Subgroups — #13268533

Order 1792.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11726798	C₂	1	4	yes	maximal
2	#11725819	C₂	1	2	yes	maximal
3	C₇×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#89471	—	2	2	no	—
5	#89454	—	2	2	no	—
6	#89454	C₂²	1	4	yes	—
7	C₄×C₇×C₁₆	C₂²	1	2	yes	—
8	C₇×C₈²	C₂²	1	1	yes	characteristic
9	#89299	—	2	8	no	—
10	#12521	—	7	1	no	maximal, Sylow
11	C₂×Dic₂₈	—	4	4	no	—
12	#6432	—	2	4	no	—
13	C₂×C₇×C₁₆	D₄	1	4	yes	—
14	C₄×C₇×C₈	D₄	1	1	yes	characteristic
15	C₄×C₇×C₈	D₄	1	1	yes	characteristic
16	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
17	#6392	—	2	8	no	—
18	#6392	—	4	4	no	—
19	#2252	—	7	4	no	—
20	#1713	—	7	2	no	—
21	C₈×C₁₆	D₇	1	1	yes	characteristic
22	C₂×Dic₁₄	—	4	4	no	—
23	C₂×C₇×C₈	D₈	1	2	yes	—
24	C₂×C₇×C₈	D₈	1	2	yes	—
25	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
26	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
27	C₄⋊Dic₇	—	4	4	no	—
28	C₄⋊Dic₇	—	4	4	no	—
29	Dic₂₈	—	8	8	no	—
30	C₇×C₁₆	Q₁₆	1	8	yes	complemented
31	#500	—	7	4	no	—
32	#500	—	14	2	no	—
33	#494	—	14	2	no	—
34	#366	—	14	8	no	—
35	C₄×C₁₆	C₂×D₇	1	2	yes	—
36	C₈²	C₂×D₇	1	1	yes	characteristic
37	C₂×C₄×C₇	#127	1	1	yes	characteristic
38	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
39	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
40	C₂×Dic₇	—	8	4	no	—
41	Dic₁₄	—	8	8	no	—
42	C₇×C₈	D₁₆	1	4	yes	—
43	C₇×C₈	Q₃₂	1	4	yes	—
44	C₇×C₈	C₂×Q₁₆	1	4	yes	—
45	C₂×Q₁₆	—	28	4	no	—
46	C₄⋊Q₈	—	14	4	no	—
47	C₂×C₁₆	D₂₈	1	4	yes	—
48	#107	—	14	8	no	—
49	#107	—	28	4	no	—
50	C₄×C₈	D₂₈	1	1	yes	characteristic
51	C₄×C₈	D₂₈	1	1	yes	characteristic
52	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
53	C₂²×C₇	#493	1	1	yes	characteristic
54	C₄×C₇	#494	1	2	yes	—
55	C₄×C₇	C₂×D₁₆	1	2	yes	—
56	C₄×C₇	C₂×Q₃₂	1	2	yes	—
57	Dic₇	—	16	4	no	—
58	C₂×Q₈	—	28	4	no	—
59	Q₁₆	—	56	8	no	—
60	C₂×C₈	D₅₆	1	2	yes	—
61	C₂×C₈	D₅₆	1	2	yes	—
62	C₂×C₈	C₂×D₂₈	1	2	yes	—
63	C₄⋊C₄	—	28	4	no	—
64	C₄⋊C₄	—	28	4	no	—
65	C₄²	C₂×D₂₈	1	1	yes	characteristic
66	C₁₆	Dic₂₈	1	8	yes	complemented
67	C₂×C₇	#1713	1	1	yes	characteristic
68	C₂×C₇	#2246	1	1	yes	characteristic
69	C₂×C₇	#2247	1	1	yes	characteristic
70	Q₈	—	56	8	no	—
71	C₂×C₄	—	56	4	no	—
72	C₂×C₄	#6437	1	1	yes	characteristic
73	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
74	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
75	C₈	D₁₁₂	1	4	yes	—
76	C₈	Dic₅₆	1	4	yes	—
77	C₈	C₂×Dic₂₈	1	4	yes	—
78	C₇	#12521	1	1	yes	characteristic, complemented, Sylow
79	C₂²	#89467	1	1	yes	characteristic
80	C₄	—	112	4	no	—
81	C₄	#89471	1	2	yes	—
82	C₄	C₂×D₁₁₂	1	2	yes	—
83	C₄	C₂×Dic₅₆	1	2	yes	—
84	C₂	#11725819	1	1	yes	characteristic
85	C₂	#11726814	1	1	yes	characteristic
86	C₂	#11726820	1	1	yes	characteristic

	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	#12521
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
30. C₇×C₁₆	Q₁₆	semidirect
66. C₁₆	Dic₂₈	semidirect
78. C₇	#12521	semidirect

Maximal subgroups

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

subgroup	index
#11726798	2
#11725819	2
C₇×C₈×C₁₆	2
#12521	7