Subgroups — C₂²×#4248 #10621592

Order 672.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#4248	C₂	1	12	yes	maximal, direct product factor
5	C₂²×Dic₁₄	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₂₁	C₂²	1	6	yes	—
8	C₂×C₇×Dic₃	C₂²	1	6	yes	—
9	C₂×C₃×Dic₇	C₂²	1	6	yes	—
10	#4248	C₂²	1	16	yes	direct product factor
11	C₂²×C₄×C₇	—	3	1	no	—
12	C₂²×Dic₇	—	3	1	no	—
13	C₂²×Dic₇	D₃	1	1	yes	characteristic
14	C₂×Dic₁₄	—	3	12	no	—
15	C₂²×Dic₆	—	7	1	no	maximal, Hall
16	C₂²×C₃×C₇	Q₈	1	4	yes	complemented
17	C₂²×C₃×C₇	C₂³	1	3	yes	—
18	Dic₂₁	C₂³	1	4	yes	—
19	C₃×Dic₇	C₂³	1	4	yes	—
20	C₇×Dic₃	C₂³	1	4	yes	—
21	C₂³×C₇	C₂×D₃	1	1	yes	characteristic
22	C₂×C₄×C₇	—	3	6	no	—
23	C₂×Dic₇	—	3	6	no	—
24	C₂×Dic₇	C₂×D₃	1	6	yes	—
25	Dic₁₄	—	3	16	no	—
26	C₂²×C₃×C₄	—	7	1	no	—
27	C₂²×Dic₃	—	7	1	no	—
28	C₂²×Dic₃	D₇	1	1	yes	characteristic
29	C₂×Dic₆	—	7	12	no	—
30	C₂×C₃×C₇	C₂×Q₈	1	6	yes	complemented
31	C₂×C₃×C₇	C₂⁴	1	1	yes	characteristic
32	C₂²×Q₈	—	21	1	no	Sylow
33	C₂²×C₇	Dic₆	1	4	yes	complemented
34	C₂²×C₇	C₂²×D₃	1	3	yes	—
35	C₄×C₇	—	3	4	no	—
36	Dic₇	—	3	4	no	—
37	Dic₇	C₂²×D₃	1	4	yes	—
38	C₂³×C₃	C₂×D₇	1	1	yes	characteristic
39	C₂×C₃×C₄	—	7	6	no	—
40	C₂×Dic₃	—	7	6	no	—
41	C₂×Dic₃	C₂×D₇	1	6	yes	—
42	Dic₆	—	7	16	no	—
43	C₃×C₇	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
44	C₂×Q₈	—	21	12	no	—
45	C₂²×C₄	—	3	1	no	—
46	C₂²×C₄	—	7	1	no	—
47	C₂²×C₄	—	21	1	no	—
48	C₂×C₇	C₂×Dic₆	1	6	yes	complemented
49	C₂×C₇	C₂³×D₃	1	1	yes	characteristic
50	C₂²×C₃	Dic₁₄	1	4	yes	complemented
51	C₂²×C₃	C₂²×D₇	1	3	yes	—
52	C₃×C₄	—	7	4	no	—
53	Dic₃	—	7	4	no	—
54	Dic₃	C₂²×D₇	1	4	yes	—
55	C₂³	D₃×D₇	1	1	yes	characteristic
56	Q₈	—	21	16	no	—
57	C₂×C₄	—	3	6	no	—
58	C₂×C₄	—	7	6	no	—
59	C₂×C₄	—	21	6	no	—
60	C₇	C₂²×Dic₆	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₃	C₂×Dic₁₄	1	6	yes	complemented
62	C₂×C₃	C₂³×D₇	1	1	yes	characteristic
63	C₂²	#4248	1	4	yes	direct product factor
64	C₂²	C₂×D₃×D₇	1	3	yes	—
65	C₄	—	3	4	no	—
66	C₄	—	7	4	no	—
67	C₄	—	21	4	no	—
68	C₃	C₂²×Dic₁₄	1	1	yes	characteristic, complemented, Sylow
69	C₂	C₂×#4248	1	6	yes	direct product factor
70	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, prime radical	C₂³×C₃×C₇	C₂²
hypocenter	C₃×C₇	C₂²×Q₈
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
4. C₂×#4248	C₂	direct (N × H)
10. #4248	C₂²	direct (N × H)
16. C₂²×C₃×C₇	Q₈	semidirect
30. C₂×C₃×C₇	C₂×Q₈	semidirect
33. C₂²×C₇	Dic₆	semidirect
43. C₃×C₇	C₂²×Q₈	semidirect
48. C₂×C₇	C₂×Dic₆	semidirect
50. C₂²×C₃	Dic₁₄	semidirect
60. C₇	C₂²×Dic₆	semidirect
61. C₂×C₃	C₂×Dic₁₄	semidirect
63. C₂²	#4248	direct (N × H)
68. C₃	C₂²×Dic₁₄	semidirect
69. C₂	C₂×#4248	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₂₁	2
C₂²×C₇×Dic₃	2
C₂²×C₃×Dic₇	2
C₂×#4248	2
C₂²×Dic₁₄	3
C₂²×Dic₆	7

Subgroups — C22×#4248 #10621592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#4248 #10621592