Subgroups — #31808716

Order 17792.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₂₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#22580790	C₂	1	2	yes	maximal, complemented
3	#22580788	C₂	1	2	yes	maximal, complemented
4	#22580787	C₂	1	1	yes	maximal, characteristic, complemented
5	#508×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₁₂	—	2	1	no	—
7	D₂₂₂₄	—	2	1	no	—
8	D₂₂₂₄	C₂²	1	2	yes	—
9	C₁₃₉⋊C₃₂	C₂²	1	2	yes	complemented
10	#135×C₁₃₉	—	2	1	no	—
11	Q₃₂×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
12	SD₃₂×C₁₃₉	—	2	1	no	—
13	D₁₆×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×D₅₅₆	—	4	1	no	—
16	D₁₁₁₂	—	2	2	no	—
17	D₁₁₁₂	—	4	1	no	—
18	#41×C₁₃₉	—	4	1	no	—
19	Q₁₆×C₁₃₉	—	2	1	no	—
20	SD₁₆×C₁₃₉	—	4	1	no	—
21	D₈×C₁₃₉	—	2	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	D₅₅₆	—	4	2	no	—
27	D₅₅₆	—	8	1	no	—
28	Q₈×C₁₃₉	—	4	1	no	—
29	D₄×C₁₃₉	—	4	1	no	—
30	D₄×C₁₃₉	—	8	1	no	—
31	C₂×C₄×C₁₃₉	—	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	—	8	1	no	—
36	C₂²×C₁₃₉	D₁₆	1	1	yes	characteristic
37	C₂×D₁₃₉	—	8	2	no	—
38	C₂×D₁₃₉	—	16	2	no	—
39	C₄×C₁₃₉	—	8	1	no	—
40	C₄×C₁₃₉	D₁₆	1	1	yes	characteristic
41	C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
42	C₂×C₁₃₉	—	2	1	no	—
43	C₂×C₁₃₉	—	16	1	no	—
44	C₂×C₁₃₉	C₂×D₁₆	1	1	yes	characteristic
45	D₁₃₉	—	16	2	no	—
46	C₁₃₉	#2263	1	1	yes	characteristic, complemented, Sylow
47	#2263	—	139	1	no	maximal, Sylow
48	#508	D₁₃₉	1	1	yes	characteristic, complemented
49	C₂×D₁₆	—	139	1	no	—
50	SD₆₄	—	139	2	no	—
51	D₃₂	—	139	2	no	—
52	M₆₄	—	139	1	no	—
53	#135	—	2	1	no	—
54	C₂×D₈	—	278	1	no	—
55	Q₃₂	C₂×D₁₃₉	1	1	yes	characteristic, complemented
56	SD₃₂	—	2	1	no	—
57	D₁₆	—	278	1	no	—
58	D₁₆	—	139	2	no	—
59	D₁₆	C₂×D₁₃₉	1	1	yes	characteristic, complemented
60	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
61	C₃₂	—	139	2	no	—
62	#41	—	4	1	no	—
63	C₂×D₄	—	556	1	no	—
64	Q₁₆	—	2	1	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	1	no	—
67	D₈	—	278	2	no	—
68	D₈	—	556	1	no	—
69	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
70	C₁₆	C₂²⋊D₁₃₉	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₃₉	1	1	yes	characteristic
72	C₂³	—	1112	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	8	1	no	—
76	D₄	—	556	2	no	—
77	D₄	—	1112	1	no	—
78	C₂×C₄	—	8	1	no	—
79	C₂×C₄	#14555770	1	1	yes	characteristic
80	C₈	#14555770	1	1	yes	characteristic
81	C₈	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
82	C₂²	—	8	1	no	—
83	C₂²	—	1112	2	no	—
84	C₂²	—	2224	2	no	—
85	C₂²	#17984783	1	1	yes	characteristic
86	C₄	—	8	1	no	—
87	C₄	#17984783	1	1	yes	characteristic
88	C₄	C₂×#14555770	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	16	1	no	—
91	C₂	—	2224	2	no	—
92	C₂	C₂×#17984783	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#17984783
commutator, cosocle	C₁₆×C₁₃₉	C₂³
Frattini	C₁₆	C₂²×D₁₃₉
Fitting	#508×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂×D₁₆
hypercenter	#508	D₁₃₉
hypocenter	C₁₃₉	#2263
LCS	G ▷ C₁₆×C₁₃₉ ▷ C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₂₂₄	C₂	semidirect
2. #22580790	C₂	semidirect
3. #22580788	C₂	semidirect
4. #22580787	C₂	semidirect
5. #508×C₁₃₉	C₂	semidirect
9. C₁₃₉⋊C₃₂	C₂²	semidirect
11. Q₃₂×C₁₃₉	C₂²	semidirect
13. D₁₆×C₁₃₉	C₂²	semidirect
46. C₁₃₉	#2263	semidirect
48. #508	D₁₃₉	semidirect
55. Q₃₂	C₂×D₁₃₉	semidirect
59. D₁₆	C₂×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₂₂₄	2
#22580790	2
#22580788	2
#22580787	2
#508×C₁₃₉	2
#2263	139