Subgroups — C₂²×C₇⋊(C₄²⋊C₃) #13060768

Order 1344.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇⋊(C₄²⋊C₃)	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₄²×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₇⋊(C₄²⋊C₃)	C₂²	1	1	yes	characteristic, direct product factor
5	C₂³×C₄×C₇	—	3	1	no	—
6	C₂×C₄²×C₇	—	3	3	no	—
7	C₂×C₄²×C₇	C₂×C₃	1	3	yes	complemented
8	C₂²×C₄²⋊C₃	—	7	1	no	maximal, Hall
9	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	3	no	—
10	C₂⁴×C₇	A₄	1	1	yes	characteristic
11	C₂²×C₄×C₇	—	3	6	no	—
12	C₂²×C₄×C₇	—	3	2	no	—
13	C₂²×C₄×C₇	—	3	3	no	—
14	C₂²×C₄×C₇	—	3	3	no	—
15	C₄²×C₇	—	3	3	no	—
16	C₄²×C₇	—	3	2	no	—
17	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
18	C₂×C₄²⋊C₃	—	7	3	no	—
19	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
20	C₂²×ΓL(1,𝔽₈)	—	16	1	no	—
21	C₂²×C₄²	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
22	C₂³×C₇	—	3	3	no	—
23	C₂³×C₇	—	3	1	no	—
24	C₂³×C₇	C₂×A₄	1	3	yes	—
25	C₂×C₄×C₇	—	3	6	no	—
26	C₂×C₄×C₇	—	3	6	no	—
27	C₂×C₄×C₇	—	3	6	no	—
28	C₂×C₄×C₇	—	3	6	no	—
29	C₂×C₄×C₇	—	3	1	no	—
30	C₂×C₄×C₇	—	3	3	no	—
31	C₂²×A₄	—	28	1	no	—
32	C₄²⋊C₃	—	7	1	no	—
33	C₂×ΓL(1,𝔽₈)	—	16	3	no	—
34	C₂³×C₄	—	3	1	no	—
35	C₂×C₄²	—	3	3	no	—
36	C₂×C₄²	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
37	C₂²×C₇	—	3	3	no	—
38	C₂²×C₇	—	3	2	no	—
39	C₂²×C₇	—	3	3	no	—
40	C₂²×C₇	—	3	3	no	—
41	C₂²×C₇	C₄²⋊C₃	1	1	yes	characteristic, complemented
42	C₂²×C₇	C₂²×A₄	1	1	yes	characteristic
43	C₄×C₇	—	3	2	no	—
44	C₄×C₇	—	3	6	no	—
45	C₂×A₄	—	28	3	no	—
46	ΓL(1,𝔽₈)	—	16	1	no	Hall
47	C₂⁴	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
48	C₂²×C₄	—	3	6	no	—
49	C₂²×C₄	—	3	2	no	—
50	C₂²×C₄	—	3	3	no	—
51	C₂²×C₄	—	3	3	no	—
52	C₄²	—	3	3	no	—
53	C₄²	—	3	2	no	—
54	C₄²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
55	C₂×C₇	—	3	1	no	—
56	C₂×C₇	—	3	3	no	—
57	C₂×C₇	C₂×C₄²⋊C₃	1	3	yes	complemented
58	C₂²×C₃	—	112	1	no	—
59	A₄	—	28	1	no	—
60	C₂³	—	3	3	no	—
61	C₂³	—	3	1	no	—
62	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	3	yes	—
63	C₂×C₄	—	3	6	no	—
64	C₂×C₄	—	3	6	no	—
65	C₂×C₄	—	3	6	no	—
66	C₂×C₄	—	3	6	no	—
67	C₂×C₄	—	3	1	no	—
68	C₂×C₄	—	3	3	no	—
69	C₇	C₂²×C₄²⋊C₃	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	112	3	no	—
71	C₂²	—	3	3	no	—
72	C₂²	—	3	2	no	—
73	C₂²	—	3	3	no	—
74	C₂²	—	3	3	no	—
75	C₂²	C₇⋊(C₄²⋊C₃)	1	1	yes	characteristic, direct product factor
76	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
77	C₄	—	3	2	no	—
78	C₄	—	3	6	no	—
79	C₃	—	112	1	no	Sylow
80	C₂	—	3	1	no	—
81	C₂	—	3	3	no	—
82	C₂	C₂×C₇⋊(C₄²⋊C₃)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	C₇⋊(C₄²⋊C₃)
commutator, cosocle, hypocenter	C₄²×C₇	C₂²×C₃
Frattini	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₂²×C₄²×C₇	C₃
socle	C₂⁴×C₇	A₄
LCS	G ▷ C₄²×C₇
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×C₇⋊(C₄²⋊C₃)	C₂	direct (N × H)
2. C₂²×C₄²×C₇	C₃	semidirect
4. C₇⋊(C₄²⋊C₃)	C₂²	direct (N × H)
7. C₂×C₄²×C₇	C₂×C₃	semidirect
17. C₄²×C₇	C₂²×C₃	semidirect
21. C₂²×C₄²	ΓL(1,𝔽₈)	semidirect
36. C₂×C₄²	C₂×ΓL(1,𝔽₈)	semidirect
41. C₂²×C₇	C₄²⋊C₃	semidirect
54. C₄²	C₂²×ΓL(1,𝔽₈)	semidirect
57. C₂×C₇	C₂×C₄²⋊C₃	semidirect
69. C₇	C₂²×C₄²⋊C₃	semidirect
75. C₂²	C₇⋊(C₄²⋊C₃)	direct (N × H)
82. C₂	C₂×C₇⋊(C₄²⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇⋊(C₄²⋊C₃)	2
C₂²×C₄²×C₇	3
C₂²×C₂²⋊ΓL(1,𝔽₈)	4
C₂²×C₄²⋊C₃	7

Subgroups — C22×C7⋊(C42⋊C3) #13060768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇⋊(C₄²⋊C₃) #13060768