Subgroups — #26369823

Order 12032.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252363	C₂	1	1	yes	maximal, characteristic
2	#20245294	C₂	1	2	yes	maximal, complemented
3	#20245218	C₂	1	2	yes	maximal, complemented
4	#20245215	C₂	1	2	yes	maximal, complemented
5	#20245211	C₂	1	2	yes	maximal, complemented
6	#20245193	C₂	1	2	yes	maximal, complemented
7	#20245191	C₂	1	1	yes	maximal, characteristic, complemented
8	#20244983	C₂	1	2	yes	maximal, complemented
9	C₄×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689657	C₄	1	1	yes	characteristic
11	C₂×#13093963	C₂²	1	2	yes	—
12	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
13	#15689253	C₄	1	1	yes	characteristic
14	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
15	#15689202	C₄	1	1	yes	characteristic
16	#15689194	C₄	1	2	yes	—
17	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
18	C₈×C₂²⋊D₄₇	C₂²	1	2	yes	—
19	#15689182	C₄	1	2	yes	—
20	#15689180	C₄	1	1	yes	characteristic
21	#15689077	C₂²	1	2	yes	—
22	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
23	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
24	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
25	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
26	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
27	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
28	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
29	#15688899	C₂²	1	2	yes	—
30	#15688898	C₂²	1	2	yes	—
31	C₁₆×Dic₄₇	C₂²	1	2	yes	—
32	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#346	C₂²	1	2	yes	—
35	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#10625324	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₄₇	C₂³	1	2	yes	—
39	#13094018	C₈	1	2	yes	—
40	#13094016	C₈	1	1	yes	characteristic
41	#13094012	C₈	1	1	yes	characteristic
42	#13094007	C₈	1	2	yes	—
43	#13094003	C₈	1	1	yes	characteristic
44	#13093998	C₈	1	1	yes	characteristic
45	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
46	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
48	#13093981	C₂×C₄	1	2	yes	—
49	#13093963	—	2	4	no	—
50	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
51	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
52	C₁₆×D₄₇	—	2	4	no	—
53	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
54	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
55	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
56	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
57	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
58	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
59	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
60	#13093935	C₂×C₄	1	2	yes	—
61	#13093934	C₂×C₄	1	2	yes	—
62	C₈×Dic₄₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
66	M₃₂×C₄₇	—	2	4	no	—
67	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
68	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
69	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₄₇	C₂×C₈	1	2	yes	—
77	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
87	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
91	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
92	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
93	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
94	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
95	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
96	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
97	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
98	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
99	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
100	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
101	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
104	#10625324	—	2	4	no	—
105	C₈×D₄₇	—	2	2	no	—
106	C₈×D₄₇	—	2	2	no	—
107	C₁₆×C₄₇	—	2	2	no	—
108	C₁₆×C₄₇	#41	1	4	yes	—
109	C₄₇⋊C₁₆	—	2	2	no	—
110	C₄₇⋊C₁₆	—	2	2	no	—
111	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
112	C₂²×D₄₇	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	4	1	no	—
116	C₂×C₄×C₄₇	—	4	1	no	—
117	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
123	C₂²⋊D₄₇	—	4	2	no	—
124	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
125	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
126	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
127	D₁₈₈	—	4	1	no	—
128	C₄×D₄₇	—	2	2	no	—
129	C₄×D₄₇	—	2	2	no	—
130	Dic₉₄	—	4	1	no	—
131	C₈×C₄₇	—	2	1	no	—
132	C₈×C₄₇	—	2	1	no	—
133	C₈×C₄₇	#131	1	2	yes	—
134	C₈×C₄₇	C₂×#41	1	2	yes	—
135	C₄₇⋊C₈	—	2	2	no	—
136	C₄₇⋊C₈	—	2	2	no	—
137	#33481	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
142	C₂×D₄₇	—	2	2	no	—
143	C₂×D₄₇	—	2	2	no	—
144	C₂×D₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	1	no	—
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
151	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
152	Dic₄₇	—	2	2	no	—
153	Dic₄₇	—	2	2	no	—
154	Dic₄₇	—	2	2	no	—
155	#2966	—	47	1	no	—
156	#2183	—	47	2	no	—
157	#2169	—	47	2	no	—
158	#2168	—	47	2	no	—
159	#2168	—	47	2	no	—
160	#2164	—	47	2	no	—
161	#2163	—	47	1	no	—
162	#2108	—	47	2	no	—
163	C₄×M₃₂	D₄₇	1	1	yes	characteristic, complemented
164	C₂×C₄₇	—	4	1	no	—
165	C₂×C₄₇	C₈×#41	1	1	yes	characteristic
166	C₂×C₄₇	#3407	1	1	yes	characteristic
167	C₂×C₄₇	#3407	1	1	yes	characteristic
168	D₄₇	—	4	2	no	—
169	#529	—	47	1	no	—
170	C₂×M₃₂	—	47	2	no	—
171	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
172	C₂²×C₁₆	—	47	2	no	—
173	#446	—	47	1	no	—
174	C₈×Q₈	—	47	1	no	—
175	#436	—	47	1	no	—
176	#435	—	47	2	no	—
177	C₈×D₄	—	47	2	no	—
178	C₈×D₄	—	47	1	no	—
179	#433	—	47	2	no	—
180	#432	—	47	1	no	—
181	#405	—	47	2	no	—
182	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
183	C₄⋊C₁₆	—	47	2	no	—
184	C₄⋊C₁₆	—	47	2	no	—
185	C₄⋊C₁₆	—	47	2	no	—
186	C₂²⋊C₁₆	—	47	1	no	—
187	C₂²⋊C₁₆	—	47	1	no	—
188	C₂²⋊C₁₆	—	47	2	no	—
189	C₂²⋊C₁₆	—	47	2	no	—
190	#346	—	47	2	no	—
191	#346	—	47	2	no	—
192	#346	C₂×D₄₇	1	2	yes	—
193	C₄×C₁₆	—	47	2	no	—
194	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
195	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
196	C₄₇	#33481	1	1	yes	characteristic, complemented, Sylow
197	C₂×M₁₆	—	47	2	no	—
198	C₂²×C₈	—	47	2	no	—
199	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
201	#126	—	47	2	no	—
202	#125	—	47	1	no	—
203	#124	—	47	1	no	—
204	#123	—	47	2	no	—
205	C₂²⋊Q₈	—	47	1	no	—
206	#121	—	47	1	no	—
207	C₄×Q₈	—	47	1	no	—
208	C₄×D₄	—	47	2	no	—
209	C₄×D₄	—	47	1	no	—
210	#117	—	47	2	no	—
211	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
212	M₃₂	—	2	4	no	—
213	M₃₂	—	94	4	no	—
214	C₂×C₁₆	—	47	2	no	—
215	C₂×C₁₆	—	47	2	no	—
216	C₂×C₁₆	—	94	4	no	—
217	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
218	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
219	C₄⋊C₈	—	47	2	no	—
220	C₄⋊C₈	—	47	2	no	—
221	C₄⋊C₈	—	47	1	no	—
222	C₄⋊C₈	—	47	1	no	—
223	C₂²⋊C₈	—	47	2	no	—
224	C₂²⋊C₈	—	47	2	no	—
225	C₂²⋊C₈	—	47	2	no	—
226	#97	—	47	2	no	—
227	#97	—	47	2	no	—
228	C₄×C₈	—	47	2	no	—
229	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
230	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₂×Q₈	—	47	1	no	—
234	C₂×D₄	—	47	2	no	—
235	C₂×D₄	—	47	1	no	—
236	C₂²×C₄	—	47	2	no	—
237	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	M₁₆	—	94	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	47	2	no	—
244	C₂×C₈	—	47	2	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
251	C₄⋊C₄	—	47	2	no	—
252	C₄⋊C₄	—	47	2	no	—
253	C₄⋊C₄	—	47	2	no	—
254	C₄⋊C₄	—	47	2	no	—
255	C₄⋊C₄	—	47	1	no	—
256	C₄⋊C₄	—	47	1	no	—
257	C₂²⋊C₄	—	47	2	no	—
258	C₂²⋊C₄	—	47	2	no	—
259	C₂²⋊C₄	—	47	2	no	—
260	C₂²⋊C₄	—	47	2	no	—
261	C₂²⋊C₄	—	47	2	no	—
262	C₄²	—	47	2	no	—
263	C₄²	C₈×D₄₇	1	1	yes	characteristic
264	C₄²	C₈×D₄₇	1	1	yes	characteristic
265	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
266	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
267	C₁₆	—	2	2	no	—
268	C₁₆	—	94	2	no	—
269	C₁₆	—	94	2	no	—
270	C₁₆	#10625350	1	4	yes	—
271	C₂³	—	47	2	no	—
272	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
273	Q₈	—	188	1	no	—
274	D₄	—	188	1	no	—
275	D₄	—	188	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	47	2	no	—
281	C₂×C₄	—	47	2	no	—
282	C₂×C₄	—	47	2	no	—
283	C₂×C₄	—	94	2	no	—
284	C₂×C₄	—	94	2	no	—
285	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
287	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
289	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	94	2	no	—
294	C₈	—	94	2	no	—
295	C₈	#13094030	1	2	yes	—
296	C₈	C₂×#10625350	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	94	2	no	—
301	C₂²	—	94	2	no	—
302	C₂²	—	94	2	no	—
303	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	94	2	no	—
310	C₄	—	94	2	no	—
311	C₄	—	94	2	no	—
312	C₄	C₄×#10625350	1	1	yes	characteristic
313	C₄	C₂×#13094030	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	188	2	no	—
316	C₂	C₈×#10625350	1	1	yes	characteristic
317	C₂	#20256490	1	1	yes	characteristic
318	C₂	#20256490	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄×M₃₂×C₄₇	C₂
prime radical	#13093998	C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₄×M₃₂	D₄₇
hypocenter	C₄₇	#33481
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20245294	C₂	semidirect
3. #20245218	C₂	semidirect
4. #20245215	C₂	semidirect
5. #20245211	C₂	semidirect
6. #20245193	C₂	semidirect
7. #20245191	C₂	semidirect
8. #20244983	C₂	semidirect
9. C₄×M₃₂×C₄₇	C₂	semidirect
163. C₄×M₃₂	D₄₇	semidirect
196. C₄₇	#33481	semidirect

Maximal subgroups

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

subgroup	index
#20252363	2
#20245294	2
#20245218	2
#20245215	2
#20245211	2
#20245193	2
#20245191	2
#20244983	2
C₄×M₃₂×C₄₇	2
#33481	47