Subgroups — #26370930

Order 12032.

598 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	#20253900	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245547	C₂	1	2	yes	maximal, complemented
4	#20245543	C₂	1	2	yes	maximal, complemented
5	#20245533	C₂	1	2	yes	maximal, complemented
6	#20245523	C₂	1	2	yes	maximal, complemented
7	#20245177	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245176	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245143	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245104	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#2140	C₂	1	1	yes	maximal, characteristic
12	#15689676	—	2	1	no	—
13	C₂×D₄×Dic₄₇	—	2	1	no	—
14	C₂×#13093964	C₂²	1	1	yes	characteristic
15	#15689527	C₂²	1	1	yes	characteristic, complemented
16	#15689527	C₂²	1	1	yes	characteristic, complemented
17	#15689524	C₂²	1	1	yes	characteristic, complemented
18	#15689524	C₂²	1	1	yes	characteristic, complemented
19	#15689523	—	2	1	no	—
20	#15689520	—	2	1	no	—
21	D₈×Dic₄₇	—	2	1	no	—
22	D₈×Dic₄₇	—	2	1	no	—
23	D₈×Dic₄₇	C₂²	1	2	yes	—
24	D₈×Dic₄₇	C₂²	1	2	yes	—
25	#15689161	C₂²	1	1	yes	characteristic
26	#15689159	—	2	1	no	—
27	C₂×#13093958	C₂²	1	1	yes	characteristic
28	C₂×#13093945	—	2	1	no	—
29	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
30	#15688951	C₂²	1	2	yes	complemented
31	#15688950	C₂²	1	2	yes	complemented
32	#15688949	C₂²	1	2	yes	complemented
33	#15688948	C₂²	1	2	yes	complemented
34	Dic₄₇⋊C₁₆	C₂²	1	2	yes	complemented
35	Dic₄₇⋊C₁₆	C₂²	1	2	yes	complemented
36	#15688924	C₂²	1	2	yes	—
37	#15688924	C₂²	1	2	yes	—
38	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#427	C₂²	1	1	yes	characteristic
41	C₄₇×#357	C₂²	1	2	yes	—
42	C₄₇×#357	C₂²	1	2	yes	—
43	C₂³×Dic₄₇	—	4	1	no	—
44	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
45	#13094023	—	2	1	no	—
46	#13094023	—	2	1	no	—
47	#13094015	—	2	1	no	—
48	#13094015	—	2	1	no	—
49	#13094001	—	4	1	no	—
50	#13094001	—	4	1	no	—
51	C₄×Dic₉₄	—	4	1	no	—
52	C₄×Dic₉₄	—	4	1	no	—
53	D₄×Dic₄₇	—	2	2	no	—
54	D₄×Dic₄₇	—	2	2	no	—
55	D₄×Dic₄₇	—	4	1	no	—
56	D₄×Dic₄₇	—	4	1	no	—
57	#13093983	—	2	1	no	—
58	C₂×C₄⋊Dic₄₇	—	2	1	no	—
59	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
60	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
61	#13093964	—	2	2	no	—
62	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
63	#13093958	—	2	1	no	—
64	#13093958	C₂³	1	1	yes	characteristic
65	#13093958	C₂³	1	1	yes	characteristic
66	#13093958	C₂³	1	2	yes	—
67	#13093957	—	2	1	no	—
68	#13093957	—	2	1	no	—
69	#13093956	—	2	1	no	—
70	#13093955	—	2	1	no	—
71	#13093955	—	2	1	no	—
72	#13093945	—	2	2	no	—
73	#13093945	—	2	2	no	—
74	C₈×Dic₄₇	D₄	1	1	yes	characteristic
75	C₈×Dic₄₇	D₄	1	1	yes	characteristic
76	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
77	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
78	C₂²×D₄×C₄₇	—	2	1	no	—
79	C₂×D₈×C₄₇	—	2	1	no	—
80	C₂×D₈×C₄₇	—	2	2	no	—
81	C₂×D₈×C₄₇	Q₈	1	4	yes	complemented
82	C₂×D₈×C₄₇	C₂³	1	2	yes	—
83	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
84	#117×C₄₇	—	2	1	no	—
85	M₃₂×C₄₇	—	2	2	no	—
86	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
87	#107×C₄₇	C₂³	1	1	yes	characteristic
88	#107×C₄₇	C₂³	1	1	yes	characteristic
89	#106×C₄₇	—	2	1	no	—
90	C₂⁴×C₄₇	—	4	1	no	—
91	C₂×D₄×C₄₇	—	2	4	no	—
92	C₂×D₄×C₄₇	—	2	2	no	—
93	C₂×D₄×C₄₇	—	4	1	no	—
94	C₂×D₄×C₄₇	—	4	2	no	—
95	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×Dic₄₇	—	4	1	no	—
97	C₂²×Dic₄₇	—	4	2	no	—
98	C₂²×Dic₄₇	—	4	2	no	—
99	C₂²×Dic₄₇	—	8	4	no	—
100	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
101	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
102	C₂×Dic₉₄	—	4	1	no	—
103	C₂×Dic₉₄	—	4	1	no	—
104	D₈×C₄₇	—	2	4	no	—
105	D₈×C₄₇	—	2	2	no	—
106	D₈×C₄₇	C₂×Q₈	1	4	yes	—
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₄₇	#41	1	2	yes	—
110	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₄₇	—	2	1	no	—
112	C₄⋊C₄×C₄₇	—	2	1	no	—
113	C₂²⋊C₄×C₄₇	—	4	1	no	—
114	C₄²×C₄₇	—	4	1	no	—
115	C₂²⋊Dic₄₇	—	4	1	no	—
116	C₂²⋊Dic₄₇	—	4	2	no	—
117	C₂²⋊Dic₄₇	—	4	2	no	—
118	C₄⋊Dic₄₇	—	2	2	no	—
119	C₄⋊Dic₄₇	—	2	1	no	—
120	C₄⋊Dic₄₇	—	2	1	no	—
121	C₄⋊Dic₄₇	—	4	1	no	—
122	C₄⋊Dic₄₇	—	4	1	no	—
123	C₄⋊Dic₄₇	—	4	1	no	—
124	Dic₄₇⋊C₄	—	4	1	no	—
125	Dic₄₇⋊C₄	—	4	1	no	—
126	Dic₄₇⋊C₄	—	4	1	no	—
127	Dic₄₇⋊C₄	—	4	1	no	—
128	C₄×Dic₄₇	—	4	1	no	—
129	C₄×Dic₄₇	D₈	1	1	yes	characteristic
130	C₄×Dic₄₇	D₈	1	1	yes	characteristic
131	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
132	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄₇⋊C₈	—	2	2	no	—
134	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
136	C₁₆×C₄₇	—	2	2	no	—
137	C₄₇⋊C₁₆	—	2	2	no	—
138	C₂³×C₄₇	—	4	2	no	—
139	C₂³×C₄₇	—	4	4	no	—
140	C₂³×C₄₇	—	8	2	no	—
141	C₂³×C₄₇	—	8	2	no	—
142	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
143	D₄×C₄₇	—	2	4	no	—
144	D₄×C₄₇	—	4	2	no	—
145	D₄×C₄₇	—	4	4	no	—
146	C₂×C₄×C₄₇	—	4	1	no	—
147	C₂×C₄×C₄₇	—	4	1	no	—
148	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
149	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₄₇	#137	1	2	yes	—
151	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₄₇	—	2	2	no	—
153	C₂×Dic₄₇	—	4	1	no	—
154	C₂×Dic₄₇	—	4	1	no	—
155	C₂×Dic₄₇	—	4	1	no	—
156	C₂×Dic₄₇	—	4	2	no	—
157	C₂×Dic₄₇	—	8	4	no	—
158	C₂×Dic₄₇	—	8	2	no	—
159	C₂×Dic₄₇	—	8	4	no	—
160	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
161	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
162	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
163	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
164	Dic₉₄	—	8	2	no	—
165	Dic₉₄	—	8	2	no	—
166	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
167	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
168	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
169	C₄₇⋊C₈	—	2	1	no	—
170	C₄₇⋊C₈	—	2	1	no	—
171	#33627	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	—	2	1	no	—
173	C₂²×C₄₇	—	2	1	no	—
174	C₂²×C₄₇	—	4	4	no	—
175	C₂²×C₄₇	—	8	8	no	—
176	C₂²×C₄₇	—	8	2	no	—
177	C₂²×C₄₇	—	8	2	no	—
178	C₂²×C₄₇	#474	1	2	yes	—
179	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
180	C₄×C₄₇	—	8	1	no	—
181	C₄×C₄₇	—	8	1	no	—
182	C₄×C₄₇	#474	1	2	yes	—
183	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
185	Dic₄₇	—	2	1	no	—
186	Dic₄₇	—	2	1	no	—
187	Dic₄₇	—	2	1	no	—
188	Dic₄₇	—	8	1	no	—
189	Dic₄₇	—	8	2	no	—
190	Dic₄₇	—	8	1	no	—
191	Dic₄₇	#509	1	2	yes	—
192	#3160	—	47	1	no	—
193	C₂×C₄×D₈	—	47	1	no	—
194	#2228	—	47	2	no	—
195	#2228	—	47	2	no	—
196	#2226	—	47	2	no	—
197	#2224	—	47	2	no	—
198	#2159	—	47	1	no	—
199	#2159	—	47	1	no	—
200	#2150	—	47	1	no	—
201	#2140	—	47	1	no	—
202	#2140	D₄₇	1	1	yes	characteristic
203	C₂×C₄₇	—	2	2	no	—
204	C₂×C₄₇	—	8	4	no	—
205	C₂×C₄₇	C₂×#474	1	1	yes	characteristic
206	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
207	C₂×C₄₇	#3414	1	1	yes	characteristic
208	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
209	#533	—	94	1	no	—
210	C₂×C₄×D₄	—	94	1	no	—
211	C₂×M₃₂	—	47	1	no	—
212	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
213	#500	—	47	1	no	—
214	#500	—	47	1	no	—
215	#499	—	47	1	no	—
216	#499	—	47	1	no	—
217	#499	—	94	1	no	—
218	#498	—	94	1	no	—
219	C₄×D₈	—	47	2	no	—
220	C₄×D₈	—	47	2	no	—
221	C₄×D₈	—	94	1	no	—
222	C₄×D₈	—	94	1	no	—
223	#427	—	47	1	no	—
224	#427	—	94	1	no	—
225	#427	C₂×D₄₇	1	1	yes	characteristic
226	C₂×#107	—	47	1	no	—
227	C₂×#102	—	94	1	no	—
228	C₂×C₄×C₈	—	47	1	no	—
229	#367	—	47	2	no	—
230	#367	—	47	2	no	—
231	#366	—	47	2	no	—
232	#366	—	47	2	no	—
233	C₄⋊C₁₆	—	47	2	no	—
234	C₄⋊C₁₆	—	47	2	no	—
235	#357	—	47	2	no	—
236	#357	—	47	2	no	—
237	#357	C₂×D₄₇	1	2	yes	—
238	#357	C₂×D₄₇	1	2	yes	—
239	C₄₇	#33627	1	1	yes	characteristic, complemented, Sylow
240	C₂²×D₄	—	2	1	no	—
241	C₂³×C₄	—	188	1	no	—
242	C₂×D₈	—	2	1	no	—
243	C₂×D₈	—	2	2	no	—
244	C₂×D₈	Dic₉₄	1	4	yes	complemented
245	C₂×D₈	C₂²×D₄₇	1	2	yes	—
246	C₂²×C₈	—	47	1	no	—
247	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
248	C₄⋊Q₈	—	94	1	no	—
249	C₄⋊Q₈	—	94	1	no	—
250	#125	—	94	1	no	—
251	#125	—	94	1	no	—
252	C₂²⋊Q₈	—	188	1	no	—
253	C₂²⋊Q₈	—	188	1	no	—
254	C₄×Q₈	—	188	1	no	—
255	C₄×Q₈	—	188	1	no	—
256	C₄×D₄	—	94	2	no	—
257	C₄×D₄	—	94	2	no	—
258	C₄×D₄	—	188	1	no	—
259	C₄×D₄	—	188	1	no	—
260	#117	—	2	1	no	—
261	#117	—	94	1	no	—
262	C₂×C₄⋊C₄	—	94	1	no	—
263	C₂×C₂²⋊C₄	—	188	1	no	—
264	C₂×C₄²	—	47	1	no	—
265	M₃₂	—	2	2	no	—
266	M₃₂	—	94	2	no	—
267	C₂×C₁₆	—	47	2	no	—
268	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
269	#107	—	47	1	no	—
270	#107	—	47	1	no	—
271	#107	—	47	2	no	—
272	#107	—	94	1	no	—
273	#107	—	94	1	no	—
274	#107	—	94	1	no	—
275	#107	C₂²×D₄₇	1	1	yes	characteristic
276	#107	C₂²×D₄₇	1	1	yes	characteristic
277	#106	—	2	1	no	—
278	#106	—	94	1	no	—
279	#106	—	94	1	no	—
280	#106	—	94	1	no	—
281	#102	—	94	2	no	—
282	#102	—	94	2	no	—
283	C₄×C₈	—	47	1	no	—
284	C₄×C₈	—	47	1	no	—
285	C₄×C₈	—	47	1	no	—
286	C₄×C₈	—	47	1	no	—
287	C₂⁴	—	4	1	no	—
288	C₂×Q₈	—	188	1	no	—
289	C₂×Q₈	—	188	1	no	—
290	C₂×D₄	—	2	4	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	4	1	no	—
293	C₂×D₄	—	4	2	no	—
294	C₂²×C₄	—	47	1	no	—
295	C₂²×C₄	—	47	1	no	—
296	C₂²×C₄	—	376	4	no	—
297	C₂²×C₄	—	188	2	no	—
298	C₂²×C₄	—	188	2	no	—
299	C₂²×C₄	—	188	1	no	—
300	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
301	D₈	—	2	4	no	—
302	D₈	—	2	2	no	—
303	D₈	C₂×Dic₉₄	1	4	yes	—
304	C₂×C₈	—	47	1	no	—
305	C₂×C₈	—	47	1	no	—
306	C₂×C₈	—	94	2	no	—
307	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
308	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
309	C₂×C₈	#10625352	1	2	yes	—
310	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	94	2	no	—
314	C₄⋊C₄	—	94	1	no	—
315	C₄⋊C₄	—	94	1	no	—
316	C₄⋊C₄	—	188	1	no	—
317	C₄⋊C₄	—	188	1	no	—
318	C₄⋊C₄	—	188	1	no	—
319	C₄⋊C₄	—	188	1	no	—
320	C₄⋊C₄	—	188	1	no	—
321	C₄⋊C₄	—	188	1	no	—
322	C₄⋊C₄	—	188	1	no	—
323	C₂²⋊C₄	—	4	1	no	—
324	C₂²⋊C₄	—	188	2	no	—
325	C₂²⋊C₄	—	188	2	no	—
326	C₂²⋊C₄	—	188	1	no	—
327	C₄²	—	4	1	no	—
328	C₄²	—	47	1	no	—
329	C₄²	—	47	1	no	—
330	C₄²	—	47	1	no	—
331	C₄²	—	47	1	no	—
332	C₄²	—	188	1	no	—
333	C₁₆	—	2	2	no	—
334	C₁₆	—	94	2	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	4	no	—
337	C₂³	—	8	2	no	—
338	C₂³	—	8	2	no	—
339	C₂³	D₈×D₄₇	1	1	yes	characteristic
340	Q₈	—	376	2	no	—
341	Q₈	—	376	2	no	—
342	D₄	—	2	4	no	—
343	D₄	—	4	4	no	—
344	D₄	—	4	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	47	1	no	—
348	C₂×C₄	—	47	1	no	—
349	C₂×C₄	—	47	1	no	—
350	C₂×C₄	—	47	1	no	—
351	C₂×C₄	—	94	2	no	—
352	C₂×C₄	—	376	4	no	—
353	C₂×C₄	—	376	4	no	—
354	C₂×C₄	—	376	2	no	—
355	C₂×C₄	—	188	1	no	—
356	C₂×C₄	—	188	1	no	—
357	C₂×C₄	—	188	1	no	—
358	C₂×C₄	—	188	2	no	—
359	C₂×C₄	#13094001	1	2	yes	—
360	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
361	C₂×C₄	#13094053	1	2	yes	—
362	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
363	C₈	—	94	1	no	—
364	C₈	—	94	1	no	—
365	C₈	#13094001	1	2	yes	—
366	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
367	C₈	C₂×#10625352	1	1	yes	characteristic
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	4	4	no	—
371	C₂²	—	8	8	no	—
372	C₂²	—	8	2	no	—
373	C₂²	—	8	2	no	—
374	C₂²	#15689401	1	2	yes	—
375	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	—	47	2	no	—
379	C₄	—	94	1	no	—
380	C₄	—	94	1	no	—
381	C₄	—	94	1	no	—
382	C₄	—	376	1	no	—
383	C₄	—	376	2	no	—
384	C₄	—	376	1	no	—
385	C₄	#15689401	1	2	yes	—
386	C₄	C₂×#13094001	1	1	yes	characteristic
387	C₄	C₂×#13094053	1	1	yes	characteristic
388	C₂	—	2	2	no	—
389	C₂	—	8	4	no	—
390	C₂	C₂×#15689401	1	1	yes	characteristic
391	C₂	#509×D₄₇	1	1	yes	characteristic
392	C₂	#20256520	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2140	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₈×C₄₇	C₂²
hypercenter	#2140	D₄₇
hypocenter	C₄₇	#33627
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2140

Complements

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

N	H ≅ G/N	type
1. #20253900	C₂	semidirect
3. #20245547	C₂	semidirect
4. #20245543	C₂	semidirect
5. #20245533	C₂	semidirect
6. #20245523	C₂	semidirect
7. #20245177	C₂	semidirect
8. #20245176	C₂	semidirect
9. #20245143	C₂	semidirect
15. #15689527	C₂²	semidirect
16. #15689527	C₂²	semidirect
17. #15689524	C₂²	semidirect
18. #15689524	C₂²	semidirect
30. #15688951	C₂²	semidirect
31. #15688950	C₂²	semidirect
32. #15688949	C₂²	semidirect
33. #15688948	C₂²	semidirect
34. Dic₄₇⋊C₁₆	C₂²	semidirect
35. Dic₄₇⋊C₁₆	C₂²	semidirect
81. C₂×D₈×C₄₇	Q₈	semidirect
239. C₄₇	#33627	semidirect
244. C₂×D₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20253900	2
C₂×D₈×Dic₄₇	2
#20245547	2
#20245543	2
#20245533	2
#20245523	2
#20245177	2
#20245176	2
#20245143	2
#20245104	2
C₄₇×#2140	2
#33627	47