Subgroups — C₂³×#14555774 #31821124

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	C₂³×D₅₅₆	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14555774	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₁₃₉	—	2	1	no	—
6	C₂²×Dic₂₇₈	—	2	7	no	—
7	C₂²×Dic₂₇₈	C₂²	1	14	yes	complemented
8	C₂²×D₅₅₆	—	2	7	no	—
9	C₂²×D₅₅₆	C₂²	1	14	yes	—
10	C₂³×Dic₁₃₉	—	2	1	no	—
11	C₂×#14555774	C₂²	1	112	yes	direct product factor
12	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃₉	C₂²	1	14	yes	complemented
14	C₂³×D₁₃₉	—	2	14	no	—
15	C₂³×D₁₃₉	—	4	8	no	—
16	C₂×Dic₂₇₈	—	2	42	no	—
17	C₂×Dic₂₇₈	C₂³	1	28	yes	complemented
18	C₂×D₅₅₆	—	2	42	no	—
19	C₂×D₅₅₆	C₂³	1	28	yes	—
20	C₂²×Dic₁₃₉	—	2	14	no	—
21	#14555774	C₂³	1	64	yes	direct product factor
22	C₂⁴×C₁₃₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃₉	D₄	1	7	yes	—
24	C₂²×C₄×C₁₃₉	C₂³	1	7	yes	—
25	C₂×C₈×C₁₃₉	C₂³	1	28	yes	complemented
26	C₂²×D₁₃₉	—	2	28	no	—
27	C₂²×D₁₃₉	—	4	56	no	—
28	Dic₂₇₈	—	2	28	no	—
29	Dic₂₇₈	C₂⁴	1	8	yes	complemented
30	D₅₅₆	—	2	28	no	—
31	D₅₅₆	C₂⁴	1	8	yes	—
32	C₂×Dic₁₃₉	—	2	28	no	—
33	C₂³×C₁₃₉	SD₁₆	1	8	yes	complemented
34	C₂³×C₁₃₉	C₂×D₄	1	7	yes	—
35	C₂×C₄×C₁₃₉	C₂×D₄	1	21	yes	—
36	C₂×C₄×C₁₃₉	C₂⁴	1	7	yes	—
37	C₈×C₁₃₉	C₂⁴	1	8	yes	complemented
38	C₂²×C₁₃₉	C₂×SD₁₆	1	28	yes	complemented
39	C₂²×C₁₃₉	C₂²×D₄	1	7	yes	—
40	C₂×D₁₃₉	—	2	8	no	—
41	C₂×D₁₃₉	—	4	56	no	—
42	C₄×C₁₃₉	C₂²×D₄	1	7	yes	—
43	C₄×C₁₃₉	C₂⁵	1	1	yes	characteristic
44	Dic₁₃₉	—	2	8	no	—
45	C₂×C₁₃₉	C₂²×SD₁₆	1	14	yes	complemented
46	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
47	D₁₃₉	—	4	8	no	—
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	—	139	1	no	—
52	C₂²×SD₁₆	—	139	28	no	—
53	C₂³×C₈	D₁₃₉	1	1	yes	characteristic, complemented
54	C₂⁵	—	278	1	no	—
55	C₂²×Q₈	—	278	7	no	—
56	C₂²×Q₈	—	139	14	no	—
57	C₂²×D₄	—	278	7	no	—
58	C₂²×D₄	—	139	14	no	—
59	C₂³×C₄	—	278	1	no	—
60	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
61	C₂×SD₁₆	—	139	112	no	—
62	C₂²×C₈	C₂×D₁₃₉	1	14	yes	complemented
63	C₂⁴	—	278	14	no	—
64	C₂⁴	—	556	8	no	—
65	C₂⁴	D₅₅₆	1	1	yes	characteristic
66	C₂×Q₈	—	278	42	no	—
67	C₂×Q₈	—	139	28	no	—
68	C₂×D₄	—	278	42	no	—
69	C₂×D₄	—	139	28	no	—
70	C₂²×C₄	—	278	14	no	—
71	C₂²×C₄	D₅₅₆	1	7	yes	—
72	C₂²×C₄	C₂²×D₁₃₉	1	7	yes	—
73	SD₁₆	—	139	64	no	—
74	C₂×C₈	C₂²×D₁₃₉	1	28	yes	complemented
75	C₂³	—	278	28	no	—
76	C₂³	—	556	56	no	—
77	C₂³	#14555774	1	8	yes	direct product factor
78	C₂³	C₂×D₅₅₆	1	7	yes	—
79	Q₈	—	278	28	no	—
80	Q₈	—	139	8	no	—
81	D₄	—	278	28	no	—
82	D₄	—	139	8	no	—
83	C₂×C₄	—	278	28	no	—
84	C₂×C₄	C₂×D₅₅₆	1	21	yes	—
85	C₂×C₄	C₂³×D₁₃₉	1	7	yes	—
86	C₈	C₂³×D₁₃₉	1	8	yes	complemented
87	C₂²	—	278	8	no	—
88	C₂²	—	556	56	no	—
89	C₂²	C₂×#14555774	1	28	yes	direct product factor
90	C₂²	C₂²×D₅₅₆	1	7	yes	—
91	C₄	—	278	8	no	—
92	C₄	C₂²×D₅₅₆	1	7	yes	—
93	C₄	C₂⁴×D₁₃₉	1	1	yes	characteristic
94	C₂	—	556	8	no	—
95	C₂	C₂²×#14555774	1	14	yes	direct product factor
96	C₂	C₂³×D₅₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₅₅₆
commutator, cosocle	C₄×C₁₃₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₃₉
Fitting	C₂³×C₈×C₁₃₉	C₂
socle	C₂⁴×C₁₃₉	D₄
prime radical	C₂³×D₅₅₆	C₂
hypercenter	C₂³×C₈	D₁₃₉
hypocenter	C₁₃₉	C₂³×SD₁₆
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₇₈	C₂	semidirect
3. C₂²×#14555774	C₂	direct (N × H)
4. C₂³×C₈×C₁₃₉	C₂	semidirect
7. C₂²×Dic₂₇₈	C₂²	semidirect
11. C₂×#14555774	C₂²	direct (N × H)
13. C₂²×C₈×C₁₃₉	C₂²	semidirect
17. C₂×Dic₂₇₈	C₂³	semidirect
21. #14555774	C₂³	direct (N × H)
25. C₂×C₈×C₁₃₉	C₂³	semidirect
29. Dic₂₇₈	C₂⁴	semidirect
33. C₂³×C₁₃₉	SD₁₆	semidirect
37. C₈×C₁₃₉	C₂⁴	semidirect
38. C₂²×C₁₃₉	C₂×SD₁₆	semidirect
45. C₂×C₁₃₉	C₂²×SD₁₆	semidirect
48. C₁₃₉	C₂³×SD₁₆	semidirect
53. C₂³×C₈	D₁₃₉	semidirect
62. C₂²×C₈	C₂×D₁₃₉	semidirect
74. C₂×C₈	C₂²×D₁₃₉	semidirect
77. C₂³	#14555774	direct (N × H)
86. C₈	C₂³×D₁₃₉	semidirect
89. C₂²	C₂×#14555774	direct (N × H)
95. C₂	C₂²×#14555774	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
C₂³×D₅₅₆	2
C₂²×#14555774	2
C₂³×C₈×C₁₃₉	2
C₂³×SD₁₆	139

Subgroups — C23×#14555774 #31821124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14555774 #31821124