Subgroups — D₇×#41⋊C₃ #10621529

Order 672.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#41⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#66662	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×#41	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₃×C₄×D₇	—	4	1	no	maximal
6	C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
7	C₇×#41	C₂×C₃	1	1	yes	characteristic, complemented
8	#1161	C₂×C₃	1	1	yes	characteristic, complemented
9	Q₈×D₇	C₂×C₃	1	1	yes	characteristic
10	#1159	—	3	1	no	—
11	D₄×D₇	—	3	1	no	—
12	#1157	—	3	1	no	—
13	C₂×C₄×D₇	—	3	1	no	—
14	C₂×#41⋊C₃	—	7	1	no	maximal, Hall
15	C₂×C₃×D₇	—	4	1	no	—
16	C₃×C₄×C₇	—	4	1	no	—
17	C₃×Dic₇	—	4	1	no	—
18	C₂²×D₇	—	3	1	no	—
19	C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
20	C₇×D₄	—	3	1	no	—
21	C₂×C₄×C₇	—	3	1	no	—
22	C₂²⋊D₇	—	3	2	no	—
23	C₂×Dic₇	—	3	1	no	—
24	D₂₈	—	3	1	no	—
25	C₄×D₇	—	3	1	no	—
26	C₄×D₇	—	3	1	no	—
27	C₄×D₇	—	3	1	no	—
28	C₄×D₇	A₄	1	1	yes	characteristic
29	Dic₁₄	—	3	1	no	—
30	#41⋊C₃	—	7	1	no	—
31	#41⋊C₃	D₇	1	1	yes	characteristic, direct product factor
32	C₂×SL(2,𝔽₃)	—	7	1	no	—
33	C₂×C₃×C₇	—	4	1	no	—
34	C₃×D₇	—	4	2	no	—
35	C₂×#41	—	7	1	no	Sylow
36	C₂²×C₇	—	3	1	no	—
37	C₂×D₇	—	3	1	no	—
38	C₂×D₇	—	6	2	no	—
39	C₂×D₇	C₂×A₄	1	1	yes	characteristic
40	C₄×C₇	—	3	1	no	—
41	C₄×C₇	C₂×A₄	1	1	yes	characteristic
42	Dic₇	—	3	1	no	—
43	Dic₇	C₂×A₄	1	1	yes	characteristic
44	C₂×C₃×C₄	—	28	1	no	—
45	SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic, complemented
46	C₃×C₇	—	4	1	no	Hall
47	#41	—	7	1	no	—
48	#41	—	21	1	no	—
49	#41	—	21	1	no	—
50	#41	C₃×D₇	1	1	yes	characteristic, complemented
51	C₂×Q₈	—	7	1	no	—
52	C₂×D₄	—	21	1	no	—
53	C₂²×C₄	—	21	1	no	—
54	C₂×C₇	—	6	1	no	—
55	C₂×C₇	C₂²×A₄	1	1	yes	characteristic
56	D₇	—	6	1	no	—
57	D₇	#41⋊C₃	1	2	yes	direct product factor
58	C₂²×C₃	—	28	1	no	—
59	C₃×C₄	—	4	1	no	—
60	C₃×C₄	—	28	1	no	—
61	C₂³	—	21	1	no	—
62	Q₈	—	21	1	no	—
63	Q₈	C₂×C₃×D₇	1	1	yes	characteristic
64	D₄	—	3	1	no	—
65	D₄	—	21	1	no	—
66	D₄	—	21	2	no	—
67	C₂×C₄	—	3	1	no	—
68	C₂×C₄	—	7	1	no	—
69	C₂×C₄	—	21	1	no	—
70	C₂×C₄	—	21	1	no	—
71	C₂×C₄	—	21	1	no	—
72	C₂×C₄	—	21	1	no	—
73	C₇	C₂×#41⋊C₃	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	—	4	1	no	—
75	C₂×C₃	—	28	2	no	—
76	C₂²	—	3	1	no	—
77	C₂²	—	7	1	no	—
78	C₂²	—	21	1	no	—
79	C₂²	—	42	2	no	—
80	C₄	—	3	1	no	—
81	C₄	—	7	1	no	—
82	C₄	—	21	1	no	—
83	C₄	A₄×D₇	1	1	yes	characteristic
84	C₃	—	4	1	no	Sylow
85	C₂	—	6	1	no	—
86	C₂	—	7	2	no	—
87	C₂	—	42	1	no	—
88	C₂	C₂×A₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	A₄×D₇
commutator, cosocle, hypocenter	C₇×Q₈	C₂²×C₃
Frattini	C₂	C₂×A₄×D₇
Fitting	C₇×#41	C₂×C₃
socle	C₂×C₇	C₂²×A₄
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₇×#41⋊C₃	C₂	semidirect
2. D₇×SL(2,𝔽₃)	C₂	semidirect
3. #66662	C₂	semidirect
4. D₇×#41	C₃	semidirect
6. C₇×SL(2,𝔽₃)	C₂²	semidirect
7. C₇×#41	C₂×C₃	semidirect
8. #1161	C₂×C₃	semidirect
31. #41⋊C₃	D₇	direct (N × H)
45. SL(2,𝔽₃)	C₂×D₇	semidirect
50. #41	C₃×D₇	semidirect
57. D₇	#41⋊C₃	direct (N × H)
73. C₇	C₂×#41⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₇×#41⋊C₃	2
D₇×SL(2,𝔽₃)	2
#66662	2
D₇×#41	3
C₃×C₄×D₇	4
C₂×#41⋊C₃	7

Subgroups — D7×#41⋊C3 #10621529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#41⋊C₃ #10621529