Subgroups — #26224462

Order 12032.

730 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253733	C₂	1	1	yes	maximal, characteristic
2	#20252322	C₂	1	1	yes	maximal, characteristic, complemented
3	#20244368	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244221	C₂	1	1	yes	maximal, characteristic
5	#20244102	C₂	1	4	yes	maximal, complemented
6	#20243528	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243223	C₂	1	4	yes	maximal, complemented
8	#20243074	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2931	C₂	1	1	yes	maximal, characteristic
10	#15689680	C₂²	1	1	yes	characteristic
11	#15689491	C₄	1	2	yes	complemented
12	#15689487	C₄	1	2	yes	complemented
13	#15689482	C₄	1	4	yes	complemented
14	#15689253	—	2	2	no	—
15	C₈×Dic₉₄	—	2	2	no	—
16	#15689149	C₂²	1	2	yes	—
17	#15689147	C₂²	1	1	yes	characteristic
18	C₂×#13093951	C₂²	1	4	yes	—
19	#15689119	C₂²	1	4	yes	—
20	#15689078	C₂²	1	2	yes	—
21	C₂×#13093935	C₂²	1	1	yes	characteristic
22	C₂×#13093934	C₂²	1	2	yes	—
23	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
24	#15688969	C₂²	1	1	yes	characteristic
25	#15688887	C₂²	1	4	yes	—
26	#15688884	C₂²	1	4	yes	—
27	C₂×#131×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#532	C₂²	1	1	yes	characteristic
29	C₄₇×#436	—	2	2	no	—
30	C₈×D₄×C₄₇	—	2	2	no	—
31	C₄₇×#423	C₂²	1	1	yes	characteristic
32	C₄₇×#408	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#10625329	C₂³	1	2	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	#13094024	C₂×C₄	1	2	yes	—
37	#13094016	C₂×C₄	1	2	yes	—
38	#13094003	—	2	2	no	—
39	C₄×Dic₉₄	—	2	2	no	—
40	#13093983	C₂³	1	2	yes	—
41	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
42	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
43	Dic₄₇⋊C₈	—	2	4	no	—
44	#13093951	—	2	8	no	—
45	#13093948	C₂×C₄	1	8	yes	—
46	#13093945	C₂×C₄	1	8	yes	—
47	#13093935	C₂×C₄	1	2	yes	—
48	#13093935	C₂×C₄	1	2	yes	—
49	#13093934	—	2	2	no	—
50	#13093934	—	2	4	no	—
51	C₈×Dic₄₇	—	2	2	no	—
52	C₈×Dic₄₇	—	2	4	no	—
53	C₄₇⋊#95	—	2	2	no	—
54	C₂×#41×C₄₇	C₂³	1	2	yes	—
55	#131×C₄₇	—	2	8	no	—
56	C₂×M₁₆×C₄₇	—	2	2	no	—
57	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₄₇	—	2	2	no	—
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
62	#127×C₄₇	C₂×C₄	1	1	yes	characteristic
63	#124×C₄₇	C₂×C₄	1	2	yes	—
64	#121×C₄₇	—	2	2	no	—
65	C₄×D₄×C₄₇	—	2	2	no	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₄₇	—	2	2	no	—
68	C₂²⋊C₈×C₄₇	—	2	4	no	—
69	C₄×C₈×C₄₇	D₄	1	2	yes	—
70	C₄×C₈×C₄₇	D₄	1	2	yes	—
71	#41×C₄₇	—	2	4	no	—
72	#41×C₄₇	—	2	4	no	—
73	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂×D₄×C₄₇	—	2	2	no	—
75	C₂×D₄×C₄₇	—	2	2	no	—
76	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	—	2	2	no	—
78	C₂²×C₄×C₄₇	#41	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₇	—	2	2	no	—
81	C₂×Dic₉₄	—	2	2	no	—
82	M₁₆×C₄₇	—	2	4	no	—
83	M₁₆×C₄₇	—	4	4	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	4	4	no	—
87	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	#41	1	4	yes	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₂²⋊C₄×C₄₇	—	2	4	no	—
93	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₄₇	C₂×D₄	1	2	yes	—
96	C₂²⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
99	Dic₄₇⋊C₄	—	2	4	no	—
100	C₄×Dic₄₇	—	2	2	no	—
101	C₄×Dic₄₇	—	2	2	no	—
102	C₄×Dic₄₇	#41	1	4	yes	—
103	#10625329	—	2	4	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
110	Q₈×C₄₇	—	2	4	no	—
111	D₄×C₄₇	—	2	4	no	—
112	D₄×C₄₇	—	4	4	no	—
113	D₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
124	C₂×Dic₄₇	—	2	4	no	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	Dic₉₄	—	4	4	no	—
128	C₈×C₄₇	—	2	2	no	—
129	C₈×C₄₇	—	4	2	no	—
130	C₈×C₄₇	#124	1	2	yes	—
131	C₈×C₄₇	#124	1	2	yes	—
132	C₄₇⋊C₈	—	2	4	no	—
133	C₄₇⋊C₈	—	4	2	no	—
134	#21661	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	4	4	no	—
137	C₂²×C₄₇	—	4	2	no	—
138	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₄₇	#529	1	2	yes	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	4	2	no	—
143	C₄×C₄₇	C₄×#41	1	2	yes	—
144	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
145	C₄×C₄₇	#532	1	1	yes	characteristic
146	Dic₄₇	—	4	4	no	—
147	Dic₄₇	—	4	4	no	—
148	#3135	—	47	1	no	—
149	#2960	—	47	1	no	—
150	#2931	D₄₇	1	1	yes	characteristic
151	#1995	—	47	1	no	—
152	#1963	—	47	1	no	—
153	#1938	—	47	4	no	—
154	#1838	—	47	1	no	—
155	#1764	—	47	4	no	—
156	#1726	—	47	1	no	—
157	C₂×C₄₇	—	2	2	no	—
158	C₂×C₄₇	—	8	2	no	—
159	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
160	C₂×C₄₇	#2955	1	2	yes	—
161	C₂×#131	C₂×D₄₇	1	2	yes	—
162	#533	—	47	1	no	—
163	#532	C₂×D₄₇	1	1	yes	characteristic
164	#491	—	47	2	no	—
165	#490	—	47	2	no	—
166	#489	—	47	4	no	—
167	#446	—	94	2	no	—
168	C₈×Q₈	—	94	2	no	—
169	#436	—	2	2	no	—
170	C₈×D₄	—	2	2	no	—
171	#424	—	47	2	no	—
172	#423	—	47	1	no	—
173	#423	C₂×D₄₇	1	1	yes	characteristic
174	C₂×#104	—	47	4	no	—
175	#416	—	47	4	no	—
176	#408	C₂×D₄₇	1	2	yes	—
177	#405	—	47	2	no	—
178	C₂×#97	—	47	1	no	—
179	C₂×#97	—	47	2	no	—
180	C₂×C₄×C₈	—	47	2	no	—
181	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
182	#376	—	47	1	no	—
183	#341	—	47	4	no	—
184	#339	—	47	4	no	—
185	C₄₇	#21661	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	C₂²×D₄₇	1	2	yes	—
187	#131	—	2	8	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	47	2	no	—
190	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	47	2	no	—
193	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₄⋊Q₈	—	47	2	no	—
196	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic
197	#127	C₄×D₄₇	1	1	yes	characteristic
198	#125	—	47	2	no	—
199	#124	C₄×D₄₇	1	2	yes	—
200	C₂²⋊Q₈	—	94	2	no	—
201	#121	—	2	2	no	—
202	C₄×Q₈	—	94	2	no	—
203	C₄×D₄	—	2	2	no	—
204	#117	—	47	2	no	—
205	C₂×C₄²	—	47	2	no	—
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	94	4	no	—
209	C₄⋊C₈	—	94	2	no	—
210	#104	—	94	8	no	—
211	#103	—	47	8	no	—
212	#102	—	47	8	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	#97	—	47	2	no	—
215	#97	—	47	2	no	—
216	#97	—	94	4	no	—
217	#97	—	94	2	no	—
218	C₄×C₈	—	94	4	no	—
219	C₄×C₈	—	94	2	no	—
220	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
221	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
222	#95	—	94	2	no	—
223	#41	—	2	4	no	—
224	#41	—	2	4	no	—
225	C₂×Q₈	—	94	2	no	—
226	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	#10625350	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	4	4	no	—
236	M₁₆	—	94	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	4	no	—
240	C₂×C₈	—	47	4	no	—
241	C₂×C₈	—	94	2	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	94	2	no	—
244	C₂×C₈	#10625352	1	4	yes	—
245	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
246	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	47	4	no	—
250	C₄⋊C₄	—	94	4	no	—
251	C₄⋊C₄	—	94	2	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	94	2	no	—
254	C₄²	—	47	4	no	—
255	C₄²	—	94	2	no	—
256	C₄²	—	94	2	no	—
257	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×#10625350	1	1	yes	characteristic
262	Q₈	—	2	4	no	—
263	Q₈	—	188	4	no	—
264	D₄	—	2	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	—	188	4	no	—
274	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
275	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
278	C₂×C₄	#41×D₄₇	1	2	yes	—
279	C₂×C₄	C₂×#10625352	1	2	yes	—
280	C₈	—	2	2	no	—
281	C₈	—	4	2	no	—
282	C₈	—	94	4	no	—
283	C₈	—	188	2	no	—
284	C₈	#13094011	1	2	yes	—
285	C₈	#13094011	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	2	no	—
289	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
290	C₂²	#15689655	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	188	4	no	—
295	C₄	—	188	4	no	—
296	C₄	C₄×#10625352	1	2	yes	—
297	C₄	C₂×#13094011	1	1	yes	characteristic
298	C₄	#15689669	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	8	2	no	—
301	C₂	C₄×#13094011	1	1	yes	characteristic
302	C₂	#20252289	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2931	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₄₇×#532	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2931	D₄₇
hypocenter	C₄₇	#21661
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. #20252322	C₂	semidirect
3. #20244368	C₂	semidirect
5. #20244102	C₂	semidirect
6. #20243528	C₂	semidirect
7. #20243223	C₂	semidirect
8. #20243074	C₂	semidirect
11. #15689491	C₄	semidirect
12. #15689487	C₄	semidirect
13. #15689482	C₄	semidirect
185. C₄₇	#21661	semidirect

Maximal subgroups

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

subgroup	index
#20253733	2
#20252322	2
#20244368	2
#20244221	2
#20244102	2
#20243528	2
#20243223	2
#20243074	2
C₄₇×#2931	2
#21661	47