²	—	47	4	no	—
281	C₄²	—	94	8	no	—
282	C₄²	—	94	8	no	—
283	C₄²	—	94	4	no	—
284	C₄²	—	94	2	no	—
285	C₄²	—	94	2	no	—
286	C₄²	C₂×C₄×D₄₇	1	2	yes	—
287	C₄²	C₂×C₄×D₄₇	1	2	yes	—
288	C₄²	C₂²×Dic₄₇	1	2	yes	—
289	C₄²	C₂²×Dic₄₇	1	2	yes	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	4	no	—
293	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
294	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	47	4	no	—
302	C₂×C₄	—	47	8	no	—
303	C₂×C₄	—	47	4	no	—
304	C₂×C₄	—	94	4	no	—
305	C₂×C₄	—	94	2	no	—
306	C₂×C₄	—	94	2	no	—
307	C₂×C₄	—	94	4	no	—
308	C₂×C₄	—	94	4	no	—
309	C₂×C₄	—	188	8	no	—
310	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
311	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
312	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
313	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
315	C₂×C₄	#41×D₄₇	1	2	yes	—
316	C₂×C₄	#41×D₄₇	1	2	yes	—
317	C₈	—	2	4	no	—
318	C₈	—	2	4	no	—
319	C₈	—	94	4	no	—
320	C₈	—	94	4	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
326	C₂²	#117×D₄₇	1	1	yes	characteristic
327	C₂²	#117×D₄₇	1	1	yes	characteristic
328	C₂²	#131×D₄₇	1	4	yes	—
329	C₄	—	2	4	no	—
330	C₄	—	2	4	no	—
331	C₄	—	2	2	no	—
332	C₄	—	47	8	no	—
333	C₄	—	94	4	no	—
334	C₄	—	188	2	no	—
335	C₄	—	188	2	no	—
336	C₄	#15689591	1	2	yes	—
337	C₄	#41×Dic₄₇	1	2	yes	—
338	C₂	—	4	2	no	—
339	C₂	#117×Dic₄₇	1	1	yes	characteristic
340	C₂	#131×Dic₄₇	1	2	yes	—
341	C₂	#433×D₄₇	1	2	yes	—
342	C₂	#20252029	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#433	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#433	D₄₇
hypocenter	C₄₇	C₄×#433
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
3. #20243247	C₂	semidirect
4. C₄⋊C₈×Dic₄₇	C₂	semidirect
7. #20243141	C₂	semidirect
8. #20243139	C₂	semidirect
9. #97×Dic₄₇	C₂	semidirect
10. C₄×C₈×Dic₄₇	C₂	semidirect
14. #15689184	C₄	semidirect
34. C₄₇×#433	C₄	semidirect
53. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
55. #13093935	C₂×C₄	semidirect
57. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
72. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
74. #97×C₄₇	C₂×C₄	semidirect
75. C₄×C₈×C₄₇	C₂×C₄	semidirect
166. Dic₄₇	#433	direct (N × H)
187. #433	Dic₄₇	direct (N × H)
210. C₄₇	C₄×#433	semidirect
235. C₄⋊C₈	C₂×Dic₄₇	semidirect
240. #97	C₂×Dic₄₇	semidirect
245. C₄×C₈	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689184	2
#117×Dic₄₇	2
#20243247	2
C₄⋊C₈×Dic₄₇	2
#20243156	2
C₂²⋊C₈×Dic₄₇	2
#20243141	2
#20243139	2
#97×Dic₄₇	2
C₄×C₈×Dic₄₇	2
C₂×C₄₇×#433	2
C₄×#433	47

Subgroups — #433×Dic47 #26214425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #433×Dic₄₇ #26214425