Subgroups — #26371449

Order 12032.

556 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252241	C₂	1	1	yes	maximal, characteristic
2	#20245593	C₂	1	2	yes	maximal, complemented
3	#20245542	C₂	1	2	yes	maximal, complemented
4	#20245430	C₂	1	2	yes	maximal, complemented
5	#20245316	C₂	1	2	yes	maximal
6	#20245250	C₂	1	2	yes	maximal, complemented
7	C₄×#13093967	C₂	1	2	yes	maximal, complemented
8	#20245195	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3177	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
11	#15689597	—	2	1	no	—
12	C₂×#13093967	C₂²	1	2	yes	—
13	C₂×#13093967	C₂²	1	2	yes	—
14	#15689236	—	2	1	no	—
15	#15689236	—	2	1	no	—
16	C₄×D₃₇₆	C₂²	1	2	yes	—
17	C₄×D₃₇₆	C₂²	1	2	yes	—
18	#15689161	C₂²	1	1	yes	characteristic
19	#15689129	—	2	1	no	—
20	#15688950	C₂²	1	2	yes	complemented
21	#15688948	C₂²	1	2	yes	complemented
22	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
23	#15688927	C₂²	1	2	yes	complemented
24	#15688927	C₂²	1	2	yes	complemented
25	#15688923	C₂²	1	2	yes	—
26	#15688923	C₂²	1	2	yes	—
27	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
28	#15688899	C₂²	1	1	yes	characteristic, complemented
29	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₄₇×#541	—	2	1	no	—
31	C₄₇×#499	C₂²	1	1	yes	characteristic, complemented
32	C₄₇×#494	C₂²	1	1	yes	characteristic, complemented
33	C₄₇×#491	—	2	1	no	—
34	C₄₇×#470	—	2	1	no	—
35	C₄₇×#467	C₂²	1	2	yes	—
36	C₄₇×#441	—	2	1	no	—
37	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	complemented
38	C₄₇×#405	C₂²	1	1	yes	characteristic
39	C₂²×D₁₈₈	—	2	1	no	—
40	C₂×D₃₇₆	—	2	1	no	—
41	C₂×D₃₇₆	—	2	4	no	—
42	C₂×D₃₇₆	C₂³	1	2	yes	—
43	C₄×D₁₈₈	—	2	2	no	—
44	C₄×D₁₈₈	—	2	2	no	—
45	#13093986	—	4	2	no	—
46	#13093983	—	2	1	no	—
47	C₂²⋊C₄×D₄₇	—	4	2	no	—
48	#13093967	—	2	4	no	—
49	#13093967	—	2	4	no	—
50	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
51	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
52	#13093958	C₂³	1	2	yes	—
53	#13093956	—	2	1	no	—
54	#13093946	—	2	2	no	—
55	#13093946	—	2	2	no	—
56	C₂×Q₁₆×C₄₇	—	2	1	no	—
57	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
58	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₄₇	—	2	1	no	—
61	#126×C₄₇	—	4	2	no	—
62	#125×C₄₇	—	2	1	no	—
63	#125×C₄₇	—	4	2	no	—
64	C₂²⋊Q₈×C₄₇	—	2	2	no	—
65	C₄×Q₈×C₄₇	—	2	2	no	—
66	#117×C₄₇	D₄	1	1	yes	characteristic
67	#107×C₄₇	C₂³	1	2	yes	—
68	#106×C₄₇	—	2	1	no	—
69	#103×C₄₇	—	2	2	no	—
70	#103×C₄₇	—	2	2	no	—
71	#97×C₄₇	D₄	1	1	yes	characteristic
72	#97×C₄₇	D₄	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂³×D₄₇	—	4	1	no	—
76	C₂×Q₈×C₄₇	—	2	2	no	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×D₁₈₈	—	2	4	no	—
79	C₂×D₁₈₈	—	2	2	no	—
80	C₂×D₁₈₈	—	4	2	no	—
81	C₂×D₁₈₈	—	4	1	no	—
82	C₂×C₄×D₄₇	—	4	2	no	—
83	C₂×C₄×D₄₇	—	4	2	no	—
84	Q₁₆×C₄₇	—	2	2	no	—
85	Q₁₆×C₄₇	—	2	2	no	—
86	Q₁₆×C₄₇	—	4	2	no	—
87	M₁₆×C₄₇	—	4	1	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	—	4	2	no	—
95	C₄⋊C₄×C₄₇	—	4	2	no	—
96	C₄⋊C₄×C₄₇	—	4	2	no	—
97	C₄⋊C₄×C₄₇	—	4	2	no	—
98	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
99	C₂²⋊C₄×C₄₇	—	4	2	no	—
100	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
101	C₄²×C₄₇	—	4	2	no	—
102	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂²⋊Dic₄₇	—	4	1	no	—
105	(C₂×D₄₇)⋊C₄	—	4	2	no	—
106	(C₂×D₄₇)⋊C₄	—	4	2	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄×Dic₄₇	—	4	1	no	—
109	D₃₇₆	—	2	4	no	—
110	D₃₇₆	—	2	2	no	—
111	D₃₇₆	#41	1	4	yes	—
112	C₄₇⋊C₁₆	—	2	2	no	—
113	C₄₇⋊C₁₆	—	2	2	no	—
114	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₄₇	—	4	2	no	—
116	C₂²×D₄₇	—	4	4	no	—
117	C₂²×D₄₇	—	16	2	no	—
118	Q₈×C₄₇	—	4	2	no	—
119	Q₈×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	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	2	yes	—
126	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₄₇	—	4	2	no	—
129	D₁₈₈	—	2	4	no	—
130	D₁₈₈	—	4	4	no	—
131	D₁₈₈	—	4	2	no	—
132	C₄×D₄₇	—	16	2	no	—
133	C₄×D₄₇	—	16	2	no	—
134	C₈×C₄₇	—	2	1	no	—
135	C₈×C₄₇	—	2	1	no	—
136	C₈×C₄₇	—	2	1	no	—
137	C₈×C₄₇	C₂×#41	1	2	yes	—
138	#33683	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₄₇	—	4	4	no	—
143	C₂×D₄₇	—	8	2	no	—
144	C₂×D₄₇	—	8	2	no	—
145	C₂×D₄₇	—	16	4	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	8	2	no	—
150	C₄×C₄₇	—	8	2	no	—
151	C₄×C₄₇	#525	1	1	yes	characteristic
152	C₄×C₄₇	#575	1	1	yes	characteristic
153	Dic₄₇	—	8	2	no	—
154	#3177	D₄₇	1	1	yes	characteristic, complemented
155	#2948	—	47	1	no	—
156	#2237	—	47	2	no	—
157	#2228	—	47	2	no	—
158	#2208	—	47	2	no	—
159	#2187	—	47	2	no	—
160	#2175	—	47	2	no	—
161	C₄×SD₃₂	—	47	2	no	—
162	#2164	—	47	1	no	—
163	C₂×C₄₇	—	4	1	no	—
164	C₂×C₄₇	#3100	1	1	yes	characteristic
165	C₂×C₄₇	#3416	1	2	yes	—
166	D₄₇	—	8	4	no	—
167	C₂²×D₈	—	47	1	no	—
168	#541	—	2	1	no	—
169	#518	—	94	1	no	—
170	C₂×SD₃₂	—	47	2	no	—
171	C₂×SD₃₂	—	47	2	no	—
172	#499	C₂×D₄₇	1	1	yes	characteristic, complemented
173	#494	C₂×D₄₇	1	1	yes	characteristic, complemented
174	#491	—	2	1	no	—
175	#470	—	2	1	no	—
176	#467	C₂×D₄₇	1	2	yes	—
177	#442	—	94	1	no	—
178	#442	—	94	1	no	—
179	#441	—	2	1	no	—
180	C₄×Q₁₆	C₂×D₄₇	1	2	yes	complemented
181	C₄×D₈	—	47	2	no	—
182	C₄×D₈	—	47	2	no	—
183	#427	—	47	1	no	—
184	#418	—	94	1	no	—
185	#405	C₂×D₄₇	1	1	yes	characteristic
186	#367	—	47	2	no	—
187	#366	—	47	2	no	—
188	C₄⋊C₁₆	—	47	2	no	—
189	#358	—	47	2	no	—
190	#358	—	47	2	no	—
191	#357	—	47	2	no	—
192	#357	—	47	2	no	—
193	C₂²⋊C₁₆	—	47	2	no	—
194	#346	—	47	1	no	—
195	C₄×C₁₆	—	47	1	no	—
196	C₄₇	#33683	1	1	yes	characteristic, complemented, Sylow
197	C₂²×D₄	—	94	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
200	C₂×D₈	—	47	2	no	—
201	C₂×D₈	—	94	4	no	—
202	C₂×D₈	—	94	1	no	—
203	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
205	C₄⋊Q₈	—	2	1	no	—
206	#126	—	4	2	no	—
207	#125	—	2	1	no	—
208	#125	—	4	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×D₄	—	94	2	no	—
212	C₄×D₄	—	94	2	no	—
213	C₄×D₄	—	188	2	no	—
214	#117	—	94	1	no	—
215	#117	C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	188	2	no	—
217	SD₃₂	—	94	4	no	—
218	SD₃₂	—	94	4	no	—
219	C₂×C₁₆	—	47	2	no	—
220	C₂×C₁₆	—	47	2	no	—
221	#107	—	47	2	no	—
222	#107	C₂²×D₄₇	1	2	yes	—
223	#106	—	2	1	no	—
224	#106	—	94	1	no	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	#102	—	94	2	no	—
228	#102	—	94	2	no	—
229	#97	C₂²⋊D₄₇	1	1	yes	characteristic
230	#97	C₂²⋊D₄₇	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₂⁴	—	188	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×D₄	—	94	4	no	—
236	C₂×D₄	—	94	2	no	—
237	C₂×D₄	—	188	2	no	—
238	C₂×D₄	—	188	1	no	—
239	C₂²×C₄	—	188	2	no	—
240	C₂²×C₄	—	188	2	no	—
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	4	2	no	—
245	D₈	—	47	4	no	—
246	D₈	—	94	4	no	—
247	D₈	—	94	2	no	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	94	2	no	—
260	C₄⋊C₄	#10625334	1	2	yes	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	188	1	no	—
263	C₂²⋊C₄	—	188	2	no	—
264	C₂²⋊C₄	—	188	2	no	—
265	C₂²⋊C₄	#10625334	1	2	yes	—
266	C₄²	—	4	2	no	—
267	C₄²	—	188	1	no	—
268	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
270	C₁₆	—	94	2	no	—
271	C₁₆	—	94	2	no	—
272	C₂³	—	188	4	no	—
273	C₂³	—	188	2	no	—
274	C₂³	—	752	2	no	—
275	C₂³	C₂×#10625334	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	D₄	—	94	4	no	—
279	D₄	—	188	4	no	—
280	D₄	—	188	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	—	752	2	no	—
287	C₂×C₄	—	752	2	no	—
288	C₂×C₄	C₂×#10625334	1	2	yes	—
289	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
290	C₂×C₄	C₂×#10625334	1	2	yes	—
291	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#41×D₄₇	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	376	2	no	—
299	C₂²	—	376	2	no	—
300	C₂²	—	188	4	no	—
301	C₂²	—	752	4	no	—
302	C₂²	C₂²×#10625334	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	376	2	no	—
309	C₄	#15689630	1	1	yes	characteristic
310	C₄	#15689921	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	376	4	no	—
313	C₂	#20253384	1	1	yes	characteristic
314	C₂	#20256532	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3177	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₃₇₆	C₂²
hypercenter	#3177	D₄₇
hypocenter	C₄₇	#33683
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3177

Complements

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

N	H ≅ G/N	type
2. #20245593	C₂	semidirect
3. #20245542	C₂	semidirect
4. #20245430	C₂	semidirect
6. #20245250	C₂	semidirect
7. C₄×#13093967	C₂	semidirect
8. #20245195	C₂	semidirect
9. C₄₇×#3177	C₂	semidirect
20. #15688950	C₂²	semidirect
21. #15688948	C₂²	semidirect
22. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
23. #15688927	C₂²	semidirect
24. #15688927	C₂²	semidirect
28. #15688899	C₂²	semidirect
29. C₄×C₄₇⋊C₁₆	C₂²	semidirect
31. C₄₇×#499	C₂²	semidirect
32. C₄₇×#494	C₂²	semidirect
37. C₄×Q₁₆×C₄₇	C₂²	semidirect
154. #3177	D₄₇	semidirect
172. #499	C₂×D₄₇	semidirect
173. #494	C₂×D₄₇	semidirect
180. C₄×Q₁₆	C₂×D₄₇	semidirect
196. C₄₇	#33683	semidirect

Maximal subgroups

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

subgroup	index
#20252241	2
#20245593	2
#20245542	2
#20245430	2
#20245316	2
#20245250	2
C₄×#13093967	2
#20245195	2
C₄₇×#3177	2
#33683	47