Subgroups — #26200290

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252848	C₂	1	2	yes	maximal, complemented
2	C₂×#15689202	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242946	C₂	1	4	yes	maximal, complemented
4	#20242898	C₂	1	2	yes	maximal, complemented
5	#20242864	C₂	1	2	yes	maximal, complemented
6	#20242144	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242065	C₂	1	2	yes	maximal, complemented
8	C₄₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094049	—	2	2	no	—
10	C₂²×#10625324	—	2	1	no	—
11	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	#15689335	—	2	4	no	—
13	#15689325	C₂²	1	4	yes	—
14	#15689315	C₂²	1	4	yes	—
15	#15689202	—	2	1	no	—
16	#15689202	—	2	1	no	—
17	#15689202	C₂²	1	2	yes	—
18	#15689202	C₂²	1	2	yes	—
19	#15689147	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
21	#15689124	—	2	2	no	—
22	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
23	#15688874	C₂²	1	2	yes	complemented
24	#15688874	C₂²	1	2	yes	complemented
25	#15688869	C₂²	1	4	yes	complemented
26	#15688867	C₂²	1	4	yes	complemented
27	C₄₇×#532	C₂²	1	2	yes	—
28	C₄₇×#500	C₂²	1	1	yes	characteristic, complemented
29	C₄₇×#493	C₂²	1	1	yes	characteristic, complemented
30	C₄₇×#486	C₂²	1	2	yes	complemented
31	C₄₇×#466	—	2	1	no	—
32	C₄₇×#466	—	2	1	no	—
33	C₄×D₈×C₄₇	—	2	1	no	—
34	C₄×D₈×C₄₇	—	2	1	no	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₄₇	—	2	1	no	—
38	#13094049	—	4	8	no	—
39	C₂×#10625336	—	2	4	no	—
40	C₂×#10625334	—	2	4	no	—
41	C₂×#10625329	—	2	2	no	—
42	C₂×#10625324	—	2	2	no	—
43	C₂×#10625324	—	2	2	no	—
44	C₂×#10625324	—	4	4	no	—
45	C₂²×C₄₇⋊C₈	—	2	1	no	—
46	C₄×D₁₈₈	—	2	1	no	—
47	C₄×D₁₈₈	—	2	1	no	—
48	C₄×D₁₈₈	D₄	1	2	yes	—
49	C₄×D₁₈₈	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
51	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
52	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
54	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
55	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
56	#13093948	—	2	4	no	—
57	#13093945	—	2	4	no	—
58	(C₂×D₄₇)⋊C₈	—	2	2	no	—
59	(C₂×D₄₇)⋊C₈	—	2	2	no	—
60	C₂×#41×C₄₇	—	2	2	no	—
61	C₂×D₈×C₄₇	—	2	1	no	—
62	C₂×D₈×C₄₇	—	2	1	no	—
63	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
65	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
66	#127×C₄₇	C₂³	1	2	yes	—
67	#124×C₄₇	—	2	2	no	—
68	#121×C₄₇	—	2	2	no	—
69	#121×C₄₇	—	2	2	no	—
70	C₄×D₄×C₄₇	—	2	2	no	—
71	C₄×D₄×C₄₇	—	2	2	no	—
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	#107×C₄₇	—	2	1	no	—
74	#107×C₄₇	—	2	1	no	—
75	#102×C₄₇	—	2	2	no	—
76	#102×C₄₇	—	2	2	no	—
77	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
79	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
81	C₂³×D₄₇	—	2	1	no	—
82	#41×C₄₇	—	4	4	no	—
83	#41×C₄₇	—	4	4	no	—
84	C₂×Q₈×C₄₇	—	2	2	no	—
85	C₂×D₄×C₄₇	—	2	2	no	—
86	C₂×D₄×C₄₇	—	2	2	no	—
87	C₂×D₄×C₄₇	—	2	2	no	—
88	C₂×D₄×C₄₇	—	4	2	no	—
89	C₂²×C₄×C₄₇	—	4	2	no	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
93	C₂²×Dic₄₇	—	2	1	no	—
94	C₂×D₁₈₈	—	2	4	no	—
95	C₂×D₁₈₈	—	2	1	no	—
96	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
97	C₂×C₄×D₄₇	—	2	2	no	—
98	C₂×C₄×D₄₇	—	2	2	no	—
99	C₂×C₄×D₄₇	—	4	2	no	—
100	C₂×C₄×D₄₇	—	4	2	no	—
101	D₈×C₄₇	—	4	2	no	—
102	D₈×C₄₇	—	4	2	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	4	1	no	—
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	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₂²⋊C₄×C₄₇	—	4	2	no	—
112	C₂²⋊C₄×C₄₇	—	4	2	no	—
113	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₇	#41	1	1	yes	characteristic
116	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
117	#10625336	—	4	8	no	—
118	#10625334	—	4	8	no	—
119	(C₂×D₄₇)⋊C₄	—	2	2	no	—
120	(C₂×D₄₇)⋊C₄	—	2	2	no	—
121	C₄⋊Dic₄₇	—	2	1	no	—
122	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
123	#10625329	—	4	4	no	—
124	C₂×C₄₇⋊C₈	—	2	2	no	—
125	C₂×C₄₇⋊C₈	—	2	4	no	—
126	C₂×C₄₇⋊C₈	—	4	2	no	—
127	#10625324	—	4	4	no	—
128	#10625324	—	4	4	no	—
129	C₂³×C₄₇	—	4	2	no	—
130	C₂³×C₄₇	#121	1	1	yes	characteristic
131	C₂²×D₄₇	—	2	2	no	—
132	C₂²×D₄₇	—	4	2	no	—
133	C₂²×D₄₇	—	4	2	no	—
134	C₂²×D₄₇	—	4	2	no	—
135	Q₈×C₄₇	—	4	2	no	—
136	Q₈×C₄₇	—	4	2	no	—
137	D₄×C₄₇	—	4	2	no	—
138	D₄×C₄₇	—	4	2	no	—
139	D₄×C₄₇	—	4	2	no	—
140	D₄×C₄₇	—	4	2	no	—
141	D₄×C₄₇	—	8	2	no	—
142	D₄×C₄₇	—	8	4	no	—
143	C₂×C₄×C₄₇	—	2	1	no	—
144	C₂×C₄×C₄₇	—	2	1	no	—
145	C₂×C₄×C₄₇	—	4	1	no	—
146	C₂×C₄×C₄₇	—	4	2	no	—
147	C₂×C₄×C₄₇	—	8	4	no	—
148	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
149	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
150	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
151	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
152	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₄₇	—	2	2	no	—
156	C₂×Dic₄₇	—	4	2	no	—
157	D₁₈₈	—	2	4	no	—
158	D₁₈₈	—	4	2	no	—
159	C₄×D₄₇	—	4	4	no	—
160	C₄×D₄₇	—	4	2	no	—
161	C₄×D₄₇	—	4	2	no	—
162	C₈×C₄₇	—	2	1	no	—
163	C₈×C₄₇	—	2	1	no	—
164	C₈×C₄₇	—	2	1	no	—
165	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	complemented
166	C₄₇⋊C₈	—	4	2	no	—
167	C₄₇⋊C₈	—	4	4	no	—
168	#19280	—	47	1	no	maximal, Sylow
169	C₂²×C₄₇	—	2	1	no	—
170	C₂²×C₄₇	—	2	1	no	—
171	C₂²×C₄₇	—	8	2	no	—
172	C₂²×C₄₇	—	8	2	no	—
173	C₂²×C₄₇	—	8	2	no	—
174	C₂²×C₄₇	#469	1	2	yes	—
175	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
176	C₂×D₄₇	—	4	2	no	—
177	C₂×D₄₇	—	4	2	no	—
178	C₂×D₄₇	—	4	2	no	—
179	C₂×D₄₇	—	8	4	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	2	1	no	—
182	C₄×C₄₇	—	2	1	no	—
183	C₄×C₄₇	—	8	2	no	—
184	C₄×C₄₇	#469	1	2	yes	—
185	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
186	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₄×C₄₇	#575	1	1	yes	characteristic
188	Dic₄₇	—	4	2	no	—
189	#3169	D₄₇	1	1	yes	characteristic, complemented
190	#3036	—	47	2	no	—
191	C₂×#436	—	47	1	no	—
192	#1701	—	47	4	no	—
193	#1693	—	47	2	no	—
194	#1687	—	47	2	no	—
195	#1566	—	47	1	no	—
196	#1547	—	47	2	no	—
197	C₂×C₄₇	—	2	2	no	—
198	C₂×C₄₇	—	16	2	no	—
199	C₂×C₄₇	#3036	1	1	yes	characteristic
200	C₂×C₄₇	C₂×#469	1	1	yes	characteristic
201	C₂×C₄₇	#3067	1	1	yes	characteristic
202	D₄₇	—	8	2	no	—
203	C₂×AGL(1,ℤ/8)	—	94	2	no	—
204	C₂²×M₁₆	—	94	1	no	—
205	#532	C₂×D₄₇	1	2	yes	—
206	C₂×C₄×D₄	—	47	1	no	—
207	#500	C₂×D₄₇	1	1	yes	characteristic, complemented
208	#493	C₂×D₄₇	1	1	yes	characteristic, complemented
209	#486	C₂×D₄₇	1	2	yes	complemented
210	#466	—	2	1	no	—
211	#466	—	2	1	no	—
212	#463	—	94	4	no	—
213	#461	—	47	4	no	—
214	#459	—	47	4	no	—
215	C₄×D₈	—	2	1	no	—
216	C₄×D₈	—	2	1	no	—
217	#436	—	47	2	no	—
218	#436	—	47	2	no	—
219	#436	—	94	1	no	—
220	#436	—	94	1	no	—
221	#423	—	47	2	no	—
222	C₂×C₄⋊C₈	—	47	1	no	—
223	#417	—	94	2	no	—
224	C₂×C₂²⋊C₈	—	94	1	no	—
225	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
226	#334	—	47	2	no	—
227	#334	—	47	2	no	—
228	#332	—	47	4	no	—
229	#331	—	47	4	no	—
230	C₄₇	#19280	1	1	yes	characteristic, complemented, Sylow
231	C₂×#41	—	2	2	no	—
232	C₂²×D₄	—	47	1	no	—
233	C₂³×C₄	—	94	1	no	—
234	AGL(1,ℤ/8)	—	188	8	no	—
235	C₂×SD₁₆	—	94	4	no	—
236	C₂×D₈	—	2	1	no	—
237	C₂×D₈	—	2	1	no	—
238	C₂×D₈	—	94	4	no	—
239	C₂×M₁₆	—	94	2	no	—
240	C₂×M₁₆	—	94	2	no	—
241	C₂×M₁₆	—	94	2	no	—
242	C₂×M₁₆	—	188	4	no	—
243	C₂²×C₈	—	94	1	no	—
244	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
245	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
246	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
247	#127	C₂²×D₄₇	1	2	yes	—
248	#124	—	2	2	no	—
249	#121	—	2	2	no	—
250	#121	—	2	2	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	—	2	2	no	—
253	C₄×D₄	—	47	2	no	—
254	C₄×D₄	—	47	2	no	—
255	C₄×D₄	—	94	1	no	—
256	C₄×D₄	—	94	1	no	—
257	C₂×C₄⋊C₄	—	47	1	no	—
258	C₂×C₂²⋊C₄	—	94	1	no	—
259	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	C₄⋊C₈	—	47	2	no	—
263	C₄⋊C₈	—	47	4	no	—
264	C₄⋊C₈	—	94	2	no	—
265	C₄⋊C₈	—	94	1	no	—
266	#103	—	94	4	no	—
267	#102	—	2	2	no	—
268	#102	—	2	2	no	—
269	#102	—	94	4	no	—
270	C₂²⋊C₈	—	94	2	no	—
271	C₂²⋊C₈	—	94	2	no	—
272	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
273	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
276	C₂⁴	—	94	1	no	—
277	#41	—	4	4	no	—
278	#41	—	4	4	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	47	2	no	—
285	C₂×D₄	—	94	1	no	—
286	C₂×D₄	—	94	4	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	94	1	no	—
289	C₂²×C₄	—	94	2	no	—
290	C₂²×C₄	—	94	2	no	—
291	C₂²×C₄	—	188	2	no	—
292	C₂²×C₄	—	188	2	no	—
293	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
294	C₂²×C₄	#10625352	1	1	yes	characteristic
295	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
296	SD₁₆	—	188	8	no	—
297	D₈	—	4	2	no	—
298	D₈	—	4	2	no	—
299	D₈	—	188	8	no	—
300	M₁₆	—	188	4	no	—
301	M₁₆	—	188	4	no	—
302	M₁₆	—	188	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	94	4	no	—
306	C₂×C₈	—	94	2	no	—
307	C₂×C₈	—	188	2	no	—
308	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
310	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
311	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	47	2	no	—
315	C₄⋊C₄	—	94	1	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	94	2	no	—
319	C₂²⋊C₄	—	94	2	no	—
320	C₄²	D₄×D₄₇	1	1	yes	characteristic
321	C₄²	#10625352	1	1	yes	characteristic
322	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
323	C₄²	C₂³×D₄₇	1	1	yes	characteristic
324	C₂³	—	4	2	no	—
325	C₂³	—	94	2	no	—
326	C₂³	—	188	2	no	—
327	C₂³	—	188	2	no	—
328	C₂³	—	188	2	no	—
329	C₂³	#13093995	1	1	yes	characteristic
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	2	no	—
336	D₄	—	8	2	no	—
337	D₄	—	8	4	no	—
338	D₄	—	94	4	no	—
339	D₄	—	188	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	8	4	no	—
345	C₂×C₄	—	94	2	no	—
346	C₂×C₄	—	188	4	no	—
347	C₂×C₄	—	188	2	no	—
348	C₂×C₄	—	188	2	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	#13093995	1	1	yes	characteristic
351	C₂×C₄	#13093995	1	1	yes	characteristic
352	C₂×C₄	#13093995	1	1	yes	characteristic
353	C₂×C₄	#13094048	1	4	yes	—
354	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
356	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	188	4	no	—
361	C₈	—	188	2	no	—
362	C₈	#13094049	1	2	yes	complemented
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	376	4	no	—
369	C₂²	—	188	2	no	—
370	C₂²	—	188	2	no	—
371	C₂²	—	188	2	no	—
372	C₂²	#15689373	1	2	yes	—
373	C₂²	C₂×#13093995	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	188	2	no	—
379	C₄	#15689373	1	2	yes	—
380	C₄	C₂×#13094048	1	2	yes	—
381	C₄	C₂×#13094049	1	1	yes	characteristic
382	C₄	#15689921	1	1	yes	characteristic
383	C₂	—	2	2	no	—
384	C₂	—	16	2	no	—
385	C₂	—	376	2	no	—
386	C₂	#20252848	1	1	yes	characteristic
387	C₂	C₂×#15689373	1	1	yes	characteristic
388	C₂	#20253104	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3169	C₂
socle	C₂²×C₄₇	C₂×#121
hypercenter	#3169	D₄₇
hypocenter	C₄₇	#19280
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20252848	C₂	semidirect
2. C₂×#15689202	C₂	semidirect
3. #20242946	C₂	semidirect
4. #20242898	C₂	semidirect
5. #20242864	C₂	semidirect
6. #20242144	C₂	semidirect
7. #20242065	C₂	semidirect
8. C₄₇×#3169	C₂	semidirect
23. #15688874	C₂²	semidirect
24. #15688874	C₂²	semidirect
25. #15688869	C₂²	semidirect
26. #15688867	C₂²	semidirect
28. C₄₇×#500	C₂²	semidirect
29. C₄₇×#493	C₂²	semidirect
30. C₄₇×#486	C₂²	semidirect
165. C₈×C₄₇	AGL(1,ℤ/8)	semidirect
189. #3169	D₄₇	semidirect
207. #500	C₂×D₄₇	semidirect
208. #493	C₂×D₄₇	semidirect
209. #486	C₂×D₄₇	semidirect
230. C₄₇	#19280	semidirect
362. C₈	#13094049	semidirect

Maximal subgroups

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

subgroup	index
#20252848	2
C₂×#15689202	2
#20242946	2
#20242898	2
#20242864	2
#20242144	2
#20242065	2
C₄₇×#3169	2
#19280	47