Subgroups — #26373873

Order 12032.

502 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255819	C₂	1	1	yes	maximal, characteristic, complemented
2	#20255736	C₂	1	1	yes	maximal, characteristic
3	#20245663	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245591	C₂	1	2	yes	maximal, complemented
5	#20245580	C₂	1	2	yes	maximal, complemented
6	#20245531	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245526	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245501	C₂	1	2	yes	maximal, complemented
9	C₄×#13093969	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245176	C₂	1	2	yes	maximal, complemented
11	C₄₇×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689843	—	2	1	no	—
13	#15689777	—	2	1	no	—
14	C₂×#13093969	C₂²	1	1	yes	characteristic
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689528	C₂²	1	1	yes	characteristic
17	#15689465	—	2	2	no	—
18	#15689437	—	2	2	no	—
19	#15689409	—	2	1	no	—
20	#15689404	—	2	1	no	—
21	#15689364	C₂²	1	2	yes	—
22	#15689359	C₂²	1	2	yes	—
23	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
24	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
25	#15689161	C₂²	1	2	yes	—
26	C₂×#13093958	C₂²	1	2	yes	—
27	#15688951	C₂²	1	2	yes	—
28	#15688951	C₂²	1	1	yes	characteristic
29	#15688949	C₂²	1	2	yes	—
30	#15688949	C₂²	1	2	yes	—
31	#15688949	C₂²	1	2	yes	—
32	#15688928	C₂²	1	2	yes	—
33	#15688928	C₂²	1	1	yes	characteristic
34	#15688925	C₂²	1	2	yes	—
35	#15688925	C₂²	1	1	yes	characteristic
36	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
37	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#10625352	—	4	1	no	—
42	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
43	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
44	#13094024	—	2	1	no	—
45	#13094024	—	2	1	no	—
46	#13094017	—	4	2	no	—
47	#13094011	—	4	1	no	—
48	#13094008	—	4	2	no	—
49	#13094004	—	4	1	no	—
50	#13094003	—	2	2	no	—
51	#13093998	—	2	2	no	—
52	C₄×Dic₉₄	—	2	1	no	—
53	#13093989	—	4	1	no	—
54	C₄×D₁₈₈	—	2	1	no	—
55	#13093983	—	2	2	no	—
56	C₂×C₄⋊Dic₄₇	—	2	2	no	—
57	#13093969	—	2	2	no	—
58	#13093958	—	2	1	no	—
59	#13093958	—	2	1	no	—
60	#13093958	—	2	2	no	—
61	#13093958	C₂³	1	1	yes	characteristic
62	#13093958	C₂³	1	2	yes	—
63	#13093958	C₂³	1	2	yes	—
64	#13093958	C₂³	1	1	yes	characteristic
65	#13093956	—	2	2	no	—
66	#13093949	—	2	1	no	—
67	#13093949	—	2	2	no	—
68	#13093946	—	2	2	no	—
69	#13093946	—	2	1	no	—
70	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
71	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
72	M₃₂×C₄₇	—	2	4	no	—
73	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
74	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
77	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂×C₂²⋊D₄₇	—	4	2	no	—
82	C₂²×Dic₄₇	—	4	2	no	—
83	#10625352	—	8	4	no	—
84	C₂×D₁₈₈	—	2	1	no	—
85	C₂×C₄×D₄₇	—	4	1	no	—
86	C₂×Dic₉₄	—	2	1	no	—
87	C₂×Dic₉₄	—	4	1	no	—
88	C₂×Dic₉₄	—	4	1	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
95	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
96	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²⋊Dic₄₇	—	4	2	no	—
98	C₂²⋊Dic₄₇	—	4	2	no	—
99	(C₂×D₄₇)⋊C₄	—	4	1	no	—
100	(C₂×D₄₇)⋊C₄	—	4	2	no	—
101	C₄⋊Dic₄₇	—	2	1	no	—
102	C₄⋊Dic₄₇	—	2	2	no	—
103	C₄⋊Dic₄₇	—	2	1	no	—
104	C₄⋊Dic₄₇	—	2	2	no	—
105	C₄⋊Dic₄₇	—	4	2	no	—
106	C₄⋊Dic₄₇	—	4	1	no	—
107	C₄⋊Dic₄₇	—	4	1	no	—
108	Dic₄₇⋊C₄	—	4	2	no	—
109	Dic₄₇⋊C₄	—	4	1	no	—
110	C₄×Dic₄₇	—	4	2	no	—
111	C₄×Dic₄₇	—	4	1	no	—
112	Dic₁₈₈	—	2	1	no	—
113	Dic₁₈₈	—	2	1	no	—
114	D₃₇₆	—	2	1	no	—
115	D₃₇₆	—	2	1	no	—
116	C₁₆×C₄₇	—	2	2	no	—
117	C₁₆×C₄₇	—	2	1	no	—
118	C₁₆×C₄₇	#41	1	2	yes	complemented
119	C₂³×C₄₇	C₂×D₈	1	2	yes	—
120	C₂²×D₄₇	—	4	1	no	—
121	D₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₄₇	—	8	4	no	—
129	C₂×Dic₄₇	—	4	1	no	—
130	C₂×Dic₄₇	—	4	1	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	1	no	—
134	C₂×Dic₄₇	—	8	4	no	—
135	D₁₈₈	—	4	1	no	—
136	D₁₈₈	—	4	1	no	—
137	C₄×D₄₇	—	8	2	no	—
138	Dic₉₄	—	4	1	no	—
139	Dic₉₄	—	4	1	no	—
140	Dic₉₄	—	8	2	no	—
141	Dic₉₄	—	8	2	no	—
142	C₈×C₄₇	—	2	2	no	—
143	C₈×C₄₇	—	2	1	no	—
144	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
146	#33920	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₄₇	—	8	1	no	—
152	C₂×D₄₇	—	8	1	no	—
153	C₂×D₄₇	—	8	1	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	#509	1	2	yes	—
157	C₄×C₄₇	#547	1	1	yes	characteristic
158	C₄×C₄₇	#578	1	1	yes	characteristic
159	Dic₄₇	—	8	1	no	—
160	Dic₄₇	—	8	2	no	—
161	Dic₄₇	—	8	2	no	—
162	Dic₄₇	—	8	1	no	—
163	Dic₄₇	—	8	1	no	—
164	#3342	—	47	1	no	—
165	#3334	—	47	1	no	—
166	#2252	—	47	1	no	—
167	#2236	—	47	2	no	—
168	#2234	—	47	2	no	—
169	#2225	—	47	1	no	—
170	#2224	—	47	1	no	—
171	#2220	—	47	2	no	—
172	C₄×SD₃₂	—	47	1	no	—
173	#2169	D₄₇	1	1	yes	characteristic, complemented
174	#2159	—	47	2	no	—
175	C₂×C₄₇	—	4	2	no	—
176	C₂×C₄₇	#3334	1	1	yes	characteristic
177	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
178	C₂×C₄₇	#3417	1	1	yes	characteristic
179	D₄₇	—	16	1	no	—
180	#556	—	94	1	no	—
181	#547	—	94	1	no	—
182	C₂×SD₃₂	—	47	1	no	—
183	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
184	#500	—	47	1	no	—
185	#500	—	47	1	no	—
186	#485	—	94	2	no	—
187	#480	—	94	2	no	—
188	#475	—	94	1	no	—
189	#474	—	94	1	no	—
190	#467	—	47	2	no	—
191	#466	—	47	2	no	—
192	C₄×Q₁₆	—	47	1	no	—
193	C₄×D₈	—	47	1	no	—
194	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
195	#427	—	47	2	no	—
196	C₂×#107	—	47	2	no	—
197	#367	—	47	1	no	—
198	#367	—	47	2	no	—
199	#366	—	47	2	no	—
200	#366	—	47	2	no	—
201	#366	—	47	2	no	—
202	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
203	#358	—	47	1	no	—
204	#358	—	47	2	no	—
205	#357	—	47	1	no	—
206	#357	—	47	2	no	—
207	C₂²⋊C₁₆	C₂×D₄₇	1	2	yes	—
208	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
209	C₄₇	#33920	1	1	yes	characteristic, complemented, Sylow
210	C₂×#41	—	188	1	no	—
211	C₂×Q₁₆	—	47	1	no	—
212	C₂×D₈	—	47	1	no	—
213	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
214	C₄⋊Q₈	—	94	1	no	—
215	C₄⋊Q₈	—	94	1	no	—
216	#126	—	188	2	no	—
217	#124	—	188	1	no	—
218	#123	—	188	2	no	—
219	C₂²⋊Q₈	—	94	2	no	—
220	C₂²⋊Q₈	—	188	1	no	—
221	#121	—	94	2	no	—
222	C₄×Q₈	—	94	1	no	—
223	C₄×Q₈	—	188	1	no	—
224	C₄×D₄	—	94	1	no	—
225	C₄×D₄	D₁₈₈	1	1	yes	characteristic
226	#117	—	94	2	no	—
227	C₂×C₄⋊C₄	—	94	2	no	—
228	SD₃₂	—	94	2	no	—
229	M₃₂	—	2	4	no	—
230	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
232	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
233	#107	—	47	2	no	—
234	#107	—	47	1	no	—
235	#107	—	47	1	no	—
236	#107	—	47	2	no	—
237	#107	—	94	2	no	—
238	#107	—	94	1	no	—
239	#107	—	94	1	no	—
240	#106	—	94	2	no	—
241	C₄⋊C₈	D₁₈₈	1	1	yes	characteristic
242	#103	—	94	1	no	—
243	#103	—	94	2	no	—
244	#102	—	94	1	no	—
245	#102	—	94	2	no	—
246	C₂²⋊C₈	D₁₈₈	1	2	yes	—
247	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
248	#41	—	376	4	no	—
249	C₂×Q₈	—	94	1	no	—
250	C₂×Q₈	—	188	1	no	—
251	C₂×Q₈	—	188	1	no	—
252	C₂×D₄	—	94	1	no	—
253	C₂×D₄	—	188	2	no	—
254	C₂×D₄	D₃₇₆	1	1	yes	characteristic
255	C₂²×C₄	—	188	2	no	—
256	C₂²×C₄	—	188	1	no	—
257	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
258	Q₁₆	—	94	1	no	—
259	Q₁₆	—	94	1	no	—
260	D₈	—	94	1	no	—
261	D₈	—	94	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
265	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
266	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
267	C₄⋊C₄	—	94	1	no	—
268	C₄⋊C₄	—	94	2	no	—
269	C₄⋊C₄	—	94	1	no	—
270	C₄⋊C₄	—	94	2	no	—
271	C₄⋊C₄	—	188	1	no	—
272	C₄⋊C₄	—	188	2	no	—
273	C₄⋊C₄	—	188	2	no	—
274	C₄⋊C₄	—	188	1	no	—
275	C₄⋊C₄	—	188	1	no	—
276	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
277	C₂²⋊C₄	—	188	2	no	—
278	C₂²⋊C₄	—	188	2	no	—
279	C₂²⋊C₄	—	188	1	no	—
280	C₂²⋊C₄	—	188	2	no	—
281	C₂²⋊C₄	D₃₇₆	1	2	yes	—
282	C₄²	—	188	2	no	—
283	C₄²	—	188	1	no	—
284	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	#10625352	1	2	yes	complemented
288	C₂³	—	188	1	no	—
289	C₂³	C₂×D₃₇₆	1	2	yes	—
290	Q₈	—	376	2	no	—
291	Q₈	—	376	2	no	—
292	Q₈	—	188	1	no	—
293	Q₈	—	188	1	no	—
294	D₄	—	2	2	no	—
295	D₄	—	376	4	no	—
296	D₄	—	188	1	no	—
297	D₄	—	188	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	376	4	no	—
301	C₂×C₄	—	376	2	no	—
302	C₂×C₄	—	188	1	no	—
303	C₂×C₄	—	188	2	no	—
304	C₂×C₄	—	188	1	no	—
305	C₂×C₄	—	188	1	no	—
306	C₂×C₄	—	188	2	no	—
307	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
308	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
309	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	C₂×#10625352	1	1	yes	characteristic
314	C₈	#13094071	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	376	1	no	—
319	C₂²	—	376	1	no	—
320	C₂²	—	376	1	no	—
321	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	—	376	1	no	—
325	C₄	—	376	2	no	—
326	C₄	—	376	2	no	—
327	C₄	—	376	1	no	—
328	C₄	—	376	1	no	—
329	C₄	#15689561	1	2	yes	—
330	C₄	#15689777	1	1	yes	characteristic
331	C₄	#15689940	1	1	yes	characteristic
332	C₂	—	4	2	no	—
333	C₂	—	752	1	no	—
334	C₂	#20255736	1	1	yes	characteristic
335	C₂	C₂×#15689561	1	1	yes	characteristic
336	C₂	#20256541	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2169	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20255736	C₂
hypercenter	#2169	D₄₇
hypocenter	C₄₇	#33920
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #2169

Complements

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

N	H ≅ G/N	type
1. #20255819	C₂	semidirect
3. #20245663	C₂	semidirect
4. #20245591	C₂	semidirect
5. #20245580	C₂	semidirect
6. #20245531	C₂	semidirect
7. #20245526	C₂	semidirect
8. #20245501	C₂	semidirect
9. C₄×#13093969	C₂	semidirect
10. #20245176	C₂	semidirect
11. C₄₇×#2169	C₂	semidirect
118. C₁₆×C₄₇	#41	semidirect
173. #2169	D₄₇	semidirect
209. C₄₇	#33920	semidirect
287. C₁₆	#10625352	semidirect

Maximal subgroups

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

subgroup	index
#20255819	2
#20255736	2
#20245663	2
#20245591	2
#20245580	2
#20245531	2
#20245526	2
#20245501	2
C₄×#13093969	2
#20245176	2
C₄₇×#2169	2
#33920	47