Subgroups — #26121889

Order 12032.

918 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688877	C₂	1	2	yes	maximal
3	C₂×#15688877	C₂	1	2	yes	maximal
4	C₄×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
9	C₂×C₄×C₄₇⋊C₈	—	2	4	no	—
10	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	complemented
11	C₂×C₄₇⋊#95	C₄	1	2	yes	—
12	C₂×C₄₇⋊#95	C₄	1	2	yes	—
13	#15688877	—	2	4	no	—
14	#15688877	—	2	4	no	—
15	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#336	C₄	1	4	yes	complemented
18	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	1	no	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
28	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	C₄×C₄₇⋊C₈	—	2	8	no	—
36	C₄₇⋊#95	—	2	4	no	—
37	C₄₇⋊#95	C₈	1	8	yes	complemented
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	C₂²×C₈×C₄₇	—	2	2	no	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
58	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
60	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	2	4	no	—
76	C₂²×Dic₄₇	—	2	4	no	—
77	C₂²×Dic₄₇	C₂×C₈	1	8	yes	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	4	no	—
81	C₂×C₈×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	—	4	2	no	—
83	C₂×C₈×C₄₇	—	4	2	no	—
84	C₂×C₈×C₄₇	—	4	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₄×Dic₄₇	—	4	2	no	—
89	C₄×Dic₄₇	—	4	2	no	—
90	C₄×Dic₄₇	—	4	2	no	—
91	C₄×Dic₄₇	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	2	4	no	—
100	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
101	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
102	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
103	C₂³×C₄₇	#117	1	2	yes	—
104	C₂³×C₄₇	C₄×D₄	1	2	yes	—
105	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
107	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
108	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
109	C₂³×C₄₇	#131	1	2	yes	—
110	C₂³×C₄₇	#131	1	2	yes	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
118	C₂×C₄×C₄₇	#97	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#117	1	2	yes	—
126	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#121	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₄₇	#123	1	2	yes	—
132	C₂×C₄×C₄₇	#123	1	2	yes	—
133	C₂×C₄×C₄₇	#124	1	2	yes	—
134	C₂×C₄×C₄₇	#125	1	2	yes	—
135	C₂×C₄×C₄₇	#126	1	2	yes	—
136	C₂×C₄×C₄₇	#126	1	2	yes	—
137	C₂×Dic₄₇	—	2	8	no	—
138	C₂×Dic₄₇	—	2	8	no	—
139	C₂×Dic₄₇	—	2	8	no	—
140	C₂×Dic₄₇	—	4	4	no	—
141	C₂×Dic₄₇	—	4	4	no	—
142	C₂×Dic₄₇	—	4	4	no	—
143	C₂×Dic₄₇	—	4	4	no	—
144	C₈×C₄₇	—	4	4	no	—
145	C₄₇⋊C₈	—	2	4	no	—
146	C₄₇⋊C₈	—	2	4	no	—
147	#10797	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	#376	1	1	yes	characteristic
149	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₄₇	#387	1	2	yes	—
152	C₂²×C₄₇	#388	1	2	yes	—
153	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
155	C₂²×C₄₇	#405	1	1	yes	characteristic
156	C₂²×C₄₇	#405	1	1	yes	characteristic
157	C₂²×C₄₇	#431	1	4	yes	—
158	C₂²×C₄₇	#433	1	4	yes	—
159	C₂²×C₄₇	C₈×D₄	1	4	yes	—
160	C₂²×C₄₇	C₈×D₄	1	2	yes	—
161	C₂²×C₄₇	#435	1	4	yes	—
162	C₂²×C₄₇	#436	1	2	yes	—
163	C₂²×C₄₇	C₈×Q₈	1	2	yes	—
164	C₂²×C₄₇	#446	1	2	yes	—
165	C₄×C₄₇	—	2	4	no	—
166	C₄×C₄₇	—	2	4	no	—
167	C₄×C₄₇	#380	1	1	yes	characteristic
168	C₄×C₄₇	#381	1	1	yes	characteristic
169	C₄×C₄₇	#382	1	1	yes	characteristic
170	C₄×C₄₇	#383	1	1	yes	characteristic
171	C₄×C₄₇	#387	1	2	yes	—
172	C₄×C₄₇	#388	1	2	yes	—
173	Dic₄₇	—	4	8	no	—
174	C₂²×C₄×C₈	—	47	1	no	—
175	C₂×#336	—	47	2	no	—
176	C₂×#336	—	47	2	no	—
177	C₂×#336	D₄₇	1	1	yes	characteristic
178	C₄×#95	—	47	1	no	—
179	C₂×C₄₇	#1434	1	1	yes	characteristic
180	C₂×C₄₇	#1744	1	1	yes	characteristic
181	C₂×C₄₇	#1746	1	1	yes	characteristic
182	C₂×C₄₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₄₇	#1752	1	1	yes	characteristic
184	C₂×C₄₇	C₈×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄₇	#1770	1	1	yes	characteristic
186	C₂×C₄₇	#1916	1	4	yes	—
187	C₂×C₄₇	#1923	1	4	yes	—
188	C₂³×C₈	—	47	1	no	—
189	C₂³×C₈	—	47	1	no	—
190	C₂³×C₈	C₂×D₄₇	1	2	yes	—
191	C₂²×C₄²	—	47	2	no	—
192	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
193	C₂×C₄×C₈	—	47	4	no	—
194	C₂×C₄×C₈	—	94	4	no	—
195	C₂×#95	—	47	2	no	—
196	C₂×#95	—	47	2	no	—
197	#336	—	94	4	no	—
198	#336	—	94	4	no	—
199	#336	Dic₄₇	1	4	yes	complemented
200	C₄₇	#10797	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₄	—	47	4	no	—
202	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
203	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
204	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	—	47	2	no	—
212	C₂²×C₈	—	47	2	no	—
213	C₂²×C₈	—	47	2	no	—
214	C₂²×C₈	—	47	4	no	—
215	C₂²×C₈	—	47	2	no	—
216	C₂²×C₈	—	94	2	no	—
217	C₂²×C₈	—	94	2	no	—
218	C₂²×C₈	—	94	2	no	—
219	C₂²×C₈	—	94	1	no	—
220	C₂²×C₈	—	94	1	no	—
221	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	—	94	2	no	—
225	C₂×C₄²	—	94	2	no	—
226	C₂×C₄²	—	94	2	no	—
227	C₂×C₄²	—	94	2	no	—
228	C₂×C₄²	—	94	2	no	—
229	C₂×C₄²	—	94	2	no	—
230	C₂×C₄²	C₄×D₄₇	1	2	yes	—
231	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
232	C₄×C₈	—	94	8	no	—
233	#95	—	47	8	no	—
234	#95	—	94	4	no	—
235	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	47	8	no	—
241	C₂²×C₄	—	94	4	no	—
242	C₂²×C₄	—	94	4	no	—
243	C₂²×C₄	—	94	4	no	—
244	C₂²×C₄	—	94	4	no	—
245	C₂²×C₄	—	94	4	no	—
246	C₂²×C₄	—	94	4	no	—
247	C₂²×C₄	C₈×D₄₇	1	4	yes	—
248	C₂²×C₄	#10625324	1	4	yes	—
249	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
250	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
251	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
252	C₂²×C₄	#10625350	1	2	yes	—
253	C₂²×C₄	#10625350	1	2	yes	—
254	C₂²×C₄	D₄×D₄₇	1	2	yes	—
255	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
256	C₂²×C₄	#10625352	1	2	yes	—
257	C₂²×C₄	#10625352	1	1	yes	characteristic
258	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
259	C₂²×C₄	#10625354	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	47	4	no	—
269	C₂×C₈	—	47	4	no	—
270	C₂×C₈	—	94	2	no	—
271	C₂×C₈	—	94	2	no	—
272	C₂×C₈	—	94	2	no	—
273	C₂×C₈	—	94	2	no	—
274	C₂×C₈	—	94	4	no	—
275	C₂×C₈	—	94	4	no	—
276	C₂×C₈	—	94	4	no	—
277	C₂×C₈	—	94	4	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	4	no	—
281	C₄²	—	188	2	no	—
282	C₄²	—	188	2	no	—
283	C₄²	—	188	2	no	—
284	C₄²	—	188	2	no	—
285	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
286	C₂³	#13093981	1	2	yes	—
287	C₂³	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
288	C₂³	#13093986	1	1	yes	characteristic
289	C₂³	#13093989	1	1	yes	characteristic
290	C₂³	C₂×C₈×D₄₇	1	2	yes	—
291	C₂³	C₂×#10625324	1	2	yes	—
292	C₂³	#13094032	1	2	yes	—
293	C₂³	#13094032	1	2	yes	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	94	8	no	—
301	C₂×C₄	—	94	8	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	—	188	4	no	—
307	C₂×C₄	C₈×Dic₄₇	1	4	yes	—
308	C₂×C₄	#13093934	1	4	yes	—
309	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
310	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
311	C₂×C₄	#13093980	1	1	yes	characteristic
312	C₂×C₄	#13093981	1	1	yes	characteristic
313	C₂×C₄	#13093981	1	1	yes	characteristic
314	C₂×C₄	#13093982	1	1	yes	characteristic
315	C₂×C₄	#13093983	1	2	yes	—
316	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
317	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
318	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
319	C₂×C₄	#13093997	1	2	yes	—
320	C₂×C₄	#13094002	1	2	yes	—
321	C₂×C₄	#13094006	1	2	yes	—
322	C₂×C₄	#13094009	1	2	yes	—
323	C₂×C₄	#13094010	1	2	yes	—
324	C₂×C₄	#13094014	1	2	yes	—
325	C₂×C₄	#13094017	1	2	yes	—
326	C₂×C₄	#13094019	1	2	yes	—
327	C₈	—	4	4	no	—
328	C₈	—	94	4	no	—
329	C₈	—	94	4	no	—
330	C₂²	#15688969	1	1	yes	characteristic
331	C₂²	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
332	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
333	C₂²	#15689005	1	2	yes	—
334	C₂²	#15689012	1	2	yes	—
335	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
336	C₂²	C₂×#13093934	1	1	yes	characteristic
337	C₂²	#15689081	1	1	yes	characteristic
338	C₂²	#15689081	1	1	yes	characteristic
339	C₂²	#15689174	1	4	yes	—
340	C₂²	#15689185	1	4	yes	—
341	C₂²	(C₂²⋊D₄₇)⋊C₈	1	4	yes	—
342	C₂²	#15689188	1	2	yes	—
343	C₂²	#15689191	1	4	yes	—
344	C₂²	#15689200	1	2	yes	—
345	C₂²	#15689247	1	2	yes	—
346	C₂²	#15689250	1	2	yes	—
347	C₄	—	2	4	no	—
348	C₄	—	2	4	no	—
349	C₄	—	188	8	no	—
350	C₄	#15688982	1	1	yes	characteristic
351	C₄	#15688985	1	1	yes	characteristic
352	C₄	#15688987	1	1	yes	characteristic
353	C₄	#15688990	1	1	yes	characteristic
354	C₄	#15689009	1	2	yes	—
355	C₄	#15689016	1	2	yes	—
356	C₂	#20241613	1	1	yes	characteristic
357	C₂	#20243133	1	1	yes	characteristic
358	C₂	#20243141	1	1	yes	characteristic
359	C₂	#20243162	1	1	yes	characteristic
360	C₂	#20243167	1	1	yes	characteristic
361	C₂	#20243253	1	1	yes	characteristic
362	C₂	#20243258	1	1	yes	characteristic
363	C₂	#20243943	1	4	yes	—
364	C₂	#20243992	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₂×C₄₇×#336	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	#10797
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
17. C₄₇×#336	C₄	semidirect
37. C₄₇⋊#95	C₈	semidirect
199. #336	Dic₄₇	semidirect
200. C₄₇	#10797	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₄₇⋊C₈	2
C₂×#15688877	2
C₄×C₄₇⋊#95	2
C₂×C₄₇×#336	2
#10797	47