Subgroups — #423×Dic₄₇ #26214182

Order 12032.

1382 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₄₇	C₂	1	2	yes	maximal
3	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20243243	C₂	1	4	yes	maximal, complemented
5	C₄⋊C₈×Dic₄₇	C₂	1	4	yes	maximal, complemented
6	#20243081	C₂	1	2	yes	maximal
7	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₄₇	C₄	1	4	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	#15689147	C₄	1	8	yes	complemented
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
14	M₁₆×Dic₄₇	—	2	8	no	—
15	C₂×#13093934	C₂²	1	4	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
17	C₄²×Dic₄₇	C₄	1	2	yes	—
18	C₄²×Dic₄₇	C₄	1	2	yes	—
19	C₄²×Dic₄₇	C₂²	1	2	yes	—
20	C₄²×Dic₄₇	C₂²	1	2	yes	—
21	#15688877	C₂²	1	8	yes	—
22	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#423	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
26	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
27	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	—	2	4	no	—
30	C₂×#10625329	C₂×C₄	1	8	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	16	no	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
42	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
44	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
46	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
47	#13093934	—	2	8	no	—
48	C₈×Dic₄₇	—	2	8	no	—
49	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₄₇	—	2	4	no	—
52	C₂×M₁₆×C₄₇	—	2	4	no	—
53	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
60	C₄⋊C₈×C₄₇	C₂×C₄	1	16	yes	complemented
61	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	2	8	no	—
76	C₂²×Dic₄₇	—	2	8	no	—
77	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
78	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
80	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
81	M₁₆×C₄₇	—	2	16	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
85	C₄²×C₄₇	C₄²	1	4	yes	—
86	C₄²×C₄₇	C₄²	1	4	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
89	C₄×Dic₄₇	—	2	4	no	—
90	C₄×Dic₄₇	—	2	8	no	—
91	C₄×Dic₄₇	—	2	2	no	—
92	C₄×Dic₄₇	—	2	2	no	—
93	C₄×Dic₄₇	—	2	8	no	—
94	C₄×Dic₄₇	—	2	8	no	—
95	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
96	C₄×Dic₄₇	M₁₆	1	16	yes	complemented
97	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
98	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
99	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
100	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
101	#10625329	—	2	16	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
105	C₂³×C₄₇	—	2	4	no	—
106	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
107	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₄₇	#117	1	2	yes	—
109	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	16	no	—
112	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₇	#117	1	2	yes	—
117	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
118	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
120	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
121	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
122	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₄₇	—	2	16	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	2	16	no	—
133	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
136	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
137	C₈×C₄₇	—	2	8	no	—
138	C₄₇⋊C₈	—	2	8	no	—
139	C₄×#423	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	4	no	—
142	C₂²×C₄₇	—	2	4	no	—
143	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
144	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
145	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
147	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
148	C₄×C₄₇	—	2	4	no	—
149	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
150	C₄×C₄₇	C₄×M₁₆	1	8	yes	complemented
151	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
154	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
155	Dic₄₇	—	2	4	no	—
156	Dic₄₇	—	2	8	no	—
157	Dic₄₇	#423	1	8	yes	direct product factor
158	C₂×#423	—	47	1	no	—
159	C₂×#423	D₄₇	1	1	yes	characteristic
160	C₂×C₄×M₁₆	—	47	2	no	—
161	C₂×C₄³	—	47	1	no	—
162	#1767	—	47	4	no	—
163	C₄×C₄⋊C₈	—	47	4	no	—
164	#1728	—	47	2	no	—
165	C₂×C₄₇	—	2	4	no	—
166	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄₇	C₂×C₄×M₁₆	1	2	yes	—
168	C₂×C₄₇	C₂×#423	1	2	yes	—
169	C₂×C₄₇	#2920	1	2	yes	—
170	C₂²×M₁₆	—	47	2	no	—
171	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
172	C₂²×C₄²	—	47	1	no	—
173	C₂²×C₄²	—	47	4	no	—
174	C₂²×C₄²	—	47	1	no	—
175	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
176	#423	—	47	8	no	—
177	#423	Dic₄₇	1	8	yes	direct product factor
178	C₂×C₄⋊C₈	—	47	4	no	—
179	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
180	C₄×M₁₆	—	94	8	no	—
181	C₂×#97	—	47	4	no	—
182	C₂×C₄×C₈	—	47	4	no	—
183	C₄³	—	47	2	no	—
184	C₄³	—	47	2	no	—
185	C₄³	—	47	2	no	—
186	C₄³	—	47	2	no	—
187	#336	—	47	8	no	—
188	C₄₇	C₄×#423	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	—	47	2	no	—
190	C₂³×C₄	—	47	2	no	—
191	C₂³×C₄	C₄×D₄₇	1	2	yes	—
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×M₁₆	—	2	4	no	—
194	C₂×M₁₆	—	2	4	no	—
195	C₂×M₁₆	—	47	8	no	—
196	C₂×M₁₆	—	94	4	no	—
197	C₂×M₁₆	—	94	4	no	—
198	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
199	C₂²×C₈	—	47	4	no	—
200	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
201	C₂×C₄²	—	47	2	no	—
202	C₂×C₄²	—	47	4	no	—
203	C₂×C₄²	—	47	4	no	—
204	C₂×C₄²	—	47	4	no	—
205	C₂×C₄²	—	47	1	no	—
206	C₂×C₄²	—	47	1	no	—
207	C₂×C₄²	—	47	4	no	—
208	C₂×C₄²	—	47	4	no	—
209	C₂×C₄²	—	47	2	no	—
210	C₂×C₄²	—	47	4	no	—
211	C₂×C₄²	—	47	2	no	—
212	C₂×C₄²	—	94	16	no	—
213	C₂×C₄²	—	94	2	no	—
214	C₂×C₄²	—	94	2	no	—
215	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
216	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₄₇	1	4	yes	—
218	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
219	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
220	C₄⋊C₈	—	47	16	no	—
221	C₄⋊C₈	C₂×Dic₄₇	1	16	yes	complemented
222	#97	—	94	8	no	—
223	C₄×C₈	—	94	8	no	—
224	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	47	2	no	—
227	C₂²×C₄	—	47	8	no	—
228	C₂²×C₄	—	47	2	no	—
229	C₂²×C₄	—	47	4	no	—
230	C₂²×C₄	—	94	8	no	—
231	C₂²×C₄	—	94	8	no	—
232	C₂²×C₄	—	94	4	no	—
233	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
234	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
235	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
238	C₂²×C₄	D₄×D₄₇	1	2	yes	—
239	C₂²×C₄	#10625352	1	2	yes	—
240	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
241	C₂²×C₄	#10625354	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
243	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
244	M₁₆	—	2	16	no	—
245	M₁₆	—	94	16	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	47	8	no	—
249	C₂×C₈	—	94	4	no	—
250	C₂×C₈	—	94	4	no	—
251	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
252	C₄²	—	47	8	no	—
253	C₄²	—	47	16	no	—
254	C₄²	—	47	2	no	—
255	C₄²	—	47	2	no	—
256	C₄²	—	47	2	no	—
257	C₄²	—	47	2	no	—
258	C₄²	—	94	8	no	—
259	C₄²	—	94	8	no	—
260	C₄²	—	94	4	no	—
261	C₄²	—	94	8	no	—
262	C₄²	—	94	2	no	—
263	C₄²	—	94	2	no	—
264	C₄²	C₄×Dic₄₇	1	4	yes	—
265	C₄²	C₄×Dic₄₇	1	4	yes	—
266	C₄²	C₂×C₄×D₄₇	1	4	yes	—
267	C₄²	C₂²×Dic₄₇	1	4	yes	—
268	C₂³	—	2	4	no	—
269	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
270	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
271	C₂³	#13093982	1	2	yes	—
272	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	16	no	—
275	C₂×C₄	—	47	4	no	—
276	C₂×C₄	—	47	8	no	—
277	C₂×C₄	—	47	4	no	—
278	C₂×C₄	—	47	8	no	—
279	C₂×C₄	—	94	16	no	—
280	C₂×C₄	—	94	16	no	—
281	C₂×C₄	—	94	4	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
285	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
287	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
288	C₂×C₄	#13093982	1	2	yes	—
289	C₂×C₄	#13093986	1	4	yes	—
290	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
291	C₂×C₄	#13093989	1	4	yes	—
292	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
293	C₂×C₄	M₁₆×D₄₇	1	8	yes	—
294	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
297	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
299	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
300	C₈	—	2	8	no	—
301	C₈	—	94	8	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	4	no	—
305	C₂²	C₄⋊C₄×Dic₄₇	1	4	yes	—
306	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
307	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
308	C₂²	C₂×#13093982	1	1	yes	characteristic
309	C₂²	C₂×M₁₆×D₄₇	1	4	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	47	8	no	—
312	C₄	—	94	4	no	—
313	C₄	—	94	8	no	—
314	C₄	C₄⋊C₄×Dic₄₇	1	4	yes	—
315	C₄	M₁₆×Dic₄₇	1	8	yes	complemented
316	C₄	C₂×#13093986	1	1	yes	characteristic
317	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
318	C₄	C₂×#13093989	1	1	yes	characteristic
319	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
320	C₂	—	2	4	no	—
321	C₂	C₂×C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
322	C₂	C₂×M₁₆×Dic₄₇	1	2	yes	—
323	C₂	#423×D₄₇	1	2	yes	—
324	C₂	#20252020	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	C₄×#423
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #20243243	C₂	semidirect
5. C₄⋊C₈×Dic₄₇	C₂	semidirect
12. #15689147	C₄	semidirect
24. C₄₇×#423	C₄	semidirect
46. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
60. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
96. C₄×Dic₄₇	M₁₆	semidirect
150. C₄×C₄₇	C₄×M₁₆	semidirect
157. Dic₄₇	#423	direct (N × H)
177. #423	Dic₄₇	direct (N × H)
188. C₄₇	C₄×#423	semidirect
221. C₄⋊C₈	C₂×Dic₄₇	semidirect
315. C₄	M₁₆×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×M₁₆×Dic₄₇	2
C₂×C₄²×Dic₄₇	2
#20243243	2
C₄⋊C₈×Dic₄₇	2
#20243081	2
C₂×C₄₇×#423	2
C₄×#423	47

Subgroups — #423×Dic47 #26214182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #423×Dic₄₇ #26214182