Subgroups — #26371334

Order 12032.

542 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253975	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245587	C₂	1	2	yes	maximal
4	#20245527	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245439	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245438	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245323	C₂	1	2	yes	maximal
8	#20245258	C₂	1	2	yes	maximal, complemented
9	#20245256	C₂	1	2	yes	maximal, complemented
10	#20245190	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₇	C₂²	1	1	yes	characteristic
13	#15689678	—	2	1	no	—
14	C₄×Q₈×D₄₇	—	2	1	no	—
15	C₂×Dic₃₇₆	C₂²	1	2	yes	—
16	C₂×#13093970	C₂²	1	2	yes	—
17	#15689503	C₂²	1	1	yes	characteristic
18	#15689475	—	2	1	no	—
19	#15689365	—	2	1	no	—
20	#15689365	C₂²	1	2	yes	—
21	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
22	Q₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
23	#15689217	—	2	1	no	—
24	#15689217	C₂²	1	2	yes	—
25	#15689217	C₂²	1	1	yes	characteristic
26	C₄×#10625337	—	2	1	no	—
27	#15689215	—	2	1	no	—
28	#107×D₄₇	C₂²	1	1	yes	characteristic
29	#103×D₄₇	—	2	1	no	—
30	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
31	#15688950	C₂²	1	2	yes	—
32	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
33	#15688928	C₂²	1	2	yes	—
34	#15688926	C₂²	1	2	yes	—
35	#15688926	C₂²	1	2	yes	—
36	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
37	#15688898	C₂²	1	2	yes	—
38	C₄₇×#500	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#367	C₂²	1	2	yes	—
41	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄₇×#358	C₂²	1	2	yes	—
43	C₂×Q₈×D₄₇	—	2	1	no	—
44	C₂×Dic₁₈₈	—	2	1	no	—
45	C₂×Dic₁₈₈	C₂³	1	2	yes	—
46	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
47	C₂×#10625337	—	2	1	no	—
48	Q₁₆×D₄₇	—	2	2	no	—
49	Q₁₆×D₄₇	—	2	2	no	—
50	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
51	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
52	#13094024	—	2	1	no	—
53	#13094015	—	4	1	no	—
54	#13094004	—	2	2	no	—
55	#13094000	—	4	1	no	—
56	Q₈×Dic₄₇	—	2	1	no	—
57	Q₈×Dic₄₇	—	4	1	no	—
58	C₄×Dic₉₄	—	2	1	no	—
59	C₄×Dic₉₄	—	4	1	no	—
60	#13093989	—	2	2	no	—
61	#13093989	—	2	1	no	—
62	#13093989	—	4	1	no	—
63	#13093982	—	4	1	no	—
64	C₄⋊C₄×D₄₇	—	2	1	no	—
65	C₄⋊C₄×D₄₇	—	4	1	no	—
66	C₄²×D₄₇	—	4	1	no	—
67	C₄²×D₄₇	D₄	1	1	yes	characteristic
68	Dic₃₇₆	—	2	4	no	—
69	#13093970	—	2	4	no	—
70	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
71	#13093958	C₂³	1	1	yes	characteristic
72	#13093957	—	2	1	no	—
73	#13093949	—	2	1	no	—
74	#13093948	—	2	1	no	—
75	#13093947	—	2	1	no	—
76	#13093947	—	2	2	no	—
77	#13093947	—	2	2	no	—
78	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
79	C₈×Dic₄₇	D₄	1	1	yes	characteristic
80	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
81	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₄₇	—	2	1	no	—
83	C₄×Q₈×C₄₇	—	2	1	no	—
84	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
85	#107×C₄₇	—	2	1	no	—
86	#107×C₄₇	C₂³	1	2	yes	—
87	#107×C₄₇	C₂³	1	1	yes	characteristic
88	#103×C₄₇	—	2	1	no	—
89	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
90	C₂×Q₈×C₄₇	—	2	1	no	—
91	C₂×Q₈×C₄₇	—	4	1	no	—
92	Q₈×D₄₇	—	2	4	no	—
93	Q₈×D₄₇	—	4	2	no	—
94	C₂×C₄×D₄₇	—	4	1	no	—
95	C₂×C₄×D₄₇	—	4	1	no	—
96	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
97	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
98	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₉₄	—	2	1	no	—
100	C₂×Dic₉₄	—	2	2	no	—
101	C₂×Dic₉₄	—	4	1	no	—
102	Q₁₆×C₄₇	—	2	1	no	—
103	Q₁₆×C₄₇	#41	1	2	yes	—
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₇	—	2	1	no	—
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₄⋊C₄×C₄₇	—	4	1	no	—
109	C₄⋊C₄×C₄₇	—	4	1	no	—
110	C₄²×C₄₇	—	4	1	no	—
111	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
112	#10625337	—	2	4	no	—
113	(C₂×D₄₇)⋊C₄	—	4	2	no	—
114	C₄⋊Dic₄₇	—	2	1	no	—
115	C₄⋊Dic₄₇	—	4	1	no	—
116	C₄⋊Dic₄₇	—	4	1	no	—
117	C₄⋊Dic₄₇	—	4	1	no	—
118	C₄⋊Dic₄₇	—	4	2	no	—
119	Dic₄₇⋊C₄	—	4	1	no	—
120	Dic₄₇⋊C₄	—	4	1	no	—
121	Dic₄₇⋊C₄	—	4	1	no	—
122	Dic₄₇⋊C₄	—	4	2	no	—
123	C₄×Dic₄₇	—	4	1	no	—
124	C₄×Dic₄₇	—	4	1	no	—
125	C₄×Dic₄₇	—	4	1	no	—
126	C₄×Dic₄₇	—	4	2	no	—
127	C₄×Dic₄₇	D₈	1	1	yes	characteristic
128	C₄×Dic₄₇	D₈	1	1	yes	characteristic
129	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
132	Dic₁₈₈	—	2	1	no	—
133	Dic₁₈₈	—	2	2	no	—
134	Dic₁₈₈	—	2	2	no	—
135	Dic₁₈₈	—	4	2	no	—
136	Dic₁₈₈	#41	1	2	yes	—
137	C₈×D₄₇	—	2	1	no	—
138	C₈×D₄₇	—	2	1	no	—
139	C₈×D₄₇	—	4	1	no	—
140	C₁₆×C₄₇	—	2	2	no	—
141	C₄₇⋊C₁₆	—	2	2	no	—
142	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
143	Q₈×C₄₇	—	2	2	no	—
144	Q₈×C₄₇	—	4	1	no	—
145	Q₈×C₄₇	—	8	2	no	—
146	C₂×C₄×C₄₇	—	4	1	no	—
147	C₂×C₄×C₄₇	—	4	1	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₄₇	—	4	1	no	—
153	C₂×Dic₄₇	—	4	1	no	—
154	C₂×Dic₄₇	—	4	2	no	—
155	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
156	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
157	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
158	C₄×D₄₇	—	2	2	no	—
159	C₄×D₄₇	—	2	1	no	—
160	C₄×D₄₇	—	2	1	no	—
161	C₄×D₄₇	—	2	2	no	—
162	C₄×D₄₇	—	4	4	no	—
163	C₄×D₄₇	—	8	2	no	—
164	Dic₉₄	—	2	2	no	—
165	Dic₉₄	—	4	4	no	—
166	Dic₉₄	—	4	1	no	—
167	Dic₉₄	—	4	2	no	—
168	Dic₉₄	—	4	2	no	—
169	C₈×C₄₇	—	2	1	no	—
170	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
171	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
172	C₄₇⋊C₈	—	2	1	no	—
173	C₄₇⋊C₈	—	2	1	no	—
174	#33673	—	47	1	no	maximal, Sylow
175	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
176	C₂×D₄₇	—	2	1	no	—
177	C₂×D₄₇	—	2	1	no	—
178	C₂×D₄₇	#510	1	2	yes	—
179	C₄×C₄₇	—	2	1	no	—
180	C₄×C₄₇	—	4	2	no	—
181	C₄×C₄₇	—	8	1	no	—
182	C₄×C₄₇	—	8	2	no	—
183	C₄×C₄₇	#510	1	2	yes	—
184	C₄×C₄₇	#525	1	1	yes	characteristic
185	C₄×C₄₇	#575	1	1	yes	characteristic
186	Dic₄₇	—	2	1	no	—
187	Dic₄₇	—	2	1	no	—
188	Dic₄₇	—	2	1	no	—
189	Dic₄₇	—	4	2	no	—
190	Dic₄₇	—	8	2	no	—
191	Dic₄₇	—	8	1	no	—
192	#3170	—	47	1	no	—
193	C₂×C₄×Q₁₆	—	47	1	no	—
194	#2235	—	47	2	no	—
195	#2225	—	47	1	no	—
196	#2225	D₄₇	1	1	yes	characteristic, complemented
197	#2209	—	47	1	no	—
198	#2209	—	47	1	no	—
199	#2188	—	47	2	no	—
200	#2176	—	47	2	no	—
201	#2176	—	47	2	no	—
202	#2163	—	47	1	no	—
203	C₂×C₄₇	#3100	1	1	yes	characteristic
204	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
205	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
206	D₄₇	—	2	2	no	—
207	C₂²×Q₁₆	—	47	1	no	—
208	#533	—	94	1	no	—
209	C₂×C₄×Q₈	—	94	1	no	—
210	C₂×Q₃₂	—	47	2	no	—
211	C₂×Q₃₂	—	47	2	no	—
212	#500	C₂×D₄₇	1	1	yes	characteristic
213	#494	—	47	1	no	—
214	#487	—	94	1	no	—
215	#467	—	47	2	no	—
216	#467	—	94	1	no	—
217	C₄×Q₁₆	—	47	1	no	—
218	C₄×Q₁₆	—	47	2	no	—
219	C₄×Q₁₆	—	47	1	no	—
220	C₄×Q₁₆	—	47	1	no	—
221	C₄×Q₁₆	—	94	1	no	—
222	C₄×Q₁₆	—	94	1	no	—
223	C₄×Q₁₆	—	94	1	no	—
224	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
225	C₂×#107	—	47	1	no	—
226	C₂×#103	—	94	1	no	—
227	C₂×C₄×C₈	—	47	1	no	—
228	#367	—	47	2	no	—
229	#367	C₂×D₄₇	1	2	yes	—
230	C₄⋊C₁₆	—	47	1	no	—
231	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
232	#358	—	47	2	no	—
233	#358	—	47	2	no	—
234	#358	—	47	2	no	—
235	#358	C₂×D₄₇	1	2	yes	—
236	C₂²⋊C₁₆	—	47	2	no	—
237	#346	—	47	2	no	—
238	C₄₇	#33673	1	1	yes	characteristic, complemented, Sylow
239	C₂²×Q₈	—	94	1	no	—
240	C₂×Q₁₆	—	47	1	no	—
241	C₂×Q₁₆	—	47	2	no	—
242	C₂×Q₁₆	—	94	1	no	—
243	C₂×Q₁₆	—	94	2	no	—
244	C₂×Q₁₆	—	94	2	no	—
245	C₂×Q₁₆	—	94	1	no	—
246	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
247	C₂²×C₈	—	47	1	no	—
248	C₂²×C₈	—	47	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	94	1	no	—
251	#125	—	188	1	no	—
252	C₂²⋊Q₈	—	94	2	no	—
253	C₂²⋊Q₈	—	188	1	no	—
254	C₄×Q₈	—	2	1	no	—
255	C₄×Q₈	—	94	1	no	—
256	C₄×Q₈	—	94	2	no	—
257	C₄×Q₈	—	94	1	no	—
258	C₄×Q₈	—	94	1	no	—
259	C₄×Q₈	—	188	1	no	—
260	C₄×Q₈	—	188	1	no	—
261	C₄×Q₈	—	188	1	no	—
262	#117	—	188	1	no	—
263	C₂×C₄⋊C₄	—	94	1	no	—
264	C₂×C₄⋊C₄	—	188	1	no	—
265	C₂×C₄²	—	47	1	no	—
266	C₂×C₄²	—	188	1	no	—
267	Q₃₂	—	94	4	no	—
268	Q₃₂	—	94	4	no	—
269	C₂×C₁₆	—	47	2	no	—
270	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
271	#107	—	2	1	no	—
272	#107	—	47	1	no	—
273	#107	—	94	1	no	—
274	#107	C₂²×D₄₇	1	1	yes	characteristic
275	#107	C₂²×D₄₇	1	2	yes	—
276	#103	—	2	1	no	—
277	#103	—	94	1	no	—
278	#103	—	94	2	no	—
279	#103	—	94	2	no	—
280	#103	—	94	1	no	—
281	#103	—	94	1	no	—
282	C₄×C₈	—	47	1	no	—
283	C₄×C₈	—	47	1	no	—
284	C₄×C₈	—	47	1	no	—
285	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂×Q₈	—	94	4	no	—
289	C₂×Q₈	—	94	1	no	—
290	C₂×Q₈	—	94	2	no	—
291	C₂×Q₈	—	188	1	no	—
292	C₂×Q₈	—	188	2	no	—
293	C₂²×C₄	—	47	1	no	—
294	C₂²×C₄	—	47	1	no	—
295	C₂²×C₄	—	47	1	no	—
296	C₂²×C₄	—	188	1	no	—
297	C₂²×C₄	—	188	1	no	—
298	Q₁₆	—	2	1	no	—
299	Q₁₆	—	47	2	no	—
300	Q₁₆	—	94	4	no	—
301	Q₁₆	—	94	2	no	—
302	Q₁₆	—	94	1	no	—
303	Q₁₆	—	94	2	no	—
304	Q₁₆	—	188	2	no	—
305	Q₁₆	#10625354	1	2	yes	—
306	C₂×C₈	—	47	1	no	—
307	C₂×C₈	—	47	1	no	—
308	C₂×C₈	—	94	1	no	—
309	C₂×C₈	—	94	1	no	—
310	C₂×C₈	—	188	1	no	—
311	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
312	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	94	1	no	—
318	C₄⋊C₄	—	188	1	no	—
319	C₄⋊C₄	—	188	1	no	—
320	C₄⋊C₄	—	188	2	no	—
321	C₄⋊C₄	—	188	1	no	—
322	C₄⋊C₄	—	188	2	no	—
323	C₄⋊C₄	—	188	1	no	—
324	C₄⋊C₄	—	188	1	no	—
325	C₄⋊C₄	—	188	1	no	—
326	C₂²⋊C₄	—	188	2	no	—
327	C₄²	—	4	1	no	—
328	C₄²	—	47	1	no	—
329	C₄²	—	47	1	no	—
330	C₄²	—	47	1	no	—
331	C₄²	—	188	1	no	—
332	C₄²	—	188	1	no	—
333	C₄²	—	188	2	no	—
334	C₄²	—	188	1	no	—
335	C₄²	D₄×D₄₇	1	1	yes	characteristic
336	C₁₆	—	2	2	no	—
337	C₁₆	—	94	2	no	—
338	C₂³	—	47	1	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	4	1	no	—
341	Q₈	—	8	2	no	—
342	Q₈	—	94	2	no	—
343	Q₈	—	188	4	no	—
344	Q₈	—	188	2	no	—
345	Q₈	—	188	2	no	—
346	Q₈	—	188	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	47	1	no	—
351	C₂×C₄	—	47	1	no	—
352	C₂×C₄	—	47	1	no	—
353	C₂×C₄	—	94	2	no	—
354	C₂×C₄	—	94	2	no	—
355	C₂×C₄	—	94	1	no	—
356	C₂×C₄	—	94	1	no	—
357	C₂×C₄	—	376	2	no	—
358	C₂×C₄	—	188	4	no	—
359	C₂×C₄	—	188	1	no	—
360	C₂×C₄	—	188	2	no	—
361	C₂×C₄	—	188	1	no	—
362	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
363	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
364	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
365	C₈	—	2	1	no	—
366	C₈	—	94	1	no	—
367	C₈	—	94	1	no	—
368	C₈	#41×D₄₇	1	1	yes	characteristic
369	C₈	C₂×#10625354	1	1	yes	characteristic
370	C₂²	—	47	2	no	—
371	C₂²	—	94	1	no	—
372	C₂²	—	94	1	no	—
373	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	94	1	no	—
379	C₄	—	94	1	no	—
380	C₄	—	94	1	no	—
381	C₄	—	376	1	no	—
382	C₄	—	376	2	no	—
383	C₄	—	188	2	no	—
384	C₄	#15689562	1	2	yes	—
385	C₄	#15689636	1	1	yes	characteristic
386	C₄	#15689922	1	1	yes	characteristic
387	C₂	—	94	2	no	—
388	C₂	#20253390	1	1	yes	characteristic
389	C₂	#510×D₄₇	1	1	yes	characteristic
390	C₂	C₂×#15689562	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2225	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₄₇	C₂×D₈
hypercenter	#2225	D₄₇
hypocenter	C₄₇	#33673
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
4. #20245527	C₂	semidirect
5. #20245439	C₂	semidirect
6. #20245438	C₂	semidirect
8. #20245258	C₂	semidirect
9. #20245256	C₂	semidirect
11. C₄₇×#2225	C₂	semidirect
196. #2225	D₄₇	semidirect
238. C₄₇	#33673	semidirect

Maximal subgroups

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

subgroup	index
#20253975	2
C₄×Q₁₆×D₄₇	2
#20245587	2
#20245527	2
#20245439	2
#20245438	2
#20245323	2
#20245258	2
#20245256	2
#20245190	2
C₄₇×#2225	2
#33673	47