Subgroups — #26203939

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253946	C₂	1	2	yes	maximal, complemented
2	#20253945	C₂	1	2	yes	maximal, complemented
3	#20253899	C₂	1	1	yes	maximal, characteristic, complemented
4	#20253815	C₂	1	1	yes	maximal, characteristic
5	#20242988	C₂	1	2	yes	maximal, complemented
6	#20242986	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242985	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242977	C₂	1	1	yes	maximal, characteristic, complemented
9	#20242974	C₂	1	2	yes	maximal, complemented
10	C₈²×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094043	—	2	2	no	—
13	#15689678	C₂²	1	2	yes	—
14	#15689671	C₂²	1	2	yes	—
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689524	—	2	2	no	—
17	#15689521	C₂²	1	2	yes	—
18	#15689519	C₂²	1	2	yes	—
19	#15689519	C₂²	1	2	yes	—
20	#15689506	—	2	2	no	—
21	#15689503	C₂²	1	1	yes	characteristic
22	#15689495	C₂²	1	2	yes	—
23	#15689495	C₂²	1	2	yes	—
24	#15689474	C₂²	1	2	yes	—
25	#15689473	C₂²	1	2	yes	—
26	#15689469	C₂²	1	2	yes	—
27	#15689351	—	2	4	no	—
28	#15689349	—	2	4	no	—
29	#15689214	—	2	4	no	—
30	SD₁₆×Dic₄₇	—	2	4	no	—
31	#15689160	—	2	2	no	—
32	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
35	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₇×#500	C₂²	1	1	yes	characteristic
39	C₄₇×#493	C₂²	1	1	yes	characteristic
40	C₄₇×#492	C₂²	1	2	yes	—
41	C₄₇×#486	C₂²	1	2	yes	—
42	C₈²×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×#10625352	—	2	2	no	—
44	#13094043	—	4	8	no	—
45	C₂×Dic₁₈₈	—	2	2	no	—
46	C₂×#10625335	—	2	4	no	—
47	C₂×#10625335	—	2	4	no	—
48	C₂×C₈×D₄₇	D₄	1	2	yes	—
49	C₂×C₈×D₄₇	D₄	1	2	yes	—
50	C₂×C₈×D₄₇	D₄	1	2	yes	—
51	#13094024	C₂³	1	2	yes	—
52	#13094024	C₂³	1	2	yes	—
53	#13094015	—	2	2	no	—
54	#13094011	—	2	2	no	—
55	#13094004	—	2	2	no	—
56	#13094000	—	2	2	no	—
57	#13093995	—	2	2	no	—
58	#13093995	—	2	2	no	—
59	Q₈×Dic₄₇	—	2	2	no	—
60	#13093989	—	2	2	no	—
61	D₄×Dic₄₇	—	2	2	no	—
62	D₄×Dic₄₇	—	2	2	no	—
63	#13093982	—	2	2	no	—
64	C₄²×D₄₇	C₂³	1	1	yes	characteristic
65	#13093958	—	2	2	no	—
66	#13093956	—	2	4	no	—
67	#13093955	—	2	4	no	—
68	#13093955	—	2	4	no	—
69	#13093949	—	2	4	no	—
70	#13093948	—	2	4	no	—
71	#13093945	—	2	4	no	—
72	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₈×Dic₄₇	D₄	1	2	yes	—
76	C₈×Dic₄₇	D₄	1	2	yes	—
77	C₈×Dic₄₇	D₄	1	2	yes	—
78	C₂×SD₁₆×C₄₇	—	2	4	no	—
79	C₂×D₈×C₄₇	—	2	2	no	—
80	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
81	#127×C₄₇	C₂³	1	2	yes	—
82	#107×C₄₇	—	2	2	no	—
83	#102×C₄₇	—	2	4	no	—
84	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
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×C₂²⋊D₄₇	—	4	2	no	—
92	C₂²×Dic₄₇	—	4	2	no	—
93	#10625352	—	4	8	no	—
94	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₉₄	—	2	2	no	—
98	C₂×Dic₉₄	—	2	2	no	—
99	SD₁₆×C₄₇	—	4	8	no	—
100	D₈×C₄₇	—	4	4	no	—
101	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₄₇	—	4	2	no	—
108	C₂²⋊Dic₄₇	—	4	2	no	—
109	#10625335	—	4	8	no	—
110	#10625335	—	4	8	no	—
111	(C₂×D₄₇)⋊C₄	—	4	2	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	C₄⋊Dic₄₇	—	4	2	no	—
117	Dic₄₇⋊C₄	—	4	2	no	—
118	C₄×Dic₄₇	—	4	2	no	—
119	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
120	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
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	C₈×D₄₇	—	2	4	no	—
127	C₈×D₄₇	—	2	4	no	—
128	C₈×D₄₇	SD₁₆	1	8	yes	complemented
129	C₂³×C₄₇	—	4	2	no	—
130	C₂²×D₄₇	#127	1	1	yes	characteristic
131	Q₈×C₄₇	—	4	4	no	—
132	D₄×C₄₇	—	4	4	no	—
133	D₄×C₄₇	—	4	4	no	—
134	D₄×C₄₇	—	4	4	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₄₇	—	8	4	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	8	4	no	—
143	C₂×Dic₄₇	#127	1	1	yes	characteristic
144	C₂×Dic₄₇	#127	1	1	yes	characteristic
145	C₂×Dic₄₇	#127	1	1	yes	characteristic
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	C₂×SD₁₆	1	4	yes	—
149	Dic₉₄	—	4	4	no	—
150	Dic₉₄	—	4	4	no	—
151	C₈×C₄₇	—	2	2	no	—
152	C₈×C₄₇	C₂×SD₁₆	1	4	yes	—
153	C₈×C₄₇	#135	1	4	yes	complemented
154	C₄₇⋊C₈	—	2	2	no	—
155	C₄₇⋊C₈	C₂×SD₁₆	1	4	yes	—
156	C₄₇⋊C₈	#135	1	4	yes	complemented
157	#19596	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	8	2	no	—
159	C₂²×C₄₇	—	8	2	no	—
160	C₂²×C₄₇	—	8	2	no	—
161	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
162	C₂×D₄₇	—	2	1	no	—
163	C₂×D₄₇	—	2	1	no	—
164	C₂×D₄₇	#492	1	2	yes	—
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	C₂²×SD₁₆	1	2	yes	—
167	C₄×C₄₇	C₂×#135	1	2	yes	—
168	C₄×C₄₇	C₂×#135	1	2	yes	—
169	Dic₄₇	—	2	1	no	—
170	Dic₄₇	—	2	1	no	—
171	Dic₄₇	—	8	2	no	—
172	Dic₄₇	—	8	2	no	—
173	Dic₄₇	#492	1	2	yes	—
174	#3167	—	47	2	no	—
175	#3167	—	47	2	no	—
176	#3160	—	47	1	no	—
177	#3147	—	47	1	no	—
178	#1708	—	47	2	no	—
179	#1707	—	47	1	no	—
180	#1707	—	47	1	no	—
181	#1706	—	47	1	no	—
182	#1706	D₄₇	1	1	yes	characteristic, complemented
183	#1705	—	47	2	no	—
184	C₂×C₈²	—	47	1	no	—
185	C₂×C₄₇	—	16	2	no	—
186	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
187	C₂×C₄₇	#3147	1	1	yes	characteristic
188	C₂×C₄₇	#3147	1	1	yes	characteristic
189	D₄₇	—	2	2	no	—
190	C₂×#135	—	94	2	no	—
191	#533	—	47	2	no	—
192	#532	—	47	2	no	—
193	#500	—	47	1	no	—
194	#500	C₂×D₄₇	1	1	yes	characteristic
195	#499	—	94	2	no	—
196	#498	—	47	2	no	—
197	#498	—	47	2	no	—
198	#498	—	47	2	no	—
199	#495	—	94	2	no	—
200	#494	—	47	1	no	—
201	#493	C₂×D₄₇	1	1	yes	characteristic
202	#492	—	47	2	no	—
203	#492	—	47	2	no	—
204	#492	C₂×D₄₇	1	2	yes	—
205	#487	—	47	2	no	—
206	#487	—	47	2	no	—
207	#486	—	47	2	no	—
208	#486	C₂×D₄₇	1	2	yes	—
209	#465	—	94	4	no	—
210	#465	—	94	4	no	—
211	C₄×SD₁₆	—	94	4	no	—
212	C₄×SD₁₆	—	94	4	no	—
213	#427	—	94	2	no	—
214	C₂×C₄×C₈	—	47	1	no	—
215	C₂×C₄×C₈	—	47	1	no	—
216	C₂×C₄×C₈	—	47	1	no	—
217	C₈²	—	47	1	no	—
218	C₈²	—	47	1	no	—
219	C₈²	—	47	1	no	—
220	C₈²	C₂×D₄₇	1	1	yes	characteristic
221	C₄₇	#19596	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	—	94	2	no	—
223	#135	—	188	8	no	—
224	C₂×Q₁₆	—	94	2	no	—
225	C₂×SD₁₆	—	2	4	no	—
226	C₂×SD₁₆	—	94	4	no	—
227	C₂×SD₁₆	—	94	4	no	—
228	C₂×D₈	—	2	2	no	—
229	C₂²×C₈	—	47	2	no	—
230	C₂²×C₈	—	47	2	no	—
231	C₂²×C₈	—	47	2	no	—
232	C₄⋊Q₈	—	47	2	no	—
233	C₄⋊Q₈	—	47	2	no	—
234	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
235	#127	C₂²×D₄₇	1	2	yes	—
236	#125	—	94	2	no	—
237	#124	—	94	2	no	—
238	C₂²⋊Q₈	—	94	2	no	—
239	C₂²⋊Q₈	—	94	2	no	—
240	#121	—	94	2	no	—
241	#121	—	94	2	no	—
242	C₄×Q₈	—	94	2	no	—
243	C₄×Q₈	—	94	2	no	—
244	C₄×D₄	—	94	2	no	—
245	C₄×D₄	—	94	2	no	—
246	#117	—	94	2	no	—
247	C₂×C₄²	—	47	1	no	—
248	#107	—	2	2	no	—
249	#107	—	94	2	no	—
250	#106	—	94	4	no	—
251	#106	—	94	4	no	—
252	#106	—	94	4	no	—
253	#103	—	94	4	no	—
254	#103	—	94	4	no	—
255	#102	—	2	4	no	—
256	#102	—	94	4	no	—
257	C₄×C₈	—	47	2	no	—
258	C₄×C₈	—	47	2	no	—
259	C₄×C₈	—	47	2	no	—
260	C₄×C₈	—	47	1	no	—
261	C₄×C₈	—	47	1	no	—
262	C₄×C₈	—	47	1	no	—
263	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
266	#41	—	188	8	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	94	2	no	—
269	C₂×Q₈	—	94	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	188	2	no	—
274	C₂²×C₄	—	47	1	no	—
275	C₂²×C₄	—	47	1	no	—
276	C₂²×C₄	—	47	1	no	—
277	C₂²×C₄	—	188	2	no	—
278	Q₁₆	—	188	4	no	—
279	SD₁₆	—	4	8	no	—
280	SD₁₆	—	188	8	no	—
281	SD₁₆	—	188	8	no	—
282	D₈	—	4	4	no	—
283	C₂×C₈	—	47	2	no	—
284	C₂×C₈	—	47	2	no	—
285	C₂×C₈	—	47	8	no	—
286	C₂×C₈	—	47	2	no	—
287	C₂×C₈	—	94	4	no	—
288	C₂×C₈	—	94	4	no	—
289	C₂×C₈	D₄×D₄₇	1	2	yes	—
290	C₂×C₈	D₄×D₄₇	1	2	yes	—
291	C₂×C₈	D₄×D₄₇	1	2	yes	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	94	2	no	—
295	C₄⋊C₄	—	94	2	no	—
296	C₄⋊C₄	—	94	2	no	—
297	C₄⋊C₄	—	94	2	no	—
298	C₄⋊C₄	—	188	2	no	—
299	C₄⋊C₄	—	188	2	no	—
300	C₂²⋊C₄	—	188	2	no	—
301	C₂²⋊C₄	—	188	2	no	—
302	C₂²⋊C₄	—	188	2	no	—
303	C₄²	—	47	1	no	—
304	C₄²	—	47	1	no	—
305	C₄²	—	47	1	no	—
306	C₄²	—	188	2	no	—
307	C₄²	C₂³×D₄₇	1	1	yes	characteristic
308	C₂³	—	4	2	no	—
309	C₂³	—	47	1	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	188	4	no	—
312	Q₈	—	188	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	4	no	—
315	D₄	—	4	4	no	—
316	D₄	—	376	4	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	47	1	no	—
319	C₂×C₄	—	47	4	no	—
320	C₂×C₄	—	47	1	no	—
321	C₂×C₄	—	47	1	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	376	4	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	47	4	no	—
332	C₈	—	47	4	no	—
333	C₈	—	94	2	no	—
334	C₈	SD₁₆×D₄₇	1	4	yes	—
335	C₈	#13094043	1	4	yes	complemented
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	47	2	no	—
340	C₂²	—	94	1	no	—
341	C₂²	—	94	1	no	—
342	C₂²	#127×D₄₇	1	1	yes	characteristic
343	C₄	—	8	2	no	—
344	C₄	—	47	2	no	—
345	C₄	—	94	1	no	—
346	C₄	—	94	1	no	—
347	C₄	—	376	2	no	—
348	C₄	—	376	2	no	—
349	C₄	C₂×SD₁₆×D₄₇	1	2	yes	—
350	C₄	#135×D₄₇	1	2	yes	—
351	C₄	C₂×#13094043	1	2	yes	—
352	C₂	—	16	2	no	—
353	C₂	—	94	2	no	—
354	C₂	#492×D₄₇	1	1	yes	characteristic
355	C₂	#20253813	1	1	yes	characteristic
356	C₂	#20253815	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₄₇
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1706	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	#20253815	C₂
hypercenter	#1706	D₄₇
hypocenter	C₄₇	#19596
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1706

Complements

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

N	H ≅ G/N	type
1. #20253946	C₂	semidirect
2. #20253945	C₂	semidirect
3. #20253899	C₂	semidirect
5. #20242988	C₂	semidirect
6. #20242986	C₂	semidirect
7. #20242985	C₂	semidirect
8. #20242977	C₂	semidirect
9. #20242974	C₂	semidirect
10. C₈²×D₄₇	C₂	semidirect
11. C₄₇×#1706	C₂	semidirect
128. C₈×D₄₇	SD₁₆	semidirect
153. C₈×C₄₇	#135	semidirect
156. C₄₇⋊C₈	#135	semidirect
182. #1706	D₄₇	semidirect
221. C₄₇	#19596	semidirect
335. C₈	#13094043	semidirect

Maximal subgroups

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

subgroup	index
#20253946	2
#20253945	2
#20253899	2
#20253815	2
#20242988	2
#20242986	2
#20242985	2
#20242977	2
#20242974	2
C₈²×D₄₇	2
C₄₇×#1706	2
#19596	47