Subgroups — #13268537

Order 1792.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11726800	C₂	1	2	yes	maximal
2	#11726798	C₂	1	2	yes	maximal
3	#11725818	C₂	1	2	yes	maximal
4	C₇×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#89454	C₂²	1	4	yes	—
6	#89453	—	2	4	no	—
7	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
8	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₇×C₈²	C₂²	1	1	yes	characteristic
10	#89300	—	2	4	no	—
11	#89299	—	2	4	no	—
12	#12522	—	7	1	no	maximal, Sylow
13	#6432	—	2	4	no	—
14	C₂×C₇×C₁₆	D₄	1	2	yes	—
15	C₂×C₇×C₁₆	D₄	1	2	yes	—
16	C₄×C₇×C₈	D₄	1	2	yes	—
17	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
18	#6392	—	2	4	no	—
19	#6392	—	2	4	no	—
20	#6390	—	4	8	no	—
21	#2254	—	7	2	no	—
22	#2252	—	7	2	no	—
23	#1710	—	7	2	no	—
24	C₈×C₁₆	D₇	1	1	yes	characteristic
25	C₂×Dic₁₄	—	4	4	no	—
26	C₂×C₇×C₈	D₈	1	4	yes	—
27	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
28	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
29	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
30	C₄⋊Dic₇	—	4	4	no	—
31	C₄⋊Dic₇	—	4	4	no	—
32	C₇×C₁₆	Q₁₆	1	4	yes	—
33	C₇×C₁₆	Q₁₆	1	4	yes	—
34	#500	—	7	4	no	—
35	#487	—	14	4	no	—
36	#367	—	14	4	no	—
37	#366	—	14	4	no	—
38	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
39	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
40	C₈²	C₂×D₇	1	1	yes	characteristic
41	C₂×C₄×C₇	#127	1	1	yes	characteristic
42	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
43	C₂×Dic₇	—	8	4	no	—
44	Dic₁₄	—	8	8	no	—
45	C₇×C₈	—	2	4	no	—
46	C₇×C₈	C₂×Q₁₆	1	2	yes	—
47	C₇×C₈	C₂×Q₁₆	1	2	yes	—
48	C₄⋊Q₈	—	14	4	no	—
49	C₂×C₁₆	D₂₈	1	2	yes	—
50	C₂×C₁₆	D₂₈	1	2	yes	—
51	#107	—	14	4	no	—
52	#107	—	14	4	no	—
53	#103	—	28	8	no	—
54	C₄×C₈	D₂₈	1	2	yes	—
55	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
56	C₂²×C₇	#493	1	1	yes	characteristic
57	C₄×C₇	#494	1	1	yes	characteristic
58	C₄×C₇	#494	1	1	yes	characteristic
59	C₄×C₇	#508	1	4	yes	—
60	Dic₇	—	16	4	no	—
61	C₂×Q₈	—	28	4	no	—
62	C₂×C₈	D₅₆	1	4	yes	—
63	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
64	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
65	C₄⋊C₄	—	28	4	no	—
66	C₄⋊C₄	—	28	4	no	—
67	C₄²	C₂×D₂₈	1	1	yes	characteristic
68	C₁₆	Dic₂₈	1	4	yes	—
69	C₁₆	Dic₂₈	1	4	yes	—
70	C₂×C₇	#1713	1	1	yes	characteristic
71	C₂×C₇	#2249	1	2	yes	—
72	Q₈	—	56	8	no	—
73	C₂×C₄	—	56	4	no	—
74	C₂×C₄	#6437	1	1	yes	characteristic
75	C₂×C₄	C₂×D₅₆	1	2	yes	—
76	C₈	—	2	4	no	—
77	C₈	C₂×Dic₂₈	1	2	yes	—
78	C₈	C₂×Dic₂₈	1	2	yes	—
79	C₇	#12522	1	1	yes	characteristic, complemented, Sylow
80	C₂²	#89467	1	1	yes	characteristic
81	C₄	—	112	4	no	—
82	C₄	#89471	1	1	yes	characteristic
83	C₄	#89471	1	1	yes	characteristic
84	C₄	#89676	1	4	yes	—
85	C₂	#11725819	1	1	yes	characteristic
86	C₂	#11726816	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	#12522
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
79. C₇	#12522	semidirect

Maximal subgroups

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

subgroup	index
#11726800	2
#11726798	2
#11725818	2
C₇×C₈×C₁₆	2
#12522	7