Subgroups — #31806003

Order 17792.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581351	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581305	C₂	1	2	yes	maximal, complemented
3	#22581299	C₂	1	2	yes	maximal
4	C₈×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	#17984842	C₂²	1	2	yes	—
7	#17984774	—	2	2	no	—
8	#17984765	—	2	4	no	—
9	#17984763	—	2	4	no	—
10	C₄×C₁₃₉⋊C₈	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₃₉	—	2	2	no	—
12	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	#127×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₈	—	2	2	no	—
16	C₂×C₁₃₉⋊C₈	D₄	1	4	yes	—
17	C₄⋊Dic₁₃₉	—	2	2	no	—
18	C₄⋊Dic₁₃₉	—	2	2	no	—
19	C₂×Q₈×C₁₃₉	—	2	1	no	—
20	C₂×D₄×C₁₃₉	—	2	1	no	—
21	C₂×D₄×C₁₃₉	—	2	2	no	—
22	SD₁₆×C₁₃₉	—	4	4	no	—
23	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₃₉	—	2	2	no	—
25	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
26	Dic₂₇₈	—	4	4	no	—
27	C₂×Dic₁₃₉	—	4	2	no	—
28	C₁₃₉⋊C₈	—	2	4	no	—
29	C₂³×C₁₃₉	—	4	1	no	—
30	Q₈×C₁₃₉	—	4	2	no	—
31	D₄×C₁₃₉	—	4	2	no	—
32	D₄×C₁₃₉	—	4	4	no	—
33	C₂×C₄×C₁₃₉	—	4	1	no	—
34	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₃₉	SD₁₆	1	4	yes	—
37	C₂²×C₁₃₉	—	8	1	no	—
38	C₂²×C₁₃₉	—	8	2	no	—
39	C₂²×C₁₃₉	#127	1	1	yes	characteristic
40	C₄×C₁₃₉	—	8	1	no	—
41	C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
42	C₄×C₁₃₉	#135	1	4	yes	—
43	Dic₁₃₉	—	8	2	no	—
44	C₂×C₁₃₉	—	16	1	no	—
45	C₂×C₁₃₉	#492	1	1	yes	characteristic
46	C₂×C₁₃₉	#495	1	2	yes	—
47	C₁₃₉	#1708	1	1	yes	characteristic, complemented, Sylow
48	#1708	—	139	1	no	maximal, Sylow
49	#498	—	139	1	no	—
50	#492	D₁₃₉	1	1	yes	characteristic
51	#487	—	139	2	no	—
52	#486	—	139	2	no	—
53	C₈²	—	139	1	no	—
54	C₂×SD₁₆	—	2	2	no	—
55	C₄⋊Q₈	—	139	2	no	—
56	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
57	#127	C₂×D₁₃₉	1	1	yes	characteristic
58	#106	—	278	2	no	—
59	#103	—	278	4	no	—
60	#102	—	278	4	no	—
61	C₄×C₈	—	139	2	no	—
62	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
63	C₂×Q₈	—	2	1	no	—
64	C₂×Q₈	—	278	2	no	—
65	C₂×D₄	—	2	2	no	—
66	C₂×D₄	—	2	1	no	—
67	SD₁₆	—	4	4	no	—
68	C₂×C₈	—	139	4	no	—
69	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	278	2	no	—
72	C₄⋊C₄	—	278	2	no	—
73	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
74	C₂³	—	4	1	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	556	4	no	—
77	D₄	—	4	2	no	—
78	D₄	—	4	4	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	556	2	no	—
81	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
83	C₈	—	278	4	no	—
84	C₈	#14555773	1	4	yes	—
85	C₂²	—	8	1	no	—
86	C₂²	—	8	2	no	—
87	C₂²	#17984838	1	1	yes	characteristic
88	C₄	—	8	1	no	—
89	C₄	—	1112	2	no	—
90	C₄	C₂×#14555773	1	2	yes	—
91	C₄	#17984863	1	4	yes	—
92	C₂	—	16	1	no	—
93	C₂	#22581325	1	1	yes	characteristic
94	C₂	#22581334	1	2	yes	—

	subgroup	G/H
center	C₂²	#17984838
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#492×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#127
prime radical	#127×C₁₃₉	C₂²
hypercenter	#492	D₁₃₉
hypocenter	C₁₃₉	#1708
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. #22581351	C₂	semidirect
2. #22581305	C₂	semidirect
4. C₈×C₁₃₉⋊C₈	C₂	semidirect
47. C₁₃₉	#1708	semidirect

Maximal subgroups

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

subgroup	index
#22581351	2
#22581305	2
#22581299	2
C₈×C₁₃₉⋊C₈	2
#492×C₁₃₉	2
#1708	139