Subgroups — C₂×C₃×#1132 #10621213

Order 672.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#1132	C₂	1	4	yes	maximal, direct product factor
5	C₂×#1132	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×D₇	—	2	1	no	—
7	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₃×Dic₇	—	2	1	no	—
9	C₃×D₂₈	—	2	1	no	—
10	C₃×D₂₈	C₂²	1	2	yes	—
11	C₃×Dic₁₄	—	2	1	no	—
12	C₃×Dic₁₄	C₂²	1	2	yes	complemented
13	C₃×C₇×C₈	C₂²	1	2	yes	complemented
14	C₂×D₂₈	C₂×C₃	1	1	yes	characteristic
15	C₂×Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
16	C₂×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
17	#1132	C₂×C₃	1	4	yes	direct product factor
18	C₂×C₃×SD₁₆	—	7	1	no	maximal, Hall
19	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
20	C₂×C₃×D₇	—	2	2	no	—
21	C₂×C₃×D₇	—	4	2	no	—
22	C₃×C₄×C₇	D₄	1	1	yes	characteristic
23	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₃×Dic₇	—	2	2	no	—
25	C₂²×D₇	—	2	1	no	—
26	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
27	C₂×Dic₇	—	2	1	no	—
28	D₂₈	—	2	1	no	—
29	D₂₈	C₂²×C₃	1	2	yes	—
30	Dic₁₄	—	2	1	no	—
31	Dic₁₄	C₂²×C₃	1	2	yes	complemented
32	C₇×C₈	C₂²×C₃	1	2	yes	complemented
33	C₂×C₃×Q₈	—	7	1	no	—
34	C₂×C₃×D₄	—	7	1	no	—
35	C₃×SD₁₆	—	7	4	no	—
36	C₂×C₃×C₈	D₇	1	1	yes	characteristic, complemented
37	C₂×C₃×C₇	SD₁₆	1	2	yes	complemented
38	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
39	C₃×D₇	—	4	2	no	—
40	C₂×SD₁₆	—	7	1	no	Sylow
41	C₂²×C₇	C₃×D₄	1	1	yes	characteristic
42	C₂×D₇	—	2	2	no	—
43	C₂×D₇	—	4	2	no	—
44	C₄×C₇	C₃×D₄	1	1	yes	characteristic
45	C₄×C₇	C₂³×C₃	1	1	yes	characteristic
46	Dic₇	—	2	2	no	—
47	C₂³×C₃	—	14	1	no	—
48	C₃×Q₈	—	7	2	no	—
49	C₃×Q₈	—	14	1	no	—
50	C₃×D₄	—	7	2	no	—
51	C₃×D₄	—	14	1	no	—
52	C₂×C₃×C₄	—	14	1	no	—
53	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
54	C₃×C₈	C₂×D₇	1	2	yes	complemented
55	C₃×C₇	C₂×SD₁₆	1	1	yes	characteristic, complemented, Hall
56	C₂×Q₈	—	7	1	no	—
57	C₂×D₄	—	7	1	no	—
58	SD₁₆	—	7	4	no	—
59	C₂×C₈	C₃×D₇	1	1	yes	characteristic, complemented
60	C₂×C₇	C₃×SD₁₆	1	2	yes	complemented
61	C₂×C₇	C₂×C₃×D₄	1	1	yes	characteristic
62	D₇	—	4	2	no	—
63	C₂²×C₃	—	14	2	no	—
64	C₂²×C₃	—	28	2	no	—
65	C₂²×C₃	D₂₈	1	1	yes	characteristic
66	C₃×C₄	—	14	2	no	—
67	C₃×C₄	D₂₈	1	1	yes	characteristic
68	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
69	C₂³	—	14	1	no	—
70	Q₈	—	7	2	no	—
71	Q₈	—	14	1	no	—
72	D₄	—	7	2	no	—
73	D₄	—	14	1	no	—
74	C₂×C₄	—	14	1	no	—
75	C₂×C₄	C₂×C₃×D₇	1	1	yes	characteristic
76	C₈	C₂×C₃×D₇	1	2	yes	complemented
77	C₇	C₂×C₃×SD₁₆	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₃	—	28	2	no	—
79	C₂×C₃	#1132	1	2	yes	direct product factor
80	C₂×C₃	C₂×D₂₈	1	1	yes	characteristic
81	C₂²	—	14	2	no	—
82	C₂²	—	28	2	no	—
83	C₂²	C₃×D₂₈	1	1	yes	characteristic
84	C₄	—	14	2	no	—
85	C₄	C₃×D₂₈	1	1	yes	characteristic
86	C₄	C₂²×C₃×D₇	1	1	yes	characteristic
87	C₃	C₂×#1132	1	1	yes	characteristic, direct product factor, Sylow
88	C₂	—	28	2	no	—
89	C₂	C₃×#1132	1	2	yes	direct product factor
90	C₂	C₂×C₃×D₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₂₈
commutator, cosocle	C₄×C₇	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×C₈	C₂
socle	C₂²×C₃×C₇	D₄
prime radical	C₂×C₃×D₂₈	C₂
hypercenter	C₂×C₃×C₈	D₇
hypocenter	C₇	C₂×C₃×SD₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×Dic₁₄	C₂	semidirect
3. C₂×C₃×C₇×C₈	C₂	semidirect
4. C₃×#1132	C₂	direct (N × H)
5. C₂×#1132	C₃	direct (N × H)
12. C₃×Dic₁₄	C₂²	semidirect
13. C₃×C₇×C₈	C₂²	semidirect
15. C₂×Dic₁₄	C₂×C₃	semidirect
16. C₂×C₇×C₈	C₂×C₃	semidirect
17. #1132	C₂×C₃	direct (N × H)
31. Dic₁₄	C₂²×C₃	semidirect
32. C₇×C₈	C₂²×C₃	semidirect
36. C₂×C₃×C₈	D₇	semidirect
37. C₂×C₃×C₇	SD₁₆	semidirect
54. C₃×C₈	C₂×D₇	semidirect
55. C₃×C₇	C₂×SD₁₆	semidirect
59. C₂×C₈	C₃×D₇	semidirect
60. C₂×C₇	C₃×SD₁₆	semidirect
76. C₈	C₂×C₃×D₇	semidirect
77. C₇	C₂×C₃×SD₁₆	semidirect
79. C₂×C₃	#1132	direct (N × H)
87. C₃	C₂×#1132	direct (N × H)
89. C₂	C₃×#1132	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×D₂₈	2
C₂×C₃×Dic₁₄	2
C₂×C₃×C₇×C₈	2
C₃×#1132	2
C₂×#1132	3
C₂×C₃×SD₁₆	7

Subgroups — C2×C3×#1132 #10621213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#1132 #10621213