Subgroups — #31806541

Order 17792.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22580977	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22580717	C₂	1	4	yes	maximal
4	#423×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555769	C₂²	1	2	yes	—
6	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₁₃₉⋊C₈)	—	2	2	no	—
8	C₄×C₁₃₉⋊C₈	C₄	1	2	yes	—
9	C₄×C₁₃₉⋊C₈	C₄	1	2	yes	—
10	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₃₉	—	2	2	no	—
13	#14555769	—	2	4	no	—
14	C₂×C₁₃₉⋊C₈	—	2	2	no	—
15	C₂×C₁₃₉⋊C₈	—	2	2	no	—
16	C₂×C₁₃₉⋊C₈	—	2	2	no	—
17	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₁₃₉	Q₈	1	2	yes	—
19	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
20	M₁₆×C₁₃₉	—	2	4	no	—
21	C₂×C₈×C₁₃₉	—	2	2	no	—
22	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₃₉	D₄	1	2	yes	—
25	C₁₃₉⋊C₈	—	2	4	no	—
26	C₁₃₉⋊C₈	—	4	2	no	—
27	C₂³×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₁₃₉	—	2	4	no	—
29	C₂×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₉	#41	1	2	yes	—
34	C₂×C₄×C₁₃₉	#41	1	2	yes	—
35	C₈×C₁₃₉	—	4	2	no	—
36	C₂²×C₁₃₉	—	2	2	no	—
37	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₁₃₉	#125	1	2	yes	—
39	C₄×C₁₃₉	—	2	2	no	—
40	C₄×C₁₃₉	#108	1	4	yes	—
41	C₄×C₁₃₉	#117	1	2	yes	—
42	C₄×C₁₃₉	#124	1	1	yes	characteristic
43	C₄×C₁₃₉	#124	1	1	yes	characteristic
44	C₂×C₁₃₉	—	2	2	no	—
45	C₂×C₁₃₉	#382	1	1	yes	characteristic
46	C₂×C₁₃₉	C₂×#108	1	2	yes	—
47	C₁₃₉	#1837	1	1	yes	characteristic, complemented, Sylow
48	#1837	—	139	1	no	maximal, Sylow
49	#423	—	139	1	no	—
50	#423	D₁₃₉	1	1	yes	characteristic
51	C₂×C₄×C₈	—	139	1	no	—
52	#341	—	139	4	no	—
53	C₂×M₁₆	—	139	2	no	—
54	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
55	C₂²×C₈	—	139	2	no	—
56	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	278	2	no	—
59	C₄×C₈	—	139	2	no	—
60	C₄×C₈	—	139	2	no	—
61	C₂²×C₄	Dic₂₇₈	1	2	yes	—
62	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
63	M₁₆	—	2	4	no	—
64	M₁₆	—	278	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	278	2	no	—
67	C₂×C₈	—	278	2	no	—
68	C₂×C₈	—	278	2	no	—
69	C₂×C₈	—	139	4	no	—
70	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
71	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
72	C₄²	C₂²⋊D₁₃₉	1	2	yes	—
73	C₂³	C₂×Dic₂₇₈	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	Dic₁₃₉⋊C₄	1	2	yes	—
76	C₂×C₄	Dic₁₃₉⋊C₄	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
79	C₂×C₄	#14555784	1	2	yes	—
80	C₂×C₄	#14555786	1	2	yes	—
81	C₈	—	4	2	no	—
82	C₈	—	278	4	no	—
83	C₈	—	556	2	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
86	C₂²	#17984833	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	#17984777	1	4	yes	—
89	C₄	#17984800	1	2	yes	—
90	C₄	#17984829	1	1	yes	characteristic
91	C₄	#17984831	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#22580818	1	1	yes	characteristic
94	C₂	C₂×#17984777	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#423×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#423	D₁₃₉
hypocenter	C₁₃₉	#1837
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
47. C₁₃₉	#1837	semidirect

Maximal subgroups

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

subgroup	index
#22580977	2
C₂×C₄×C₁₃₉⋊C₈	2
#22580717	2
#423×C₁₃₉	2
#1837	139