Subgroups — #13285002

Order 1792.

2286 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#89707	C₂	1	14	yes	maximal
3	C₂²×C₄²×C₇	C₂²	1	7	yes	—
4	C₂²×C₄⋊Dic₇	C₂²	1	28	yes	—
5	C₄³×C₇	C₄	1	8	yes	complemented
6	#89707	—	2	56	no	—
7	#14692	—	7	1	no	maximal, Sylow
8	C₂³×C₄×C₇	C₂³	1	7	yes	—
9	C₂³×Dic₇	C₂³	1	8	yes	—
10	C₂×C₄²×C₇	C₂×C₄	1	28	yes	—
11	C₂×C₄²×C₇	D₄	1	28	yes	—
12	C₂×C₄²×C₇	Q₈	1	28	yes	—
13	C₂×C₄⋊Dic₇	—	2	168	no	—
14	C₂×#384	—	7	14	no	—
15	C₂×C₄³	D₇	1	1	yes	characteristic
16	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
17	C₂²×C₄×C₇	C₂²×C₄	1	14	yes	—
18	C₂²×C₄×C₇	C₂×D₄	1	42	yes	—
19	C₂²×C₄×C₇	C₂×Q₈	1	42	yes	—
20	C₂²×Dic₇	—	2	56	no	—
21	C₄²×C₇	C₄⋊C₄	1	112	yes	complemented
22	C₄⋊Dic₇	—	4	112	no	—
23	C₂²×C₄⋊C₄	—	7	28	no	—
24	C₂²×C₄²	C₂×D₇	1	7	yes	—
25	#384	—	14	56	no	—
26	C₄³	Dic₇	1	8	yes	complemented
27	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
28	C₂³×C₇	C₂²×D₄	1	7	yes	—
29	C₂³×C₇	C₂²×Q₈	1	7	yes	—
30	C₂×C₄×C₇	C₂×C₄⋊C₄	1	84	yes	—
31	C₂×C₄×C₇	#127	1	28	yes	—
32	C₂×C₄×C₇	C₄⋊Q₈	1	84	yes	—
33	C₂×Dic₇	—	4	56	no	—
34	C₂³×C₄	—	7	8	no	—
35	C₂³×C₄	C₂²×D₇	1	7	yes	—
36	C₂×C₄⋊C₄	—	14	168	no	—
37	C₂×C₄²	Dic₁₄	1	28	yes	—
38	C₂×C₄²	D₂₈	1	28	yes	—
39	C₂×C₄²	C₂×Dic₇	1	28	yes	—
40	C₂²×C₇	C₂²×C₄⋊C₄	1	7	yes	—
41	C₂²×C₇	C₂×#127	1	7	yes	—
42	C₂²×C₇	C₂×C₄⋊Q₈	1	21	yes	—
43	C₄×C₇	#384	1	56	yes	complemented
44	Dic₇	—	8	8	no	—
45	C₂⁴	C₂³×D₇	1	1	yes	characteristic
46	C₂²×C₄	—	14	56	no	—
47	C₂²×C₄	C₂×Dic₁₄	1	42	yes	—
48	C₂²×C₄	C₂×D₂₈	1	42	yes	—
49	C₂²×C₄	C₂²×Dic₇	1	14	yes	—
50	C₄⋊C₄	—	28	112	no	—
51	C₄²	C₄⋊Dic₇	1	112	yes	complemented
52	C₂×C₇	C₂×#384	1	7	yes	—
53	C₂×C₇	#2867	1	1	yes	characteristic
54	C₂×C₇	#2868	1	7	yes	—
55	C₂³	C₂²×Dic₁₄	1	7	yes	—
56	C₂³	C₂²×D₂₈	1	7	yes	—
57	C₂³	C₂³×Dic₇	1	1	yes	characteristic
58	C₂×C₄	—	28	56	no	—
59	C₂×C₄	#6432	1	84	yes	—
60	C₂×C₄	#6437	1	28	yes	—
61	C₂×C₄	C₂×C₄⋊Dic₇	1	84	yes	—
62	C₇	#14692	1	1	yes	characteristic, complemented, Sylow
63	C₂²	C₂×#6432	1	21	yes	—
64	C₂²	C₂×#6437	1	7	yes	—
65	C₂²	C₂²×C₄⋊Dic₇	1	7	yes	—
66	C₄	—	56	8	no	—
67	C₄	#89707	1	56	yes	complemented
68	C₂	#11730756	1	7	yes	—
69	C₂	#11730763	1	1	yes	characteristic
70	C₂	C₂×#89707	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₇
commutator	C₂³×C₇	C₂³×C₄
Fitting	C₂×C₄³×C₇	C₂
socle, prime radical, cosocle	C₂⁴×C₇	C₂⁴
hypercenter	C₂×C₄³	D₇
hypocenter	C₇	#14692
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
5. C₄³×C₇	C₄	semidirect
21. C₄²×C₇	C₄⋊C₄	semidirect
26. C₄³	Dic₇	semidirect
43. C₄×C₇	#384	semidirect
51. C₄²	C₄⋊Dic₇	semidirect
62. C₇	#14692	semidirect
67. C₄	#89707	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄³×C₇	2
C₂×#89707	2
#14692	7