Subgroups — C₂²×#4246 #10621590

Order 672.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#4246	C₂	1	12	yes	maximal, direct product factor
5	C₂²×D₂₈	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂²×D₂₁	—	2	4	no	—
8	C₂²×D₂₁	C₂²	1	6	yes	complemented
9	C₂²×C₃×D₇	—	2	4	no	—
10	C₂²×C₃×D₇	C₂²	1	6	yes	complemented
11	C₂×C₇×Dic₃	C₂²	1	6	yes	complemented
12	#4246	C₂²	1	16	yes	direct product factor
13	C₂³×D₇	—	3	1	no	—
14	C₂³×D₇	D₃	1	1	yes	characteristic, complemented
15	C₂²×C₄×C₇	—	3	1	no	—
16	C₂×D₂₈	—	3	12	no	—
17	C₂²×#67	—	7	1	no	maximal, Hall
18	C₂²×C₃×C₇	D₄	1	4	yes	complemented
19	C₂²×C₃×C₇	C₂³	1	3	yes	—
20	C₂×D₂₁	—	2	12	no	—
21	C₂×D₂₁	C₂³	1	4	yes	complemented
22	C₂×C₃×D₇	—	2	12	no	—
23	C₂×C₃×D₇	C₂³	1	4	yes	complemented
24	C₇×Dic₃	C₂³	1	4	yes	complemented
25	C₂³×C₇	C₂×D₃	1	1	yes	characteristic
26	C₂²×D₇	—	2	4	no	—
27	C₂²×D₇	—	3	6	no	—
28	C₂²×D₇	—	6	4	no	—
29	C₂²×D₇	C₂×D₃	1	6	yes	complemented
30	C₂×C₄×C₇	—	3	6	no	—
31	D₂₈	—	3	16	no	—
32	C₂⁴×C₃	—	7	1	no	—
33	C₂³×D₃	—	7	1	no	—
34	C₂×#67	—	7	12	no	—
35	C₂²×Dic₃	D₇	1	1	yes	characteristic, complemented
36	C₂×C₃×C₇	C₂×D₄	1	6	yes	complemented
37	C₂×C₃×C₇	C₂⁴	1	1	yes	characteristic
38	D₂₁	—	2	4	no	—
39	C₃×D₇	—	2	4	no	—
40	C₂²×D₄	—	21	1	no	Sylow
41	C₂²×C₇	#67	1	4	yes	complemented
42	C₂²×C₇	C₂²×D₃	1	3	yes	—
43	C₂×D₇	—	2	12	no	—
44	C₂×D₇	—	3	4	no	—
45	C₂×D₇	—	6	12	no	—
46	C₂×D₇	C₂²×D₃	1	4	yes	complemented
47	C₄×C₇	—	3	4	no	—
48	C₂³×C₃	—	7	6	no	—
49	C₂³×C₃	—	14	4	no	—
50	C₂³×C₃	C₂×D₇	1	1	yes	characteristic
51	C₂²×D₃	—	7	6	no	—
52	C₂²×D₃	—	14	4	no	—
53	#67	—	7	16	no	—
54	C₂×Dic₃	C₂×D₇	1	6	yes	complemented
55	C₃×C₇	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
56	C₂⁴	—	7	1	no	—
57	C₂⁴	—	21	1	no	—
58	C₂×D₄	—	21	12	no	—
59	C₂²×C₄	—	3	1	no	—
60	C₂×C₇	C₂×#67	1	6	yes	complemented
61	C₂×C₇	C₂³×D₃	1	1	yes	characteristic
62	D₇	—	2	4	no	—
63	D₇	—	6	4	no	—
64	C₂²×C₃	—	7	4	no	—
65	C₂²×C₃	—	14	12	no	—
66	C₂²×C₃	D₂₈	1	4	yes	complemented
67	C₂²×C₃	C₂²×D₇	1	3	yes	—
68	C₂×D₃	—	7	4	no	—
69	C₂×D₃	—	14	12	no	—
70	Dic₃	C₂²×D₇	1	4	yes	complemented
71	C₂³	—	7	6	no	—
72	C₂³	—	14	4	no	—
73	C₂³	—	21	6	no	—
74	C₂³	—	42	4	no	—
75	C₂³	D₃×D₇	1	1	yes	characteristic
76	D₄	—	21	16	no	—
77	C₂×C₄	—	3	6	no	—
78	C₇	C₂²×#67	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	14	4	no	—
80	C₂×C₃	C₂×D₂₈	1	6	yes	complemented
81	C₂×C₃	C₂³×D₇	1	1	yes	characteristic
82	D₃	—	14	4	no	—
83	C₂²	—	7	4	no	—
84	C₂²	—	14	12	no	—
85	C₂²	—	21	4	no	—
86	C₂²	—	42	12	no	—
87	C₂²	#4246	1	4	yes	direct product factor
88	C₂²	C₂×D₃×D₇	1	3	yes	—
89	C₄	—	3	4	no	—
90	C₃	C₂²×D₂₈	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	14	4	no	—
92	C₂	—	42	4	no	—
93	C₂	C₂×#4246	1	6	yes	direct product factor
94	C₂	C₂²×D₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₃×D₇
commutator, cosocle	C₂×C₃×C₇	C₂⁴
Frattini	C₂	C₂²×D₃×D₇
Fitting, socle	C₂³×C₃×C₇	C₂²
hypocenter	C₃×C₇	C₂²×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂³×D₂₁	C₂	semidirect
2. C₂³×C₃×D₇	C₂	semidirect
3. C₂²×C₇×Dic₃	C₂	semidirect
4. C₂×#4246	C₂	direct (N × H)
8. C₂²×D₂₁	C₂²	semidirect
10. C₂²×C₃×D₇	C₂²	semidirect
11. C₂×C₇×Dic₃	C₂²	semidirect
12. #4246	C₂²	direct (N × H)
14. C₂³×D₇	D₃	semidirect
18. C₂²×C₃×C₇	D₄	semidirect
21. C₂×D₂₁	C₂³	semidirect
23. C₂×C₃×D₇	C₂³	semidirect
24. C₇×Dic₃	C₂³	semidirect
29. C₂²×D₇	C₂×D₃	semidirect
35. C₂²×Dic₃	D₇	semidirect
36. C₂×C₃×C₇	C₂×D₄	semidirect
41. C₂²×C₇	#67	semidirect
46. C₂×D₇	C₂²×D₃	semidirect
54. C₂×Dic₃	C₂×D₇	semidirect
55. C₃×C₇	C₂²×D₄	semidirect
60. C₂×C₇	C₂×#67	semidirect
66. C₂²×C₃	D₂₈	semidirect
70. Dic₃	C₂²×D₇	semidirect
78. C₇	C₂²×#67	semidirect
80. C₂×C₃	C₂×D₂₈	semidirect
87. C₂²	#4246	direct (N × H)
90. C₃	C₂²×D₂₈	semidirect
93. C₂	C₂×#4246	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×D₂₁	2
C₂³×C₃×D₇	2
C₂²×C₇×Dic₃	2
C₂×#4246	2
C₂²×D₂₈	3
C₂²×#67	7

Subgroups — C22×#4246 #10621590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#4246 #10621590