Subgroups — C₃×#6507 #10621255

Order 672.

136 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×Dic₇⋊C₄	C₂	1	4	yes	maximal
4	C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
5	#6507	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₇×Q₈	—	2	2	no	—
7	C₂×C₃×C₄×C₇	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₃×Dic₇	C₂²	1	2	yes	—
10	C₂×C₃×Dic₇	C₂²	1	2	yes	—
11	C₃×Dic₁₄	—	2	2	no	—
12	C₂×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
13	C₂×Dic₁₄	C₂×C₃	1	1	yes	characteristic
14	Dic₇⋊C₄	C₂×C₃	1	4	yes	—
15	C₄×Dic₇	C₂×C₃	1	1	yes	characteristic
16	C₃×C₄⋊Q₈	—	7	1	no	maximal, Hall
17	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
18	C₃×C₄×C₇	—	2	2	no	—
19	C₃×C₄×C₇	D₄	1	2	yes	—
20	C₃×Dic₇	—	2	2	no	—
21	C₃×Dic₇	Q₈	1	4	yes	complemented
22	C₇×Q₈	—	2	2	no	—
23	C₂×C₄×C₇	C₂²×C₃	1	2	yes	—
24	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
25	C₂×Dic₇	C₂²×C₃	1	2	yes	—
26	C₂×Dic₇	C₂²×C₃	1	2	yes	—
27	Dic₁₄	—	2	2	no	—
28	C₂×C₃×Q₈	—	7	1	no	—
29	C₂×C₃×Q₈	D₇	1	1	yes	characteristic
30	C₃×C₄⋊C₄	—	7	4	no	—
31	C₃×C₄²	—	7	1	no	—
32	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₃×C₇	C₂×Q₈	1	2	yes	—
34	C₄⋊Q₈	—	7	1	no	Sylow
35	C₂²×C₇	C₂³×C₃	1	1	yes	characteristic
36	C₄×C₇	—	2	2	no	—
37	C₄×C₇	C₃×D₄	1	2	yes	—
38	Dic₇	—	2	2	no	—
39	Dic₇	C₃×Q₈	1	4	yes	complemented
40	C₃×Q₈	—	2	2	no	—
41	C₃×Q₈	—	14	2	no	—
42	C₂×C₃×C₄	—	7	2	no	—
43	C₂×C₃×C₄	—	7	2	no	—
44	C₂×C₃×C₄	C₂×D₇	1	2	yes	—
45	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
46	C₃×C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
47	C₂×Q₈	—	7	1	no	—
48	C₂×Q₈	C₃×D₇	1	1	yes	characteristic
49	C₄⋊C₄	—	7	4	no	—
50	C₄²	—	7	1	no	—
51	C₂×C₇	C₂×C₃×D₄	1	1	yes	characteristic
52	C₂×C₇	C₂×C₃×Q₈	1	2	yes	—
53	C₂²×C₃	C₂²×D₇	1	1	yes	characteristic
54	C₃×C₄	—	2	2	no	—
55	C₃×C₄	—	7	4	no	—
56	C₃×C₄	—	14	2	no	—
57	C₃×C₄	C₂²⋊D₇	1	2	yes	—
58	Q₈	—	2	2	no	—
59	Q₈	—	14	2	no	—
60	C₂×C₄	—	7	2	no	—
61	C₂×C₄	—	7	2	no	—
62	C₂×C₄	C₂×C₃×D₇	1	2	yes	—
63	C₂×C₄	C₂×C₃×D₇	1	1	yes	characteristic
64	C₇	C₃×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	Q₈×D₇	1	2	yes	—
66	C₂×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
67	C₂²	C₂²×C₃×D₇	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	—	7	4	no	—
70	C₄	—	14	2	no	—
71	C₄	C₃×C₂²⋊D₇	1	2	yes	—
72	C₃	#6507	1	1	yes	characteristic, direct product factor, Sylow
73	C₂	C₃×Q₈×D₇	1	2	yes	—
74	C₂	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
5. #6507	C₃	direct (N × H)
21. C₃×Dic₇	Q₈	semidirect
39. Dic₇	C₃×Q₈	semidirect
46. C₃×C₇	C₄⋊Q₈	semidirect
64. C₇	C₃×C₄⋊Q₈	semidirect
72. C₃	#6507	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×Dic₇⋊C₄	2
C₃×C₄×Dic₇	2
#6507	3
C₃×C₄⋊Q₈	7

Subgroups — C3×#6507 #10621255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#6507 #10621255