Subgroups — #26186193

Order 12032.

698 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253396	C₂	1	1	yes	maximal, characteristic
2	#20252318	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#20242242	C₂	1	4	yes	maximal, complemented
5	#20242197	C₂	1	4	yes	maximal, complemented
6	#20242164	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241792	C₂	1	1	yes	maximal, characteristic, complemented
8	#20241681	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1486	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄₇⋊C₈	—	2	1	no	—
11	#15689680	C₂²	1	1	yes	characteristic
12	#15689453	C₄	1	2	yes	—
13	#15689413	C₄	1	2	yes	—
14	#15689328	C₄	1	2	yes	—
15	#15689274	C₄	1	2	yes	—
16	#15689250	—	2	1	no	—
17	#15689250	C₂²	1	2	yes	complemented
18	#15689247	—	2	1	no	—
19	#15689247	C₂²	1	2	yes	complemented
20	C₄×#10625335	C₂²	1	4	yes	—
21	D₄×C₄₇⋊C₈	—	2	1	no	—
22	D₄×C₄₇⋊C₈	—	2	1	no	—
23	D₄×C₄₇⋊C₈	C₂²	1	2	yes	—
24	D₄×C₄₇⋊C₈	C₂²	1	2	yes	—
25	#15689176	C₂²	1	1	yes	characteristic
26	#15689151	—	2	1	no	—
27	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
29	#15689081	—	2	1	no	—
30	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
31	#15688875	C₂²	1	4	yes	complemented
32	#15688855	C₂²	1	4	yes	complemented
33	#15688845	C₂²	1	2	yes	complemented
34	#15688844	C₂²	1	2	yes	complemented
35	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
36	C₄₇×#431	C₂²	1	1	yes	characteristic
37	C₄₇×#423	C₂²	1	1	yes	characteristic
38	C₄₇×D₄⋊C₈	C₂²	1	4	yes	—
39	C₂×#10625335	C₂×C₄	1	4	yes	—
40	C₂²×C₄₇⋊C₈	—	2	1	no	—
41	C₂²×C₄₇⋊C₈	—	2	2	no	—
42	C₂²×C₄₇⋊C₈	—	2	2	no	—
43	C₂²×C₄₇⋊C₈	—	4	4	no	—
44	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
45	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
46	#13094024	C₂×C₄	1	1	yes	characteristic
47	#13094024	C₂×C₄	1	1	yes	characteristic
48	#13094016	—	2	1	no	—
49	#13094003	—	2	1	no	—
50	#13094003	C₂×C₄	1	2	yes	—
51	C₄×Dic₉₄	—	2	1	no	—
52	C₄×Dic₉₄	C₂³	1	2	yes	—
53	#13093983	—	2	1	no	—
54	#13093955	C₂×C₄	1	4	yes	—
55	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
56	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
57	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
58	Dic₄₇⋊C₈	—	2	2	no	—
59	Dic₄₇⋊C₈	—	2	2	no	—
60	#13093947	C₂×C₄	1	4	yes	—
61	#13093945	C₂×C₄	1	4	yes	—
62	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
63	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
64	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
65	#13093934	—	2	2	no	—
66	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
67	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
68	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
69	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₈×Dic₄₇	—	2	2	no	—
72	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
73	C₂³×C₄×C₄₇	—	2	1	no	—
74	C₂×M₁₆×C₄₇	—	2	1	no	—
75	C₄×D₄×C₄₇	—	2	1	no	—
76	C₄×D₄×C₄₇	—	2	1	no	—
77	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
78	C₄×D₄×C₄₇	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
81	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₄₇	—	2	1	no	—
83	C₄⋊C₈×C₄₇	—	2	1	no	—
84	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
85	C₂²⋊C₈×C₄₇	—	2	1	no	—
86	C₄×C₈×C₄₇	C₂³	1	2	yes	—
87	C₂⁴×C₄₇	—	2	1	no	—
88	C₂×D₄×C₄₇	—	2	4	no	—
89	C₂×D₄×C₄₇	—	2	1	no	—
90	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₄₇	—	2	1	no	—
92	C₂²×C₄×C₄₇	—	2	2	no	—
93	C₂²×C₄×C₄₇	—	2	2	no	—
94	C₂²×C₄×C₄₇	—	4	2	no	—
95	C₂²×C₄×C₄₇	—	4	2	no	—
96	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
99	C₂×Dic₉₄	—	2	1	no	—
100	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
101	M₁₆×C₄₇	—	4	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	2	1	no	—
105	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
106	C₂²⋊C₄×C₄₇	—	2	2	no	—
107	C₂²⋊C₄×C₄₇	—	2	2	no	—
108	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₇	#41	1	1	yes	characteristic
111	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
112	C₂²⋊Dic₄₇	—	2	2	no	—
113	#10625335	—	2	8	no	—
114	C₄⋊Dic₄₇	—	2	1	no	—
115	C₄⋊Dic₄₇	—	2	1	no	—
116	C₄⋊Dic₄₇	—	2	1	no	—
117	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
118	Dic₄₇⋊C₄	—	2	2	no	—
119	Dic₄₇⋊C₄	—	2	2	no	—
120	C₄×Dic₄₇	—	2	2	no	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	2	2	no	—
124	C₂×C₄₇⋊C₈	—	4	4	no	—
125	C₂×C₄₇⋊C₈	—	4	4	no	—
126	C₂×C₄₇⋊C₈	—	4	2	no	—
127	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
128	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₂³×C₄₇	—	2	4	no	—
133	C₂³×C₄₇	—	2	2	no	—
134	C₂³×C₄₇	—	4	2	no	—
135	C₂³×C₄₇	—	4	2	no	—
136	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
137	D₄×C₄₇	—	2	4	no	—
138	D₄×C₄₇	—	2	2	no	—
139	D₄×C₄₇	#131	1	4	yes	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	4	1	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	—	4	4	no	—
147	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄×C₄₇	#137	1	2	yes	—
152	C₂×C₄×C₄₇	C₂³×C₄	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	—	2	2	no	—
157	Dic₉₄	—	2	4	no	—
158	Dic₉₄	—	4	2	no	—
159	C₈×C₄₇	—	4	2	no	—
160	C₈×C₄₇	—	4	2	no	—
161	C₄₇⋊C₈	—	2	4	no	—
162	C₄₇⋊C₈	—	2	1	no	—
163	C₄₇⋊C₈	—	2	1	no	—
164	C₄₇⋊C₈	—	2	1	no	—
165	C₄₇⋊C₈	—	2	1	no	—
166	C₄₇⋊C₈	—	4	2	no	—
167	#17905	—	47	1	no	maximal, Sylow
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	—	2	1	no	—
170	C₂²×C₄₇	—	2	4	no	—
171	C₂²×C₄₇	—	4	8	no	—
172	C₂²×C₄₇	—	4	2	no	—
173	C₂²×C₄₇	—	4	2	no	—
174	C₂²×C₄₇	#440	1	2	yes	—
175	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	—	4	2	no	—
180	C₄×C₄₇	#440	1	2	yes	—
181	C₄×C₄₇	C₂×#131	1	2	yes	—
182	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
184	Dic₄₇	—	4	2	no	—
185	Dic₄₇	—	4	2	no	—
186	#3101	—	47	1	no	—
187	#2960	—	47	1	no	—
188	C₂×C₈×D₄	—	47	1	no	—
189	#1585	—	47	4	no	—
190	SD₁₆⋊C₈	—	47	4	no	—
191	#1571	—	47	1	no	—
192	#1486	D₄₇	1	1	yes	characteristic
193	#1481	—	47	1	no	—
194	#1452	—	47	1	no	—
195	C₂×C₄₇	—	2	2	no	—
196	C₂×C₄₇	—	4	4	no	—
197	C₂×C₄₇	#2929	1	1	yes	characteristic
198	C₂×C₄₇	C₂×#440	1	1	yes	characteristic
199	C₂×C₄₇	#2955	1	1	yes	characteristic
200	C₂³×C₈	—	94	1	no	—
201	#533	—	47	1	no	—
202	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
203	#483	—	47	2	no	—
204	#476	—	47	2	no	—
205	#461	—	47	2	no	—
206	#450	—	47	2	no	—
207	#446	—	47	2	no	—
208	#446	—	94	1	no	—
209	C₈×Q₈	—	47	2	no	—
210	C₈×Q₈	—	94	1	no	—
211	C₄×SD₁₆	—	47	4	no	—
212	C₈×D₄	—	47	2	no	—
213	C₈×D₄	—	47	2	no	—
214	C₈×D₄	—	94	1	no	—
215	C₈×D₄	—	94	1	no	—
216	#431	—	47	1	no	—
217	#431	C₂×D₄₇	1	1	yes	characteristic
218	#424	—	94	1	no	—
219	#423	C₂×D₄₇	1	1	yes	characteristic
220	C₂×C₄⋊C₈	—	47	1	no	—
221	C₂×C₂²⋊C₈	—	94	1	no	—
222	#405	—	94	1	no	—
223	C₂×C₄×C₈	—	47	1	no	—
224	#335	—	47	4	no	—
225	Q₈⋊C₈	—	47	4	no	—
226	D₄⋊C₈	C₂×D₄₇	1	4	yes	—
227	#322	—	47	2	no	—
228	#322	—	47	2	no	—
229	C₄₇	#17905	1	1	yes	characteristic, complemented, Sylow
230	C₂²×D₄	C₄×D₄₇	1	1	yes	characteristic
231	C₂³×C₄	—	2	1	no	—
232	C₂×SD₁₆	—	47	4	no	—
233	C₂×M₁₆	—	2	1	no	—
234	C₂²×C₈	—	47	1	no	—
235	C₂²×C₈	—	47	1	no	—
236	C₂²×C₈	—	94	2	no	—
237	C₂²×C₈	—	94	2	no	—
238	C₂²×C₈	—	94	1	no	—
239	C₂²×C₈	—	188	4	no	—
240	C₄⋊Q₈	—	47	1	no	—
241	C₄⋊Q₈	—	47	1	no	—
242	#125	—	94	1	no	—
243	C₂²⋊Q₈	—	47	2	no	—
244	C₂²⋊Q₈	—	94	1	no	—
245	C₄×Q₈	—	47	2	no	—
246	C₄×Q₈	—	94	1	no	—
247	C₄×D₄	—	2	1	no	—
248	C₄×D₄	—	2	1	no	—
249	C₄×D₄	C₄×D₄₇	1	2	yes	—
250	C₄×D₄	C₂²×D₄₇	1	2	yes	—
251	#117	—	94	1	no	—
252	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
253	C₂×C₂²⋊C₄	—	2	1	no	—
254	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
255	#106	—	47	4	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	47	2	no	—
259	C₄⋊C₈	—	47	2	no	—
260	C₄⋊C₈	—	94	2	no	—
261	C₄⋊C₈	—	94	2	no	—
262	C₄⋊C₈	—	94	1	no	—
263	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
264	#103	—	47	4	no	—
265	#102	—	47	4	no	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	47	2	no	—
268	C₂²⋊C₈	—	94	2	no	—
269	C₂²⋊C₈	—	94	2	no	—
270	#97	—	94	2	no	—
271	C₄×C₈	—	47	1	no	—
272	C₄×C₈	—	47	1	no	—
273	C₄×C₈	—	47	2	no	—
274	C₄×C₈	—	47	1	no	—
275	C₄×C₈	—	47	1	no	—
276	C₄×C₈	—	94	2	no	—
277	C₄×C₈	C₂²×D₄₇	1	2	yes	—
278	C₂⁴	—	2	1	no	—
279	C₂×Q₈	—	47	2	no	—
280	C₂×Q₈	—	94	1	no	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
291	C₂²×C₄	#10625352	1	1	yes	characteristic
292	SD₁₆	—	94	8	no	—
293	M₁₆	—	4	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	47	4	no	—
297	C₂×C₈	—	47	1	no	—
298	C₂×C₈	—	47	1	no	—
299	C₂×C₈	—	47	1	no	—
300	C₂×C₈	—	47	1	no	—
301	C₂×C₈	—	94	2	no	—
302	C₂×C₈	—	94	2	no	—
303	C₂×C₈	—	94	2	no	—
304	C₂×C₈	—	188	4	no	—
305	C₂×C₈	—	188	4	no	—
306	C₂×C₈	—	188	2	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	47	2	no	—
309	C₄⋊C₄	—	94	2	no	—
310	C₄⋊C₄	—	94	1	no	—
311	C₄⋊C₄	—	94	1	no	—
312	C₄⋊C₄	—	94	1	no	—
313	C₄⋊C₄	—	94	2	no	—
314	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	94	2	no	—
318	C₄²	—	94	2	no	—
319	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
320	C₄²	D₄×D₄₇	1	1	yes	characteristic
321	C₄²	#10625352	1	1	yes	characteristic
322	C₄²	C₂³×D₄₇	1	1	yes	characteristic
323	C₂³	—	2	4	no	—
324	C₂³	—	2	2	no	—
325	C₂³	—	4	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
328	Q₈	—	94	4	no	—
329	Q₈	—	188	2	no	—
330	D₄	—	2	4	no	—
331	D₄	—	2	2	no	—
332	D₄	#13094032	1	4	yes	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	94	2	no	—
341	C₂×C₄	—	94	2	no	—
342	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
343	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
344	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
345	C₂×C₄	#13094048	1	2	yes	—
346	C₂×C₄	#13094053	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
349	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	94	4	no	—
353	C₈	—	94	1	no	—
354	C₈	—	94	1	no	—
355	C₈	—	94	1	no	—
356	C₈	—	94	1	no	—
357	C₈	—	188	2	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	4	no	—
361	C₂²	—	4	8	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	4	2	no	—
364	C₂²	#15689220	1	2	yes	—
365	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	188	2	no	—
371	C₄	—	188	2	no	—
372	C₄	#15689220	1	2	yes	—
373	C₄	C₂×#13094032	1	2	yes	—
374	C₄	C₂×#13094048	1	1	yes	characteristic
375	C₄	C₂×#13094053	1	1	yes	characteristic
376	C₂	—	2	2	no	—
377	C₂	—	4	4	no	—
378	C₂	#20252078	1	1	yes	characteristic
379	C₂	C₂×#15689220	1	1	yes	characteristic
380	C₂	#20252287	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1486	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1486	D₄₇
hypocenter	C₄₇	#17905
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1486

Complements

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

N	H ≅ G/N	type
2. #20252318	C₂	semidirect
4. #20242242	C₂	semidirect
5. #20242197	C₂	semidirect
6. #20242164	C₂	semidirect
7. #20241792	C₂	semidirect
8. #20241681	C₂	semidirect
17. #15689250	C₂²	semidirect
19. #15689247	C₂²	semidirect
31. #15688875	C₂²	semidirect
32. #15688855	C₂²	semidirect
33. #15688845	C₂²	semidirect
34. #15688844	C₂²	semidirect
229. C₄₇	#17905	semidirect

Maximal subgroups

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

subgroup	index
#20253396	2
#20252318	2
C₂×D₄×C₄₇⋊C₈	2
#20242242	2
#20242197	2
#20242164	2
#20241792	2
#20241681	2
C₄₇×#1486	2
#17905	47