Subgroups — C₄₇×#8270 #26047968

Order 12032.

342 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1452	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#1276	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1276	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#431	C₂²	1	1	yes	characteristic
9	C₄₇×#431	C₂²	1	1	yes	characteristic
10	C₄₇×#431	C₂²	1	1	yes	characteristic
11	C₄₇×#431	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄³×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#336	C₄	1	2	yes	—
16	C₄₇×#336	C₄	1	2	yes	—
17	C₄₇×#335	C₄	1	2	yes	complemented
18	C₄₇×#335	C₄	1	2	yes	complemented
19	C₄₇×#334	C₄	1	2	yes	complemented
20	C₄₇×#334	C₄	1	2	yes	complemented
21	C₄₇×#322	—	2	1	no	—
22	C₄₇×#322	—	2	1	no	—
23	C₄₇×#322	—	2	1	no	—
24	C₈²×C₄₇	—	2	1	no	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₇	—	2	1	no	—
29	C₂×C₄²×C₄₇	—	2	1	no	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
33	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₂²⋊C₈×C₄₇	—	2	1	no	—
37	C₂²⋊C₈×C₄₇	—	2	1	no	—
38	C₂²⋊C₈×C₄₇	—	2	1	no	—
39	C₂²⋊C₈×C₄₇	—	2	1	no	—
40	C₄×C₈×C₄₇	—	2	1	no	—
41	C₄×C₈×C₄₇	—	2	1	no	—
42	C₄×C₈×C₄₇	—	2	1	no	—
43	C₄×C₈×C₄₇	—	2	1	no	—
44	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₄₇	Q₈	1	2	yes	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	—	2	2	no	—
60	C₂×C₈×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
64	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₄₇	—	2	4	no	—
70	C₄²×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	4	2	no	—
73	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
77	C₂³×C₄₇	#117	1	1	yes	characteristic
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	—	4	2	no	—
83	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#131	1	2	yes	—
90	C₂×C₄×C₄₇	#131	1	2	yes	—
91	C₈×C₄₇	—	2	1	no	—
92	C₈×C₄₇	—	2	1	no	—
93	C₈×C₄₇	—	2	1	no	—
94	C₈×C₄₇	—	2	1	no	—
95	C₈×C₄₇	—	2	2	no	—
96	C₈×C₄₇	—	2	2	no	—
97	C₈×C₄₇	—	4	2	no	—
98	C₈×C₄₇	—	4	2	no	—
99	C₈×C₄₇	—	4	2	no	—
100	C₈×C₄₇	—	4	2	no	—
101	#8270	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₄₇	—	2	1	no	—
103	C₂²×C₄₇	—	2	1	no	—
104	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₄₇	#433	1	2	yes	—
106	C₄×C₄₇	—	2	1	no	—
107	C₄×C₄₇	—	2	1	no	—
108	C₄×C₄₇	—	4	4	no	—
109	C₄×C₄₇	#405	1	1	yes	characteristic
110	C₄×C₄₇	#405	1	1	yes	characteristic
111	C₄×C₄₇	#424	1	2	yes	—
112	C₄×C₄₇	#443	1	2	yes	—
113	C₄×C₄₇	#444	1	2	yes	—
114	#1744	C₂×C₄₇	1	1	yes	characteristic
115	#1564	C₂×C₄₇	1	1	yes	characteristic
116	#1564	C₂×C₄₇	1	1	yes	characteristic
117	#1452	C₂×C₄₇	1	1	yes	characteristic
118	#1450	C₂×C₄₇	1	1	yes	characteristic
119	#1276	C₂×C₄₇	1	1	yes	characteristic
120	#1276	C₂×C₄₇	1	1	yes	characteristic
121	C₂×C₄₇	—	2	2	no	—
122	C₂×C₄₇	#1768	1	1	yes	characteristic
123	C₂×C₄₇	#1772	1	1	yes	characteristic
124	C₂×C₄₇	#1773	1	1	yes	characteristic
125	#431	C₂²×C₄₇	1	1	yes	characteristic
126	#431	C₂²×C₄₇	1	1	yes	characteristic
127	#431	C₂²×C₄₇	1	1	yes	characteristic
128	#431	C₂²×C₄₇	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
131	C₄³	C₂²×C₄₇	1	1	yes	characteristic
132	#336	C₄×C₄₇	1	2	yes	—
133	#336	C₄×C₄₇	1	2	yes	—
134	#335	C₄×C₄₇	1	2	yes	complemented
135	#335	C₄×C₄₇	1	2	yes	complemented
136	#334	C₄×C₄₇	1	2	yes	complemented
137	#334	C₄×C₄₇	1	2	yes	complemented
138	#322	—	2	1	no	—
139	#322	—	2	1	no	—
140	#322	—	2	1	no	—
141	C₈²	—	2	1	no	—
142	C₄₇	#8270	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
144	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
145	C₂²×C₈	C₂×C₄×C₄₇	1	2	yes	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
149	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
150	C₄⋊C₈	C₂×C₄×C₄₇	1	2	yes	—
151	C₄⋊C₈	C₂×C₄×C₄₇	1	2	yes	—
152	C₄⋊C₈	C₂×C₄×C₄₇	1	2	yes	—
153	C₄⋊C₈	C₂×C₄×C₄₇	1	2	yes	—
154	C₂²⋊C₈	—	2	1	no	—
155	C₂²⋊C₈	—	2	1	no	—
156	C₂²⋊C₈	—	2	1	no	—
157	C₂²⋊C₈	—	2	1	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
163	C₄×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
164	C₄×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
165	C₄×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
166	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
167	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
168	C₄×C₈	Q₈×C₄₇	1	2	yes	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
171	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
172	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	C₄²×C₄₇	1	1	yes	characteristic
182	C₂×C₈	C₄²×C₄₇	1	1	yes	characteristic
183	C₂×C₈	C₄²×C₄₇	1	1	yes	characteristic
184	C₂×C₈	C₄²×C₄₇	1	1	yes	characteristic
185	C₂×C₈	C₄⋊C₄×C₄₇	1	2	yes	—
186	C₂×C₈	C₄⋊C₄×C₄₇	1	2	yes	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	C₂²×C₄×C₄₇	1	1	yes	characteristic
192	C₄²	C₂²×C₄×C₄₇	1	1	yes	characteristic
193	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
194	C₄²	C₂×Q₈×C₄₇	1	1	yes	characteristic
195	C₂³	#117×C₄₇	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	C₂×C₄²×C₄₇	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
203	C₂×C₄	C₄×D₄×C₄₇	1	1	yes	characteristic
204	C₂×C₄	C₄×D₄×C₄₇	1	1	yes	characteristic
205	C₂×C₄	C₄×Q₈×C₄₇	1	1	yes	characteristic
206	C₂×C₄	C₄×Q₈×C₄₇	1	1	yes	characteristic
207	C₂×C₄	#131×C₄₇	1	2	yes	—
208	C₂×C₄	#131×C₄₇	1	2	yes	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	2	2	no	—
214	C₈	—	2	2	no	—
215	C₈	—	4	2	no	—
216	C₈	—	4	2	no	—
217	C₈	—	4	2	no	—
218	C₈	—	4	2	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₄×C₄⋊C₄×C₄₇	1	1	yes	characteristic
222	C₂²	C₄₇×#433	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	4	no	—
226	C₄	C₄₇×#405	1	1	yes	characteristic
227	C₄	C₄₇×#405	1	1	yes	characteristic
228	C₄	C₄₇×#424	1	2	yes	—
229	C₄	C₄₇×#443	1	2	yes	—
230	C₄	C₄₇×#444	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	C₄₇×#1768	1	1	yes	characteristic
233	C₂	C₄₇×#1772	1	1	yes	characteristic
234	C₂	C₄₇×#1773	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₄₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₄₇
Frattini, cosocle	C₂×C₄²	C₂³×C₄₇
socle	C₂²×C₄₇	C₄×C₄⋊C₄
prime radical	C₂³×C₄₇	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₄₇ ◁ C₂×C₄×C₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₄₇×#335	C₄	semidirect
18. C₄₇×#335	C₄	semidirect
19. C₄₇×#334	C₄	semidirect
20. C₄₇×#334	C₄	semidirect
101. #8270	C₄₇	direct (N × H)
134. #335	C₄×C₄₇	semidirect
135. #335	C₄×C₄₇	semidirect
136. #334	C₄×C₄₇	semidirect
137. #334	C₄×C₄₇	semidirect
142. C₄₇	#8270	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#1744	2
C₄₇×#1564	2
C₄₇×#1452	2
C₄₇×#1450	2
C₄₇×#1276	2
#8270	47

Subgroups — C47×#8270 #26047968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#8270 #26047968