Subgroups — #26118897

Order 12032.

562 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689526	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689414	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689410	C₂	1	1	yes	maximal, characteristic
4	#20244181	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688871	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241222	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094024	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
10	#15689526	—	2	2	no	—
11	#15689526	—	2	2	no	—
12	#15689414	—	2	2	no	—
13	#15689414	—	2	2	no	—
14	#15689410	—	2	2	no	—
15	#15689410	—	2	2	no	—
16	C₂×#13093957	—	2	2	no	—
17	C₂×#13093956	—	2	1	no	—
18	C₂×#13093955	—	2	1	no	—
19	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#13093946	—	2	1	no	—
21	C₂×#13093944	—	2	1	no	—
22	C₂×(C₂×D₄₇)⋊C₈	—	2	2	no	—
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	#15688871	—	2	2	no	—
25	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#333	—	2	2	no	—
29	C₂²×Dic₉₄	—	2	1	no	—
30	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₇	—	2	1	no	—
32	C₂²×C₄₇⋊C₈	—	2	1	no	—
33	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
34	#13094024	—	2	1	no	—
35	#13094024	—	2	1	no	—
36	#13094024	—	2	1	no	—
37	#13094024	D₄	1	2	yes	—
38	C₄×D₁₈₈	—	2	1	no	—
39	C₄×D₁₈₈	—	2	1	no	—
40	C₄×D₁₈₈	—	2	1	no	—
41	C₄×D₁₈₈	—	2	1	no	—
42	C₂×C₄⋊Dic₄₇	—	2	1	no	—
43	C₂×C₄⋊Dic₄₇	—	2	1	no	—
44	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
45	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
46	#13093957	—	2	8	no	—
47	#13093956	—	2	4	no	—
48	#13093955	—	2	4	no	—
49	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
50	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
51	#13093946	—	2	4	no	—
52	#13093944	—	2	4	no	—
53	(C₂×D₄₇)⋊C₈	—	4	4	no	—
54	C₄×C₄₇⋊C₈	—	2	1	no	—
55	C₄×C₄₇⋊C₈	—	2	1	no	—
56	C₂²×Q₈×C₄₇	—	2	1	no	—
57	C₂²×C₈×C₄₇	—	2	1	no	—
58	C₂²×C₈×C₄₇	—	2	1	no	—
59	C₄⋊Q₈×C₄₇	—	2	1	no	—
60	C₄⋊Q₈×C₄₇	—	2	1	no	—
61	C₄⋊Q₈×C₄₇	—	2	1	no	—
62	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₇	—	2	1	no	—
67	C₄⋊C₈×C₄₇	—	2	1	no	—
68	C₄⋊C₈×C₄₇	—	2	1	no	—
69	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
70	C₂³×D₄₇	—	2	1	no	—
71	C₂×Q₈×C₄₇	—	2	2	no	—
72	C₂×Q₈×C₄₇	—	4	1	no	—
73	C₂×Q₈×C₄₇	—	8	2	no	—
74	C₂²×C₄×C₄₇	—	4	1	no	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	1	no	—
79	C₂²×Dic₄₇	—	4	1	no	—
80	C₂×D₁₈₈	—	2	1	no	—
81	C₂×D₁₈₈	—	2	1	no	—
82	C₂×D₁₈₈	—	2	1	no	—
83	C₂×D₁₈₈	—	2	1	no	—
84	C₂×D₁₈₈	—	2	1	no	—
85	C₂×D₁₈₈	Q₁₆	1	2	yes	—
86	C₂×C₄×D₄₇	—	4	1	no	—
87	C₂×C₄×D₄₇	—	4	1	no	—
88	C₂×C₄×D₄₇	—	4	2	no	—
89	C₂×C₄×D₄₇	—	4	2	no	—
90	C₂×Dic₉₄	—	2	2	no	—
91	C₂×Dic₉₄	—	4	1	no	—
92	C₂×Dic₉₄	—	8	2	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	4	1	no	—
95	C₂×C₈×C₄₇	—	4	1	no	—
96	C₂×C₈×C₄₇	—	4	1	no	—
97	C₂×C₈×C₄₇	—	4	2	no	—
98	C₄⋊C₄×C₄₇	—	2	2	no	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	—	4	1	no	—
101	C₄⋊C₄×C₄₇	—	4	1	no	—
102	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
106	(C₂×D₄₇)⋊C₄	—	4	1	no	—
107	(C₂×D₄₇)⋊C₄	—	4	1	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	2	1	no	—
110	C₄⋊Dic₄₇	—	2	1	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄⋊Dic₄₇	—	4	1	no	—
113	C₄⋊Dic₄₇	—	4	1	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	4	1	no	—
118	C₂×C₄₇⋊C₈	—	4	1	no	—
119	C₂×C₄₇⋊C₈	SD₁₆	1	4	yes	—
120	C₂³×C₄₇	#121	1	1	yes	characteristic
121	C₂²×D₄₇	—	2	2	no	—
122	C₂²×D₄₇	—	4	1	no	—
123	C₂²×D₄₇	—	4	2	no	—
124	C₂²×D₄₇	—	4	1	no	—
125	C₂²×D₄₇	—	4	2	no	—
126	Q₈×C₄₇	—	8	2	no	—
127	Q₈×C₄₇	—	8	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	8	2	no	—
131	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
133	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
141	C₂×C₄×C₄₇	#137	1	2	yes	—
142	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	4	1	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	8	2	no	—
147	D₁₈₈	—	2	2	no	—
148	D₁₈₈	—	2	2	no	—
149	D₁₈₈	—	4	1	no	—
150	D₁₈₈	—	4	1	no	—
151	C₄×D₄₇	—	8	2	no	—
152	C₄×D₄₇	—	8	1	no	—
153	C₄×D₄₇	—	8	1	no	—
154	Dic₉₄	—	8	2	no	—
155	Dic₉₄	—	8	2	no	—
156	C₈×C₄₇	—	4	2	no	—
157	C₈×C₄₇	—	8	1	no	—
158	C₄₇⋊C₈	—	2	4	no	—
159	C₄₇⋊C₈	—	4	2	no	—
160	#10278	—	47	1	no	maximal, Sylow
161	C₂²×C₄₇	#392	1	1	yes	characteristic
162	C₂²×C₄₇	#461	1	2	yes	—
163	C₂²×C₄₇	#467	1	1	yes	characteristic
164	C₂²×C₄₇	#470	1	1	yes	characteristic
165	C₂²×C₄₇	#472	1	2	yes	—
166	C₂×D₄₇	—	4	4	no	—
167	C₂×D₄₇	—	4	2	no	—
168	C₂×D₄₇	—	8	2	no	—
169	C₂×D₄₇	—	8	1	no	—
170	C₂×D₄₇	—	8	1	no	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	—	8	2	no	—
174	C₄×C₄₇	#448	1	2	yes	—
175	C₄×C₄₇	#451	1	1	yes	characteristic
176	C₄×C₄₇	#451	1	1	yes	characteristic
177	C₄×C₄₇	#452	1	1	yes	characteristic
178	C₄×C₄₇	#452	1	1	yes	characteristic
179	C₄×C₄₇	#492	1	1	yes	characteristic
180	C₄×C₄₇	#497	1	1	yes	characteristic
181	Dic₄₇	—	8	1	no	—
182	Dic₄₇	—	8	2	no	—
183	C₂×#500	—	47	1	no	—
184	C₂×#476	—	47	1	no	—
185	C₂×#476	—	47	1	no	—
186	#1954	—	47	1	no	—
187	C₂×#333	—	47	1	no	—
188	C₂×#333	D₄₇	1	1	yes	characteristic, complemented
189	#1277	—	47	1	no	—
190	C₂×C₄₇	#1675	1	2	yes	—
191	C₂×C₄₇	#1695	1	2	yes	—
192	C₂×C₄₇	#2001	1	1	yes	characteristic
193	C₂×C₄₇	#2013	1	1	yes	characteristic
194	C₂×C₄₇	#2018	1	1	yes	characteristic
195	D₄₇	—	8	2	no	—
196	C₂×C₄⋊Q₈	—	47	1	no	—
197	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
198	C₂×C₄×D₄	—	47	1	no	—
199	#500	—	94	2	no	—
200	#500	—	94	2	no	—
201	#476	—	94	2	no	—
202	#476	—	94	2	no	—
203	#476	—	94	2	no	—
204	#476	—	94	2	no	—
205	C₂×#107	—	94	2	no	—
206	C₂×#106	—	94	1	no	—
207	C₂×#106	—	94	1	no	—
208	C₂×C₄⋊C₈	—	47	1	no	—
209	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
210	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
211	C₂×#102	—	94	1	no	—
212	C₂×#102	—	94	1	no	—
213	C₂×C₂²⋊C₈	—	94	2	no	—
214	C₂×C₄×C₈	—	47	1	no	—
215	#333	—	2	2	no	—
216	#333	—	94	2	no	—
217	C₄₇	#10278	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	—	2	1	no	—
219	C₂²×Q₈	—	94	1	no	—
220	C₂²×D₄	—	47	1	no	—
221	C₂³×C₄	—	94	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	47	2	no	—
225	C₂²×C₈	—	94	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	47	2	no	—
230	C₄⋊Q₈	—	94	1	no	—
231	C₄⋊Q₈	—	94	1	no	—
232	C₄⋊Q₈	—	94	1	no	—
233	C₄⋊Q₈	D₁₈₈	1	2	yes	—
234	C₄×D₄	—	94	1	no	—
235	C₄×D₄	—	94	1	no	—
236	C₄×D₄	—	94	1	no	—
237	C₄×D₄	—	94	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	47	1	no	—
241	C₂×C₄⋊C₄	—	94	1	no	—
242	C₂×C₄⋊C₄	—	94	1	no	—
243	C₂×C₂²⋊C₄	—	94	1	no	—
244	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
245	#107	—	94	8	no	—
246	#106	—	94	4	no	—
247	#106	—	94	4	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	47	2	no	—
252	C₄⋊C₈	—	94	1	no	—
253	C₄⋊C₈	C₂²⋊D₄₇	1	2	yes	—
254	#102	—	94	4	no	—
255	#102	—	94	4	no	—
256	C₂²⋊C₈	—	188	4	no	—
257	C₄×C₈	—	94	1	no	—
258	C₄×C₈	—	94	1	no	—
259	C₂⁴	—	94	1	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	8	2	no	—
263	C₂×Q₈	—	94	2	no	—
264	C₂×Q₈	—	376	2	no	—
265	C₂×Q₈	—	188	1	no	—
266	C₂×D₄	—	47	2	no	—
267	C₂×D₄	—	94	1	no	—
268	C₂×D₄	—	94	1	no	—
269	C₂×D₄	—	94	1	no	—
270	C₂×D₄	—	94	1	no	—
271	C₂×D₄	—	94	1	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	94	1	no	—
274	C₂²×C₄	—	188	1	no	—
275	C₂²×C₄	—	188	1	no	—
276	C₂²×C₄	—	188	1	no	—
277	C₂²×C₄	—	188	2	no	—
278	C₂²×C₄	—	188	2	no	—
279	C₂²×C₄	#10625350	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
281	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	—	47	4	no	—
288	C₂×C₈	—	94	2	no	—
289	C₂×C₈	—	94	2	no	—
290	C₂×C₈	—	94	2	no	—
291	C₂×C₈	—	188	1	no	—
292	C₂×C₈	—	188	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	94	2	no	—
298	C₄⋊C₄	—	94	1	no	—
299	C₄⋊C₄	—	94	1	no	—
300	C₄⋊C₄	—	94	2	no	—
301	C₄⋊C₄	—	188	1	no	—
302	C₄⋊C₄	—	188	1	no	—
303	C₂²⋊C₄	—	188	1	no	—
304	C₂²⋊C₄	—	188	1	no	—
305	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
306	C₄²	D₄×D₄₇	1	1	yes	characteristic
307	C₄²	D₄×D₄₇	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
309	C₂³	—	94	2	no	—
310	C₂³	—	188	1	no	—
311	C₂³	—	188	2	no	—
312	C₂³	—	188	1	no	—
313	C₂³	—	188	2	no	—
314	C₂³	#13093997	1	1	yes	characteristic
315	Q₈	—	8	2	no	—
316	Q₈	—	8	2	no	—
317	Q₈	—	376	2	no	—
318	Q₈	—	376	2	no	—
319	D₄	—	94	2	no	—
320	D₄	—	94	2	no	—
321	D₄	—	188	1	no	—
322	D₄	—	188	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	376	2	no	—
327	C₂×C₄	—	376	1	no	—
328	C₂×C₄	—	376	1	no	—
329	C₂×C₄	—	376	2	no	—
330	C₂×C₄	—	188	1	no	—
331	C₂×C₄	—	188	2	no	—
332	C₂×C₄	—	188	2	no	—
333	C₂×C₄	#13093992	1	1	yes	characteristic
334	C₂×C₄	#13093994	1	1	yes	characteristic
335	C₂×C₄	#13093994	1	1	yes	characteristic
336	C₂×C₄	#13093997	1	1	yes	characteristic
337	C₂×C₄	#13093998	1	1	yes	characteristic
338	C₂×C₄	#13094020	1	1	yes	characteristic
339	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
340	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
341	C₂×C₄	#13094045	1	1	yes	characteristic
342	C₂×C₄	#13094053	1	1	yes	characteristic
343	C₂×C₄	#13094056	1	2	yes	—
344	C₂×C₄	#13094056	1	1	yes	characteristic
345	C₈	—	4	2	no	—
346	C₈	—	8	1	no	—
347	C₈	—	94	4	no	—
348	C₈	—	188	2	no	—
349	C₂²	—	376	1	no	—
350	C₂²	—	376	2	no	—
351	C₂²	—	376	1	no	—
352	C₂²	—	188	4	no	—
353	C₂²	—	188	2	no	—
354	C₂²	#15689032	1	1	yes	characteristic
355	C₂²	#15689327	1	2	yes	—
356	C₂²	#15689363	1	1	yes	characteristic
357	C₂²	#15689380	1	1	yes	characteristic
358	C₂²	#15689392	1	2	yes	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	8	2	no	—
362	C₄	—	376	2	no	—
363	C₄	—	376	1	no	—
364	C₄	#15689263	1	2	yes	—
365	C₄	#15689277	1	1	yes	characteristic
366	C₄	#15689280	1	1	yes	characteristic
367	C₄	#15689282	1	1	yes	characteristic
368	C₄	#15689288	1	1	yes	characteristic
369	C₄	#15689494	1	1	yes	characteristic
370	C₄	#15689516	1	1	yes	characteristic
371	C₂	—	376	2	no	—
372	C₂	#20242786	1	2	yes	—
373	C₂	#20242910	1	2	yes	—
374	C₂	#20244397	1	1	yes	characteristic
375	C₂	#20244472	1	1	yes	characteristic
376	C₂	#20244496	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093997
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#333	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	C₂×#333	D₄₇
hypocenter	C₄₇	#10278
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
1. C₂×#15689526	C₂	semidirect
5. C₂×#15688871	C₂	semidirect
6. #20241222	C₂	semidirect
7. C₂×C₄₇×#333	C₂	semidirect
188. C₂×#333	D₄₇	semidirect
217. C₄₇	#10278	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689526	2
C₂×#15689414	2
C₂×#15689410	2
#20244181	2
C₂×#15688871	2
#20241222	2
C₂×C₄₇×#333	2
#10278	47