Subgroups — C₂³×C₄⋊C₄×D₄₇ #27122831

Order 12032.

19742 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₄₇	C₂	1	2	yes	maximal
2	C₂⁴×C₄×D₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×Dic₄₇⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄⋊C₄×D₄₇	C₂	1	56	yes	maximal, direct product factor
6	C₂³×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₄₇	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₄₇	C₂²	1	2	yes	—
9	C₂⁴×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₄₇	—	2	32	no	—
11	C₂³×C₄×D₄₇	C₄	1	32	yes	complemented
12	C₂³×C₄×D₄₇	C₂²	1	56	yes	—
13	C₂³×C₄×D₄₇	C₂²	1	28	yes	—
14	C₂²×C₄⋊Dic₄₇	C₂²	1	28	yes	complemented
15	C₂²×Dic₄₇⋊C₄	C₂²	1	56	yes	complemented
16	C₂×C₄⋊C₄×D₄₇	C₂²	1	448	yes	direct product factor
17	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
18	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	28	yes	complemented
20	C₂⁴×D₄₇	C₂×C₄	1	16	yes	—
21	C₂⁴×D₄₇	D₄	1	16	yes	—
22	C₂⁴×D₄₇	Q₈	1	16	yes	—
23	C₂⁴×D₄₇	C₂³	1	14	yes	—
24	C₂³×Dic₄₇	—	2	16	no	—
25	C₂³×Dic₄₇	C₂×C₄	1	16	yes	complemented
26	C₂³×Dic₄₇	C₂³	1	28	yes	—
27	C₂³×Dic₄₇	C₂³	1	14	yes	—
28	C₂²×C₄×D₄₇	—	2	448	no	—
29	C₂²×C₄×D₄₇	C₂×C₄	1	448	yes	complemented
30	C₂²×C₄×D₄₇	C₂³	1	224	yes	—
31	C₂²×C₄×D₄₇	C₂³	1	112	yes	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	112	yes	complemented
33	C₂×Dic₄₇⋊C₄	C₂³	1	224	yes	complemented
34	C₄⋊C₄×D₄₇	C₂³	1	512	yes	direct product factor
35	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	—	2	16	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	complemented
38	C₂³×C₄×C₄₇	C₂³	1	28	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	14	yes	—
40	C₂×C₄⋊C₄×C₄₇	C₂³	1	112	yes	complemented
41	C₂⁴×C₄₇	C₂²×C₄	1	8	yes	—
42	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
43	C₂⁴×C₄₇	C₂×Q₈	1	8	yes	—
44	C₂⁴×C₄₇	C₂⁴	1	7	yes	—
45	C₂³×D₄₇	C₄⋊C₄	1	256	yes	direct product factor
46	C₂³×D₄₇	C₂²×C₄	1	112	yes	—
47	C₂³×D₄₇	C₂×D₄	1	112	yes	—
48	C₂³×D₄₇	C₂×Q₈	1	112	yes	—
49	C₂³×D₄₇	C₂⁴	1	28	yes	—
50	C₂²×C₄×C₄₇	—	2	112	no	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	112	yes	complemented
52	C₂²×C₄×C₄₇	C₂⁴	1	56	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	28	yes	—
54	C₂²×Dic₄₇	—	2	112	no	—
55	C₂²×Dic₄₇	C₂²×C₄	1	112	yes	complemented
56	C₂²×Dic₄₇	C₂⁴	1	56	yes	—
57	C₂²×Dic₄₇	C₂⁴	1	28	yes	—
58	C₂×C₄×D₄₇	—	2	896	no	—
59	C₂×C₄×D₄₇	C₂²×C₄	1	896	yes	complemented
60	C₂×C₄×D₄₇	C₂⁴	1	128	yes	—
61	C₂×C₄×D₄₇	C₂⁴	1	64	yes	—
62	C₄⋊C₄×C₄₇	C₂⁴	1	64	yes	complemented
63	C₄⋊Dic₄₇	C₂⁴	1	64	yes	complemented
64	Dic₄₇⋊C₄	C₂⁴	1	128	yes	complemented
65	C₂³×C₄₇	C₂×C₄⋊C₄	1	64	yes	complemented
66	C₂³×C₄₇	C₂³×C₄	1	28	yes	—
67	C₂³×C₄₇	C₂²×D₄	1	28	yes	—
68	C₂³×C₄₇	C₂²×Q₈	1	28	yes	—
69	C₂³×C₄₇	C₂⁵	1	7	yes	—
70	C₂²×D₄₇	C₂×C₄⋊C₄	1	896	yes	direct product factor
71	C₂²×D₄₇	C₂³×C₄	1	112	yes	—
72	C₂²×D₄₇	C₂²×D₄	1	112	yes	—
73	C₂²×D₄₇	C₂²×Q₈	1	112	yes	—
74	C₂²×D₄₇	C₂⁵	1	8	yes	—
75	C₂×C₄×C₄₇	—	2	112	no	—
76	C₂×C₄×C₄₇	C₂³×C₄	1	112	yes	complemented
77	C₂×C₄×C₄₇	C₂⁵	1	16	yes	—
78	C₂×C₄×C₄₇	C₂⁵	1	8	yes	—
79	C₂×Dic₄₇	—	2	112	no	—
80	C₂×Dic₄₇	C₂³×C₄	1	112	yes	complemented
81	C₂×Dic₄₇	C₂⁵	1	16	yes	—
82	C₂×Dic₄₇	C₂⁵	1	8	yes	—
83	C₄×D₄₇	—	2	256	no	—
84	C₄×D₄₇	C₂³×C₄	1	256	yes	complemented
85	C₂⁴×C₄⋊C₄	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	C₂²×C₄⋊C₄	1	112	yes	complemented
87	C₂²×C₄₇	C₂⁴×C₄	1	14	yes	—
88	C₂²×C₄₇	C₂³×D₄	1	14	yes	—
89	C₂²×C₄₇	C₂³×Q₈	1	14	yes	—
90	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
91	C₂×D₄₇	C₂²×C₄⋊C₄	1	448	yes	direct product factor
92	C₂×D₄₇	C₂⁴×C₄	1	16	yes	—
93	C₂×D₄₇	C₂³×D₄	1	16	yes	—
94	C₂×D₄₇	C₂³×Q₈	1	16	yes	—
95	C₄×C₄₇	—	2	16	no	—
96	C₄×C₄₇	C₂⁴×C₄	1	16	yes	complemented
97	Dic₄₇	—	2	16	no	—
98	Dic₄₇	C₂⁴×C₄	1	16	yes	complemented
99	C₂⁵×C₄	—	47	1	no	—
100	C₂⁵×C₄	—	47	2	no	—
101	C₂³×C₄⋊C₄	—	47	2	no	—
102	C₂³×C₄⋊C₄	—	47	1	no	—
103	C₂³×C₄⋊C₄	—	47	56	no	—
104	C₂³×C₄⋊C₄	D₄₇	1	1	yes	characteristic, direct product factor
105	C₂×C₄₇	C₂³×C₄⋊C₄	1	28	yes	complemented
106	C₂×C₄₇	C₂⁵×C₄	1	1	yes	characteristic
107	C₂×C₄₇	C₂⁴×D₄	1	1	yes	characteristic
108	C₂×C₄₇	C₂⁴×Q₈	1	1	yes	characteristic
109	D₄₇	C₂³×C₄⋊C₄	1	32	yes	direct product factor
110	C₂⁶	—	47	1	no	—
111	C₂⁴×C₄	—	47	1	no	—
112	C₂⁴×C₄	—	47	32	no	—
113	C₂⁴×C₄	—	47	28	no	—
114	C₂⁴×C₄	—	47	2	no	—
115	C₂⁴×C₄	—	47	56	no	—
116	C₂⁴×C₄	—	94	32	no	—
117	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
118	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
119	C₂²×C₄⋊C₄	—	47	56	no	—
120	C₂²×C₄⋊C₄	—	47	28	no	—
121	C₂²×C₄⋊C₄	—	47	448	no	—
122	C₂²×C₄⋊C₄	C₂×D₄₇	1	28	yes	direct product factor
123	C₄₇	C₂⁴×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	—	47	16	no	—
125	C₂⁵	—	47	16	no	—
126	C₂⁵	—	47	16	no	—
127	C₂⁵	—	47	14	no	—
128	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
129	C₂³×C₄	—	2	16	no	—
130	C₂³×C₄	—	47	16	no	—
131	C₂³×C₄	—	47	14	no	—
132	C₂³×C₄	—	47	448	no	—
133	C₂³×C₄	—	47	112	no	—
134	C₂³×C₄	—	47	28	no	—
135	C₂³×C₄	—	47	224	no	—
136	C₂³×C₄	—	94	448	no	—
137	C₂³×C₄	—	94	16	no	—
138	C₂³×C₄	C₄×D₄₇	1	16	yes	complemented
139	C₂³×C₄	C₂²×D₄₇	1	28	yes	—
140	C₂³×C₄	C₂²×D₄₇	1	14	yes	—
141	C₂×C₄⋊C₄	—	47	224	no	—
142	C₂×C₄⋊C₄	—	47	112	no	—
143	C₂×C₄⋊C₄	—	47	512	no	—
144	C₂×C₄⋊C₄	C₂²×D₄₇	1	112	yes	direct product factor
145	C₂⁴	—	47	256	no	—
146	C₂⁴	—	47	112	no	—
147	C₂⁴	—	47	112	no	—
148	C₂⁴	—	47	112	no	—
149	C₂⁴	—	47	28	no	—
150	C₂⁴	C₂×C₄×D₄₇	1	8	yes	—
151	C₂⁴	D₄×D₄₇	1	8	yes	—
152	C₂⁴	Q₈×D₄₇	1	8	yes	—
153	C₂⁴	C₂³×D₄₇	1	7	yes	—
154	C₂²×C₄	—	2	112	no	—
155	C₂²×C₄	—	47	112	no	—
156	C₂²×C₄	—	47	28	no	—
157	C₂²×C₄	—	47	896	no	—
158	C₂²×C₄	—	47	64	no	—
159	C₂²×C₄	—	47	56	no	—
160	C₂²×C₄	—	47	128	no	—
161	C₂²×C₄	—	94	896	no	—
162	C₂²×C₄	—	94	112	no	—
163	C₂²×C₄	C₂×C₄×D₄₇	1	112	yes	complemented
164	C₂²×C₄	C₂³×D₄₇	1	56	yes	—
165	C₂²×C₄	C₂³×D₄₇	1	28	yes	—
166	C₄⋊C₄	—	47	128	no	—
167	C₄⋊C₄	—	47	64	no	—
168	C₄⋊C₄	C₂³×D₄₇	1	64	yes	direct product factor
169	C₂³	—	47	896	no	—
170	C₂³	—	47	112	no	—
171	C₂³	—	47	112	no	—
172	C₂³	—	47	112	no	—
173	C₂³	—	47	8	no	—
174	C₂³	C₄⋊C₄×D₄₇	1	64	yes	direct product factor
175	C₂³	C₂²×C₄×D₄₇	1	28	yes	—
176	C₂³	C₂×D₄×D₄₇	1	28	yes	—
177	C₂³	C₂×Q₈×D₄₇	1	28	yes	—
178	C₂³	C₂⁴×D₄₇	1	7	yes	—
179	C₂×C₄	—	2	112	no	—
180	C₂×C₄	—	47	112	no	—
181	C₂×C₄	—	47	8	no	—
182	C₂×C₄	—	47	256	no	—
183	C₂×C₄	—	47	16	no	—
184	C₂×C₄	—	94	256	no	—
185	C₂×C₄	—	94	112	no	—
186	C₂×C₄	C₂²×C₄×D₄₇	1	112	yes	complemented
187	C₂×C₄	C₂⁴×D₄₇	1	16	yes	—
188	C₂×C₄	C₂⁴×D₄₇	1	8	yes	—
189	C₂²	—	47	448	no	—
190	C₂²	—	47	16	no	—
191	C₂²	—	47	16	no	—
192	C₂²	—	47	16	no	—
193	C₂²	C₂×C₄⋊C₄×D₄₇	1	112	yes	direct product factor
194	C₂²	C₂³×C₄×D₄₇	1	14	yes	—
195	C₂²	C₂²×D₄×D₄₇	1	14	yes	—
196	C₂²	C₂²×Q₈×D₄₇	1	14	yes	—
197	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
198	C₄	—	2	16	no	—
199	C₄	—	47	16	no	—
200	C₄	—	94	16	no	—
201	C₄	C₂³×C₄×D₄₇	1	16	yes	complemented
202	C₂	—	47	32	no	—
203	C₂	C₂²×C₄⋊C₄×D₄₇	1	28	yes	direct product factor
204	C₂	C₂⁴×C₄×D₄₇	1	1	yes	characteristic
205	C₂	C₂³×D₄×D₄₇	1	1	yes	characteristic
206	C₂	C₂³×Q₈×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₄₇
commutator	C₂×C₄₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂³×C₄⋊C₄×C₄₇	C₂
socle	C₂⁵×C₄₇	C₂³
prime radical	C₂⁵×D₄₇	C₂²
cosocle	C₂²×C₄₇	C₂⁶
hypercenter	C₂³×C₄⋊C₄	D₄₇
hypocenter	C₄₇	C₂⁴×C₄⋊C₄
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂³×C₄⋊Dic₄₇	C₂	semidirect
4. C₂³×Dic₄₇⋊C₄	C₂	semidirect
5. C₂²×C₄⋊C₄×D₄₇	C₂	direct (N × H)
6. C₂³×C₄⋊C₄×C₄₇	C₂	semidirect
11. C₂³×C₄×D₄₇	C₄	semidirect
14. C₂²×C₄⋊Dic₄₇	C₂²	semidirect
15. C₂²×Dic₄₇⋊C₄	C₂²	semidirect
16. C₂×C₄⋊C₄×D₄₇	C₂²	direct (N × H)
19. C₂²×C₄⋊C₄×C₄₇	C₂²	semidirect
25. C₂³×Dic₄₇	C₂×C₄	semidirect
29. C₂²×C₄×D₄₇	C₂×C₄	semidirect
32. C₂×C₄⋊Dic₄₇	C₂³	semidirect
33. C₂×Dic₄₇⋊C₄	C₂³	semidirect
34. C₄⋊C₄×D₄₇	C₂³	direct (N × H)
37. C₂³×C₄×C₄₇	C₂×C₄	semidirect
40. C₂×C₄⋊C₄×C₄₇	C₂³	semidirect
45. C₂³×D₄₇	C₄⋊C₄	direct (N × H)
51. C₂²×C₄×C₄₇	C₂²×C₄	semidirect
55. C₂²×Dic₄₇	C₂²×C₄	semidirect
59. C₂×C₄×D₄₇	C₂²×C₄	semidirect
62. C₄⋊C₄×C₄₇	C₂⁴	semidirect
63. C₄⋊Dic₄₇	C₂⁴	semidirect
64. Dic₄₇⋊C₄	C₂⁴	semidirect
65. C₂³×C₄₇	C₂×C₄⋊C₄	semidirect
70. C₂²×D₄₇	C₂×C₄⋊C₄	direct (N × H)
76. C₂×C₄×C₄₇	C₂³×C₄	semidirect
80. C₂×Dic₄₇	C₂³×C₄	semidirect
84. C₄×D₄₇	C₂³×C₄	semidirect
86. C₂²×C₄₇	C₂²×C₄⋊C₄	semidirect
91. C₂×D₄₇	C₂²×C₄⋊C₄	direct (N × H)
96. C₄×C₄₇	C₂⁴×C₄	semidirect
98. Dic₄₇	C₂⁴×C₄	semidirect
104. C₂³×C₄⋊C₄	D₄₇	direct (N × H)
105. C₂×C₄₇	C₂³×C₄⋊C₄	semidirect
109. D₄₇	C₂³×C₄⋊C₄	direct (N × H)
122. C₂²×C₄⋊C₄	C₂×D₄₇	direct (N × H)
123. C₄₇	C₂⁴×C₄⋊C₄	semidirect
138. C₂³×C₄	C₄×D₄₇	semidirect
144. C₂×C₄⋊C₄	C₂²×D₄₇	direct (N × H)
163. C₂²×C₄	C₂×C₄×D₄₇	semidirect
168. C₄⋊C₄	C₂³×D₄₇	direct (N × H)
174. C₂³	C₄⋊C₄×D₄₇	direct (N × H)
186. C₂×C₄	C₂²×C₄×D₄₇	semidirect
193. C₂²	C₂×C₄⋊C₄×D₄₇	direct (N × H)
201. C₄	C₂³×C₄×D₄₇	semidirect
203. C₂	C₂²×C₄⋊C₄×D₄₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×D₄₇	2
C₂³×C₄⋊Dic₄₇	2
C₂³×Dic₄₇⋊C₄	2
C₂²×C₄⋊C₄×D₄₇	2
C₂³×C₄⋊C₄×C₄₇	2
C₂⁴×C₄⋊C₄	47

Subgroups — C23×C4⋊C4×D47 #27122831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊C₄×D₄₇ #27122831