Subgroups — #26373202

Order 12032.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255818	C₂	1	1	yes	maximal, characteristic, complemented
2	#20255109	C₂	1	1	yes	maximal, characteristic
3	#20245617	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245598	C₂	1	2	yes	maximal, complemented
5	#20245480	C₂	1	2	yes	maximal, complemented
6	#20245439	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245329	C₂	1	2	yes	maximal, complemented
8	C₄×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245207	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245092	C₂	1	2	yes	maximal, complemented
11	C₄₇×#2224	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094043	C₂²	1	2	yes	—
13	#15689847	—	2	1	no	—
14	#15689777	—	2	1	no	—
15	C₂×Dic₃₇₆	C₂²	1	2	yes	—
16	C₂×#13093968	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
18	#15689503	C₂²	1	1	yes	characteristic
19	#15689466	—	2	2	no	—
20	#15689405	—	2	1	no	—
21	#15689365	C₂²	1	2	yes	—
22	#15689365	C₂²	1	2	yes	—
23	#15689328	—	2	1	no	—
24	#15689286	—	2	2	no	—
25	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
26	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
27	#15689160	C₂²	1	2	yes	—
28	#15688950	C₂²	1	1	yes	characteristic
29	#15688928	C₂²	1	2	yes	—
30	#15688926	C₂²	1	2	yes	—
31	#15688926	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×#366	C₂²	1	2	yes	—
39	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄₇×#357	C₂²	1	2	yes	—
41	C₂×#10625352	—	2	2	no	—
42	#13094043	—	2	4	no	—
43	#13094043	—	4	2	no	—
44	C₂×Dic₁₈₈	—	2	1	no	—
45	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₈₈	C₂³	1	2	yes	—
47	C₂×#10625335	—	2	2	no	—
48	C₂×C₈×D₄₇	C₂³	1	2	yes	—
49	#13094024	—	2	1	no	—
50	#13094019	—	4	2	no	—
51	#13094013	—	4	1	no	—
52	#13094008	—	4	2	no	—
53	#13094004	—	2	2	no	—
54	#13094004	—	2	2	no	—
55	#13093998	—	4	1	no	—
56	Q₈×Dic₄₇	—	4	1	no	—
57	C₄×Dic₉₄	—	2	1	no	—
58	C₄×D₁₈₈	D₄	1	1	yes	characteristic
59	#13093982	—	2	2	no	—
60	C₂×C₄⋊Dic₄₇	—	4	1	no	—
61	Dic₃₇₆	—	4	2	no	—
62	#13093968	—	2	2	no	—
63	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
65	C₁₆×D₄₇	—	4	2	no	—
66	#13093958	C₂³	1	1	yes	characteristic
67	#13093955	—	2	2	no	—
68	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
69	#13093949	—	2	1	no	—
70	#13093947	—	2	2	no	—
71	#13093947	—	2	2	no	—
72	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
73	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₄₇	—	2	1	no	—
75	C₄×D₄×C₄₇	—	2	1	no	—
76	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
77	#107×C₄₇	—	2	1	no	—
78	#107×C₄₇	C₂³	1	2	yes	—
79	#107×C₄₇	C₂³	1	1	yes	characteristic
80	#102×C₄₇	—	2	1	no	—
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₇	—	4	1	no	—
83	C₂×D₄×C₄₇	—	2	1	no	—
84	C₂²×C₄×C₄₇	—	4	1	no	—
85	C₂×C₂²⋊D₄₇	—	4	2	no	—
86	C₂²×Dic₄₇	—	4	2	no	—
87	#10625352	—	4	4	no	—
88	#10625352	—	8	2	no	—
89	C₂×D₁₈₈	D₈	1	1	yes	characteristic
90	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
91	C₂×Dic₉₄	—	2	1	no	—
92	C₂×Dic₉₄	—	2	2	no	—
93	D₈×C₄₇	—	2	1	no	—
94	D₈×C₄₇	#41	1	2	yes	—
95	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	2	1	no	—
99	C₄⋊C₄×C₄₇	—	4	1	no	—
100	C₂²⋊C₄×C₄₇	—	4	1	no	—
101	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
102	#10625335	—	4	4	no	—
103	(C₂×D₄₇)⋊C₄	—	4	2	no	—
104	(C₂×D₄₇)⋊C₄	—	4	2	no	—
105	(C₂×D₄₇)⋊C₄	—	4	2	no	—
106	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
107	C₄⋊Dic₄₇	—	2	1	no	—
108	C₄⋊Dic₄₇	—	4	1	no	—
109	C₄⋊Dic₄₇	—	4	1	no	—
110	C₄⋊Dic₄₇	—	4	2	no	—
111	C₄⋊Dic₄₇	—	4	2	no	—
112	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
113	Dic₄₇⋊C₄	—	4	1	no	—
114	C₄×Dic₄₇	—	4	1	no	—
115	C₄×Dic₄₇	—	4	2	no	—
116	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
117	Dic₁₈₈	—	2	1	no	—
118	Dic₁₈₈	—	2	2	no	—
119	Dic₁₈₈	—	2	1	no	—
120	Dic₁₈₈	—	2	2	no	—
121	Dic₁₈₈	—	8	1	no	—
122	C₈×D₄₇	—	2	2	no	—
123	C₈×D₄₇	—	2	2	no	—
124	C₁₆×C₄₇	—	2	2	no	—
125	C₄₇⋊C₁₆	—	2	1	no	—
126	C₄₇⋊C₁₆	—	2	1	no	—
127	C₂³×C₄₇	—	4	1	no	—
128	C₂²×D₄₇	C₂×D₈	1	2	yes	—
129	Q₈×C₄₇	—	8	2	no	—
130	D₄×C₄₇	—	2	2	no	—
131	D₄×C₄₇	—	4	1	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	—	8	2	no	—
135	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₂²⋊D₄₇	—	8	4	no	—
139	C₂×Dic₄₇	—	4	1	no	—
140	C₂×Dic₄₇	—	4	1	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	8	4	no	—
143	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
144	D₁₈₈	—	2	2	no	—
145	C₄×D₄₇	—	2	2	no	—
146	C₄×D₄₇	—	2	2	no	—
147	Dic₉₄	—	4	2	no	—
148	Dic₉₄	—	4	1	no	—
149	Dic₉₄	—	4	2	no	—
150	Dic₉₄	—	4	1	no	—
151	C₈×C₄₇	—	2	1	no	—
152	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
153	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
154	C₄₇⋊C₈	—	2	2	no	—
155	#33869	—	47	1	no	maximal, Sylow
156	C₂²×C₄₇	—	4	2	no	—
157	C₂²×C₄₇	—	8	1	no	—
158	C₂²×C₄₇	—	8	1	no	—
159	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
160	C₂×D₄₇	—	2	2	no	—
161	C₂×D₄₇	—	2	2	no	—
162	C₂×D₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	8	1	no	—
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	#508	1	2	yes	—
167	C₄×C₄₇	#546	1	1	yes	characteristic
168	C₄×C₄₇	#577	1	1	yes	characteristic
169	Dic₄₇	—	2	2	no	—
170	Dic₄₇	—	8	1	no	—
171	Dic₄₇	—	8	1	no	—
172	Dic₄₇	—	8	2	no	—
173	#3342	—	47	1	no	—
174	#3283	—	47	1	no	—
175	#2241	—	47	1	no	—
176	#2237	—	47	2	no	—
177	#2224	D₄₇	1	1	yes	characteristic, complemented
178	#2216	—	47	2	no	—
179	#2209	—	47	1	no	—
180	#2189	—	47	2	no	—
181	C₄×SD₃₂	—	47	1	no	—
182	D₄×C₁₆	—	47	1	no	—
183	#2138	—	47	2	no	—
184	C₂×C₄₇	—	8	2	no	—
185	C₂×C₄₇	#3294	1	1	yes	characteristic
186	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
187	C₂×C₄₇	#3417	1	1	yes	characteristic
188	D₄₇	—	4	2	no	—
189	C₂×#135	—	47	2	no	—
190	#556	—	94	1	no	—
191	#547	—	94	1	no	—
192	C₂×Q₃₂	—	47	2	no	—
193	C₂×SD₃₂	—	47	1	no	—
194	C₂²×C₁₆	—	47	2	no	—
195	#500	C₂×D₄₇	1	1	yes	characteristic
196	#494	—	47	1	no	—
197	#485	—	94	2	no	—
198	#475	—	94	1	no	—
199	#467	—	47	2	no	—
200	#467	—	47	2	no	—
201	#461	—	94	1	no	—
202	#452	—	94	2	no	—
203	C₄×Q₁₆	—	47	1	no	—
204	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
205	C₈×D₄	—	47	1	no	—
206	#427	—	47	2	no	—
207	#367	—	47	1	no	—
208	#366	C₂×D₄₇	1	2	yes	—
209	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
210	#358	—	47	2	no	—
211	#358	—	47	2	no	—
212	#358	—	47	1	no	—
213	#358	—	47	2	no	—
214	#357	—	47	1	no	—
215	#357	C₂×D₄₇	1	2	yes	—
216	C₂²⋊C₁₆	—	47	2	no	—
217	C₄×C₁₆	—	47	1	no	—
218	C₄₇	#33869	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	94	2	no	—
220	#135	—	94	4	no	—
221	#135	—	188	2	no	—
222	C₂×Q₁₆	—	47	2	no	—
223	C₂×Q₁₆	—	47	1	no	—
224	C₂×Q₁₆	—	94	1	no	—
225	C₂×SD₁₆	—	94	2	no	—
226	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
227	C₂²×C₈	—	47	2	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	94	1	no	—
230	#126	—	188	2	no	—
231	#124	—	188	1	no	—
232	#123	—	188	2	no	—
233	C₂²⋊Q₈	—	94	2	no	—
234	C₂²⋊Q₈	—	94	2	no	—
235	#121	—	188	1	no	—
236	C₄×Q₈	—	94	1	no	—
237	C₄×Q₈	—	188	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	47	1	no	—
240	#117	—	94	2	no	—
241	C₂×C₄⋊C₄	—	188	1	no	—
242	Q₃₂	—	188	2	no	—
243	SD₃₂	—	94	2	no	—
244	C₂×C₁₆	—	47	1	no	—
245	C₂×C₁₆	—	47	1	no	—
246	C₂×C₁₆	—	188	2	no	—
247	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
248	#107	—	2	1	no	—
249	#107	—	47	1	no	—
250	#107	C₂²×D₄₇	1	2	yes	—
251	#107	C₂²×D₄₇	1	1	yes	characteristic
252	#106	—	94	2	no	—
253	C₄⋊C₈	—	47	1	no	—
254	#103	—	94	2	no	—
255	#103	—	94	2	no	—
256	#103	—	94	1	no	—
257	#102	—	2	1	no	—
258	C₂²⋊C₈	—	47	2	no	—
259	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
260	#41	—	376	2	no	—
261	#41	—	188	4	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	94	1	no	—
264	C₂×Q₈	—	94	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	47	1	no	—
267	C₂×D₄	—	188	2	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	47	2	no	—
270	C₂²×C₄	—	188	2	no	—
271	Q₁₆	—	94	1	no	—
272	Q₁₆	—	94	2	no	—
273	Q₁₆	—	94	1	no	—
274	Q₁₆	—	94	2	no	—
275	Q₁₆	—	376	1	no	—
276	SD₁₆	—	188	4	no	—
277	D₈	—	2	1	no	—
278	D₈	#10625354	1	2	yes	—
279	C₂×C₈	—	47	2	no	—
280	C₂×C₈	—	94	2	no	—
281	C₂×C₈	—	94	2	no	—
282	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
283	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	47	1	no	—
288	C₄⋊C₄	—	94	1	no	—
289	C₄⋊C₄	—	188	2	no	—
290	C₄⋊C₄	—	188	2	no	—
291	C₄⋊C₄	—	188	1	no	—
292	C₄⋊C₄	—	188	1	no	—
293	C₄⋊C₄	—	188	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	47	2	no	—
296	C₂²⋊C₄	—	188	2	no	—
297	C₂²⋊C₄	—	188	2	no	—
298	C₂²⋊C₄	—	188	2	no	—
299	C₄²	—	188	2	no	—
300	C₄²	—	188	1	no	—
301	C₄²	D₄×D₄₇	1	1	yes	characteristic
302	C₁₆	—	2	2	no	—
303	C₁₆	—	94	1	no	—
304	C₁₆	—	94	1	no	—
305	C₂³	—	4	1	no	—
306	C₂³	—	47	2	no	—
307	Q₈	—	8	2	no	—
308	Q₈	—	188	2	no	—
309	Q₈	—	188	1	no	—
310	Q₈	—	188	2	no	—
311	Q₈	—	188	1	no	—
312	D₄	—	2	2	no	—
313	D₄	—	4	1	no	—
314	D₄	—	94	2	no	—
315	D₄	—	376	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	47	2	no	—
320	C₂×C₄	—	94	2	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	376	4	no	—
323	C₂×C₄	—	188	1	no	—
324	C₂×C₄	—	188	1	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
327	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	94	2	no	—
331	C₈	C₂×#10625354	1	1	yes	characteristic
332	C₈	#13094070	1	1	yes	characteristic
333	C₂²	—	4	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	94	2	no	—
337	C₂²	—	94	2	no	—
338	C₂²	—	94	2	no	—
339	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	2	no	—
343	C₄	—	94	2	no	—
344	C₄	—	376	1	no	—
345	C₄	—	376	2	no	—
346	C₄	—	376	1	no	—
347	C₄	#15689552	1	2	yes	—
348	C₄	#15689764	1	1	yes	characteristic
349	C₄	#15689932	1	1	yes	characteristic
350	C₂	—	8	2	no	—
351	C₂	—	188	2	no	—
352	C₂	#20255198	1	1	yes	characteristic
353	C₂	C₂×#15689552	1	1	yes	characteristic
354	C₂	#20256540	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×#2224	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20255109	C₂
hypercenter	#2224	D₄₇
hypocenter	C₄₇	#33869
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2224

Complements

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

N	H ≅ G/N	type
1. #20255818	C₂	semidirect
3. #20245617	C₂	semidirect
4. #20245598	C₂	semidirect
5. #20245480	C₂	semidirect
6. #20245439	C₂	semidirect
7. #20245329	C₂	semidirect
8. C₄×#13093968	C₂	semidirect
9. #20245207	C₂	semidirect
10. #20245092	C₂	semidirect
11. C₄₇×#2224	C₂	semidirect
177. #2224	D₄₇	semidirect
218. C₄₇	#33869	semidirect

Maximal subgroups

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

subgroup	index
#20255818	2
#20255109	2
#20245617	2
#20245598	2
#20245480	2
#20245439	2
#20245329	2
C₄×#13093968	2
#20245207	2
#20245092	2
C₄₇×#2224	2
#33869	47