Subgroups — #26373917

Order 12032.

482 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255812	C₂	1	1	yes	maximal, characteristic
2	#20255780	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245661	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245599	C₂	1	2	yes	maximal, complemented
5	#20245567	C₂	1	2	yes	maximal, complemented
6	#20245541	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245500	C₂	1	2	yes	maximal, complemented
8	C₄×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245219	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245175	C₂	1	2	yes	maximal, complemented
11	C₄₇×#2226	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689838	—	2	1	no	—
13	#15689834	—	2	1	no	—
14	C₂×#13093968	C₂²	1	1	yes	characteristic
15	C₂×#13093963	C₂²	1	2	yes	—
16	#15689528	C₂²	1	1	yes	characteristic
17	#15689460	—	2	2	no	—
18	#15689445	—	2	2	no	—
19	#15689415	—	2	1	no	—
20	#15689413	—	2	1	no	—
21	#15689365	C₂²	1	2	yes	—
22	#15689356	C₂²	1	2	yes	—
23	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
24	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
25	#15689160	C₂²	1	2	yes	—
26	#107×D₄₇	C₂²	1	2	yes	—
27	#15688950	C₂²	1	1	yes	characteristic
28	#15688949	C₂²	1	2	yes	—
29	#15688948	C₂²	1	2	yes	—
30	#15688948	C₂²	1	2	yes	—
31	#15688928	C₂²	1	2	yes	—
32	#15688926	C₂²	1	1	yes	characteristic
33	#15688924	C₂²	1	1	yes	characteristic
34	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₄₇×#500	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#367	C₂²	1	2	yes	—
39	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄₇×#357	C₂²	1	2	yes	—
41	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₄₇	C₂³	1	2	yes	—
43	#13094024	—	2	1	no	—
44	#13094015	—	4	1	no	—
45	#13094010	—	4	2	no	—
46	#13094004	—	2	2	no	—
47	#13094000	—	4	2	no	—
48	#13093998	—	4	1	no	—
49	#13093995	—	2	2	no	—
50	C₄×Dic₉₄	—	2	1	no	—
51	C₄×D₁₈₈	D₄	1	1	yes	characteristic
52	#13093986	—	4	1	no	—
53	#13093983	—	4	1	no	—
54	#13093982	—	2	2	no	—
55	C₄⋊C₄×D₄₇	—	2	2	no	—
56	#13093968	—	2	2	no	—
57	#13093963	—	2	4	no	—
58	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	#13093958	—	2	1	no	—
61	#13093958	C₂³	1	1	yes	characteristic
62	#13093958	C₂³	1	2	yes	—
63	#13093957	—	2	2	no	—
64	#13093955	—	2	2	no	—
65	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
66	#13093949	—	2	1	no	—
67	#13093947	—	2	2	no	—
68	#13093945	—	2	2	no	—
69	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
70	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₄₇	—	2	1	no	—
72	C₄×D₄×C₄₇	—	2	1	no	—
73	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
74	#107×C₄₇	—	2	1	no	—
75	#107×C₄₇	C₂³	1	2	yes	—
76	#107×C₄₇	C₂³	1	1	yes	characteristic
77	#102×C₄₇	—	2	1	no	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₇	—	4	1	no	—
80	C₂×D₄×C₄₇	—	2	1	no	—
81	C₂²×C₄×C₄₇	—	4	1	no	—
82	C₂×C₂²⋊D₄₇	—	4	2	no	—
83	C₂×D₁₈₈	D₈	1	1	yes	characteristic
84	C₂×C₄×D₄₇	—	4	2	no	—
85	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
86	C₂×Dic₉₄	—	2	1	no	—
87	C₂×Dic₉₄	—	4	1	no	—
88	D₈×C₄₇	—	2	1	no	—
89	D₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	2	1	no	—
94	C₄⋊C₄×C₄₇	—	4	1	no	—
95	C₂²⋊C₄×C₄₇	—	4	1	no	—
96	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²⋊Dic₄₇	—	4	2	no	—
98	(C₂×D₄₇)⋊C₄	—	4	2	no	—
99	(C₂×D₄₇)⋊C₄	—	4	2	no	—
100	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄⋊Dic₄₇	—	2	1	no	—
103	C₄⋊Dic₄₇	—	4	1	no	—
104	C₄⋊Dic₄₇	—	4	2	no	—
105	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
106	Dic₄₇⋊C₄	—	4	1	no	—
107	Dic₄₇⋊C₄	—	4	2	no	—
108	C₄×Dic₄₇	—	4	1	no	—
109	C₄×Dic₄₇	—	4	2	no	—
110	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
111	Dic₁₈₈	—	2	1	no	—
112	Dic₁₈₈	—	2	1	no	—
113	C₈×D₄₇	—	2	2	no	—
114	C₈×D₄₇	—	2	2	no	—
115	C₁₆×C₄₇	—	2	2	no	—
116	C₄₇⋊C₁₆	—	2	1	no	—
117	C₄₇⋊C₁₆	#41	1	2	yes	—
118	C₂³×C₄₇	—	4	1	no	—
119	C₂²×D₄₇	C₂×D₈	1	2	yes	—
120	Q₈×C₄₇	—	16	1	no	—
121	D₄×C₄₇	—	4	1	no	—
122	D₄×C₄₇	—	4	1	no	—
123	C₂×C₄×C₄₇	—	4	1	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	8	2	no	—
126	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₄₇	—	16	2	no	—
130	C₂×Dic₄₇	—	4	1	no	—
131	C₂×Dic₄₇	—	4	1	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
134	D₁₈₈	—	2	2	no	—
135	C₄×D₄₇	—	2	2	no	—
136	C₄×D₄₇	—	2	2	no	—
137	C₄×D₄₇	—	16	2	no	—
138	Dic₉₄	—	4	1	no	—
139	Dic₉₄	—	4	1	no	—
140	Dic₉₄	—	16	1	no	—
141	C₈×C₄₇	—	2	1	no	—
142	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄₇⋊C₈	—	2	2	no	—
145	#33924	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	8	1	no	—
147	C₂²×C₄₇	—	8	1	no	—
148	C₂²×C₄₇	—	8	1	no	—
149	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₄₇	—	2	2	no	—
151	C₂×D₄₇	—	2	2	no	—
152	C₂×D₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	8	1	no	—
155	C₄×C₄₇	—	8	2	no	—
156	C₄×C₄₇	#510	1	2	yes	—
157	C₄×C₄₇	#547	1	1	yes	characteristic
158	C₄×C₄₇	#578	1	1	yes	characteristic
159	Dic₄₇	—	2	2	no	—
160	Dic₄₇	—	8	1	no	—
161	Dic₄₇	—	8	2	no	—
162	Dic₄₇	—	8	1	no	—
163	#3341	—	47	1	no	—
164	#3338	—	47	1	no	—
165	#2252	—	47	1	no	—
166	#2237	—	47	2	no	—
167	#2232	—	47	2	no	—
168	#2227	—	47	1	no	—
169	#2226	D₄₇	1	1	yes	characteristic, complemented
170	#2220	—	47	2	no	—
171	C₄×SD₃₂	—	47	1	no	—
172	#2169	—	47	1	no	—
173	#2159	—	47	2	no	—
174	C₂×C₄₇	—	16	1	no	—
175	C₂×C₄₇	#3334	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
177	C₂×C₄₇	#3415	1	1	yes	characteristic
178	D₄₇	—	4	2	no	—
179	#555	—	94	1	no	—
180	#554	—	94	1	no	—
181	C₂×SD₃₂	—	47	1	no	—
182	C₂×M₃₂	—	47	2	no	—
183	#500	—	47	1	no	—
184	#500	C₂×D₄₇	1	1	yes	characteristic
185	#484	—	94	2	no	—
186	#482	—	94	2	no	—
187	#477	—	94	1	no	—
188	#476	—	94	1	no	—
189	#467	—	47	2	no	—
190	#466	—	47	2	no	—
191	C₄×Q₁₆	—	47	1	no	—
192	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
193	C₈×D₄	—	47	1	no	—
194	#427	—	47	2	no	—
195	C₂×#107	—	47	2	no	—
196	#367	—	47	1	no	—
197	#367	C₂×D₄₇	1	2	yes	—
198	#366	—	47	2	no	—
199	#366	—	47	2	no	—
200	#366	—	47	2	no	—
201	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
202	#358	—	47	2	no	—
203	#358	—	47	1	no	—
204	#357	—	47	1	no	—
205	#357	C₂×D₄₇	1	2	yes	—
206	C₂²⋊C₁₆	—	47	2	no	—
207	C₄×C₁₆	—	47	1	no	—
208	C₄₇	#33924	1	1	yes	characteristic, complemented, Sylow
209	C₂×Q₁₆	—	47	1	no	—
210	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
211	C₂²×C₈	—	47	2	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	94	1	no	—
214	#125	—	188	1	no	—
215	#124	—	188	2	no	—
216	C₂²⋊Q₈	—	94	2	no	—
217	C₂²⋊Q₈	—	188	2	no	—
218	#121	—	94	2	no	—
219	#121	—	188	1	no	—
220	C₄×Q₈	—	94	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	47	1	no	—
223	C₄×D₄	—	188	1	no	—
224	#117	—	94	2	no	—
225	#117	—	188	1	no	—
226	C₂×C₄⋊C₄	—	94	2	no	—
227	SD₃₂	—	94	2	no	—
228	M₃₂	—	94	4	no	—
229	C₂×C₁₆	—	47	1	no	—
230	C₂×C₁₆	—	47	1	no	—
231	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
232	#107	—	2	1	no	—
233	#107	—	47	1	no	—
234	#107	—	47	2	no	—
235	#107	—	94	2	no	—
236	#107	—	94	1	no	—
237	#107	C₂²×D₄₇	1	2	yes	—
238	#107	C₂²×D₄₇	1	1	yes	characteristic
239	#106	—	94	2	no	—
240	C₄⋊C₈	—	47	1	no	—
241	#103	—	94	2	no	—
242	#103	—	94	1	no	—
243	#102	—	2	1	no	—
244	#102	—	94	2	no	—
245	C₂²⋊C₈	—	47	2	no	—
246	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	94	1	no	—
249	C₂×Q₈	—	188	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	47	1	no	—
252	C₂×D₄	—	188	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	47	2	no	—
255	C₂²×C₄	—	188	2	no	—
256	Q₁₆	—	94	1	no	—
257	Q₁₆	—	94	1	no	—
258	D₈	—	2	1	no	—
259	D₈	—	2	1	no	—
260	C₂×C₈	—	47	2	no	—
261	C₂×C₈	—	94	2	no	—
262	C₂×C₈	—	94	2	no	—
263	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	47	1	no	—
269	C₄⋊C₄	—	94	2	no	—
270	C₄⋊C₄	—	94	1	no	—
271	C₄⋊C₄	—	188	2	no	—
272	C₄⋊C₄	—	188	2	no	—
273	C₄⋊C₄	—	188	1	no	—
274	C₄⋊C₄	—	188	1	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	47	2	no	—
277	C₂²⋊C₄	—	188	2	no	—
278	C₂²⋊C₄	—	188	2	no	—
279	C₂²⋊C₄	—	188	2	no	—
280	C₄²	—	188	2	no	—
281	C₄²	—	188	1	no	—
282	C₄²	D₄×D₄₇	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	47	2	no	—
285	C₁₆	—	94	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	47	2	no	—
288	Q₈	—	16	1	no	—
289	Q₈	—	188	1	no	—
290	Q₈	—	188	1	no	—
291	Q₈	—	752	1	no	—
292	D₄	—	4	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	94	2	no	—
295	D₄	—	752	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	47	2	no	—
300	C₂×C₄	—	94	2	no	—
301	C₂×C₄	—	94	2	no	—
302	C₂×C₄	—	188	2	no	—
303	C₂×C₄	—	188	1	no	—
304	C₂×C₄	—	188	1	no	—
305	C₂×C₄	—	752	2	no	—
306	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
307	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	94	2	no	—
311	C₈	#41×D₄₇	1	1	yes	characteristic
312	C₈	#13094072	1	1	yes	characteristic
313	C₂²	—	8	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	94	2	no	—
317	C₂²	—	94	2	no	—
318	C₂²	—	94	2	no	—
319	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	94	2	no	—
324	C₄	—	376	1	no	—
325	C₄	—	376	2	no	—
326	C₄	—	376	1	no	—
327	C₄	#15689562	1	2	yes	—
328	C₄	#15689775	1	1	yes	characteristic
329	C₄	#15689939	1	1	yes	characteristic
330	C₂	—	16	1	no	—
331	C₂	—	188	2	no	—
332	C₂	#20255734	1	1	yes	characteristic
333	C₂	C₂×#15689562	1	1	yes	characteristic
334	C₂	#20256528	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×#2226	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20255812	C₂
hypercenter	#2226	D₄₇
hypocenter	C₄₇	#33924
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

Complements

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

N	H ≅ G/N	type
2. #20255780	C₂	semidirect
3. #20245661	C₂	semidirect
4. #20245599	C₂	semidirect
5. #20245567	C₂	semidirect
6. #20245541	C₂	semidirect
7. #20245500	C₂	semidirect
8. C₄×#13093968	C₂	semidirect
9. #20245219	C₂	semidirect
10. #20245175	C₂	semidirect
11. C₄₇×#2226	C₂	semidirect
169. #2226	D₄₇	semidirect
208. C₄₇	#33924	semidirect

Maximal subgroups

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

subgroup	index
#20255812	2
#20255780	2
#20245661	2
#20245599	2
#20245567	2
#20245541	2
#20245500	2
C₄×#13093968	2
#20245219	2
#20245175	2
C₄₇×#2226	2
#33924	47