Subgroups — #26126147

Order 12032.

362 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#10625352	C₂	1	1	yes	maximal, characteristic
2	#20245185	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241368	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241366	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10625352	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₄₇	—	2	1	no	—
10	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
11	#15689247	C₄	1	1	yes	characteristic
12	D₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₂²⋊D₄₇	—	2	1	no	—
14	(C₂²⋊D₄₇)⋊C₈	—	2	1	no	—
15	#15689175	C₄	1	1	yes	characteristic
16	#15689173	—	2	1	no	—
17	C₂×C₈×Dic₄₇	—	2	1	no	—
18	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
20	Dic₄₇⋊C₁₆	—	2	1	no	—
21	Dic₄₇⋊C₁₆	—	2	1	no	—
22	(C₂×D₄₇)⋊C₁₆	—	2	1	no	—
23	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625352	C₈	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	—	2	1	no	—
29	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
30	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
31	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
32	#13093989	C₈	1	1	yes	characteristic
33	D₄×Dic₄₇	C₈	1	1	yes	characteristic
34	C₄×C₂²⋊D₄₇	—	2	1	no	—
35	(C₂²⋊D₄₇)⋊C₄	—	2	1	no	—
36	#13093982	C₈	1	1	yes	characteristic
37	#13093980	—	2	1	no	—
38	C₂×C₄×Dic₄₇	—	2	1	no	—
39	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄₇⋊C₁₆	—	2	1	no	—
41	C₂×C₄₇⋊C₁₆	—	2	1	no	—
42	C₁₆×D₄₇	—	4	2	no	—
43	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
44	Dic₄₇⋊C₈	—	2	1	no	—
45	Dic₄₇⋊C₈	—	2	1	no	—
46	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
47	(C₂×D₄₇)⋊C₈	—	2	1	no	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₄₇	—	2	1	no	—
50	C₈×Dic₄₇	—	2	1	no	—
51	C₈×Dic₄₇	—	2	1	no	—
52	C₈×Dic₄₇	D₄	1	1	yes	characteristic
53	C₈×Dic₄₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	—	2	1	no	—
55	C₄×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₄₇	—	2	1	no	—
57	C₂×C₁₆×C₄₇	—	2	1	no	—
58	C₄⋊C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₈×C₄₇	—	2	1	no	—
60	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂×C₂²⋊D₄₇	—	2	1	no	—
64	C₂²×Dic₄₇	—	2	1	no	—
65	C₂²×Dic₄₇	—	2	1	no	—
66	#10625352	—	2	2	no	—
67	#10625352	C₁₆	1	4	yes	complemented
68	C₂×C₄×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₄₇	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₄₇	M₁₆	1	1	yes	characteristic
71	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₄₇	—	2	1	no	—
73	C₂×C₈×C₄₇	—	4	1	no	—
74	C₂×C₈×C₄₇	—	4	1	no	—
75	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
78	C₂²⋊C₄×C₄₇	—	2	1	no	—
79	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²⋊Dic₄₇	—	2	1	no	—
81	(C₂×D₄₇)⋊C₄	—	2	1	no	—
82	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
83	Dic₄₇⋊C₄	—	2	1	no	—
84	Dic₄₇⋊C₄	—	2	1	no	—
85	C₄×Dic₄₇	—	2	1	no	—
86	C₄×Dic₄₇	—	2	1	no	—
87	C₄×Dic₄₇	—	2	1	no	—
88	C₄×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
90	C₄×Dic₄₇	M₁₆	1	1	yes	characteristic
91	C₂×C₄₇⋊C₈	—	2	1	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₈×D₄₇	—	2	1	no	—
96	C₈×D₄₇	—	2	1	no	—
97	C₈×D₄₇	—	4	1	no	—
98	C₁₆×C₄₇	—	4	1	no	—
99	C₁₆×C₄₇	—	4	1	no	—
100	C₄₇⋊C₁₆	—	4	1	no	—
101	C₄₇⋊C₁₆	—	4	1	no	—
102	C₂³×C₄₇	—	2	1	no	—
103	C₂²×D₄₇	C₂²⋊C₈	1	1	yes	characteristic
104	D₄×C₄₇	—	2	1	no	—
105	D₄×C₄₇	C₂×C₁₆	1	2	yes	complemented
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	—	4	1	no	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂²⋊D₄₇	—	2	4	no	—
113	C₂×Dic₄₇	—	2	1	no	—
114	C₂×Dic₄₇	—	2	4	no	—
115	C₂×Dic₄₇	—	2	1	no	—
116	C₂×Dic₄₇	—	4	2	no	—
117	C₂×Dic₄₇	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×Dic₄₇	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×Dic₄₇	C₂²⋊C₈	1	1	yes	characteristic
120	C₄×D₄₇	—	2	1	no	—
121	C₄×D₄₇	—	2	1	no	—
122	C₄×D₄₇	—	4	1	no	—
123	C₄×D₄₇	C₂×C₁₆	1	2	yes	—
124	C₄×D₄₇	M₃₂	1	2	yes	—
125	Dic₉₄	—	2	1	no	—
126	Dic₉₄	C₂×C₁₆	1	2	yes	complemented
127	C₈×C₄₇	—	2	1	no	—
128	C₈×C₄₇	—	4	1	no	—
129	C₈×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₈×C₄₇	#137	1	1	yes	characteristic
131	C₄₇⋊C₈	—	2	1	no	—
132	C₄₇⋊C₈	—	2	1	no	—
133	C₄₇⋊C₈	—	4	1	no	—
134	#11621	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	4	1	no	—
137	C₂²×C₄₇	—	4	1	no	—
138	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₂×D₄₇	—	2	1	no	—
140	C₂×D₄₇	—	2	1	no	—
141	C₂×D₄₇	C₂²⋊C₁₆	1	2	yes	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	4	1	no	—
144	C₄×C₄₇	#417	1	1	yes	characteristic
145	C₄×C₄₇	#421	1	1	yes	characteristic
146	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
147	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
148	Dic₄₇	—	2	2	no	—
149	Dic₄₇	—	2	1	no	—
150	Dic₄₇	—	2	1	no	—
151	Dic₄₇	—	4	1	no	—
152	Dic₄₇	C₂²⋊C₁₆	1	2	yes	—
153	C₈×#41	—	47	1	no	—
154	#2162	—	47	1	no	—
155	C₂×C₄⋊C₁₆	—	47	1	no	—
156	Q₈⋊C₁₆	—	47	1	no	—
157	Q₈⋊C₁₆	—	47	1	no	—
158	D₄⋊C₁₆	—	47	1	no	—
159	D₄⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
160	C₂×C₄₇	—	4	2	no	—
161	C₂×C₄₇	#1477	1	1	yes	characteristic
162	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
163	C₂×C₄₇	#2117	1	1	yes	characteristic
164	D₄₇	—	2	2	no	—
165	C₄×#41	—	47	1	no	—
166	C₂²×C₁₆	—	94	1	no	—
167	C₈×Q₈	—	47	1	no	—
168	C₈×Q₈	—	47	1	no	—
169	C₈×D₄	—	47	1	no	—
170	C₈×D₄	—	94	1	no	—
171	C₈×D₄	—	94	1	no	—
172	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
173	#431	—	47	1	no	—
174	#431	—	94	1	no	—
175	C₂×C₄×C₈	—	47	1	no	—
176	C₂×C₄×C₈	—	94	1	no	—
177	C₄⋊C₁₆	—	47	1	no	—
178	C₄⋊C₁₆	—	94	1	no	—
179	C₄⋊C₁₆	—	94	1	no	—
180	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
181	C₂²⋊C₁₆	—	94	1	no	—
182	C₄×C₁₆	—	47	1	no	—
183	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
184	C₄₇	#11621	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	47	1	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	47	1	no	—
188	C₂²×C₈	—	47	1	no	—
189	C₂²×C₈	—	94	1	no	—
190	C₄×Q₈	—	47	1	no	—
191	C₄×Q₈	—	47	1	no	—
192	C₄×D₄	—	47	1	no	—
193	C₄×D₄	—	94	1	no	—
194	C₄×D₄	—	94	1	no	—
195	C₄×D₄	C₄×D₄₇	1	1	yes	characteristic
196	#117	—	47	1	no	—
197	#117	—	94	1	no	—
198	C₂×C₄²	—	47	1	no	—
199	C₂×C₄²	—	94	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	—	94	1	no	—
203	C₂×C₁₆	—	94	1	no	—
204	C₂×C₁₆	—	188	2	no	—
205	C₄⋊C₈	—	47	1	no	—
206	C₄⋊C₈	—	94	1	no	—
207	C₄⋊C₈	—	94	1	no	—
208	C₄⋊C₈	C₄×D₄₇	1	1	yes	characteristic
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	94	1	no	—
211	C₂²⋊C₈	—	94	1	no	—
212	C₄×C₈	—	47	1	no	—
213	C₄×C₈	—	47	1	no	—
214	C₄×C₈	—	47	1	no	—
215	C₄×C₈	—	94	1	no	—
216	C₄×C₈	—	94	1	no	—
217	C₄×C₈	—	94	1	no	—
218	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	#41	—	47	4	no	—
220	#41	—	94	2	no	—
221	C₂×Q₈	—	47	1	no	—
222	C₂×D₄	—	94	1	no	—
223	C₂×D₄	C₈×D₄₇	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	47	1	no	—
226	C₂²×C₄	—	47	1	no	—
227	C₂²×C₄	—	47	1	no	—
228	C₂²×C₄	—	94	1	no	—
229	C₂²×C₄	—	94	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	47	1	no	—
234	C₂×C₈	—	47	1	no	—
235	C₂×C₈	—	94	1	no	—
236	C₂×C₈	—	94	1	no	—
237	C₂×C₈	—	94	1	no	—
238	C₂×C₈	—	188	1	no	—
239	C₂×C₈	—	188	2	no	—
240	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
241	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
242	C₄⋊C₄	—	47	1	no	—
243	C₄⋊C₄	—	94	1	no	—
244	C₄⋊C₄	—	94	1	no	—
245	C₄⋊C₄	C₈×D₄₇	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	94	1	no	—
248	C₂²⋊C₄	—	94	1	no	—
249	C₄²	—	47	1	no	—
250	C₄²	—	47	1	no	—
251	C₄²	—	47	1	no	—
252	C₄²	—	94	1	no	—
253	C₄²	—	94	1	no	—
254	C₄²	—	94	1	no	—
255	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
256	C₁₆	—	4	1	no	—
257	C₁₆	—	4	1	no	—
258	C₁₆	—	188	1	no	—
259	C₁₆	—	188	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	47	1	no	—
262	Q₈	—	47	2	no	—
263	Q₈	—	94	1	no	—
264	D₄	—	2	1	no	—
265	D₄	—	94	4	no	—
266	D₄	C₁₆×D₄₇	1	2	yes	complemented
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	47	1	no	—
271	C₂×C₄	—	47	2	no	—
272	C₂×C₄	—	47	2	no	—
273	C₂×C₄	—	47	1	no	—
274	C₂×C₄	—	47	1	no	—
275	C₂×C₄	—	94	4	no	—
276	C₂×C₄	—	94	1	no	—
277	C₂×C₄	—	94	1	no	—
278	C₂×C₄	—	94	1	no	—
279	C₂×C₄	—	94	1	no	—
280	C₂×C₄	—	188	1	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	4	1	no	—
287	C₈	—	94	1	no	—
288	C₈	—	94	1	no	—
289	C₈	—	188	1	no	—
290	C₈	#13094048	1	1	yes	characteristic
291	C₈	#13094053	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	47	2	no	—
296	C₂²	—	94	1	no	—
297	C₂²	—	94	1	no	—
298	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	47	2	no	—
302	C₄	—	94	2	no	—
303	C₄	—	94	1	no	—
304	C₄	—	94	1	no	—
305	C₄	—	188	1	no	—
306	C₄	#15689121	1	1	yes	characteristic
307	C₄	#15689138	1	1	yes	characteristic
308	C₄	C₂×C₁₆×D₄₇	1	1	yes	characteristic
309	C₄	M₃₂×D₄₇	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	—	94	2	no	—
312	C₂	#20241772	1	1	yes	characteristic
313	C₂	C₂²⋊C₁₆×D₄₇	1	1	yes	characteristic
314	C₂	#20245014	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×D₄⋊C₁₆	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂×#10625352	C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	D₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#11621
LCS	G ▷ 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. #20245185	C₂	semidirect
3. C₄⋊C₁₆×D₄₇	C₂	semidirect
4. #20241368	C₂	semidirect
5. #20241366	C₂	semidirect
6. D₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
7. C₄₇×D₄⋊C₁₆	C₂	semidirect
67. #10625352	C₁₆	semidirect
105. D₄×C₄₇	C₂×C₁₆	semidirect
126. Dic₉₄	C₂×C₁₆	semidirect
159. D₄⋊C₁₆	D₄₇	semidirect
184. C₄₇	#11621	semidirect
266. D₄	C₁₆×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₈×#10625352	2
#20245185	2
C₄⋊C₁₆×D₄₇	2
#20241368	2
#20241366	2
D₄⋊(C₄₇⋊C₁₆)	2
C₄₇×D₄⋊C₁₆	2
#11621	47