Subgroups — #13268540

Order 1792.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11726800	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	#89300	—	2	8	no	—
10	#12523	—	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	2	yes	—
15	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
16	#6392	—	2	8	no	—
17	#6392	—	4	4	no	—
18	#2254	—	7	4	no	—
19	#1713	—	7	2	no	—
20	C₈×C₁₆	D₇	1	1	yes	characteristic
21	C₂×Dic₁₄	—	4	4	no	—
22	C₂×C₇×C₈	D₈	1	4	yes	—
23	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
24	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
25	C₄⋊Dic₇	—	4	4	no	—
26	C₄⋊Dic₇	—	4	4	no	—
27	Dic₂₈	—	8	8	no	—
28	C₇×C₁₆	Q₁₆	1	8	yes	complemented
29	#500	—	7	4	no	—
30	#500	—	14	2	no	—
31	#494	—	14	2	no	—
32	#367	—	14	8	no	—
33	C₄×C₁₆	C₂×D₇	1	2	yes	—
34	C₈²	C₂×D₇	1	1	yes	characteristic
35	C₂×C₄×C₇	#127	1	1	yes	characteristic
36	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
37	C₂×Dic₇	—	8	4	no	—
38	Dic₁₄	—	8	8	no	—
39	C₇×C₈	SD₃₂	1	8	yes	—
40	C₇×C₈	C₂×Q₁₆	1	4	yes	—
41	C₂×Q₁₆	—	28	4	no	—
42	C₄⋊Q₈	—	14	4	no	—
43	C₂×C₁₆	D₂₈	1	4	yes	—
44	#107	—	14	8	no	—
45	#107	—	28	4	no	—
46	C₄×C₈	D₂₈	1	2	yes	—
47	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
48	C₂²×C₇	#493	1	1	yes	characteristic
49	C₄×C₇	#494	1	2	yes	—
50	C₄×C₇	C₂×SD₃₂	1	4	yes	—
51	Dic₇	—	16	4	no	—
52	C₂×Q₈	—	28	4	no	—
53	Q₁₆	—	56	8	no	—
54	C₂×C₈	D₅₆	1	4	yes	—
55	C₂×C₈	C₂×D₂₈	1	2	yes	—
56	C₄⋊C₄	—	28	4	no	—
57	C₄⋊C₄	—	28	4	no	—
58	C₄²	C₂×D₂₈	1	1	yes	characteristic
59	C₁₆	Dic₂₈	1	8	yes	complemented
60	C₂×C₇	#1713	1	1	yes	characteristic
61	C₂×C₇	#2248	1	2	yes	—
62	Q₈	—	56	8	no	—
63	C₂×C₄	—	56	4	no	—
64	C₂×C₄	#6437	1	1	yes	characteristic
65	C₂×C₄	C₂×D₅₆	1	2	yes	—
66	C₈	#6374	1	8	yes	—
67	C₈	C₂×Dic₂₈	1	4	yes	—
68	C₇	#12523	1	1	yes	characteristic, complemented, Sylow
69	C₂²	#89467	1	1	yes	characteristic
70	C₄	—	112	4	no	—
71	C₄	#89471	1	2	yes	—
72	C₄	C₂×#6374	1	4	yes	—
73	C₂	#11725819	1	1	yes	characteristic
74	C₂	#11726815	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	#12523
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
28. C₇×C₁₆	Q₁₆	semidirect
59. C₁₆	Dic₂₈	semidirect
68. C₇	#12523	semidirect

Maximal subgroups

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

subgroup	index
#11726800	2
#11725819	2
C₇×C₈×C₁₆	2
#12523	7