Subgroups — #26119090

Order 12032.

622 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689474	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689470	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689470	C₂	1	1	yes	maximal, characteristic
4	#20244349	C₂	1	1	yes	maximal, characteristic, complemented
5	#20244194	C₂	1	1	yes	maximal, characteristic
6	#20244190	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094024	C₂²	1	1	yes	characteristic
9	C₂×#13094024	C₂²	1	1	yes	characteristic
10	C₂×#13094021	C₂²	1	1	yes	characteristic
11	#15689474	—	2	2	no	—
12	#15689474	—	2	2	no	—
13	#15689470	—	2	2	no	—
14	#15689470	—	2	2	no	—
15	#15689470	—	2	2	no	—
16	#15689470	—	2	2	no	—
17	C₂×#13093949	—	2	2	no	—
18	C₂×#13093949	—	2	2	no	—
19	C₂×#13093946	—	2	2	no	—
20	C₂×#13093946	—	2	2	no	—
21	C₂×#13093946	—	2	2	no	—
22	#15688993	C₂²	1	1	yes	characteristic
23	#15688886	—	2	2	no	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×Dic₉₄	—	2	1	no	—
28	C₂²×Dic₉₄	—	2	1	no	—
29	C₂²×D₁₈₈	—	2	1	no	—
30	C₂²×D₁₈₈	—	2	1	no	—
31	C₂²×D₁₈₈	—	2	1	no	—
32	#13094024	—	2	1	no	—
33	#13094024	—	2	1	no	—
34	#13094024	—	2	1	no	—
35	#13094024	—	2	1	no	—
36	#13094024	—	2	1	no	—
37	#13094024	—	2	1	no	—
38	#13094024	D₄	1	2	yes	—
39	#13094024	D₄	1	2	yes	—
40	#13094021	—	2	1	no	—
41	#13094021	—	2	1	no	—
42	#13094021	—	2	1	no	—
43	#13094021	D₄	1	2	yes	—
44	C₂×C₄⋊Dic₄₇	—	2	1	no	—
45	C₂×C₄⋊Dic₄₇	—	2	1	no	—
46	C₂×C₄⋊Dic₄₇	—	2	1	no	—
47	C₂×C₄⋊Dic₄₇	—	2	1	no	—
48	C₂×C₄⋊Dic₄₇	—	2	1	no	—
49	C₂×C₄⋊Dic₄₇	—	2	1	no	—
50	C₂×C₄⋊Dic₄₇	—	2	1	no	—
51	#13093949	—	2	8	no	—
52	#13093949	—	4	4	no	—
53	#13093946	—	2	8	no	—
54	#13093946	—	2	8	no	—
55	#13093946	—	4	4	no	—
56	C₂²×C₈×C₄₇	D₄	1	2	yes	—
57	C₂²×C₈×C₄₇	D₄	1	2	yes	—
58	C₂²×C₈×C₄₇	D₄	1	2	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₄₇	—	2	1	no	—
61	C₄×C₈×C₄₇	—	2	1	no	—
62	C₄×C₈×C₄₇	—	2	1	no	—
63	C₄×C₈×C₄₇	—	2	1	no	—
64	C₄×C₈×C₄₇	—	2	1	no	—
65	C₄×C₈×C₄₇	—	2	1	no	—
66	C₂³×D₄₇	—	4	1	no	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	4	1	no	—
71	C₂²×Dic₄₇	—	4	1	no	—
72	C₂²×Dic₄₇	—	4	1	no	—
73	C₂×D₁₈₈	—	2	2	no	—
74	C₂×D₁₈₈	—	2	2	no	—
75	C₂×D₁₈₈	—	2	2	no	—
76	C₂×D₁₈₈	—	4	2	no	—
77	C₂×D₁₈₈	—	4	2	no	—
78	C₂×D₁₈₈	—	4	2	no	—
79	C₂×D₁₈₈	—	4	2	no	—
80	C₂×D₁₈₈	—	4	2	no	—
81	C₂×D₁₈₈	—	4	2	no	—
82	C₂×D₁₈₈	—	4	1	no	—
83	C₂×D₁₈₈	—	4	1	no	—
84	C₂×D₁₈₈	—	4	1	no	—
85	C₂×Dic₉₄	—	2	2	no	—
86	C₂×Dic₉₄	—	2	2	no	—
87	C₂×Dic₉₄	—	4	2	no	—
88	C₂×Dic₉₄	—	4	2	no	—
89	C₂×Dic₉₄	—	4	2	no	—
90	C₂×Dic₉₄	—	4	2	no	—
91	C₂×Dic₉₄	—	4	1	no	—
92	C₂×Dic₉₄	—	4	1	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	2	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	#41	1	1	yes	characteristic
106	C₄⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄⋊Dic₄₇	—	4	1	no	—
111	C₄⋊Dic₄₇	—	4	1	no	—
112	C₄⋊Dic₄₇	—	4	1	no	—
113	C₄⋊Dic₄₇	—	4	1	no	—
114	C₄⋊Dic₄₇	—	4	2	no	—
115	C₄⋊Dic₄₇	—	4	2	no	—
116	C₄⋊Dic₄₇	—	4	2	no	—
117	C₂³×C₄₇	#127	1	1	yes	characteristic
118	C₂²×D₄₇	—	4	2	no	—
119	C₂²×D₄₇	—	8	1	no	—
120	C₂²×D₄₇	—	8	1	no	—
121	C₂²×D₄₇	—	8	1	no	—
122	C₂²×D₄₇	—	8	1	no	—
123	C₂²×D₄₇	—	8	1	no	—
124	C₂²×D₄₇	—	8	1	no	—
125	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#135	1	2	yes	—
132	C₂×C₄×C₄₇	#135	1	2	yes	—
133	C₂×C₄×C₄₇	#135	1	2	yes	—
134	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₄×C₄₇	#137	1	2	yes	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	8	1	no	—
140	C₂×Dic₄₇	—	8	1	no	—
141	C₂×Dic₄₇	—	8	1	no	—
142	C₂×Dic₄₇	—	8	1	no	—
143	C₂×Dic₄₇	—	8	1	no	—
144	C₂×Dic₄₇	—	8	1	no	—
145	C₂×Dic₄₇	—	8	1	no	—
146	D₁₈₈	—	4	4	no	—
147	D₁₈₈	—	4	4	no	—
148	D₁₈₈	—	4	4	no	—
149	D₁₈₈	—	8	2	no	—
150	D₁₈₈	—	8	2	no	—
151	D₁₈₈	—	8	2	no	—
152	Dic₉₄	—	4	4	no	—
153	Dic₉₄	—	4	4	no	—
154	Dic₉₄	—	8	2	no	—
155	Dic₉₄	—	8	2	no	—
156	C₈×C₄₇	—	4	2	no	—
157	C₈×C₄₇	—	4	2	no	—
158	C₈×C₄₇	—	4	2	no	—
159	#10311	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	#392	1	1	yes	characteristic
161	C₂²×C₄₇	#495	1	1	yes	characteristic
162	C₂²×C₄₇	#495	1	1	yes	characteristic
163	C₂²×C₄₇	#495	1	1	yes	characteristic
164	C₂²×C₄₇	#496	1	1	yes	characteristic
165	C₂²×C₄₇	#496	1	1	yes	characteristic
166	C₂²×C₄₇	#497	1	1	yes	characteristic
167	C₂×D₄₇	—	8	2	no	—
168	C₂×D₄₇	—	8	2	no	—
169	C₂×D₄₇	—	8	2	no	—
170	C₂×D₄₇	—	16	1	no	—
171	C₂×D₄₇	—	16	1	no	—
172	C₂×D₄₇	—	16	1	no	—
173	C₂×D₄₇	—	16	1	no	—
174	C₄×C₄₇	#449	1	2	yes	—
175	C₄×C₄₇	#449	1	2	yes	—
176	C₄×C₄₇	#452	1	2	yes	—
177	C₄×C₄₇	#463	1	2	yes	—
178	C₄×C₄₇	#463	1	2	yes	—
179	C₄×C₄₇	#464	1	2	yes	—
180	Dic₄₇	—	8	2	no	—
181	Dic₄₇	—	8	2	no	—
182	Dic₄₇	—	16	1	no	—
183	C₂×#487	—	47	1	no	—
184	C₂×#486	—	47	1	no	—
185	C₂×#486	—	47	1	no	—
186	#1990	—	47	1	no	—
187	#1957	—	47	1	no	—
188	#1956	—	47	1	no	—
189	#1273	D₄₇	1	1	yes	characteristic, complemented
190	C₂×C₄₇	#1711	1	1	yes	characteristic
191	C₂×C₄₇	#1717	1	1	yes	characteristic
192	C₂×C₄₇	#1719	1	1	yes	characteristic
193	C₂×C₄₇	#1720	1	1	yes	characteristic
194	C₂×C₄₇	#2012	1	1	yes	characteristic
195	C₂×C₄₇	#2012	1	1	yes	characteristic
196	C₂×C₄₇	#2015	1	1	yes	characteristic
197	D₄₇	—	16	2	no	—
198	C₂×C₄⋊Q₈	—	47	1	no	—
199	C₂×C₄⋊Q₈	—	47	1	no	—
200	C₂×#127	—	47	1	no	—
201	#487	—	94	2	no	—
202	#487	—	94	2	no	—
203	#486	—	94	2	no	—
204	#486	—	94	2	no	—
205	#486	—	94	2	no	—
206	#486	—	94	2	no	—
207	C₂×#103	—	94	2	no	—
208	C₂×#103	—	94	2	no	—
209	C₂×#102	—	94	2	no	—
210	C₂×#102	—	94	2	no	—
211	C₂×#102	—	94	2	no	—
212	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
214	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
215	#384	—	47	1	no	—
216	#340	—	94	2	no	—
217	C₄₇	#10311	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	—	94	1	no	—
219	C₂²×Q₈	—	94	1	no	—
220	C₂²×D₄	—	94	1	no	—
221	C₂²×D₄	—	94	1	no	—
222	C₂²×D₄	—	94	1	no	—
223	C₂²×C₈	D₁₈₈	1	2	yes	—
224	C₂²×C₈	D₁₈₈	1	2	yes	—
225	C₂²×C₈	D₁₈₈	1	2	yes	—
226	C₄⋊Q₈	—	47	2	no	—
227	C₄⋊Q₈	—	47	2	no	—
228	C₄⋊Q₈	—	94	1	no	—
229	C₄⋊Q₈	—	94	1	no	—
230	C₄⋊Q₈	—	94	1	no	—
231	C₄⋊Q₈	—	94	1	no	—
232	C₄⋊Q₈	—	94	1	no	—
233	C₄⋊Q₈	—	94	1	no	—
234	#127	—	47	2	no	—
235	#127	—	94	1	no	—
236	#127	—	94	1	no	—
237	#127	—	94	1	no	—
238	C₂×C₄⋊C₄	—	94	1	no	—
239	C₂×C₄⋊C₄	—	94	1	no	—
240	C₂×C₄⋊C₄	—	94	1	no	—
241	C₂×C₄⋊C₄	—	94	1	no	—
242	C₂×C₄⋊C₄	—	94	1	no	—
243	C₂×C₄⋊C₄	—	94	1	no	—
244	C₂×C₄⋊C₄	—	94	1	no	—
245	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
246	#103	—	94	8	no	—
247	#103	—	188	4	no	—
248	#102	—	94	8	no	—
249	#102	—	94	8	no	—
250	#102	—	188	4	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	2	1	no	—
257	C₂⁴	—	188	1	no	—
258	C₂×Q₈	—	94	2	no	—
259	C₂×Q₈	—	94	2	no	—
260	C₂×Q₈	—	188	2	no	—
261	C₂×Q₈	—	188	2	no	—
262	C₂×Q₈	—	188	2	no	—
263	C₂×Q₈	—	188	2	no	—
264	C₂×Q₈	—	188	1	no	—
265	C₂×Q₈	—	188	1	no	—
266	C₂×D₄	—	94	2	no	—
267	C₂×D₄	—	94	2	no	—
268	C₂×D₄	—	94	2	no	—
269	C₂×D₄	—	188	1	no	—
270	C₂×D₄	—	188	1	no	—
271	C₂×D₄	—	188	1	no	—
272	C₂×D₄	—	188	2	no	—
273	C₂×D₄	—	188	2	no	—
274	C₂×D₄	—	188	2	no	—
275	C₂×D₄	—	188	2	no	—
276	C₂×D₄	—	188	2	no	—
277	C₂×D₄	—	188	2	no	—
278	C₂²×C₄	—	188	1	no	—
279	C₂²×C₄	—	188	1	no	—
280	C₂²×C₄	—	188	1	no	—
281	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₄⋊C₄	—	94	2	no	—
294	C₄⋊C₄	—	94	2	no	—
295	C₄⋊C₄	—	94	2	no	—
296	C₄⋊C₄	—	94	2	no	—
297	C₄⋊C₄	—	188	2	no	—
298	C₄⋊C₄	—	188	2	no	—
299	C₄⋊C₄	—	188	2	no	—
300	C₄⋊C₄	—	188	1	no	—
301	C₄⋊C₄	—	188	1	no	—
302	C₄⋊C₄	—	188	1	no	—
303	C₄⋊C₄	—	188	1	no	—
304	C₄²	D₄×D₄₇	1	1	yes	characteristic
305	C₄²	D₄×D₄₇	1	1	yes	characteristic
306	C₄²	D₄×D₄₇	1	1	yes	characteristic
307	C₄²	#10625354	1	1	yes	characteristic
308	C₂³	—	376	1	no	—
309	C₂³	—	376	1	no	—
310	C₂³	—	376	1	no	—
311	C₂³	—	376	1	no	—
312	C₂³	—	376	1	no	—
313	C₂³	—	376	1	no	—
314	C₂³	—	188	2	no	—
315	C₂³	#13094021	1	1	yes	characteristic
316	Q₈	—	376	2	no	—
317	Q₈	—	376	2	no	—
318	Q₈	—	188	4	no	—
319	Q₈	—	188	4	no	—
320	D₄	—	376	2	no	—
321	D₄	—	376	2	no	—
322	D₄	—	376	2	no	—
323	D₄	—	188	4	no	—
324	D₄	—	188	4	no	—
325	D₄	—	188	4	no	—
326	C₂×C₄	—	376	1	no	—
327	C₂×C₄	—	376	1	no	—
328	C₂×C₄	—	376	1	no	—
329	C₂×C₄	—	376	1	no	—
330	C₂×C₄	—	376	1	no	—
331	C₂×C₄	—	376	1	no	—
332	C₂×C₄	—	376	1	no	—
333	C₂×C₄	—	188	2	no	—
334	C₂×C₄	—	188	2	no	—
335	C₂×C₄	#13093994	1	1	yes	characteristic
336	C₂×C₄	#13093994	1	1	yes	characteristic
337	C₂×C₄	#13093994	1	1	yes	characteristic
338	C₂×C₄	#13093999	1	1	yes	characteristic
339	C₂×C₄	#13093999	1	1	yes	characteristic
340	C₂×C₄	#13093999	1	1	yes	characteristic
341	C₂×C₄	#13094046	1	2	yes	—
342	C₂×C₄	#13094046	1	2	yes	—
343	C₂×C₄	#13094046	1	2	yes	—
344	C₂×C₄	#13094052	1	2	yes	—
345	C₂×C₄	#13094052	1	2	yes	—
346	C₂×C₄	#13094057	1	2	yes	—
347	C₈	—	4	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	4	2	no	—
350	C₂²	—	376	2	no	—
351	C₂²	—	376	2	no	—
352	C₂²	—	376	2	no	—
353	C₂²	—	752	1	no	—
354	C₂²	—	752	1	no	—
355	C₂²	—	752	1	no	—
356	C₂²	—	752	1	no	—
357	C₂²	#15689033	1	1	yes	characteristic
358	C₂²	#15689507	1	1	yes	characteristic
359	C₂²	#15689507	1	1	yes	characteristic
360	C₂²	#15689507	1	1	yes	characteristic
361	C₂²	#15689512	1	1	yes	characteristic
362	C₂²	#15689512	1	1	yes	characteristic
363	C₂²	#15689517	1	1	yes	characteristic
364	C₄	—	376	2	no	—
365	C₄	—	376	2	no	—
366	C₄	—	752	1	no	—
367	C₄	#15689270	1	2	yes	—
368	C₄	#15689270	1	2	yes	—
369	C₄	#15689287	1	2	yes	—
370	C₄	#15689339	1	2	yes	—
371	C₄	#15689339	1	2	yes	—
372	C₄	#15689346	1	2	yes	—
373	C₂	—	752	2	no	—
374	C₂	#20243000	1	1	yes	characteristic
375	C₂	#20243027	1	1	yes	characteristic
376	C₂	#20243041	1	1	yes	characteristic
377	C₂	#20243048	1	1	yes	characteristic
378	C₂	#20244468	1	1	yes	characteristic
379	C₂	#20244468	1	1	yes	characteristic
380	C₂	#20244485	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094021
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1273	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#10311
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
1. C₂×#15689474	C₂	semidirect
4. #20244349	C₂	semidirect
6. #20244190	C₂	semidirect
7. C₄₇×#1273	C₂	semidirect
189. #1273	D₄₇	semidirect
217. C₄₇	#10311	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689474	2
C₂×#15689470	2
#20244349	2
#20244194	2
#20244190	2
C₄₇×#1273	2
#10311	47