Subgroups — C₂²×#4247 #10621591

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×D₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#4247	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₂²⋊D₇	—	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×D₃×C₇	—	2	4	no	—
10	C₂²×D₃×C₇	C₂²	1	6	yes	complemented
11	C₂×C₃×Dic₇	C₂²	1	6	yes	complemented
12	#4247	C₂²	1	16	yes	direct product factor
13	C₂⁴×C₇	—	3	1	no	—
14	C₂³×D₇	—	3	1	no	—
15	C₂×C₂²⋊D₇	—	3	12	no	—
16	C₂²×Dic₇	D₃	1	1	yes	characteristic, complemented
17	C₂²×D₁₂	—	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×D₃×C₇	—	2	12	no	—
23	C₂×D₃×C₇	C₂³	1	4	yes	complemented
24	C₃×Dic₇	C₂³	1	4	yes	complemented
25	C₂³×C₇	—	3	6	no	—
26	C₂³×C₇	—	6	4	no	—
27	C₂³×C₇	C₂×D₃	1	1	yes	characteristic
28	C₂²×D₇	—	3	6	no	—
29	C₂²×D₇	—	6	4	no	—
30	C₂²⋊D₇	—	3	16	no	—
31	C₂×Dic₇	C₂×D₃	1	6	yes	complemented
32	C₂³×D₃	—	7	1	no	—
33	C₂³×D₃	D₇	1	1	yes	characteristic, complemented
34	C₂²×C₃×C₄	—	7	1	no	—
35	C₂×D₁₂	—	7	12	no	—
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	D₃×C₇	—	2	4	no	—
40	C₂²×D₄	—	21	1	no	Sylow
41	C₂²×C₇	—	3	4	no	—
42	C₂²×C₇	—	6	12	no	—
43	C₂²×C₇	D₁₂	1	4	yes	complemented
44	C₂²×C₇	C₂²×D₃	1	3	yes	—
45	C₂×D₇	—	3	4	no	—
46	C₂×D₇	—	6	12	no	—
47	Dic₇	C₂²×D₃	1	4	yes	complemented
48	C₂³×C₃	C₂×D₇	1	1	yes	characteristic
49	C₂²×D₃	—	2	4	no	—
50	C₂²×D₃	—	7	6	no	—
51	C₂²×D₃	—	14	4	no	—
52	C₂²×D₃	C₂×D₇	1	6	yes	complemented
53	C₂×C₃×C₄	—	7	6	no	—
54	D₁₂	—	7	16	no	—
55	C₃×C₇	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
56	C₂⁴	—	3	1	no	—
57	C₂⁴	—	21	1	no	—
58	C₂×D₄	—	21	12	no	—
59	C₂²×C₄	—	7	1	no	—
60	C₂×C₇	—	6	4	no	—
61	C₂×C₇	C₂×D₁₂	1	6	yes	complemented
62	C₂×C₇	C₂³×D₃	1	1	yes	characteristic
63	D₇	—	6	4	no	—
64	C₂²×C₃	C₂²⋊D₇	1	4	yes	complemented
65	C₂²×C₃	C₂²×D₇	1	3	yes	—
66	C₂×D₃	—	2	12	no	—
67	C₂×D₃	—	7	4	no	—
68	C₂×D₃	—	14	12	no	—
69	C₂×D₃	C₂²×D₇	1	4	yes	complemented
70	C₃×C₄	—	7	4	no	—
71	C₂³	—	3	6	no	—
72	C₂³	—	6	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	—	7	6	no	—
78	C₇	C₂²×D₁₂	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	C₂×C₂²⋊D₇	1	6	yes	complemented
80	C₂×C₃	C₂³×D₇	1	1	yes	characteristic
81	D₃	—	2	4	no	—
82	D₃	—	14	4	no	—
83	C₂²	—	3	4	no	—
84	C₂²	—	6	12	no	—
85	C₂²	—	21	4	no	—
86	C₂²	—	42	12	no	—
87	C₂²	#4247	1	4	yes	direct product factor
88	C₂²	C₂×D₃×D₇	1	3	yes	—
89	C₄	—	7	4	no	—
90	C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	6	4	no	—
92	C₂	—	42	4	no	—
93	C₂	C₂×#4247	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×D₃×C₇	C₂	semidirect
3. C₂²×C₃×Dic₇	C₂	semidirect
4. C₂×#4247	C₂	direct (N × H)
8. C₂²×D₂₁	C₂²	semidirect
10. C₂²×D₃×C₇	C₂²	semidirect
11. C₂×C₃×Dic₇	C₂²	semidirect
12. #4247	C₂²	direct (N × H)
16. C₂²×Dic₇	D₃	semidirect
18. C₂²×C₃×C₇	D₄	semidirect
21. C₂×D₂₁	C₂³	semidirect
23. C₂×D₃×C₇	C₂³	semidirect
24. C₃×Dic₇	C₂³	semidirect
31. C₂×Dic₇	C₂×D₃	semidirect
33. C₂³×D₃	D₇	semidirect
36. C₂×C₃×C₇	C₂×D₄	semidirect
43. C₂²×C₇	D₁₂	semidirect
47. Dic₇	C₂²×D₃	semidirect
52. C₂²×D₃	C₂×D₇	semidirect
55. C₃×C₇	C₂²×D₄	semidirect
61. C₂×C₇	C₂×D₁₂	semidirect
64. C₂²×C₃	C₂²⋊D₇	semidirect
69. C₂×D₃	C₂²×D₇	semidirect
78. C₇	C₂²×D₁₂	semidirect
79. C₂×C₃	C₂×C₂²⋊D₇	semidirect
87. C₂²	#4247	direct (N × H)
90. C₃	C₂²×C₂²⋊D₇	semidirect
93. C₂	C₂×#4247	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×D₃×C₇	2
C₂²×C₃×Dic₇	2
C₂×#4247	2
C₂²×C₂²⋊D₇	3
C₂²×D₁₂	7

Subgroups — C22×#4247 #10621591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#4247 #10621591