Subgroups — C₂²×C₄²×C₈×C₄₇ #72201367

Order 24064.

13188 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₈×C₄₇	C₂	1	6	yes	maximal
2	C₂²×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×C₈×C₄₇	C₂	1	24	yes	maximal, direct product factor
4	C₂⁴×C₈×C₄₇	C₂²	1	4	yes	—
5	C₂³×C₄²×C₄₇	C₄	1	4	yes	—
6	C₂³×C₄²×C₄₇	C₂²	1	3	yes	—
7	C₂²×C₄×C₈×C₄₇	C₄	1	96	yes	direct product factor
8	C₂²×C₄×C₈×C₄₇	C₂²	1	72	yes	—
9	C₂×C₄³×C₄₇	C₄	1	12	yes	—
10	C₂×C₄³×C₄₇	C₂²	1	12	yes	—
11	C₄²×C₈×C₄₇	C₂²	1	64	yes	direct product factor
12	C₂⁴×C₄×C₄₇	C₂×C₄	1	6	yes	—
13	C₂⁴×C₄×C₄₇	C₂³	1	1	yes	characteristic
14	C₂³×C₈×C₄₇	C₂×C₄	1	96	yes	—
15	C₂³×C₈×C₄₇	C₂³	1	24	yes	—
16	C₂²×C₄²×C₄₇	C₈	1	64	yes	direct product factor
17	C₂²×C₄²×C₄₇	C₂×C₄	1	18	yes	—
18	C₂²×C₄²×C₄₇	C₂×C₄	1	48	yes	—
19	C₂²×C₄²×C₄₇	C₂×C₄	1	48	yes	—
20	C₂²×C₄²×C₄₇	C₂³	1	18	yes	—
21	C₂×C₄×C₈×C₄₇	C₂×C₄	1	576	yes	direct product factor
22	C₂×C₄×C₈×C₄₇	C₂³	1	96	yes	—
23	C₄³×C₄₇	C₂×C₄	1	48	yes	—
24	C₄³×C₄₇	C₂³	1	16	yes	—
25	C₂⁵×C₄₇	C₂²×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	C₄²	1	48	yes	—
27	C₂³×C₄×C₄₇	C₂×C₈	1	96	yes	—
28	C₂³×C₄×C₄₇	C₂²×C₄	1	3	yes	—
29	C₂³×C₄×C₄₇	C₂²×C₄	1	24	yes	—
30	C₂³×C₄×C₄₇	C₂²×C₄	1	36	yes	—
31	C₂³×C₄×C₄₇	C₂⁴	1	3	yes	—
32	C₂²×C₈×C₄₇	C₄²	1	256	yes	direct product factor
33	C₂²×C₈×C₄₇	C₂²×C₄	1	288	yes	—
34	C₂²×C₈×C₄₇	C₂⁴	1	16	yes	—
35	C₂×C₄²×C₄₇	C₄²	1	144	yes	—
36	C₂×C₄²×C₄₇	C₂×C₈	1	384	yes	direct product factor
37	C₂×C₄²×C₄₇	C₂²×C₄	1	36	yes	—
38	C₂×C₄²×C₄₇	C₂²×C₄	1	64	yes	—
39	C₂×C₄²×C₄₇	C₂²×C₄	1	144	yes	—
40	C₂×C₄²×C₄₇	C₂⁴	1	12	yes	—
41	C₄×C₈×C₄₇	C₂²×C₄	1	384	yes	direct product factor
42	C₂⁴×C₄₇	C₂×C₄²	1	12	yes	—
43	C₂⁴×C₄₇	C₂²×C₈	1	16	yes	—
44	C₂⁴×C₄₇	C₂³×C₄	1	3	yes	—
45	C₂²×C₄×C₄₇	C₄×C₈	1	384	yes	direct product factor
46	C₂²×C₄×C₄₇	C₂×C₄²	1	36	yes	—
47	C₂²×C₄×C₄₇	C₂×C₄²	1	64	yes	—
48	C₂²×C₄×C₄₇	C₂×C₄²	1	144	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₈	1	288	yes	—
50	C₂²×C₄×C₄₇	C₂³×C₄	1	3	yes	—
51	C₂²×C₄×C₄₇	C₂³×C₄	1	24	yes	—
52	C₂²×C₄×C₄₇	C₂³×C₄	1	36	yes	—
53	C₂²×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂×C₄²	1	384	yes	direct product factor
55	C₂×C₈×C₄₇	C₂³×C₄	1	96	yes	—
56	C₄²×C₄₇	C₂×C₄²	1	144	yes	—
57	C₄²×C₄₇	C₂²×C₈	1	256	yes	direct product factor
58	C₄²×C₄₇	C₂³×C₄	1	48	yes	—
59	C₂²×C₄²×C₈	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₄₇	C₄³	1	16	yes	—
61	C₂³×C₄₇	C₂×C₄×C₈	1	96	yes	—
62	C₂³×C₄₇	C₂²×C₄²	1	18	yes	—
63	C₂³×C₄₇	C₂³×C₈	1	24	yes	—
64	C₂³×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₄³	1	48	yes	—
66	C₂×C₄×C₄₇	C₂×C₄×C₈	1	576	yes	direct product factor
67	C₂×C₄×C₄₇	C₂²×C₄²	1	18	yes	—
68	C₂×C₄×C₄₇	C₂²×C₄²	1	48	yes	—
69	C₂×C₄×C₄₇	C₂²×C₄²	1	48	yes	—
70	C₂×C₄×C₄₇	C₂³×C₈	1	96	yes	—
71	C₂×C₄×C₄₇	C₂⁴×C₄	1	6	yes	—
72	C₈×C₄₇	C₂²×C₄²	1	64	yes	direct product factor
73	C₂³×C₄×C₈	C₂×C₄₇	1	6	yes	—
74	C₂²×C₄³	C₂×C₄₇	1	1	yes	characteristic
75	C₂×C₄²×C₈	C₂×C₄₇	1	24	yes	direct product factor
76	C₂²×C₄₇	C₄²×C₈	1	64	yes	direct product factor
77	C₂²×C₄₇	C₂×C₄³	1	12	yes	—
78	C₂²×C₄₇	C₂²×C₄×C₈	1	72	yes	—
79	C₂²×C₄₇	C₂³×C₄²	1	3	yes	—
80	C₂²×C₄₇	C₂⁴×C₈	1	4	yes	—
81	C₄×C₄₇	C₂×C₄³	1	12	yes	—
82	C₄×C₄₇	C₂²×C₄×C₈	1	96	yes	direct product factor
83	C₄×C₄₇	C₂³×C₄²	1	4	yes	—
84	C₂⁴×C₈	C₂²×C₄₇	1	4	yes	—
85	C₂³×C₄²	C₄×C₄₇	1	4	yes	—
86	C₂³×C₄²	C₂²×C₄₇	1	3	yes	—
87	C₂²×C₄×C₈	C₄×C₄₇	1	96	yes	direct product factor
88	C₂²×C₄×C₈	C₂²×C₄₇	1	72	yes	—
89	C₂×C₄³	C₄×C₄₇	1	12	yes	—
90	C₂×C₄³	C₂²×C₄₇	1	12	yes	—
91	C₄²×C₈	C₂²×C₄₇	1	64	yes	direct product factor
92	C₂×C₄₇	C₂×C₄²×C₈	1	24	yes	direct product factor
93	C₂×C₄₇	C₂²×C₄³	1	1	yes	characteristic
94	C₂×C₄₇	C₂³×C₄×C₈	1	6	yes	—
95	C₂⁴×C₄	C₂×C₄×C₄₇	1	6	yes	—
96	C₂⁴×C₄	C₂³×C₄₇	1	1	yes	characteristic
97	C₂³×C₈	C₂×C₄×C₄₇	1	96	yes	—
98	C₂³×C₈	C₂³×C₄₇	1	24	yes	—
99	C₂²×C₄²	C₈×C₄₇	1	64	yes	direct product factor
100	C₂²×C₄²	C₂×C₄×C₄₇	1	18	yes	—
101	C₂²×C₄²	C₂×C₄×C₄₇	1	48	yes	—
102	C₂²×C₄²	C₂×C₄×C₄₇	1	48	yes	—
103	C₂²×C₄²	C₂³×C₄₇	1	18	yes	—
104	C₂×C₄×C₈	C₂×C₄×C₄₇	1	576	yes	direct product factor
105	C₂×C₄×C₈	C₂³×C₄₇	1	96	yes	—
106	C₄³	C₂×C₄×C₄₇	1	48	yes	—
107	C₄³	C₂³×C₄₇	1	16	yes	—
108	C₄₇	C₂²×C₄²×C₈	1	1	yes	characteristic, direct product factor, Sylow
109	C₂⁵	C₂²×C₄×C₄₇	1	1	yes	characteristic
110	C₂³×C₄	C₄²×C₄₇	1	48	yes	—
111	C₂³×C₄	C₂×C₈×C₄₇	1	96	yes	—
112	C₂³×C₄	C₂²×C₄×C₄₇	1	3	yes	—
113	C₂³×C₄	C₂²×C₄×C₄₇	1	24	yes	—
114	C₂³×C₄	C₂²×C₄×C₄₇	1	36	yes	—
115	C₂³×C₄	C₂⁴×C₄₇	1	3	yes	—
116	C₂²×C₈	C₄²×C₄₇	1	256	yes	direct product factor
117	C₂²×C₈	C₂²×C₄×C₄₇	1	288	yes	—
118	C₂²×C₈	C₂⁴×C₄₇	1	16	yes	—
119	C₂×C₄²	C₄²×C₄₇	1	144	yes	—
120	C₂×C₄²	C₂×C₈×C₄₇	1	384	yes	direct product factor
121	C₂×C₄²	C₂²×C₄×C₄₇	1	36	yes	—
122	C₂×C₄²	C₂²×C₄×C₄₇	1	64	yes	—
123	C₂×C₄²	C₂²×C₄×C₄₇	1	144	yes	—
124	C₂×C₄²	C₂⁴×C₄₇	1	12	yes	—
125	C₄×C₈	C₂²×C₄×C₄₇	1	384	yes	direct product factor
126	C₂⁴	C₂×C₄²×C₄₇	1	12	yes	—
127	C₂⁴	C₂²×C₈×C₄₇	1	16	yes	—
128	C₂⁴	C₂³×C₄×C₄₇	1	3	yes	—
129	C₂²×C₄	C₄×C₈×C₄₇	1	384	yes	direct product factor
130	C₂²×C₄	C₂×C₄²×C₄₇	1	36	yes	—
131	C₂²×C₄	C₂×C₄²×C₄₇	1	64	yes	—
132	C₂²×C₄	C₂×C₄²×C₄₇	1	144	yes	—
133	C₂²×C₄	C₂²×C₈×C₄₇	1	288	yes	—
134	C₂²×C₄	C₂³×C₄×C₄₇	1	3	yes	—
135	C₂²×C₄	C₂³×C₄×C₄₇	1	24	yes	—
136	C₂²×C₄	C₂³×C₄×C₄₇	1	36	yes	—
137	C₂²×C₄	C₂⁵×C₄₇	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄²×C₄₇	1	384	yes	direct product factor
139	C₂×C₈	C₂³×C₄×C₄₇	1	96	yes	—
140	C₄²	C₂×C₄²×C₄₇	1	144	yes	—
141	C₄²	C₂²×C₈×C₄₇	1	256	yes	direct product factor
142	C₄²	C₂³×C₄×C₄₇	1	48	yes	—
143	C₂³	C₄³×C₄₇	1	16	yes	—
144	C₂³	C₂×C₄×C₈×C₄₇	1	96	yes	—
145	C₂³	C₂²×C₄²×C₄₇	1	18	yes	—
146	C₂³	C₂³×C₈×C₄₇	1	24	yes	—
147	C₂³	C₂⁴×C₄×C₄₇	1	1	yes	characteristic
148	C₂×C₄	C₄³×C₄₇	1	48	yes	—
149	C₂×C₄	C₂×C₄×C₈×C₄₇	1	576	yes	direct product factor
150	C₂×C₄	C₂²×C₄²×C₄₇	1	18	yes	—
151	C₂×C₄	C₂²×C₄²×C₄₇	1	48	yes	—
152	C₂×C₄	C₂²×C₄²×C₄₇	1	48	yes	—
153	C₂×C₄	C₂³×C₈×C₄₇	1	96	yes	—
154	C₂×C₄	C₂⁴×C₄×C₄₇	1	6	yes	—
155	C₈	C₂²×C₄²×C₄₇	1	64	yes	direct product factor
156	C₂²	C₄²×C₈×C₄₇	1	64	yes	direct product factor
157	C₂²	C₂×C₄³×C₄₇	1	12	yes	—
158	C₂²	C₂²×C₄×C₈×C₄₇	1	72	yes	—
159	C₂²	C₂³×C₄²×C₄₇	1	3	yes	—
160	C₂²	C₂⁴×C₈×C₄₇	1	4	yes	—
161	C₄	C₂×C₄³×C₄₇	1	12	yes	—
162	C₄	C₂²×C₄×C₈×C₄₇	1	96	yes	direct product factor
163	C₄	C₂³×C₄²×C₄₇	1	4	yes	—
164	C₂	C₂×C₄²×C₈×C₄₇	1	24	yes	direct product factor
165	C₂	C₂²×C₄³×C₄₇	1	1	yes	characteristic
166	C₂	C₂³×C₄×C₈×C₄₇	1	6	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₄	C₂⁵×C₄₇
socle, prime radical	C₂⁵×C₄₇	C₂²×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄²×C₈×C₄₇	C₂	direct (N × H)
7. C₂²×C₄×C₈×C₄₇	C₄	direct (N × H)
11. C₄²×C₈×C₄₇	C₂²	direct (N × H)
16. C₂²×C₄²×C₄₇	C₈	direct (N × H)
21. C₂×C₄×C₈×C₄₇	C₂×C₄	direct (N × H)
32. C₂²×C₈×C₄₇	C₄²	direct (N × H)
36. C₂×C₄²×C₄₇	C₂×C₈	direct (N × H)
41. C₄×C₈×C₄₇	C₂²×C₄	direct (N × H)
45. C₂²×C₄×C₄₇	C₄×C₈	direct (N × H)
54. C₂×C₈×C₄₇	C₂×C₄²	direct (N × H)
57. C₄²×C₄₇	C₂²×C₈	direct (N × H)
59. C₂²×C₄²×C₈	C₄₇	direct (N × H)
66. C₂×C₄×C₄₇	C₂×C₄×C₈	direct (N × H)
72. C₈×C₄₇	C₂²×C₄²	direct (N × H)
75. C₂×C₄²×C₈	C₂×C₄₇	direct (N × H)
76. C₂²×C₄₇	C₄²×C₈	direct (N × H)
82. C₄×C₄₇	C₂²×C₄×C₈	direct (N × H)
87. C₂²×C₄×C₈	C₄×C₄₇	direct (N × H)
91. C₄²×C₈	C₂²×C₄₇	direct (N × H)
92. C₂×C₄₇	C₂×C₄²×C₈	direct (N × H)
99. C₂²×C₄²	C₈×C₄₇	direct (N × H)
104. C₂×C₄×C₈	C₂×C₄×C₄₇	direct (N × H)
108. C₄₇	C₂²×C₄²×C₈	direct (N × H)
116. C₂²×C₈	C₄²×C₄₇	direct (N × H)
120. C₂×C₄²	C₂×C₈×C₄₇	direct (N × H)
125. C₄×C₈	C₂²×C₄×C₄₇	direct (N × H)
129. C₂²×C₄	C₄×C₈×C₄₇	direct (N × H)
138. C₂×C₈	C₂×C₄²×C₄₇	direct (N × H)
141. C₄²	C₂²×C₈×C₄₇	direct (N × H)
149. C₂×C₄	C₂×C₄×C₈×C₄₇	direct (N × H)
155. C₈	C₂²×C₄²×C₄₇	direct (N × H)
156. C₂²	C₄²×C₈×C₄₇	direct (N × H)
162. C₄	C₂²×C₄×C₈×C₄₇	direct (N × H)
164. C₂	C₂×C₄²×C₈×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
C₂³×C₄×C₈×C₄₇	2
C₂²×C₄³×C₄₇	2
C₂×C₄²×C₈×C₄₇	2
C₂²×C₄²×C₈	47

Subgroups — C22×C42×C8×C47 #72201367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄²×C₈×C₄₇ #72201367