Subgroups — C₂×C₃×#1144 #10621254

Order 672.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#1144	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#1144	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₇×Q₈	—	2	1	no	—
7	C₃×C₇×Q₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₇	—	2	1	no	—
9	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₃×Dic₇	—	2	1	no	—
11	C₃×Dic₁₄	—	2	1	no	—
12	C₃×Dic₁₄	C₂²	1	2	yes	—
13	C₃×C₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
15	C₂×Dic₁₄	C₂×C₃	1	1	yes	characteristic
16	#1144	C₂×C₃	1	4	yes	direct product factor
17	C₂×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×Q₁₆	—	7	1	no	maximal, Hall
19	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
20	C₃×C₄×C₇	—	2	2	no	—
21	C₃×C₄×C₇	D₄	1	1	yes	characteristic
22	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₃×Dic₇	—	2	2	no	—
24	C₇×Q₈	—	2	1	no	—
25	C₇×Q₈	C₂²×C₃	1	2	yes	—
26	C₂×C₄×C₇	—	2	1	no	—
27	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
28	C₂×Dic₇	—	2	1	no	—
29	Dic₁₄	—	2	1	no	—
30	Dic₁₄	C₂²×C₃	1	2	yes	—
31	C₇⋊C₈	C₂²×C₃	1	2	yes	—
32	C₂×C₃×Q₈	—	7	1	no	—
33	C₂×C₃×Q₈	D₇	1	1	yes	characteristic
34	C₃×Q₁₆	—	7	4	no	—
35	C₂×C₃×C₈	—	7	1	no	—
36	C₂×C₃×C₇	Q₁₆	1	2	yes	complemented
37	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Q₁₆	—	7	1	no	Sylow
39	C₂²×C₇	C₃×D₄	1	1	yes	characteristic
40	C₄×C₇	—	2	2	no	—
41	C₄×C₇	C₃×D₄	1	1	yes	characteristic
42	C₄×C₇	C₂³×C₃	1	1	yes	characteristic
43	Dic₇	—	2	2	no	—
44	C₃×Q₈	—	2	1	no	—
45	C₃×Q₈	—	7	2	no	—
46	C₃×Q₈	—	14	1	no	—
47	C₃×Q₈	C₂×D₇	1	2	yes	—
48	C₂×C₃×C₄	—	2	1	no	—
49	C₂×C₃×C₄	—	14	1	no	—
50	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
51	C₃×C₈	—	7	2	no	—
52	C₃×C₇	C₂×Q₁₆	1	1	yes	characteristic, complemented, Hall
53	C₂×Q₈	—	7	1	no	—
54	C₂×Q₈	C₃×D₇	1	1	yes	characteristic
55	Q₁₆	—	7	4	no	—
56	C₂×C₈	—	7	1	no	—
57	C₂×C₇	C₃×Q₁₆	1	2	yes	complemented
58	C₂×C₇	C₂×C₃×D₄	1	1	yes	characteristic
59	C₂²×C₃	C₂²⋊D₇	1	1	yes	characteristic
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	—	14	2	no	—
62	C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
63	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
64	Q₈	—	2	1	no	—
65	Q₈	—	7	2	no	—
66	Q₈	—	14	1	no	—
67	Q₈	C₂×C₃×D₇	1	2	yes	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	14	1	no	—
70	C₂×C₄	C₂×C₃×D₇	1	1	yes	characteristic
71	C₈	—	7	2	no	—
72	C₇	C₂×C₃×Q₁₆	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	#1144	1	2	yes	direct product factor
74	C₂×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
75	C₂²	C₃×C₂²⋊D₇	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	14	2	no	—
78	C₄	C₃×C₂²⋊D₇	1	1	yes	characteristic
79	C₄	C₂²×C₃×D₇	1	1	yes	characteristic
80	C₃	C₂×#1144	1	1	yes	characteristic, direct product factor, Sylow
81	C₂	C₃×#1144	1	2	yes	direct product factor
82	C₂	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×Q₈	C₂
socle, prime radical	C₂²×C₃×C₇	D₄
hypercenter	C₂×C₃×Q₈	D₇
hypocenter	C₇	C₂×C₃×Q₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₈

Complements

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

N	H ≅ G/N	type
3. C₃×#1144	C₂	direct (N × H)
5. C₂×#1144	C₃	direct (N × H)
16. #1144	C₂×C₃	direct (N × H)
36. C₂×C₃×C₇	Q₁₆	semidirect
52. C₃×C₇	C₂×Q₁₆	semidirect
57. C₂×C₇	C₃×Q₁₆	semidirect
72. C₇	C₂×C₃×Q₁₆	semidirect
73. C₂×C₃	#1144	direct (N × H)
80. C₃	C₂×#1144	direct (N × H)
81. C₂	C₃×#1144	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×C₇×Q₈	2
C₂×C₃×Dic₁₄	2
C₃×#1144	2
C₂×C₃×C₇⋊C₈	2
C₂×#1144	3
C₂×C₃×Q₁₆	7

Subgroups — C2×C3×#1144 #10621254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#1144 #10621254