Subgroups — C₂²×#10596874 #72243271

Order 2352.

540 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#3861	C₂	1	1	yes	maximal, characteristic
2	C₂×#10596874	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₇⋊Dic₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×#3861	C₄	1	4	yes	complemented
5	C₂²×#3861	C₂²	1	3	yes	—
6	#10596874	C₂²	1	4	yes	direct product factor
7	C₂³×C₇²	C₂×C₃	1	1	yes	characteristic
8	C₂×C₇⋊Dic₇	C₂×C₃	1	6	yes	complemented
9	C₂²×ΓL(1,𝔽₈)⋊C₄	—	7	2	no	maximal
10	C₂²×ΓL(1,𝔽₈)⋊C₄	—	7	6	no	maximal
11	C₂×#3861	C₂×C₄	1	6	yes	complemented
12	C₂×#3861	C₂³	1	1	yes	characteristic
13	C₂²×C₇²	C₃×C₄	1	4	yes	complemented
14	C₂²×C₇²	C₂²×C₃	1	3	yes	—
15	C₇⋊Dic₇	C₂²×C₃	1	4	yes	complemented
16	C₂³×ΓL(1,𝔽₈)	—	7	2	no	—
17	C₂³×ΓL(1,𝔽₈)	—	14	6	no	—
18	C₂×ΓL(1,𝔽₈)⋊C₄	—	7	12	no	—
19	C₂×ΓL(1,𝔽₈)⋊C₄	—	7	36	no	—
20	#3861	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
21	C₂²×Dic₇	—	7	2	no	—
22	C₂²×Dic₇	—	7	6	no	—
23	C₂×C₇²	C₂×C₃×C₄	1	6	yes	complemented
24	C₂×C₇²	C₂³×C₃	1	1	yes	characteristic
25	C₂²×ΓL(1,𝔽₈)	—	7	8	no	—
26	C₂²×ΓL(1,𝔽₈)	—	7	6	no	—
27	C₂²×ΓL(1,𝔽₈)	—	14	24	no	—
28	C₂²×ΓL(1,𝔽₈)	—	14	18	no	—
29	ΓL(1,𝔽₈)⋊C₄	—	7	8	no	—
30	ΓL(1,𝔽₈)⋊C₄	—	7	24	no	—
31	C₂³×C₇	AGL(1,𝔽₇)	1	2	yes	—
32	C₂³×C₇	AGL(1,𝔽₇)	1	6	yes	—
33	C₂×Dic₇	—	7	12	no	—
34	C₂×Dic₇	—	7	36	no	—
35	C₇²	C₂²×C₃×C₄	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₃×C₄	—	49	1	no	Hall
37	C₂×ΓL(1,𝔽₈)	—	7	12	no	—
38	C₂×ΓL(1,𝔽₈)	—	7	2	no	—
39	C₂×ΓL(1,𝔽₈)	—	14	36	no	—
40	C₂×ΓL(1,𝔽₈)	—	14	6	no	—
41	C₂²×C₇	ΓL(1,𝔽₈)⋊C₄	1	8	yes	complemented
42	C₂²×C₇	ΓL(1,𝔽₈)⋊C₄	1	24	yes	complemented
43	C₂²×C₇	C₂×AGL(1,𝔽₇)	1	6	yes	—
44	C₂²×C₇	C₂×AGL(1,𝔽₇)	1	18	yes	—
45	Dic₇	—	7	8	no	—
46	Dic₇	—	7	24	no	—
47	C₂³×C₃	—	98	1	no	—
48	C₂×C₃×C₄	—	49	6	no	—
49	ΓL(1,𝔽₈)	—	7	2	no	—
50	ΓL(1,𝔽₈)	—	14	6	no	—
51	C₂²×C₄	—	49	1	no	Sylow
52	C₂×C₇	C₂×ΓL(1,𝔽₈)⋊C₄	1	12	yes	complemented
53	C₂×C₇	C₂×ΓL(1,𝔽₈)⋊C₄	1	36	yes	complemented
54	C₂×C₇	C₂²×AGL(1,𝔽₇)	1	2	yes	—
55	C₂×C₇	C₂²×AGL(1,𝔽₇)	1	6	yes	—
56	C₂²×C₃	—	98	4	no	—
57	C₂²×C₃	—	98	3	no	—
58	C₃×C₄	—	49	4	no	—
59	C₂³	#64424	1	1	yes	characteristic
60	C₂×C₄	—	49	6	no	—
61	C₇	C₂²×ΓL(1,𝔽₈)⋊C₄	1	2	yes	complemented
62	C₇	C₂²×ΓL(1,𝔽₈)⋊C₄	1	6	yes	complemented
63	C₂×C₃	—	98	1	no	—
64	C₂×C₃	—	98	6	no	—
65	C₂²	#10596874	1	4	yes	direct product factor
66	C₂²	C₂×#64424	1	3	yes	—
67	C₄	—	49	4	no	—
68	C₃	—	98	1	no	Sylow
69	C₂	C₂×#10596874	1	6	yes	direct product factor
70	C₂	C₂²×#64424	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	#64424
commutator, hypocenter	C₇²	C₂²×C₃×C₄
Frattini	C₂	C₂²×#64424
Fitting, socle	C₂³×C₇²	C₂×C₃
prime radical	C₂³×#3861	C₂
cosocle	C₂×C₇²	C₂³×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×#10596874	C₂	direct (N × H)
3. C₂²×C₇⋊Dic₇	C₃	semidirect
4. C₂²×#3861	C₄	semidirect
6. #10596874	C₂²	direct (N × H)
8. C₂×C₇⋊Dic₇	C₂×C₃	semidirect
11. C₂×#3861	C₂×C₄	semidirect
13. C₂²×C₇²	C₃×C₄	semidirect
15. C₇⋊Dic₇	C₂²×C₃	semidirect
20. #3861	C₂²×C₄	semidirect
23. C₂×C₇²	C₂×C₃×C₄	semidirect
35. C₇²	C₂²×C₃×C₄	semidirect
41. C₂²×C₇	ΓL(1,𝔽₈)⋊C₄	semidirect
42. C₂²×C₇	ΓL(1,𝔽₈)⋊C₄	semidirect
52. C₂×C₇	C₂×ΓL(1,𝔽₈)⋊C₄	semidirect
53. C₂×C₇	C₂×ΓL(1,𝔽₈)⋊C₄	semidirect
61. C₇	C₂²×ΓL(1,𝔽₈)⋊C₄	semidirect
62. C₇	C₂²×ΓL(1,𝔽₈)⋊C₄	semidirect
65. C₂²	#10596874	direct (N × H)
69. C₂	C₂×#10596874	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#3861	2
C₂×#10596874	2
C₂²×C₇⋊Dic₇	3
C₂²×ΓL(1,𝔽₈)⋊C₄	7

Subgroups — C22×#10596874 #72243271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10596874 #72243271