Subgroups — #26371725

Order 12032.

858 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	C₂×#15689557	C₂	1	2	yes	maximal, complemented
2	C₂×#15689552	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20253868	C₂	1	2	yes	maximal, complemented
5	#20245652	C₂	1	2	yes	maximal, complemented
6	#20245651	C₂	1	2	yes	maximal, complemented
7	#20245649	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245646	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245636	C₂	1	1	yes	maximal, characteristic, complemented
10	#20244983	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2246	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094048	—	2	2	no	—
13	C₂×#13094043	C₂²	1	2	yes	—
14	C₂×D₈×D₄₇	C₂²	1	2	yes	—
15	#15689686	—	2	2	no	—
16	#15689557	—	2	8	no	—
17	#15689552	—	2	4	no	—
18	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic, complemented
19	C₂×#13093969	C₂²	1	2	yes	complemented
20	C₂×#13093968	C₂²	1	2	yes	complemented
21	C₂×#13093968	C₂²	1	2	yes	complemented
22	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
23	C₂×#13093965	C₂²	1	2	yes	—
24	C₂×#13093965	C₂²	1	2	yes	complemented
25	D₁₆×D₄₇	—	2	4	no	—
26	C₂×#13093963	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
28	#15689511	—	2	2	no	—
29	#15689510	—	2	2	no	—
30	#15689507	C₂²	1	2	yes	complemented
31	#15689506	C₂²	1	2	yes	complemented
32	#15689498	C₂²	1	2	yes	complemented
33	#15689077	C₂²	1	1	yes	characteristic
34	#15688899	C₂²	1	1	yes	characteristic, complemented
35	#15688898	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
37	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
39	C₂×D₁₆×C₄₇	C₂²	1	2	yes	complemented
40	C₄₇×#493	C₂²	1	2	yes	complemented
41	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
42	C₂×#10625352	—	2	2	no	—
43	C₂×D₄×D₄₇	—	2	2	no	—
44	#13094048	—	4	8	no	—
45	#13094043	—	2	4	no	—
46	#13094043	—	2	4	no	—
47	C₂×Dic₁₈₈	C₂³	1	2	yes	—
48	C₂×#10625325	—	2	2	no	—
49	C₂×#10625335	—	2	2	no	—
50	C₂×#10625335	—	2	2	no	—
51	C₂×D₃₇₆	C₂³	1	2	yes	—
52	C₂×#10625334	—	2	2	no	—
53	C₂×#10625334	—	2	2	no	—
54	D₈×D₄₇	—	2	4	no	—
55	D₈×D₄₇	—	2	4	no	—
56	C₂×#10625324	D₄	1	1	yes	characteristic
57	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
58	#13094020	—	2	2	no	—
59	#13094012	—	2	2	no	—
60	#13094011	—	2	2	no	—
61	#13093996	—	4	4	no	—
62	#13093992	—	4	4	no	—
63	#13093981	D₄	1	1	yes	characteristic
64	Dic₃₇₆	—	2	2	no	—
65	#13093969	—	2	4	no	—
66	#13093968	—	2	4	no	—
67	#13093968	—	2	4	no	—
68	D₇₅₂	—	2	2	no	—
69	#13093965	—	2	4	no	—
70	#13093965	—	2	4	no	—
71	#13093963	D₄	1	4	yes	complemented
72	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₁₆×D₄₇	D₄	1	4	yes	complemented
75	#13093935	D₄	1	1	yes	characteristic
76	#13093934	D₄	1	1	yes	characteristic
77	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
78	C₂×D₈×C₄₇	—	2	2	no	—
79	C₂×D₈×C₄₇	C₂³	1	2	yes	—
80	C₂×D₈×C₄₇	C₂³	1	2	yes	—
81	#127×C₄₇	—	2	2	no	—
82	D₁₆×C₄₇	—	2	2	no	—
83	D₁₆×C₄₇	—	2	2	no	—
84	D₁₆×C₄₇	—	2	4	no	—
85	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
88	C₂³×D₄₇	—	4	2	no	—
89	C₂×D₄×C₄₇	—	2	2	no	—
90	C₂×D₄×C₄₇	—	2	2	no	—
91	C₂×D₄×C₄₇	—	4	4	no	—
92	C₂×C₂²⋊D₄₇	—	4	2	no	—
93	C₂×C₂²⋊D₄₇	—	4	2	no	—
94	C₂×C₂²⋊D₄₇	—	4	4	no	—
95	C₂²×Dic₄₇	—	4	2	no	—
96	#10625352	—	4	4	no	—
97	#10625352	—	4	4	no	—
98	D₄×D₄₇	—	4	4	no	—
99	D₄×D₄₇	—	4	4	no	—
100	C₂×D₁₈₈	—	2	2	no	—
101	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×Dic₉₄	—	2	2	no	—
103	D₈×C₄₇	—	2	2	no	—
104	D₈×C₄₇	—	2	2	no	—
105	D₈×C₄₇	—	2	2	no	—
106	D₈×C₄₇	—	2	2	no	—
107	D₈×C₄₇	—	4	4	no	—
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
110	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₂²⋊Dic₄₇	—	4	4	no	—
112	#10625335	—	4	4	no	—
113	#10625335	—	4	4	no	—
114	#10625334	—	4	4	no	—
115	#10625334	—	4	4	no	—
116	(C₂×D₄₇)⋊C₄	—	4	4	no	—
117	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
118	Dic₄₇⋊C₄	D₈	1	2	yes	—
119	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
122	Dic₁₈₈	—	2	2	no	—
123	Dic₁₈₈	—	2	2	no	—
124	D₃₇₆	—	2	2	no	—
125	D₃₇₆	—	2	2	no	—
126	#10625325	—	4	4	no	—
127	#10625324	D₈	1	4	yes	—
128	C₈×D₄₇	C₂×D₄	1	2	yes	—
129	C₈×D₄₇	C₂×D₄	1	2	yes	—
130	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
131	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
132	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
133	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
134	C₂³×C₄₇	—	4	2	no	—
135	C₂³×C₄₇	—	4	2	no	—
136	C₂²×D₄₇	—	4	2	no	—
137	C₂²×D₄₇	—	8	4	no	—
138	C₂²×D₄₇	—	16	4	no	—
139	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
140	D₄×C₄₇	—	4	2	no	—
141	D₄×C₄₇	—	4	2	no	—
142	D₄×C₄₇	—	4	2	no	—
143	D₄×C₄₇	—	4	2	no	—
144	D₄×C₄₇	—	16	4	no	—
145	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₄₇	—	8	4	no	—
148	C₂²⋊D₄₇	—	8	4	no	—
149	C₂²⋊D₄₇	—	16	4	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	—	8	4	no	—
152	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
153	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
154	D₁₈₈	—	4	2	no	—
155	D₁₈₈	—	4	2	no	—
156	C₄×D₄₇	#127	1	2	yes	—
157	C₄×D₄₇	C₂×D₈	1	2	yes	—
158	Dic₉₄	—	4	2	no	—
159	Dic₉₄	—	4	2	no	—
160	C₈×C₄₇	C₂×D₈	1	2	yes	—
161	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
163	C₄₇⋊C₈	#127	1	2	yes	—
164	C₄₇⋊C₈	C₂×D₈	1	2	yes	—
165	#33720	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	—	8	2	no	—
167	C₂²×C₄₇	—	8	2	no	—
168	C₂²×C₄₇	—	8	4	no	—
169	C₂²×C₄₇	—	8	4	no	—
170	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₄₇	—	2	2	no	—
172	C₂×D₄₇	—	8	2	no	—
173	C₂×D₄₇	—	8	4	no	—
174	C₂×D₄₇	—	16	8	no	—
175	C₂×D₄₇	#493	1	2	yes	—
176	C₄×C₄₇	—	2	2	no	—
177	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
178	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
179	Dic₄₇	—	2	2	no	—
180	Dic₄₇	—	8	2	no	—
181	Dic₄₇	#493	1	2	yes	—
182	C₂×#509	—	47	2	no	—
183	C₂×#508	—	47	1	no	—
184	C₂²×D₁₆	—	47	1	no	—
185	#3155	—	47	2	no	—
186	#2250	—	47	2	no	—
187	#2250	—	47	2	no	—
188	#2249	—	47	1	no	—
189	#2249	—	47	1	no	—
190	#2246	—	47	1	no	—
191	#2246	D₄₇	1	1	yes	characteristic, complemented
192	#2108	—	47	1	no	—
193	C₂×C₄₇	—	16	2	no	—
194	C₂×C₄₇	—	16	2	no	—
195	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
196	C₂×C₄₇	#3415	1	2	yes	—
197	D₄₇	—	2	2	no	—
198	D₄₇	—	16	2	no	—
199	C₂×AGL(1,ℤ/8)	—	94	2	no	—
200	C₂×#135	—	47	2	no	—
201	C₂²×D₈	—	47	2	no	—
202	#535	—	94	2	no	—
203	#509	—	94	8	no	—
204	#508	—	94	4	no	—
205	C₂×Q₃₂	—	47	1	no	—
206	C₂×SD₃₂	—	47	2	no	—
207	C₂×SD₃₂	—	47	2	no	—
208	C₂×SD₃₂	—	47	2	no	—
209	C₂×D₁₆	—	47	2	no	—
210	C₂×D₁₆	—	47	2	no	—
211	C₂×D₁₆	—	47	1	no	—
212	C₂×D₁₆	—	94	4	no	—
213	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic
214	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
215	C₂×D₁₆	C₂×D₄₇	1	2	yes	complemented
216	C₂×M₃₂	—	47	1	no	—
217	C₂²×C₁₆	—	47	1	no	—
218	#496	—	94	2	no	—
219	#496	—	94	2	no	—
220	#495	—	47	2	no	—
221	#495	—	47	2	no	—
222	#493	—	47	2	no	—
223	#493	C₂×D₄₇	1	2	yes	complemented
224	#405	—	47	1	no	—
225	#346	—	47	1	no	—
226	#346	—	47	1	no	—
227	C₄×C₁₆	—	47	1	no	—
228	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
229	C₄₇	#33720	1	1	yes	characteristic, complemented, Sylow
230	C₂×#41	—	94	2	no	—
231	C₂²×D₄	—	94	2	no	—
232	AGL(1,ℤ/8)	—	188	8	no	—
233	#135	—	94	4	no	—
234	#135	—	94	4	no	—
235	C₂×Q₁₆	—	47	2	no	—
236	C₂×SD₁₆	—	94	2	no	—
237	C₂×SD₁₆	—	94	2	no	—
238	C₂×SD₁₆	—	94	2	no	—
239	C₂×D₈	—	2	2	no	—
240	C₂×D₈	—	47	2	no	—
241	C₂×D₈	—	94	4	no	—
242	C₂×D₈	—	94	4	no	—
243	C₂×D₈	—	94	2	no	—
244	C₂×D₈	—	94	2	no	—
245	C₂×D₈	C₂²×D₄₇	1	2	yes	—
246	C₂×D₈	C₂²×D₄₇	1	2	yes	—
247	C₂×M₁₆	—	47	1	no	—
248	C₂²×C₈	—	47	1	no	—
249	#127	—	2	2	no	—
250	#127	—	94	2	no	—
251	#124	—	94	2	no	—
252	#124	—	94	2	no	—
253	#121	—	188	4	no	—
254	#120	—	188	4	no	—
255	#117	—	47	1	no	—
256	Q₃₂	—	94	2	no	—
257	SD₃₂	—	94	4	no	—
258	SD₃₂	—	94	4	no	—
259	SD₃₂	—	94	4	no	—
260	D₁₆	—	2	2	no	—
261	D₁₆	—	2	2	no	—
262	D₁₆	—	2	4	no	—
263	D₁₆	—	94	4	no	—
264	D₁₆	—	94	4	no	—
265	D₁₆	—	94	2	no	—
266	M₃₂	—	47	4	no	—
267	C₂×C₁₆	—	47	1	no	—
268	C₂×C₁₆	—	47	1	no	—
269	C₂×C₁₆	—	47	4	no	—
270	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
271	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
272	#97	—	47	1	no	—
273	#97	—	47	1	no	—
274	C₄×C₈	—	47	1	no	—
275	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
276	C₂⁴	—	188	2	no	—
277	#41	—	188	4	no	—
278	#41	—	188	4	no	—
279	C₂×Q₈	—	94	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	4	4	no	—
283	C₂×D₄	—	94	2	no	—
284	C₂×D₄	—	188	4	no	—
285	C₂×D₄	—	188	4	no	—
286	C₂×D₄	—	188	2	no	—
287	C₂×D₄	—	188	4	no	—
288	C₂×D₄	—	188	2	no	—
289	C₂²×C₄	—	47	1	no	—
290	C₂²×C₄	—	188	2	no	—
291	Q₁₆	—	94	2	no	—
292	Q₁₆	—	94	2	no	—
293	SD₁₆	—	188	4	no	—
294	SD₁₆	—	188	4	no	—
295	SD₁₆	—	188	4	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	D₈	—	2	2	no	—
299	D₈	—	2	2	no	—
300	D₈	—	4	4	no	—
301	D₈	—	94	2	no	—
302	D₈	—	94	2	no	—
303	D₈	—	188	4	no	—
304	D₈	—	188	4	no	—
305	M₁₆	—	47	4	no	—
306	C₂×C₈	—	47	1	no	—
307	C₂×C₈	—	47	1	no	—
308	C₂×C₈	—	47	2	no	—
309	C₂×C₈	—	47	2	no	—
310	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
311	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
312	C₄⋊C₄	—	47	2	no	—
313	C₂²⋊C₄	—	47	2	no	—
314	C₂²⋊C₄	—	188	4	no	—
315	C₂²⋊C₄	—	188	4	no	—
316	C₄²	—	47	1	no	—
317	C₄²	D₄×D₄₇	1	1	yes	characteristic
318	C₁₆	—	47	2	no	—
319	C₁₆	—	47	2	no	—
320	C₁₆	D₄×D₄₇	1	2	yes	—
321	C₁₆	D₄×D₄₇	1	2	yes	—
322	C₂³	—	4	2	no	—
323	C₂³	—	4	2	no	—
324	C₂³	—	47	1	no	—
325	C₂³	—	376	4	no	—
326	C₂³	—	188	2	no	—
327	C₂³	—	752	4	no	—
328	Q₈	—	188	2	no	—
329	Q₈	—	188	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	16	4	no	—
335	D₄	—	376	4	no	—
336	D₄	—	376	4	no	—
337	D₄	—	188	2	no	—
338	D₄	—	188	2	no	—
339	D₄	—	752	4	no	—
340	C₂×C₄	—	47	1	no	—
341	C₂×C₄	—	47	2	no	—
342	C₂×C₄	—	47	2	no	—
343	C₂×C₄	—	47	2	no	—
344	C₂×C₄	—	376	4	no	—
345	C₂×C₄	—	188	2	no	—
346	C₂×C₄	D₈×D₄₇	1	2	yes	—
347	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
348	C₈	—	47	2	no	—
349	C₈	—	47	2	no	—
350	C₈	D₈×D₄₇	1	2	yes	—
351	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
352	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
353	C₂²	—	8	4	no	—
354	C₂²	—	8	4	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	47	2	no	—
358	C₂²	—	94	2	no	—
359	C₂²	—	376	4	no	—
360	C₂²	—	376	2	no	—
361	C₂²	—	752	8	no	—
362	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
363	C₄	—	2	2	no	—
364	C₄	—	47	2	no	—
365	C₄	—	94	2	no	—
366	C₄	—	376	2	no	—
367	C₄	#127×D₄₇	1	1	yes	characteristic
368	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
369	C₂	—	16	2	no	—
370	C₂	—	16	2	no	—
371	C₂	—	94	2	no	—
372	C₂	—	752	2	no	—
373	C₂	#493×D₄₇	1	1	yes	characteristic
374	C₂	#20256525	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2246	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#2246	D₄₇
hypocenter	C₄₇	#33720
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

Complements

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

N	H ≅ G/N	type
1. C₂×#15689557	C₂	semidirect
2. C₂×#15689552	C₂	semidirect
3. C₂×D₁₆×D₄₇	C₂	semidirect
4. #20253868	C₂	semidirect
5. #20245652	C₂	semidirect
6. #20245651	C₂	semidirect
7. #20245649	C₂	semidirect
8. #20245646	C₂	semidirect
9. #20245636	C₂	semidirect
10. #20244983	C₂	semidirect
11. C₄₇×#2246	C₂	semidirect
18. C₂×Dic₃₇₆	C₂²	semidirect
19. C₂×#13093969	C₂²	semidirect
20. C₂×#13093968	C₂²	semidirect
21. C₂×#13093968	C₂²	semidirect
24. C₂×#13093965	C₂²	semidirect
30. #15689507	C₂²	semidirect
31. #15689506	C₂²	semidirect
32. #15689498	C₂²	semidirect
34. #15688899	C₂²	semidirect
35. #15688898	C₂²	semidirect
36. C₁₆×Dic₄₇	C₂²	semidirect
38. C₂×D₁₆×C₄₇	C₂²	semidirect
39. C₂×D₁₆×C₄₇	C₂²	semidirect
40. C₄₇×#493	C₂²	semidirect
41. C₄×C₁₆×C₄₇	C₂²	semidirect
71. #13093963	D₄	semidirect
74. C₁₆×D₄₇	D₄	semidirect
191. #2246	D₄₇	semidirect
214. C₂×D₁₆	C₂×D₄₇	semidirect
215. C₂×D₁₆	C₂×D₄₇	semidirect
223. #493	C₂×D₄₇	semidirect
228. C₄×C₁₆	C₂×D₄₇	semidirect
229. C₄₇	#33720	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689557	2
C₂×#15689552	2
C₂×D₁₆×D₄₇	2
#20253868	2
#20245652	2
#20245651	2
#20245649	2
#20245646	2
#20245636	2
#20244983	2
C₄₇×#2246	2
#33720	47