Subgroups — C₈×SD₃₂×C₄₇ #26051739

Order 12032.

242 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×SD₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1372	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1332	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#1331	C₂	1	1	yes	maximal, characteristic
7	C₈×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×SD₃₂×C₄₇	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₄₇	—	2	1	no	—
10	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
11	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₈×D₄×C₄₇	—	2	1	no	—
13	C₄₇×#367	C₄	1	1	yes	characteristic
14	C₄₇×#358	C₄	1	1	yes	characteristic
15	C₄₇×#357	C₄	1	1	yes	characteristic
16	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#335	C₂²	1	1	yes	characteristic
19	C₄₇×#335	C₂²	1	1	yes	characteristic
20	C₄₇×Q₈⋊C₈	—	2	1	no	—
21	C₄₇×D₄⋊C₈	—	2	1	no	—
22	C₈²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	—	4	1	no	—
26	C₄×Q₈×C₄₇	—	2	1	no	—
27	C₄×D₄×C₄₇	—	2	1	no	—
28	SD₃₂×C₄₇	C₈	1	4	yes	direct product factor
29	C₂×C₁₆×C₄₇	—	2	1	no	—
30	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
32	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
33	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₄₇	—	2	1	no	—
35	C₄⋊C₈×C₄₇	—	2	1	no	—
36	C₄⋊C₈×C₄₇	—	4	1	no	—
37	#103×C₄₇	—	2	1	no	—
38	#102×C₄₇	—	2	1	no	—
39	C₂²⋊C₈×C₄₇	—	4	1	no	—
40	C₄×C₈×C₄₇	—	4	1	no	—
41	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₄₇	—	2	1	no	—
45	C₂×D₄×C₄₇	—	2	1	no	—
46	C₂²×C₄×C₄₇	—	4	1	no	—
47	Q₁₆×C₄₇	—	2	1	no	—
48	Q₁₆×C₄₇	C₂×C₈	1	2	yes	complemented
49	D₈×C₄₇	—	2	1	no	—
50	D₈×C₄₇	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₄₇	—	2	1	no	—
52	C₂×C₈×C₄₇	—	4	1	no	—
53	C₂×C₈×C₄₇	—	4	1	no	—
54	C₂×C₈×C₄₇	—	8	2	no	—
55	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
56	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	—	2	1	no	—
60	C₄⋊C₄×C₄₇	—	2	1	no	—
61	C₄⋊C₄×C₄₇	—	4	1	no	—
62	C₂²⋊C₄×C₄₇	—	4	1	no	—
63	C₄²×C₄₇	—	4	1	no	—
64	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₁₆×C₄₇	—	2	2	no	—
66	C₁₆×C₄₇	—	2	1	no	—
67	C₁₆×C₄₇	C₂×C₈	1	2	yes	complemented
68	C₂³×C₄₇	—	4	1	no	—
69	Q₈×C₄₇	—	2	2	no	—
70	Q₈×C₄₇	—	4	1	no	—
71	D₄×C₄₇	—	2	2	no	—
72	D₄×C₄₇	—	4	1	no	—
73	C₂×C₄×C₄₇	—	4	1	no	—
74	C₂×C₄×C₄₇	—	4	1	no	—
75	C₂×C₄×C₄₇	—	4	1	no	—
76	C₂×C₄×C₄₇	—	8	1	no	—
77	C₂×C₄×C₄₇	—	8	1	no	—
78	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
81	C₈×C₄₇	—	2	2	no	—
82	C₈×C₄₇	—	2	1	no	—
83	C₈×C₄₇	—	2	1	no	—
84	C₈×C₄₇	—	8	1	no	—
85	C₈×C₄₇	—	8	1	no	—
86	C₈×C₄₇	SD₃₂	1	2	yes	direct product factor
87	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
88	C₈×C₄₇	#131	1	1	yes	characteristic
89	C₈×SD₃₂	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₄₇	—	4	2	no	—
91	C₂²×C₄₇	—	8	1	no	—
92	C₂²×C₄₇	—	8	1	no	—
93	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
94	C₄×C₄₇	—	2	1	no	—
95	C₄×C₄₇	—	4	2	no	—
96	C₄×C₄₇	—	8	1	no	—
97	C₄×C₄₇	—	8	1	no	—
98	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
99	C₄×C₄₇	#443	1	1	yes	characteristic
100	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
101	C₄×C₄₇	#508	1	1	yes	characteristic
102	C₄×SD₃₂	C₂×C₄₇	1	1	yes	characteristic
103	C₈×Q₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
104	C₈×D₈	C₂×C₄₇	1	1	yes	characteristic
105	#1372	C₂×C₄₇	1	1	yes	characteristic, complemented
106	#1332	C₂×C₄₇	1	1	yes	characteristic, complemented
107	#1331	C₂×C₄₇	1	1	yes	characteristic
108	C₈×C₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
109	C₂×C₄₇	—	8	2	no	—
110	C₂×C₄₇	C₈×D₈	1	1	yes	characteristic
111	C₂×C₄₇	C₄×SD₃₂	1	1	yes	characteristic
112	C₂×C₄₇	#2178	1	1	yes	characteristic
113	C₂×SD₃₂	C₄×C₄₇	1	1	yes	characteristic
114	C₈×Q₈	—	2	1	no	—
115	C₄×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic
116	C₄×D₈	C₂²×C₄₇	1	1	yes	characteristic
117	C₈×D₄	—	2	1	no	—
118	#367	C₄×C₄₇	1	1	yes	characteristic
119	#358	C₄×C₄₇	1	1	yes	characteristic
120	#357	C₄×C₄₇	1	1	yes	characteristic
121	C₄×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
122	C₄×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
123	#335	C₂²×C₄₇	1	1	yes	characteristic
124	#335	C₂²×C₄₇	1	1	yes	characteristic
125	Q₈⋊C₈	—	2	1	no	—
126	D₄⋊C₈	—	2	1	no	—
127	C₈²	C₂²×C₄₇	1	1	yes	characteristic
128	C₄₇	C₈×SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×Q₁₆	C₂×C₄×C₄₇	1	1	yes	characteristic
130	C₂×D₈	C₂×C₄×C₄₇	1	1	yes	characteristic
131	C₂²×C₈	—	4	1	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×D₄	—	2	1	no	—
134	SD₃₂	C₈×C₄₇	1	4	yes	direct product factor
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	C₂×C₄×C₄₇	1	1	yes	characteristic
137	C₂×C₁₆	C₂×C₄×C₄₇	1	1	yes	characteristic
138	#107	C₂×C₄×C₄₇	1	1	yes	characteristic
139	#107	C₂×C₄×C₄₇	1	1	yes	characteristic
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	4	1	no	—
143	#103	—	2	1	no	—
144	#102	—	2	1	no	—
145	C₂²⋊C₈	—	4	1	no	—
146	C₄×C₈	—	4	1	no	—
147	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
148	C₄×C₈	D₄×C₄₇	1	1	yes	characteristic
149	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
150	C₂×Q₈	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂²×C₄	—	4	1	no	—
153	Q₁₆	—	2	1	no	—
154	Q₁₆	C₂×C₈×C₄₇	1	2	yes	complemented
155	D₈	—	2	1	no	—
156	D₈	C₂×C₈×C₄₇	1	2	yes	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	8	2	no	—
161	C₂×C₈	D₈×C₄₇	1	1	yes	characteristic
162	C₂×C₈	D₈×C₄₇	1	1	yes	characteristic
163	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
164	C₂×C₈	#41×C₄₇	1	1	yes	characteristic
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	4	1	no	—
168	C₂²⋊C₄	—	4	1	no	—
169	C₄²	—	4	1	no	—
170	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
171	C₁₆	—	2	2	no	—
172	C₁₆	—	2	1	no	—
173	C₁₆	C₂×C₈×C₄₇	1	2	yes	complemented
174	C₂³	—	4	1	no	—
175	Q₈	—	2	2	no	—
176	Q₈	—	4	1	no	—
177	D₄	—	2	2	no	—
178	D₄	—	4	1	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	8	1	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	C₄×D₄×C₄₇	1	1	yes	characteristic
185	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
186	C₂×C₄	#135×C₄₇	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	8	1	no	—
191	C₈	—	8	1	no	—
192	C₈	SD₃₂×C₄₇	1	2	yes	direct product factor
193	C₈	C₂²×C₈×C₄₇	1	1	yes	characteristic
194	C₈	#131×C₄₇	1	1	yes	characteristic
195	C₂²	—	4	2	no	—
196	C₂²	—	8	1	no	—
197	C₂²	—	8	1	no	—
198	C₂²	C₄×D₈×C₄₇	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	4	2	no	—
201	C₄	—	8	1	no	—
202	C₄	—	8	1	no	—
203	C₄	C₈×D₄×C₄₇	1	1	yes	characteristic
204	C₄	C₄₇×#443	1	1	yes	characteristic
205	C₄	C₂×SD₃₂×C₄₇	1	1	yes	characteristic
206	C₄	C₄₇×#508	1	1	yes	characteristic
207	C₂	—	8	2	no	—
208	C₂	C₈×D₈×C₄₇	1	1	yes	characteristic
209	C₂	C₄×SD₃₂×C₄₇	1	1	yes	characteristic
210	C₂	C₄₇×#2178	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₄₇	D₈
commutator	C₈	C₂²×C₈×C₄₇
Frattini, cosocle	C₄×C₈	C₂³×C₄₇
socle	C₂²×C₄₇	C₄×D₈
prime radical	C₂×D₈×C₄₇	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₄₇ ◁ C₄×C₈×C₄₇ ◁ C₈²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×Q₁₆×C₄₇	C₂	semidirect
4. C₄₇×#1372	C₂	semidirect
5. C₄₇×#1332	C₂	semidirect
7. C₈×C₁₆×C₄₇	C₂	semidirect
28. SD₃₂×C₄₇	C₈	direct (N × H)
48. Q₁₆×C₄₇	C₂×C₈	semidirect
67. C₁₆×C₄₇	C₂×C₈	semidirect
86. C₈×C₄₇	SD₃₂	direct (N × H)
89. C₈×SD₃₂	C₄₇	direct (N × H)
103. C₈×Q₁₆	C₂×C₄₇	semidirect
105. #1372	C₂×C₄₇	semidirect
106. #1332	C₂×C₄₇	semidirect
108. C₈×C₁₆	C₂×C₄₇	semidirect
128. C₄₇	C₈×SD₃₂	direct (N × H)
134. SD₃₂	C₈×C₄₇	direct (N × H)
154. Q₁₆	C₂×C₈×C₄₇	semidirect
173. C₁₆	C₂×C₈×C₄₇	semidirect
192. C₈	SD₃₂×C₄₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×SD₃₂×C₄₇	2
C₈×Q₁₆×C₄₇	2
C₈×D₈×C₄₇	2
C₄₇×#1372	2
C₄₇×#1332	2
C₄₇×#1331	2
C₈×C₁₆×C₄₇	2
C₈×SD₃₂	47

Subgroups — C8×SD32×C47 #26051739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×SD₃₂×C₄₇ #26051739