Subgroups — C₂³×#14523235 #30653448

Order 16256.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅₀₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523235	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₁₂₇	—	2	1	no	—
6	C₂²×D₅₀₈	—	2	7	no	—
7	C₂²×D₅₀₈	C₂²	1	14	yes	—
8	C₂×#14523235	C₂²	1	112	yes	direct product factor
9	C₂²×C₁₂₇⋊C₈	C₂²	1	14	yes	complemented
10	C₂²×Q₈×C₁₂₇	—	2	7	no	—
11	C₂²×Q₈×C₁₂₇	C₂²	1	14	yes	complemented
12	C₂³×C₄×C₁₂₇	—	2	1	no	—
13	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂³×D₁₂₇	—	2	14	no	—
15	C₂³×D₁₂₇	—	4	8	no	—
16	C₂×D₅₀₈	—	2	42	no	—
17	C₂×D₅₀₈	C₂³	1	28	yes	—
18	#14523235	C₂³	1	64	yes	direct product factor
19	C₂×C₁₂₇⋊C₈	C₂³	1	28	yes	complemented
20	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂×Q₈×C₁₂₇	—	2	42	no	—
22	C₂×Q₈×C₁₂₇	C₂³	1	28	yes	complemented
23	C₂²×C₄×C₁₂₇	—	2	14	no	—
24	C₂²×C₄×C₁₂₇	D₄	1	7	yes	—
25	C₂²×C₄×C₁₂₇	C₂³	1	7	yes	—
26	C₂²×D₁₂₇	—	2	28	no	—
27	C₂²×D₁₂₇	—	4	56	no	—
28	D₅₀₈	—	2	28	no	—
29	D₅₀₈	C₂⁴	1	8	yes	—
30	C₁₂₇⋊C₈	C₂⁴	1	8	yes	complemented
31	C₂³×C₁₂₇	SD₁₆	1	8	yes	complemented
32	C₂³×C₁₂₇	C₂×D₄	1	7	yes	—
33	Q₈×C₁₂₇	—	2	28	no	—
34	Q₈×C₁₂₇	C₂⁴	1	8	yes	complemented
35	C₂×C₄×C₁₂₇	—	2	28	no	—
36	C₂×C₄×C₁₂₇	C₂×D₄	1	21	yes	—
37	C₂×C₄×C₁₂₇	C₂⁴	1	7	yes	—
38	C₂²×C₁₂₇	C₂×SD₁₆	1	28	yes	complemented
39	C₂²×C₁₂₇	C₂²×D₄	1	7	yes	—
40	C₂×D₁₂₇	—	2	8	no	—
41	C₂×D₁₂₇	—	4	56	no	—
42	C₄×C₁₂₇	—	2	8	no	—
43	C₄×C₁₂₇	C₂²×D₄	1	7	yes	—
44	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
45	C₂×C₁₂₇	C₂²×SD₁₆	1	14	yes	complemented
46	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
47	D₁₂₇	—	4	8	no	—
48	C₂³×SD₁₆	—	127	1	no	maximal, Sylow
49	C₁₂₇	C₂³×SD₁₆	1	1	yes	characteristic, complemented, Sylow
50	C₂³×Q₈	D₁₂₇	1	1	yes	characteristic, complemented
51	C₂³×D₄	—	127	1	no	—
52	C₂²×SD₁₆	—	127	28	no	—
53	C₂³×C₈	—	127	1	no	—
54	C₂⁵	—	254	1	no	—
55	C₂²×Q₈	—	2	7	no	—
56	C₂²×Q₈	C₂×D₁₂₇	1	14	yes	complemented
57	C₂²×D₄	—	127	14	no	—
58	C₂²×D₄	—	254	7	no	—
59	C₂³×C₄	—	2	1	no	—
60	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂×SD₁₆	—	127	112	no	—
62	C₂²×C₈	—	127	14	no	—
63	C₂⁴	—	508	8	no	—
64	C₂⁴	—	254	14	no	—
65	C₂⁴	C₂²⋊D₁₂₇	1	1	yes	characteristic
66	C₂×Q₈	—	2	42	no	—
67	C₂×Q₈	C₂²×D₁₂₇	1	28	yes	complemented
68	C₂×D₄	—	127	28	no	—
69	C₂×D₄	—	254	42	no	—
70	C₂²×C₄	—	2	14	no	—
71	C₂²×C₄	C₂²⋊D₁₂₇	1	7	yes	—
72	C₂²×C₄	C₂²×D₁₂₇	1	7	yes	—
73	SD₁₆	—	127	64	no	—
74	C₂×C₈	—	127	28	no	—
75	C₂³	—	508	56	no	—
76	C₂³	—	254	28	no	—
77	C₂³	#14523235	1	8	yes	direct product factor
78	C₂³	C₂×C₂²⋊D₁₂₇	1	7	yes	—
79	Q₈	—	2	28	no	—
80	Q₈	C₂³×D₁₂₇	1	8	yes	complemented
81	D₄	—	127	8	no	—
82	D₄	—	254	28	no	—
83	C₂×C₄	—	2	28	no	—
84	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	21	yes	—
85	C₂×C₄	C₂³×D₁₂₇	1	7	yes	—
86	C₈	—	127	8	no	—
87	C₂²	—	508	56	no	—
88	C₂²	—	254	8	no	—
89	C₂²	C₂×#14523235	1	28	yes	direct product factor
90	C₂²	C₂²×C₂²⋊D₁₂₇	1	7	yes	—
91	C₄	—	2	8	no	—
92	C₄	C₂²×C₂²⋊D₁₂₇	1	7	yes	—
93	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
94	C₂	—	508	8	no	—
95	C₂	C₂²×#14523235	1	14	yes	direct product factor
96	C₂	C₂³×C₂²⋊D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₁₂₇
commutator, cosocle	C₄×C₁₂₇	C₂⁵
Frattini	C₄	C₂⁴×D₁₂₇
Fitting	C₂³×Q₈×C₁₂₇	C₂
socle	C₂⁴×C₁₂₇	D₄
prime radical	C₂³×D₅₀₈	C₂
hypercenter	C₂³×Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂³×SD₁₆
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#14523235	C₂	direct (N × H)
3. C₂³×C₁₂₇⋊C₈	C₂	semidirect
4. C₂³×Q₈×C₁₂₇	C₂	semidirect
8. C₂×#14523235	C₂²	direct (N × H)
9. C₂²×C₁₂₇⋊C₈	C₂²	semidirect
11. C₂²×Q₈×C₁₂₇	C₂²	semidirect
18. #14523235	C₂³	direct (N × H)
19. C₂×C₁₂₇⋊C₈	C₂³	semidirect
22. C₂×Q₈×C₁₂₇	C₂³	semidirect
30. C₁₂₇⋊C₈	C₂⁴	semidirect
31. C₂³×C₁₂₇	SD₁₆	semidirect
34. Q₈×C₁₂₇	C₂⁴	semidirect
38. C₂²×C₁₂₇	C₂×SD₁₆	semidirect
45. C₂×C₁₂₇	C₂²×SD₁₆	semidirect
49. C₁₂₇	C₂³×SD₁₆	semidirect
50. C₂³×Q₈	D₁₂₇	semidirect
56. C₂²×Q₈	C₂×D₁₂₇	semidirect
67. C₂×Q₈	C₂²×D₁₂₇	semidirect
77. C₂³	#14523235	direct (N × H)
80. Q₈	C₂³×D₁₂₇	semidirect
89. C₂²	C₂×#14523235	direct (N × H)
95. C₂	C₂²×#14523235	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×D₅₀₈	2
C₂²×#14523235	2
C₂³×C₁₂₇⋊C₈	2
C₂³×Q₈×C₁₂₇	2
C₂³×SD₁₆	127

Subgroups — C23×#14523235 #30653448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523235 #30653448