Subgroups — #26207860

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256378	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₃₇₆	C₂	1	1	yes	maximal, characteristic
3	#20255668	C₂	1	2	yes	maximal, complemented
4	#20242982	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242938	C₂	1	2	yes	maximal, complemented
6	#20242783	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242659	C₂	1	2	yes	maximal, complemented
8	#20242650	C₂	1	2	yes	maximal, complemented
9	#20242272	C₂	1	1	yes	maximal, characteristic
10	C₈×#10625337	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1670	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689847	C₂²	1	2	yes	—
13	Q₈×D₁₈₈	C₂²	1	2	yes	—
14	#15689528	—	2	1	no	—
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689519	C₂²	1	2	yes	—
17	#15689508	—	2	1	no	—
18	#15689475	C₂²	1	2	yes	—
19	#15689467	—	2	2	no	—
20	#15689404	—	2	2	no	—
21	#15689404	C₂²	1	2	yes	—
22	#15689382	—	2	2	no	—
23	#15689347	—	2	2	no	—
24	#15689334	C₂²	1	2	yes	—
25	#15689325	C₂²	1	2	yes	—
26	#15689280	—	2	2	no	—
27	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
28	Q₁₆×Dic₄₇	—	2	1	no	—
29	C₄×#10625337	C₂²	1	1	yes	characteristic
30	C₄×D₃₇₆	—	2	1	no	—
31	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
32	#15689200	C₂²	1	2	yes	—
33	#15689130	—	2	2	no	—
34	#15688875	C₂²	1	1	yes	characteristic
35	#15688870	C₂²	1	2	yes	—
36	Q₈⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
37	#15688853	C₂²	1	1	yes	characteristic
38	#15688850	C₂²	1	2	yes	—
39	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₄₇×#494	C₂²	1	1	yes	characteristic
41	C₄₇×#475	C₂²	1	2	yes	—
42	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
43	C₄₇×#335	C₂²	1	1	yes	characteristic
44	C₄₇×#332	C₂²	1	2	yes	—
45	C₂×Q₈×D₄₇	—	2	2	no	—
46	C₂×Dic₁₈₈	—	2	2	no	—
47	C₂×#10625337	D₄	1	1	yes	characteristic
48	C₂×#10625336	—	2	2	no	—
49	C₂×D₃₇₆	D₄	1	1	yes	characteristic
50	C₂×#10625324	—	2	2	no	—
51	#13094024	—	2	2	no	—
52	#13094024	C₂³	1	2	yes	—
53	#13094019	—	4	2	no	—
54	#13094013	—	2	2	no	—
55	#13094004	—	2	2	no	—
56	#13094004	—	2	2	no	—
57	#13094004	—	4	2	no	—
58	Q₈×Dic₄₇	—	2	2	no	—
59	C₄×Dic₉₄	C₂³	1	1	yes	characteristic
60	C₄×D₁₈₈	—	2	2	no	—
61	C₄×D₁₈₈	C₂³	1	2	yes	—
62	#13093982	—	2	2	no	—
63	#13093958	—	2	1	no	—
64	#13093958	—	2	1	no	—
65	#13093958	—	2	2	no	—
66	#13093958	D₄	1	1	yes	characteristic
67	#13093957	D₄	1	1	yes	characteristic
68	#13093955	—	4	2	no	—
69	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
70	Dic₄₇⋊C₈	—	2	1	no	—
71	#13093949	—	2	2	no	—
72	#13093948	—	2	2	no	—
73	#13093948	D₄	1	1	yes	characteristic
74	#13093947	—	2	2	no	—
75	#13093947	—	2	2	no	—
76	#13093947	D₄	1	1	yes	characteristic
77	#13093946	—	2	2	no	—
78	#13093946	D₄	1	2	yes	—
79	(C₂×D₄₇)⋊C₈	—	2	2	no	—
80	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₄₇	—	2	1	no	—
83	C₂×Q₁₆×C₄₇	—	2	1	no	—
84	C₂×Q₁₆×C₄₇	—	2	1	no	—
85	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
86	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
87	#106×C₄₇	—	2	2	no	—
88	C₄⋊C₈×C₄₇	—	2	1	no	—
89	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
90	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
91	#103×C₄₇	—	2	2	no	—
92	C₄×C₈×C₄₇	—	2	1	no	—
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₇	—	2	2	no	—
95	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	Q₈×D₄₇	—	4	4	no	—
97	Q₈×D₄₇	—	8	2	no	—
98	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
99	C₂×C₄×D₄₇	—	2	2	no	—
100	C₂×C₄×D₄₇	—	4	2	no	—
101	C₂×Dic₉₄	—	2	2	no	—
102	C₂×Dic₉₄	—	4	2	no	—
103	C₂×Dic₉₄	C₂×D₄	1	1	yes	characteristic
104	Q₁₆×C₄₇	—	4	2	no	—
105	Q₁₆×C₄₇	—	8	1	no	—
106	C₂×C₈×C₄₇	—	2	2	no	—
107	C₂×C₈×C₄₇	—	2	1	no	—
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₄⋊C₄×C₄₇	—	2	1	no	—
111	C₄⋊C₄×C₄₇	—	2	2	no	—
112	C₄⋊C₄×C₄₇	—	2	2	no	—
113	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₇	—	2	1	no	—
115	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
116	#10625337	—	2	2	no	—
117	#10625336	—	8	2	no	—
118	(C₂×D₄₇)⋊C₄	—	2	2	no	—
119	(C₂×D₄₇)⋊C₄	—	4	2	no	—
120	(C₂×D₄₇)⋊C₄	—	4	2	no	—
121	(C₂×D₄₇)⋊C₄	—	4	2	no	—
122	C₄⋊Dic₄₇	—	2	2	no	—
123	C₄⋊Dic₄₇	—	2	2	no	—
124	C₄⋊Dic₄₇	—	4	2	no	—
125	C₄⋊Dic₄₇	—	4	2	no	—
126	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
127	C₄⋊Dic₄₇	C₂×D₄	1	1	yes	characteristic
128	Dic₄₇⋊C₄	—	2	1	no	—
129	C₄×Dic₄₇	—	2	1	no	—
130	C₄×Dic₄₇	—	4	2	no	—
131	C₂×C₄₇⋊C₈	—	2	1	no	—
132	C₂×C₄₇⋊C₈	—	2	1	no	—
133	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
135	Dic₁₈₈	—	8	2	no	—
136	D₃₇₆	—	2	2	no	—
137	#10625324	—	8	2	no	—
138	C₂²×D₄₇	—	2	2	no	—
139	Q₈×C₄₇	—	2	1	no	—
140	Q₈×C₄₇	—	4	2	no	—
141	Q₈×C₄₇	—	4	2	no	—
142	Q₈×C₄₇	#135	1	2	yes	—
143	C₂×C₄×C₄₇	—	2	1	no	—
144	C₂×C₄×C₄₇	—	2	1	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×Dic₄₇	—	2	1	no	—
150	C₂×Dic₄₇	—	2	1	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	—	4	2	no	—
153	D₁₈₈	—	2	2	no	—
154	D₁₈₈	—	2	2	no	—
155	C₄×D₄₇	—	4	2	no	—
156	C₄×D₄₇	—	4	2	no	—
157	C₄×D₄₇	—	8	4	no	—
158	Dic₉₄	—	2	1	no	—
159	Dic₉₄	—	2	1	no	—
160	Dic₉₄	—	4	2	no	—
161	Dic₉₄	—	4	2	no	—
162	Dic₉₄	—	8	4	no	—
163	C₈×C₄₇	—	2	1	no	—
164	C₈×C₄₇	—	4	2	no	—
165	C₈×C₄₇	—	4	1	no	—
166	C₈×C₄₇	#137	1	2	yes	—
167	C₄₇⋊C₈	—	2	2	no	—
168	C₄₇⋊C₈	—	2	1	no	—
169	C₄₇⋊C₈	—	2	1	no	—
170	C₄₇⋊C₈	—	4	1	no	—
171	#19907	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	D₄²	1	1	yes	characteristic
173	C₂×D₄₇	—	4	2	no	—
174	C₂×D₄₇	—	4	2	no	—
175	C₂×D₄₇	—	4	2	no	—
176	C₄×C₄₇	—	2	2	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	4	1	no	—
179	C₄×C₄₇	—	8	2	no	—
180	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
181	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
182	C₄×C₄₇	#576	1	1	yes	characteristic
183	C₄×C₄₇	#577	1	1	yes	characteristic
184	Dic₄₇	—	4	1	no	—
185	Dic₄₇	—	4	2	no	—
186	Dic₄₇	—	4	1	no	—
187	Dic₄₇	—	8	2	no	—
188	#3393	—	47	1	no	—
189	Q₈×D₈	—	47	1	no	—
190	#3328	—	47	2	no	—
191	#1707	—	47	1	no	—
192	#1699	—	47	2	no	—
193	#1674	—	47	1	no	—
194	#1670	D₄₇	1	1	yes	characteristic, complemented
195	#1652	—	47	2	no	—
196	#1651	—	47	2	no	—
197	#1589	—	47	1	no	—
198	C₈×Q₁₆	—	47	1	no	—
199	C₂×C₄₇	#3278	1	1	yes	characteristic
200	C₂×C₄₇	#3280	1	1	yes	characteristic
201	C₂×C₄₇	#3293	1	1	yes	characteristic
202	D₄₇	—	8	2	no	—
203	#556	—	47	2	no	—
204	D₄×Q₈	—	47	2	no	—
205	#500	—	47	1	no	—
206	#500	—	94	1	no	—
207	#498	—	47	2	no	—
208	#495	—	94	1	no	—
209	#494	C₂×D₄₇	1	1	yes	characteristic
210	#487	—	47	2	no	—
211	#485	—	94	2	no	—
212	#475	C₂×D₄₇	1	2	yes	—
213	#474	—	47	2	no	—
214	#474	—	94	2	no	—
215	#470	—	94	2	no	—
216	#464	—	94	2	no	—
217	#462	—	47	2	no	—
218	#461	—	47	2	no	—
219	#451	—	94	2	no	—
220	C₈×Q₈	—	47	1	no	—
221	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
222	C₄×Q₁₆	—	47	1	no	—
223	C₄×Q₁₆	—	94	1	no	—
224	C₄×D₈	—	47	1	no	—
225	C₄×D₈	—	94	1	no	—
226	#436	—	47	2	no	—
227	#419	—	94	2	no	—
228	#335	—	47	1	no	—
229	#335	C₂×D₄₇	1	1	yes	characteristic
230	#332	—	47	2	no	—
231	#332	C₂×D₄₇	1	2	yes	—
232	Q₈⋊C₈	—	47	1	no	—
233	Q₈⋊C₈	—	47	1	no	—
234	D₄⋊C₈	—	47	2	no	—
235	C₈²	—	47	1	no	—
236	C₄₇	#19907	1	1	yes	characteristic, complemented, Sylow
237	C₂²×Q₈	—	94	2	no	—
238	C₂×Q₁₆	—	2	1	no	—
239	C₂×Q₁₆	—	2	1	no	—
240	C₂×Q₁₆	—	47	1	no	—
241	C₂×Q₁₆	—	94	2	no	—
242	C₂×SD₁₆	—	94	2	no	—
243	C₂×D₈	—	47	1	no	—
244	C₂×M₁₆	—	94	2	no	—
245	C₄⋊Q₈	—	47	2	no	—
246	C₄⋊Q₈	—	94	2	no	—
247	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
248	#126	—	188	2	no	—
249	#124	—	94	2	no	—
250	C₂²⋊Q₈	—	94	2	no	—
251	C₂²⋊Q₈	—	94	2	no	—
252	C₂²⋊Q₈	—	188	2	no	—
253	C₄×Q₈	—	47	1	no	—
254	C₄×Q₈	—	94	2	no	—
255	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
256	C₄×D₄	—	47	2	no	—
257	C₄×D₄	—	94	2	no	—
258	#117	—	94	2	no	—
259	#107	—	47	1	no	—
260	#107	—	47	1	no	—
261	#107	—	94	2	no	—
262	#107	—	94	1	no	—
263	#107	—	94	1	no	—
264	#106	—	2	2	no	—
265	#106	—	188	2	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	47	1	no	—
268	C₄⋊C₈	—	94	1	no	—
269	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
270	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
271	#103	—	2	2	no	—
272	#103	—	47	1	no	—
273	#103	—	47	1	no	—
274	#103	—	94	2	no	—
275	#103	—	94	2	no	—
276	#103	—	94	2	no	—
277	#103	—	94	2	no	—
278	#102	—	47	2	no	—
279	#102	—	94	2	no	—
280	C₂²⋊C₈	—	94	2	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	47	1	no	—
283	C₄×C₈	—	47	1	no	—
284	C₄×C₈	—	94	1	no	—
285	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	47	1	no	—
288	C₂×Q₈	—	94	2	no	—
289	C₂×Q₈	—	376	2	no	—
290	C₂×Q₈	—	188	2	no	—
291	C₂×Q₈	—	188	4	no	—
292	C₂×Q₈	D₄×D₄₇	1	1	yes	characteristic
293	C₂×D₄	—	47	2	no	—
294	C₂²×C₄	—	94	2	no	—
295	C₂²×C₄	—	188	2	no	—
296	Q₁₆	—	4	2	no	—
297	Q₁₆	—	8	1	no	—
298	Q₁₆	—	94	2	no	—
299	Q₁₆	—	376	2	no	—
300	SD₁₆	—	376	2	no	—
301	D₈	—	94	2	no	—
302	M₁₆	—	376	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	47	1	no	—
306	C₂×C₈	—	47	1	no	—
307	C₂×C₈	—	94	1	no	—
308	C₂×C₈	—	94	1	no	—
309	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
310	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	47	2	no	—
315	C₄⋊C₄	—	47	1	no	—
316	C₄⋊C₄	—	94	1	no	—
317	C₄⋊C₄	—	94	2	no	—
318	C₄⋊C₄	—	94	2	no	—
319	C₄⋊C₄	—	188	2	no	—
320	C₄⋊C₄	—	188	2	no	—
321	C₄⋊C₄	D₄×D₄₇	1	1	yes	characteristic
322	C₂²⋊C₄	—	94	2	no	—
323	C₂²⋊C₄	—	188	2	no	—
324	C₂²⋊C₄	—	188	2	no	—
325	C₂²⋊C₄	—	188	2	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	94	1	no	—
328	C₄²	—	188	2	no	—
329	C₄²	C₂³×D₄₇	1	1	yes	characteristic
330	C₂³	—	94	2	no	—
331	Q₈	—	2	1	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	94	1	no	—
335	Q₈	—	94	1	no	—
336	Q₈	—	376	4	no	—
337	Q₈	—	188	2	no	—
338	Q₈	—	188	2	no	—
339	Q₈	#13094047	1	2	yes	—
340	D₄	—	94	2	no	—
341	D₄	—	94	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	94	1	no	—
346	C₂×C₄	—	94	1	no	—
347	C₂×C₄	—	94	2	no	—
348	C₂×C₄	—	376	4	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	—	188	2	no	—
351	C₂×C₄	—	188	2	no	—
352	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
354	C₂×C₄	#13094070	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	94	2	no	—
359	C₈	—	94	1	no	—
360	C₈	—	94	1	no	—
361	C₈	—	188	1	no	—
362	C₈	#13094053	1	2	yes	—
363	C₂²	—	188	2	no	—
364	C₂²	—	188	2	no	—
365	C₂²	—	188	2	no	—
366	C₂²	#15689757	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	8	2	no	—
371	C₄	—	376	2	no	—
372	C₄	—	188	2	no	—
373	C₄	—	188	1	no	—
374	C₄	—	188	1	no	—
375	C₄	C₂×#13094047	1	1	yes	characteristic
376	C₄	C₂×#13094053	1	1	yes	characteristic
377	C₄	#15689925	1	1	yes	characteristic
378	C₄	#15689933	1	1	yes	characteristic
379	C₂	—	376	2	no	—
380	C₂	#20255057	1	1	yes	characteristic
381	C₂	#20255070	1	1	yes	characteristic
382	C₂	#20255191	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689757
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1670	C₂
socle	C₂²×C₄₇	D₄²
prime radical	C₂×D₃₇₆	D₄
hypercenter	#1670	D₄₇
hypocenter	C₄₇	#19907
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1670

Complements

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

N	H ≅ G/N	type
3. #20255668	C₂	semidirect
4. #20242982	C₂	semidirect
5. #20242938	C₂	semidirect
6. #20242783	C₂	semidirect
7. #20242659	C₂	semidirect
8. #20242650	C₂	semidirect
10. C₈×#10625337	C₂	semidirect
11. C₄₇×#1670	C₂	semidirect
194. #1670	D₄₇	semidirect
236. C₄₇	#19907	semidirect

Maximal subgroups

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

subgroup	index
#20256378	2
Q₈×D₃₇₆	2
#20255668	2
#20242982	2
#20242938	2
#20242783	2
#20242659	2
#20242650	2
#20242272	2
C₈×#10625337	2
C₄₇×#1670	2
#19907	47