Subgroups — C₁₆×C₂²⋊Dic₄₇ #26131357

Order 12032.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₄₇	C₂	1	2	yes	maximal, complemented
3	C₈×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20241465	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
9	C₄×C₂²⋊Dic₄₇	C₄	1	1	yes	characteristic
10	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	complemented
11	C₁₆×Dic₄₇	—	2	4	no	—
12	#15688877	C₄	1	2	yes	—
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₄₇	—	2	4	no	—
15	C₂²×C₁₆×C₄₇	C₄	1	4	yes	complemented
16	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₂²⋊Dic₄₇	C₈	1	2	yes	—
20	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
24	C₂²⋊(C₄₇⋊C₈)	C₈	1	4	yes	—
25	C₈×Dic₄₇	—	2	2	no	—
26	C₈×Dic₄₇	—	2	2	no	—
27	C₄₇⋊#95	C₈	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₁₆×C₄₇	—	2	2	no	—
36	C₂×C₁₆×C₄₇	—	2	8	no	—
37	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
40	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₄₇	C₂×C₈	1	4	yes	—
48	C₂×C₈×C₄₇	—	2	2	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	—	2	4	no	—
52	C₂×C₈×C₄₇	C₄²	1	2	yes	—
53	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₄₇	#41	1	2	yes	—
57	C₂²⋊Dic₄₇	C₁₆	1	8	yes	direct product factor
58	C₄×Dic₄₇	—	2	2	no	—
59	C₄×Dic₄₇	—	2	2	no	—
60	C₂×C₄₇⋊C₈	—	2	2	no	—
61	C₂×C₄₇⋊C₈	—	2	2	no	—
62	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
63	C₁₆×C₄₇	—	2	2	no	—
64	C₁₆×C₄₇	C₂²⋊C₄	1	4	yes	complemented
65	C₄₇⋊C₁₆	—	2	4	no	—
66	C₂³×C₄₇	—	2	2	no	—
67	C₂³×C₄₇	—	2	2	no	—
68	C₂³×C₄₇	C₄×C₈	1	2	yes	—
69	C₂³×C₄₇	C₂×C₁₆	1	4	yes	—
70	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
76	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	#131	1	2	yes	—
80	C₂×Dic₄₇	—	2	4	no	—
81	C₂×Dic₄₇	—	2	4	no	—
82	C₂×Dic₄₇	C₂×C₁₆	1	8	yes	complemented
83	C₈×C₄₇	—	2	2	no	—
84	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₄₇	#117	1	1	yes	characteristic
86	C₈×C₄₇	C₄×D₄	1	2	yes	—
87	C₄₇⋊C₈	—	2	4	no	—
88	C₁₆×C₂²⋊C₄	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	—	2	2	no	—
90	C₂²×C₄₇	—	2	8	no	—
91	C₂²×C₄₇	—	2	2	no	—
92	C₂²×C₄₇	C₄×C₁₆	1	4	yes	complemented
93	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂²×C₄₇	C₈×D₄	1	2	yes	—
95	C₂²×C₄₇	C₂²×C₁₆	1	2	yes	—
96	C₂²×C₄₇	#504	1	2	yes	—
97	C₄×C₄₇	—	2	2	no	—
98	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₄₇	#405	1	1	yes	characteristic
100	C₄×C₄₇	C₈×D₄	1	2	yes	—
101	Dic₄₇	—	2	8	no	—
102	C₂³×C₁₆	D₄₇	1	1	yes	characteristic
103	C₂×C₂²⋊C₁₆	—	47	1	no	—
104	C₂×C₄×C₁₆	—	47	2	no	—
105	C₈×C₂²⋊C₄	—	47	1	no	—
106	#1374	—	47	2	no	—
107	C₂×C₄₇	—	2	4	no	—
108	C₂×C₄₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄₇	C₂×C₄×C₁₆	1	1	yes	characteristic
110	C₂×C₄₇	#2108	1	1	yes	characteristic
111	C₂×C₄₇	D₄×C₁₆	1	4	yes	—
112	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
113	C₂²×C₁₆	—	2	4	no	—
114	C₂²×C₁₆	—	47	2	no	—
115	C₂²×C₁₆	Dic₄₇	1	4	yes	complemented
116	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
117	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
118	C₂×C₂²⋊C₈	—	47	1	no	—
119	C₂×C₄×C₈	—	47	2	no	—
120	C₄×C₂²⋊C₄	—	47	1	no	—
121	C₂²⋊C₁₆	—	47	4	no	—
122	C₄×C₁₆	—	94	4	no	—
123	#336	—	47	2	no	—
124	C₄₇	C₁₆×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	47	2	no	—
129	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
130	C₂²×C₈	C₄×D₄₇	1	2	yes	—
131	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
132	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	—	47	2	no	—
134	C₂×C₄²	—	47	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	8	no	—
137	C₂×C₁₆	—	47	4	no	—
138	C₂×C₁₆	—	94	2	no	—
139	C₂×C₁₆	—	94	2	no	—
140	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
141	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
142	C₂×C₁₆	C₂²⋊D₄₇	1	4	yes	—
143	C₂²⋊C₈	—	47	4	no	—
144	C₄×C₈	—	94	2	no	—
145	C₄×C₈	—	94	2	no	—
146	#95	—	47	2	no	—
147	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	47	4	no	—
151	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
152	C₂²×C₄	C₈×D₄₇	1	2	yes	—
153	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	47	4	no	—
160	C₂×C₈	—	94	2	no	—
161	C₂×C₈	—	94	2	no	—
162	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
164	C₂×C₈	#10625350	1	2	yes	—
165	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
167	C₂²⋊C₄	—	47	8	no	—
168	C₄²	—	94	2	no	—
169	C₄²	—	94	2	no	—
170	C₁₆	—	2	2	no	—
171	C₁₆	—	94	4	no	—
172	C₁₆	C₂²⋊Dic₄₇	1	4	yes	direct product factor
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	C₈×Dic₄₇	1	2	yes	—
176	C₂³	C₁₆×D₄₇	1	4	yes	—
177	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	47	8	no	—
183	C₂×C₄	—	94	4	no	—
184	C₂×C₄	—	94	4	no	—
185	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
186	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
187	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
188	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
189	C₂×C₄	#13094030	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	—	94	4	no	—
192	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
193	C₈	#13093983	1	1	yes	characteristic
194	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
195	C₂²	—	2	8	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₁₆×Dic₄₇	1	4	yes	complemented
199	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
200	C₂²	C₈×C₂²⋊D₄₇	1	2	yes	—
201	C₂²	C₂×C₁₆×D₄₇	1	2	yes	—
202	C₂²	#15689539	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	94	8	no	—
205	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
206	C₄	#15689078	1	1	yes	characteristic
207	C₄	C₈×C₂²⋊D₄₇	1	2	yes	—
208	C₂	—	2	4	no	—
209	C₂	C₈×C₂²⋊Dic₄₇	1	1	yes	characteristic
210	C₂	C₂×C₁₆×Dic₄₇	1	1	yes	characteristic
211	C₂	#20244984	1	1	yes	characteristic
212	C₂	C₁₆×C₂²⋊D₄₇	1	4	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₄₇
commutator	C₂×C₄₇	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂³×C₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂³×C₁₆	D₄₇
hypocenter	C₄₇	C₁₆×C₂²⋊C₄
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₆×Dic₄₇	C₂	semidirect
4. #20241465	C₂	semidirect
10. C₂²⋊(C₄₇⋊C₁₆)	C₄	semidirect
15. C₂²×C₁₆×C₄₇	C₄	semidirect
57. C₂²⋊Dic₄₇	C₁₆	direct (N × H)
64. C₁₆×C₄₇	C₂²⋊C₄	semidirect
82. C₂×Dic₄₇	C₂×C₁₆	semidirect
92. C₂²×C₄₇	C₄×C₁₆	semidirect
115. C₂²×C₁₆	Dic₄₇	semidirect
124. C₄₇	C₁₆×C₂²⋊C₄	semidirect
172. C₁₆	C₂²⋊Dic₄₇	direct (N × H)
198. C₂²	C₁₆×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂×C₁₆×Dic₄₇	2
C₈×C₂²⋊Dic₄₇	2
#20241465	2
C₂³×C₁₆×C₄₇	2
C₁₆×C₂²⋊C₄	47

Subgroups — C16×C22⋊Dic47 #26131357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×C₂²⋊Dic₄₇ #26131357