Subgroups — D₇×#41 #6552

Order 224.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	#1161	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1159	C₂	1	3	yes	maximal, complemented
5	D₄×D₇	C₂	1	3	yes	maximal, complemented
6	#1157	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×D₇	C₂	1	3	yes	maximal, complemented
8	C₂²×D₇	C₂²	1	3	yes	—
9	C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
10	C₇×D₄	C₂²	1	3	yes	complemented
11	C₂×C₄×C₇	C₂²	1	3	yes	complemented
12	C₂²⋊D₇	C₂²	1	6	yes	complemented
13	C₂×Dic₇	C₂²	1	3	yes	complemented
14	D₂₈	C₂²	1	3	yes	complemented
15	C₄×D₇	C₂²	1	1	yes	characteristic
16	C₄×D₇	C₂²	1	3	yes	—
17	C₄×D₇	C₂²	1	3	yes	complemented
18	C₄×D₇	C₂²	1	3	yes	complemented
19	Dic₁₄	C₂²	1	3	yes	complemented
20	C₂×#41	—	7	1	no	maximal, Sylow
21	C₂²×C₇	C₂³	1	3	yes	—
22	C₂×D₇	—	2	6	no	—
23	C₂×D₇	C₂³	1	3	yes	—
24	C₂×D₇	C₂³	1	1	yes	characteristic
25	C₄×C₇	C₂³	1	3	yes	—
26	C₄×C₇	C₂³	1	1	yes	characteristic
27	Dic₇	C₂³	1	3	yes	—
28	Dic₇	C₂³	1	1	yes	characteristic
29	#41	—	7	1	no	—
30	#41	—	7	3	no	—
31	#41	—	7	3	no	—
32	#41	D₇	1	1	yes	characteristic, direct product factor
33	C₂×Q₈	—	7	1	no	—
34	C₂×D₄	—	7	3	no	—
35	C₂²×C₄	—	7	3	no	—
36	C₂×C₇	—	2	3	no	—
37	C₂×C₇	C₂⁴	1	1	yes	characteristic
38	D₇	—	2	3	no	—
39	D₇	#41	1	2	yes	direct product factor
40	C₂³	—	7	3	no	—
41	Q₈	—	7	3	no	—
42	Q₈	C₂×D₇	1	1	yes	characteristic, complemented
43	D₄	—	7	6	no	—
44	D₄	—	7	3	no	—
45	D₄	C₂×D₇	1	3	yes	complemented
46	C₂×C₄	—	7	3	no	—
47	C₂×C₄	—	7	3	no	—
48	C₂×C₄	—	7	3	no	—
49	C₂×C₄	—	7	1	no	—
50	C₂×C₄	—	7	3	no	—
51	C₂×C₄	C₂×D₇	1	3	yes	complemented
52	C₇	C₂×#41	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	7	1	no	—
54	C₂²	—	7	3	no	—
55	C₂²	—	14	6	no	—
56	C₂²	C₂²×D₇	1	3	yes	—
57	C₄	—	7	1	no	—
58	C₄	—	7	3	no	—
59	C₄	C₂²×D₇	1	3	yes	—
60	C₄	C₂²×D₇	1	1	yes	characteristic
61	C₂	—	2	3	no	—
62	C₂	—	7	2	no	—
63	C₂	—	14	3	no	—
64	C₂	C₂³×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₇
commutator, socle, cosocle	C₂×C₇	C₂⁴
Frattini	C₂	C₂³×D₇
Fitting	C₇×#41	C₂
hypercenter	#41	D₇
hypocenter	C₇	C₂×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₇×#41	C₂	semidirect
2. #1161	C₂	semidirect
3. Q₈×D₇	C₂	semidirect
4. #1159	C₂	semidirect
5. D₄×D₇	C₂	semidirect
6. #1157	C₂	semidirect
7. C₂×C₄×D₇	C₂	semidirect
9. C₇×Q₈	C₂²	semidirect
10. C₇×D₄	C₂²	semidirect
11. C₂×C₄×C₇	C₂²	semidirect
12. C₂²⋊D₇	C₂²	semidirect
13. C₂×Dic₇	C₂²	semidirect
14. D₂₈	C₂²	semidirect
17. C₄×D₇	C₂²	semidirect
18. C₄×D₇	C₂²	semidirect
19. Dic₁₄	C₂²	semidirect
32. #41	D₇	direct (N × H)
39. D₇	#41	direct (N × H)
42. Q₈	C₂×D₇	semidirect
45. D₄	C₂×D₇	semidirect
51. C₂×C₄	C₂×D₇	semidirect
52. C₇	C₂×#41	semidirect

Maximal subgroups

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

subgroup	index
C₇×#41	2
#1161	2
Q₈×D₇	2
#1159	2
D₄×D₇	2
#1157	2
C₂×C₄×D₇	2
C₂×#41	7

Subgroups — D7×#41 #6552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#41 #6552