Subgroups — #26125983

Order 12032.

450 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×D₄×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245142	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13093963	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241344	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241342	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	—	2	1	no	—
9	C₄×D₄×D₄₇	C₄	1	1	yes	characteristic
10	C₂×#13093963	—	2	1	no	—
11	C₂×#13093963	—	2	1	no	—
12	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
13	D₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
14	#15689188	C₄	1	1	yes	characteristic
15	C₈×C₂²⋊D₄₇	—	2	1	no	—
16	(C₂²⋊D₄₇)⋊C₈	—	2	1	no	—
17	C₄⋊C₈×D₄₇	C₄	1	1	yes	characteristic
18	C₂²⋊C₈×D₄₇	—	2	1	no	—
19	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
20	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
21	Dic₄₇⋊C₁₆	—	2	1	no	—
22	#15688898	—	2	1	no	—
23	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
24	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
26	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×D₄×D₄₇	C₈	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	—	2	1	no	—
29	C₂²×C₄₇⋊C₈	—	2	1	no	—
30	C₂×C₈×D₄₇	—	2	1	no	—
31	C₂×C₈×D₄₇	—	2	1	no	—
32	C₂×C₈×D₄₇	—	2	1	no	—
33	C₂×C₈×D₄₇	—	4	2	no	—
34	C₂×C₈×D₄₇	—	4	2	no	—
35	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
36	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
37	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
38	D₄×Dic₄₇	C₈	1	1	yes	characteristic
39	#13093986	C₈	1	1	yes	characteristic
40	C₄×C₂²⋊D₄₇	—	2	1	no	—
41	(C₂²⋊D₄₇)⋊C₄	—	2	1	no	—
42	C₄⋊C₄×D₄₇	C₈	1	1	yes	characteristic
43	C₂²⋊C₄×D₄₇	—	2	1	no	—
44	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
45	#13093963	—	2	4	no	—
46	#13093963	—	4	2	no	—
47	C₂×C₄₇⋊C₁₆	—	2	1	no	—
48	C₂×C₄₇⋊C₁₆	—	2	1	no	—
49	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
50	Dic₄₇⋊C₈	—	2	1	no	—
51	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
52	(C₂×D₄₇)⋊C₈	—	2	1	no	—
53	(C₂×D₄₇)⋊C₈	—	2	1	no	—
54	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₄₇	D₄	1	1	yes	characteristic
56	C₈×Dic₄₇	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	—	2	1	no	—
58	C₄×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₄₇	—	2	1	no	—
60	C₂×C₁₆×C₄₇	—	2	1	no	—
61	C₄⋊C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₈×C₄₇	—	2	1	no	—
63	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂³×D₄₇	—	2	1	no	—
65	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	—	2	1	no	—
67	C₂×C₂²⋊D₄₇	—	2	1	no	—
68	C₂²×Dic₄₇	—	2	1	no	—
69	D₄×D₄₇	—	2	2	no	—
70	D₄×D₄₇	—	2	2	no	—
71	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₄₇	—	2	1	no	—
73	C₂×C₄×D₄₇	—	2	1	no	—
74	C₂×C₄×D₄₇	—	2	1	no	—
75	C₂×C₄×D₄₇	—	4	2	no	—
76	C₂×C₄×D₄₇	—	4	2	no	—
77	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₄₇	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₄₇	M₁₆	1	1	yes	characteristic
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	—	4	1	no	—
82	C₂×C₈×C₄₇	—	4	1	no	—
83	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
86	C₂²⋊C₄×C₄₇	—	2	1	no	—
87	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²⋊Dic₄₇	—	2	1	no	—
89	(C₂×D₄₇)⋊C₄	—	2	1	no	—
90	(C₂×D₄₇)⋊C₄	—	2	1	no	—
91	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
92	Dic₄₇⋊C₄	—	2	1	no	—
93	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
95	C₄×Dic₄₇	M₁₆	1	1	yes	characteristic
96	C₂×C₄₇⋊C₈	—	2	1	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₈×D₄₇	—	2	2	no	—
101	C₈×D₄₇	—	4	2	no	—
102	C₈×D₄₇	—	4	1	no	—
103	C₈×D₄₇	—	4	1	no	—
104	C₈×D₄₇	—	4	2	no	—
105	C₈×D₄₇	D₈	1	2	yes	—
106	C₈×D₄₇	SD₁₆	1	2	yes	—
107	C₁₆×C₄₇	—	2	2	no	—
108	C₁₆×C₄₇	—	4	1	no	—
109	C₄₇⋊C₁₆	—	2	2	no	—
110	C₄₇⋊C₁₆	—	4	1	no	—
111	C₂³×C₄₇	—	2	1	no	—
112	C₂²×D₄₇	—	2	4	no	—
113	C₂²×D₄₇	—	2	1	no	—
114	C₂²×D₄₇	—	4	2	no	—
115	C₂²×D₄₇	—	4	2	no	—
116	C₂²×D₄₇	C₂²⋊C₈	1	1	yes	characteristic
117	D₄×C₄₇	—	2	1	no	—
118	D₄×C₄₇	M₃₂	1	2	yes	complemented
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	4	1	no	—
121	C₂×C₄×C₄₇	—	4	1	no	—
122	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
125	C₂²⋊D₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	2	1	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	C₂²⋊C₈	1	1	yes	characteristic
129	C₂×Dic₄₇	C₂²⋊C₈	1	1	yes	characteristic
130	C₂×Dic₄₇	C₂²⋊C₈	1	1	yes	characteristic
131	D₁₈₈	—	2	1	no	—
132	D₁₈₈	M₃₂	1	2	yes	complemented
133	C₄×D₄₇	—	2	1	no	—
134	C₄×D₄₇	—	2	1	no	—
135	C₄×D₄₇	—	4	1	no	—
136	C₄×D₄₇	—	4	2	no	—
137	C₄×D₄₇	—	4	1	no	—
138	C₄×D₄₇	—	4	1	no	—
139	C₄×D₄₇	—	4	2	no	—
140	C₄×D₄₇	#102	1	2	yes	—
141	C₄×D₄₇	#104	1	2	yes	—
142	C₈×C₄₇	—	2	1	no	—
143	C₈×C₄₇	—	4	1	no	—
144	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
145	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
146	C₄₇⋊C₈	—	2	1	no	—
147	C₄₇⋊C₈	—	4	1	no	—
148	C₄₇⋊C₈	#102	1	2	yes	—
149	#11583	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	—	4	1	no	—
152	C₂²×C₄₇	—	4	1	no	—
153	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₂×D₄₇	—	2	1	no	—
155	C₂×D₄₇	—	2	1	no	—
156	C₂×D₄₇	—	2	2	no	—
157	C₂×D₄₇	—	4	8	no	—
158	C₂×D₄₇	—	4	1	no	—
159	C₂×D₄₇	—	4	1	no	—
160	C₂×D₄₇	D₄⋊C₈	1	2	yes	—
161	C₄×C₄₇	—	2	1	no	—
162	C₄×C₄₇	—	4	1	no	—
163	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
164	C₄×C₄₇	C₂×#104	1	1	yes	characteristic
165	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
166	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
167	Dic₄₇	—	2	1	no	—
168	Dic₄₇	—	2	1	no	—
169	Dic₄₇	—	4	1	no	—
170	Dic₄₇	D₄⋊C₈	1	2	yes	—
171	C₂×C₈×D₄	—	47	1	no	—
172	#2150	—	47	1	no	—
173	C₄×M₃₂	—	47	1	no	—
174	D₄⋊C₁₆	—	47	1	no	—
175	D₄⋊C₁₆	—	47	1	no	—
176	D₄⋊C₁₆	—	47	1	no	—
177	D₄⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
178	C₂×C₄₇	—	4	2	no	—
179	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
180	C₂×C₄₇	#2112	1	1	yes	characteristic
181	C₂×C₄₇	#2117	1	1	yes	characteristic
182	D₄₇	—	2	2	no	—
183	D₄₇	—	4	2	no	—
184	C₂³×C₈	—	94	1	no	—
185	C₂×C₄×D₄	—	47	1	no	—
186	C₂×M₃₂	—	94	1	no	—
187	C₂×M₃₂	—	94	1	no	—
188	C₈×D₄	—	47	1	no	—
189	C₈×D₄	—	47	1	no	—
190	C₈×D₄	—	47	1	no	—
191	C₈×D₄	—	94	1	no	—
192	C₈×D₄	—	94	1	no	—
193	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
194	C₂×C₄⋊C₈	—	47	1	no	—
195	C₂×C₂²⋊C₈	—	94	1	no	—
196	C₂×C₄×C₈	—	47	1	no	—
197	C₄⋊C₁₆	—	47	1	no	—
198	C₄⋊C₁₆	—	94	1	no	—
199	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
200	#346	—	94	1	no	—
201	C₄×C₁₆	—	47	1	no	—
202	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
203	C₄₇	#11583	1	1	yes	characteristic, complemented, Sylow
204	C₂²×D₄	—	47	1	no	—
205	C₂³×C₄	—	94	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	47	1	no	—
208	C₂²×C₈	—	47	1	no	—
209	C₂²×C₈	—	94	1	no	—
210	C₂²×C₈	—	94	1	no	—
211	C₂²×C₈	—	94	1	no	—
212	C₂²×C₈	—	94	1	no	—
213	C₂²×C₈	—	188	2	no	—
214	C₂²×C₈	—	188	2	no	—
215	C₄×D₄	—	47	1	no	—
216	C₄×D₄	—	47	1	no	—
217	C₄×D₄	—	47	1	no	—
218	C₄×D₄	—	94	1	no	—
219	C₄×D₄	—	94	1	no	—
220	C₄×D₄	C₄×D₄₇	1	1	yes	characteristic
221	C₂×C₄⋊C₄	—	47	1	no	—
222	C₂×C₂²⋊C₄	—	94	1	no	—
223	C₂×C₄²	—	47	1	no	—
224	M₃₂	—	94	4	no	—
225	M₃₂	—	188	2	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	94	1	no	—
229	C₂×C₁₆	—	94	1	no	—
230	C₄⋊C₈	—	47	1	no	—
231	C₄⋊C₈	—	94	1	no	—
232	C₄⋊C₈	C₄×D₄₇	1	1	yes	characteristic
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	94	1	no	—
235	C₂²⋊C₈	—	94	1	no	—
236	C₂²⋊C₈	—	94	1	no	—
237	C₄×C₈	—	47	1	no	—
238	C₄×C₈	—	47	1	no	—
239	C₄×C₈	—	47	1	no	—
240	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
241	C₂⁴	—	94	1	no	—
242	C₂×D₄	—	47	1	no	—
243	C₂×D₄	—	94	1	no	—
244	C₂×D₄	—	94	2	no	—
245	C₂×D₄	—	94	2	no	—
246	C₂×D₄	C₈×D₄₇	1	1	yes	characteristic
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	47	1	no	—
249	C₂²×C₄	—	47	1	no	—
250	C₂²×C₄	—	47	1	no	—
251	C₂²×C₄	—	94	1	no	—
252	C₂²×C₄	—	94	1	no	—
253	C₂²×C₄	—	94	1	no	—
254	C₂²×C₄	—	94	1	no	—
255	C₂²×C₄	—	188	2	no	—
256	C₂²×C₄	—	188	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	47	1	no	—
261	C₂×C₈	—	47	2	no	—
262	C₂×C₈	—	47	2	no	—
263	C₂×C₈	—	47	1	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	—	94	1	no	—
266	C₂×C₈	—	188	2	no	—
267	C₂×C₈	—	188	2	no	—
268	C₂×C₈	—	188	2	no	—
269	C₂×C₈	—	188	1	no	—
270	C₂×C₈	—	188	1	no	—
271	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
272	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
273	C₄⋊C₄	—	47	1	no	—
274	C₄⋊C₄	—	94	1	no	—
275	C₄⋊C₄	C₈×D₄₇	1	1	yes	characteristic
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	94	1	no	—
278	C₂²⋊C₄	—	94	1	no	—
279	C₂²⋊C₄	—	94	1	no	—
280	C₄²	—	47	1	no	—
281	C₄²	—	47	1	no	—
282	C₄²	—	47	1	no	—
283	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	94	2	no	—
287	C₁₆	—	188	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	47	1	no	—
290	C₂³	—	94	4	no	—
291	C₂³	—	94	1	no	—
292	C₂³	—	188	2	no	—
293	C₂³	—	188	2	no	—
294	D₄	—	2	1	no	—
295	D₄	—	47	2	no	—
296	D₄	—	94	4	no	—
297	D₄	—	94	1	no	—
298	D₄	#13093963	1	2	yes	complemented
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	47	1	no	—
303	C₂×C₄	—	47	2	no	—
304	C₂×C₄	—	47	2	no	—
305	C₂×C₄	—	47	1	no	—
306	C₂×C₄	—	47	1	no	—
307	C₂×C₄	—	94	1	no	—
308	C₂×C₄	—	94	1	no	—
309	C₂×C₄	—	94	1	no	—
310	C₂×C₄	—	188	1	no	—
311	C₂×C₄	—	188	2	no	—
312	C₂×C₄	—	188	2	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	—	188	1	no	—
315	C₂×C₄	—	188	1	no	—
316	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
318	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	4	1	no	—
321	C₈	—	47	2	no	—
322	C₈	—	94	1	no	—
323	C₈	—	188	1	no	—
324	C₈	D₈×D₄₇	1	1	yes	characteristic
325	C₈	SD₁₆×D₄₇	1	1	yes	characteristic
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	47	2	no	—
330	C₂²	—	94	1	no	—
331	C₂²	—	94	1	no	—
332	C₂²	—	94	2	no	—
333	C₂²	—	188	8	no	—
334	C₂²	—	188	1	no	—
335	C₂²	—	188	1	no	—
336	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	47	2	no	—
340	C₄	—	94	1	no	—
341	C₄	—	94	1	no	—
342	C₄	—	188	1	no	—
343	C₄	#102×D₄₇	1	1	yes	characteristic
344	C₄	#104×D₄₇	1	1	yes	characteristic
345	C₄	M₃₂×D₄₇	1	1	yes	characteristic
346	C₄	C₂×#13093963	1	1	yes	characteristic
347	C₂	—	4	2	no	—
348	C₂	—	94	2	no	—
349	C₂	—	188	2	no	—
350	C₂	D₄⋊C₈×D₄₇	1	1	yes	characteristic
351	C₂	#20244997	1	1	yes	characteristic
352	C₂	#20245014	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×D₄⋊C₁₆	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂×D₄×D₄₇	C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	D₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#11583
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #20245142	C₂	semidirect
3. C₄×#13093963	C₂	semidirect
4. #20241344	C₂	semidirect
5. D₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
6. #20241342	C₂	semidirect
7. C₄₇×D₄⋊C₁₆	C₂	semidirect
20. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
23. C₄×C₄₇⋊C₁₆	C₂²	semidirect
25. C₄₇×C₄⋊C₁₆	C₂²	semidirect
26. C₄×C₁₆×C₄₇	C₂²	semidirect
118. D₄×C₄₇	M₃₂	semidirect
132. D₁₈₈	M₃₂	semidirect
177. D₄⋊C₁₆	D₄₇	semidirect
199. C₄⋊C₁₆	C₂×D₄₇	semidirect
202. C₄×C₁₆	C₂×D₄₇	semidirect
203. C₄₇	#11583	semidirect
298. D₄	#13093963	semidirect

Maximal subgroups

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

subgroup	index
C₈×D₄×D₄₇	2
#20245142	2
C₄×#13093963	2
#20241344	2
D₄⋊(C₄₇⋊C₁₆)	2
#20241342	2
C₄₇×D₄⋊C₁₆	2
#11583	47