Subgroups — C₄×C₇×C₈×C₉² #72195909

Order 18144.

1012 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇×C₉²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈×C₉²	C₂	1	2	yes	maximal
3	C₃×C₄×C₇×C₈×C₉	C₃	1	4	yes	maximal
4	C₂×C₄×C₇×C₉²	C₄	1	2	yes	—
5	C₂×C₄×C₇×C₉²	C₂²	1	1	yes	characteristic
6	C₇×C₈×C₉²	C₄	1	4	yes	direct product factor
7	C₃×C₄²×C₇×C₉	C₂×C₃	1	4	yes	—
8	C₂×C₃×C₇×C₈×C₉	C₂×C₃	1	8	yes	—
9	C₄×C₈×C₉²	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₇×C₉²	C₂×C₄	1	1	yes	characteristic
11	C₄×C₇×C₉²	C₈	1	4	yes	direct product factor
12	C₄×C₇×C₉²	C₂×C₄	1	2	yes	—
13	C₃²×C₄×C₇×C₈	C₃²	1	1	yes	characteristic
14	C₄×C₇×C₈×C₉	C₉	1	12	yes	direct product factor
15	C₂×C₃×C₄×C₇×C₉	C₃×C₄	1	8	yes	—
16	C₂×C₃×C₄×C₇×C₉	C₂²×C₃	1	4	yes	—
17	C₃×C₇×C₈×C₉	C₃×C₄	1	16	yes	—
18	C₂×C₈×C₉²	C₂×C₇	1	2	yes	—
19	C₄²×C₉²	C₂×C₇	1	1	yes	characteristic
20	C₂×C₇×C₉²	C₄²	1	1	yes	characteristic
21	C₂×C₇×C₉²	C₂×C₈	1	2	yes	—
22	C₂×C₃²×C₇×C₈	C₂×C₃²	1	2	yes	—
23	C₃²×C₄²×C₇	C₂×C₃²	1	1	yes	characteristic
24	C₂×C₇×C₈×C₉	C₂×C₉	1	24	yes	—
25	C₄²×C₇×C₉	C₂×C₉	1	12	yes	—
26	C₃×C₄×C₈×C₉	C₃×C₇	1	4	yes	—
27	C₂²×C₃×C₇×C₉	C₂×C₃×C₄	1	4	yes	—
28	C₃×C₄×C₇×C₉	C₃×C₈	1	16	yes	—
29	C₃×C₄×C₇×C₉	C₂×C₃×C₄	1	8	yes	—
30	C₃×C₄×C₇×C₈	C₃×C₉	1	4	yes	—
31	C₂×C₄×C₉²	C₄×C₇	1	2	yes	—
32	C₂×C₄×C₉²	C₂²×C₇	1	1	yes	characteristic
33	C₈×C₉²	C₄×C₇	1	4	yes	direct product factor
34	C₇×C₉²	C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
35	C₂×C₃²×C₄×C₇	C₃²×C₄	1	2	yes	—
36	C₂×C₃²×C₄×C₇	C₂²×C₃²	1	1	yes	characteristic
37	C₃²×C₇×C₈	C₃²×C₄	1	4	yes	—
38	C₂×C₄×C₇×C₉	C₄×C₉	1	24	yes	—
39	C₂×C₄×C₇×C₉	C₂²×C₉	1	12	yes	—
40	C₇×C₈×C₉	C₄×C₉	1	48	yes	direct product factor
41	C₂×C₃×C₈×C₉	C₂×C₃×C₇	1	8	yes	—
42	C₃×C₄²×C₉	C₂×C₃×C₇	1	4	yes	—
43	C₂×C₃×C₇×C₉	C₃×C₄²	1	4	yes	—
44	C₂×C₃×C₇×C₉	C₂×C₃×C₈	1	8	yes	—
45	C₂×C₃×C₇×C₈	C₂×C₃×C₉	1	8	yes	—
46	C₃×C₄²×C₇	C₂×C₃×C₉	1	4	yes	—
47	C₂²×C₉²	C₂×C₄×C₇	1	1	yes	characteristic
48	C₄×C₉²	C₇×C₈	1	4	yes	direct product factor
49	C₄×C₉²	C₂×C₄×C₇	1	2	yes	—
50	C₃²×C₄×C₈	C₃²×C₇	1	1	yes	characteristic
51	C₄×C₈×C₉	C₇×C₉	1	12	yes	direct product factor
52	C₂²×C₃²×C₇	C₂×C₃²×C₄	1	1	yes	characteristic
53	C₃²×C₄×C₇	C₃²×C₈	1	4	yes	—
54	C₃²×C₄×C₇	C₂×C₃²×C₄	1	2	yes	—
55	C₂²×C₇×C₉	C₂×C₄×C₉	1	12	yes	—
56	C₄×C₇×C₉	C₈×C₉	1	48	yes	direct product factor
57	C₄×C₇×C₉	C₂×C₄×C₉	1	24	yes	—
58	C₄×C₇×C₈	C₉²	1	1	yes	characteristic, direct product factor, Hall
59	C₂×C₃×C₄×C₉	C₃×C₄×C₇	1	8	yes	—
60	C₂×C₃×C₄×C₉	C₂²×C₃×C₇	1	4	yes	—
61	C₃×C₈×C₉	C₃×C₄×C₇	1	16	yes	—
62	C₃×C₇×C₉	C₃×C₄×C₈	1	4	yes	—
63	C₂×C₃×C₄×C₇	C₃×C₄×C₉	1	8	yes	—
64	C₂×C₃×C₄×C₇	C₂²×C₃×C₉	1	4	yes	—
65	C₃×C₇×C₈	C₃×C₄×C₉	1	16	yes	—
66	C₂×C₉²	C₄²×C₇	1	1	yes	characteristic
67	C₂×C₉²	C₂×C₇×C₈	1	2	yes	—
68	C₂×C₃²×C₈	C₂×C₃²×C₇	1	2	yes	—
69	C₃²×C₄²	C₂×C₃²×C₇	1	1	yes	characteristic
70	C₂×C₈×C₉	C₂×C₇×C₉	1	24	yes	—
71	C₄²×C₉	C₂×C₇×C₉	1	12	yes	—
72	C₂×C₃²×C₇	C₃²×C₄²	1	1	yes	characteristic
73	C₂×C₃²×C₇	C₂×C₃²×C₈	1	2	yes	—
74	C₂×C₇×C₉	C₄²×C₉	1	12	yes	—
75	C₂×C₇×C₉	C₂×C₈×C₉	1	24	yes	—
76	C₂×C₇×C₈	C₂×C₉²	1	2	yes	—
77	C₄²×C₇	C₂×C₉²	1	1	yes	characteristic
78	C₂²×C₃×C₉	C₂×C₃×C₄×C₇	1	4	yes	—
79	C₃×C₄×C₉	C₃×C₇×C₈	1	16	yes	—
80	C₃×C₄×C₉	C₂×C₃×C₄×C₇	1	8	yes	—
81	C₃×C₄×C₈	C₃×C₇×C₉	1	4	yes	—
82	C₂²×C₃×C₇	C₂×C₃×C₄×C₉	1	4	yes	—
83	C₃×C₄×C₇	C₃×C₈×C₉	1	16	yes	—
84	C₃×C₄×C₇	C₂×C₃×C₄×C₉	1	8	yes	—
85	C₉²	C₄×C₇×C₈	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×C₃²×C₄	C₃²×C₄×C₇	1	2	yes	—
87	C₂×C₃²×C₄	C₂²×C₃²×C₇	1	1	yes	characteristic
88	C₃²×C₈	C₃²×C₄×C₇	1	4	yes	—
89	C₂×C₄×C₉	C₄×C₇×C₉	1	24	yes	—
90	C₂×C₄×C₉	C₂²×C₇×C₉	1	12	yes	—
91	C₈×C₉	C₄×C₇×C₉	1	48	yes	direct product factor
92	C₃²×C₇	C₃²×C₄×C₈	1	1	yes	characteristic
93	C₇×C₉	C₄×C₈×C₉	1	12	yes	direct product factor
94	C₂×C₄×C₇	C₄×C₉²	1	2	yes	—
95	C₂×C₄×C₇	C₂²×C₉²	1	1	yes	characteristic
96	C₇×C₈	C₄×C₉²	1	4	yes	direct product factor
97	C₂×C₃×C₉	C₃×C₄²×C₇	1	4	yes	—
98	C₂×C₃×C₉	C₂×C₃×C₇×C₈	1	8	yes	—
99	C₂×C₃×C₈	C₂×C₃×C₇×C₉	1	8	yes	—
100	C₃×C₄²	C₂×C₃×C₇×C₉	1	4	yes	—
101	C₂×C₃×C₇	C₃×C₄²×C₉	1	4	yes	—
102	C₂×C₃×C₇	C₂×C₃×C₈×C₉	1	8	yes	—
103	C₂²×C₃²	C₂×C₃²×C₄×C₇	1	1	yes	characteristic
104	C₃²×C₄	C₃²×C₇×C₈	1	4	yes	—
105	C₃²×C₄	C₂×C₃²×C₄×C₇	1	2	yes	—
106	C₂²×C₉	C₂×C₄×C₇×C₉	1	12	yes	—
107	C₄×C₉	C₇×C₈×C₉	1	48	yes	direct product factor
108	C₄×C₉	C₂×C₄×C₇×C₉	1	24	yes	—
109	C₄×C₈	C₇×C₉²	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×C₇	C₂×C₄×C₉²	1	1	yes	characteristic
111	C₄×C₇	C₈×C₉²	1	4	yes	direct product factor
112	C₄×C₇	C₂×C₄×C₉²	1	2	yes	—
113	C₃×C₉	C₃×C₄×C₇×C₈	1	4	yes	—
114	C₂×C₃×C₄	C₃×C₄×C₇×C₉	1	8	yes	—
115	C₂×C₃×C₄	C₂²×C₃×C₇×C₉	1	4	yes	—
116	C₃×C₈	C₃×C₄×C₇×C₉	1	16	yes	—
117	C₃×C₇	C₃×C₄×C₈×C₉	1	4	yes	—
118	C₂×C₃²	C₃²×C₄²×C₇	1	1	yes	characteristic
119	C₂×C₃²	C₂×C₃²×C₇×C₈	1	2	yes	—
120	C₂×C₉	C₄²×C₇×C₉	1	12	yes	—
121	C₂×C₉	C₂×C₇×C₈×C₉	1	24	yes	—
122	C₂×C₈	C₂×C₇×C₉²	1	2	yes	—
123	C₄²	C₂×C₇×C₉²	1	1	yes	characteristic
124	C₂×C₇	C₄²×C₉²	1	1	yes	characteristic
125	C₂×C₇	C₂×C₈×C₉²	1	2	yes	—
126	C₂²×C₃	C₂×C₃×C₄×C₇×C₉	1	4	yes	—
127	C₃×C₄	C₃×C₇×C₈×C₉	1	16	yes	—
128	C₃×C₄	C₂×C₃×C₄×C₇×C₉	1	8	yes	—
129	C₃²	C₃²×C₄×C₇×C₈	1	1	yes	characteristic
130	C₉	C₄×C₇×C₈×C₉	1	12	yes	direct product factor
131	C₂×C₄	C₄×C₇×C₉²	1	2	yes	—
132	C₂×C₄	C₂²×C₇×C₉²	1	1	yes	characteristic
133	C₈	C₄×C₇×C₉²	1	4	yes	direct product factor
134	C₇	C₄×C₈×C₉²	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×C₃	C₂×C₃×C₇×C₈×C₉	1	8	yes	—
136	C₂×C₃	C₃×C₄²×C₇×C₉	1	4	yes	—
137	C₂²	C₂×C₄×C₇×C₉²	1	1	yes	characteristic
138	C₄	C₇×C₈×C₉²	1	4	yes	direct product factor
139	C₄	C₂×C₄×C₇×C₉²	1	2	yes	—
140	C₃	C₃×C₄×C₇×C₈×C₉	1	4	yes	—
141	C₂	C₂×C₇×C₈×C₉²	1	2	yes	—
142	C₂	C₄²×C₇×C₉²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃²×C₄	C₂²×C₃²×C₇
socle, prime radical	C₂²×C₃²×C₇	C₂×C₃²×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₇×C₈×C₉²	C₄	direct (N × H)
9. C₄×C₈×C₉²	C₇	direct (N × H)
11. C₄×C₇×C₉²	C₈	direct (N × H)
14. C₄×C₇×C₈×C₉	C₉	direct (N × H)
33. C₈×C₉²	C₄×C₇	direct (N × H)
34. C₇×C₉²	C₄×C₈	direct (N × H)
40. C₇×C₈×C₉	C₄×C₉	direct (N × H)
48. C₄×C₉²	C₇×C₈	direct (N × H)
51. C₄×C₈×C₉	C₇×C₉	direct (N × H)
56. C₄×C₇×C₉	C₈×C₉	direct (N × H)
58. C₄×C₇×C₈	C₉²	direct (N × H)
85. C₉²	C₄×C₇×C₈	direct (N × H)
91. C₈×C₉	C₄×C₇×C₉	direct (N × H)
93. C₇×C₉	C₄×C₈×C₉	direct (N × H)
96. C₇×C₈	C₄×C₉²	direct (N × H)
107. C₄×C₉	C₇×C₈×C₉	direct (N × H)
109. C₄×C₈	C₇×C₉²	direct (N × H)
111. C₄×C₇	C₈×C₉²	direct (N × H)
130. C₉	C₄×C₇×C₈×C₉	direct (N × H)
133. C₈	C₄×C₇×C₉²	direct (N × H)
134. C₇	C₄×C₈×C₉²	direct (N × H)
138. C₄	C₇×C₈×C₉²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₇×C₉²	2
C₂×C₇×C₈×C₉²	2
C₃×C₄×C₇×C₈×C₉	3
C₄×C₈×C₉²	7

Subgroups — C4×C7×C8×C92 #72195909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇×C₈×C₉² #72195909