Subgroups — #31807509

Order 17792.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581753	C₂	1	3	yes	maximal, complemented
2	#22580921	C₂	1	3	yes	maximal, complemented
3	#338×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14555785	C₂²	1	3	yes	—
5	#17984875	—	2	6	no	—
6	#17984870	—	2	6	no	—
7	#17984864	—	2	6	no	—
8	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	(C₂×D₁₃₉)⋊C₈	—	2	6	no	—
10	C₂×M₁₆×C₁₃₉	C₂²	1	3	yes	—
11	#14555785	—	2	6	no	—
12	#14555785	—	2	6	no	—
13	C₂×Dic₂₇₈	D₄	1	3	yes	—
14	C₂×D₅₅₆	D₄	1	3	yes	—
15	C₂×C₂²⋊D₁₃₉	—	2	3	no	—
16	C₂×C₄×D₁₃₉	—	2	3	no	—
17	Dic₅₅₆	—	2	6	no	—
18	#14555774	—	2	12	no	—
19	D₁₁₁₂	—	2	6	no	—
20	C₂×C₁₃₉⋊C₈	—	2	3	no	—
21	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₃₉	—	2	6	no	—
23	C₂×C₈×C₁₃₉	D₄	1	6	yes	—
24	C₂²×D₁₃₉	—	2	3	no	—
25	Dic₂₇₈	—	2	6	no	—
26	D₅₅₆	—	2	6	no	—
27	C₂²⋊D₁₃₉	—	4	6	no	—
28	C₂×Dic₁₃₉	—	2	3	no	—
29	C₄×D₁₃₉	—	4	3	no	—
30	C₄×D₁₃₉	—	4	3	no	—
31	C₁₃₉⋊C₈	—	4	1	no	—
32	C₂³×C₁₃₉	#41	1	1	yes	characteristic
33	C₂×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
35	C₈×C₁₃₉	—	2	6	no	—
36	C₂²×C₁₃₉	—	4	1	no	—
37	C₂²×C₁₃₉	#121	1	3	yes	—
38	C₂×D₁₃₉	—	4	6	no	—
39	C₂×D₁₃₉	—	4	3	no	—
40	C₄×C₁₃₉	#120	1	3	yes	—
41	C₄×C₁₃₉	#127	1	1	yes	characteristic
42	Dic₁₃₉	—	4	3	no	—
43	C₂×C₁₃₉	—	2	3	no	—
44	C₂×C₁₃₉	#392	1	1	yes	characteristic
45	D₁₃₉	—	8	3	no	—
46	C₁₃₉	#2017	1	1	yes	characteristic, complemented, Sylow
47	#2017	—	139	1	no	maximal, Sylow
48	#575	—	139	3	no	—
49	#408	—	139	3	no	—
50	#338	D₁₃₉	1	1	yes	characteristic, complemented
51	C₂×#41	—	139	3	no	—
52	#137	—	278	6	no	—
53	AGL(1,ℤ/8)	—	278	6	no	—
54	#135	—	278	6	no	—
55	C₂×M₁₆	C₂×D₁₃₉	1	3	yes	—
56	C₂²×C₈	—	139	1	no	—
57	C₂²⋊C₈	—	278	6	no	—
58	#41	—	278	6	no	—
59	#41	—	278	6	no	—
60	C₂×Q₈	—	139	3	no	—
61	C₂×D₄	—	278	3	no	—
62	C₂×D₄	—	139	3	no	—
63	C₂²×C₄	—	278	3	no	—
64	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
65	Q₁₆	—	278	6	no	—
66	SD₁₆	—	278	12	no	—
67	D₈	—	278	6	no	—
68	M₁₆	—	2	6	no	—
69	C₂×C₈	—	278	3	no	—
70	C₂×C₈	D₅₅₆	1	6	yes	—
71	C₂³	—	278	3	no	—
72	C₂³	#14555786	1	1	yes	characteristic
73	Q₈	—	278	6	no	—
74	D₄	—	278	6	no	—
75	D₄	—	556	6	no	—
76	C₂×C₄	—	278	3	no	—
77	C₂×C₄	—	556	3	no	—
78	C₂×C₄	—	556	3	no	—
79	C₂×C₄	D₄×D₁₃₉	1	3	yes	—
80	C₂×C₄	C₂×D₅₅₆	1	3	yes	—
81	C₈	—	2	6	no	—
82	C₈	—	556	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	556	6	no	—
85	C₂²	—	556	3	no	—
86	C₂²	#17984817	1	3	yes	—
87	C₄	—	556	3	no	—
88	C₄	#17984812	1	3	yes	—
89	C₄	#17984839	1	1	yes	characteristic
90	C₂	—	2	3	no	—
91	C₂	—	1112	3	no	—
92	C₂	#22580863	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984839
commutator, cosocle	C₂²×C₄×C₁₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#338×C₁₃₉	C₂
socle	C₂×C₁₃₉	#392
hypercenter	#338	D₁₃₉
hypocenter	C₁₃₉	#2017
LCS	G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #22581753	C₂	semidirect
2. #22580921	C₂	semidirect
3. #338×C₁₃₉	C₂	semidirect
46. C₁₃₉	#2017	semidirect
50. #338	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581753	2
#22580921	2
#338×C₁₃₉	2
#2017	139