Subgroups — #26131542

Order 12032.

428 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689239	C₂	1	1	yes	maximal, characteristic
2	#20245138	C₂	1	1	yes	maximal, characteristic
3	#20245123	C₂	1	1	yes	maximal, characteristic
4	M₃₂×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243269	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241499	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094046	C₂²	1	1	yes	characteristic
9	C₂×#13094032	—	2	1	no	—
10	C₂×#13093964	C₂²	1	1	yes	characteristic
11	#15689239	—	2	1	no	—
12	#15689239	—	2	1	no	—
13	#15689239	C₄	1	2	yes	complemented
14	#15689239	C₄	1	2	yes	complemented
15	C₂×#13093960	C₂²	1	1	yes	characteristic
16	C₂×#13093950	—	2	1	no	—
17	#15689079	—	2	1	no	—
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	#15688940	C₄	1	2	yes	complemented
20	#15688937	C₄	1	2	yes	complemented
21	#15688932	C₄	1	4	yes	complemented
22	#15688898	—	2	1	no	—
23	C₁₆×Dic₄₇	—	2	1	no	—
24	#15688882	—	2	1	no	—
25	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#427	C₂²	1	1	yes	characteristic
28	C₂×#10625350	—	2	1	no	—
29	#13094046	—	2	1	no	—
30	#13094046	—	2	1	no	—
31	#13094046	C₂×C₄	1	2	yes	—
32	#13094046	C₂×C₄	1	2	yes	—
33	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
34	C₂×#10625325	—	2	1	no	—
35	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
36	#13094032	—	2	2	no	—
37	#13094032	—	2	2	no	—
38	#13094032	—	4	1	no	—
39	#13094032	—	4	1	no	—
40	C₂×#10625329	—	2	1	no	—
41	C₂×#10625324	—	4	1	no	—
42	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₈×D₄₇	—	4	1	no	—
44	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
45	#13093964	—	2	1	no	—
46	#13093964	C₂×C₄	1	2	yes	—
47	C₂×C₄₇⋊C₁₆	—	2	1	no	—
48	#13093960	—	2	1	no	—
49	#13093960	C₂×C₄	1	2	yes	—
50	#13093950	—	2	2	no	—
51	#13093950	—	2	2	no	—
52	#13093935	—	4	1	no	—
53	C₄×C₄₇⋊C₈	—	4	1	no	—
54	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₄₇	D₄	1	1	yes	characteristic
57	C₈×Dic₄₇	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₄₇	—	2	1	no	—
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	#117×C₄₇	—	2	1	no	—
61	M₃₂×C₄₇	—	2	1	no	—
62	M₃₂×C₄₇	C₂×C₄	1	2	yes	—
63	C₂×C₁₆×C₄₇	—	2	1	no	—
64	#108×C₄₇	—	2	1	no	—
65	#108×C₄₇	C₂×C₄	1	2	yes	—
66	#107×C₄₇	—	2	1	no	—
67	#106×C₄₇	C₂×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₇	—	4	1	no	—
70	C₂²×Dic₄₇	—	2	1	no	—
71	#10625350	—	2	2	no	—
72	#10625350	—	2	2	no	—
73	#10625350	—	4	1	no	—
74	#10625350	—	4	1	no	—
75	C₂×D₁₈₈	—	2	1	no	—
76	C₂×C₄×D₄₇	—	4	1	no	—
77	C₂×Dic₉₄	—	2	1	no	—
78	M₁₆×C₄₇	—	2	2	no	—
79	M₁₆×C₄₇	—	4	1	no	—
80	C₂×C₈×C₄₇	—	4	1	no	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
86	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
87	C₄⋊C₄×C₄₇	—	2	2	no	—
88	C₂²⋊C₄×C₄₇	—	4	1	no	—
89	C₄²×C₄₇	—	4	1	no	—
90	C₄×Dic₄₇	—	2	1	no	—
91	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
93	#10625329	—	2	2	no	—
94	#10625329	—	4	1	no	—
95	C₂×C₄₇⋊C₈	—	4	1	no	—
96	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
99	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
100	C₂×C₄₇⋊C₈	SD₁₆	1	1	yes	characteristic
101	C₂×C₄₇⋊C₈	SD₁₆	1	1	yes	characteristic
102	Dic₁₈₈	—	2	1	no	—
103	Dic₁₈₈	C₄²	1	2	yes	—
104	D₃₇₆	—	2	1	no	—
105	D₃₇₆	C₄²	1	2	yes	—
106	#10625325	—	2	2	no	—
107	#10625325	—	2	2	no	—
108	#10625324	—	4	2	no	—
109	#10625324	—	4	2	no	—
110	C₈×D₄₇	—	4	2	no	—
111	C₈×D₄₇	—	4	2	no	—
112	C₁₆×C₄₇	—	2	2	no	—
113	C₄₇⋊C₁₆	—	2	2	no	—
114	C₂³×C₄₇	#135	1	1	yes	characteristic
115	C₂²×D₄₇	—	4	1	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
123	C₂²⋊D₄₇	—	4	2	no	—
124	C₂²⋊D₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	2	1	no	—
126	C₂×Dic₄₇	—	2	1	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	1	no	—
129	D₁₈₈	—	2	2	no	—
130	D₁₈₈	—	4	1	no	—
131	C₄×D₄₇	—	4	2	no	—
132	C₄×D₄₇	—	4	2	no	—
133	Dic₉₄	—	2	2	no	—
134	Dic₉₄	—	4	1	no	—
135	C₈×C₄₇	—	4	2	no	—
136	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
137	C₈×C₄₇	#117	1	1	yes	characteristic
138	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
139	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
140	C₄₇⋊C₈	—	4	2	no	—
141	C₄₇⋊C₈	#102	1	1	yes	characteristic
142	C₄₇⋊C₈	#102	1	1	yes	characteristic
143	C₄₇⋊C₈	#102	1	1	yes	characteristic
144	C₄₇⋊C₈	#102	1	1	yes	characteristic
145	#12646	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	4	1	no	—
147	C₂²×C₄₇	#416	1	1	yes	characteristic
148	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
149	C₂²×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
150	C₂×D₄₇	—	4	2	no	—
151	C₂×D₄₇	—	8	2	no	—
152	C₄×C₄₇	—	8	2	no	—
153	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
155	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
156	C₄×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
157	Dic₄₇	—	2	2	no	—
158	Dic₄₇	—	4	1	no	—
159	Dic₄₇	—	4	2	no	—
160	C₂×#443	—	47	1	no	—
161	#2148	—	47	1	no	—
162	#2145	—	47	1	no	—
163	C₄×M₃₂	—	47	1	no	—
164	#1772	—	47	1	no	—
165	#1389	D₄₇	1	1	yes	characteristic, complemented
166	#1388	—	47	1	no	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	—	2	1	no	—
169	C₂×C₄₇	—	2	1	no	—
170	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
171	D₄₇	—	8	2	no	—
172	C₂×#135	—	47	1	no	—
173	C₂×#131	—	94	1	no	—
174	C₂×M₃₂	—	47	1	no	—
175	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
176	#443	—	47	2	no	—
177	#443	—	47	2	no	—
178	#443	—	94	1	no	—
179	#443	—	94	1	no	—
180	C₂×#108	—	47	1	no	—
181	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
182	#427	C₂×D₄₇	1	1	yes	characteristic
183	C₂×#104	—	94	1	no	—
184	#405	—	94	1	no	—
185	C₂×C₄×C₈	—	47	1	no	—
186	#362	—	47	2	no	—
187	#361	—	47	2	no	—
188	#360	—	47	4	no	—
189	#346	—	94	1	no	—
190	C₄×C₁₆	—	94	1	no	—
191	#339	—	94	1	no	—
192	C₄₇	#12646	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	94	1	no	—
194	#135	—	47	2	no	—
195	#135	—	47	2	no	—
196	#135	—	94	1	no	—
197	#135	—	94	1	no	—
198	C₂×Q₁₆	—	47	1	no	—
199	C₂×SD₁₆	—	94	1	no	—
200	C₂×D₈	—	47	1	no	—
201	#131	—	94	2	no	—
202	#131	—	94	2	no	—
203	#131	—	188	1	no	—
204	#131	—	188	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	94	1	no	—
207	C₂×M₁₆	—	188	1	no	—
208	C₂²×C₈	—	47	1	no	—
209	C₂²×C₈	—	188	1	no	—
210	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
211	#117	—	2	1	no	—
212	C₂×C₄²	—	47	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	47	2	no	—
215	M₃₂	—	94	1	no	—
216	M₃₂	C₄×D₄₇	1	2	yes	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	94	1	no	—
219	#108	—	2	1	no	—
220	#108	—	47	2	no	—
221	#108	—	94	1	no	—
222	#108	C₄×D₄₇	1	2	yes	—
223	#107	—	2	1	no	—
224	#106	C₄×D₄₇	1	2	yes	—
225	#104	—	94	2	no	—
226	#104	—	94	2	no	—
227	#97	—	188	1	no	—
228	C₄×C₈	—	47	1	no	—
229	C₄×C₈	—	47	1	no	—
230	C₄×C₈	—	47	1	no	—
231	C₄×C₈	—	47	1	no	—
232	C₄×C₈	—	188	1	no	—
233	#41	—	94	2	no	—
234	#41	—	94	2	no	—
235	#41	—	188	1	no	—
236	#41	—	188	1	no	—
237	C₂×Q₈	—	94	1	no	—
238	C₂×D₄	—	94	1	no	—
239	C₂×D₄	—	188	1	no	—
240	C₂²×C₄	—	94	1	no	—
241	C₂²×C₄	—	188	1	no	—
242	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
243	Q₁₆	—	47	2	no	—
244	Q₁₆	—	94	1	no	—
245	SD₁₆	—	94	2	no	—
246	SD₁₆	—	94	2	no	—
247	D₈	—	47	2	no	—
248	D₈	—	94	1	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	94	2	no	—
252	M₁₆	—	188	2	no	—
253	M₁₆	—	188	2	no	—
254	M₁₆	—	188	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	47	1	no	—
257	C₂×C₈	—	47	1	no	—
258	C₂×C₈	—	47	1	no	—
259	C₂×C₈	—	47	1	no	—
260	C₂×C₈	—	47	1	no	—
261	C₂×C₈	—	47	1	no	—
262	C₂×C₈	—	188	2	no	—
263	C₂×C₈	—	188	2	no	—
264	C₂×C₈	—	188	1	no	—
265	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
268	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
269	C₂×C₈	#10625352	1	1	yes	characteristic
270	C₂×C₈	#10625354	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	47	1	no	—
275	C₄²	—	47	1	no	—
276	C₄²	—	94	1	no	—
277	C₁₆	—	2	2	no	—
278	C₁₆	—	94	2	no	—
279	C₂³	—	188	1	no	—
280	C₂³	#13094045	1	1	yes	characteristic
281	Q₈	—	94	2	no	—
282	Q₈	—	188	1	no	—
283	D₄	—	94	2	no	—
284	D₄	—	188	2	no	—
285	D₄	—	188	2	no	—
286	D₄	—	188	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	94	1	no	—
289	C₂×C₄	—	94	1	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	—	188	1	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
296	C₂×C₄	#13093986	1	1	yes	characteristic
297	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
298	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
299	C₂×C₄	#13094047	1	1	yes	characteristic
300	C₈	—	4	2	no	—
301	C₈	—	47	1	no	—
302	C₈	—	47	1	no	—
303	C₈	—	47	1	no	—
304	C₈	—	47	1	no	—
305	C₈	—	188	2	no	—
306	C₈	C₄²×D₄₇	1	1	yes	characteristic
307	C₈	#13093982	1	1	yes	characteristic
308	C₈	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
309	C₈	#13093986	1	1	yes	characteristic
310	C₂²	—	4	1	no	—
311	C₂²	—	376	2	no	—
312	C₂²	—	188	2	no	—
313	C₂²	#15689118	1	1	yes	characteristic
314	C₂²	#15689203	1	1	yes	characteristic
315	C₂²	#15689210	1	1	yes	characteristic
316	C₄	—	8	2	no	—
317	C₄	—	94	2	no	—
318	C₄	—	188	1	no	—
319	C₄	—	188	2	no	—
320	C₄	#15688970	1	1	yes	characteristic
321	C₄	#102×D₄₇	1	1	yes	characteristic
322	C₄	#15689205	1	1	yes	characteristic
323	C₄	#15689208	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	376	2	no	—
328	C₂	#20243202	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#102×D₄₇
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1389	C₂
socle	C₂×C₄₇	C₄×#102
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1389	D₄₇
hypocenter	C₄₇	#12646
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1389

Complements

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

N	H ≅ G/N	type
4. M₃₂×Dic₄₇	C₂	semidirect
5. #20243269	C₂	semidirect
6. #20241499	C₂	semidirect
7. C₄₇×#1389	C₂	semidirect
13. #15689239	C₄	semidirect
14. #15689239	C₄	semidirect
19. #15688940	C₄	semidirect
20. #15688937	C₄	semidirect
21. #15688932	C₄	semidirect
165. #1389	D₄₇	semidirect
192. C₄₇	#12646	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689239	2
#20245138	2
#20245123	2
M₃₂×Dic₄₇	2
#20243269	2
#20241499	2
C₄₇×#1389	2
#12646	47