Subgroups — #26371633

Order 12032.

578 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	#20253899	C₂	1	1	yes	maximal, characteristic
2	#494×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245632	C₂	1	4	yes	maximal, complemented
4	#20245617	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245616	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245615	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245106	C₂	1	4	yes	maximal
8	C₄×#13093963	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₄₇	—	2	1	no	—
11	C₂×Q₁₆×D₄₇	C₂²	1	2	yes	—
12	#15689678	—	2	1	no	—
13	C₄⋊Q₈×D₄₇	—	2	1	no	—
14	C₂×#13093963	C₂²	1	2	yes	—
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689524	—	2	1	no	—
17	#15689524	C₂²	1	2	yes	—
18	#15689519	—	2	1	no	—
19	#15689503	C₂²	1	1	yes	characteristic
20	#15689502	—	2	1	no	—
21	#15689501	—	2	1	no	—
22	#15689501	C₂²	1	2	yes	—
23	#15689160	—	2	1	no	—
24	#15689160	C₂²	1	2	yes	—
25	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
26	#15688928	C₂²	1	4	yes	—
27	#15688927	C₂²	1	4	yes	—
28	#15688926	C₂²	1	4	yes	—
29	#15688898	C₂²	1	2	yes	—
30	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₄₇×#500	C₂²	1	1	yes	characteristic
32	C₄₇×#494	C₂²	1	1	yes	characteristic
33	C₄₇×#358	C₂²	1	4	yes	—
34	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₄₇	—	2	2	no	—
36	C₂×Dic₁₈₈	—	2	1	no	—
37	C₂×Dic₁₈₈	C₂³	1	2	yes	—
38	C₂×#10625337	—	2	2	no	—
39	C₂×#10625337	—	2	2	no	—
40	Q₁₆×D₄₇	—	2	8	no	—
41	Q₁₆×D₄₇	—	4	4	no	—
42	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
43	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
44	#13094024	—	2	1	no	—
45	#13094024	—	2	1	no	—
46	#13094023	—	4	1	no	—
47	#13094023	—	4	1	no	—
48	#13094022	—	2	2	no	—
49	#13094015	—	2	2	no	—
50	#13094004	—	4	1	no	—
51	#13094000	—	4	1	no	—
52	Q₈×Dic₄₇	—	4	1	no	—
53	#13093989	—	4	1	no	—
54	#13093982	—	2	2	no	—
55	C₄⋊C₄×D₄₇	—	4	1	no	—
56	C₄⋊C₄×D₄₇	—	4	1	no	—
57	C₄²×D₄₇	D₄	1	1	yes	characteristic
58	#13093963	—	2	4	no	—
59	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
60	#13093958	—	2	1	no	—
61	#13093958	C₂³	1	2	yes	—
62	#13093955	—	2	2	no	—
63	#13093955	—	2	2	no	—
64	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₄₇	D₄	1	1	yes	characteristic
66	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₄₇	—	2	1	no	—
68	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₄₇	—	2	1	no	—
70	C₄⋊Q₈×C₄₇	—	2	1	no	—
71	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
72	#107×C₄₇	—	2	1	no	—
73	#107×C₄₇	C₂³	1	2	yes	—
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₄₇	—	2	2	no	—
76	C₂×Q₈×C₄₇	—	4	1	no	—
77	Q₈×D₄₇	—	4	4	no	—
78	Q₈×D₄₇	—	4	4	no	—
79	C₂×C₄×D₄₇	—	4	2	no	—
80	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
81	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
82	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₉₄	—	2	2	no	—
84	C₂×Dic₉₄	—	4	1	no	—
85	C₂×Dic₉₄	—	4	2	no	—
86	Q₁₆×C₄₇	—	2	4	no	—
87	Q₁₆×C₄₇	—	4	2	no	—
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	4	1	no	—
92	C₄⋊C₄×C₄₇	—	4	1	no	—
93	C₄⋊C₄×C₄₇	—	4	1	no	—
94	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
95	#10625337	—	4	4	no	—
96	#10625337	—	4	4	no	—
97	(C₂×D₄₇)⋊C₄	—	4	2	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	4	1	no	—
100	C₄⋊Dic₄₇	—	4	1	no	—
101	C₄⋊Dic₄₇	—	4	1	no	—
102	C₄⋊Dic₄₇	—	4	2	no	—
103	Dic₄₇⋊C₄	—	4	2	no	—
104	Dic₄₇⋊C₄	—	4	2	no	—
105	Dic₄₇⋊C₄	—	4	2	no	—
106	C₄×Dic₄₇	—	4	2	no	—
107	C₄×Dic₄₇	D₈	1	1	yes	characteristic
108	C₄×Dic₄₇	D₈	1	1	yes	characteristic
109	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
112	Dic₁₈₈	—	2	4	no	—
113	Dic₁₈₈	—	4	2	no	—
114	C₈×D₄₇	SD₁₆	1	4	yes	—
115	C₈×D₄₇	#41	1	4	yes	—
116	C₁₆×C₄₇	—	2	2	no	—
117	C₄₇⋊C₁₆	—	2	2	no	—
118	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₄₇	—	4	2	no	—
120	Q₈×C₄₇	—	4	2	no	—
121	Q₈×C₄₇	—	8	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
132	C₄×D₄₇	—	2	2	no	—
133	C₄×D₄₇	—	2	2	no	—
134	C₄×D₄₇	—	8	4	no	—
135	C₄×D₄₇	#124	1	2	yes	—
136	C₄×D₄₇	C₂×SD₁₆	1	2	yes	—
137	Dic₉₄	—	4	2	no	—
138	Dic₉₄	—	4	2	no	—
139	Dic₉₄	—	8	2	no	—
140	Dic₉₄	—	8	4	no	—
141	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
142	C₈×C₄₇	C₂×#41	1	2	yes	—
143	C₄₇⋊C₈	#124	1	2	yes	—
144	C₄₇⋊C₈	C₂×SD₁₆	1	2	yes	—
145	#33705	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₄₇	—	2	1	no	—
148	C₂×D₄₇	—	2	1	no	—
149	C₂×D₄₇	#486	1	2	yes	—
150	C₄×C₄₇	—	8	2	no	—
151	C₄×C₄₇	—	8	2	no	—
152	C₄×C₄₇	#510	1	2	yes	—
153	C₄×C₄₇	#510	1	2	yes	—
154	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
155	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
156	Dic₄₇	—	2	1	no	—
157	Dic₄₇	—	2	1	no	—
158	Dic₄₇	—	8	2	no	—
159	Dic₄₇	—	8	2	no	—
160	Dic₄₇	#486	1	2	yes	—
161	#3160	—	47	1	no	—
162	C₂×#494	—	47	1	no	—
163	#2245	—	47	4	no	—
164	#2241	—	47	1	no	—
165	#2241	—	47	1	no	—
166	#2241	—	47	1	no	—
167	#2241	D₄₇	1	1	yes	characteristic, complemented
168	#2141	—	47	4	no	—
169	C₄×M₃₂	—	47	1	no	—
170	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
171	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
172	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
173	D₄₇	—	2	2	no	—
174	C₂²×Q₁₆	—	47	2	no	—
175	C₂²×Q₁₆	—	94	1	no	—
176	#533	—	94	1	no	—
177	C₂×C₄⋊Q₈	—	94	1	no	—
178	C₂×M₃₂	—	47	2	no	—
179	#500	—	47	1	no	—
180	#500	C₂×D₄₇	1	1	yes	characteristic
181	#499	—	47	2	no	—
182	#499	—	94	1	no	—
183	#498	—	94	1	no	—
184	#494	—	47	2	no	—
185	#494	—	47	1	no	—
186	#494	—	94	1	no	—
187	#494	—	94	1	no	—
188	#494	C₂×D₄₇	1	1	yes	characteristic
189	#427	—	47	2	no	—
190	#427	—	94	1	no	—
191	C₂×C₄×C₈	—	47	1	no	—
192	#358	—	47	4	no	—
193	#358	—	47	4	no	—
194	#358	—	47	4	no	—
195	#358	C₂×D₄₇	1	4	yes	—
196	#346	—	47	2	no	—
197	C₄×C₁₆	—	47	1	no	—
198	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
199	C₄₇	#33705	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	94	2	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	47	2	no	—
203	C₂×Q₁₆	—	94	8	no	—
204	C₂×Q₁₆	—	94	2	no	—
205	C₂×Q₁₆	—	94	2	no	—
206	C₂×Q₁₆	—	94	1	no	—
207	C₂×Q₁₆	—	188	4	no	—
208	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
209	C₂²×C₈	—	47	1	no	—
210	C₂²×C₈	—	47	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	94	2	no	—
214	C₄⋊Q₈	—	94	1	no	—
215	C₄⋊Q₈	—	94	1	no	—
216	C₄⋊Q₈	—	188	1	no	—
217	C₄⋊Q₈	—	188	1	no	—
218	#125	—	94	2	no	—
219	C₂²⋊Q₈	—	188	1	no	—
220	C₂²⋊Q₈	—	188	1	no	—
221	C₄×Q₈	—	188	1	no	—
222	C₄×Q₈	—	188	1	no	—
223	#117	—	94	2	no	—
224	C₂×C₄⋊C₄	—	188	1	no	—
225	C₂×C₄⋊C₄	—	188	1	no	—
226	C₂×C₄²	—	47	1	no	—
227	M₃₂	—	94	4	no	—
228	C₂×C₁₆	—	47	2	no	—
229	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
230	#107	—	2	1	no	—
231	#107	—	47	2	no	—
232	#107	—	94	1	no	—
233	#107	C₂²×D₄₇	1	2	yes	—
234	#106	—	94	2	no	—
235	#106	—	94	2	no	—
236	C₄×C₈	—	47	1	no	—
237	C₄×C₈	—	47	1	no	—
238	C₄×C₈	—	47	1	no	—
239	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	94	2	no	—
243	C₂×Q₈	—	188	2	no	—
244	C₂×Q₈	—	188	4	no	—
245	C₂×Q₈	—	188	4	no	—
246	C₂×Q₈	—	188	1	no	—
247	C₂²×C₄	—	47	1	no	—
248	C₂²×C₄	—	47	1	no	—
249	C₂²×C₄	—	47	1	no	—
250	C₂²×C₄	—	188	2	no	—
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	4	2	no	—
253	Q₁₆	—	94	4	no	—
254	Q₁₆	—	188	4	no	—
255	Q₁₆	—	188	4	no	—
256	Q₁₆	—	188	2	no	—
257	C₂×C₈	—	47	1	no	—
258	C₂×C₈	—	47	1	no	—
259	C₂×C₈	—	47	4	no	—
260	C₂×C₈	—	47	4	no	—
261	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	94	2	no	—
268	C₄⋊C₄	—	188	2	no	—
269	C₄⋊C₄	—	188	2	no	—
270	C₄⋊C₄	—	188	2	no	—
271	C₄⋊C₄	—	188	2	no	—
272	C₄⋊C₄	—	188	1	no	—
273	C₄⋊C₄	—	188	1	no	—
274	C₄⋊C₄	—	188	1	no	—
275	C₂²⋊C₄	—	188	2	no	—
276	C₄²	—	47	1	no	—
277	C₄²	—	47	1	no	—
278	C₄²	—	47	1	no	—
279	C₄²	—	188	2	no	—
280	C₄²	D₄×D₄₇	1	1	yes	characteristic
281	C₁₆	—	2	2	no	—
282	C₁₆	—	94	2	no	—
283	C₂³	—	47	1	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	376	4	no	—
288	Q₈	—	376	2	no	—
289	Q₈	—	188	2	no	—
290	Q₈	—	188	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	47	1	no	—
294	C₂×C₄	—	47	2	no	—
295	C₂×C₄	—	47	2	no	—
296	C₂×C₄	—	47	1	no	—
297	C₂×C₄	—	47	1	no	—
298	C₂×C₄	—	94	2	no	—
299	C₂×C₄	—	94	2	no	—
300	C₂×C₄	—	376	4	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	—	188	2	no	—
303	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
304	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
306	C₈	—	47	2	no	—
307	C₈	—	47	2	no	—
308	C₈	SD₁₆×D₄₇	1	2	yes	—
309	C₈	#41×D₄₇	1	2	yes	—
310	C₂²	—	47	2	no	—
311	C₂²	—	94	1	no	—
312	C₂²	—	94	1	no	—
313	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	47	2	no	—
317	C₄	—	94	1	no	—
318	C₄	—	94	1	no	—
319	C₄	—	376	2	no	—
320	C₄	—	376	2	no	—
321	C₄	#15689562	1	2	yes	—
322	C₄	#15689565	1	2	yes	—
323	C₄	#124×D₄₇	1	1	yes	characteristic
324	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
325	C₂	—	94	2	no	—
326	C₂	#486×D₄₇	1	1	yes	characteristic
327	C₂	C₂×#15689562	1	1	yes	characteristic
328	C₂	C₂×#15689565	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2241	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₄₇	C₂×D₈
hypercenter	#2241	D₄₇
hypocenter	C₄₇	#33705
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
3. #20245632	C₂	semidirect
4. #20245617	C₂	semidirect
5. #20245616	C₂	semidirect
6. #20245615	C₂	semidirect
9. C₄₇×#2241	C₂	semidirect
167. #2241	D₄₇	semidirect
199. C₄₇	#33705	semidirect

Maximal subgroups

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

subgroup	index
#20253899	2
#494×D₄₇	2
#20245632	2
#20245617	2
#20245616	2
#20245615	2
#20245106	2
C₄×#13093963	2
C₄₇×#2241	2
#33705	47