Subgroups — #26188089

Order 12032.

578 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253971	C₂	1	1	yes	maximal, characteristic
2	#20252330	C₂	1	2	yes	maximal
3	#20242298	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242297	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242262	C₂	1	2	yes	maximal, complemented
6	#20242259	C₂	1	2	yes	maximal, complemented
7	#20242206	C₂	1	1	yes	maximal, characteristic, complemented
8	#334×D₄₇	C₂	1	1	yes	maximal, characteristic
9	#20241789	C₂	1	2	yes	maximal
10	#20241674	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689677	C₂²	1	2	yes	—
13	#15689527	C₄	1	1	yes	characteristic
14	#15689501	C₄	1	1	yes	characteristic
15	#15689472	C₄	1	2	yes	—
16	#15689365	C₄	1	1	yes	characteristic
17	#15689363	C₄	1	2	yes	—
18	#15689361	C₄	1	1	yes	characteristic
19	#15689254	C₂²	1	2	yes	—
20	#15689253	—	2	2	no	—
21	#15689253	C₂²	1	2	yes	—
22	#15689252	—	2	2	no	—
23	#15689247	C₂²	1	2	yes	—
24	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
25	C₄×#10625337	C₂²	1	2	yes	—
26	#15689174	C₂²	1	1	yes	characteristic
27	#15689148	—	2	2	no	—
28	C₄⋊C₈×D₄₇	C₂²	1	2	yes	—
29	#15689077	—	2	2	no	—
30	C₄×#10625324	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
32	#15688874	C₂²	1	1	yes	characteristic
33	#15688873	C₂²	1	2	yes	—
34	#15688855	C₂²	1	2	yes	—
35	Q₈⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
36	#15688853	C₂²	1	2	yes	—
37	#15688845	C₂²	1	1	yes	characteristic
38	#15688844	C₂²	1	1	yes	characteristic
39	#15688844	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₄₇	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
42	C₄₇×#334	C₂²	1	1	yes	characteristic
43	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₄₇×#322	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#10625337	C₂×C₄	1	2	yes	—
47	C₂×#10625324	D₄	1	2	yes	—
48	C₂×C₈×D₄₇	—	2	2	no	—
49	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
51	#13094023	C₂×C₄	1	2	yes	—
52	#13094022	C₂×C₄	1	2	yes	—
53	#13094014	—	2	2	no	—
54	#13094004	C₂×C₄	1	2	yes	—
55	#13094002	—	2	2	no	—
56	#13094000	C₂×C₄	1	2	yes	—
57	C₄×Dic₉₄	—	2	2	no	—
58	C₄×Dic₉₄	C₂³	1	2	yes	—
59	#13093981	—	2	2	no	—
60	C₄²×D₄₇	C₂³	1	1	yes	characteristic
61	#13093958	C₂×C₄	1	1	yes	characteristic
62	#13093957	C₂×C₄	1	2	yes	—
63	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
64	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
65	Dic₄₇⋊C₈	—	2	2	no	—
66	Dic₄₇⋊C₈	—	2	2	no	—
67	Dic₄₇⋊C₈	—	2	1	no	—
68	Dic₄₇⋊C₈	—	2	2	no	—
69	#13093949	C₂×C₄	1	2	yes	—
70	#13093948	C₂×C₄	1	2	yes	—
71	#13093947	C₂×C₄	1	2	yes	—
72	(C₂×D₄₇)⋊C₈	—	2	1	no	—
73	#13093935	—	2	2	no	—
74	#13093934	—	2	2	no	—
75	#13093934	D₄	1	2	yes	—
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₈×Dic₄₇	—	2	2	no	—
80	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
82	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
84	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₄₇	—	2	2	no	—
86	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
87	#103×C₄₇	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₄₇	—	2	2	no	—
89	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
96	C₂×Dic₉₄	—	2	2	no	—
97	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
98	Q₁₆×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	1	no	—
101	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₄₇	—	2	2	no	—
107	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
108	#10625337	—	4	2	no	—
109	(C₂×D₄₇)⋊C₄	—	2	2	no	—
110	(C₂×D₄₇)⋊C₄	—	2	2	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
113	Dic₄₇⋊C₄	—	2	2	no	—
114	Dic₄₇⋊C₄	—	2	2	no	—
115	Dic₄₇⋊C₄	—	2	2	no	—
116	Dic₄₇⋊C₄	—	2	2	no	—
117	Dic₄₇⋊C₄	—	2	2	no	—
118	C₄×Dic₄₇	—	2	2	no	—
119	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₄₇	#41	1	1	yes	characteristic
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	2	1	no	—
124	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
127	Dic₁₈₈	—	2	2	no	—
128	#10625324	—	2	4	no	—
129	C₈×D₄₇	—	2	2	no	—
130	C₈×D₄₇	—	4	1	no	—
131	C₈×D₄₇	—	4	4	no	—
132	C₂²×D₄₇	C₄×D₄	1	1	yes	characteristic
133	Q₈×C₄₇	—	2	2	no	—
134	Q₈×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
143	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
145	C₄×D₄₇	—	2	1	no	—
146	C₄×D₄₇	—	2	1	no	—
147	C₄×D₄₇	—	4	1	no	—
148	C₄×D₄₇	#137	1	4	yes	—
149	Dic₉₄	—	2	2	no	—
150	Dic₉₄	—	2	2	no	—
151	Dic₉₄	—	8	2	no	—
152	C₈×C₄₇	—	2	1	no	—
153	C₈×C₄₇	—	2	2	no	—
154	C₈×C₄₇	—	4	1	no	—
155	C₈×C₄₇	—	4	2	no	—
156	C₈×C₄₇	#131	1	2	yes	—
157	C₄₇⋊C₈	—	2	1	no	—
158	C₄₇⋊C₈	—	2	1	no	—
159	C₄₇⋊C₈	—	2	2	no	—
160	C₄₇⋊C₈	—	4	1	no	—
161	C₄₇⋊C₈	—	4	2	no	—
162	#18072	—	47	1	no	maximal, Sylow
163	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×D₄₇	—	2	1	no	—
165	C₂×D₄₇	—	2	1	no	—
166	C₂×D₄₇	#441	1	2	yes	—
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	4	2	no	—
169	C₄×C₄₇	—	4	2	no	—
170	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
171	C₄×C₄₇	#568	1	1	yes	characteristic
172	C₄×C₄₇	C₂×#137	1	2	yes	—
173	Dic₄₇	—	2	1	no	—
174	Dic₄₇	—	2	1	no	—
175	Dic₄₇	—	4	2	no	—
176	Dic₄₇	—	4	2	no	—
177	Dic₄₇	#441	1	2	yes	—
178	#3170	—	47	1	no	—
179	#2962	—	47	2	no	—
180	#1593	—	47	1	no	—
181	#1593	—	47	1	no	—
182	#1587	—	47	2	no	—
183	#1587	—	47	2	no	—
184	#1579	—	47	1	no	—
185	#1579	D₄₇	1	1	yes	characteristic, complemented
186	C₂×#334	—	47	1	no	—
187	#1481	—	47	2	no	—
188	#1451	—	47	1	no	—
189	C₂×C₄₇	#2935	1	1	yes	characteristic
190	C₂×C₄₇	C₂×#441	1	1	yes	characteristic
191	C₂×C₄₇	#2957	1	1	yes	characteristic
192	D₄₇	—	2	2	no	—
193	#533	—	47	2	no	—
194	#500	—	47	1	no	—
195	#494	—	47	1	no	—
196	#487	—	47	2	no	—
197	#467	—	47	1	no	—
198	#467	—	47	1	no	—
199	#467	—	47	2	no	—
200	#446	—	47	2	no	—
201	#446	—	47	2	no	—
202	#446	—	94	2	no	—
203	#446	—	94	2	no	—
204	C₈×Q₈	—	47	2	no	—
205	C₈×Q₈	C₂×D₄₇	1	2	yes	—
206	C₄×Q₁₆	—	47	2	no	—
207	C₄×Q₁₆	—	47	1	no	—
208	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
209	#431	—	47	1	no	—
210	#424	—	94	2	no	—
211	C₂×C₄⋊C₈	—	47	2	no	—
212	#405	—	94	2	no	—
213	C₄×M₁₆	—	47	1	no	—
214	C₂×C₄×C₈	—	47	1	no	—
215	#334	—	47	2	no	—
216	#334	—	47	1	no	—
217	#334	C₂×D₄₇	1	1	yes	characteristic
218	Q₈⋊C₈	—	47	2	no	—
219	Q₈⋊C₈	—	47	2	no	—
220	Q₈⋊C₈	—	47	2	no	—
221	Q₈⋊C₈	C₂×D₄₇	1	2	yes	—
222	#322	—	47	1	no	—
223	#322	—	47	1	no	—
224	#322	—	47	1	no	—
225	#322	C₂×D₄₇	1	1	yes	characteristic
226	C₄₇	#18072	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₁₆	—	47	2	no	—
228	C₂×Q₁₆	—	47	1	no	—
229	C₂×Q₁₆	C₄×D₄₇	1	1	yes	characteristic
230	C₂×M₁₆	—	47	2	no	—
231	C₂²×C₈	—	47	1	no	—
232	C₂²×C₈	—	47	1	no	—
233	C₂²×C₈	—	94	2	no	—
234	C₄⋊Q₈	—	47	2	no	—
235	C₄⋊Q₈	—	47	2	no	—
236	#125	—	94	2	no	—
237	C₂²⋊Q₈	—	47	2	no	—
238	C₂²⋊Q₈	—	47	2	no	—
239	C₂²⋊Q₈	—	94	2	no	—
240	C₄×Q₈	—	47	2	no	—
241	C₄×Q₈	—	94	2	no	—
242	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
243	#117	—	94	2	no	—
244	C₂×C₄²	—	47	1	no	—
245	#107	—	47	1	no	—
246	#107	—	47	2	no	—
247	#107	C₄×D₄₇	1	1	yes	characteristic
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	—	47	2	no	—
250	C₄⋊C₈	—	94	2	no	—
251	C₄⋊C₈	—	94	2	no	—
252	C₄⋊C₈	—	94	1	no	—
253	C₄⋊C₈	—	94	2	no	—
254	C₄⋊C₈	—	94	2	no	—
255	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
256	#103	—	47	2	no	—
257	#103	—	47	2	no	—
258	#103	—	47	2	no	—
259	#103	C₄×D₄₇	1	2	yes	—
260	C₂²⋊C₈	—	94	1	no	—
261	#97	—	47	2	no	—
262	#97	—	94	2	no	—
263	#97	—	94	2	no	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	—	47	1	no	—
266	C₄×C₈	—	47	1	no	—
267	C₄×C₈	—	47	1	no	—
268	C₄×C₈	—	47	1	no	—
269	C₄×C₈	—	47	1	no	—
270	C₄×C₈	—	94	2	no	—
271	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
274	C₂×Q₈	—	47	2	no	—
275	C₂×Q₈	—	94	2	no	—
276	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
277	C₂²×C₄	—	47	1	no	—
278	C₂²×C₄	—	47	1	no	—
279	C₂²×C₄	—	47	1	no	—
280	Q₁₆	—	2	2	no	—
281	Q₁₆	—	94	2	no	—
282	Q₁₆	—	188	2	no	—
283	M₁₆	—	94	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	47	1	no	—
287	C₂×C₈	—	47	1	no	—
288	C₂×C₈	—	47	2	no	—
289	C₂×C₈	—	94	2	no	—
290	C₂×C₈	—	94	2	no	—
291	C₂×C₈	—	94	1	no	—
292	C₂×C₈	—	188	1	no	—
293	C₂×C₈	—	188	4	no	—
294	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
296	C₂×C₈	D₄×D₄₇	1	2	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	47	2	no	—
299	C₄⋊C₄	—	94	2	no	—
300	C₄⋊C₄	—	94	2	no	—
301	C₄⋊C₄	—	94	2	no	—
302	C₄⋊C₄	—	94	2	no	—
303	C₄⋊C₄	—	94	2	no	—
304	C₄⋊C₄	—	94	2	no	—
305	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
306	C₂²⋊C₄	—	94	2	no	—
307	C₂²⋊C₄	—	94	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	47	1	no	—
310	C₄²	—	47	1	no	—
311	C₄²	—	47	1	no	—
312	C₄²	—	94	2	no	—
313	C₄²	C₂³×D₄₇	1	1	yes	characteristic
314	C₂³	—	47	1	no	—
315	Q₈	—	2	2	no	—
316	Q₈	—	2	2	no	—
317	Q₈	—	94	2	no	—
318	Q₈	—	94	2	no	—
319	Q₈	—	376	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	47	1	no	—
323	C₂×C₄	—	47	4	no	—
324	C₂×C₄	—	47	1	no	—
325	C₂×C₄	—	47	1	no	—
326	C₂×C₄	—	94	1	no	—
327	C₂×C₄	—	94	1	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	94	2	no	—
330	C₂×C₄	—	188	1	no	—
331	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
333	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	2	no	—
338	C₈	—	94	1	no	—
339	C₈	—	94	1	no	—
340	C₈	—	94	2	no	—
341	C₈	—	188	1	no	—
342	C₈	—	188	2	no	—
343	C₈	#13094030	1	2	yes	—
344	C₂²	—	47	2	no	—
345	C₂²	—	94	1	no	—
346	C₂²	—	94	1	no	—
347	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	47	2	no	—
352	C₄	—	94	1	no	—
353	C₄	—	94	1	no	—
354	C₄	—	188	2	no	—
355	C₄	—	188	2	no	—
356	C₄	C₂×#13094030	1	1	yes	characteristic
357	C₄	#15689888	1	1	yes	characteristic
358	C₄	#137×D₄₇	1	2	yes	—
359	C₂	—	94	2	no	—
360	C₂	#20252127	1	1	yes	characteristic
361	C₂	#441×D₄₇	1	1	yes	characteristic
362	C₂	#20252302	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×#1579	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂²×D₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1579	D₄₇
hypocenter	C₄₇	#18072
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #20242298	C₂	semidirect
4. #20242297	C₂	semidirect
5. #20242262	C₂	semidirect
6. #20242259	C₂	semidirect
7. #20242206	C₂	semidirect
11. C₄₇×#1579	C₂	semidirect
185. #1579	D₄₇	semidirect
226. C₄₇	#18072	semidirect

Maximal subgroups

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

subgroup	index
#20253971	2
#20252330	2
#20242298	2
#20242297	2
#20242262	2
#20242259	2
#20242206	2
#334×D₄₇	2
#20241789	2
#20241674	2
C₄₇×#1579	2
#18072	47