Subgroups — C₄²×C₅²×C₇×C₈ #72199821

Order 22400.

3264 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	C₄³×C₅²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅²×C₇×C₈	C₂	1	6	yes	maximal
3	C₂×C₄²×C₅²×C₇	C₄	1	4	yes	—
4	C₂×C₄²×C₅²×C₇	C₂²	1	3	yes	—
5	C₂²×C₅²×C₇×C₈	C₂²	1	4	yes	—
6	C₄×C₅²×C₇×C₈	C₄	1	24	yes	direct product factor
7	C₄²×C₅×C₇×C₈	C₅	1	6	yes	maximal, direct product factor
8	C₄²×C₅²×C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₄×C₅²×C₇	C₂×C₄	1	6	yes	—
10	C₂²×C₄×C₅²×C₇	C₂³	1	1	yes	characteristic
11	C₄²×C₅²×C₇	C₈	1	16	yes	direct product factor
12	C₄²×C₅²×C₇	C₂×C₄	1	12	yes	—
13	C₂×C₅²×C₇×C₈	C₂×C₄	1	24	yes	—
14	C₄³×C₅×C₇	C₂×C₅	1	6	yes	—
15	C₂×C₄×C₅×C₇×C₈	C₂×C₅	1	36	yes	—
16	C₂×C₄×C₅²×C₈	C₂×C₇	1	6	yes	—
17	C₄³×C₅²	C₂×C₇	1	1	yes	characteristic
18	C₂³×C₅²×C₇	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅²×C₇	C₄²	1	12	yes	—
20	C₂×C₄×C₅²×C₇	C₂×C₈	1	24	yes	—
21	C₂×C₄×C₅²×C₇	C₂²×C₄	1	6	yes	—
22	C₅²×C₇×C₈	C₄²	1	16	yes	direct product factor
23	C₂²×C₅×C₇×C₈	C₂²×C₅	1	24	yes	—
24	C₂×C₄²×C₅×C₇	C₄×C₅	1	24	yes	—
25	C₂×C₄²×C₅×C₇	C₂²×C₅	1	18	yes	—
26	C₄×C₅×C₇×C₈	C₄×C₅	1	144	yes	direct product factor
27	C₄²×C₇×C₈	C₅²	1	1	yes	characteristic, direct product factor, Hall
28	C₂²×C₅²×C₈	C₂²×C₇	1	4	yes	—
29	C₂×C₄²×C₅²	C₄×C₇	1	4	yes	—
30	C₂×C₄²×C₅²	C₂²×C₇	1	3	yes	—
31	C₄×C₅²×C₈	C₄×C₇	1	24	yes	direct product factor
32	C₂²×C₅²×C₇	C₂×C₄²	1	3	yes	—
33	C₂²×C₅²×C₇	C₂²×C₈	1	4	yes	—
34	C₄×C₅²×C₇	C₄×C₈	1	24	yes	direct product factor
35	C₄×C₅²×C₇	C₂×C₄²	1	4	yes	—
36	C₄²×C₅×C₈	C₅×C₇	1	6	yes	direct product factor
37	C₂²×C₄×C₅×C₇	C₂×C₄×C₅	1	36	yes	—
38	C₂²×C₄×C₅×C₇	C₂³×C₅	1	6	yes	—
39	C₂×C₅×C₇×C₈	C₂×C₄×C₅	1	144	yes	—
40	C₄²×C₅×C₇	C₅×C₈	1	96	yes	direct product factor
41	C₄²×C₅×C₇	C₂×C₄×C₅	1	72	yes	—
42	C₂×C₄×C₇×C₈	C₂×C₅²	1	6	yes	—
43	C₄³×C₇	C₂×C₅²	1	1	yes	characteristic
44	C₂²×C₄×C₅²	C₂×C₄×C₇	1	6	yes	—
45	C₂²×C₄×C₅²	C₂³×C₇	1	1	yes	characteristic
46	C₂×C₅²×C₈	C₂×C₄×C₇	1	24	yes	—
47	C₄²×C₅²	C₇×C₈	1	16	yes	direct product factor
48	C₄²×C₅²	C₂×C₄×C₇	1	12	yes	—
49	C₂×C₅²×C₇	C₄³	1	1	yes	characteristic
50	C₂×C₅²×C₇	C₂×C₄×C₈	1	6	yes	—
51	C₂×C₄×C₅×C₈	C₂×C₅×C₇	1	36	yes	—
52	C₄³×C₅	C₂×C₅×C₇	1	6	yes	—
53	C₂³×C₅×C₇	C₂²×C₄×C₅	1	6	yes	—
54	C₂×C₄×C₅×C₇	C₄²×C₅	1	72	yes	—
55	C₂×C₄×C₅×C₇	C₂×C₅×C₈	1	144	yes	—
56	C₂×C₄×C₅×C₇	C₂²×C₄×C₅	1	36	yes	—
57	C₅×C₇×C₈	C₄²×C₅	1	96	yes	direct product factor
58	C₂²×C₇×C₈	C₂²×C₅²	1	4	yes	—
59	C₂×C₄²×C₇	C₄×C₅²	1	4	yes	—
60	C₂×C₄²×C₇	C₂²×C₅²	1	3	yes	—
61	C₄×C₇×C₈	C₄×C₅²	1	24	yes	direct product factor
62	C₂³×C₅²	C₂²×C₄×C₇	1	1	yes	characteristic
63	C₂×C₄×C₅²	C₄²×C₇	1	12	yes	—
64	C₂×C₄×C₅²	C₂×C₇×C₈	1	24	yes	—
65	C₂×C₄×C₅²	C₂²×C₄×C₇	1	6	yes	—
66	C₅²×C₈	C₄²×C₇	1	16	yes	direct product factor
67	C₅²×C₇	C₄²×C₈	1	1	yes	characteristic, direct product factor, Hall
68	C₂²×C₅×C₈	C₂²×C₅×C₇	1	24	yes	—
69	C₂×C₄²×C₅	C₄×C₅×C₇	1	24	yes	—
70	C₂×C₄²×C₅	C₂²×C₅×C₇	1	18	yes	—
71	C₄×C₅×C₈	C₄×C₅×C₇	1	144	yes	direct product factor
72	C₂²×C₅×C₇	C₂×C₄²×C₅	1	18	yes	—
73	C₂²×C₅×C₇	C₂²×C₅×C₈	1	24	yes	—
74	C₄×C₅×C₇	C₄×C₅×C₈	1	144	yes	direct product factor
75	C₄×C₅×C₇	C₂×C₄²×C₅	1	24	yes	—
76	C₄²×C₈	C₅²×C₇	1	1	yes	characteristic, direct product factor, Sylow
77	C₂²×C₄×C₇	C₂×C₄×C₅²	1	6	yes	—
78	C₂²×C₄×C₇	C₂³×C₅²	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×C₄×C₅²	1	24	yes	—
80	C₄²×C₇	C₅²×C₈	1	16	yes	direct product factor
81	C₄²×C₇	C₂×C₄×C₅²	1	12	yes	—
82	C₂²×C₅²	C₂×C₄²×C₇	1	3	yes	—
83	C₂²×C₅²	C₂²×C₇×C₈	1	4	yes	—
84	C₄×C₅²	C₄×C₇×C₈	1	24	yes	direct product factor
85	C₄×C₅²	C₂×C₄²×C₇	1	4	yes	—
86	C₂²×C₄×C₅	C₂×C₄×C₅×C₇	1	36	yes	—
87	C₂²×C₄×C₅	C₂³×C₅×C₇	1	6	yes	—
88	C₂×C₅×C₈	C₂×C₄×C₅×C₇	1	144	yes	—
89	C₄²×C₅	C₅×C₇×C₈	1	96	yes	direct product factor
90	C₄²×C₅	C₂×C₄×C₅×C₇	1	72	yes	—
91	C₂×C₅×C₇	C₄³×C₅	1	6	yes	—
92	C₂×C₅×C₇	C₂×C₄×C₅×C₈	1	36	yes	—
93	C₂×C₄×C₈	C₂×C₅²×C₇	1	6	yes	—
94	C₄³	C₂×C₅²×C₇	1	1	yes	characteristic
95	C₂³×C₇	C₂²×C₄×C₅²	1	1	yes	characteristic
96	C₂×C₄×C₇	C₄²×C₅²	1	12	yes	—
97	C₂×C₄×C₇	C₂×C₅²×C₈	1	24	yes	—
98	C₂×C₄×C₇	C₂²×C₄×C₅²	1	6	yes	—
99	C₇×C₈	C₄²×C₅²	1	16	yes	direct product factor
100	C₂×C₅²	C₄³×C₇	1	1	yes	characteristic
101	C₂×C₅²	C₂×C₄×C₇×C₈	1	6	yes	—
102	C₂³×C₅	C₂²×C₄×C₅×C₇	1	6	yes	—
103	C₂×C₄×C₅	C₄²×C₅×C₇	1	72	yes	—
104	C₂×C₄×C₅	C₂×C₅×C₇×C₈	1	144	yes	—
105	C₂×C₄×C₅	C₂²×C₄×C₅×C₇	1	36	yes	—
106	C₅×C₈	C₄²×C₅×C₇	1	96	yes	direct product factor
107	C₅×C₇	C₄²×C₅×C₈	1	6	yes	direct product factor
108	C₂²×C₈	C₂²×C₅²×C₇	1	4	yes	—
109	C₂×C₄²	C₄×C₅²×C₇	1	4	yes	—
110	C₂×C₄²	C₂²×C₅²×C₇	1	3	yes	—
111	C₄×C₈	C₄×C₅²×C₇	1	24	yes	direct product factor
112	C₂²×C₇	C₂×C₄²×C₅²	1	3	yes	—
113	C₂²×C₇	C₂²×C₅²×C₈	1	4	yes	—
114	C₄×C₇	C₄×C₅²×C₈	1	24	yes	direct product factor
115	C₄×C₇	C₂×C₄²×C₅²	1	4	yes	—
116	C₅²	C₄²×C₇×C₈	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×C₅	C₂×C₄²×C₅×C₇	1	18	yes	—
118	C₂²×C₅	C₂²×C₅×C₇×C₈	1	24	yes	—
119	C₄×C₅	C₄×C₅×C₇×C₈	1	144	yes	direct product factor
120	C₄×C₅	C₂×C₄²×C₅×C₇	1	24	yes	—
121	C₂²×C₄	C₂×C₄×C₅²×C₇	1	6	yes	—
122	C₂²×C₄	C₂³×C₅²×C₇	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄×C₅²×C₇	1	24	yes	—
124	C₄²	C₅²×C₇×C₈	1	16	yes	direct product factor
125	C₄²	C₂×C₄×C₅²×C₇	1	12	yes	—
126	C₂×C₇	C₄³×C₅²	1	1	yes	characteristic
127	C₂×C₇	C₂×C₄×C₅²×C₈	1	6	yes	—
128	C₂×C₅	C₂×C₄×C₅×C₇×C₈	1	36	yes	—
129	C₂×C₅	C₄³×C₅×C₇	1	6	yes	—
130	C₂³	C₂²×C₄×C₅²×C₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₅²×C₇×C₈	1	24	yes	—
132	C₂×C₄	C₄²×C₅²×C₇	1	12	yes	—
133	C₂×C₄	C₂²×C₄×C₅²×C₇	1	6	yes	—
134	C₈	C₄²×C₅²×C₇	1	16	yes	direct product factor
135	C₇	C₄²×C₅²×C₈	1	1	yes	characteristic, direct product factor, Sylow
136	C₅	C₄²×C₅×C₇×C₈	1	6	yes	direct product factor
137	C₂²	C₂²×C₅²×C₇×C₈	1	4	yes	—
138	C₂²	C₂×C₄²×C₅²×C₇	1	3	yes	—
139	C₄	C₄×C₅²×C₇×C₈	1	24	yes	direct product factor
140	C₄	C₂×C₄²×C₅²×C₇	1	4	yes	—
141	C₂	C₂×C₄×C₅²×C₇×C₈	1	6	yes	—
142	C₂	C₄³×C₅²×C₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₄	C₂³×C₅²×C₇
socle, prime radical	C₂³×C₅²×C₇	C₂²×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₄×C₅²×C₇×C₈	C₄	direct (N × H)
7. C₄²×C₅×C₇×C₈	C₅	direct (N × H)
8. C₄²×C₅²×C₈	C₇	direct (N × H)
11. C₄²×C₅²×C₇	C₈	direct (N × H)
22. C₅²×C₇×C₈	C₄²	direct (N × H)
26. C₄×C₅×C₇×C₈	C₄×C₅	direct (N × H)
27. C₄²×C₇×C₈	C₅²	direct (N × H)
31. C₄×C₅²×C₈	C₄×C₇	direct (N × H)
34. C₄×C₅²×C₇	C₄×C₈	direct (N × H)
36. C₄²×C₅×C₈	C₅×C₇	direct (N × H)
40. C₄²×C₅×C₇	C₅×C₈	direct (N × H)
47. C₄²×C₅²	C₇×C₈	direct (N × H)
57. C₅×C₇×C₈	C₄²×C₅	direct (N × H)
61. C₄×C₇×C₈	C₄×C₅²	direct (N × H)
66. C₅²×C₈	C₄²×C₇	direct (N × H)
67. C₅²×C₇	C₄²×C₈	direct (N × H)
71. C₄×C₅×C₈	C₄×C₅×C₇	direct (N × H)
74. C₄×C₅×C₇	C₄×C₅×C₈	direct (N × H)
76. C₄²×C₈	C₅²×C₇	direct (N × H)
80. C₄²×C₇	C₅²×C₈	direct (N × H)
84. C₄×C₅²	C₄×C₇×C₈	direct (N × H)
89. C₄²×C₅	C₅×C₇×C₈	direct (N × H)
99. C₇×C₈	C₄²×C₅²	direct (N × H)
106. C₅×C₈	C₄²×C₅×C₇	direct (N × H)
107. C₅×C₇	C₄²×C₅×C₈	direct (N × H)
111. C₄×C₈	C₄×C₅²×C₇	direct (N × H)
114. C₄×C₇	C₄×C₅²×C₈	direct (N × H)
116. C₅²	C₄²×C₇×C₈	direct (N × H)
119. C₄×C₅	C₄×C₅×C₇×C₈	direct (N × H)
124. C₄²	C₅²×C₇×C₈	direct (N × H)
134. C₈	C₄²×C₅²×C₇	direct (N × H)
135. C₇	C₄²×C₅²×C₈	direct (N × H)
136. C₅	C₄²×C₅×C₇×C₈	direct (N × H)
139. C₄	C₄×C₅²×C₇×C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄³×C₅²×C₇	2
C₂×C₄×C₅²×C₇×C₈	2
C₄²×C₅×C₇×C₈	5
C₄²×C₅²×C₈	7

Subgroups — C42×C52×C7×C8 #72199821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₅²×C₇×C₈ #72199821