Subgroups — C₂³×C₄×C₇₁⋊C₈ #32860832

Order 18176.

4638 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇₁⋊C₈	C₂	1	2	yes	maximal
2	C₂²×C₄×C₇₁⋊C₈	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄²×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₇₁⋊C₈	C₄	1	32	yes	direct product factor
5	C₂³×C₇₁⋊C₈	C₂²	1	28	yes	—
6	C₂×C₄×C₇₁⋊C₈	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₄×C₇₁	C₄	1	2	yes	—
8	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₇₁	C₄	1	14	yes	—
10	C₂²×C₄²×C₇₁	C₂²	1	14	yes	—
11	C₂²×C₇₁⋊C₈	C₂×C₄	1	224	yes	direct product factor
12	C₂²×C₇₁⋊C₈	C₂³	1	56	yes	—
13	C₄×C₇₁⋊C₈	C₂³	1	64	yes	direct product factor
14	C₂⁵×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₇₁	C₈	1	32	yes	complemented
16	C₂³×C₄×C₇₁	C₂×C₄	1	7	yes	—
17	C₂³×C₄×C₇₁	C₂×C₄	1	16	yes	—
18	C₂³×C₄×C₇₁	C₂×C₄	1	28	yes	—
19	C₂³×C₄×C₇₁	C₂³	1	7	yes	—
20	C₂×C₄²×C₇₁	C₂×C₄	1	84	yes	—
21	C₂×C₄²×C₇₁	C₂³	1	28	yes	—
22	C₂×C₇₁⋊C₈	C₂²×C₄	1	224	yes	direct product factor
23	C₂×C₇₁⋊C₈	C₂⁴	1	16	yes	—
24	C₂⁴×C₇₁	C₄²	1	8	yes	—
25	C₂⁴×C₇₁	C₂×C₈	1	16	yes	—
26	C₂⁴×C₇₁	C₂²×C₄	1	7	yes	—
27	C₂²×C₄×C₇₁	C₄²	1	56	yes	—
28	C₂²×C₄×C₇₁	C₂×C₈	1	224	yes	complemented
29	C₂²×C₄×C₇₁	C₂²×C₄	1	56	yes	—
30	C₂²×C₄×C₇₁	C₂²×C₄	1	21	yes	—
31	C₂²×C₄×C₇₁	C₂²×C₄	1	56	yes	—
32	C₂²×C₄×C₇₁	C₂⁴	1	7	yes	—
33	C₄²×C₇₁	C₂²×C₄	1	56	yes	—
34	C₄²×C₇₁	C₂⁴	1	8	yes	—
35	C₂³×C₇₁	C₄×C₈	1	64	yes	complemented
36	C₂³×C₇₁	C₂×C₄²	1	28	yes	—
37	C₂³×C₇₁	C₂²×C₈	1	56	yes	—
38	C₂³×C₇₁	C₂³×C₄	1	7	yes	—
39	C₂×C₄×C₇₁	C₂×C₄²	1	84	yes	—
40	C₂×C₄×C₇₁	C₂²×C₈	1	224	yes	complemented
41	C₂×C₄×C₇₁	C₂³×C₄	1	28	yes	—
42	C₂×C₄×C₇₁	C₂³×C₄	1	16	yes	—
43	C₂×C₄×C₇₁	C₂³×C₄	1	7	yes	—
44	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
45	C₇₁⋊C₈	C₂³×C₄	1	32	yes	direct product factor
46	C₂²×C₇₁	C₂×C₄×C₈	1	112	yes	complemented
47	C₂²×C₇₁	C₂²×C₄²	1	14	yes	—
48	C₂²×C₇₁	C₂³×C₈	1	28	yes	—
49	C₂²×C₇₁	C₂⁴×C₄	1	1	yes	characteristic
50	C₄×C₇₁	C₂²×C₄²	1	14	yes	—
51	C₄×C₇₁	C₂³×C₈	1	32	yes	complemented
52	C₄×C₇₁	C₂⁴×C₄	1	2	yes	—
53	C₂³×C₄×C₈	—	71	1	no	maximal, Sylow
54	C₂×C₇₁	C₂²×C₄×C₈	1	28	yes	complemented
55	C₂×C₇₁	C₂³×C₄²	1	1	yes	characteristic
56	C₂×C₇₁	C₂⁴×C₈	1	2	yes	—
57	C₂⁴×C₈	—	71	2	no	—
58	C₂³×C₄²	D₇₁	1	1	yes	characteristic
59	C₂²×C₄×C₈	—	71	28	no	—
60	C₇₁	C₂³×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
61	C₂⁴×C₄	Dic₇₁	1	2	yes	—
62	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
63	C₂³×C₈	—	71	32	no	—
64	C₂³×C₈	—	71	28	no	—
65	C₂²×C₄²	Dic₇₁	1	14	yes	—
66	C₂²×C₄²	C₂×D₇₁	1	14	yes	—
67	C₂×C₄×C₈	—	71	112	no	—
68	C₂⁵	C₂×Dic₇₁	1	1	yes	characteristic
69	C₂³×C₄	C₇₁⋊C₈	1	32	yes	direct product factor
70	C₂³×C₄	C₄×D₇₁	1	16	yes	—
71	C₂³×C₄	C₂×Dic₇₁	1	7	yes	—
72	C₂³×C₄	C₂×Dic₇₁	1	28	yes	—
73	C₂³×C₄	C₂²×D₇₁	1	7	yes	—
74	C₂²×C₈	—	71	224	no	—
75	C₂²×C₈	—	71	56	no	—
76	C₂×C₄²	C₂×Dic₇₁	1	84	yes	—
77	C₂×C₄²	C₂²×D₇₁	1	28	yes	—
78	C₄×C₈	—	71	64	no	—
79	C₂⁴	C₄×Dic₇₁	1	8	yes	—
80	C₂⁴	C₂×C₇₁⋊C₈	1	16	yes	—
81	C₂⁴	C₂²×Dic₇₁	1	7	yes	—
82	C₂²×C₄	C₄×Dic₇₁	1	56	yes	—
83	C₂²×C₄	C₂×C₇₁⋊C₈	1	224	yes	direct product factor
84	C₂²×C₄	C₂×C₄×D₇₁	1	56	yes	—
85	C₂²×C₄	C₂²×Dic₇₁	1	21	yes	—
86	C₂²×C₄	C₂²×Dic₇₁	1	56	yes	—
87	C₂²×C₄	C₂³×D₇₁	1	7	yes	—
88	C₂×C₈	—	71	224	no	—
89	C₂×C₈	—	71	16	no	—
90	C₄²	C₂²×Dic₇₁	1	56	yes	—
91	C₄²	C₂³×D₇₁	1	8	yes	—
92	C₂³	C₄×C₇₁⋊C₈	1	64	yes	direct product factor
93	C₂³	C₂×C₄×Dic₇₁	1	28	yes	—
94	C₂³	C₂²×C₇₁⋊C₈	1	56	yes	—
95	C₂³	C₂³×Dic₇₁	1	7	yes	—
96	C₂×C₄	C₂×C₄×Dic₇₁	1	84	yes	—
97	C₂×C₄	C₂²×C₇₁⋊C₈	1	224	yes	direct product factor
98	C₂×C₄	C₂²×C₄×D₇₁	1	28	yes	—
99	C₂×C₄	C₂³×Dic₇₁	1	7	yes	—
100	C₂×C₄	C₂³×Dic₇₁	1	16	yes	—
101	C₂×C₄	C₂⁴×D₇₁	1	1	yes	characteristic
102	C₈	—	71	32	no	—
103	C₂²	C₂×C₄×C₇₁⋊C₈	1	112	yes	direct product factor
104	C₂²	C₂²×C₄×Dic₇₁	1	14	yes	—
105	C₂²	C₂³×C₇₁⋊C₈	1	28	yes	—
106	C₂²	C₂⁴×Dic₇₁	1	1	yes	characteristic
107	C₄	C₂²×C₄×Dic₇₁	1	14	yes	—
108	C₄	C₂³×C₇₁⋊C₈	1	32	yes	direct product factor
109	C₄	C₂³×C₄×D₇₁	1	2	yes	—
110	C₂	C₂²×C₄×C₇₁⋊C₈	1	28	yes	direct product factor
111	C₂	C₂³×C₄×Dic₇₁	1	1	yes	characteristic
112	C₂	C₂⁴×C₇₁⋊C₈	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂³×C₄²	D₇₁
commutator, hypocenter	C₇₁	C₂³×C₄×C₈
Frattini	C₂×C₄	C₂⁴×D₇₁
Fitting	C₂³×C₄²×C₇₁	C₂
socle, prime radical	C₂⁵×C₇₁	C₂×C₄
cosocle	C₂×C₄×C₇₁	C₂⁵
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₇₁⋊C₈	C₂	direct (N × H)
4. C₂³×C₇₁⋊C₈	C₄	direct (N × H)
6. C₂×C₄×C₇₁⋊C₈	C₂²	direct (N × H)
11. C₂²×C₇₁⋊C₈	C₂×C₄	direct (N × H)
13. C₄×C₇₁⋊C₈	C₂³	direct (N × H)
15. C₂³×C₄×C₇₁	C₈	semidirect
22. C₂×C₇₁⋊C₈	C₂²×C₄	direct (N × H)
28. C₂²×C₄×C₇₁	C₂×C₈	semidirect
35. C₂³×C₇₁	C₄×C₈	semidirect
40. C₂×C₄×C₇₁	C₂²×C₈	semidirect
45. C₇₁⋊C₈	C₂³×C₄	direct (N × H)
46. C₂²×C₇₁	C₂×C₄×C₈	semidirect
51. C₄×C₇₁	C₂³×C₈	semidirect
54. C₂×C₇₁	C₂²×C₄×C₈	semidirect
60. C₇₁	C₂³×C₄×C₈	semidirect
69. C₂³×C₄	C₇₁⋊C₈	direct (N × H)
83. C₂²×C₄	C₂×C₇₁⋊C₈	direct (N × H)
92. C₂³	C₄×C₇₁⋊C₈	direct (N × H)
97. C₂×C₄	C₂²×C₇₁⋊C₈	direct (N × H)
103. C₂²	C₂×C₄×C₇₁⋊C₈	direct (N × H)
108. C₄	C₂³×C₇₁⋊C₈	direct (N × H)
110. C₂	C₂²×C₄×C₇₁⋊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
C₂⁴×C₇₁⋊C₈	2
C₂²×C₄×C₇₁⋊C₈	2
C₂³×C₄²×C₇₁	2
C₂³×C₄×C₈	71

Subgroups — C23×C4×C71⋊C8 #32860832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×C₇₁⋊C₈ #32860832