Subgroups — C₄₇×#11631 #26051329

Order 12032.

422 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#2162	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×D₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₄₇	—	2	1	no	—
6	C₂×C₄×D₄×C₄₇	C₄	1	1	yes	characteristic
7	C₈×D₄×C₄₇	—	2	1	no	—
8	C₈×D₄×C₄₇	—	2	1	no	—
9	C₈×D₄×C₄₇	C₄	1	2	yes	—
10	C₈×D₄×C₄₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×C₄⋊C₁₆	—	2	1	no	—
15	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	complemented
16	C₄₇×C₂²⋊C₁₆	—	2	1	no	—
17	C₄₇×C₂²⋊C₁₆	—	2	1	no	—
18	C₄₇×#346	—	2	1	no	—
19	C₄×C₁₆×C₄₇	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₄₇	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₄₇	—	2	1	no	—
22	C₂²×C₈×C₄₇	—	2	1	no	—
23	C₂²×C₈×C₄₇	—	2	2	no	—
24	C₂²×C₈×C₄₇	—	2	2	no	—
25	C₂²×C₈×C₄₇	—	4	4	no	—
26	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₄×D₄×C₄₇	—	2	1	no	—
29	C₄×D₄×C₄₇	—	2	1	no	—
30	C₄×D₄×C₄₇	C₈	1	2	yes	—
31	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₄₇	C₈	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₄₇	—	2	2	no	—
36	C₂×C₁₆×C₄₇	—	2	2	no	—
37	C₄⋊C₈×C₄₇	—	2	1	no	—
38	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₂²⋊C₈×C₄₇	—	2	2	no	—
40	C₂²⋊C₈×C₄₇	—	2	2	no	—
41	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	C₂⁴×C₄₇	—	2	1	no	—
46	C₂×D₄×C₄₇	—	2	4	no	—
47	C₂×D₄×C₄₇	—	2	1	no	—
48	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₄₇	—	2	1	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	—	4	1	no	—
60	C₂×C₈×C₄₇	—	4	4	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₂×C₈×C₄₇	—	4	4	no	—
63	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₄₇	—	2	1	no	—
68	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
69	C₂²⋊C₄×C₄₇	—	2	2	no	—
70	C₂²⋊C₄×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
74	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₁₆×C₄₇	—	4	2	no	—
76	C₁₆×C₄₇	—	4	2	no	—
77	C₂³×C₄₇	—	2	4	no	—
78	C₂³×C₄₇	—	2	2	no	—
79	C₂³×C₄₇	—	4	2	no	—
80	C₂³×C₄₇	—	4	2	no	—
81	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
82	D₄×C₄₇	—	2	4	no	—
83	D₄×C₄₇	—	2	2	no	—
84	D₄×C₄₇	M₃₂	1	4	yes	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	4	4	no	—
91	C₂×C₄×C₄₇	—	4	2	no	—
92	C₂×C₄×C₄₇	—	4	2	no	—
93	C₂×C₄×C₄₇	—	4	2	no	—
94	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
100	C₈×C₄₇	—	2	1	no	—
101	C₈×C₄₇	—	2	1	no	—
102	C₈×C₄₇	—	2	1	no	—
103	C₈×C₄₇	—	4	2	no	—
104	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
105	#11631	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₄₇	—	2	1	no	—
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	—	2	4	no	—
109	C₂²×C₄₇	—	4	8	no	—
110	C₂²×C₄₇	—	4	2	no	—
111	C₂²×C₄₇	—	4	2	no	—
112	C₂²×C₄₇	#350	1	2	yes	—
113	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₄×C₄₇	—	2	1	no	—
115	C₄×C₄₇	—	2	1	no	—
116	C₄×C₄₇	—	2	1	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	—	4	2	no	—
119	C₄×C₄₇	#418	1	1	yes	characteristic
120	C₄×C₄₇	#421	1	1	yes	characteristic
121	C₄×C₄₇	C₂×M₃₂	1	2	yes	—
122	C₂×C₈×D₄	C₂×C₄₇	1	1	yes	characteristic
123	#2163	C₂×C₄₇	1	1	yes	characteristic, complemented
124	#2162	C₂×C₄₇	1	1	yes	characteristic, complemented
125	D₄⋊C₁₆	C₂×C₄₇	1	4	yes	complemented
126	C₂×C₄₇	—	2	2	no	—
127	C₂×C₄₇	—	4	4	no	—
128	C₂×C₄₇	#1478	1	1	yes	characteristic
129	C₂×C₄₇	#2112	1	1	yes	characteristic
130	C₂×C₄₇	C₂×#350	1	1	yes	characteristic
131	C₂³×C₈	—	2	1	no	—
132	C₂×C₄×D₄	C₄×C₄₇	1	1	yes	characteristic
133	C₈×D₄	—	2	1	no	—
134	C₈×D₄	—	2	1	no	—
135	C₈×D₄	C₄×C₄₇	1	2	yes	—
136	C₈×D₄	C₂²×C₄₇	1	2	yes	—
137	C₂×C₄⋊C₈	C₄×C₄₇	1	1	yes	characteristic
138	C₂×C₂²⋊C₈	—	2	1	no	—
139	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
140	C₄⋊C₁₆	—	2	1	no	—
141	C₄⋊C₁₆	C₂²×C₄₇	1	2	yes	complemented
142	C₂²⋊C₁₆	—	2	1	no	—
143	C₂²⋊C₁₆	—	2	1	no	—
144	#346	—	2	1	no	—
145	C₄×C₁₆	C₂²×C₄₇	1	2	yes	complemented
146	C₄₇	#11631	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×D₄	C₈×C₄₇	1	1	yes	characteristic
148	C₂³×C₄	—	2	1	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	4	4	no	—
153	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
154	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	—	2	1	no	—
157	C₄×D₄	C₈×C₄₇	1	2	yes	—
158	C₄×D₄	C₂×C₄×C₄₇	1	2	yes	—
159	C₂×C₄⋊C₄	C₈×C₄₇	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	2	1	no	—
161	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	—	2	2	no	—
164	C₄⋊C₈	—	2	1	no	—
165	C₄⋊C₈	C₂×C₄×C₄₇	1	2	yes	—
166	C₂²⋊C₈	—	2	2	no	—
167	C₂²⋊C₈	—	2	2	no	—
168	C₄×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
169	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
170	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
171	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
172	C₂⁴	—	2	1	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	C₂×C₈×C₄₇	1	2	yes	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₈×C₄₇	1	1	yes	characteristic
183	C₂²×C₄	M₁₆×C₄₇	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	4	no	—
190	C₂×C₈	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
192	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
193	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	C₂×C₈×C₄₇	1	2	yes	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₄²	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
199	C₄²	C₂×C₈×C₄₇	1	1	yes	characteristic
200	C₄²	M₁₆×C₄₇	1	1	yes	characteristic
201	C₄²	C₂²×C₄×C₄₇	1	1	yes	characteristic
202	C₁₆	—	4	2	no	—
203	C₁₆	—	4	2	no	—
204	C₂³	—	2	4	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	4	2	no	—
207	C₂³	—	4	2	no	—
208	C₂³	C₂²⋊C₈×C₄₇	1	1	yes	characteristic
209	D₄	—	2	4	no	—
210	D₄	—	2	2	no	—
211	D₄	M₃₂×C₄₇	1	4	yes	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₂²⋊C₈×C₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂²⋊C₈×C₄₇	1	1	yes	characteristic
223	C₂×C₄	C₂²⋊C₈×C₄₇	1	1	yes	characteristic
224	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₈×C₄₇	1	1	yes	characteristic
226	C₂×C₄	C₂×M₁₆×C₄₇	1	1	yes	characteristic
227	C₈	—	2	1	no	—
228	C₈	—	2	1	no	—
229	C₈	—	2	1	no	—
230	C₈	—	4	2	no	—
231	C₈	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	4	8	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₄₇×#350	1	2	yes	—
239	C₂²	C₂×C₂²⋊C₈×C₄₇	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	2	no	—
245	C₄	C₄₇×#418	1	1	yes	characteristic
246	C₄	C₄₇×#421	1	1	yes	characteristic
247	C₄	C₂×M₃₂×C₄₇	1	2	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	4	4	no	—
250	C₂	C₄₇×#1478	1	1	yes	characteristic
251	C₂	C₄₇×#2112	1	1	yes	characteristic
252	C₂	C₂×C₄₇×#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₄₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₄₇
Frattini, cosocle	C₄×C₈	C₂³×C₄₇
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₄₇	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₄₇ ◁ C₂×C₄×C₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₇×#2163	C₂	semidirect
3. C₄₇×#2162	C₂	semidirect
4. C₄₇×D₄⋊C₁₆	C₂	semidirect
15. C₄₇×C₄⋊C₁₆	C₂²	semidirect
19. C₄×C₁₆×C₄₇	C₂²	semidirect
105. #11631	C₄₇	direct (N × H)
123. #2163	C₂×C₄₇	semidirect
124. #2162	C₂×C₄₇	semidirect
125. D₄⋊C₁₆	C₂×C₄₇	semidirect
141. C₄⋊C₁₆	C₂²×C₄₇	semidirect
145. C₄×C₁₆	C₂²×C₄₇	semidirect
146. C₄₇	#11631	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₄₇	2
C₄₇×#2163	2
C₄₇×#2162	2
C₄₇×D₄⋊C₁₆	2
#11631	47

Subgroups — C47×#11631 #26051329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#11631 #26051329