Subgroups — #31806005

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	#22581326	C₂	1	1	yes	maximal, characteristic
3	#22581304	C₂	1	2	yes	maximal, complemented
4	#22581300	C₂	1	2	yes	maximal
5	C₈²×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555774	—	2	2	no	—
7	#17984842	C₂²	1	2	yes	—
8	#17984842	C₂²	1	1	yes	characteristic
9	#17984839	C₂²	1	1	yes	characteristic
10	#17984774	—	2	2	no	—
11	#17984767	—	2	4	no	—
12	#17984764	—	2	4	no	—
13	C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₈	—	2	1	no	—
16	C₂×Dic₂₇₈	—	2	2	no	—
17	C₂×D₅₅₆	—	2	1	no	—
18	C₂×D₅₅₆	—	2	2	no	—
19	#14555774	—	4	4	no	—
20	C₄⋊Dic₁₃₉	—	2	2	no	—
21	C₄⋊Dic₁₃₉	—	2	2	no	—
22	C₄⋊Dic₁₃₉	—	2	2	no	—
23	C₂×C₈×C₁₃₉	D₄	1	4	yes	—
24	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
25	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
26	C₂²×D₁₃₉	—	4	1	no	—
27	Dic₂₇₈	—	4	2	no	—
28	Dic₂₇₈	—	4	4	no	—
29	D₅₅₆	—	4	2	no	—
30	D₅₅₆	—	4	4	no	—
31	C₂×Dic₁₃₉	—	4	1	no	—
32	C₂×Dic₁₃₉	—	4	2	no	—
33	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
35	C₈×C₁₃₉	—	2	4	no	—
36	C₈×C₁₃₉	SD₁₆	1	4	yes	—
37	C₂²×C₁₃₉	#127	1	1	yes	characteristic
38	C₂×D₁₃₉	—	8	1	no	—
39	C₂×D₁₃₉	—	8	2	no	—
40	C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
41	C₄×C₁₃₉	#135	1	4	yes	—
42	Dic₁₃₉	—	8	1	no	—
43	Dic₁₃₉	—	8	2	no	—
44	C₂×C₁₃₉	#492	1	1	yes	characteristic
45	C₂×C₁₃₉	#495	1	2	yes	—
46	D₁₃₉	—	16	1	no	—
47	C₁₃₉	#1708	1	1	yes	characteristic, complemented, Sylow
48	#1708	—	139	1	no	maximal, Sylow
49	#498	—	139	1	no	—
50	#492	—	139	1	no	—
51	#487	—	139	2	no	—
52	#486	—	139	2	no	—
53	C₈²	D₁₃₉	1	1	yes	characteristic, complemented
54	C₂×SD₁₆	—	278	2	no	—
55	C₄⋊Q₈	—	139	2	no	—
56	C₄⋊Q₈	—	139	1	no	—
57	#127	—	139	1	no	—
58	#106	—	278	2	no	—
59	#103	—	278	4	no	—
60	#102	—	278	4	no	—
61	C₄×C₈	C₂×D₁₃₉	1	2	yes	—
62	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
63	C₂×Q₈	—	278	2	no	—
64	C₂×Q₈	—	278	1	no	—
65	C₂×D₄	—	278	2	no	—
66	C₂×D₄	—	278	1	no	—
67	SD₁₆	—	556	4	no	—
68	C₂×C₈	D₅₅₆	1	4	yes	—
69	C₂×C₈	D₅₅₆	1	2	yes	—
70	C₄⋊C₄	—	278	2	no	—
71	C₄⋊C₄	—	278	2	no	—
72	C₄⋊C₄	—	278	2	no	—
73	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
74	C₂³	—	556	1	no	—
75	Q₈	—	556	2	no	—
76	Q₈	—	556	4	no	—
77	D₄	—	556	2	no	—
78	D₄	—	556	4	no	—
79	C₂×C₄	—	556	1	no	—
80	C₂×C₄	—	556	2	no	—
81	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
82	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	#14555774	1	4	yes	—
85	C₂²	—	1112	1	no	—
86	C₂²	—	1112	2	no	—
87	C₂²	#17984839	1	1	yes	characteristic
88	C₄	—	1112	1	no	—
89	C₄	—	1112	2	no	—
90	C₄	C₂×#14555774	1	2	yes	—
91	C₄	#17984864	1	4	yes	—
92	C₂	—	2224	1	no	—
93	C₂	#22581326	1	1	yes	characteristic
94	C₂	#22581337	1	2	yes	—

	subgroup	G/H
center	C₂²	#17984839
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	C₈²×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#127
prime radical	#17984839	C₂²
hypercenter	C₈²	D₁₃₉
hypocenter	C₁₃₉	#1708
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #22581351	C₂	semidirect
3. #22581304	C₂	semidirect
5. C₈²×C₁₃₉	C₂	semidirect
47. C₁₃₉	#1708	semidirect
53. C₈²	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581351	2
#22581326	2
#22581304	2
#22581300	2
C₈²×C₁₃₉	2
#1708	139