Subgroups — C₄×C₇⋊(C₄²⋊C₃) #13055059

Order 1344.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇⋊(C₄²⋊C₃)	C₂	1	1	yes	maximal, characteristic
2	C₄³×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₇⋊(C₄²⋊C₃)	C₄	1	1	yes	characteristic, direct product factor
5	C₂×C₄²×C₇	—	3	1	no	—
6	C₂×C₄²×C₇	—	3	1	no	—
7	C₂×C₄²×C₇	C₂×C₃	1	1	yes	characteristic
8	C₄×C₄²⋊C₃	—	7	1	no	maximal, Hall
9	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
10	C₂²×C₄×C₇	—	3	1	no	—
11	C₂²×C₄×C₇	—	3	1	no	—
12	C₂²×C₄×C₇	A₄	1	1	yes	characteristic
13	C₄²×C₇	—	3	1	no	—
14	C₄²×C₇	—	3	4	no	—
15	C₄²×C₇	—	3	2	no	—
16	C₄²×C₇	—	3	2	no	—
17	C₄²×C₇	C₃×C₄	1	1	yes	characteristic, complemented
18	C₂×C₄²⋊C₃	—	7	1	no	—
19	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
20	C₄×ΓL(1,𝔽₈)	—	16	1	no	—
21	C₄³	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
22	C₂³×C₇	C₂×A₄	1	1	yes	characteristic
23	C₂×C₄×C₇	—	3	4	no	—
24	C₂×C₄×C₇	—	3	2	no	—
25	C₂×C₄×C₇	—	3	2	no	—
26	C₂×C₄×C₇	—	3	2	no	—
27	C₂×C₄×C₇	—	3	1	no	—
28	C₂×C₄×C₇	—	3	1	no	—
29	C₂×C₄×C₇	—	3	1	no	—
30	C₂×C₄×C₇	—	3	1	no	—
31	C₄×A₄	—	28	1	no	—
32	C₄²⋊C₃	—	7	1	no	—
33	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
34	C₂×C₄²	—	3	1	no	—
35	C₂×C₄²	—	3	1	no	—
36	C₂×C₄²	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	C₂²×C₇	—	3	1	no	—
38	C₂²×C₇	—	3	1	no	—
39	C₂²×C₇	C₄×A₄	1	1	yes	characteristic
40	C₄×C₇	—	3	4	no	—
41	C₄×C₇	—	3	2	no	—
42	C₄×C₇	—	3	2	no	—
43	C₄×C₇	—	3	1	no	—
44	C₄×C₇	C₄²⋊C₃	1	1	yes	characteristic, complemented
45	C₂×A₄	—	28	1	no	—
46	ΓL(1,𝔽₈)	—	16	1	no	Hall
47	C₂²×C₄	—	3	1	no	—
48	C₂²×C₄	—	3	1	no	—
49	C₂²×C₄	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
50	C₄²	—	3	1	no	—
51	C₄²	—	3	4	no	—
52	C₄²	—	3	2	no	—
53	C₄²	—	3	2	no	—
54	C₄²	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
55	C₂×C₇	—	3	1	no	—
56	C₂×C₇	—	3	1	no	—
57	C₂×C₇	C₂×C₄²⋊C₃	1	1	yes	characteristic
58	A₄	—	28	1	no	—
59	C₃×C₄	—	112	1	no	—
60	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
61	C₂×C₄	—	3	2	no	—
62	C₂×C₄	—	3	4	no	—
63	C₂×C₄	—	3	2	no	—
64	C₂×C₄	—	3	2	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₄	—	3	1	no	—
67	C₂×C₄	—	3	1	no	—
68	C₂×C₄	—	3	1	no	—
69	C₇	C₄×C₄²⋊C₃	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	112	1	no	—
71	C₂²	—	3	1	no	—
72	C₂²	—	3	1	no	—
73	C₂²	C₄×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
74	C₄	—	3	2	no	—
75	C₄	—	3	2	no	—
76	C₄	—	3	4	no	—
77	C₄	—	3	1	no	—
78	C₄	C₇⋊(C₄²⋊C₃)	1	1	yes	characteristic, direct product factor
79	C₃	—	112	1	no	Sylow
80	C₂	—	3	1	no	—
81	C₂	—	3	1	no	—
82	C₂	C₂×C₇⋊(C₄²⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	C₇⋊(C₄²⋊C₃)
commutator, hypocenter	C₄²×C₇	C₃×C₄
Frattini	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₄³×C₇	C₃
socle	C₂³×C₇	C₂×A₄
prime radical	C₂×C₇⋊(C₄²⋊C₃)	C₂
cosocle	C₂×C₄²×C₇	C₂×C₃
LCS	G ▷ C₄²×C₇
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₄³×C₇	C₃	semidirect
4. C₇⋊(C₄²⋊C₃)	C₄	direct (N × H)
17. C₄²×C₇	C₃×C₄	semidirect
21. C₄³	ΓL(1,𝔽₈)	semidirect
44. C₄×C₇	C₄²⋊C₃	semidirect
54. C₄²	C₄×ΓL(1,𝔽₈)	semidirect
69. C₇	C₄×C₄²⋊C₃	semidirect
78. C₄	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
C₄³×C₇	3
C₄×C₂²⋊ΓL(1,𝔽₈)	4
C₄×C₄²⋊C₃	7

Subgroups — C4×C7⋊(C42⋊C3) #13055059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇⋊(C₄²⋊C₃) #13055059