Subgroups — #31808719

Order 17792.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581385	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₂₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#22580792	C₂	1	2	yes	maximal, complemented
4	D₄₄₄₈	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984864	—	2	1	no	—
7	C₂×D₁₁₁₂	—	2	1	no	—
8	Dic₁₁₁₂	C₂²	1	1	yes	characteristic, complemented
9	#17984787	—	2	1	no	—
10	D₂₂₂₄	—	2	1	no	—
11	D₂₂₂₄	C₂²	1	2	yes	—
12	D₂₂₂₄	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₃₉	C₂²	1	2	yes	complemented
15	#14555785	—	4	1	no	—
16	C₂×D₅₅₆	—	4	1	no	—
17	Dic₅₅₆	—	2	1	no	—
18	#14555774	—	4	1	no	—
19	D₁₁₁₂	—	2	2	no	—
20	D₁₁₁₂	—	2	1	no	—
21	D₁₁₁₂	—	4	1	no	—
22	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
23	C₁₆×C₁₃₉	D₄	1	1	yes	characteristic
24	C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
25	C₂²×D₁₃₉	—	8	1	no	—
26	Dic₂₇₈	—	4	1	no	—
27	D₅₅₆	—	4	2	no	—
28	D₅₅₆	—	4	1	no	—
29	D₅₅₆	—	8	1	no	—
30	C₂²⋊D₁₃₉	—	8	1	no	—
31	C₄×D₁₃₉	—	8	1	no	—
32	C₂×C₄×C₁₃₉	D₈	1	1	yes	characteristic
33	C₈×C₁₃₉	D₈	1	1	yes	characteristic
34	C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₃₉	D₁₆	1	1	yes	characteristic
36	C₂×D₁₃₉	—	8	1	no	—
37	C₂×D₁₃₉	—	8	2	no	—
38	C₂×D₁₃₉	—	16	2	no	—
39	C₄×C₁₃₉	D₁₆	1	1	yes	characteristic
40	C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
41	Dic₁₃₉	—	8	1	no	—
42	C₂×C₁₃₉	—	2	1	no	—
43	C₂×C₁₃₉	C₂×D₁₆	1	1	yes	characteristic
44	D₁₃₉	—	16	1	no	—
45	D₁₃₉	—	16	2	no	—
46	C₁₃₉	#2263	1	1	yes	characteristic, complemented, Sylow
47	#2263	—	139	1	no	maximal, Sylow
48	#508	—	139	1	no	—
49	C₂×D₁₆	—	139	1	no	—
50	SD₆₄	—	139	2	no	—
51	D₃₂	—	139	2	no	—
52	M₆₄	D₁₃₉	1	1	yes	characteristic, complemented
53	#135	—	278	1	no	—
54	C₂×D₈	—	278	1	no	—
55	Q₃₂	—	139	1	no	—
56	SD₃₂	—	278	1	no	—
57	D₁₆	—	278	1	no	—
58	D₁₆	—	139	2	no	—
59	D₁₆	—	139	1	no	—
60	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
61	C₃₂	C₂×D₁₃₉	1	2	yes	complemented
62	#41	—	556	1	no	—
63	C₂×D₄	—	556	1	no	—
64	Q₁₆	—	278	1	no	—
65	SD₁₆	—	556	1	no	—
66	D₈	—	278	2	no	—
67	D₈	—	278	1	no	—
68	D₈	—	556	1	no	—
69	C₂×C₈	D₅₅₆	1	1	yes	characteristic
70	C₁₆	D₅₅₆	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₃₉	1	1	yes	characteristic
72	C₂³	—	1112	1	no	—
73	Q₈	—	556	1	no	—
74	D₄	—	556	2	no	—
75	D₄	—	556	1	no	—
76	D₄	—	1112	1	no	—
77	D₄	—	1112	1	no	—
78	C₂×C₄	—	1112	1	no	—
79	C₂×C₄	D₁₁₁₂	1	1	yes	characteristic
80	C₈	D₁₁₁₂	1	1	yes	characteristic
81	C₈	C₂×D₅₅₆	1	1	yes	characteristic
82	C₂²	—	1112	1	no	—
83	C₂²	—	1112	2	no	—
84	C₂²	—	2224	2	no	—
85	C₂²	D₂₂₂₄	1	1	yes	characteristic
86	C₄	—	1112	1	no	—
87	C₄	D₂₂₂₄	1	1	yes	characteristic
88	C₄	C₂×D₁₁₁₂	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	2224	2	no	—
91	C₂	—	2224	1	no	—
92	C₂	C₂×D₂₂₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₂₂₄
commutator, cosocle	C₁₆×C₁₃₉	C₂³
Frattini	C₁₆	C₂²×D₁₃₉
Fitting	M₆₄×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂×D₁₆
hypercenter	M₆₄	D₁₃₉
hypocenter	C₁₃₉	#2263
LCS	G ▷ C₁₆×C₁₃₉ ▷ C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. #22581385	C₂	semidirect
2. C₂×D₂₂₂₄	C₂	semidirect
3. #22580792	C₂	semidirect
4. D₄₄₄₈	C₂	semidirect
5. M₆₄×C₁₃₉	C₂	semidirect
8. Dic₁₁₁₂	C₂²	semidirect
12. D₂₂₂₄	C₂²	semidirect
14. C₃₂×C₁₃₉	C₂²	semidirect
46. C₁₃₉	#2263	semidirect
52. M₆₄	D₁₃₉	semidirect
61. C₃₂	C₂×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581385	2
C₂×D₂₂₂₄	2
#22580792	2
D₄₄₄₈	2
M₆₄×C₁₃₉	2
#2263	139