Subgroups — #26186054

Order 12032.

666 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252353	C₂	1	2	yes	maximal, complemented
2	C₄×#13094047	C₂	1	1	yes	maximal, characteristic
3	#20242303	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242301	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242297	C₂	1	1	yes	maximal, characteristic, complemented
6	#20242293	C₂	1	2	yes	maximal, complemented
7	#20242288	C₂	1	2	yes	maximal, complemented
8	#20242133	C₂	1	1	yes	maximal, characteristic, complemented
9	#20241769	C₂	1	2	yes	maximal, complemented
10	#322×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094047	C₄	1	1	yes	characteristic
13	C₄×#10625354	C₂²	1	2	yes	—
14	#15689254	C₂²	1	2	yes	—
15	Q₁₆×Dic₄₇	C₄	1	1	yes	characteristic
16	C₄×#10625336	C₂²	1	2	yes	—
17	#15689208	C₄	1	2	yes	—
18	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
19	#15689205	C₄	1	1	yes	characteristic
20	#15689202	C₂²	1	2	yes	—
21	#15689200	C₂²	1	2	yes	—
22	#15689196	—	2	2	no	—
23	#15689193	—	2	2	no	—
24	#15689175	—	2	2	no	—
25	#15689175	C₂²	1	2	yes	—
26	#15689160	C₄	1	1	yes	characteristic
27	#15689118	C₄	1	2	yes	—
28	#15689075	—	2	2	no	—
29	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
32	#15688875	C₂²	1	2	yes	—
33	#15688874	C₂²	1	1	yes	characteristic
34	Q₈⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
35	#15688852	C₂²	1	2	yes	—
36	#15688850	C₂²	1	2	yes	—
37	#15688845	C₂²	1	1	yes	characteristic
38	#15688845	C₂²	1	1	yes	characteristic
39	#15688844	C₂²	1	1	yes	characteristic
40	C₄₇×#446	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×#10625354	C₂×C₄	1	2	yes	—
46	#13094047	—	2	4	no	—
47	C₂×#10625336	C₂×C₄	1	2	yes	—
48	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
49	C₂×#10625324	—	2	2	no	—
50	C₂×#10625324	—	2	2	no	—
51	C₂×C₈×D₄₇	—	2	1	no	—
52	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₄₇	D₄	1	2	yes	—
55	Q₈×Dic₄₇	C₂×C₄	1	2	yes	—
56	C₄×D₁₈₈	—	2	2	no	—
57	C₄×D₁₈₈	C₂³	1	2	yes	—
58	#13093986	—	2	2	no	—
59	#13093986	C₂×C₄	1	2	yes	—
60	#13093982	—	2	2	no	—
61	#13093982	C₂×C₄	1	2	yes	—
62	C₄²×D₄₇	—	2	2	no	—
63	C₄²×D₄₇	C₂³	1	1	yes	characteristic
64	#13093958	C₂×C₄	1	1	yes	characteristic
65	#13093955	C₂×C₄	1	2	yes	—
66	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
67	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
68	#13093948	C₂×C₄	1	2	yes	—
69	#13093946	C₂×C₄	1	2	yes	—
70	#13093944	C₂×C₄	1	2	yes	—
71	(C₂×D₄₇)⋊C₈	—	2	2	no	—
72	(C₂×D₄₇)⋊C₈	—	2	2	no	—
73	(C₂×D₄₇)⋊C₈	—	2	2	no	—
74	#13093935	—	2	2	no	—
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₈×Dic₄₇	—	2	2	no	—
79	C₈×Dic₄₇	—	2	1	no	—
80	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₄₇	D₄	1	2	yes	—
83	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
85	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₄₇	—	2	2	no	—
87	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
88	#103×C₄₇	C₂×C₄	1	2	yes	—
89	#97×C₄₇	—	2	2	no	—
90	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
94	#10625354	—	2	4	no	—
95	#10625354	—	2	4	no	—
96	C₂×D₁₈₈	—	2	2	no	—
97	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₄₇	—	2	2	no	—
99	C₂×C₄×D₄₇	—	2	2	no	—
100	C₂×C₄×D₄₇	—	2	2	no	—
101	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
104	Q₁₆×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	2	2	no	—
106	C₂×C₈×C₄₇	—	2	1	no	—
107	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
112	C₄²×C₄₇	—	2	2	no	—
113	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
114	#10625336	—	2	4	no	—
115	(C₂×D₄₇)⋊C₄	—	2	2	no	—
116	(C₂×D₄₇)⋊C₄	—	2	2	no	—
117	(C₂×D₄₇)⋊C₄	—	2	2	no	—
118	C₄⋊Dic₄₇	—	2	2	no	—
119	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
120	C₄×Dic₄₇	—	2	2	no	—
121	C₄×Dic₄₇	—	2	2	no	—
122	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₄₇	#41	1	1	yes	characteristic
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	2	1	no	—
127	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
130	D₃₇₆	—	2	2	no	—
131	#10625324	—	4	4	no	—
132	#10625324	—	4	4	no	—
133	C₈×D₄₇	—	2	2	no	—
134	C₈×D₄₇	—	4	2	no	—
135	C₈×D₄₇	—	4	2	no	—
136	C₈×D₄₇	M₁₆	1	4	yes	complemented
137	C₂²×D₄₇	—	2	2	no	—
138	C₂²×D₄₇	C₄×D₄	1	1	yes	characteristic
139	Q₈×C₄₇	—	2	2	no	—
140	Q₈×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
150	D₁₈₈	—	2	2	no	—
151	D₁₈₈	—	2	2	no	—
152	D₁₈₈	—	4	4	no	—
153	C₄×D₄₇	—	2	2	no	—
154	C₄×D₄₇	—	2	2	no	—
155	C₄×D₄₇	—	4	2	no	—
156	C₄×D₄₇	—	4	2	no	—
157	C₄×D₄₇	—	4	4	no	—
158	C₄×D₄₇	—	4	4	no	—
159	C₄×D₄₇	C₂×M₁₆	1	2	yes	—
160	C₄×D₄₇	#131	1	2	yes	—
161	C₈×C₄₇	—	2	1	no	—
162	C₈×C₄₇	—	2	2	no	—
163	C₈×C₄₇	—	4	1	no	—
164	C₈×C₄₇	—	4	2	no	—
165	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
166	C₄₇⋊C₈	—	2	1	no	—
167	C₄₇⋊C₈	—	2	2	no	—
168	C₄₇⋊C₈	—	4	1	no	—
169	C₄₇⋊C₈	—	4	2	no	—
170	C₄₇⋊C₈	C₂×M₁₆	1	2	yes	—
171	#17890	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₄₇	—	2	1	no	—
174	C₂×D₄₇	—	2	1	no	—
175	C₂×D₄₇	—	4	2	no	—
176	C₂×D₄₇	—	4	2	no	—
177	C₂×D₄₇	—	4	2	no	—
178	C₂×D₄₇	#436	1	2	yes	—
179	C₄×C₄₇	—	2	1	no	—
180	C₄×C₄₇	—	4	2	no	—
181	C₄×C₄₇	—	4	2	no	—
182	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
183	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
184	C₄×C₄₇	#575	1	2	yes	—
185	Dic₄₇	—	2	1	no	—
186	Dic₄₇	—	4	2	no	—
187	Dic₄₇	#436	1	2	yes	—
188	Dic₄₇	#444	1	2	yes	—
189	#2965	—	47	2	no	—
190	C₄×#135	—	47	1	no	—
191	#1594	—	47	1	no	—
192	#1594	—	47	1	no	—
193	#1593	—	47	1	no	—
194	#1593	D₄₇	1	1	yes	characteristic, complemented
195	#1592	—	47	2	no	—
196	#1592	—	47	2	no	—
197	#1564	—	47	1	no	—
198	#1476	—	47	2	no	—
199	C₂×#322	—	47	1	no	—
200	C₂×C₄₇	C₂×#436	1	1	yes	characteristic
201	C₂×C₄₇	#2943	1	1	yes	characteristic
202	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
203	D₄₇	—	2	2	no	—
204	D₄₇	—	8	2	no	—
205	C₂×#135	—	47	1	no	—
206	C₄×#41	—	47	2	no	—
207	#446	—	47	2	no	—
208	#446	C₂×D₄₇	1	2	yes	—
209	C₄×Q₁₆	—	47	1	no	—
210	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
211	C₄×SD₁₆	—	47	2	no	—
212	C₄×SD₁₆	—	47	2	no	—
213	C₄×D₈	—	47	1	no	—
214	C₄×D₈	—	47	1	no	—
215	#436	—	47	2	no	—
216	#436	—	47	2	no	—
217	#435	—	94	2	no	—
218	#435	—	94	2	no	—
219	#431	—	47	2	no	—
220	#431	—	94	2	no	—
221	#427	—	47	1	no	—
222	#416	—	47	2	no	—
223	C₄×M₁₆	—	94	2	no	—
224	C₂×C₄×C₈	—	47	1	no	—
225	C₂×C₄×C₈	—	47	1	no	—
226	C₂×C₄×C₈	—	47	1	no	—
227	#335	—	47	2	no	—
228	#334	—	47	1	no	—
229	#334	C₂×D₄₇	1	1	yes	characteristic
230	Q₈⋊C₈	—	47	2	no	—
231	Q₈⋊C₈	C₂×D₄₇	1	2	yes	—
232	D₄⋊C₈	—	47	2	no	—
233	D₄⋊C₈	—	47	2	no	—
234	#322	—	47	1	no	—
235	#322	—	47	1	no	—
236	#322	—	47	1	no	—
237	#322	C₂×D₄₇	1	1	yes	characteristic
238	C₄₇	#17890	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	47	2	no	—
240	#135	—	94	4	no	—
241	C₂×Q₁₆	C₄×D₄₇	1	1	yes	characteristic
242	C₂×SD₁₆	—	47	2	no	—
243	C₂×D₈	—	47	1	no	—
244	C₂×M₁₆	—	94	2	no	—
245	C₂×M₁₆	—	94	2	no	—
246	C₂²×C₈	—	47	1	no	—
247	C₂²×C₈	—	47	1	no	—
248	C₂²×C₈	—	47	2	no	—
249	C₂²×C₈	—	94	1	no	—
250	C₄×Q₈	—	47	2	no	—
251	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
252	C₄×D₄	—	47	2	no	—
253	C₄×D₄	—	47	2	no	—
254	C₄×D₄	—	94	2	no	—
255	C₄×D₄	—	94	2	no	—
256	#117	—	47	2	no	—
257	#117	—	94	2	no	—
258	C₂×C₄²	—	47	1	no	—
259	C₂×C₄²	—	94	2	no	—
260	#107	—	47	1	no	—
261	#107	C₄×D₄₇	1	1	yes	characteristic
262	#106	—	47	2	no	—
263	C₄⋊C₈	—	2	2	no	—
264	C₄⋊C₈	—	47	2	no	—
265	C₄⋊C₈	—	94	2	no	—
266	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
267	#103	—	47	2	no	—
268	#103	C₄×D₄₇	1	2	yes	—
269	#102	—	47	2	no	—
270	#102	—	47	2	no	—
271	C₂²⋊C₈	—	94	2	no	—
272	C₂²⋊C₈	—	94	2	no	—
273	C₂²⋊C₈	—	94	2	no	—
274	#97	—	2	2	no	—
275	#97	—	94	2	no	—
276	C₄×C₈	—	47	1	no	—
277	C₄×C₈	—	47	1	no	—
278	C₄×C₈	—	47	2	no	—
279	C₄×C₈	—	47	1	no	—
280	C₄×C₈	—	47	1	no	—
281	C₄×C₈	—	47	1	no	—
282	C₄×C₈	—	94	1	no	—
283	C₄×C₈	—	94	2	no	—
284	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
287	#41	—	94	4	no	—
288	#41	—	94	4	no	—
289	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
290	C₂×D₄	—	47	2	no	—
291	C₂×D₄	—	94	2	no	—
292	C₂²×C₄	—	47	1	no	—
293	C₂²×C₄	—	47	1	no	—
294	C₂²×C₄	—	47	1	no	—
295	C₂²×C₄	—	94	2	no	—
296	C₂²×C₄	—	94	2	no	—
297	C₂²×C₄	—	94	2	no	—
298	Q₁₆	—	2	2	no	—
299	SD₁₆	—	94	4	no	—
300	D₈	—	94	2	no	—
301	M₁₆	—	188	4	no	—
302	M₁₆	—	188	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	47	1	no	—
306	C₂×C₈	—	47	4	no	—
307	C₂×C₈	—	47	1	no	—
308	C₂×C₈	—	47	2	no	—
309	C₂×C₈	—	94	2	no	—
310	C₂×C₈	—	94	2	no	—
311	C₂×C₈	—	94	1	no	—
312	C₂×C₈	—	188	2	no	—
313	C₂×C₈	—	188	2	no	—
314	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
315	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
316	C₂×C₈	D₄×D₄₇	1	2	yes	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	47	2	no	—
319	C₄⋊C₄	—	94	2	no	—
320	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
321	C₂²⋊C₄	—	94	2	no	—
322	C₂²⋊C₄	—	94	2	no	—
323	C₂²⋊C₄	—	94	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	47	1	no	—
326	C₄²	—	47	1	no	—
327	C₄²	—	47	1	no	—
328	C₄²	—	94	2	no	—
329	C₄²	—	94	2	no	—
330	C₄²	C₂³×D₄₇	1	1	yes	characteristic
331	C₂³	—	47	1	no	—
332	C₂³	—	94	2	no	—
333	Q₈	—	2	2	no	—
334	Q₈	—	2	2	no	—
335	D₄	—	94	2	no	—
336	D₄	—	94	2	no	—
337	D₄	—	188	4	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	47	1	no	—
341	C₂×C₄	—	47	1	no	—
342	C₂×C₄	—	47	2	no	—
343	C₂×C₄	—	47	2	no	—
344	C₂×C₄	—	47	1	no	—
345	C₂×C₄	—	94	2	no	—
346	C₂×C₄	—	94	2	no	—
347	C₂×C₄	—	94	2	no	—
348	C₂×C₄	—	188	4	no	—
349	C₂×C₄	—	188	4	no	—
350	C₂×C₄	—	188	2	no	—
351	C₂×C₄	—	188	2	no	—
352	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
354	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	2	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	2	no	—
359	C₈	—	47	2	no	—
360	C₈	—	94	1	no	—
361	C₈	—	94	2	no	—
362	C₈	—	188	1	no	—
363	C₈	—	188	2	no	—
364	C₈	M₁₆×D₄₇	1	2	yes	—
365	C₂²	—	47	2	no	—
366	C₂²	—	94	1	no	—
367	C₂²	—	94	1	no	—
368	C₂²	—	188	2	no	—
369	C₂²	—	188	2	no	—
370	C₂²	—	188	2	no	—
371	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	4	2	no	—
374	C₄	—	4	2	no	—
375	C₄	—	47	2	no	—
376	C₄	—	47	2	no	—
377	C₄	—	94	1	no	—
378	C₄	—	188	2	no	—
379	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
380	C₄	#131×D₄₇	1	1	yes	characteristic
381	C₄	#15689922	1	2	yes	—
382	C₂	—	94	2	no	—
383	C₂	—	376	2	no	—
384	C₂	#436×D₄₇	1	1	yes	characteristic
385	C₂	#20252187	1	1	yes	characteristic
386	C₂	#444×D₄₇	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×#1593	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094047	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1593	D₄₇
hypocenter	C₄₇	#17890
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
1. #20252353	C₂	semidirect
3. #20242303	C₂	semidirect
4. #20242301	C₂	semidirect
5. #20242297	C₂	semidirect
6. #20242293	C₂	semidirect
7. #20242288	C₂	semidirect
8. #20242133	C₂	semidirect
9. #20241769	C₂	semidirect
10. #322×D₄₇	C₂	semidirect
11. C₄₇×#1593	C₂	semidirect
136. C₈×D₄₇	M₁₆	semidirect
194. #1593	D₄₇	semidirect
238. C₄₇	#17890	semidirect

Maximal subgroups

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

subgroup	index
#20252353	2
C₄×#13094047	2
#20242303	2
#20242301	2
#20242297	2
#20242293	2
#20242288	2
#20242133	2
#20241769	2
#322×D₄₇	2
C₄₇×#1593	2
#17890	47