Subgroups — #26369693

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256521	C₂	1	1	yes	maximal, characteristic
2	#20252300	C₂	1	2	yes	maximal, complemented
3	#20245274	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245273	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245272	C₂	1	2	yes	maximal, complemented
6	#20245270	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245266	C₂	1	2	yes	maximal, complemented
8	#20245181	C₂	1	1	yes	maximal, characteristic, complemented
9	#359×D₄₇	C₂	1	2	yes	maximal, complemented
10	#20244983	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2178	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₄₇	C₂²	1	2	yes	—
13	#15689888	—	2	2	no	—
14	#15689565	C₄	1	2	yes	complemented
15	#15689562	C₄	1	2	yes	complemented
16	#15689560	C₄	1	2	yes	complemented
17	#15689557	C₄	1	2	yes	complemented
18	#15689555	C₂²	1	1	yes	characteristic
19	#15689553	C₂²	1	2	yes	—
20	C₂×#13093963	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
22	#15689243	—	2	2	no	—
23	#15689242	—	2	2	no	—
24	#15689241	C₂²	1	2	yes	complemented
25	#15689240	C₂²	1	2	yes	complemented
26	#15689239	C₂²	1	2	yes	complemented
27	#108×D₄₇	C₂²	1	2	yes	—
28	#15689138	—	2	2	no	—
29	#15689077	C₂²	1	1	yes	characteristic
30	#15688953	C₂²	1	1	yes	characteristic, complemented
31	#15688952	C₂²	1	2	yes	complemented
32	#15688931	C₂²	1	2	yes	complemented
33	#15688930	C₂²	1	2	yes	complemented
34	#15688929	C₂²	1	2	yes	complemented
35	#15688899	C₂²	1	1	yes	characteristic, complemented
36	#15688898	C₂²	1	1	yes	characteristic, complemented
37	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
38	C₄₇×#508	C₂²	1	1	yes	characteristic
39	C₄₇×#443	C₂²	1	2	yes	complemented
40	C₄₇×#368	C₂²	1	1	yes	characteristic, complemented
41	C₄₇×#359	C₂²	1	2	yes	complemented
42	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
43	#41×D₄₇	—	2	2	no	—
44	#13094047	C₂×C₄	1	2	yes	—
45	#13094046	C₂³	1	2	yes	—
46	#13094045	—	2	2	no	—
47	#13094044	—	2	2	no	—
48	#13094043	C₂×C₄	1	2	yes	—
49	Q₁₆×D₄₇	C₂×C₄	1	2	yes	—
50	SD₁₆×D₄₇	—	2	2	no	—
51	D₈×D₄₇	C₂×C₄	1	2	yes	—
52	#13094032	—	2	2	no	—
53	#13094032	—	4	2	no	—
54	#13094031	—	2	2	no	—
55	#13094030	—	2	2	no	—
56	#13094030	—	4	2	no	—
57	C₂×#10625324	—	4	2	no	—
58	C₂×#10625324	D₄	1	1	yes	characteristic
59	M₁₆×D₄₇	—	2	2	no	—
60	M₁₆×D₄₇	—	4	2	no	—
61	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
62	#13093981	D₄	1	1	yes	characteristic
63	Dic₃₇₆	C₂×C₄	1	1	yes	characteristic
64	#13093970	C₂×C₄	1	2	yes	—
65	#13093969	C₂×C₄	1	2	yes	—
66	#13093968	C₂×C₄	1	2	yes	—
67	#13093967	C₂×C₄	1	2	yes	—
68	D₇₅₂	C₂×C₄	1	1	yes	characteristic
69	#13093965	C₂×C₄	1	2	yes	—
70	#13093963	C₂×C₄	1	2	yes	—
71	#13093963	C₂×C₄	1	2	yes	—
72	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₁₆×D₄₇	—	2	2	no	—
75	#13093960	C₂³	1	2	yes	—
76	#13093959	—	2	2	no	—
77	#13093952	—	2	2	no	—
78	#13093951	—	2	2	no	—
79	#13093950	—	2	2	no	—
80	#13093935	D₄	1	1	yes	characteristic
81	#13093934	D₄	1	1	yes	characteristic
82	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
83	#135×C₄₇	C₂³	1	2	yes	—
84	#131×C₄₇	—	2	2	no	—
85	Q₃₂×C₄₇	C₂×C₄	1	1	yes	characteristic
86	SD₃₂×C₄₇	C₂×C₄	1	2	yes	—
87	D₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
89	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
90	#108×C₄₇	C₂³	1	2	yes	—
91	#104×C₄₇	—	2	2	no	—
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	#41×C₄₇	—	2	2	no	—
94	#10625354	—	2	2	no	—
95	Q₈×D₄₇	—	2	2	no	—
96	#10625352	—	2	2	no	—
97	#10625352	—	4	2	no	—
98	D₄×D₄₇	—	2	2	no	—
99	D₄×D₄₇	—	4	2	no	—
100	#10625350	—	2	2	no	—
101	#10625350	—	4	2	no	—
102	C₂×C₄×D₄₇	—	4	2	no	—
103	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
104	Q₁₆×C₄₇	C₂²×C₄	1	2	yes	—
105	SD₁₆×C₄₇	—	2	2	no	—
106	D₈×C₄₇	C₂²×C₄	1	2	yes	—
107	M₁₆×C₄₇	—	2	2	no	—
108	M₁₆×C₄₇	—	4	2	no	—
109	C₂×C₈×C₄₇	—	4	2	no	—
110	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	#10625337	—	2	2	no	—
114	#10625336	—	2	2	no	—
115	#10625335	—	2	2	no	—
116	#10625334	—	2	2	no	—
117	(C₂×D₄₇)⋊C₄	—	2	1	no	—
118	Dic₄₇⋊C₄	—	2	1	no	—
119	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
120	#10625329	—	2	2	no	—
121	#10625329	—	4	2	no	—
122	C₂×C₄₇⋊C₈	—	4	2	no	—
123	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
125	Dic₁₈₈	C₂²×C₄	1	2	yes	—
126	D₃₇₆	C₂²×C₄	1	2	yes	—
127	#10625325	—	2	2	no	—
128	#10625324	—	4	2	no	—
129	#10625324	—	4	2	no	—
130	#10625324	—	4	2	no	—
131	#10625324	D₈	1	2	yes	—
132	#10625324	D₈	1	2	yes	—
133	C₈×D₄₇	—	4	2	no	—
134	C₈×D₄₇	—	4	2	no	—
135	C₈×D₄₇	—	4	2	no	—
136	C₈×D₄₇	C₂²×C₄	1	2	yes	—
137	C₈×D₄₇	#41	1	2	yes	—
138	C₁₆×C₄₇	—	2	1	no	—
139	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
140	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
141	C₄₇⋊C₁₆	—	2	1	no	—
142	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
143	C₂²×D₄₇	—	4	2	no	—
144	C₂²×D₄₇	#135	1	1	yes	characteristic
145	Q₈×C₄₇	—	2	2	no	—
146	D₄×C₄₇	—	2	2	no	—
147	D₄×C₄₇	—	4	2	no	—
148	C₂×C₄×C₄₇	—	2	1	no	—
149	C₂×C₄×C₄₇	—	4	2	no	—
150	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₄₇	—	4	2	no	—
152	C₂²⋊D₄₇	—	4	2	no	—
153	C₂²⋊D₄₇	—	4	2	no	—
154	C₂×Dic₄₇	—	2	1	no	—
155	C₂×Dic₄₇	—	4	2	no	—
156	C₂×Dic₄₇	#135	1	1	yes	characteristic
157	D₁₈₈	—	2	2	no	—
158	D₁₈₈	—	4	2	no	—
159	C₄×D₄₇	—	4	2	no	—
160	C₄×D₄₇	—	4	2	no	—
161	C₄×D₄₇	—	4	2	no	—
162	C₄×D₄₇	C₄×D₄	1	2	yes	—
163	C₄×D₄₇	C₂×D₈	1	2	yes	—
164	Dic₉₄	—	2	2	no	—
165	Dic₉₄	—	4	2	no	—
166	C₈×C₄₇	—	4	2	no	—
167	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
168	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
169	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
170	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
171	C₄₇⋊C₈	—	4	2	no	—
172	C₄₇⋊C₈	C₄×D₄	1	2	yes	—
173	C₄₇⋊C₈	C₂×D₈	1	2	yes	—
174	#33464	—	47	1	no	maximal, Sylow
175	C₂²×C₄₇	—	4	2	no	—
176	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
177	C₂×D₄₇	—	4	1	no	—
178	C₂×D₄₇	—	4	2	no	—
179	C₂×D₄₇	—	16	2	no	—
180	C₂×D₄₇	C₄×D₈	1	2	yes	—
181	C₄×C₄₇	—	4	1	no	—
182	C₄×C₄₇	—	4	2	no	—
183	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
185	Dic₄₇	—	4	1	no	—
186	Dic₄₇	—	4	2	no	—
187	Dic₄₇	C₄×D₈	1	2	yes	—
188	#3414	—	47	1	no	—
189	#2957	—	47	2	no	—
190	#2179	—	47	1	no	—
191	#2179	—	47	2	no	—
192	#2179	—	47	1	no	—
193	#2178	—	47	2	no	—
194	#2178	—	47	1	no	—
195	#2178	D₄₇	1	1	yes	characteristic, complemented
196	#2161	—	47	1	no	—
197	C₂×#359	—	47	2	no	—
198	#2108	—	47	1	no	—
199	C₂×C₄₇	—	2	1	no	—
200	C₂×C₄₇	—	8	2	no	—
201	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
202	D₄₇	—	2	2	no	—
203	D₄₇	—	8	2	no	—
204	C₂×#135	—	47	2	no	—
205	#568	—	94	2	no	—
206	#510	—	47	2	no	—
207	#510	—	47	2	no	—
208	#509	—	47	2	no	—
209	#509	—	47	2	no	—
210	#508	—	47	2	no	—
211	#508	—	47	1	no	—
212	#508	C₂×D₄₇	1	1	yes	characteristic
213	C₂×M₃₂	—	47	1	no	—
214	C₂²×C₁₆	—	47	1	no	—
215	#444	—	94	2	no	—
216	#444	—	94	2	no	—
217	#443	—	47	2	no	—
218	#443	—	47	2	no	—
219	#443	—	47	2	no	—
220	#443	C₂×D₄₇	1	2	yes	complemented
221	C₂×#108	—	47	2	no	—
222	#421	—	94	2	no	—
223	#405	—	47	1	no	—
224	#368	—	47	1	no	—
225	#368	—	47	2	no	—
226	#368	C₂×D₄₇	1	1	yes	characteristic, complemented
227	#359	—	47	2	no	—
228	#359	—	47	2	no	—
229	#359	—	47	2	no	—
230	#359	C₂×D₄₇	1	2	yes	complemented
231	#346	—	47	1	no	—
232	#346	—	47	1	no	—
233	C₄×C₁₆	—	47	1	no	—
234	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
235	C₄₇	#33464	1	1	yes	characteristic, complemented, Sylow
236	C₂×#41	—	94	2	no	—
237	#135	—	47	2	no	—
238	#135	—	47	2	no	—
239	#135	—	47	2	no	—
240	#135	—	94	2	no	—
241	#135	—	94	2	no	—
242	#135	C₂²×D₄₇	1	2	yes	—
243	C₂×Q₁₆	—	47	2	no	—
244	C₂×SD₁₆	—	94	2	no	—
245	C₂×D₈	—	47	2	no	—
246	#131	—	2	2	no	—
247	#131	—	94	2	no	—
248	#131	—	94	2	no	—
249	#131	—	94	2	no	—
250	#131	—	188	2	no	—
251	#131	—	188	2	no	—
252	C₂×M₁₆	—	47	1	no	—
253	C₂×M₁₆	—	94	2	no	—
254	C₂×M₁₆	—	188	2	no	—
255	C₂×M₁₆	—	188	2	no	—
256	C₂²×C₈	—	47	1	no	—
257	#117	—	47	1	no	—
258	Q₃₂	—	47	2	no	—
259	Q₃₂	—	47	1	no	—
260	Q₃₂	C₄×D₄₇	1	1	yes	characteristic
261	SD₃₂	—	47	2	no	—
262	SD₃₂	—	47	2	no	—
263	SD₃₂	—	47	2	no	—
264	SD₃₂	C₄×D₄₇	1	2	yes	—
265	D₁₆	—	47	2	no	—
266	D₁₆	—	47	1	no	—
267	D₁₆	C₄×D₄₇	1	1	yes	characteristic
268	M₃₂	—	47	2	no	—
269	M₃₂	—	47	2	no	—
270	C₂×C₁₆	—	47	1	no	—
271	C₂×C₁₆	—	47	1	no	—
272	C₂×C₁₆	—	94	2	no	—
273	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
274	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
275	#108	—	47	2	no	—
276	#108	—	94	2	no	—
277	#108	C₂²×D₄₇	1	2	yes	—
278	#104	—	2	2	no	—
279	#104	—	94	2	no	—
280	#104	—	94	2	no	—
281	#104	—	94	2	no	—
282	#97	—	47	1	no	—
283	#97	—	47	1	no	—
284	C₄×C₈	—	47	1	no	—
285	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
286	#41	—	2	2	no	—
287	#41	—	94	2	no	—
288	#41	—	94	2	no	—
289	#41	—	94	2	no	—
290	#41	—	188	2	no	—
291	#41	—	188	2	no	—
292	C₂×Q₈	—	94	2	no	—
293	C₂×D₄	—	94	2	no	—
294	C₂×D₄	—	188	2	no	—
295	C₂²×C₄	—	47	1	no	—
296	C₂²×C₄	—	188	2	no	—
297	Q₁₆	—	47	2	no	—
298	Q₁₆	—	94	2	no	—
299	Q₁₆	C₂×C₄×D₄₇	1	2	yes	—
300	SD₁₆	—	2	2	no	—
301	SD₁₆	—	94	2	no	—
302	SD₁₆	—	94	2	no	—
303	SD₁₆	—	94	2	no	—
304	D₈	—	47	2	no	—
305	D₈	—	94	2	no	—
306	D₈	C₂×C₄×D₄₇	1	2	yes	—
307	M₁₆	—	2	2	no	—
308	M₁₆	—	4	2	no	—
309	M₁₆	—	47	2	no	—
310	M₁₆	—	47	2	no	—
311	M₁₆	—	94	2	no	—
312	M₁₆	—	188	2	no	—
313	M₁₆	—	188	2	no	—
314	M₁₆	—	188	2	no	—
315	M₁₆	—	188	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	—	47	1	no	—
318	C₂×C₈	—	47	1	no	—
319	C₂×C₈	—	47	2	no	—
320	C₂×C₈	—	47	2	no	—
321	C₂×C₈	—	188	2	no	—
322	C₂×C₈	—	188	2	no	—
323	C₂×C₈	—	188	2	no	—
324	C₂×C₈	—	188	2	no	—
325	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
326	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
327	C₄⋊C₄	—	94	1	no	—
328	C₂²⋊C₄	—	94	1	no	—
329	C₄²	—	47	1	no	—
330	C₄²	D₄×D₄₇	1	1	yes	characteristic
331	C₁₆	—	2	1	no	—
332	C₁₆	—	47	2	no	—
333	C₁₆	—	94	1	no	—
334	C₁₆	C₂×C₄×D₄₇	1	1	yes	characteristic
335	C₁₆	C₂×C₄×D₄₇	1	1	yes	characteristic
336	C₂³	—	47	1	no	—
337	C₂³	—	188	2	no	—
338	Q₈	—	2	2	no	—
339	Q₈	—	94	2	no	—
340	Q₈	—	188	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	4	2	no	—
343	D₄	—	94	2	no	—
344	D₄	—	188	2	no	—
345	D₄	—	188	2	no	—
346	D₄	—	188	2	no	—
347	D₄	—	188	2	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	47	1	no	—
351	C₂×C₄	—	47	2	no	—
352	C₂×C₄	—	47	2	no	—
353	C₂×C₄	—	94	1	no	—
354	C₂×C₄	—	188	2	no	—
355	C₂×C₄	—	188	2	no	—
356	C₂×C₄	—	188	2	no	—
357	C₂×C₄	—	188	2	no	—
358	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	47	2	no	—
361	C₈	—	47	2	no	—
362	C₈	—	188	2	no	—
363	C₈	D₈×D₄₇	1	1	yes	characteristic
364	C₈	D₈×D₄₇	1	1	yes	characteristic
365	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
366	C₈	#41×D₄₇	1	1	yes	characteristic
367	C₂²	—	4	2	no	—
368	C₂²	—	47	2	no	—
369	C₂²	—	188	1	no	—
370	C₂²	—	188	2	no	—
371	C₂²	—	752	2	no	—
372	C₂²	#135×D₄₇	1	1	yes	characteristic
373	C₄	—	4	1	no	—
374	C₄	—	4	2	no	—
375	C₄	—	47	2	no	—
376	C₄	—	188	1	no	—
377	C₄	—	188	2	no	—
378	C₄	C₄×D₄×D₄₇	1	1	yes	characteristic
379	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
380	C₂	—	2	1	no	—
381	C₂	—	8	2	no	—
382	C₂	—	94	2	no	—
383	C₂	—	376	2	no	—
384	C₂	C₄×D₈×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2178	C₂
socle	C₂×C₄₇	C₂×C₄×D₈
prime radical	#20256521	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2178	D₄₇
hypocenter	C₄₇	#33464
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
2. #20252300	C₂	semidirect
3. #20245274	C₂	semidirect
4. #20245273	C₂	semidirect
5. #20245272	C₂	semidirect
6. #20245270	C₂	semidirect
7. #20245266	C₂	semidirect
8. #20245181	C₂	semidirect
9. #359×D₄₇	C₂	semidirect
10. #20244983	C₂	semidirect
11. C₄₇×#2178	C₂	semidirect
14. #15689565	C₄	semidirect
15. #15689562	C₄	semidirect
16. #15689560	C₄	semidirect
17. #15689557	C₄	semidirect
24. #15689241	C₂²	semidirect
25. #15689240	C₂²	semidirect
26. #15689239	C₂²	semidirect
30. #15688953	C₂²	semidirect
31. #15688952	C₂²	semidirect
32. #15688931	C₂²	semidirect
33. #15688930	C₂²	semidirect
34. #15688929	C₂²	semidirect
35. #15688899	C₂²	semidirect
36. #15688898	C₂²	semidirect
37. C₁₆×Dic₄₇	C₂²	semidirect
39. C₄₇×#443	C₂²	semidirect
40. C₄₇×#368	C₂²	semidirect
41. C₄₇×#359	C₂²	semidirect
42. C₄×C₁₆×C₄₇	C₂²	semidirect
195. #2178	D₄₇	semidirect
220. #443	C₂×D₄₇	semidirect
226. #368	C₂×D₄₇	semidirect
230. #359	C₂×D₄₇	semidirect
234. C₄×C₁₆	C₂×D₄₇	semidirect
235. C₄₇	#33464	semidirect

Maximal subgroups

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

subgroup	index
#20256521	2
#20252300	2
#20245274	2
#20245273	2
#20245272	2
#20245270	2
#20245266	2
#20245181	2
#359×D₄₇	2
#20244983	2
C₄₇×#2178	2
#33464	47