Subgroups — #120×C₄₇⋊C₈ #26221774

Order 12032.

1786 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×C₄₇⋊C₈	C₂	1	3	yes	maximal, complemented
3	#20243681	C₂	1	3	yes	maximal, complemented
4	#20243568	C₂	1	3	yes	maximal, complemented
5	#20243298	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊C₄×C₄₇⋊C₈	C₂	1	3	yes	maximal, complemented
7	C₄×#120×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	—	2	12	no	—
9	C₂³×C₄₇⋊C₈	C₂²	1	3	yes	—
10	C₂³×C₄₇⋊C₈	C₂²	1	3	yes	—
11	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
12	D₄×C₄₇⋊C₈	—	2	12	no	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	complemented
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
15	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
16	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	3	yes	complemented
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	complemented
18	#15688878	C₂²	1	6	yes	complemented
19	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#120×C₄₇	C₄	1	1	yes	characteristic
21	C₂×C₄×D₄×C₄₇	C₂²	1	3	yes	—
22	C₄₇×#386	C₄	1	3	yes	—
23	C₄₇×#385	C₄	1	3	yes	—
24	C₄₇×#379	C₄	1	1	yes	characteristic
25	C₄×C₂²⋊C₄×C₄₇	C₂²	1	3	yes	—
26	C₂²×C₄₇⋊C₈	—	2	32	no	—
27	C₂²×C₄₇⋊C₈	—	2	6	no	—
28	C₂²×C₄₇⋊C₈	—	2	12	no	—
29	C₂²×C₄₇⋊C₈	—	2	12	no	—
30	C₂²×C₄₇⋊C₈	—	2	6	no	—
31	C₂²×C₄₇⋊C₈	D₄	1	12	yes	complemented
32	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
36	C₄⋊(C₄₇⋊C₈)	—	2	6	no	—
37	C₂²⋊(C₄₇⋊C₈)	—	2	6	no	—
38	C₂²⋊(C₄₇⋊C₈)	—	2	6	no	—
39	C₂²⋊(C₄₇⋊C₈)	—	2	6	no	—
40	C₄×C₄₇⋊C₈	—	2	6	no	—
41	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×D₄×C₄₇	C₂×C₄	1	3	yes	—
43	C₂³×C₄×C₄₇	—	2	6	no	—
44	C₂³×C₄×C₄₇	—	2	6	no	—
45	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
48	#120×C₄₇	C₈	1	8	yes	complemented
49	C₄×D₄×C₄₇	—	2	6	no	—
50	C₄×D₄×C₄₇	—	2	6	no	—
51	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
52	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
53	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
54	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
55	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
57	#95×C₄₇	C₂×C₄	1	6	yes	—
58	C₂⁴×C₄₇	—	2	6	no	—
59	C₂⁴×C₄₇	—	2	6	no	—
60	C₂⁴×C₄₇	C₂×C₈	1	4	yes	complemented
61	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂⁴×C₄₇	C₂²×C₄	1	3	yes	—
63	C₂×D₄×C₄₇	—	2	6	no	—
64	C₂×D₄×C₄₇	—	2	6	no	—
65	C₂×D₄×C₄₇	C₂×C₈	1	12	yes	complemented
66	C₂²×C₄×C₄₇	—	2	24	no	—
67	C₂²×C₄×C₄₇	—	2	8	no	—
68	C₂²×C₄×C₄₇	—	2	6	no	—
69	C₂²×C₄×C₄₇	—	2	12	no	—
70	C₂²×C₄×C₄₇	—	2	6	no	—
71	C₂²×C₄×C₄₇	—	2	6	no	—
72	C₂²×C₄×C₄₇	—	2	3	no	—
73	C₂²×C₄×C₄₇	—	2	3	no	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
79	C₂²×C₄×C₄₇	#41	1	6	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₄₇	—	2	6	no	—
82	C₂²⋊C₄×C₄₇	—	2	6	no	—
83	C₂²⋊C₄×C₄₇	—	2	6	no	—
84	C₂²⋊C₄×C₄₇	—	2	6	no	—
85	C₂²⋊C₄×C₄₇	C₂×C₈	1	12	yes	complemented
86	C₄²×C₄₇	—	2	3	no	—
87	C₄²×C₄₇	—	2	3	no	—
88	C₂×C₄₇⋊C₈	—	2	24	no	—
89	C₂×C₄₇⋊C₈	—	2	6	no	—
90	C₂×C₄₇⋊C₈	—	2	24	no	—
91	C₂×C₄₇⋊C₈	—	2	3	no	—
92	C₂×C₄₇⋊C₈	—	2	3	no	—
93	C₂×C₄₇⋊C₈	—	2	6	no	—
94	C₂×C₄₇⋊C₈	—	4	4	no	—
95	C₂×C₄₇⋊C₈	C₂×D₄	1	6	yes	—
96	C₂×C₄₇⋊C₈	C₂×D₄	1	6	yes	—
97	C₂³×C₄₇	—	2	8	no	—
98	C₂³×C₄₇	—	2	24	no	—
99	C₂³×C₄₇	—	2	6	no	—
100	C₂³×C₄₇	—	2	6	no	—
101	C₂³×C₄₇	—	2	24	no	—
102	C₂³×C₄₇	—	2	3	no	—
103	C₂³×C₄₇	—	2	3	no	—
104	C₂³×C₄₇	C₄×D₄	1	6	yes	—
105	C₂³×C₄₇	C₂²×C₈	1	6	yes	—
106	C₂³×C₄₇	C₂²×C₈	1	2	yes	complemented
107	C₂³×C₄₇	#131	1	6	yes	—
108	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	D₄×C₄₇	—	2	24	no	—
110	C₂×C₄×C₄₇	—	2	24	no	—
111	C₂×C₄×C₄₇	—	2	3	no	—
112	C₂×C₄×C₄₇	—	2	6	no	—
113	C₂×C₄×C₄₇	—	2	3	no	—
114	C₂×C₄×C₄₇	—	2	6	no	—
115	C₂×C₄×C₄₇	—	2	6	no	—
116	C₂×C₄×C₄₇	—	2	12	no	—
117	C₂×C₄×C₄₇	—	2	3	no	—
118	C₂×C₄×C₄₇	—	2	3	no	—
119	C₂×C₄×C₄₇	—	2	6	no	—
120	C₂×C₄×C₄₇	—	4	1	no	—
121	C₂×C₄×C₄₇	—	4	3	no	—
122	C₂×C₄×C₄₇	C₄×D₄	1	6	yes	—
123	C₂×C₄×C₄₇	C₂²×C₈	1	6	yes	complemented
124	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
125	C₂×C₄×C₄₇	C₂×#41	1	3	yes	—
126	C₄₇⋊C₈	—	2	6	no	—
127	C₄₇⋊C₈	—	4	1	no	—
128	C₄₇⋊C₈	—	4	3	no	—
129	C₄₇⋊C₈	#120	1	4	yes	direct product factor
130	C₈×#120	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	32	no	—
132	C₂²×C₄₇	—	2	12	no	—
133	C₂²×C₄₇	—	2	6	no	—
134	C₂²×C₄₇	—	2	12	no	—
135	C₂²×C₄₇	—	2	6	no	—
136	C₂²×C₄₇	—	2	6	no	—
137	C₂²×C₄₇	—	4	3	no	—
138	C₂²×C₄₇	—	4	1	no	—
139	C₂²×C₄₇	C₈×D₄	1	12	yes	complemented
140	C₂²×C₄₇	C₂×C₄×D₄	1	3	yes	—
141	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#131	1	3	yes	—
143	C₄×C₄₇	—	2	6	no	—
144	C₄×C₄₇	—	2	6	no	—
145	C₄×C₄₇	—	4	3	no	—
146	C₄×C₄₇	—	4	1	no	—
147	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
148	C₄×C₄₇	#525	1	3	yes	—
149	C₂⁴×C₈	—	47	1	no	—
150	C₂×C₈×D₄	—	47	3	no	—
151	C₄×#120	D₄₇	1	1	yes	characteristic
152	#1869	—	47	3	no	—
153	#1850	—	47	3	no	—
154	#1779	—	47	1	no	—
155	C₈×C₂²⋊C₄	—	47	3	no	—
156	C₂×C₄₇	—	2	12	no	—
157	C₂×C₄₇	—	4	2	no	—
158	C₂×C₄₇	C₄×#120	1	1	yes	characteristic
159	C₂×C₄₇	C₂×C₈×D₄	1	3	yes	—
160	C₂×C₄₇	#2929	1	3	yes	—
161	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
162	C₂³×C₈	—	47	3	no	—
163	C₂³×C₈	—	47	3	no	—
164	C₂³×C₈	—	47	1	no	—
165	C₂³×C₈	—	94	12	no	—
166	C₂×#120	Dic₄₇	1	1	yes	characteristic
167	C₂×C₄×D₄	C₂×D₄₇	1	3	yes	—
168	C₈×D₄	—	94	12	no	—
169	C₂×C₄⋊C₈	—	47	3	no	—
170	C₂×C₂²⋊C₈	—	47	3	no	—
171	C₂×C₂²⋊C₈	—	47	3	no	—
172	C₂×C₂²⋊C₈	—	47	3	no	—
173	C₂×C₄×C₈	—	47	3	no	—
174	#386	Dic₄₇	1	3	yes	—
175	#385	Dic₄₇	1	3	yes	—
176	#379	Dic₄₇	1	1	yes	characteristic
177	C₄×C₂²⋊C₄	C₂×D₄₇	1	3	yes	—
178	#336	—	47	6	no	—
179	C₄₇	C₈×#120	1	1	yes	characteristic, complemented, Sylow
180	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂²×D₄	C₂×Dic₄₇	1	3	yes	—
182	C₂³×C₄	—	2	6	no	—
183	C₂³×C₄	—	2	6	no	—
184	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
185	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
187	C₂²×C₈	—	47	1	no	—
188	C₂²×C₈	—	47	12	no	—
189	C₂²×C₈	—	47	3	no	—
190	C₂²×C₈	—	47	1	no	—
191	C₂²×C₈	—	47	3	no	—
192	C₂²×C₈	—	94	12	no	—
193	C₂²×C₈	—	94	12	no	—
194	C₂²×C₈	—	94	6	no	—
195	C₂²×C₈	—	94	32	no	—
196	C₂²×C₈	—	94	6	no	—
197	#120	C₄₇⋊C₈	1	8	yes	direct product factor
198	C₄×D₄	—	2	6	no	—
199	C₄×D₄	—	2	6	no	—
200	C₂×C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
201	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	3	yes	—
202	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	3	yes	—
203	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	3	yes	—
204	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	3	yes	—
205	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
206	C₄⋊C₈	—	94	6	no	—
207	C₂²⋊C₈	—	94	6	no	—
208	C₂²⋊C₈	—	94	6	no	—
209	C₂²⋊C₈	—	94	6	no	—
210	C₄×C₈	—	94	6	no	—
211	#95	C₂×Dic₄₇	1	6	yes	—
212	C₂⁴	—	2	6	no	—
213	C₂⁴	—	2	6	no	—
214	C₂⁴	C₂×C₄₇⋊C₈	1	4	yes	complemented
215	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
216	C₂⁴	C₂²×Dic₄₇	1	3	yes	—
217	C₂×D₄	—	2	6	no	—
218	C₂×D₄	—	2	6	no	—
219	C₂×D₄	C₂×C₄₇⋊C₈	1	12	yes	complemented
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	6	no	—
223	C₂²×C₄	—	2	12	no	—
224	C₂²×C₄	—	2	6	no	—
225	C₂²×C₄	—	2	6	no	—
226	C₂²×C₄	—	2	3	no	—
227	C₂²×C₄	—	2	3	no	—
228	C₂²×C₄	D₄×D₄₇	1	6	yes	—
229	C₂²×C₄	#10625352	1	6	yes	—
230	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
231	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
233	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
234	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
235	C₂×C₈	—	47	6	no	—
236	C₂×C₈	—	47	6	no	—
237	C₂×C₈	—	94	24	no	—
238	C₂×C₈	—	94	24	no	—
239	C₂×C₈	—	94	6	no	—
240	C₂×C₈	—	94	3	no	—
241	C₂×C₈	—	94	3	no	—
242	C₂×C₈	—	94	6	no	—
243	C₂×C₈	—	188	4	no	—
244	C₄⋊C₄	—	2	6	no	—
245	C₂²⋊C₄	—	2	6	no	—
246	C₂²⋊C₄	—	2	6	no	—
247	C₂²⋊C₄	—	2	6	no	—
248	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	12	yes	complemented
249	C₄²	—	2	3	no	—
250	C₄²	—	2	3	no	—
251	C₂³	—	2	8	no	—
252	C₂³	—	2	24	no	—
253	C₂³	—	2	6	no	—
254	C₂³	—	2	6	no	—
255	C₂³	—	2	24	no	—
256	C₂³	—	2	3	no	—
257	C₂³	—	2	3	no	—
258	C₂³	D₄×Dic₄₇	1	6	yes	—
259	C₂³	C₂²×C₄₇⋊C₈	1	6	yes	—
260	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	complemented
261	C₂³	#13094031	1	6	yes	—
262	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
263	D₄	—	2	24	no	—
264	C₂×C₄	—	2	24	no	—
265	C₂×C₄	—	2	3	no	—
266	C₂×C₄	—	2	6	no	—
267	C₂×C₄	—	2	3	no	—
268	C₂×C₄	—	2	6	no	—
269	C₂×C₄	—	2	6	no	—
270	C₂×C₄	—	2	12	no	—
271	C₂×C₄	—	2	3	no	—
272	C₂×C₄	—	2	3	no	—
273	C₂×C₄	—	2	6	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	3	no	—
276	C₂×C₄	D₄×Dic₄₇	1	6	yes	—
277	C₂×C₄	C₂²×C₄₇⋊C₈	1	6	yes	complemented
278	C₂×C₄	C₂×D₄×D₄₇	1	3	yes	—
279	C₂×C₄	C₂×#10625352	1	3	yes	—
280	C₈	—	47	4	no	—
281	C₈	—	94	6	no	—
282	C₈	—	188	1	no	—
283	C₈	—	188	3	no	—
284	C₂²	—	2	32	no	—
285	C₂²	—	2	6	no	—
286	C₂²	—	2	12	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	2	12	no	—
289	C₂²	—	2	6	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	3	no	—
292	C₂²	D₄×C₄₇⋊C₈	1	12	yes	complemented
293	C₂²	C₂×D₄×Dic₄₇	1	3	yes	—
294	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
295	C₂²	C₂×#13094031	1	3	yes	—
296	C₄	—	2	6	no	—
297	C₄	—	2	6	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	3	no	—
300	C₄	#120×D₄₇	1	1	yes	characteristic
301	C₄	#15689629	1	3	yes	—
302	C₂	—	2	12	no	—
303	C₂	—	4	2	no	—
304	C₂	#120×Dic₄₇	1	1	yes	characteristic
305	C₂	C₂×D₄×C₄₇⋊C₈	1	3	yes	—
306	C₂	#20252083	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Fitting	C₄×#120×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂×#120×C₄₇	C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#120	D₄₇
hypocenter	C₄₇	C₈×#120
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#120

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₄₇⋊C₈	C₂	semidirect
2. C₂×D₄×C₄₇⋊C₈	C₂	semidirect
3. #20243681	C₂	semidirect
4. #20243568	C₂	semidirect
5. #20243298	C₂	semidirect
6. C₂²⋊C₄×C₄₇⋊C₈	C₂	semidirect
11. C₂³×C₄₇⋊C₈	C₂²	semidirect
13. C₂×C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
16. C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	semidirect
17. C₂×C₄×C₄₇⋊C₈	C₂²	semidirect
18. #15688878	C₂²	semidirect
31. C₂²×C₄₇⋊C₈	D₄	semidirect
48. #120×C₄₇	C₈	semidirect
60. C₂⁴×C₄₇	C₂×C₈	semidirect
65. C₂×D₄×C₄₇	C₂×C₈	semidirect
85. C₂²⋊C₄×C₄₇	C₂×C₈	semidirect
106. C₂³×C₄₇	C₂²×C₈	semidirect
123. C₂×C₄×C₄₇	C₂²×C₈	semidirect
129. C₄₇⋊C₈	#120	direct (N × H)
139. C₂²×C₄₇	C₈×D₄	semidirect
179. C₄₇	C₈×#120	semidirect
197. #120	C₄₇⋊C₈	direct (N × H)
214. C₂⁴	C₂×C₄₇⋊C₈	semidirect
219. C₂×D₄	C₂×C₄₇⋊C₈	semidirect
248. C₂²⋊C₄	C₂×C₄₇⋊C₈	semidirect
260. C₂³	C₂²×C₄₇⋊C₈	semidirect
277. C₂×C₄	C₂²×C₄₇⋊C₈	semidirect
292. C₂²	D₄×C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₄₇⋊C₈	2
C₂×D₄×C₄₇⋊C₈	2
#20243681	2
#20243568	2
#20243298	2
C₂²⋊C₄×C₄₇⋊C₈	2
C₄×#120×C₄₇	2
C₈×#120	47

Subgroups — #120×C47⋊C8 #26221774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #120×C₄₇⋊C₈ #26221774