Subgroups — #26224456

Order 12032.

746 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252323	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252100	C₂	1	1	yes	maximal, characteristic
3	#20244366	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244217	C₂	1	1	yes	maximal, characteristic
5	#20244101	C₂	1	4	yes	maximal, complemented
6	#20243526	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243235	C₂	1	4	yes	maximal, complemented
8	C₄×#13093934	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3131	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094031	C₂²	1	2	yes	—
11	#15689254	—	2	2	no	—
12	Q₈×C₄₇⋊C₈	—	2	2	no	—
13	#15689201	—	2	2	no	—
14	D₄×C₄₇⋊C₈	—	2	2	no	—
15	#15689150	C₂²	1	2	yes	—
16	#15689147	C₂²	1	1	yes	characteristic
17	#15689147	C₂²	1	1	yes	characteristic
18	C₂×#13093951	C₂²	1	4	yes	—
19	#15689090	C₂²	1	2	yes	—
20	#15689079	C₂²	1	2	yes	—
21	C₂×#13093934	C₂²	1	2	yes	—
22	C₂×#13093934	C₂²	1	2	yes	—
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688888	C₂²	1	4	yes	—
26	#15688882	C₂²	1	4	yes	—
27	C₄₇×#533	C₂²	1	1	yes	characteristic
28	C₄₇×#532	C₂²	1	1	yes	characteristic
29	C₄₇×#488	C₄	1	4	yes	complemented
30	C₄₇×#487	C₄	1	2	yes	complemented
31	C₄₇×#486	C₄	1	2	yes	complemented
32	C₄₇×#416	C₂²	1	4	yes	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	#13094031	—	2	8	no	—
35	C₂×#10625329	—	2	2	no	—
36	C₂×#10625329	C₂³	1	2	yes	—
37	C₂×#10625329	C₂³	1	2	yes	—
38	C₂²×C₄₇⋊C₈	—	2	2	no	—
39	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₄₇	—	2	2	no	—
42	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
43	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
44	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
45	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
46	#13093951	—	2	8	no	—
47	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
48	#13093935	—	2	2	no	—
49	#13093934	—	2	4	no	—
50	#13093934	—	2	4	no	—
51	C₄×C₄₇⋊C₈	—	2	2	no	—
52	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
53	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
54	C₂×#41×C₄₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
58	#127×C₄₇	C₂×C₄	1	1	yes	characteristic
59	#125×C₄₇	C₂×C₄	1	2	yes	—
60	#124×C₄₇	C₂×C₄	1	2	yes	—
61	C₂²⋊Q₈×C₄₇	—	2	2	no	—
62	#121×C₄₇	—	2	2	no	—
63	C₄×Q₈×C₄₇	—	2	2	no	—
64	C₄×D₄×C₄₇	—	2	2	no	—
65	#117×C₄₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	#103×C₄₇	C₂×C₄	1	8	yes	—
68	#102×C₄₇	C₂×C₄	1	8	yes	—
69	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
71	#41×C₄₇	—	2	4	no	—
72	#41×C₄₇	—	2	4	no	—
73	C₂×Q₈×C₄₇	—	2	2	no	—
74	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂×D₄×C₄₇	—	2	2	no	—
76	C₂×D₄×C₄₇	—	2	2	no	—
77	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	#41	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	—	2	2	no	—
88	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂²⋊C₄×C₄₇	—	2	2	no	—
90	C₂²⋊C₄×C₄₇	—	2	4	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂×D₄	1	2	yes	—
95	C₄×Dic₄₇	—	2	2	no	—
96	C₄×Dic₄₇	—	4	4	no	—
97	C₄×Dic₄₇	#41	1	4	yes	—
98	#10625329	—	2	4	no	—
99	#10625329	—	2	4	no	—
100	#10625329	—	4	4	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	—	4	4	no	—
105	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
106	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄₇⋊C₈	#41	1	4	yes	—
109	C₂³×C₄₇	—	2	2	no	—
110	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
111	Q₈×C₄₇	—	2	4	no	—
112	Q₈×C₄₇	—	4	4	no	—
113	D₄×C₄₇	—	2	4	no	—
114	D₄×C₄₇	—	4	4	no	—
115	D₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₈×C₄₇	—	2	4	no	—
130	C₄₇⋊C₈	—	2	2	no	—
131	C₄₇⋊C₈	—	4	2	no	—
132	C₄₇⋊C₈	—	4	2	no	—
133	C₄₇⋊C₈	#124	1	2	yes	—
134	C₄₇⋊C₈	#124	1	2	yes	—
135	#21660	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	4	4	no	—
138	C₂²×C₄₇	—	4	2	no	—
139	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₄₇	#529	1	2	yes	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	4	4	no	—
143	C₄×C₄₇	—	4	2	no	—
144	C₄×C₄₇	#444	1	4	yes	—
145	C₄×C₄₇	C₄×#41	1	2	yes	—
146	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
147	C₄×C₄₇	#532	1	1	yes	characteristic
148	Dic₄₇	—	4	4	no	—
149	#3131	D₄₇	1	1	yes	characteristic
150	#2960	—	47	1	no	—
151	#2931	—	47	1	no	—
152	#1994	—	47	1	no	—
153	#1962	—	47	1	no	—
154	#1938	—	47	4	no	—
155	#1837	—	47	1	no	—
156	#1765	—	47	4	no	—
157	C₄×#97	—	47	1	no	—
158	C₂×C₄₇	—	2	2	no	—
159	C₂×C₄₇	—	8	2	no	—
160	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
161	C₂×C₄₇	C₂×#444	1	2	yes	—
162	C₂×#131	—	47	2	no	—
163	#533	C₂×D₄₇	1	1	yes	characteristic
164	#532	C₂×D₄₇	1	1	yes	characteristic
165	#488	Dic₄₇	1	4	yes	complemented
166	#487	Dic₄₇	1	2	yes	complemented
167	#486	Dic₄₇	1	2	yes	complemented
168	#446	—	94	2	no	—
169	C₈×Q₈	—	94	2	no	—
170	#436	—	94	2	no	—
171	C₈×D₄	—	94	2	no	—
172	#424	—	47	2	no	—
173	#423	—	47	1	no	—
174	#423	—	47	1	no	—
175	C₂×#104	—	47	4	no	—
176	#416	C₂×D₄₇	1	4	yes	—
177	#408	—	47	2	no	—
178	#405	—	47	2	no	—
179	C₂×#97	—	47	2	no	—
180	C₂×#97	—	47	2	no	—
181	C₂×C₄×C₈	—	47	1	no	—
182	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
183	C₄³	—	47	1	no	—
184	#341	—	47	4	no	—
185	#339	—	47	4	no	—
186	C₄₇	#21660	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	C₂²×D₄₇	1	2	yes	—
188	#131	—	94	8	no	—
189	C₂×M₁₆	—	47	2	no	—
190	C₂×M₁₆	—	47	2	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂²×C₈	—	47	1	no	—
193	C₂²×C₈	—	47	1	no	—
194	C₂²×C₈	—	94	2	no	—
195	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
196	C₄⋊Q₈	C₂×Dic₄₇	1	1	yes	characteristic
197	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
198	#127	C₂×Dic₄₇	1	1	yes	characteristic
199	#125	C₂×Dic₄₇	1	2	yes	—
200	#124	C₂×Dic₄₇	1	2	yes	—
201	C₂²⋊Q₈	—	2	2	no	—
202	#121	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	#117	C₂²×D₄₇	1	2	yes	—
206	C₂×C₄²	—	47	2	no	—
207	C₂×C₄²	—	94	2	no	—
208	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
209	C₄⋊C₈	—	94	4	no	—
210	C₄⋊C₈	—	94	2	no	—
211	C₄⋊C₈	—	94	2	no	—
212	#104	—	94	8	no	—
213	#103	C₂×Dic₄₇	1	8	yes	—
214	#102	C₂×Dic₄₇	1	8	yes	—
215	C₂²⋊C₈	—	94	4	no	—
216	#97	—	94	2	no	—
217	#97	—	94	4	no	—
218	#97	—	94	4	no	—
219	C₄×C₈	—	47	2	no	—
220	C₄×C₈	—	47	2	no	—
221	C₄×C₈	—	94	2	no	—
222	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
223	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
224	#41	—	2	4	no	—
225	#41	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	C₂²×Dic₄₇	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	C₂²×Dic₄₇	1	2	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	#10625352	1	2	yes	—
234	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
235	M₁₆	—	94	4	no	—
236	M₁₆	—	94	4	no	—
237	M₁₆	—	188	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	47	1	no	—
241	C₂×C₈	—	47	1	no	—
242	C₂×C₈	—	47	4	no	—
243	C₂×C₈	—	47	2	no	—
244	C₂×C₈	—	94	2	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	188	4	no	—
248	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	47	4	no	—
257	C₄²	—	94	2	no	—
258	C₄²	—	188	4	no	—
259	C₄²	D₄×D₄₇	1	2	yes	—
260	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
261	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
262	C₂³	—	2	2	no	—
263	C₂³	C₂×#10625352	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	94	4	no	—
275	C₂×C₄	—	188	4	no	—
276	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
277	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
278	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
280	C₂×C₄	#41×D₄₇	1	2	yes	—
281	C₂×C₄	#41×D₄₇	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	47	2	no	—
284	C₈	—	47	2	no	—
285	C₈	—	94	2	no	—
286	C₈	—	188	2	no	—
287	C₈	—	188	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
292	C₂²	#15689652	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	188	4	no	—
297	C₄	#15689245	1	4	yes	—
298	C₄	#41×Dic₄₇	1	2	yes	—
299	C₄	#124×D₄₇	1	1	yes	characteristic
300	C₄	#15689672	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	8	2	no	—
303	C₂	#124×Dic₄₇	1	1	yes	characteristic
304	C₂	C₂×#15689245	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#3131	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₄₇×#532	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#3131	D₄₇
hypocenter	C₄₇	#21660
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

Complements

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

N	H ≅ G/N	type
1. #20252323	C₂	semidirect
3. #20244366	C₂	semidirect
5. #20244101	C₂	semidirect
6. #20243526	C₂	semidirect
7. #20243235	C₂	semidirect
8. C₄×#13093934	C₂	semidirect
29. C₄₇×#488	C₄	semidirect
30. C₄₇×#487	C₄	semidirect
31. C₄₇×#486	C₄	semidirect
165. #488	Dic₄₇	semidirect
166. #487	Dic₄₇	semidirect
167. #486	Dic₄₇	semidirect
186. C₄₇	#21660	semidirect

Maximal subgroups

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

subgroup	index
#20252323	2
#20252100	2
#20244366	2
#20244217	2
#20244101	2
#20243526	2
#20243235	2
C₄×#13093934	2
C₄₇×#3131	2
#21660	47