Subgroups — #26203961

Order 12032.

750 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253974	C₂	1	1	yes	maximal, characteristic, complemented
2	#20253943	C₂	1	2	yes	maximal, complemented
3	#20253710	C₂	1	2	yes	maximal, complemented
4	#20242999	C₂	1	2	yes	maximal, complemented
5	#20242997	C₂	1	1	yes	maximal, characteristic, complemented
6	#20242993	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242988	C₂	1	2	yes	maximal, complemented
8	#20242986	C₂	1	1	yes	maximal, characteristic, complemented
9	#20242977	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#3169	C₂	1	1	yes	maximal, characteristic
12	#15689680	C₂²	1	2	yes	—
13	#15689528	C₂²	1	1	yes	characteristic
14	#15689519	C₂²	1	2	yes	—
15	#15689503	C₂²	1	1	yes	characteristic
16	#15689495	C₂²	1	2	yes	—
17	#15689477	—	2	4	no	—
18	#15689475	C₂²	1	2	yes	—
19	#15689474	C₂²	1	2	yes	—
20	#15689473	C₂²	1	2	yes	—
21	#15689473	C₂²	1	2	yes	—
22	#15689469	C₂²	1	2	yes	—
23	#15689469	C₂²	1	2	yes	—
24	#15689469	C₂²	1	2	yes	—
25	#15689364	—	2	2	no	—
26	#15689350	—	2	4	no	—
27	C₄×Dic₁₈₈	—	2	2	no	—
28	C₄×#10625335	—	2	4	no	—
29	#15689119	—	2	4	no	—
30	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
33	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₄₇×#532	C₂²	1	2	yes	—
37	C₄₇×#500	C₂²	1	1	yes	characteristic
38	C₄₇×#493	C₂²	1	1	yes	characteristic
39	C₄₇×#486	C₂²	1	2	yes	—
40	C₄₇×#466	—	2	2	no	—
41	C₄×D₈×C₄₇	—	2	2	no	—
42	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×Dic₁₈₈	—	2	2	no	—
44	C₂×#10625335	—	2	4	no	—
45	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
46	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
47	#13094024	C₂³	1	2	yes	—
48	#13094024	C₂³	1	2	yes	—
49	#13094016	—	2	2	no	—
50	#13094003	—	2	2	no	—
51	#13094003	—	2	2	no	—
52	C₄×Dic₉₄	—	2	2	no	—
53	C₄×Dic₉₄	—	2	2	no	—
54	#13093983	—	2	2	no	—
55	#13093958	—	2	2	no	—
56	#13093955	—	2	4	no	—
57	#13093949	—	2	4	no	—
58	#13093948	—	2	4	no	—
59	#13093948	—	2	4	no	—
60	#13093947	—	2	4	no	—
61	#13093945	—	2	4	no	—
62	#13093945	—	2	4	no	—
63	#13093945	—	2	4	no	—
64	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
65	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
66	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×#41×C₄₇	—	2	2	no	—
71	C₂×D₈×C₄₇	—	2	2	no	—
72	C₂²×C₈×C₄₇	D₄	1	2	yes	—
73	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
74	#127×C₄₇	C₂³	1	2	yes	—
75	#124×C₄₇	—	2	2	no	—
76	#121×C₄₇	—	2	2	no	—
77	#121×C₄₇	—	2	2	no	—
78	C₄×D₄×C₄₇	—	2	2	no	—
79	C₄×D₄×C₄₇	—	2	2	no	—
80	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
81	#107×C₄₇	—	2	2	no	—
82	#102×C₄₇	—	2	4	no	—
83	C₄×C₈×C₄₇	D₄	1	2	yes	—
84	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	#41×C₄₇	—	4	8	no	—
87	C₂×Q₈×C₄₇	—	2	2	no	—
88	C₂×D₄×C₄₇	—	2	2	no	—
89	C₂×D₄×C₄₇	—	2	2	no	—
90	C₂×D₄×C₄₇	—	2	2	no	—
91	C₂×D₄×C₄₇	—	4	2	no	—
92	C₂²×C₄×C₄₇	—	4	2	no	—
93	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×Dic₉₄	—	2	2	no	—
97	C₂×Dic₉₄	—	2	2	no	—
98	D₈×C₄₇	—	4	4	no	—
99	C₂×C₈×C₄₇	—	2	4	no	—
100	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₄₇	—	2	2	no	—
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₂²⋊C₄×C₄₇	—	4	2	no	—
105	C₂²⋊C₄×C₄₇	—	4	2	no	—
106	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
110	C₂²⋊Dic₄₇	—	4	2	no	—
111	#10625335	—	4	8	no	—
112	C₄⋊Dic₄₇	—	2	2	no	—
113	C₄⋊Dic₄₇	—	2	2	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	Dic₄₇⋊C₄	—	4	2	no	—
117	Dic₄₇⋊C₄	—	4	2	no	—
118	C₄×Dic₄₇	—	4	2	no	—
119	C₂×C₄₇⋊C₈	—	2	4	no	—
120	C₂×C₄₇⋊C₈	—	2	4	no	—
121	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
122	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
123	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
124	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
125	Dic₁₈₈	—	4	4	no	—
126	C₂³×C₄₇	—	4	2	no	—
127	C₂³×C₄₇	#127	1	1	yes	characteristic
128	Q₈×C₄₇	—	4	4	no	—
129	D₄×C₄₇	—	4	4	no	—
130	D₄×C₄₇	—	4	4	no	—
131	D₄×C₄₇	—	4	4	no	—
132	D₄×C₄₇	—	8	4	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	8	4	no	—
137	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#135	1	4	yes	—
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	Dic₉₄	—	4	4	no	—
147	Dic₉₄	—	4	4	no	—
148	C₈×C₄₇	—	2	2	no	—
149	C₈×C₄₇	#135	1	4	yes	—
150	C₄₇⋊C₈	—	2	2	no	—
151	C₄₇⋊C₈	—	2	2	no	—
152	C₄₇⋊C₈	—	2	2	no	—
153	C₄₇⋊C₈	#135	1	4	yes	—
154	#19600	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	—	2	1	no	—
156	C₂²×C₄₇	—	2	1	no	—
157	C₂²×C₄₇	—	8	2	no	—
158	C₂²×C₄₇	—	8	2	no	—
159	C₂²×C₄₇	—	8	2	no	—
160	C₂²×C₄₇	#495	1	2	yes	—
161	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	#495	1	2	yes	—
166	C₄×C₄₇	C₂×#135	1	2	yes	—
167	C₄×C₄₇	C₂×#135	1	2	yes	—
168	C₄×C₄₇	C₂×#135	1	2	yes	—
169	Dic₄₇	—	8	2	no	—
170	Dic₄₇	—	8	2	no	—
171	#3170	—	47	1	no	—
172	#3169	D₄₇	1	1	yes	characteristic
173	#3167	—	47	2	no	—
174	#3131	—	47	2	no	—
175	#1711	—	47	2	no	—
176	#1710	—	47	1	no	—
177	#1709	—	47	1	no	—
178	#1708	—	47	2	no	—
179	#1707	—	47	1	no	—
180	#1706	—	47	1	no	—
181	C₂×C₈²	—	47	1	no	—
182	C₂×C₄₇	—	2	2	no	—
183	C₂×C₄₇	—	16	2	no	—
184	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
185	C₂×C₄₇	#3147	1	1	yes	characteristic
186	C₂×C₄₇	#3147	1	1	yes	characteristic
187	#533	—	47	2	no	—
188	#532	C₂×D₄₇	1	2	yes	—
189	#500	—	47	1	no	—
190	#500	C₂×D₄₇	1	1	yes	characteristic
191	#498	—	47	2	no	—
192	#494	—	47	1	no	—
193	#493	C₂×D₄₇	1	1	yes	characteristic
194	#492	—	47	2	no	—
195	#488	—	94	4	no	—
196	#487	—	47	2	no	—
197	#487	—	47	2	no	—
198	#487	—	47	2	no	—
199	#487	—	47	2	no	—
200	#486	—	47	2	no	—
201	#486	—	47	2	no	—
202	#486	—	47	2	no	—
203	#486	C₂×D₄₇	1	2	yes	—
204	#467	—	94	2	no	—
205	#466	—	2	2	no	—
206	#465	—	94	4	no	—
207	C₄×Q₁₆	—	94	2	no	—
208	C₄×SD₁₆	—	94	4	no	—
209	C₄×D₈	—	2	2	no	—
210	#416	—	94	4	no	—
211	C₂×C₄×C₈	—	47	1	no	—
212	C₂×C₄×C₈	—	47	1	no	—
213	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
214	C₈²	—	47	1	no	—
215	C₈²	—	47	1	no	—
216	C₈²	—	47	1	no	—
217	C₈²	—	47	1	no	—
218	C₄₇	#19600	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	2	2	no	—
220	C₂×Q₁₆	—	94	2	no	—
221	C₂×SD₁₆	—	94	4	no	—
222	C₂×D₈	—	2	2	no	—
223	C₂²×C₈	—	47	2	no	—
224	C₂²×C₈	—	47	2	no	—
225	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
226	C₄⋊Q₈	—	47	2	no	—
227	C₄⋊Q₈	—	47	2	no	—
228	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
229	#127	C₂²×D₄₇	1	2	yes	—
230	#125	—	94	2	no	—
231	#124	—	2	2	no	—
232	C₂²⋊Q₈	—	94	2	no	—
233	C₂²⋊Q₈	—	94	2	no	—
234	#121	—	2	2	no	—
235	#121	—	2	2	no	—
236	C₄×Q₈	—	94	2	no	—
237	C₄×Q₈	—	94	2	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	—	2	2	no	—
240	#117	—	94	2	no	—
241	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
242	#107	—	2	2	no	—
243	#107	—	94	2	no	—
244	#106	—	94	4	no	—
245	#103	—	94	4	no	—
246	#103	—	94	4	no	—
247	#103	—	94	4	no	—
248	#103	—	94	4	no	—
249	#102	—	2	4	no	—
250	#102	—	94	4	no	—
251	#102	—	94	4	no	—
252	#102	—	94	4	no	—
253	C₄×C₈	—	47	2	no	—
254	C₄×C₈	—	47	2	no	—
255	C₄×C₈	—	47	1	no	—
256	C₄×C₈	—	47	1	no	—
257	C₄×C₈	—	47	1	no	—
258	C₄×C₈	—	47	1	no	—
259	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
260	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
262	#41	—	4	8	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	94	2	no	—
265	C₂×Q₈	—	94	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
274	Q₁₆	—	188	4	no	—
275	SD₁₆	—	188	8	no	—
276	D₈	—	4	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	47	2	no	—
279	C₂×C₈	—	47	2	no	—
280	C₂×C₈	—	47	2	no	—
281	C₂×C₈	—	47	2	no	—
282	C₂×C₈	—	94	4	no	—
283	C₂×C₈	—	94	4	no	—
284	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
285	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	94	2	no	—
289	C₄⋊C₄	—	94	2	no	—
290	C₄⋊C₄	—	94	2	no	—
291	C₄⋊C₄	—	94	2	no	—
292	C₄⋊C₄	—	188	2	no	—
293	C₄⋊C₄	—	188	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	188	2	no	—
297	C₄²	—	188	2	no	—
298	C₄²	D₄×D₄₇	1	1	yes	characteristic
299	C₄²	D₄×D₄₇	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
301	C₄²	C₂³×D₄₇	1	1	yes	characteristic
302	C₂³	—	4	2	no	—
303	C₂³	#13094020	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	188	4	no	—
306	Q₈	—	188	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	8	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	#13094020	1	1	yes	characteristic
318	C₂×C₄	#13094020	1	1	yes	characteristic
319	C₂×C₄	#13094020	1	1	yes	characteristic
320	C₂×C₄	#13094043	1	4	yes	—
321	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	47	4	no	—
326	C₈	—	94	2	no	—
327	C₈	—	94	2	no	—
328	C₈	—	94	2	no	—
329	C₈	#13094044	1	4	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	#15689506	1	2	yes	—
336	C₂²	C₂×#13094020	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	376	2	no	—
341	C₄	—	376	2	no	—
342	C₄	#15689506	1	2	yes	—
343	C₄	#135×D₄₇	1	2	yes	—
344	C₄	C₂×#13094043	1	2	yes	—
345	C₄	C₂×#13094044	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₂×#15689506	1	1	yes	characteristic
349	C₂	#20253812	1	1	yes	characteristic
350	C₂	#20253814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3169	C₂
socle	C₂²×C₄₇	C₂×#127
hypercenter	#3169	D₄₇
hypocenter	C₄₇	#19600
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20253974	C₂	semidirect
2. #20253943	C₂	semidirect
3. #20253710	C₂	semidirect
4. #20242999	C₂	semidirect
5. #20242997	C₂	semidirect
6. #20242993	C₂	semidirect
7. #20242988	C₂	semidirect
8. #20242986	C₂	semidirect
9. #20242977	C₂	semidirect
10. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
218. C₄₇	#19600	semidirect

Maximal subgroups

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

subgroup	index
#20253974	2
#20253943	2
#20253710	2
#20242999	2
#20242997	2
#20242993	2
#20242988	2
#20242986	2
#20242977	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#3169	2
#19600	47