Subgroups — #26199394

Order 12032.

650 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254518	C₂	1	1	yes	maximal, characteristic
2	#20254516	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689526	C₂	1	1	yes	maximal, characteristic, complemented
4	#20252099	C₂	1	1	yes	maximal, characteristic
5	#20242910	C₂	1	4	yes	maximal, complemented
6	#20242786	C₂	1	4	yes	maximal, complemented
7	#20242134	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242070	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1548	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094032	—	2	1	no	—
11	#15689673	C₂²	1	1	yes	characteristic
12	C₂×#13094024	C₂²	1	1	yes	characteristic
13	#15689526	—	2	1	no	—
14	#15689526	—	2	1	no	—
15	#15689526	C₂²	1	2	yes	—
16	#15689526	C₂²	1	2	yes	—
17	#15689521	—	2	1	no	—
18	#15689519	—	2	1	no	—
19	#15689471	—	2	1	no	—
20	#15689470	—	2	1	no	—
21	#15689442	—	2	2	no	—
22	#15689441	—	2	2	no	—
23	#15689414	C₂²	1	4	yes	—
24	#15689410	C₂²	1	4	yes	—
25	#15689200	—	2	1	no	—
26	#15689200	C₂²	1	2	yes	—
27	#15689188	—	2	1	no	—
28	#15689188	C₂²	1	2	yes	—
29	#15689176	C₂²	1	1	yes	characteristic
30	C₂×#13093957	—	2	1	no	—
31	C₂×#13093957	—	2	1	no	—
32	#15689091	—	2	1	no	—
33	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
34	#15688875	C₂²	1	2	yes	—
35	#15688873	C₂²	1	2	yes	—
36	#15688871	C₂²	1	4	yes	—
37	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#431	C₂²	1	1	yes	characteristic
39	C₄₇×#423	C₂²	1	1	yes	characteristic
40	C₄₇×#333	C₂²	1	4	yes	—
41	C₂×#10625350	—	2	1	no	—
42	C₂²×Dic₉₄	—	2	1	no	—
43	#13094032	—	4	4	no	—
44	C₂×#10625324	—	2	2	no	—
45	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
46	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
47	C₂×C₈×D₄₇	—	2	2	no	—
48	#13094024	—	2	1	no	—
49	#13094024	—	2	1	no	—
50	#13094024	—	2	1	no	—
51	#13094024	D₄	1	1	yes	characteristic
52	#13094024	C₂³	1	2	yes	—
53	#13094021	D₄	1	1	yes	characteristic
54	#13094012	—	2	1	no	—
55	#13093998	—	2	1	no	—
56	#13093998	D₄	1	2	yes	—
57	C₄×D₁₈₈	—	2	1	no	—
58	C₄×D₁₈₈	C₂³	1	2	yes	—
59	C₂×C₄⋊Dic₄₇	—	2	1	no	—
60	C₂×C₄⋊Dic₄₇	—	2	1	no	—
61	#13093957	—	2	2	no	—
62	#13093957	—	2	2	no	—
63	#13093957	—	2	2	no	—
64	#13093957	—	2	2	no	—
65	#13093956	—	2	4	no	—
66	#13093955	—	2	4	no	—
67	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
68	#13093946	—	2	4	no	—
69	#13093944	—	2	4	no	—
70	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
71	(C₂×D₄₇)⋊C₈	—	2	2	no	—
72	(C₂×D₄₇)⋊C₈	—	2	2	no	—
73	C₄×C₄₇⋊C₈	—	2	1	no	—
74	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₄₇	—	2	1	no	—
79	C₂×M₁₆×C₄₇	—	2	1	no	—
80	C₄⋊Q₈×C₄₇	—	2	1	no	—
81	C₄⋊Q₈×C₄₇	—	2	1	no	—
82	C₄⋊Q₈×C₄₇	—	2	1	no	—
83	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
86	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₄₇	—	2	1	no	—
88	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
89	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
90	C₂²⋊C₈×C₄₇	—	2	1	no	—
91	C₄×C₈×C₄₇	—	2	1	no	—
92	C₂×Q₈×C₄₇	—	2	2	no	—
93	C₂×Q₈×C₄₇	—	4	1	no	—
94	C₂×Q₈×C₄₇	—	8	2	no	—
95	C₂²×C₄×C₄₇	—	4	1	no	—
96	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
99	C₂×C₂²⋊D₄₇	—	2	2	no	—
100	C₂²×Dic₄₇	—	4	1	no	—
101	#10625350	—	4	2	no	—
102	#10625350	—	4	2	no	—
103	C₂×D₁₈₈	—	2	1	no	—
104	C₂×D₁₈₈	—	2	1	no	—
105	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
106	C₂×C₄×D₄₇	—	2	2	no	—
107	C₂×Dic₉₄	—	2	1	no	—
108	C₂×Dic₉₄	—	2	2	no	—
109	C₂×Dic₉₄	—	4	1	no	—
110	C₂×Dic₉₄	—	8	2	no	—
111	M₁₆×C₄₇	—	4	2	no	—
112	C₂×C₈×C₄₇	—	2	2	no	—
113	C₂×C₈×C₄₇	—	2	2	no	—
114	C₄⋊C₄×C₄₇	—	2	2	no	—
115	C₄⋊C₄×C₄₇	—	2	2	no	—
116	C₄⋊C₄×C₄₇	—	4	1	no	—
117	C₄⋊C₄×C₄₇	—	4	1	no	—
118	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₇	#41	1	1	yes	characteristic
121	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
122	(C₂×D₄₇)⋊C₄	—	2	2	no	—
123	(C₂×D₄₇)⋊C₄	—	2	2	no	—
124	C₄⋊Dic₄₇	—	2	1	no	—
125	C₄⋊Dic₄₇	—	2	2	no	—
126	C₄⋊Dic₄₇	—	2	2	no	—
127	C₄⋊Dic₄₇	—	4	1	no	—
128	C₄⋊Dic₄₇	—	4	1	no	—
129	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
130	C₂×C₄₇⋊C₈	—	2	2	no	—
131	C₂×C₄₇⋊C₈	—	2	2	no	—
132	C₂×C₄₇⋊C₈	SD₁₆	1	4	yes	—
133	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
137	#10625324	—	4	4	no	—
138	C₈×D₄₇	—	4	4	no	—
139	C₂³×C₄₇	#121	1	1	yes	characteristic
140	C₂²×D₄₇	—	2	2	no	—
141	Q₈×C₄₇	—	8	2	no	—
142	Q₈×C₄₇	—	8	2	no	—
143	C₂×C₄×C₄₇	—	2	2	no	—
144	C₂×C₄×C₄₇	—	2	2	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	—	8	2	no	—
147	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
149	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
151	C₂×C₄×C₄₇	#137	1	2	yes	—
152	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
155	C₂²⋊D₄₇	—	4	4	no	—
156	C₂×Dic₄₇	—	2	2	no	—
157	C₂×Dic₄₇	—	4	2	no	—
158	C₂×Dic₄₇	—	8	2	no	—
159	D₁₈₈	—	2	4	no	—
160	D₁₈₈	—	4	2	no	—
161	D₁₈₈	—	4	2	no	—
162	C₄×D₄₇	—	4	2	no	—
163	C₄×D₄₇	—	4	2	no	—
164	Dic₉₄	—	4	2	no	—
165	Dic₉₄	—	8	2	no	—
166	Dic₉₄	—	8	2	no	—
167	C₈×C₄₇	—	4	2	no	—
168	C₈×C₄₇	—	4	2	no	—
169	C₄₇⋊C₈	—	2	1	no	—
170	C₄₇⋊C₈	—	2	1	no	—
171	C₄₇⋊C₈	—	4	2	no	—
172	C₄₇⋊C₈	C₂×SD₁₆	1	2	yes	—
173	C₄₇⋊C₈	C₂×SD₁₆	1	2	yes	—
174	#19193	—	47	1	no	maximal, Sylow
175	C₂²×C₄₇	—	2	1	no	—
176	C₂²×C₄₇	—	2	1	no	—
177	C₂²×C₄₇	#461	1	2	yes	—
178	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
179	C₂×D₄₇	—	4	2	no	—
180	C₂×D₄₇	—	4	2	no	—
181	C₂×D₄₇	—	4	2	no	—
182	C₄×C₄₇	—	2	1	no	—
183	C₄×C₄₇	—	2	1	no	—
184	C₄×C₄₇	—	8	2	no	—
185	C₄×C₄₇	#461	1	2	yes	—
186	C₄×C₄₇	#472	1	2	yes	—
187	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
188	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
189	C₄×C₄₇	#578	1	2	yes	—
190	Dic₄₇	—	4	2	no	—
191	Dic₄₇	—	8	2	no	—
192	#3227	—	47	1	no	—
193	#3227	—	47	1	no	—
194	C₂×#500	—	47	1	no	—
195	#2931	—	47	1	no	—
196	#1695	—	47	4	no	—
197	#1675	—	47	4	no	—
198	#1564	—	47	1	no	—
199	#1548	—	47	1	no	—
200	#1548	D₄₇	1	1	yes	characteristic, complemented
201	C₂×C₄₇	—	2	2	no	—
202	C₂×C₄₇	C₂×#461	1	1	yes	characteristic
203	C₂×C₄₇	#3059	1	1	yes	characteristic
204	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
205	D₄₇	—	8	2	no	—
206	C₂×#131	—	94	1	no	—
207	#532	—	47	1	no	—
208	C₂×C₄⋊Q₈	—	47	1	no	—
209	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
210	#500	—	47	2	no	—
211	#500	—	47	2	no	—
212	#500	—	94	1	no	—
213	#500	—	94	1	no	—
214	#498	—	94	1	no	—
215	#498	—	94	1	no	—
216	#486	—	94	1	no	—
217	#486	—	94	1	no	—
218	#481	—	94	2	no	—
219	#481	—	94	2	no	—
220	#476	—	47	4	no	—
221	#476	—	47	4	no	—
222	#436	—	47	2	no	—
223	#436	—	94	1	no	—
224	C₈×D₄	—	47	2	no	—
225	C₈×D₄	—	94	1	no	—
226	#431	—	47	1	no	—
227	#431	C₂×D₄₇	1	1	yes	characteristic
228	C₂×#107	—	94	1	no	—
229	C₂×#107	—	94	1	no	—
230	#423	C₂×D₄₇	1	1	yes	characteristic
231	#408	—	94	1	no	—
232	C₂×C₄×C₈	—	47	1	no	—
233	#335	—	47	2	no	—
234	#334	—	47	2	no	—
235	#333	—	47	4	no	—
236	#333	C₂×D₄₇	1	4	yes	—
237	C₄₇	#19193	1	1	yes	characteristic, complemented, Sylow
238	C₂×#41	—	94	1	no	—
239	C₂²×Q₈	—	2	1	no	—
240	C₂²×Q₈	—	94	1	no	—
241	#131	—	188	4	no	—
242	C₂×M₁₆	—	2	1	no	—
243	C₂×M₁₆	—	94	2	no	—
244	C₂²×C₈	—	47	1	no	—
245	C₂²×C₈	—	47	1	no	—
246	C₂²×C₈	—	94	2	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	47	1	no	—
251	C₄⋊Q₈	—	47	2	no	—
252	C₄⋊Q₈	—	94	1	no	—
253	C₄⋊Q₈	—	94	1	no	—
254	C₄⋊Q₈	—	94	1	no	—
255	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
256	#127	—	47	1	no	—
257	#124	—	94	1	no	—
258	#121	—	47	2	no	—
259	#121	—	94	1	no	—
260	C₄×D₄	—	47	2	no	—
261	C₄×D₄	—	94	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	94	1	no	—
265	C₂×C₄⋊C₄	—	94	1	no	—
266	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
267	#107	—	94	2	no	—
268	#107	—	94	2	no	—
269	#107	—	94	2	no	—
270	#107	—	94	2	no	—
271	#106	—	94	4	no	—
272	#106	—	94	4	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	—	47	2	no	—
275	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
276	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
277	#102	—	94	4	no	—
278	#102	—	94	4	no	—
279	C₂²⋊C₈	—	2	1	no	—
280	C₂²⋊C₈	—	94	2	no	—
281	C₂²⋊C₈	—	94	2	no	—
282	C₂²⋊C₈	—	94	1	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	47	1	no	—
285	C₄×C₈	—	47	1	no	—
286	C₄×C₈	—	47	1	no	—
287	C₄×C₈	—	47	1	no	—
288	C₄×C₈	—	94	1	no	—
289	#41	—	188	2	no	—
290	#41	—	188	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	4	1	no	—
293	C₂×Q₈	—	8	2	no	—
294	C₂×Q₈	—	94	1	no	—
295	C₂×Q₈	—	94	2	no	—
296	C₂×Q₈	—	376	2	no	—
297	C₂×Q₈	—	188	1	no	—
298	C₂×D₄	—	47	2	no	—
299	C₂×D₄	—	94	2	no	—
300	C₂×D₄	—	94	1	no	—
301	C₂×D₄	—	94	1	no	—
302	C₂²×C₄	—	4	1	no	—
303	C₂²×C₄	—	94	2	no	—
304	C₂²×C₄	—	188	1	no	—
305	C₂²×C₄	#10625350	1	1	yes	characteristic
306	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
307	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
308	M₁₆	—	4	2	no	—
309	M₁₆	—	188	4	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	47	4	no	—
313	C₂×C₈	—	47	1	no	—
314	C₂×C₈	—	47	1	no	—
315	C₂×C₈	—	47	1	no	—
316	C₂×C₈	—	47	1	no	—
317	C₂×C₈	—	94	2	no	—
318	C₂×C₈	—	94	2	no	—
319	C₂×C₈	—	188	4	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	47	2	no	—
325	C₄⋊C₄	—	94	1	no	—
326	C₄⋊C₄	—	94	2	no	—
327	C₄⋊C₄	—	94	2	no	—
328	C₄⋊C₄	—	188	1	no	—
329	C₄⋊C₄	—	188	1	no	—
330	C₂²⋊C₄	—	94	2	no	—
331	C₂²⋊C₄	—	94	2	no	—
332	C₄²	#10625350	1	1	yes	characteristic
333	C₄²	D₄×D₄₇	1	1	yes	characteristic
334	C₄²	D₄×D₄₇	1	1	yes	characteristic
335	C₄²	C₂³×D₄₇	1	1	yes	characteristic
336	C₂³	—	94	2	no	—
337	C₂³	#13093997	1	1	yes	characteristic
338	Q₈	—	8	2	no	—
339	Q₈	—	8	2	no	—
340	Q₈	—	376	2	no	—
341	Q₈	—	376	2	no	—
342	Q₈	—	188	2	no	—
343	D₄	—	94	4	no	—
344	D₄	—	188	4	no	—
345	D₄	—	188	2	no	—
346	D₄	—	188	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	—	94	2	no	—
352	C₂×C₄	—	376	2	no	—
353	C₂×C₄	—	188	2	no	—
354	C₂×C₄	—	188	2	no	—
355	C₂×C₄	—	188	2	no	—
356	C₂×C₄	#13093997	1	1	yes	characteristic
357	C₂×C₄	#13093997	1	1	yes	characteristic
358	C₂×C₄	#13093997	1	1	yes	characteristic
359	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
360	C₂×C₄	#13094056	1	2	yes	—
361	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
362	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
363	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
364	C₈	—	4	2	no	—
365	C₈	—	4	2	no	—
366	C₈	—	47	2	no	—
367	C₈	—	47	2	no	—
368	C₈	—	94	1	no	—
369	C₈	—	94	1	no	—
370	C₈	—	188	2	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	188	2	no	—
374	C₂²	—	188	2	no	—
375	C₂²	—	188	2	no	—
376	C₂²	#15689327	1	2	yes	—
377	C₂²	C₂×#13093997	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	8	2	no	—
381	C₄	—	376	2	no	—
382	C₄	—	188	2	no	—
383	C₄	#15689327	1	2	yes	—
384	C₄	#15689392	1	2	yes	—
385	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
386	C₄	C₂×#13094056	1	1	yes	characteristic
387	C₄	#15689939	1	2	yes	—
388	C₂	—	2	2	no	—
389	C₂	—	376	2	no	—
390	C₂	C₂×#15689327	1	1	yes	characteristic
391	C₂	#20253042	1	1	yes	characteristic
392	C₂	C₂×#15689392	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093997
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1548	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	#15689673	C₂²
hypercenter	#1548	D₄₇
hypocenter	C₄₇	#19193
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1548

Complements

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

N	H ≅ G/N	type
3. C₂×#15689526	C₂	semidirect
5. #20242910	C₂	semidirect
6. #20242786	C₂	semidirect
7. #20242134	C₂	semidirect
8. #20242070	C₂	semidirect
9. C₄₇×#1548	C₂	semidirect
200. #1548	D₄₇	semidirect
237. C₄₇	#19193	semidirect

Maximal subgroups

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

subgroup	index
#20254518	2
#20254516	2
C₂×#15689526	2
#20252099	2
#20242910	2
#20242786	2
#20242134	2
#20242070	2
C₄₇×#1548	2
#19193	47