Subgroups — C₂×#11924213 #72228763

Order 2352.

380 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#3861	C₂	1	1	yes	maximal, characteristic
2	C₂×#10596874	C₂	1	2	yes	maximal
3	#11924213	C₂	1	4	yes	maximal, direct product factor
4	C₂×#87648	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂²×#3861	C₂²	1	1	yes	characteristic
6	C₄×#3861	C₂²	1	2	yes	—
7	#10596874	C₂²	1	4	yes	—
8	C₂×C₄×C₇²	C₂×C₃	1	1	yes	characteristic
9	C₂×C₇⋊Dic₇	C₂×C₃	1	2	yes	—
10	#87648	C₂×C₃	1	4	yes	complemented
11	C₂×ΓL(1,𝔽₈)⋊Q₈	—	7	8	no	maximal
12	C₂×#3861	Q₈	1	2	yes	complemented
13	C₂×#3861	C₂³	1	1	yes	characteristic
14	C₂²×C₇²	C₂²×C₃	1	1	yes	characteristic
15	C₄×C₇²	C₂²×C₃	1	2	yes	—
16	C₇⋊Dic₇	C₂²×C₃	1	4	yes	—
17	C₂×C₄×ΓL(1,𝔽₈)	—	7	8	no	—
18	C₂×ΓL(1,𝔽₈)⋊C₄	—	7	16	no	—
19	ΓL(1,𝔽₈)⋊Q₈	—	7	32	no	—
20	#3861	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
21	C₂×Dic₁₄	—	7	8	no	—
22	C₂×C₇²	C₃×Q₈	1	2	yes	complemented
23	C₂×C₇²	C₂³×C₃	1	1	yes	characteristic
24	C₂²×ΓL(1,𝔽₈)	—	7	8	no	—
25	C₄×ΓL(1,𝔽₈)	—	7	16	no	—
26	ΓL(1,𝔽₈)⋊C₄	—	7	32	no	—
27	C₂×C₄×C₇	AGL(1,𝔽₇)	1	8	yes	—
28	C₂×Dic₇	—	7	16	no	—
29	Dic₁₄	—	7	32	no	—
30	C₇²	C₂×C₃×Q₈	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃×Q₈	—	49	1	no	Hall
32	C₂×ΓL(1,𝔽₈)	—	7	16	no	—
33	C₂×ΓL(1,𝔽₈)	—	7	8	no	—
34	C₂²×C₇	C₂×AGL(1,𝔽₇)	1	8	yes	—
35	C₄×C₇	C₂×AGL(1,𝔽₇)	1	16	yes	—
36	Dic₇	—	7	32	no	—
37	C₃×Q₈	—	49	4	no	—
38	C₂×C₃×C₄	—	49	2	no	—
39	C₂×C₃×C₄	—	49	1	no	—
40	ΓL(1,𝔽₈)	—	7	8	no	—
41	C₂×Q₈	—	49	1	no	Sylow
42	C₂×C₇	ΓL(1,𝔽₈)⋊Q₈	1	16	yes	complemented
43	C₂×C₇	C₂²×AGL(1,𝔽₇)	1	8	yes	—
44	C₂²×C₃	—	49	1	no	—
45	C₃×C₄	—	49	4	no	—
46	C₃×C₄	—	49	2	no	—
47	Q₈	—	49	4	no	—
48	C₂×C₄	—	49	2	no	—
49	C₂×C₄	#64424	1	1	yes	characteristic
50	C₇	C₂×ΓL(1,𝔽₈)⋊Q₈	1	8	yes	complemented
51	C₂×C₃	—	49	1	no	—
52	C₂×C₃	—	49	2	no	—
53	C₂²	C₂×#64424	1	1	yes	characteristic
54	C₄	—	49	4	no	—
55	C₄	C₂×#64424	1	2	yes	—
56	C₃	—	49	1	no	Sylow
57	C₂	#11924213	1	2	yes	direct product factor
58	C₂	C₂²×#64424	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#64424
commutator, cosocle	C₂×C₇²	C₂³×C₃
Frattini	C₂	C₂²×#64424
Fitting	C₂×C₄×C₇²	C₂×C₃
socle	C₂²×C₇²	C₂²×C₃
prime radical	C₂²×#3861	C₂²
hypercenter	C₂×C₄	#64424
hypocenter	C₇²	C₂×C₃×Q₈
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. #11924213	C₂	direct (N × H)
4. C₂×#87648	C₃	semidirect
10. #87648	C₂×C₃	semidirect
12. C₂×#3861	Q₈	semidirect
20. #3861	C₂×Q₈	semidirect
22. C₂×C₇²	C₃×Q₈	semidirect
30. C₇²	C₂×C₃×Q₈	semidirect
42. C₂×C₇	ΓL(1,𝔽₈)⋊Q₈	semidirect
50. C₇	C₂×ΓL(1,𝔽₈)⋊Q₈	semidirect
57. C₂	#11924213	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×C₄×#3861	2
C₂×#10596874	2
#11924213	2
C₂×#87648	3
C₂×ΓL(1,𝔽₈)⋊Q₈	7

Subgroups — C2×#11924213 #72228763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11924213 #72228763