Subgroups — #26224237

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689201	C₂	1	1	yes	maximal, characteristic
3	#20244329	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244187	C₂	1	1	yes	maximal, characteristic
5	#20243996	C₂	1	4	yes	maximal, complemented
6	#20243508	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243172	C₂	1	4	yes	maximal, complemented
8	#20243075	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2398	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	2	yes	—
11	Q₈×C₄₇⋊C₈	—	2	4	no	—
12	#15689201	—	2	4	no	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
18	C₂×#13093935	C₂²	1	1	yes	characteristic
19	C₂×#13093935	C₂²	1	2	yes	—
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
23	#15688878	C₂²	1	4	yes	—
24	#15688878	C₂²	1	4	yes	—
25	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#400	C₄	1	2	yes	—
28	C₄₇×#397	C₄	1	2	yes	—
29	C₄₇×#394	C₄	1	2	yes	—
30	C₄₇×#393	C₄	1	2	yes	—
31	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
32	C₄₇×#376	C₂²	1	1	yes	characteristic
33	C₂×#10625329	—	2	4	no	—
34	C₂×#10625329	—	2	8	no	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
42	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
43	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
44	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
45	#13093935	—	2	2	no	—
46	#13093935	—	2	4	no	—
47	C₄×C₄₇⋊C₈	—	2	4	no	—
48	C₄×C₄₇⋊C₈	—	2	4	no	—
49	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
50	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
53	C₄×Q₈×C₄₇	—	2	2	no	—
54	C₄×Q₈×C₄₇	—	2	2	no	—
55	C₄×D₄×C₄₇	—	2	2	no	—
56	C₄×D₄×C₄₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
62	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	#95×C₄₇	C₂×C₄	1	2	yes	—
67	#95×C₄₇	C₂×C₄	1	2	yes	—
68	#95×C₄₇	C₂×C₄	1	4	yes	—
69	#95×C₄₇	C₂×C₄	1	4	yes	—
70	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂×Q₈×C₄₇	—	2	2	no	—
72	C₂×Q₈×C₄₇	—	2	4	no	—
73	C₂×D₄×C₄₇	—	2	2	no	—
74	C₂×D₄×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	—	2	2	no	—
77	C₂²×C₄×C₄₇	—	2	4	no	—
78	C₂²×C₄×C₄₇	—	2	4	no	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	—	2	4	no	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₂²⋊C₄×C₄₇	—	2	4	no	—
90	C₂²⋊C₄×C₄₇	—	2	8	no	—
91	C₂²⋊C₄×C₄₇	—	4	4	no	—
92	C₄²×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	C₂×D₄	1	2	yes	—
97	C₄²×C₄₇	#41	1	2	yes	—
98	#10625329	—	4	8	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	2	4	no	—
105	C₂×C₄₇⋊C₈	—	2	4	no	—
106	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
107	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
108	C₂×C₄₇⋊C₈	#41	1	4	yes	—
109	C₂×C₄₇⋊C₈	#41	1	4	yes	—
110	C₂³×C₄₇	—	2	2	no	—
111	C₂³×C₄₇	—	2	4	no	—
112	C₂³×C₄₇	—	2	4	no	—
113	C₂³×C₄₇	—	2	2	no	—
114	C₂³×C₄₇	—	2	2	no	—
115	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	Q₈×C₄₇	—	2	8	no	—
117	D₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	4	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	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	4	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₄₇	#131	1	4	yes	—
135	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
137	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₄₇⋊C₈	—	2	4	no	—
140	C₄₇⋊C₈	—	2	2	no	—
141	C₄₇⋊C₈	—	4	2	no	—
142	C₄₇⋊C₈	—	4	2	no	—
143	C₄₇⋊C₈	#124	1	4	yes	—
144	#21639	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	—	4	2	no	—
147	C₂²×C₄₇	—	4	4	no	—
148	C₂²×C₄₇	—	4	4	no	—
149	C₂²×C₄₇	—	4	2	no	—
150	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂²×C₄₇	C₄×#41	1	2	yes	—
152	C₂²×C₄₇	C₂×#131	1	2	yes	—
153	C₂²×C₄₇	#568	1	2	yes	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	2	4	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	4	2	no	—
158	C₄×C₄₇	—	4	4	no	—
159	C₄×C₄₇	—	4	2	no	—
160	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
161	C₄×C₄₇	#529	1	2	yes	—
162	C₄×C₄₇	#532	1	1	yes	characteristic
163	C₂×C₈×Q₈	—	47	1	no	—
164	C₂×#436	—	47	1	no	—
165	#2398	D₄₇	1	1	yes	characteristic
166	#1985	—	47	1	no	—
167	#1955	—	47	1	no	—
168	#1923	—	47	4	no	—
169	#1832	—	47	1	no	—
170	#1753	—	47	4	no	—
171	#1726	—	47	1	no	—
172	C₂×C₄₇	—	8	2	no	—
173	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
174	C₂×C₄₇	#2929	1	1	yes	characteristic
175	C₂×C₄₇	#2930	1	1	yes	characteristic
176	C₂×C₄₇	#2969	1	4	yes	—
177	C₂²×M₁₆	—	47	2	no	—
178	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
179	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
180	C₈×Q₈	—	94	4	no	—
181	#436	—	94	4	no	—
182	C₂×C₄⋊C₈	—	47	1	no	—
183	C₂×C₄⋊C₈	—	47	1	no	—
184	C₂×C₄⋊C₈	—	47	2	no	—
185	C₂×C₂²⋊C₈	—	47	2	no	—
186	C₂×C₂²⋊C₈	—	47	4	no	—
187	C₂×#97	—	47	1	no	—
188	C₂×#97	—	47	2	no	—
189	C₂×C₄×C₈	—	47	1	no	—
190	C₂×C₄×C₈	—	47	2	no	—
191	C₂×C₄×C₈	—	47	2	no	—
192	#400	Dic₄₇	1	2	yes	—
193	#397	Dic₄₇	1	2	yes	—
194	#394	Dic₄₇	1	2	yes	—
195	#393	Dic₄₇	1	2	yes	—
196	C₄×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
197	#376	C₂×D₄₇	1	1	yes	characteristic
198	#336	—	47	4	no	—
199	#336	—	47	4	no	—
200	C₄₇	#21639	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
202	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
203	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
204	C₂×M₁₆	—	94	8	no	—
205	C₂×M₁₆	—	94	4	no	—
206	C₂²×C₈	—	47	1	no	—
207	C₂²×C₈	—	47	2	no	—
208	C₂²×C₈	—	47	1	no	—
209	C₂²×C₈	—	47	2	no	—
210	C₂²×C₈	—	47	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
217	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
219	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
220	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
221	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
222	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
223	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
224	C₄⋊C₈	—	94	4	no	—
225	C₄⋊C₈	—	94	2	no	—
226	C₄⋊C₈	—	94	2	no	—
227	C₂²⋊C₈	—	94	8	no	—
228	C₂²⋊C₈	—	94	4	no	—
229	#97	—	94	4	no	—
230	#97	—	94	2	no	—
231	C₄×C₈	—	47	4	no	—
232	C₄×C₈	—	94	4	no	—
233	C₄×C₈	—	94	4	no	—
234	#95	C₂×Dic₄₇	1	2	yes	—
235	#95	C₂×Dic₄₇	1	2	yes	—
236	#95	C₂×Dic₄₇	1	4	yes	—
237	#95	C₂×Dic₄₇	1	4	yes	—
238	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	4	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
248	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
250	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
253	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
254	M₁₆	—	188	8	no	—
255	C₂×C₈	—	47	2	no	—
256	C₂×C₈	—	47	4	no	—
257	C₂×C₈	—	47	2	no	—
258	C₂×C₈	—	47	4	no	—
259	C₂×C₈	—	94	2	no	—
260	C₂×C₈	—	94	2	no	—
261	C₂×C₈	—	94	2	no	—
262	C₂×C₈	—	94	2	no	—
263	C₂×C₈	—	94	2	no	—
264	C₂×C₈	—	94	4	no	—
265	C₂×C₈	—	94	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	2	8	no	—
271	C₂²⋊C₄	—	4	4	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	D₄×D₄₇	1	2	yes	—
277	C₄²	#10625352	1	2	yes	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	4	no	—
280	C₂³	—	2	4	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	2	2	no	—
283	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
284	Q₈	—	2	8	no	—
285	D₄	—	4	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
301	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
302	C₂×C₄	#13094031	1	4	yes	—
303	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
304	C₂×C₄	#41×D₄₇	1	2	yes	—
305	C₂×C₄	#41×D₄₇	1	2	yes	—
306	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
307	C₈	—	47	4	no	—
308	C₈	—	94	4	no	—
309	C₈	—	94	2	no	—
310	C₈	—	188	2	no	—
311	C₈	—	188	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
318	C₂²	#41×Dic₄₇	1	2	yes	—
319	C₂²	C₂×#13094031	1	2	yes	—
320	C₂²	#15689890	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	4	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	4	no	—
327	C₄	#124×D₄₇	1	1	yes	characteristic
328	C₄	#15689652	1	2	yes	—
329	C₄	#15689672	1	1	yes	characteristic
330	C₂	—	8	2	no	—
331	C₂	#124×Dic₄₇	1	1	yes	characteristic
332	C₂	#20252083	1	1	yes	characteristic
333	C₂	#20252092	1	1	yes	characteristic
334	C₂	#20252393	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#2398	C₂
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2398	D₄₇
hypocenter	C₄₇	#21639
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2398

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₄₇⋊C₈	C₂	semidirect
3. #20244329	C₂	semidirect
5. #20243996	C₂	semidirect
6. #20243508	C₂	semidirect
7. #20243172	C₂	semidirect
8. #20243075	C₂	semidirect
200. C₄₇	#21639	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₄₇⋊C₈	2
C₂×#15689201	2
#20244329	2
#20244187	2
#20243996	2
#20243508	2
#20243172	2
#20243075	2
C₄₇×#2398	2
#21639	47