Subgroups — #26200294

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	#20253967	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252852	C₂	1	2	yes	maximal, complemented
3	#20242949	C₂	1	4	yes	maximal, complemented
4	#20242901	C₂	1	2	yes	maximal, complemented
5	#20242867	C₂	1	2	yes	maximal, complemented
6	#20242144	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242063	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094050	—	2	2	no	—
10	#15689673	C₂²	1	2	yes	—
11	#15689528	C₂²	1	1	yes	characteristic, complemented
12	#15689500	C₂²	1	1	yes	characteristic, complemented
13	#15689470	C₂²	1	2	yes	complemented
14	#15689359	—	2	1	no	—
15	#15689359	—	2	1	no	—
16	#15689338	—	2	4	no	—
17	#15689328	C₂²	1	4	yes	—
18	#15689318	C₂²	1	4	yes	—
19	C₄×D₃₇₆	—	2	1	no	—
20	C₄×D₃₇₆	—	2	1	no	—
21	#15689147	C₂²	1	2	yes	—
22	#15689122	—	2	2	no	—
23	#15688874	C₂²	1	2	yes	complemented
24	#15688874	C₂²	1	2	yes	complemented
25	#15688868	C₂²	1	4	yes	complemented
26	#15688866	C₂²	1	4	yes	complemented
27	C₂²×M₁₆×C₄₇	—	2	1	no	—
28	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#436	—	2	1	no	—
30	C₄₇×#436	—	2	1	no	—
31	C₄₇×#436	C₂²	1	2	yes	—
32	C₄₇×#436	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#10625350	—	2	2	no	—
37	#13094050	—	4	8	no	—
38	C₂×#10625335	—	2	4	no	—
39	C₂×D₃₇₆	—	2	1	no	—
40	C₂×D₃₇₆	—	2	1	no	—
41	C₂×#10625334	—	2	4	no	—
42	C₂×#10625329	—	2	2	no	—
43	#13094024	C₂³	1	2	yes	—
44	#13094021	C₂³	1	2	yes	—
45	#13094012	—	2	2	no	—
46	#13093998	—	2	2	no	—
47	#13093998	—	2	2	no	—
48	C₄×D₁₈₈	—	2	2	no	—
49	C₄×D₁₈₈	—	2	2	no	—
50	#13093958	—	2	1	no	—
51	#13093958	—	2	1	no	—
52	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
54	#13093947	—	2	4	no	—
55	#13093946	—	2	2	no	—
56	#13093946	—	2	2	no	—
57	#13093944	—	2	4	no	—
58	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₄₇	—	2	1	no	—
60	C₂×M₁₆×C₄₇	—	2	2	no	—
61	C₂×M₁₆×C₄₇	—	2	2	no	—
62	C₂×M₁₆×C₄₇	—	4	4	no	—
63	C₂²×C₈×C₄₇	—	2	1	no	—
64	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
66	C₄×D₄×C₄₇	—	2	1	no	—
67	C₄×D₄×C₄₇	—	2	1	no	—
68	C₄×D₄×C₄₇	D₄	1	2	yes	—
69	C₄×D₄×C₄₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₄₇	—	2	1	no	—
74	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
75	C₂²⋊C₈×C₄₇	—	2	2	no	—
76	C₂²⋊C₈×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	C₂⁴×C₄₇	—	2	1	no	—
82	C₂×D₄×C₄₇	—	2	1	no	—
83	C₂×D₄×C₄₇	—	2	4	no	—
84	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₇	—	2	1	no	—
86	C₂²×C₄×C₄₇	—	2	2	no	—
87	C₂²×C₄×C₄₇	—	2	2	no	—
88	C₂²×C₄×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	C₂×C₂²⋊D₄₇	—	4	2	no	—
94	#10625350	—	4	4	no	—
95	#10625350	—	4	4	no	—
96	C₂×D₁₈₈	—	2	2	no	—
97	C₂×D₁₈₈	—	2	2	no	—
98	C₂×D₁₈₈	—	2	2	no	—
99	C₂×C₄×D₄₇	—	4	2	no	—
100	C₂×Dic₉₄	—	2	2	no	—
101	M₁₆×C₄₇	—	4	4	no	—
102	M₁₆×C₄₇	—	4	4	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	4	1	no	—
106	C₂×C₈×C₄₇	—	4	2	no	—
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₄₇	—	2	1	no	—
112	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₄₇	—	2	2	no	—
114	C₂²⋊C₄×C₄₇	—	2	2	no	—
115	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₇	#41	1	1	yes	characteristic
118	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
119	#10625335	—	4	8	no	—
120	#10625334	—	4	8	no	—
121	(C₂×D₄₇)⋊C₄	—	4	2	no	—
122	(C₂×D₄₇)⋊C₄	—	4	2	no	—
123	C₄⋊Dic₄₇	—	2	2	no	—
124	C₄⋊Dic₄₇	—	2	2	no	—
125	#10625329	—	4	4	no	—
126	C₂×C₄₇⋊C₈	—	2	4	no	—
127	D₃₇₆	—	4	2	no	—
128	D₃₇₆	—	4	2	no	—
129	C₂³×C₄₇	—	2	2	no	—
130	C₂³×C₄₇	—	4	2	no	—
131	C₂³×C₄₇	—	4	2	no	—
132	C₂³×C₄₇	—	4	2	no	—
133	C₂³×C₄₇	#121	1	1	yes	characteristic
134	C₂²×D₄₇	—	4	2	no	—
135	D₄×C₄₇	—	2	4	no	—
136	D₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	2	1	no	—
138	C₂×C₄×C₄₇	—	2	1	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	4	1	no	—
141	C₂×C₄×C₄₇	—	4	4	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
147	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
149	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₄₇	—	8	4	no	—
153	C₂×Dic₄₇	—	4	2	no	—
154	D₁₈₈	—	4	2	no	—
155	D₁₈₈	—	4	2	no	—
156	D₁₈₈	—	4	2	no	—
157	D₁₈₈	—	4	2	no	—
158	D₁₈₈	—	8	2	no	—
159	C₄×D₄₇	—	8	4	no	—
160	Dic₉₄	—	4	2	no	—
161	Dic₉₄	—	4	2	no	—
162	C₈×C₄₇	—	2	1	no	—
163	C₈×C₄₇	—	2	1	no	—
164	C₈×C₄₇	—	2	1	no	—
165	C₈×C₄₇	—	4	2	no	—
166	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	complemented
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	—	4	2	no	—
172	C₂²×C₄₇	—	4	2	no	—
173	C₂²×C₄₇	—	4	2	no	—
174	C₂²×C₄₇	—	8	4	no	—
175	C₂²×C₄₇	#469	1	2	yes	—
176	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
177	C₂×D₄₇	—	8	2	no	—
178	C₂×D₄₇	—	8	2	no	—
179	C₂×D₄₇	—	8	2	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	2	1	no	—
182	C₄×C₄₇	—	2	1	no	—
183	C₄×C₄₇	—	4	2	no	—
184	C₄×C₄₇	#469	1	2	yes	—
185	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
187	C₄×C₄₇	#575	1	1	yes	characteristic
188	Dic₄₇	—	8	2	no	—
189	#3169	—	47	1	no	—
190	#3036	—	47	2	no	—
191	C₂×#436	D₄₇	1	1	yes	characteristic, complemented
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	—	8	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	—	16	2	no	—
203	C₂×AGL(1,ℤ/8)	—	94	2	no	—
204	C₂²×M₁₆	—	2	1	no	—
205	#532	—	47	2	no	—
206	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
207	#500	—	47	1	no	—
208	#493	—	47	1	no	—
209	#486	—	47	2	no	—
210	#466	—	94	1	no	—
211	#466	—	94	1	no	—
212	#463	—	94	4	no	—
213	#461	—	47	4	no	—
214	#459	—	47	4	no	—
215	C₄×D₈	—	94	1	no	—
216	C₄×D₈	—	94	1	no	—
217	#436	—	2	1	no	—
218	#436	—	2	1	no	—
219	#436	C₂×D₄₇	1	2	yes	—
220	#436	C₂×D₄₇	1	2	yes	—
221	#423	—	47	2	no	—
222	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
223	#417	—	94	2	no	—
224	C₂×C₂²⋊C₈	—	2	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	—	94	2	no	—
232	C₂²×D₄	D₁₈₈	1	1	yes	characteristic
233	C₂³×C₄	—	2	1	no	—
234	AGL(1,ℤ/8)	—	188	8	no	—
235	C₂×SD₁₆	—	94	4	no	—
236	C₂×D₈	—	94	4	no	—
237	C₂×D₈	—	94	1	no	—
238	C₂×D₈	—	94	1	no	—
239	C₂×M₁₆	—	2	2	no	—
240	C₂×M₁₆	—	2	2	no	—
241	C₂×M₁₆	—	4	4	no	—
242	C₂×M₁₆	—	94	2	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
245	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
246	C₄⋊Q₈	—	47	2	no	—
247	#127	—	47	2	no	—
248	#124	—	94	2	no	—
249	#121	—	94	2	no	—
250	#121	—	94	2	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	—	94	2	no	—
254	C₄×D₄	—	94	2	no	—
255	C₄×D₄	D₁₈₈	1	2	yes	—
256	C₄×D₄	C₂²×D₄₇	1	2	yes	—
257	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
258	C₂×C₂²⋊C₄	—	2	1	no	—
259	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
260	#107	—	94	1	no	—
261	#107	—	94	1	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	47	4	no	—
264	C₄⋊C₈	—	94	2	no	—
265	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
266	#103	—	94	4	no	—
267	#102	—	94	2	no	—
268	#102	—	94	2	no	—
269	#102	—	94	4	no	—
270	C₂²⋊C₈	—	2	2	no	—
271	C₂²⋊C₈	—	2	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	—	2	1	no	—
277	#41	—	188	4	no	—
278	#41	—	188	4	no	—
279	C₂×Q₈	—	94	2	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	—	94	2	no	—
283	C₂×D₄	—	94	2	no	—
284	C₂×D₄	—	94	2	no	—
285	C₂×D₄	—	188	2	no	—
286	C₂×D₄	C₂×D₁₈₈	1	2	yes	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	188	2	no	—
293	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
294	C₂²×C₄	#10625350	1	1	yes	characteristic
295	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
296	SD₁₆	—	188	8	no	—
297	D₈	—	188	2	no	—
298	D₈	—	188	8	no	—
299	D₈	—	188	2	no	—
300	M₁₆	—	4	4	no	—
301	M₁₆	—	4	4	no	—
302	M₁₆	—	188	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	4	2	no	—
307	C₂×C₈	—	94	4	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	1	no	—
313	C₄⋊C₄	—	94	2	no	—
314	C₄⋊C₄	—	94	2	no	—
315	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₂²⋊C₄	—	188	2	no	—
319	C₂²⋊C₄	—	188	2	no	—
320	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
321	C₄²	#10625350	1	1	yes	characteristic
322	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
323	C₄²	C₂³×D₄₇	1	1	yes	characteristic
324	C₂³	—	2	2	no	—
325	C₂³	—	4	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	—	4	2	no	—
328	C₂³	—	188	2	no	—
329	C₂³	#13093998	1	1	yes	characteristic
330	Q₈	—	188	2	no	—
331	Q₈	—	188	2	no	—
332	D₄	—	2	4	no	—
333	D₄	—	4	2	no	—
334	D₄	—	376	2	no	—
335	D₄	—	376	4	no	—
336	D₄	—	188	2	no	—
337	D₄	—	188	2	no	—
338	D₄	—	188	2	no	—
339	D₄	—	188	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	376	4	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	#13093998	1	1	yes	characteristic
351	C₂×C₄	#13093998	1	1	yes	characteristic
352	C₂×C₄	#13093998	1	1	yes	characteristic
353	C₂×C₄	#13094052	1	4	yes	—
354	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
356	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	4	2	no	—
361	C₈	—	188	4	no	—
362	C₈	#13094050	1	2	yes	complemented
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	8	4	no	—
369	C₂²	—	376	2	no	—
370	C₂²	—	376	2	no	—
371	C₂²	—	376	2	no	—
372	C₂²	#15689376	1	2	yes	—
373	C₂²	C₂×#13093998	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	376	2	no	—
379	C₄	#15689376	1	2	yes	—
380	C₄	C₂×#13094050	1	1	yes	characteristic
381	C₄	C₂×#13094052	1	2	yes	—
382	C₄	#15689921	1	1	yes	characteristic
383	C₂	—	2	2	no	—
384	C₂	—	8	2	no	—
385	C₂	—	752	2	no	—
386	C₂	#20252852	1	1	yes	characteristic
387	C₂	C₂×#15689376	1	1	yes	characteristic
388	C₂	#20253108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093998
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#436	C₂
socle	C₂²×C₄₇	C₂×#121
hypercenter	C₂×#436	D₄₇
hypocenter	C₄₇	#19280
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #20253967	C₂	semidirect
2. #20252852	C₂	semidirect
3. #20242949	C₂	semidirect
4. #20242901	C₂	semidirect
5. #20242867	C₂	semidirect
6. #20242144	C₂	semidirect
7. #20242063	C₂	semidirect
8. C₂×C₄₇×#436	C₂	semidirect
11. #15689528	C₂²	semidirect
12. #15689500	C₂²	semidirect
13. #15689470	C₂²	semidirect
23. #15688874	C₂²	semidirect
24. #15688874	C₂²	semidirect
25. #15688868	C₂²	semidirect
26. #15688866	C₂²	semidirect
166. C₈×C₄₇	AGL(1,ℤ/8)	semidirect
191. C₂×#436	D₄₇	semidirect
230. C₄₇	#19280	semidirect
362. C₈	#13094050	semidirect

Maximal subgroups

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

subgroup	index
#20253967	2
#20252852	2
#20242949	2
#20242901	2
#20242867	2
#20242144	2
#20242063	2
C₂×C₄₇×#436	2
#19280	47