Subgroups — #26107662

Order 12032.

836 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243368	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	Q₈⋊(C₄₇⋊C₈)	—	2	8	no	—
12	C₄₇×#533	—	2	8	no	—
13	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
15	C₂×#125×C₄₇	—	2	1	no	—
16	C₂×C₂²⋊Q₈×C₄₇	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₄₇	C₄	1	2	yes	—
18	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
19	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₂×#117×C₄₇	—	2	1	no	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
27	C₄×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×Q₈×C₄₇	—	2	1	no	—
29	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₄₇	—	2	4	no	—
34	C₄⋊Q₈×C₄₇	—	2	4	no	—
35	#125×C₄₇	—	2	8	no	—
36	C₂²⋊Q₈×C₄₇	—	2	8	no	—
37	C₂²⋊Q₈×C₄₇	—	2	4	no	—
38	C₂²⋊Q₈×C₄₇	—	2	4	no	—
39	C₄×Q₈×C₄₇	—	2	8	no	—
40	C₄×Q₈×C₄₇	—	2	4	no	—
41	C₄×Q₈×C₄₇	D₄	1	8	yes	—
42	#117×C₄₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	—	2	4	no	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Q₈×C₄₇	—	2	4	no	—
58	C₂×Q₈×C₄₇	—	2	2	no	—
59	C₂×Q₈×C₄₇	—	2	4	no	—
60	C₂×Q₈×C₄₇	—	2	4	no	—
61	C₂×Q₈×C₄₇	—	4	2	no	—
62	C₂×Q₈×C₄₇	—	4	2	no	—
63	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	4	1	no	—
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	2	8	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	—	2	8	no	—
88	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
89	C₂²⋊C₄×C₄₇	—	2	8	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	—	2	8	no	—
94	C₄²×C₄₇	C₂×D₄	1	4	yes	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂×C₄₇⋊C₈	—	4	2	no	—
99	C₂×C₄₇⋊C₈	—	4	2	no	—
100	C₂×C₄₇⋊C₈	—	4	2	no	—
101	C₂³×C₄₇	—	2	1	no	—
102	C₂³×C₄₇	—	2	1	no	—
103	C₂³×C₄₇	—	2	1	no	—
104	C₂³×C₄₇	—	2	1	no	—
105	C₂³×C₄₇	—	2	1	no	—
106	C₂³×C₄₇	—	2	1	no	—
107	C₂³×C₄₇	#103	1	2	yes	—
108	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	Q₈×C₄₇	—	2	8	no	—
110	Q₈×C₄₇	—	4	8	no	—
111	Q₈×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	4	no	—
131	C₂×C₄×C₄₇	#103	1	2	yes	—
132	C₂×C₄×C₄₇	#104	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₄₇	#120	1	2	yes	—
135	C₂×C₄×C₄₇	#120	1	2	yes	—
136	C₂×C₄×C₄₇	#131	1	2	yes	—
137	C₂×C₄×C₄₇	#131	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#137	1	2	yes	—
141	C₄₇⋊C₈	—	4	4	no	—
142	C₄₇⋊C₈	—	8	2	no	—
143	#8314	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	4	1	no	—
148	C₂²×C₄₇	—	4	1	no	—
149	C₂²×C₄₇	—	4	1	no	—
150	C₂²×C₄₇	—	4	1	no	—
151	C₂²×C₄₇	#379	1	1	yes	characteristic
152	C₂²×C₄₇	#408	1	1	yes	characteristic
153	C₂²×C₄₇	#408	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
155	C₂²×C₄₇	#419	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
157	C₂²×C₄₇	#421	1	1	yes	characteristic
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	—	4	1	no	—
162	C₄×C₄₇	—	4	4	no	—
163	C₄×C₄₇	—	4	4	no	—
164	C₄×C₄₇	#408	1	4	yes	—
165	C₄×C₄₇	#448	1	2	yes	—
166	C₄×C₄₇	#451	1	2	yes	—
167	C₂×#533	D₄₇	1	1	yes	characteristic
168	#1797	—	47	1	no	—
169	C₄×C₂²⋊C₈	—	47	1	no	—
170	C₂×Q₈⋊C₈	—	47	4	no	—
171	C₂×C₄₇	—	4	2	no	—
172	C₂×C₄₇	#1481	1	1	yes	characteristic
173	C₂×C₄₇	#1485	1	1	yes	characteristic
174	C₂×C₄₇	#1493	1	1	yes	characteristic
175	C₂×C₄₇	#1497	1	1	yes	characteristic
176	C₂×C₄₇	#1780	1	1	yes	characteristic
177	C₂×C₄₇	#1787	1	1	yes	characteristic
178	C₂×C₄₇	#1789	1	1	yes	characteristic
179	#533	—	2	8	no	—
180	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
181	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
182	C₂×#125	—	2	1	no	—
183	C₂×C₂²⋊Q₈	—	2	1	no	—
184	C₂×C₂²⋊Q₈	Dic₄₇	1	2	yes	—
185	C₂×C₄×Q₈	—	2	1	no	—
186	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
187	C₂×#117	—	2	1	no	—
188	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
189	C₂×C₄⋊C₈	—	47	2	no	—
190	C₂×C₂²⋊C₈	—	94	1	no	—
191	C₂×C₂²⋊C₈	—	94	1	no	—
192	C₂×C₄×C₈	—	47	2	no	—
193	#336	—	94	1	no	—
194	#336	—	94	1	no	—
195	Q₈⋊C₈	—	94	8	no	—
196	C₄₇	#8314	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₈	—	2	1	no	—
198	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
199	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
200	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
202	C₂²×C₈	—	94	2	no	—
203	C₂²×C₈	—	94	2	no	—
204	C₄⋊Q₈	—	2	4	no	—
205	C₄⋊Q₈	—	2	4	no	—
206	#125	—	2	8	no	—
207	C₂²⋊Q₈	—	2	8	no	—
208	C₂²⋊Q₈	—	2	4	no	—
209	C₂²⋊Q₈	—	2	4	no	—
210	C₄×Q₈	—	2	8	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	C₂²⋊D₄₇	1	8	yes	—
213	#117	—	2	8	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	—	2	4	no	—
223	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
226	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
227	C₄⋊C₈	—	94	4	no	—
228	C₂²⋊C₈	—	94	4	no	—
229	C₂²⋊C₈	—	188	2	no	—
230	C₄×C₈	—	94	4	no	—
231	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	C₂²⋊Dic₄₇	1	4	yes	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	#10625336	1	2	yes	—
250	C₂²×C₄	#10625337	1	2	yes	—
251	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂×C₈	—	94	4	no	—
258	C₂×C₈	—	188	2	no	—
259	C₂×C₈	—	188	2	no	—
260	C₂×C₈	—	188	2	no	—
261	C₂×C₈	—	188	2	no	—
262	C₂×C₈	—	188	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
270	C₂²⋊C₄	—	2	8	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	8	no	—
275	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	#13093948	1	2	yes	—
283	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
284	Q₈	—	2	8	no	—
285	Q₈	—	4	8	no	—
286	Q₈	—	4	4	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	#13093948	1	2	yes	—
307	C₂×C₄	#13093951	1	2	yes	—
308	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
309	C₂×C₄	#13093993	1	2	yes	—
310	C₂×C₄	#13093993	1	2	yes	—
311	C₂×C₄	#13094031	1	2	yes	—
312	C₂×C₄	#13094031	1	2	yes	—
313	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
314	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
315	C₂×C₄	#13094055	1	2	yes	—
316	C₈	—	376	2	no	—
317	C₈	—	188	4	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#15688979	1	1	yes	characteristic
326	C₂²	#15689090	1	1	yes	characteristic
327	C₂²	#15689090	1	1	yes	characteristic
328	C₂²	C₂×#13093948	1	1	yes	characteristic
329	C₂²	#15689132	1	1	yes	characteristic
330	C₂²	C₂×#13093951	1	1	yes	characteristic
331	C₂²	#15689140	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	4	no	—
337	C₄	—	4	4	no	—
338	C₄	#15689090	1	4	yes	—
339	C₄	#15689262	1	2	yes	—
340	C₄	#15689279	1	2	yes	—
341	C₂	—	4	2	no	—
342	C₂	#20241791	1	1	yes	characteristic
343	C₂	#20241809	1	1	yes	characteristic
344	C₂	#20241843	1	1	yes	characteristic
345	C₂	#20241861	1	1	yes	characteristic
346	C₂	#20243300	1	1	yes	characteristic
347	C₂	#20243323	1	1	yes	characteristic
348	C₂	#20243332	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#8314
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
3. C₂×Q₈⋊(C₄₇⋊C₈)	C₂	semidirect
196. C₄₇	#8314	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
C₂×Q₈⋊(C₄₇⋊C₈)	2
C₂×C₄₇×#533	2
#8314	47