Subgroups — #13268511

Order 1792.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11726820	C₂	1	4	yes	maximal
2	#11725819	C₂	1	2	yes	maximal
3	C₇×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₆	—	2	8	no	—
5	#89471	C₂²	1	4	yes	—
6	#89454	—	2	4	no	—
7	C₄×C₇×C₁₆	C₂²	1	2	yes	—
8	C₇×C₈²	C₂²	1	1	yes	characteristic
9	#12516	—	7	1	no	maximal, Sylow
10	C₂×Dic₂₈	—	2	8	no	—
11	#6432	—	2	4	no	—
12	C₂×C₇×C₁₆	D₄	1	4	yes	—
13	C₄×C₇×C₈	D₄	1	2	yes	—
14	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
15	#6392	—	4	8	no	—
16	Dic₅₆	—	4	16	no	—
17	#2247	—	7	4	no	—
18	#1713	—	7	2	no	—
19	C₈×C₁₆	D₇	1	1	yes	characteristic
20	C₂×Dic₁₄	—	4	4	no	—
21	C₂×C₇×C₈	D₈	1	4	yes	—
22	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
23	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
24	C₄⋊Dic₇	—	4	8	no	—
25	Dic₂₈	—	4	16	no	—
26	C₇×C₁₆	D₈	1	8	yes	—
27	C₂×Q₃₂	—	14	8	no	—
28	#500	—	14	4	no	—
29	#494	—	7	4	no	—
30	C₄×C₁₆	C₂×D₇	1	2	yes	—
31	C₈²	C₂×D₇	1	1	yes	characteristic
32	C₂×C₄×C₇	#127	1	1	yes	characteristic
33	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
34	C₂×Dic₇	—	8	4	no	—
35	Dic₁₄	—	8	8	no	—
36	C₇×C₈	Q₃₂	1	8	yes	complemented
37	C₇×C₈	C₂×D₈	1	4	yes	—
38	C₂×Q₁₆	—	14	8	no	—
39	C₄⋊Q₈	—	14	4	no	—
40	Q₃₂	—	28	16	no	—
41	C₂×C₁₆	D₂₈	1	4	yes	—
42	#107	—	28	8	no	—
43	C₄×C₈	D₂₈	1	2	yes	—
44	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
45	C₂²×C₇	#493	1	1	yes	characteristic
46	C₄×C₇	#493	1	2	yes	—
47	C₄×C₇	C₂×Q₃₂	1	4	yes	—
48	Dic₇	—	16	4	no	—
49	C₂×Q₈	—	28	4	no	—
50	Q₁₆	—	28	16	no	—
51	C₂×C₈	D₅₆	1	4	yes	—
52	C₂×C₈	C₂×D₂₈	1	2	yes	—
53	C₄⋊C₄	—	28	8	no	—
54	C₄²	C₂×D₂₈	1	1	yes	characteristic
55	C₁₆	D₅₆	1	8	yes	—
56	C₂×C₇	#1712	1	1	yes	characteristic
57	C₂×C₇	#2247	1	2	yes	—
58	Q₈	—	56	8	no	—
59	C₂×C₄	—	56	4	no	—
60	C₂×C₄	#6437	1	1	yes	characteristic
61	C₂×C₄	C₂×D₅₆	1	2	yes	—
62	C₈	Dic₅₆	1	8	yes	complemented
63	C₈	C₂×D₅₆	1	4	yes	—
64	C₇	#12516	1	1	yes	characteristic, complemented, Sylow
65	C₂²	#89467	1	1	yes	characteristic
66	C₄	—	112	4	no	—
67	C₄	#89467	1	2	yes	—
68	C₄	C₂×Dic₅₆	1	4	yes	—
69	C₂	#11725816	1	1	yes	characteristic
70	C₂	#11726820	1	2	yes	—

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×C₈×C₁₆	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	C₈×C₁₆	D₇
hypocenter	C₇	#12516
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
36. C₇×C₈	Q₃₂	semidirect
62. C₈	Dic₅₆	semidirect
64. C₇	#12516	semidirect

Maximal subgroups

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

subgroup	index
#11726820	2
#11725819	2
C₇×C₈×C₁₆	2
#12516	7