Subgroups — C₇×#3748 #11759293

Order 1008.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₇×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₇×#208	—	3	4	no	maximal
4	C₂²×C₃²×C₇	C₄	1	1	yes	characteristic
5	C₃²×C₄×C₇	C₄	1	1	yes	characteristic
6	C₃²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₇	D₃	1	4	yes	—
8	C₇×C₃⋊C₈	—	3	8	no	—
9	#3748	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃²×C₇	—	2	1	no	—
11	C₂×C₃²×C₇	C₂×C₄	1	1	yes	characteristic
12	C₇×M₁₆	—	9	1	no	Hall
13	C₂²×C₃×C₇	Dic₃	1	4	yes	—
14	C₃×C₄×C₇	Dic₃	1	4	yes	—
15	C₃×C₄×C₇	C₂×D₃	1	4	yes	—
16	C₂×C₃²×C₄	C₂×C₇	1	1	yes	characteristic
17	C₃⋊(C₃⋊C₈)	C₂×C₇	1	2	yes	complemented
18	C₃²×C₇	M₁₆	1	1	yes	characteristic, complemented, Hall
19	C₂×C₄×C₇	C₃⋊D₃	1	1	yes	characteristic
20	C₇×C₈	—	9	2	no	—
21	#208	—	3	4	no	—
22	C₂×C₃×C₇	—	2	4	no	—
23	C₂×C₃×C₇	C₂×Dic₃	1	4	yes	—
24	C₂²×C₃²	C₄×C₇	1	1	yes	characteristic
25	C₃²×C₄	C₄×C₇	1	1	yes	characteristic
26	C₃²×C₄	C₂²×C₇	1	1	yes	characteristic
27	C₂²×C₇	C₃⋊Dic₃	1	1	yes	characteristic
28	C₄×C₇	C₃⋊Dic₃	1	1	yes	characteristic
29	C₄×C₇	C₂×C₃⋊D₃	1	1	yes	characteristic
30	C₂×C₃×C₄	D₃×C₇	1	4	yes	—
31	C₃⋊C₈	—	3	8	no	—
32	C₃×C₇	#208	1	4	yes	complemented
33	C₂×C₃²	—	2	1	no	—
34	C₂×C₃²	C₂×C₄×C₇	1	1	yes	characteristic
35	M₁₆	—	9	1	no	Sylow
36	C₂×C₇	—	2	1	no	—
37	C₂×C₇	C₂×C₃⋊Dic₃	1	1	yes	characteristic
38	C₂²×C₃	C₇×Dic₃	1	4	yes	—
39	C₃×C₄	C₇×Dic₃	1	4	yes	—
40	C₃×C₄	C₂×D₃×C₇	1	4	yes	—
41	C₃²	C₇×M₁₆	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄	C₇×C₃⋊D₃	1	1	yes	characteristic
43	C₈	—	9	2	no	—
44	C₇	#3748	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₃	—	2	4	no	—
46	C₂×C₃	C₂×C₇×Dic₃	1	4	yes	—
47	C₂²	C₇×C₃⋊Dic₃	1	1	yes	characteristic
48	C₄	C₇×C₃⋊Dic₃	1	1	yes	characteristic
49	C₄	C₂×C₇×C₃⋊D₃	1	1	yes	characteristic
50	C₃	C₇×#208	1	4	yes	complemented
51	C₂	—	2	1	no	—
52	C₂	C₂×C₇×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂×C₄×C₇
Frattini	C₄	C₂×C₇×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₇	C₂
socle	C₂×C₃²×C₇	C₂×C₄
prime radical	C₂²×C₃²×C₇	C₄
cosocle	C₃²×C₄	C₂²×C₇
hypercenter	C₂×C₄×C₇	C₃⋊D₃
hypocenter	C₃²	C₇×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇

Complements

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

N	H ≅ G/N	type
2. C₇×C₃⋊(C₃⋊C₈)	C₂	semidirect
9. #3748	C₇	direct (N × H)
17. C₃⋊(C₃⋊C₈)	C₂×C₇	semidirect
18. C₃²×C₇	M₁₆	semidirect
32. C₃×C₇	#208	semidirect
41. C₃²	C₇×M₁₆	semidirect
44. C₇	#3748	direct (N × H)
50. C₃	C₇×#208	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₇	2
C₇×C₃⋊(C₃⋊C₈)	2
C₇×#208	3
#3748	7

Subgroups — C7×#3748 #11759293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3748 #11759293