Subgroups — #26186975

Order 12032.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253956	C₂	1	1	yes	maximal, characteristic
2	#20252538	C₂	1	2	yes	maximal, complemented
3	#20242302	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242296	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242281	C₂	1	2	yes	maximal, complemented
6	#20242208	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#10625335	C₂	1	2	yes	maximal, complemented
8	#20242142	C₂	1	1	yes	maximal, characteristic, complemented
9	#20241850	C₂	1	2	yes	maximal, complemented
10	#20241673	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1580	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689655	C₂²	1	2	yes	—
13	#15689524	C₄	1	1	yes	characteristic
14	#15689506	C₄	1	1	yes	characteristic
15	#15689476	C₄	1	2	yes	—
16	#15689365	C₄	1	1	yes	characteristic
17	#15689356	C₄	1	1	yes	characteristic
18	#15689351	C₄	1	2	yes	—
19	#15689250	C₂²	1	2	yes	—
20	C₈×Dic₉₄	C₂²	1	2	yes	—
21	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
22	C₄×#10625335	C₂²	1	2	yes	—
23	#15689201	C₂²	1	2	yes	—
24	#15689198	—	2	2	no	—
25	#15689194	—	2	2	no	—
26	#15689181	—	2	2	no	—
27	#15689174	C₂²	1	1	yes	characteristic
28	#15689174	C₂²	1	1	yes	characteristic
29	#15689149	—	2	2	no	—
30	#15689145	C₂²	1	2	yes	—
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	#15688853	C₂²	1	2	yes	—
36	D₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
37	#15688844	C₂²	1	1	yes	characteristic
38	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₄₇	C₂²	1	2	yes	—
42	C₄₇×#334	C₂²	1	1	yes	characteristic
43	C₄₇×D₄⋊C₈	C₂²	1	2	yes	—
44	C₄₇×#322	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#10625335	C₂×C₄	1	2	yes	—
47	C₂×#10625329	—	2	2	no	—
48	C₂×#10625324	—	2	2	no	—
49	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
51	#13094017	—	2	2	no	—
52	#13094015	C₂×C₄	1	2	yes	—
53	#13094011	C₂×C₄	1	2	yes	—
54	#13094006	—	2	2	no	—
55	#13094004	C₂×C₄	1	2	yes	—
56	#13093995	C₂×C₄	1	2	yes	—
57	C₄×Dic₉₄	C₂³	1	2	yes	—
58	C₄×C₂²⋊D₄₇	—	2	2	no	—
59	#13093983	—	2	2	no	—
60	C₄²×D₄₇	C₂³	1	1	yes	characteristic
61	#13093958	C₂×C₄	1	1	yes	characteristic
62	#13093955	C₂×C₄	1	2	yes	—
63	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
64	Dic₄₇⋊C₈	—	2	2	no	—
65	Dic₄₇⋊C₈	—	2	2	no	—
66	Dic₄₇⋊C₈	—	2	1	no	—
67	Dic₄₇⋊C₈	—	2	2	no	—
68	Dic₄₇⋊C₈	D₄	1	2	yes	—
69	#13093949	C₂×C₄	1	2	yes	—
70	#13093947	C₂×C₄	1	2	yes	—
71	#13093945	C₂×C₄	1	2	yes	—
72	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
73	(C₂×D₄₇)⋊C₈	—	2	1	no	—
74	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
75	#13093934	—	2	2	no	—
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×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₄₇	—	2	2	no	—
84	C₄×D₄×C₄₇	C₂³	1	2	yes	—
85	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
87	#102×C₄₇	C₂×C₄	1	2	yes	—
88	C₂²⋊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×D₄×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₄₇	—	2	2	no	—
94	C₂×C₂²⋊D₄₇	—	2	2	no	—
95	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
98	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
99	D₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	1	no	—
102	C₂×C₈×C₄₇	—	8	2	no	—
103	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
106	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
107	C₂²⋊C₄×C₄₇	—	2	2	no	—
108	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
109	C₂²⋊Dic₄₇	—	2	2	no	—
110	C₂²⋊Dic₄₇	—	2	2	no	—
111	C₂²⋊Dic₄₇	—	2	2	no	—
112	#10625335	—	2	4	no	—
113	(C₂×D₄₇)⋊C₄	—	2	2	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
117	Dic₄₇⋊C₄	—	2	2	no	—
118	Dic₄₇⋊C₄	—	2	2	no	—
119	C₄×Dic₄₇	—	2	2	no	—
120	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₄₇	#41	1	1	yes	characteristic
123	#10625329	—	8	2	no	—
124	C₂×C₄₇⋊C₈	—	2	2	no	—
125	C₂×C₄₇⋊C₈	—	2	1	no	—
126	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
129	Dic₁₈₈	—	2	2	no	—
130	#10625324	—	8	2	no	—
131	C₈×D₄₇	—	2	2	no	—
132	C₈×D₄₇	—	2	2	no	—
133	C₂³×C₄₇	—	2	2	no	—
134	C₂²×D₄₇	C₄×D₄	1	1	yes	characteristic
135	D₄×C₄₇	—	2	2	no	—
136	D₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂²⋊D₄₇	—	8	2	no	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
147	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
149	C₄×D₄₇	—	2	1	no	—
150	C₄×D₄₇	—	2	1	no	—
151	C₄×D₄₇	—	2	2	no	—
152	C₄×D₄₇	—	4	1	no	—
153	Dic₉₄	—	2	2	no	—
154	Dic₉₄	—	2	2	no	—
155	C₈×C₄₇	—	2	1	no	—
156	C₈×C₄₇	—	2	2	no	—
157	C₈×C₄₇	—	4	1	no	—
158	C₈×C₄₇	—	4	2	no	—
159	C₈×C₄₇	#131	1	2	yes	—
160	C₄₇⋊C₈	—	2	1	no	—
161	C₄₇⋊C₈	—	2	2	no	—
162	C₄₇⋊C₈	—	4	1	no	—
163	C₄₇⋊C₈	—	4	2	no	—
164	C₄₇⋊C₈	#131	1	2	yes	—
165	#17974	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	—	4	2	no	—
167	C₂²×C₄₇	—	4	2	no	—
168	C₂²×C₄₇	—	4	2	no	—
169	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×D₄₇	—	2	1	no	—
171	C₂×D₄₇	—	2	1	no	—
172	C₂×D₄₇	—	2	1	no	—
173	C₄×C₄₇	—	2	1	no	—
174	C₄×C₄₇	—	4	2	no	—
175	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
176	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
177	C₄×C₄₇	#577	1	2	yes	—
178	Dic₄₇	—	2	1	no	—
179	Dic₄₇	—	2	1	no	—
180	Dic₄₇	—	2	1	no	—
181	Dic₄₇	—	4	2	no	—
182	Dic₄₇	—	4	2	no	—
183	#3168	—	47	1	no	—
184	#2993	—	47	2	no	—
185	#1594	—	47	1	no	—
186	#1593	—	47	1	no	—
187	#1590	—	47	2	no	—
188	#1580	D₄₇	1	1	yes	characteristic, complemented
189	#1579	—	47	1	no	—
190	C₈×SD₁₆	—	47	2	no	—
191	#1566	—	47	1	no	—
192	#1495	—	47	2	no	—
193	#1450	—	47	1	no	—
194	C₂×C₄₇	—	8	2	no	—
195	C₂×C₄₇	#2931	1	1	yes	characteristic
196	C₂×C₄₇	#2952	1	1	yes	characteristic
197	C₂×C₄₇	#2957	1	1	yes	characteristic
198	D₄₇	—	4	1	no	—
199	#529	—	47	2	no	—
200	#499	—	47	1	no	—
201	#495	—	47	1	no	—
202	#488	—	47	2	no	—
203	#467	—	47	1	no	—
204	#466	—	47	1	no	—
205	#465	—	47	2	no	—
206	#446	—	47	2	no	—
207	C₈×Q₈	—	47	2	no	—
208	C₄×Q₁₆	—	47	1	no	—
209	C₄×SD₁₆	—	47	2	no	—
210	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
211	#436	—	47	2	no	—
212	#436	—	94	2	no	—
213	#435	—	94	2	no	—
214	C₈×D₄	C₂×D₄₇	1	2	yes	—
215	#433	—	94	2	no	—
216	#431	—	47	1	no	—
217	#431	—	47	1	no	—
218	#424	—	94	2	no	—
219	#423	—	47	2	no	—
220	C₂×C₄×C₈	—	47	1	no	—
221	#334	—	47	2	no	—
222	#334	—	47	1	no	—
223	#334	C₂×D₄₇	1	1	yes	characteristic
224	Q₈⋊C₈	—	47	2	no	—
225	Q₈⋊C₈	—	47	2	no	—
226	D₄⋊C₈	—	47	2	no	—
227	D₄⋊C₈	C₂×D₄₇	1	2	yes	—
228	#322	—	47	1	no	—
229	#322	C₂×D₄₇	1	1	yes	characteristic
230	C₈²	—	47	1	no	—
231	C₈²	—	47	1	no	—
232	C₄₇	#17974	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₁₆	—	47	1	no	—
234	C₂×SD₁₆	—	47	2	no	—
235	C₂×D₈	C₄×D₄₇	1	1	yes	characteristic
236	C₂×M₁₆	—	94	2	no	—
237	C₂×M₁₆	—	94	2	no	—
238	C₂²×C₈	—	2	2	no	—
239	C₂²×C₈	—	47	1	no	—
240	C₂²×C₈	—	47	1	no	—
241	#126	—	94	2	no	—
242	#125	—	47	2	no	—
243	#124	—	47	2	no	—
244	#123	—	94	2	no	—
245	C₂²⋊Q₈	—	47	2	no	—
246	#121	—	47	2	no	—
247	C₄×Q₈	—	47	2	no	—
248	C₄×D₄	—	94	2	no	—
249	C₄×D₄	C₂²×D₄₇	1	2	yes	—
250	#117	—	94	2	no	—
251	C₂×C₄²	—	47	1	no	—
252	#107	—	47	1	no	—
253	#107	C₄×D₄₇	1	1	yes	characteristic
254	#106	—	47	2	no	—
255	C₄⋊C₈	—	47	2	no	—
256	C₄⋊C₈	—	47	2	no	—
257	C₄⋊C₈	—	94	2	no	—
258	C₄⋊C₈	—	94	1	no	—
259	C₄⋊C₈	—	94	2	no	—
260	C₄⋊C₈	—	94	2	no	—
261	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
262	#103	—	47	2	no	—
263	#103	—	47	2	no	—
264	#102	—	47	2	no	—
265	#102	C₄×D₄₇	1	2	yes	—
266	C₂²⋊C₈	—	2	2	no	—
267	C₂²⋊C₈	—	47	2	no	—
268	C₂²⋊C₈	—	94	1	no	—
269	C₂²⋊C₈	—	94	2	no	—
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	1	no	—
276	C₄×C₈	—	94	2	no	—
277	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
280	C₂×Q₈	—	47	2	no	—
281	C₂×D₄	—	94	2	no	—
282	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	47	1	no	—
285	C₂²×C₄	—	47	1	no	—
286	C₂²×C₄	—	47	1	no	—
287	Q₁₆	—	94	2	no	—
288	SD₁₆	—	94	4	no	—
289	D₈	—	2	2	no	—
290	M₁₆	—	376	2	no	—
291	M₁₆	—	376	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	8	2	no	—
295	C₂×C₈	—	47	1	no	—
296	C₂×C₈	—	47	1	no	—
297	C₂×C₈	—	47	2	no	—
298	C₂×C₈	—	94	2	no	—
299	C₂×C₈	—	94	2	no	—
300	C₂×C₈	—	94	2	no	—
301	C₂×C₈	—	94	1	no	—
302	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
304	C₂×C₈	D₄×D₄₇	1	2	yes	—
305	C₄⋊C₄	—	47	2	no	—
306	C₄⋊C₄	—	94	2	no	—
307	C₄⋊C₄	—	94	2	no	—
308	C₄⋊C₄	—	94	2	no	—
309	C₄⋊C₄	—	94	2	no	—
310	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
311	C₂²⋊C₄	—	2	2	no	—
312	C₂²⋊C₄	—	94	2	no	—
313	C₂²⋊C₄	—	94	2	no	—
314	C₂²⋊C₄	—	94	2	no	—
315	C₂²⋊C₄	—	94	2	no	—
316	C₄²	—	47	1	no	—
317	C₄²	—	47	1	no	—
318	C₄²	—	47	1	no	—
319	C₄²	—	94	2	no	—
320	C₄²	C₂³×D₄₇	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	—	47	1	no	—
323	Q₈	—	94	2	no	—
324	Q₈	—	94	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	376	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	47	1	no	—
332	C₂×C₄	—	47	1	no	—
333	C₂×C₄	—	47	1	no	—
334	C₂×C₄	—	94	1	no	—
335	C₂×C₄	—	94	1	no	—
336	C₂×C₄	—	94	2	no	—
337	C₂×C₄	—	94	2	no	—
338	C₂×C₄	—	94	2	no	—
339	C₂×C₄	—	188	1	no	—
340	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
342	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	2	no	—
345	C₈	—	4	1	no	—
346	C₈	—	4	2	no	—
347	C₈	—	47	2	no	—
348	C₈	—	94	1	no	—
349	C₈	—	94	2	no	—
350	C₈	—	188	1	no	—
351	C₈	—	188	2	no	—
352	C₈	#13094030	1	2	yes	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	94	1	no	—
357	C₂²	—	94	1	no	—
358	C₂²	—	94	1	no	—
359	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	4	2	no	—
362	C₄	—	94	1	no	—
363	C₄	—	94	1	no	—
364	C₄	—	94	1	no	—
365	C₄	—	188	2	no	—
366	C₄	—	188	2	no	—
367	C₄	#131×D₄₇	1	1	yes	characteristic
368	C₄	C₂×#13094030	1	1	yes	characteristic
369	C₄	#15689933	1	2	yes	—
370	C₂	—	8	2	no	—
371	C₂	—	188	1	no	—
372	C₂	#20252095	1	1	yes	characteristic
373	C₂	#20252270	1	1	yes	characteristic
374	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×#1580	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689356	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1580	D₄₇
hypocenter	C₄₇	#17974
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1580

Complements

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

N	H ≅ G/N	type
2. #20252538	C₂	semidirect
3. #20242302	C₂	semidirect
4. #20242296	C₂	semidirect
5. #20242281	C₂	semidirect
6. #20242208	C₂	semidirect
7. C₈×#10625335	C₂	semidirect
8. #20242142	C₂	semidirect
9. #20241850	C₂	semidirect
10. #20241673	C₂	semidirect
11. C₄₇×#1580	C₂	semidirect
188. #1580	D₄₇	semidirect
232. C₄₇	#17974	semidirect

Maximal subgroups

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

subgroup	index
#20253956	2
#20252538	2
#20242302	2
#20242296	2
#20242281	2
#20242208	2
C₈×#10625335	2
#20242142	2
#20241850	2
#20241673	2
C₄₇×#1580	2
#17974	47