Subgroups — C₂⁴×C₄×C₇×C₈ #72183245

Order 3584.

37508 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	C₂⁵×C₇×C₈	C₂	1	2	yes	maximal
2	C₂⁴×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄×C₇×C₈	C₂	1	60	yes	maximal, direct product factor
4	C₂⁵×C₄×C₇	C₄	1	2	yes	—
5	C₂⁵×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂⁴×C₇×C₈	C₄	1	64	yes	direct product factor
7	C₂⁴×C₇×C₈	C₂²	1	60	yes	—
8	C₂³×C₄²×C₇	C₄	1	30	yes	—
9	C₂³×C₄²×C₇	C₂²	1	30	yes	—
10	C₂²×C₄×C₇×C₈	C₂²	1	560	yes	direct product factor
11	C₂⁴×C₄×C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂⁶×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂⁴×C₄×C₇	C₈	1	64	yes	direct product factor
14	C₂⁴×C₄×C₇	C₂×C₄	1	15	yes	—
15	C₂⁴×C₄×C₇	C₂×C₄	1	32	yes	—
16	C₂⁴×C₄×C₇	C₂×C₄	1	60	yes	—
17	C₂⁴×C₄×C₇	C₂³	1	15	yes	—
18	C₂³×C₇×C₈	C₂×C₄	1	960	yes	direct product factor
19	C₂³×C₇×C₈	C₂³	1	280	yes	—
20	C₂²×C₄²×C₇	C₂×C₄	1	420	yes	—
21	C₂²×C₄²×C₇	C₂³	1	140	yes	—
22	C₂×C₄×C₇×C₈	C₂³	1	960	yes	direct product factor
23	C₂⁵×C₈	C₂×C₇	1	2	yes	—
24	C₂⁴×C₄²	C₂×C₇	1	1	yes	characteristic
25	C₂³×C₄×C₈	C₂×C₇	1	60	yes	direct product factor
26	C₂⁵×C₇	C₄²	1	16	yes	—
27	C₂⁵×C₇	C₂×C₈	1	32	yes	—
28	C₂⁵×C₇	C₂²×C₄	1	15	yes	—
29	C₂³×C₄×C₇	C₄²	1	240	yes	—
30	C₂³×C₄×C₇	C₂×C₈	1	960	yes	direct product factor
31	C₂³×C₄×C₇	C₂²×C₄	1	280	yes	—
32	C₂³×C₄×C₇	C₂²×C₄	1	105	yes	—
33	C₂³×C₄×C₇	C₂²×C₄	1	240	yes	—
34	C₂³×C₄×C₇	C₂⁴	1	35	yes	—
35	C₂²×C₇×C₈	C₂²×C₄	1	2240	yes	direct product factor
36	C₂²×C₇×C₈	C₂⁴	1	240	yes	—
37	C₂×C₄²×C₇	C₂²×C₄	1	840	yes	—
38	C₂×C₄²×C₇	C₂⁴	1	120	yes	—
39	C₄×C₇×C₈	C₂⁴	1	256	yes	direct product factor
40	C₂⁵×C₄	C₄×C₇	1	2	yes	—
41	C₂⁵×C₄	C₂²×C₇	1	1	yes	characteristic
42	C₂⁴×C₈	C₄×C₇	1	64	yes	direct product factor
43	C₂⁴×C₈	C₂²×C₇	1	60	yes	—
44	C₂³×C₄²	C₄×C₇	1	30	yes	—
45	C₂³×C₄²	C₂²×C₇	1	30	yes	—
46	C₂²×C₄×C₈	C₂²×C₇	1	560	yes	direct product factor
47	C₂⁴×C₇	C₄×C₈	1	256	yes	direct product factor
48	C₂⁴×C₇	C₂×C₄²	1	120	yes	—
49	C₂⁴×C₇	C₂²×C₈	1	240	yes	—
50	C₂⁴×C₇	C₂³×C₄	1	35	yes	—
51	C₂²×C₄×C₇	C₂×C₄²	1	840	yes	—
52	C₂²×C₄×C₇	C₂²×C₈	1	2240	yes	direct product factor
53	C₂²×C₄×C₇	C₂³×C₄	1	280	yes	—
54	C₂²×C₄×C₇	C₂³×C₄	1	105	yes	—
55	C₂²×C₄×C₇	C₂³×C₄	1	240	yes	—
56	C₂²×C₄×C₇	C₂⁵	1	15	yes	—
57	C₂×C₇×C₈	C₂³×C₄	1	960	yes	direct product factor
58	C₂×C₇×C₈	C₂⁵	1	32	yes	—
59	C₄²×C₇	C₂³×C₄	1	240	yes	—
60	C₄²×C₇	C₂⁵	1	16	yes	—
61	C₂⁶	C₂×C₄×C₇	1	1	yes	characteristic
62	C₂⁴×C₄	C₇×C₈	1	64	yes	direct product factor
63	C₂⁴×C₄	C₂×C₄×C₇	1	15	yes	—
64	C₂⁴×C₄	C₂×C₄×C₇	1	32	yes	—
65	C₂⁴×C₄	C₂×C₄×C₇	1	60	yes	—
66	C₂⁴×C₄	C₂³×C₇	1	15	yes	—
67	C₂³×C₈	C₂×C₄×C₇	1	960	yes	direct product factor
68	C₂³×C₈	C₂³×C₇	1	280	yes	—
69	C₂²×C₄²	C₂×C₄×C₇	1	420	yes	—
70	C₂²×C₄²	C₂³×C₇	1	140	yes	—
71	C₂×C₄×C₈	C₂³×C₇	1	960	yes	direct product factor
72	C₂³×C₇	C₂×C₄×C₈	1	960	yes	direct product factor
73	C₂³×C₇	C₂²×C₄²	1	140	yes	—
74	C₂³×C₇	C₂³×C₈	1	280	yes	—
75	C₂³×C₇	C₂⁴×C₄	1	15	yes	—
76	C₂×C₄×C₇	C₂²×C₄²	1	420	yes	—
77	C₂×C₄×C₇	C₂³×C₈	1	960	yes	direct product factor
78	C₂×C₄×C₇	C₂⁴×C₄	1	15	yes	—
79	C₂×C₄×C₇	C₂⁴×C₄	1	32	yes	—
80	C₂×C₄×C₇	C₂⁴×C₄	1	60	yes	—
81	C₂×C₄×C₇	C₂⁶	1	1	yes	characteristic
82	C₇×C₈	C₂⁴×C₄	1	64	yes	direct product factor
83	C₂⁵	C₄²×C₇	1	16	yes	—
84	C₂⁵	C₂×C₇×C₈	1	32	yes	—
85	C₂⁵	C₂²×C₄×C₇	1	15	yes	—
86	C₂³×C₄	C₄²×C₇	1	240	yes	—
87	C₂³×C₄	C₂×C₇×C₈	1	960	yes	direct product factor
88	C₂³×C₄	C₂²×C₄×C₇	1	280	yes	—
89	C₂³×C₄	C₂²×C₄×C₇	1	105	yes	—
90	C₂³×C₄	C₂²×C₄×C₇	1	240	yes	—
91	C₂³×C₄	C₂⁴×C₇	1	35	yes	—
92	C₂²×C₈	C₂²×C₄×C₇	1	2240	yes	direct product factor
93	C₂²×C₈	C₂⁴×C₇	1	240	yes	—
94	C₂×C₄²	C₂²×C₄×C₇	1	840	yes	—
95	C₂×C₄²	C₂⁴×C₇	1	120	yes	—
96	C₄×C₈	C₂⁴×C₇	1	256	yes	direct product factor
97	C₂²×C₇	C₂²×C₄×C₈	1	560	yes	direct product factor
98	C₂²×C₇	C₂³×C₄²	1	30	yes	—
99	C₂²×C₇	C₂⁴×C₈	1	60	yes	—
100	C₂²×C₇	C₂⁵×C₄	1	1	yes	characteristic
101	C₄×C₇	C₂³×C₄²	1	30	yes	—
102	C₄×C₇	C₂⁴×C₈	1	64	yes	direct product factor
103	C₄×C₇	C₂⁵×C₄	1	2	yes	—
104	C₂⁴	C₄×C₇×C₈	1	256	yes	direct product factor
105	C₂⁴	C₂×C₄²×C₇	1	120	yes	—
106	C₂⁴	C₂²×C₇×C₈	1	240	yes	—
107	C₂⁴	C₂³×C₄×C₇	1	35	yes	—
108	C₂²×C₄	C₂×C₄²×C₇	1	840	yes	—
109	C₂²×C₄	C₂²×C₇×C₈	1	2240	yes	direct product factor
110	C₂²×C₄	C₂³×C₄×C₇	1	280	yes	—
111	C₂²×C₄	C₂³×C₄×C₇	1	105	yes	—
112	C₂²×C₄	C₂³×C₄×C₇	1	240	yes	—
113	C₂²×C₄	C₂⁵×C₇	1	15	yes	—
114	C₂×C₈	C₂³×C₄×C₇	1	960	yes	direct product factor
115	C₂×C₈	C₂⁵×C₇	1	32	yes	—
116	C₄²	C₂³×C₄×C₇	1	240	yes	—
117	C₄²	C₂⁵×C₇	1	16	yes	—
118	C₂×C₇	C₂³×C₄×C₈	1	60	yes	direct product factor
119	C₂×C₇	C₂⁴×C₄²	1	1	yes	characteristic
120	C₂×C₇	C₂⁵×C₈	1	2	yes	—
121	C₂³	C₂×C₄×C₇×C₈	1	960	yes	direct product factor
122	C₂³	C₂²×C₄²×C₇	1	140	yes	—
123	C₂³	C₂³×C₇×C₈	1	280	yes	—
124	C₂³	C₂⁴×C₄×C₇	1	15	yes	—
125	C₂×C₄	C₂²×C₄²×C₇	1	420	yes	—
126	C₂×C₄	C₂³×C₇×C₈	1	960	yes	direct product factor
127	C₂×C₄	C₂⁴×C₄×C₇	1	15	yes	—
128	C₂×C₄	C₂⁴×C₄×C₇	1	32	yes	—
129	C₂×C₄	C₂⁴×C₄×C₇	1	60	yes	—
130	C₂×C₄	C₂⁶×C₇	1	1	yes	characteristic
131	C₈	C₂⁴×C₄×C₇	1	64	yes	direct product factor
132	C₇	C₂⁴×C₄×C₈	1	1	yes	characteristic, direct product factor, Sylow
133	C₂²	C₂²×C₄×C₇×C₈	1	560	yes	direct product factor
134	C₂²	C₂³×C₄²×C₇	1	30	yes	—
135	C₂²	C₂⁴×C₇×C₈	1	60	yes	—
136	C₂²	C₂⁵×C₄×C₇	1	1	yes	characteristic
137	C₄	C₂³×C₄²×C₇	1	30	yes	—
138	C₄	C₂⁴×C₇×C₈	1	64	yes	direct product factor
139	C₄	C₂⁵×C₄×C₇	1	2	yes	—
140	C₂	C₂³×C₄×C₇×C₈	1	60	yes	direct product factor
141	C₂	C₂⁴×C₄²×C₇	1	1	yes	characteristic
142	C₂	C₂⁵×C₇×C₈	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂⁶×C₇
socle, prime radical	C₂⁶×C₇	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂³×C₄×C₇×C₈	C₂	direct (N × H)
6. C₂⁴×C₇×C₈	C₄	direct (N × H)
10. C₂²×C₄×C₇×C₈	C₂²	direct (N × H)
11. C₂⁴×C₄×C₈	C₇	direct (N × H)
13. C₂⁴×C₄×C₇	C₈	direct (N × H)
18. C₂³×C₇×C₈	C₂×C₄	direct (N × H)
22. C₂×C₄×C₇×C₈	C₂³	direct (N × H)
25. C₂³×C₄×C₈	C₂×C₇	direct (N × H)
30. C₂³×C₄×C₇	C₂×C₈	direct (N × H)
35. C₂²×C₇×C₈	C₂²×C₄	direct (N × H)
39. C₄×C₇×C₈	C₂⁴	direct (N × H)
42. C₂⁴×C₈	C₄×C₇	direct (N × H)
46. C₂²×C₄×C₈	C₂²×C₇	direct (N × H)
47. C₂⁴×C₇	C₄×C₈	direct (N × H)
52. C₂²×C₄×C₇	C₂²×C₈	direct (N × H)
57. C₂×C₇×C₈	C₂³×C₄	direct (N × H)
62. C₂⁴×C₄	C₇×C₈	direct (N × H)
67. C₂³×C₈	C₂×C₄×C₇	direct (N × H)
71. C₂×C₄×C₈	C₂³×C₇	direct (N × H)
72. C₂³×C₇	C₂×C₄×C₈	direct (N × H)
77. C₂×C₄×C₇	C₂³×C₈	direct (N × H)
82. C₇×C₈	C₂⁴×C₄	direct (N × H)
87. C₂³×C₄	C₂×C₇×C₈	direct (N × H)
92. C₂²×C₈	C₂²×C₄×C₇	direct (N × H)
96. C₄×C₈	C₂⁴×C₇	direct (N × H)
97. C₂²×C₇	C₂²×C₄×C₈	direct (N × H)
102. C₄×C₇	C₂⁴×C₈	direct (N × H)
104. C₂⁴	C₄×C₇×C₈	direct (N × H)
109. C₂²×C₄	C₂²×C₇×C₈	direct (N × H)
114. C₂×C₈	C₂³×C₄×C₇	direct (N × H)
118. C₂×C₇	C₂³×C₄×C₈	direct (N × H)
121. C₂³	C₂×C₄×C₇×C₈	direct (N × H)
126. C₂×C₄	C₂³×C₇×C₈	direct (N × H)
131. C₈	C₂⁴×C₄×C₇	direct (N × H)
132. C₇	C₂⁴×C₄×C₈	direct (N × H)
133. C₂²	C₂²×C₄×C₇×C₈	direct (N × H)
138. C₄	C₂⁴×C₇×C₈	direct (N × H)
140. C₂	C₂³×C₄×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
C₂⁵×C₇×C₈	2
C₂⁴×C₄²×C₇	2
C₂³×C₄×C₇×C₈	2
C₂⁴×C₄×C₈	7

Subgroups — C24×C4×C7×C8 #72183245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄×C₇×C₈ #72183245