Subgroups — C₂³×#14555773 #31821123

Order 17792.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14555773	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₈	—	2	7	no	—
6	C₂²×Dic₂₇₈	C₂²	1	14	yes	complemented
7	C₂³×Dic₁₃₉	—	2	1	no	—
8	C₂×#14555773	C₂²	1	112	yes	direct product factor
9	C₂²×C₁₃₉⋊C₈	C₂²	1	14	yes	complemented
10	C₂⁵×C₁₃₉	—	2	1	no	—
11	C₂²×D₄×C₁₃₉	—	2	7	no	—
12	C₂²×D₄×C₁₃₉	C₂²	1	14	yes	—
13	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₇₈	—	2	42	no	—
15	C₂×Dic₂₇₈	C₂³	1	28	yes	complemented
16	C₂²×Dic₁₃₉	—	2	14	no	—
17	#14555773	C₂³	1	64	yes	direct product factor
18	C₂×C₁₃₉⋊C₈	C₂³	1	28	yes	complemented
19	C₂⁴×C₁₃₉	—	2	14	no	—
20	C₂⁴×C₁₃₉	—	4	8	no	—
21	C₂⁴×C₁₃₉	D₄	1	1	yes	characteristic
22	C₂×D₄×C₁₃₉	—	2	42	no	—
23	C₂×D₄×C₁₃₉	C₂³	1	28	yes	—
24	C₂²×C₄×C₁₃₉	D₄	1	7	yes	—
25	C₂²×C₄×C₁₃₉	C₂³	1	7	yes	—
26	Dic₂₇₈	—	2	28	no	—
27	Dic₂₇₈	C₂⁴	1	8	yes	complemented
28	C₂×Dic₁₃₉	—	2	28	no	—
29	C₁₃₉⋊C₈	C₂⁴	1	8	yes	complemented
30	C₂³×C₁₃₉	—	2	28	no	—
31	C₂³×C₁₃₉	—	4	56	no	—
32	C₂³×C₁₃₉	SD₁₆	1	8	yes	complemented
33	C₂³×C₁₃₉	C₂×D₄	1	7	yes	—
34	D₄×C₁₃₉	—	2	28	no	—
35	D₄×C₁₃₉	C₂⁴	1	8	yes	—
36	C₂×C₄×C₁₃₉	C₂×D₄	1	21	yes	—
37	C₂×C₄×C₁₃₉	C₂⁴	1	7	yes	—
38	C₂²×C₁₃₉	—	2	8	no	—
39	C₂²×C₁₃₉	—	4	56	no	—
40	C₂²×C₁₃₉	C₂×SD₁₆	1	28	yes	complemented
41	C₂²×C₁₃₉	C₂²×D₄	1	7	yes	—
42	C₄×C₁₃₉	C₂²×D₄	1	7	yes	—
43	C₄×C₁₃₉	C₂⁵	1	1	yes	characteristic
44	Dic₁₃₉	—	2	8	no	—
45	C₂×C₁₃₉	—	4	8	no	—
46	C₂×C₁₃₉	C₂²×SD₁₆	1	14	yes	complemented
47	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
48	C₁₃₉	C₂³×SD₁₆	1	1	yes	characteristic, complemented, Sylow
49	C₂³×SD₁₆	—	139	1	no	maximal, Sylow
50	C₂³×Q₈	—	139	1	no	—
51	C₂³×D₄	D₁₃₉	1	1	yes	characteristic
52	C₂²×SD₁₆	—	139	28	no	—
53	C₂³×C₈	—	139	1	no	—
54	C₂⁵	—	2	1	no	—
55	C₂²×Q₈	—	278	7	no	—
56	C₂²×Q₈	—	139	14	no	—
57	C₂²×D₄	—	2	7	no	—
58	C₂²×D₄	C₂×D₁₃₉	1	14	yes	—
59	C₂³×C₄	—	278	1	no	—
60	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
61	C₂×SD₁₆	—	139	112	no	—
62	C₂²×C₈	—	139	14	no	—
63	C₂⁴	—	2	14	no	—
64	C₂⁴	—	4	8	no	—
65	C₂⁴	C₂²⋊D₁₃₉	1	1	yes	characteristic
66	C₂×Q₈	—	278	42	no	—
67	C₂×Q₈	—	139	28	no	—
68	C₂×D₄	—	2	42	no	—
69	C₂×D₄	C₂²×D₁₃₉	1	28	yes	—
70	C₂²×C₄	—	278	14	no	—
71	C₂²×C₄	C₂²⋊D₁₃₉	1	7	yes	—
72	C₂²×C₄	C₂²×D₁₃₉	1	7	yes	—
73	SD₁₆	—	139	64	no	—
74	C₂×C₈	—	139	28	no	—
75	C₂³	—	2	28	no	—
76	C₂³	—	4	56	no	—
77	C₂³	#14555773	1	8	yes	direct product factor
78	C₂³	C₂×C₂²⋊D₁₃₉	1	7	yes	—
79	Q₈	—	278	28	no	—
80	Q₈	—	139	8	no	—
81	D₄	—	2	28	no	—
82	D₄	C₂³×D₁₃₉	1	8	yes	—
83	C₂×C₄	—	278	28	no	—
84	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	21	yes	—
85	C₂×C₄	C₂³×D₁₃₉	1	7	yes	—
86	C₈	—	139	8	no	—
87	C₂²	—	2	8	no	—
88	C₂²	—	4	56	no	—
89	C₂²	C₂×#14555773	1	28	yes	direct product factor
90	C₂²	C₂²×C₂²⋊D₁₃₉	1	7	yes	—
91	C₄	—	278	8	no	—
92	C₄	C₂²×C₂²⋊D₁₃₉	1	7	yes	—
93	C₄	C₂⁴×D₁₃₉	1	1	yes	characteristic
94	C₂	—	4	8	no	—
95	C₂	C₂²×#14555773	1	14	yes	direct product factor
96	C₂	C₂³×C₂²⋊D₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₁₃₉
commutator, cosocle	C₄×C₁₃₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₃₉
Fitting, prime radical	C₂³×D₄×C₁₃₉	C₂
socle	C₂⁴×C₁₃₉	D₄
hypercenter	C₂³×D₄	D₁₃₉
hypocenter	C₁₃₉	C₂³×SD₁₆
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₇₈	C₂	semidirect
2. C₂²×#14555773	C₂	direct (N × H)
3. C₂³×C₁₃₉⋊C₈	C₂	semidirect
6. C₂²×Dic₂₇₈	C₂²	semidirect
8. C₂×#14555773	C₂²	direct (N × H)
9. C₂²×C₁₃₉⋊C₈	C₂²	semidirect
15. C₂×Dic₂₇₈	C₂³	semidirect
17. #14555773	C₂³	direct (N × H)
18. C₂×C₁₃₉⋊C₈	C₂³	semidirect
27. Dic₂₇₈	C₂⁴	semidirect
29. C₁₃₉⋊C₈	C₂⁴	semidirect
32. C₂³×C₁₃₉	SD₁₆	semidirect
40. C₂²×C₁₃₉	C₂×SD₁₆	semidirect
46. C₂×C₁₃₉	C₂²×SD₁₆	semidirect
48. C₁₃₉	C₂³×SD₁₆	semidirect
77. C₂³	#14555773	direct (N × H)
89. C₂²	C₂×#14555773	direct (N × H)
95. C₂	C₂²×#14555773	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₇₈	2
C₂²×#14555773	2
C₂³×C₁₃₉⋊C₈	2
C₂³×D₄×C₁₃₉	2
C₂³×SD₁₆	139

Subgroups — C23×#14555773 #31821123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14555773 #31821123