Subgroups — #26198793

Order 12032.

770 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	#20253816	C₂	1	1	yes	maximal, characteristic, complemented
2	#20253389	C₂	1	2	yes	maximal, complemented
3	C₂×#15689202	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242878	C₂	1	2	yes	maximal, complemented
5	#20242868	C₂	1	2	yes	maximal, complemented
6	#20242854	C₂	1	4	yes	maximal, complemented
7	#20242035	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	—	2	1	no	—
10	C₂×#13094046	—	2	1	no	—
11	C₂²×#10625324	—	2	1	no	—
12	#15689673	C₂²	1	2	yes	—
13	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
14	#15689507	—	2	1	no	—
15	#15689507	—	2	1	no	—
16	#15689503	C₂²	1	1	yes	characteristic, complemented
17	#15689500	C₂²	1	1	yes	characteristic, complemented
18	#15689496	C₂²	1	2	yes	complemented
19	#15689437	—	2	2	no	—
20	#15689404	C₂²	1	4	yes	—
21	#15689319	C₂²	1	4	yes	—
22	#15689259	—	2	2	no	—
23	C₄×D₃₇₆	—	2	4	no	—
24	#15689202	—	2	1	no	—
25	#15689202	—	2	1	no	—
26	#15689202	C₂²	1	2	yes	—
27	#15689202	C₂²	1	2	yes	—
28	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
29	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
30	#15688868	C₂²	1	4	yes	complemented
31	#15688866	C₂²	1	4	yes	complemented
32	C₄₇×#431	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#335	C₂²	1	2	yes	complemented
35	C₄₇×#334	C₂²	1	2	yes	complemented
36	C₂×#10625350	—	2	2	no	—
37	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₄₇	—	2	1	no	—
39	#13094046	—	4	4	no	—
40	#13094046	—	4	4	no	—
41	C₂×Dic₁₈₈	—	2	1	no	—
42	C₂×Dic₁₈₈	—	2	1	no	—
43	C₂×#10625325	—	2	2	no	—
44	C₂×#10625325	—	2	2	no	—
45	C₂×D₃₇₆	—	2	4	no	—
46	C₂×D₃₇₆	—	2	1	no	—
47	C₂×D₃₇₆	—	2	1	no	—
48	C₂×#10625324	—	2	2	no	—
49	C₂×#10625324	—	2	2	no	—
50	C₂×#10625324	—	4	4	no	—
51	C₂²×C₄₇⋊C₈	—	2	1	no	—
52	#13094024	C₂³	1	2	yes	—
53	#13094021	C₂³	1	2	yes	—
54	#13094012	—	2	2	no	—
55	#13093998	—	2	2	no	—
56	#13093998	—	2	2	no	—
57	C₄×D₁₈₈	—	2	1	no	—
58	C₄×D₁₈₈	—	2	1	no	—
59	C₄×D₁₈₈	—	2	2	no	—
60	C₄×D₁₈₈	—	2	2	no	—
61	C₄×D₁₈₈	D₄	1	2	yes	—
62	C₄×D₁₈₈	C₂³	1	2	yes	—
63	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
64	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
65	#13093958	—	2	4	no	—
66	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
67	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
68	#13093946	—	2	4	no	—
69	#13093946	—	2	4	no	—
70	(C₂×D₄₇)⋊C₈	—	2	2	no	—
71	(C₂×D₄₇)⋊C₈	—	2	2	no	—
72	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
73	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
76	C₂²⋊C₈×C₄₇	—	2	2	no	—
77	C₄×C₈×C₄₇	—	2	2	no	—
78	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
79	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
80	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂³×D₄₇	—	2	1	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
86	C₂×C₂²⋊D₄₇	—	4	2	no	—
87	C₂²×Dic₄₇	—	2	1	no	—
88	#10625350	—	4	4	no	—
89	#10625350	—	4	4	no	—
90	C₂×D₁₈₈	—	2	1	no	—
91	C₂×D₁₈₈	—	2	2	no	—
92	C₂×D₁₈₈	—	2	2	no	—
93	C₂×D₁₈₈	—	2	2	no	—
94	C₂×D₁₈₈	—	4	2	no	—
95	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
96	C₂×C₄×D₄₇	—	2	2	no	—
97	C₂×C₄×D₄₇	—	2	2	no	—
98	C₂×C₄×D₄₇	—	4	2	no	—
99	C₂×C₄×D₄₇	—	4	2	no	—
100	C₂×C₄×D₄₇	—	4	2	no	—
101	C₂×Dic₉₄	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	2	4	no	—
104	C₂×C₈×C₄₇	D₈	1	4	yes	complemented
105	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₄₇	#41	1	1	yes	characteristic
112	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
113	(C₂×D₄₇)⋊C₄	—	2	2	no	—
114	(C₂×D₄₇)⋊C₄	—	2	2	no	—
115	(C₂×D₄₇)⋊C₄	—	4	2	no	—
116	(C₂×D₄₇)⋊C₄	—	4	2	no	—
117	C₄⋊Dic₄₇	—	2	1	no	—
118	C₄⋊Dic₄₇	—	2	2	no	—
119	C₄⋊Dic₄₇	—	2	2	no	—
120	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	4	2	no	—
123	Dic₁₈₈	—	4	2	no	—
124	Dic₁₈₈	—	4	2	no	—
125	D₃₇₆	—	4	2	no	—
126	D₃₇₆	—	4	8	no	—
127	D₃₇₆	—	4	2	no	—
128	#10625325	—	4	4	no	—
129	#10625325	—	4	4	no	—
130	#10625324	—	4	4	no	—
131	#10625324	—	4	4	no	—
132	C₂³×C₄₇	#121	1	1	yes	characteristic
133	C₂²×D₄₇	—	2	2	no	—
134	C₂²×D₄₇	—	4	2	no	—
135	C₂²×D₄₇	—	4	2	no	—
136	C₂²×D₄₇	—	4	2	no	—
137	C₂²×D₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	4	1	no	—
140	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₄₇	—	8	4	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	—	4	2	no	—
152	D₁₈₈	—	2	4	no	—
153	D₁₈₈	—	4	2	no	—
154	D₁₈₈	—	4	2	no	—
155	D₁₈₈	—	4	2	no	—
156	D₁₈₈	—	4	4	no	—
157	D₁₈₈	—	8	2	no	—
158	C₄×D₄₇	—	4	2	no	—
159	C₄×D₄₇	—	4	2	no	—
160	C₄×D₄₇	—	4	4	no	—
161	C₄×D₄₇	—	8	4	no	—
162	Dic₉₄	—	4	2	no	—
163	Dic₉₄	—	4	2	no	—
164	C₈×C₄₇	—	2	1	no	—
165	C₈×C₄₇	—	2	1	no	—
166	C₈×C₄₇	—	4	4	no	—
167	C₈×C₄₇	C₂×D₈	1	2	yes	—
168	C₈×C₄₇	C₂×D₈	1	2	yes	—
169	C₄₇⋊C₈	—	4	2	no	—
170	#19133	—	47	1	no	maximal, Sylow
171	C₂²×C₄₇	—	2	1	no	—
172	C₂²×C₄₇	—	2	1	no	—
173	C₂²×C₄₇	#459	1	2	yes	—
174	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
175	C₂×D₄₇	—	4	2	no	—
176	C₂×D₄₇	—	4	2	no	—
177	C₂×D₄₇	—	4	2	no	—
178	C₂×D₄₇	—	8	4	no	—
179	C₂×D₄₇	—	8	2	no	—
180	C₂×D₄₇	—	8	2	no	—
181	C₂×D₄₇	—	8	2	no	—
182	C₄×C₄₇	—	2	1	no	—
183	C₄×C₄₇	—	2	1	no	—
184	C₄×C₄₇	—	2	1	no	—
185	C₄×C₄₇	#459	1	2	yes	—
186	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
187	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₄×C₄₇	#575	1	2	yes	—
189	Dic₄₇	—	4	2	no	—
190	Dic₄₇	—	8	2	no	—
191	#3147	—	47	1	no	—
192	#3100	—	47	2	no	—
193	C₂×#436	—	47	1	no	—
194	#1689	—	47	2	no	—
195	#1687	—	47	2	no	—
196	#1685	—	47	4	no	—
197	#1564	D₄₇	1	1	yes	characteristic, complemented
198	#1540	—	47	2	no	—
199	C₂×C₄₇	—	2	2	no	—
200	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
201	C₂×C₄₇	#3054	1	1	yes	characteristic
202	C₂×C₄₇	#3067	1	1	yes	characteristic
203	D₄₇	—	8	2	no	—
204	D₄₇	—	16	2	no	—
205	C₂×#135	—	94	1	no	—
206	C₂×#135	—	94	1	no	—
207	C₂²×M₁₆	—	94	1	no	—
208	#532	—	47	2	no	—
209	C₂×C₄×D₄	—	47	1	no	—
210	#495	—	94	1	no	—
211	#495	—	94	1	no	—
212	#494	—	47	1	no	—
213	#493	—	47	1	no	—
214	#492	—	47	2	no	—
215	#480	—	94	2	no	—
216	#474	—	47	4	no	—
217	#459	—	47	4	no	—
218	#447	—	94	2	no	—
219	C₄×D₈	—	94	4	no	—
220	#436	—	47	2	no	—
221	#436	—	47	2	no	—
222	#436	—	94	1	no	—
223	#436	—	94	1	no	—
224	#431	C₂×D₄₇	1	2	yes	—
225	C₂×C₄⋊C₈	—	47	1	no	—
226	C₂×C₂²⋊C₈	—	94	1	no	—
227	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
228	#335	C₂×D₄₇	1	2	yes	complemented
229	#334	C₂×D₄₇	1	2	yes	complemented
230	#332	—	47	4	no	—
231	#331	—	47	4	no	—
232	C₄₇	#19133	1	1	yes	characteristic, complemented, Sylow
233	C₂×#41	—	94	2	no	—
234	C₂²×D₄	—	47	1	no	—
235	C₂³×C₄	—	94	1	no	—
236	#135	—	188	4	no	—
237	#135	—	188	4	no	—
238	C₂×Q₁₆	—	94	1	no	—
239	C₂×Q₁₆	—	94	1	no	—
240	C₂×SD₁₆	—	94	2	no	—
241	C₂×SD₁₆	—	94	2	no	—
242	C₂×D₈	—	94	4	no	—
243	C₂×D₈	—	94	1	no	—
244	C₂×D₈	—	94	1	no	—
245	C₂×M₁₆	—	94	2	no	—
246	C₂×M₁₆	—	94	2	no	—
247	C₂×M₁₆	—	188	4	no	—
248	C₂²×C₈	—	94	1	no	—
249	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
250	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
251	C₄⋊Q₈	—	47	2	no	—
252	#127	—	47	2	no	—
253	#124	—	94	2	no	—
254	#121	—	94	2	no	—
255	#121	—	94	2	no	—
256	C₄×D₄	—	47	2	no	—
257	C₄×D₄	—	47	2	no	—
258	C₄×D₄	—	94	1	no	—
259	C₄×D₄	—	94	2	no	—
260	C₄×D₄	—	94	1	no	—
261	C₄×D₄	—	94	2	no	—
262	C₂×C₄⋊C₄	—	47	1	no	—
263	C₂×C₂²⋊C₄	—	94	1	no	—
264	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
265	#107	—	94	4	no	—
266	C₄⋊C₈	—	47	2	no	—
267	C₄⋊C₈	—	94	1	no	—
268	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
269	#102	—	94	4	no	—
270	#102	—	94	4	no	—
271	C₂²⋊C₈	—	2	2	no	—
272	C₂²⋊C₈	—	94	2	no	—
273	C₂²⋊C₈	—	94	2	no	—
274	C₄×C₈	—	2	2	no	—
275	C₄×C₈	D₁₈₈	1	1	yes	characteristic
276	C₄×C₈	D₁₈₈	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
279	C₂⁴	—	94	1	no	—
280	#41	—	188	4	no	—
281	#41	—	188	4	no	—
282	C₂×Q₈	—	94	2	no	—
283	C₂×D₄	—	47	2	no	—
284	C₂×D₄	—	94	1	no	—
285	C₂×D₄	—	94	2	no	—
286	C₂×D₄	—	94	2	no	—
287	C₂×D₄	—	94	2	no	—
288	C₂×D₄	—	188	2	no	—
289	C₂×D₄	—	188	2	no	—
290	C₂²×C₄	—	94	1	no	—
291	C₂²×C₄	—	94	2	no	—
292	C₂²×C₄	—	94	2	no	—
293	C₂²×C₄	—	188	2	no	—
294	C₂²×C₄	—	188	2	no	—
295	C₂²×C₄	—	188	2	no	—
296	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
297	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
298	C₂²×C₄	#10625354	1	1	yes	characteristic
299	Q₁₆	—	188	2	no	—
300	Q₁₆	—	188	2	no	—
301	SD₁₆	—	188	4	no	—
302	SD₁₆	—	188	4	no	—
303	D₈	—	188	2	no	—
304	D₈	—	188	8	no	—
305	D₈	—	188	2	no	—
306	M₁₆	—	188	4	no	—
307	M₁₆	—	188	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	4	no	—
310	C₂×C₈	—	94	2	no	—
311	C₂×C₈	—	188	2	no	—
312	C₂×C₈	D₃₇₆	1	4	yes	complemented
313	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
314	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
315	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
316	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
317	C₄⋊C₄	—	47	2	no	—
318	C₄⋊C₄	—	94	1	no	—
319	C₄⋊C₄	—	94	2	no	—
320	C₄⋊C₄	—	94	2	no	—
321	C₂²⋊C₄	—	94	2	no	—
322	C₂²⋊C₄	—	94	2	no	—
323	C₂²⋊C₄	—	188	2	no	—
324	C₂²⋊C₄	—	188	2	no	—
325	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
326	C₄²	D₄×D₄₇	1	1	yes	characteristic
327	C₄²	#10625354	1	1	yes	characteristic
328	C₄²	C₂³×D₄₇	1	1	yes	characteristic
329	C₂³	—	94	2	no	—
330	C₂³	—	188	2	no	—
331	C₂³	—	188	2	no	—
332	C₂³	—	188	2	no	—
333	C₂³	—	188	2	no	—
334	C₂³	#13093999	1	1	yes	characteristic
335	Q₈	—	188	2	no	—
336	Q₈	—	188	2	no	—
337	D₄	—	94	4	no	—
338	D₄	—	376	2	no	—
339	D₄	—	376	4	no	—
340	D₄	—	188	2	no	—
341	D₄	—	188	2	no	—
342	D₄	—	188	2	no	—
343	D₄	—	188	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	94	2	no	—
347	C₂×C₄	—	376	4	no	—
348	C₂×C₄	—	188	2	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	—	188	4	no	—
351	C₂×C₄	—	188	2	no	—
352	C₂×C₄	—	188	2	no	—
353	C₂×C₄	#13093999	1	1	yes	characteristic
354	C₂×C₄	#13093999	1	1	yes	characteristic
355	C₂×C₄	#13093999	1	1	yes	characteristic
356	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
357	C₂×C₄	#13094052	1	2	yes	—
358	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
359	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
360	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	4	4	no	—
364	C₈	—	188	2	no	—
365	C₈	C₂×D₃₇₆	1	2	yes	—
366	C₈	C₂×D₃₇₆	1	2	yes	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	376	4	no	—
370	C₂²	—	376	2	no	—
371	C₂²	—	376	2	no	—
372	C₂²	—	376	2	no	—
373	C₂²	—	188	2	no	—
374	C₂²	—	188	2	no	—
375	C₂²	—	188	2	no	—
376	C₂²	#15689319	1	2	yes	—
377	C₂²	C₂×#13093999	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	376	2	no	—
382	C₄	—	188	2	no	—
383	C₄	#15689319	1	2	yes	—
384	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
385	C₄	C₂×#13094052	1	1	yes	characteristic
386	C₄	#15689920	1	2	yes	—
387	C₂	—	2	2	no	—
388	C₂	—	376	2	no	—
389	C₂	—	752	2	no	—
390	C₂	C₂×#15689319	1	1	yes	characteristic
391	C₂	#20252998	1	1	yes	characteristic
392	C₂	#20253109	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093999
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	C₂×#121
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#19133
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #20253816	C₂	semidirect
2. #20253389	C₂	semidirect
3. C₂×#15689202	C₂	semidirect
4. #20242878	C₂	semidirect
5. #20242868	C₂	semidirect
6. #20242854	C₂	semidirect
7. #20242035	C₂	semidirect
8. C₄₇×#1564	C₂	semidirect
16. #15689503	C₂²	semidirect
17. #15689500	C₂²	semidirect
18. #15689496	C₂²	semidirect
30. #15688868	C₂²	semidirect
31. #15688866	C₂²	semidirect
34. C₄₇×#335	C₂²	semidirect
35. C₄₇×#334	C₂²	semidirect
104. C₂×C₈×C₄₇	D₈	semidirect
197. #1564	D₄₇	semidirect
228. #335	C₂×D₄₇	semidirect
229. #334	C₂×D₄₇	semidirect
232. C₄₇	#19133	semidirect
312. C₂×C₈	D₃₇₆	semidirect

Maximal subgroups

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

subgroup	index
#20253816	2
#20253389	2
C₂×#15689202	2
#20242878	2
#20242868	2
#20242854	2
#20242035	2
C₄₇×#1564	2
#19133	47