Subgroups — #26369389

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245192	C₂	1	2	yes	maximal, complemented
3	#20245192	C₂	1	2	yes	maximal, complemented
4	#20245188	C₂	1	2	yes	maximal, complemented
5	#20245184	C₂	1	2	yes	maximal, complemented
6	#20245143	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245002	C₂	1	1	yes	maximal, characteristic
9	#20244987	C₂	1	1	yes	maximal, characteristic, complemented
10	#20244984	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2113	C₂	1	1	yes	maximal, characteristic
12	C₂×#13094031	C₄	1	1	yes	characteristic
13	#41×Dic₄₇	C₄	1	1	yes	characteristic
14	C₂×#13093964	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#15689081	C₄	1	1	yes	characteristic
17	#15689081	C₄	1	2	yes	—
18	#15689078	C₂²	1	2	yes	—
19	#15689078	C₂²	1	1	yes	characteristic
20	M₁₆×Dic₄₇	C₄	1	2	yes	—
21	M₁₆×Dic₄₇	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
24	Dic₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
25	Dic₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
26	Dic₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
27	Dic₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
28	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
29	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
30	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
31	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
32	#15688898	C₂²	1	2	yes	—
33	#15688898	C₂²	1	2	yes	—
34	C₁₆×Dic₄₇	C₂²	1	2	yes	—
35	C₁₆×Dic₄₇	C₂²	1	2	yes	—
36	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
38	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
41	#13094031	—	2	4	no	—
42	C₂×#10625329	C₂×C₄	1	2	yes	—
43	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
46	Q₈×Dic₄₇	C₈	1	1	yes	characteristic
47	Q₈×Dic₄₇	C₈	1	1	yes	characteristic
48	D₄×Dic₄₇	C₈	1	2	yes	—
49	D₄×Dic₄₇	C₈	1	1	yes	characteristic
50	D₄×Dic₄₇	C₈	1	2	yes	—
51	D₄×Dic₄₇	C₈	1	1	yes	characteristic
52	#13093983	C₂×C₄	1	2	yes	—
53	#13093983	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
56	#13093964	—	2	2	no	—
57	C₂×C₄₇⋊C₁₆	—	2	2	no	—
58	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
61	#13093934	C₂×C₄	1	2	yes	—
62	#13093934	C₂×C₄	1	2	yes	—
63	#13093934	C₂×C₄	1	2	yes	—
64	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
66	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
67	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
69	C₈×Dic₄₇	C₂³	1	2	yes	—
70	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
71	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
72	#131×C₄₇	—	2	2	no	—
73	#131×C₄₇	—	2	2	no	—
74	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
75	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
76	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	M₃₂×C₄₇	—	2	2	no	—
79	C₂×C₁₆×C₄₇	—	2	2	no	—
80	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
83	#41×C₄₇	—	2	2	no	—
84	#41×C₄₇	—	2	2	no	—
85	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
86	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
87	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
91	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
92	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
93	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
94	M₁₆×C₄₇	—	2	2	no	—
95	M₁₆×C₄₇	—	2	4	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	#41	1	2	yes	—
101	C₂×C₈×C₄₇	#41	1	2	yes	—
102	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
103	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
104	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
105	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
106	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
107	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
108	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
109	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
110	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
111	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
112	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
113	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
114	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
116	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
117	#10625329	—	2	4	no	—
118	#10625329	M₁₆	1	4	yes	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	4	no	—
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
124	C₁₆×C₄₇	—	2	1	no	—
125	C₁₆×C₄₇	—	2	1	no	—
126	C₁₆×C₄₇	—	2	2	no	—
127	C₄₇⋊C₁₆	—	2	1	no	—
128	C₄₇⋊C₁₆	—	2	1	no	—
129	C₄₇⋊C₁₆	—	2	2	no	—
130	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
131	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
132	Q₈×C₄₇	—	2	2	no	—
133	D₄×C₄₇	—	2	4	no	—
134	D₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	#117	1	2	yes	—
138	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	—	2	4	no	—
144	C₂×Dic₄₇	—	4	1	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	C₂²×C₈	1	1	yes	characteristic
147	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
148	C₂×Dic₄₇	C₂²×C₈	1	1	yes	characteristic
149	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
150	C₂×Dic₄₇	C₂²×C₈	1	1	yes	characteristic
151	C₂×Dic₄₇	C₂²×C₈	1	1	yes	characteristic
152	C₈×C₄₇	—	2	2	no	—
153	C₈×C₄₇	#117	1	2	yes	—
154	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
155	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
156	C₄₇⋊C₈	—	2	1	no	—
157	C₄₇⋊C₈	—	2	1	no	—
158	C₄₇⋊C₈	C₂×M₁₆	1	4	yes	—
159	#33423	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	—	2	2	no	—
163	C₂²×C₄₇	—	2	2	no	—
164	C₂²×C₄₇	—	2	2	no	—
165	C₂²×C₄₇	#431	1	2	yes	—
166	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
167	C₄×C₄₇	—	2	2	no	—
168	C₄×C₄₇	#431	1	2	yes	—
169	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
170	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
171	Dic₄₇	—	2	1	no	—
172	Dic₄₇	—	2	2	no	—
173	Dic₄₇	—	2	1	no	—
174	Dic₄₇	—	2	1	no	—
175	Dic₄₇	—	2	2	no	—
176	Dic₄₇	#504	1	2	yes	—
177	C₄×#131	—	47	1	no	—
178	#2164	—	47	2	no	—
179	#2164	—	47	2	no	—
180	#2163	—	47	2	no	—
181	#2162	—	47	2	no	—
182	#2150	—	47	1	no	—
183	C₂×C₄⋊C₁₆	—	47	1	no	—
184	#2113	—	47	1	no	—
185	#2113	D₄₇	1	1	yes	characteristic
186	#2108	—	47	1	no	—
187	#2108	—	47	1	no	—
188	C₂×C₄₇	—	2	2	no	—
189	C₂×C₄₇	—	4	2	no	—
190	C₂×C₄₇	C₂×#431	1	1	yes	characteristic
191	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
192	C₂×C₄₇	#3407	1	1	yes	characteristic
193	C₂×#131	—	47	1	no	—
194	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
195	C₄×#41	—	47	1	no	—
196	C₂×M₃₂	—	47	1	no	—
197	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
198	C₂²×C₁₆	—	47	1	no	—
199	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
200	#405	—	47	1	no	—
201	#405	—	47	2	no	—
202	#405	—	47	1	no	—
203	#405	—	47	2	no	—
204	C₄×M₁₆	—	47	1	no	—
205	C₄×M₁₆	—	47	2	no	—
206	C₂×C₄×C₈	—	47	1	no	—
207	C₂×C₄×C₈	—	47	2	no	—
208	C₄⋊C₁₆	—	47	1	no	—
209	C₄⋊C₁₆	—	47	1	no	—
210	C₄⋊C₁₆	—	47	1	no	—
211	C₄⋊C₁₆	—	47	1	no	—
212	C₄⋊C₁₆	—	47	2	no	—
213	C₄⋊C₁₆	—	47	2	no	—
214	C₂²⋊C₁₆	—	47	2	no	—
215	C₂²⋊C₁₆	—	47	2	no	—
216	C₂²⋊C₁₆	C₂×D₄₇	1	2	yes	—
217	C₂²⋊C₁₆	C₂×D₄₇	1	2	yes	—
218	#346	—	47	2	no	—
219	#346	—	47	2	no	—
220	C₄×C₁₆	—	47	2	no	—
221	C₄×C₁₆	—	47	2	no	—
222	C₄₇	#33423	1	1	yes	characteristic, complemented, Sylow
223	C₂×#41	C₄×D₄₇	1	1	yes	characteristic
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	94	4	no	—
227	C₂×M₁₆	—	47	1	no	—
228	C₂×M₁₆	—	47	2	no	—
229	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
230	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
231	C₂²×C₈	—	47	2	no	—
232	C₂²×C₈	—	47	1	no	—
233	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
234	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
235	C₄×Q₈	—	47	1	no	—
236	C₄×Q₈	—	47	1	no	—
237	C₄×D₄	—	47	2	no	—
238	C₄×D₄	—	47	1	no	—
239	C₄×D₄	—	47	2	no	—
240	C₄×D₄	—	47	1	no	—
241	#117	—	47	2	no	—
242	#117	—	47	1	no	—
243	C₂×C₄²	—	47	2	no	—
244	C₂×C₄²	—	47	1	no	—
245	M₃₂	—	2	2	no	—
246	M₃₂	—	94	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	47	1	no	—
249	C₂×C₁₆	—	47	2	no	—
250	C₂×C₁₆	—	47	1	no	—
251	C₂×C₁₆	—	94	2	no	—
252	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
254	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
255	#97	—	47	2	no	—
256	#97	—	47	2	no	—
257	#97	—	47	2	no	—
258	C₄×C₈	—	47	2	no	—
259	C₄×C₈	—	47	1	no	—
260	C₄×C₈	—	47	2	no	—
261	C₄×C₈	—	47	1	no	—
262	C₄×C₈	—	47	2	no	—
263	C₄×C₈	—	47	1	no	—
264	C₄×C₈	—	47	1	no	—
265	#41	—	2	2	no	—
266	#41	—	2	2	no	—
267	C₂×Q₈	C₈×D₄₇	1	1	yes	characteristic
268	C₂×D₄	C₈×D₄₇	1	2	yes	—
269	C₂×D₄	C₈×D₄₇	1	1	yes	characteristic
270	C₂²×C₄	—	47	2	no	—
271	C₂²×C₄	—	47	1	no	—
272	C₂²×C₄	—	47	2	no	—
273	C₂²×C₄	—	47	1	no	—
274	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
275	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
276	M₁₆	—	2	2	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	47	4	no	—
279	M₁₆	—	94	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	47	2	no	—
283	C₂×C₈	—	47	1	no	—
284	C₂×C₈	—	47	1	no	—
285	C₂×C₈	—	94	4	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
289	C₂×C₈	#10625352	1	2	yes	—
290	C₂×C₈	#10625352	1	2	yes	—
291	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
292	C₄⋊C₄	—	47	2	no	—
293	C₄⋊C₄	—	47	2	no	—
294	C₄⋊C₄	—	47	1	no	—
295	C₄⋊C₄	—	47	1	no	—
296	C₂²⋊C₄	—	47	2	no	—
297	C₂²⋊C₄	—	47	2	no	—
298	C₂²⋊C₄	—	47	2	no	—
299	C₄²	—	47	2	no	—
300	C₄²	—	47	1	no	—
301	C₄²	—	47	2	no	—
302	C₄²	—	47	1	no	—
303	C₄²	—	47	1	no	—
304	C₄²	—	47	2	no	—
305	C₄²	—	47	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	94	1	no	—
310	C₁₆	—	94	1	no	—
311	C₁₆	—	94	2	no	—
312	C₂³	C₂×C₈×D₄₇	1	2	yes	—
313	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	D₄	—	2	4	no	—
316	D₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	47	1	no	—
320	C₂×C₄	—	47	1	no	—
321	C₂×C₄	—	47	2	no	—
322	C₂×C₄	—	47	1	no	—
323	C₂×C₄	—	47	2	no	—
324	C₂×C₄	—	47	1	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	4	no	—
327	C₂×C₄	—	188	1	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	#13093980	1	2	yes	—
330	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
331	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
332	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	47	4	no	—
336	C₈	—	94	1	no	—
337	C₈	—	94	1	no	—
338	C₈	#13093980	1	2	yes	—
339	C₈	C₂×#10625352	1	1	yes	characteristic
340	C₈	C₂×#10625352	1	1	yes	characteristic
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	#15689173	1	2	yes	—
347	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	47	2	no	—
350	C₄	—	94	1	no	—
351	C₄	—	94	2	no	—
352	C₄	—	94	1	no	—
353	C₄	—	94	1	no	—
354	C₄	—	94	2	no	—
355	C₄	#15689173	1	2	yes	—
356	C₄	C₂×#13093980	1	1	yes	characteristic
357	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	4	2	no	—
360	C₂	C₂×#15689173	1	1	yes	characteristic
361	C₂	#504×D₄₇	1	1	yes	characteristic
362	C₂	#20256489	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#2113	C₂
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2113	D₄₇
hypocenter	C₄₇	#33423
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
2. #20245192	C₂	semidirect
3. #20245192	C₂	semidirect
4. #20245188	C₂	semidirect
5. #20245184	C₂	semidirect
6. #20245143	C₂	semidirect
7. C₂×Dic₄₇⋊C₁₆	C₂	semidirect
9. #20244987	C₂	semidirect
10. #20244984	C₂	semidirect
222. C₄₇	#33423	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₄₇	2
#20245192	2
#20245188	2
#20245184	2
#20245143	2
C₂×Dic₄₇⋊C₁₆	2
#20245002	2
#20244987	2
#20244984	2
C₄₇×#2113	2
#33423	47