Subgroups — #26186196

Order 12032.

698 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	#20253397	C₂	1	1	yes	maximal, characteristic
2	#20252318	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689188	C₂	1	1	yes	maximal, characteristic
4	#20242244	C₂	1	4	yes	maximal, complemented
5	#20242199	C₂	1	4	yes	maximal, complemented
6	#20241812	C₂	1	1	yes	maximal, characteristic
7	#20241790	C₂	1	1	yes	maximal, characteristic, complemented
8	#20241681	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1571	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₄₇	—	2	1	no	—
11	#15689680	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
13	#15689454	C₄	1	2	yes	—
14	#15689414	C₄	1	2	yes	—
15	#15689329	C₄	1	2	yes	—
16	#15689276	C₄	1	2	yes	—
17	#15689250	—	2	1	no	—
18	#15689250	C₂²	1	2	yes	complemented
19	#15689247	—	2	1	no	—
20	#15689247	C₂²	1	2	yes	complemented
21	C₄×#10625325	C₂²	1	4	yes	—
22	#15689188	—	2	1	no	—
23	#15689188	—	2	1	no	—
24	#15689188	C₂²	1	2	yes	—
25	#15689188	C₂²	1	2	yes	—
26	#15689176	C₂²	1	1	yes	characteristic
27	#15689151	—	2	1	no	—
28	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
29	#15689081	—	2	1	no	—
30	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
31	#15688853	C₂²	1	4	yes	complemented
32	#15688850	C₂²	1	4	yes	—
33	#15688845	C₂²	1	2	yes	complemented
34	#15688844	C₂²	1	2	yes	complemented
35	C₄₇×#431	C₂²	1	1	yes	characteristic
36	C₄₇×#423	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#335	C₂²	1	4	yes	complemented
39	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×D₄₇	—	2	1	no	—
41	C₂×#10625325	C₂×C₄	1	4	yes	—
42	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
43	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₈×D₄₇	—	2	2	no	—
45	C₂×C₈×D₄₇	—	2	2	no	—
46	C₂×C₈×D₄₇	—	4	4	no	—
47	#13094024	C₂×C₄	1	1	yes	characteristic
48	#13094024	C₂×C₄	1	1	yes	characteristic
49	#13094016	—	2	1	no	—
50	#13094003	—	2	1	no	—
51	#13094003	C₂×C₄	1	2	yes	—
52	C₄×Dic₉₄	—	2	1	no	—
53	C₄×Dic₉₄	C₂³	1	2	yes	—
54	C₄×D₁₈₈	—	2	1	no	—
55	C₄×D₁₈₈	—	2	1	no	—
56	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
57	C₄×D₁₈₈	C₂³	1	2	yes	—
58	#13093983	—	2	1	no	—
59	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
61	#13093956	C₂×C₄	1	4	yes	—
62	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
63	Dic₄₇⋊C₈	—	2	2	no	—
64	Dic₄₇⋊C₈	—	2	2	no	—
65	#13093949	C₂×C₄	1	4	yes	—
66	#13093946	C₂×C₄	1	4	yes	—
67	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
68	(C₂×D₄₇)⋊C₈	—	2	2	no	—
69	(C₂×D₄₇)⋊C₈	—	2	2	no	—
70	#13093934	—	2	2	no	—
71	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
72	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
73	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×Dic₄₇	—	2	2	no	—
77	C₂×M₁₆×C₄₇	—	2	1	no	—
78	C₂²×C₈×C₄₇	—	2	1	no	—
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₄₇	—	2	1	no	—
81	C₄⋊C₈×C₄₇	—	2	1	no	—
82	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
83	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
84	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
85	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
86	C₄×C₈×C₄₇	C₂³	1	2	yes	—
87	C₂³×D₄₇	—	2	1	no	—
88	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
91	C₂²×Dic₄₇	—	2	1	no	—
92	C₂×D₁₈₈	—	2	1	no	—
93	C₂×D₁₈₈	—	2	4	no	—
94	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₄₇	—	2	2	no	—
96	C₂×C₄×D₄₇	—	2	2	no	—
97	C₂×C₄×D₄₇	—	4	2	no	—
98	C₂×C₄×D₄₇	—	4	2	no	—
99	C₂×Dic₉₄	—	2	1	no	—
100	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
101	M₁₆×C₄₇	—	4	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	4	2	no	—
105	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
106	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₇	#41	1	1	yes	characteristic
109	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
110	C₂²⋊Dic₄₇	—	2	2	no	—
111	(C₂×D₄₇)⋊C₄	—	2	2	no	—
112	(C₂×D₄₇)⋊C₄	—	2	2	no	—
113	C₄⋊Dic₄₇	—	2	1	no	—
114	C₄⋊Dic₄₇	—	2	1	no	—
115	C₄⋊Dic₄₇	—	2	1	no	—
116	C₄⋊Dic₄₇	—	2	1	no	—
117	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
118	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
119	Dic₄₇⋊C₄	—	2	2	no	—
120	Dic₄₇⋊C₄	—	2	2	no	—
121	C₄×Dic₄₇	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	2	2	no	—
124	C₂×C₄₇⋊C₈	—	2	2	no	—
125	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
129	#10625325	—	2	8	no	—
130	C₈×D₄₇	—	4	4	no	—
131	C₈×D₄₇	—	4	4	no	—
132	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
133	C₂²×D₄₇	—	2	4	no	—
134	C₂²×D₄₇	—	2	2	no	—
135	C₂²×D₄₇	—	4	2	no	—
136	C₂²×D₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	4	1	no	—
139	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₄×C₄₇	#137	1	2	yes	—
144	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	D₁₈₈	—	2	4	no	—
152	D₁₈₈	—	2	2	no	—
153	D₁₈₈	#131	1	4	yes	—
154	C₄×D₄₇	—	4	2	no	—
155	C₄×D₄₇	—	4	2	no	—
156	C₄×D₄₇	—	4	4	no	—
157	Dic₉₄	—	2	4	no	—
158	Dic₉₄	—	4	2	no	—
159	C₈×C₄₇	—	2	4	no	—
160	C₈×C₄₇	—	4	2	no	—
161	C₈×C₄₇	—	4	2	no	—
162	C₄₇⋊C₈	—	2	1	no	—
163	C₄₇⋊C₈	—	2	1	no	—
164	C₄₇⋊C₈	—	2	1	no	—
165	C₄₇⋊C₈	—	2	1	no	—
166	C₄₇⋊C₈	—	4	2	no	—
167	#17905	—	47	1	no	maximal, Sylow
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	—	2	1	no	—
170	C₂²×C₄₇	#440	1	2	yes	—
171	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₄₇	—	2	4	no	—
173	C₂×D₄₇	—	4	8	no	—
174	C₂×D₄₇	—	4	2	no	—
175	C₂×D₄₇	—	4	2	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	#440	1	2	yes	—
180	C₄×C₄₇	C₂×#131	1	2	yes	—
181	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
183	Dic₄₇	—	4	2	no	—
184	Dic₄₇	—	4	2	no	—
185	Dic₄₇	—	4	2	no	—
186	#3101	—	47	1	no	—
187	#2960	—	47	1	no	—
188	C₂×C₈×D₄	—	47	1	no	—
189	#1585	—	47	4	no	—
190	SD₁₆⋊C₈	—	47	4	no	—
191	#1571	D₄₇	1	1	yes	characteristic, complemented
192	#1486	—	47	1	no	—
193	#1481	—	47	1	no	—
194	#1452	—	47	1	no	—
195	C₂×C₄₇	—	2	2	no	—
196	C₂×C₄₇	#2929	1	1	yes	characteristic
197	C₂×C₄₇	C₂×#440	1	1	yes	characteristic
198	C₂×C₄₇	#2955	1	1	yes	characteristic
199	D₄₇	—	4	4	no	—
200	C₂³×C₈	—	94	1	no	—
201	#533	—	47	1	no	—
202	C₂×C₄×D₄	—	47	1	no	—
203	#483	—	47	2	no	—
204	#476	—	47	2	no	—
205	#461	—	47	2	no	—
206	#450	—	47	2	no	—
207	#446	—	47	2	no	—
208	#446	—	94	1	no	—
209	C₈×Q₈	—	47	2	no	—
210	C₈×Q₈	—	94	1	no	—
211	C₄×SD₁₆	—	47	4	no	—
212	C₈×D₄	—	47	2	no	—
213	C₈×D₄	—	47	2	no	—
214	C₈×D₄	—	94	1	no	—
215	C₈×D₄	—	94	1	no	—
216	#431	—	47	1	no	—
217	#431	C₂×D₄₇	1	1	yes	characteristic
218	#424	—	94	1	no	—
219	#423	C₂×D₄₇	1	1	yes	characteristic
220	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
221	C₂×C₂²⋊C₈	—	94	1	no	—
222	#405	—	94	1	no	—
223	C₂×C₄×C₈	—	47	1	no	—
224	#335	C₂×D₄₇	1	4	yes	complemented
225	Q₈⋊C₈	—	47	4	no	—
226	D₄⋊C₈	—	47	4	no	—
227	#322	—	47	2	no	—
228	#322	—	47	2	no	—
229	C₄₇	#17905	1	1	yes	characteristic, complemented, Sylow
230	C₂²×D₄	—	47	1	no	—
231	C₂³×C₄	—	94	1	no	—
232	C₂×SD₁₆	—	47	4	no	—
233	C₂×M₁₆	—	2	1	no	—
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	—	47	1	no	—
236	C₂²×C₈	—	47	1	no	—
237	C₂²×C₈	—	94	2	no	—
238	C₂²×C₈	—	94	2	no	—
239	C₂²×C₈	—	188	4	no	—
240	C₄⋊Q₈	—	47	1	no	—
241	C₄⋊Q₈	—	47	1	no	—
242	#125	—	94	1	no	—
243	C₂²⋊Q₈	—	47	2	no	—
244	C₂²⋊Q₈	—	94	1	no	—
245	C₄×Q₈	—	47	2	no	—
246	C₄×Q₈	—	94	1	no	—
247	C₄×D₄	—	47	2	no	—
248	C₄×D₄	—	47	2	no	—
249	C₄×D₄	—	94	1	no	—
250	C₄×D₄	—	94	1	no	—
251	#117	—	94	1	no	—
252	C₂×C₄⋊C₄	—	47	1	no	—
253	C₂×C₂²⋊C₄	—	94	1	no	—
254	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
255	#106	—	47	4	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	94	2	no	—
259	C₄⋊C₈	—	94	2	no	—
260	C₄⋊C₈	—	94	1	no	—
261	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
262	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
263	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
264	#103	—	47	4	no	—
265	#102	—	47	4	no	—
266	C₂²⋊C₈	—	94	2	no	—
267	C₂²⋊C₈	—	94	2	no	—
268	C₂²⋊C₈	—	94	1	no	—
269	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
270	#97	—	94	2	no	—
271	C₄×C₈	—	47	1	no	—
272	C₄×C₈	—	47	1	no	—
273	C₄×C₈	—	47	1	no	—
274	C₄×C₈	—	47	1	no	—
275	C₄×C₈	—	47	2	no	—
276	C₄×C₈	—	94	2	no	—
277	C₄×C₈	C₂²×D₄₇	1	2	yes	—
278	C₂⁴	—	94	1	no	—
279	C₂×Q₈	—	47	2	no	—
280	C₂×Q₈	—	94	1	no	—
281	C₂×D₄	—	47	2	no	—
282	C₂×D₄	—	94	4	no	—
283	C₂×D₄	—	94	1	no	—
284	C₂²×C₄	—	94	1	no	—
285	C₂²×C₄	—	94	2	no	—
286	C₂²×C₄	—	94	2	no	—
287	C₂²×C₄	—	188	2	no	—
288	C₂²×C₄	—	188	2	no	—
289	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
291	C₂²×C₄	#10625354	1	1	yes	characteristic
292	SD₁₆	—	94	8	no	—
293	M₁₆	—	4	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	47	1	no	—
298	C₂×C₈	—	47	1	no	—
299	C₂×C₈	—	47	1	no	—
300	C₂×C₈	—	47	1	no	—
301	C₂×C₈	—	94	2	no	—
302	C₂×C₈	—	94	2	no	—
303	C₂×C₈	—	94	2	no	—
304	C₂×C₈	—	188	4	no	—
305	C₂×C₈	—	188	4	no	—
306	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
307	C₄⋊C₄	—	47	2	no	—
308	C₄⋊C₄	—	47	2	no	—
309	C₄⋊C₄	—	94	2	no	—
310	C₄⋊C₄	—	94	1	no	—
311	C₄⋊C₄	—	94	1	no	—
312	C₄⋊C₄	—	94	1	no	—
313	C₄⋊C₄	—	94	1	no	—
314	C₄⋊C₄	—	94	2	no	—
315	C₂²⋊C₄	—	94	2	no	—
316	C₂²⋊C₄	—	94	2	no	—
317	C₂²⋊C₄	—	94	2	no	—
318	C₄²	—	94	2	no	—
319	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
320	C₄²	D₄×D₄₇	1	1	yes	characteristic
321	C₄²	#10625354	1	1	yes	characteristic
322	C₄²	C₂³×D₄₇	1	1	yes	characteristic
323	C₂³	—	94	4	no	—
324	C₂³	—	94	2	no	—
325	C₂³	—	188	2	no	—
326	C₂³	—	188	2	no	—
327	C₂³	#13093986	1	1	yes	characteristic
328	Q₈	—	94	4	no	—
329	Q₈	—	188	2	no	—
330	D₄	—	47	4	no	—
331	D₄	—	94	4	no	—
332	D₄	—	94	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	94	2	no	—
336	C₂×C₄	—	94	2	no	—
337	C₂×C₄	—	94	2	no	—
338	C₂×C₄	—	188	2	no	—
339	C₂×C₄	—	188	4	no	—
340	C₂×C₄	—	188	2	no	—
341	C₂×C₄	—	188	2	no	—
342	C₂×C₄	#13093986	1	1	yes	characteristic
343	C₂×C₄	#13093986	1	1	yes	characteristic
344	C₂×C₄	#13093986	1	1	yes	characteristic
345	C₂×C₄	#13094048	1	2	yes	—
346	C₂×C₄	#13094056	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
349	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	4	2	no	—
352	C₈	—	4	2	no	—
353	C₈	—	94	1	no	—
354	C₈	—	94	1	no	—
355	C₈	—	94	1	no	—
356	C₈	—	94	1	no	—
357	C₈	—	188	2	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	94	4	no	—
361	C₂²	—	188	8	no	—
362	C₂²	—	188	2	no	—
363	C₂²	—	188	2	no	—
364	C₂²	#15689222	1	2	yes	—
365	C₂²	C₂×#13093986	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	188	2	no	—
370	C₄	—	188	2	no	—
371	C₄	—	188	2	no	—
372	C₄	#15689222	1	2	yes	—
373	C₄	#131×D₄₇	1	2	yes	—
374	C₄	C₂×#13094048	1	1	yes	characteristic
375	C₄	C₂×#13094056	1	1	yes	characteristic
376	C₂	—	2	2	no	—
377	C₂	—	188	4	no	—
378	C₂	#20252081	1	1	yes	characteristic
379	C₂	C₂×#15689222	1	1	yes	characteristic
380	C₂	#20252290	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1571	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1571	D₄₇
hypocenter	C₄₇	#17905
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1571

Complements

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

N	H ≅ G/N	type
2. #20252318	C₂	semidirect
4. #20242244	C₂	semidirect
5. #20242199	C₂	semidirect
7. #20241790	C₂	semidirect
8. #20241681	C₂	semidirect
9. C₄₇×#1571	C₂	semidirect
18. #15689250	C₂²	semidirect
20. #15689247	C₂²	semidirect
31. #15688853	C₂²	semidirect
33. #15688845	C₂²	semidirect
34. #15688844	C₂²	semidirect
38. C₄₇×#335	C₂²	semidirect
191. #1571	D₄₇	semidirect
224. #335	C₂×D₄₇	semidirect
229. C₄₇	#17905	semidirect

Maximal subgroups

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

subgroup	index
#20253397	2
#20252318	2
C₂×#15689188	2
#20242244	2
#20242199	2
#20241812	2
#20241790	2
#20241681	2
C₄₇×#1571	2
#17905	47