Subgroups — C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄ #10620759

Order 672.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂	1	4	yes	maximal, direct product factor
5	C₂×Dic₇⋊C₄	C₃	1	1	yes	maximal, characteristic, complemented, Hall
6	C₂³×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
8	C₂×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
9	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
10	C₂×ΓL(1,𝔽₈)⋊C₄	C₄	1	4	yes	complemented
11	C₂×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
12	C₂×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
13	C₂²×C₄×C₇	C₂×C₃	1	1	yes	characteristic
14	C₂²×Dic₇	C₂×C₃	1	1	yes	characteristic
15	C₂²×Dic₇	C₂×C₃	1	1	yes	characteristic
16	Dic₇⋊C₄	C₂×C₃	1	4	yes	complemented
17	C₂×C₃×C₄⋊C₄	—	7	1	no	maximal, Hall
18	C₂²×ΓL(1,𝔽₈)	C₂×C₄	1	2	yes	—
19	C₂²×ΓL(1,𝔽₈)	D₄	1	2	yes	—
20	C₂²×ΓL(1,𝔽₈)	Q₈	1	2	yes	—
21	C₂²×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
22	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
23	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
24	ΓL(1,𝔽₈)⋊C₄	C₂×C₄	1	4	yes	complemented
25	C₂³×C₇	C₂²×C₃	1	1	yes	characteristic
26	C₂×C₄×C₇	—	2	2	no	—
27	C₂×C₄×C₇	C₂²×C₃	1	2	yes	—
28	C₂×Dic₇	—	2	2	no	—
29	C₂×Dic₇	C₃×C₄	1	4	yes	complemented
30	C₂×Dic₇	C₂²×C₃	1	2	yes	—
31	C₂×Dic₇	C₂²×C₃	1	2	yes	—
32	C₂²×C₃×C₄	—	7	1	no	—
33	C₂²×C₃×C₄	—	7	1	no	—
34	C₂²×C₃×C₄	—	7	1	no	—
35	C₃×C₄⋊C₄	—	7	4	no	—
36	C₂×ΓL(1,𝔽₈)	C₄⋊C₄	1	4	yes	complemented
37	C₂×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
38	C₂×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
39	C₂×ΓL(1,𝔽₈)	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄⋊C₄	—	7	1	no	Sylow
41	C₂²×C₇	C₂×C₃×C₄	1	2	yes	—
42	C₂²×C₇	C₃×D₄	1	2	yes	—
43	C₂²×C₇	C₃×Q₈	1	2	yes	—
44	C₂²×C₇	C₂³×C₃	1	1	yes	characteristic
45	C₄×C₇	—	2	2	no	—
46	Dic₇	—	2	2	no	—
47	Dic₇	C₂×C₃×C₄	1	4	yes	complemented
48	C₂³×C₃	—	7	1	no	—
49	C₂×C₃×C₄	—	7	4	no	—
50	C₂×C₃×C₄	—	7	2	no	—
51	C₂×C₃×C₄	—	7	2	no	—
52	C₂×C₃×C₄	—	7	2	no	—
53	C₂×C₃×C₄	—	14	2	no	—
54	C₂×C₃×C₄	—	14	2	no	—
55	ΓL(1,𝔽₈)	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
56	C₂²×C₄	—	7	1	no	—
57	C₂²×C₄	—	7	1	no	—
58	C₂²×C₄	AGL(1,𝔽₇)	1	1	yes	characteristic
59	C₄⋊C₄	—	7	4	no	—
60	C₂×C₇	C₃×C₄⋊C₄	1	4	yes	complemented
61	C₂×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
62	C₂×C₇	C₂×C₃×D₄	1	1	yes	characteristic
63	C₂×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
64	C₂²×C₃	—	7	2	no	—
65	C₂²×C₃	—	7	2	no	—
66	C₂²×C₃	—	7	2	no	—
67	C₂²×C₃	—	7	1	no	—
68	C₃×C₄	—	7	4	no	—
69	C₃×C₄	—	14	2	no	—
70	C₃×C₄	—	14	2	no	—
71	C₂³	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	7	4	no	—
74	C₂×C₄	—	7	2	no	—
75	C₂×C₄	—	7	2	no	—
76	C₂×C₄	—	14	2	no	—
77	C₂×C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
78	C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	7	1	no	—
80	C₂×C₃	—	7	1	no	—
81	C₂×C₃	—	7	1	no	—
82	C₂×C₃	—	7	4	no	—
83	C₂²	ΓL(1,𝔽₈)⋊Q₈	1	2	yes	—
84	C₂²	C₄×AGL(1,𝔽₇)	1	2	yes	—
85	C₂²	(C₂²⋊D₇)⋊C₃	1	2	yes	—
86	C₂²	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	7	4	no	—
89	C₄	—	14	2	no	—
90	C₃	—	7	1	no	Sylow
91	C₂	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	1	4	yes	direct product factor
92	C₂	C₂×ΓL(1,𝔽₈)⋊Q₈	1	1	yes	characteristic
93	C₂	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
94	C₂	C₂×(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂²×C₃×C₄
Frattini	C₂²	C₂²×AGL(1,𝔽₇)
Fitting	C₂²×C₄×C₇	C₂×C₃
socle	C₂³×C₇	C₂²×C₃
prime radical	C₂³×ΓL(1,𝔽₈)	C₂²
cosocle	C₂²×C₇	C₂³×C₃
hypercenter	C₂²×C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₂×C₃×C₄⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂	direct (N × H)
5. C₂×Dic₇⋊C₄	C₃	semidirect
10. C₂×ΓL(1,𝔽₈)⋊C₄	C₄	semidirect
16. Dic₇⋊C₄	C₂×C₃	semidirect
24. ΓL(1,𝔽₈)⋊C₄	C₂×C₄	semidirect
29. C₂×Dic₇	C₃×C₄	semidirect
36. C₂×ΓL(1,𝔽₈)	C₄⋊C₄	semidirect
47. Dic₇	C₂×C₃×C₄	semidirect
55. ΓL(1,𝔽₈)	C₂×C₄⋊C₄	semidirect
60. C₂×C₇	C₃×C₄⋊C₄	semidirect
78. C₇	C₂×C₃×C₄⋊C₄	semidirect
91. C₂	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×ΓL(1,𝔽₈)	2
C₂²×ΓL(1,𝔽₈)⋊C₄	2
(ΓL(1,𝔽₈)⋊C₄)⋊C₄	2
C₂×Dic₇⋊C₄	3
C₂×C₃×C₄⋊C₄	7

Subgroups — C2×(ΓL(1,𝔽8)⋊C4)⋊C4 #10620759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄ #10620759