Subgroups — C₂×#22550256 #30649146

Order 16256.

552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₀₈	C₂	1	4	yes	maximal
2	C₂×#16861209	C₂	1	2	yes	maximal
3	#22550256	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	C₂²	1	4	yes	—
6	C₂×Dic₅₀₈	—	2	16	no	—
7	C₂×Dic₅₀₈	C₂²	1	16	yes	—
8	#16861209	—	2	4	no	—
9	#16861209	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₁₂₇	—	2	4	no	—
11	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₂₇	C₂²	1	4	yes	—
14	C₂×Dic₂₅₄	—	2	4	no	—
15	C₂×Dic₂₅₄	—	2	16	no	—
16	C₂×Dic₂₅₄	C₂³	1	8	yes	—
17	C₂²×Dic₁₂₇	—	2	4	no	—
18	Dic₅₀₈	—	2	32	no	—
19	C₄⋊Dic₁₂₇	—	2	16	no	—
20	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	D₄	1	8	yes	—
23	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
24	C₄²×C₁₂₇	D₄	1	2	yes	—
25	C₄²×C₁₂₇	C₂³	1	2	yes	—
26	Dic₂₅₄	—	2	16	no	—
27	Dic₂₅₄	—	4	8	no	—
28	C₂×Dic₁₂₇	—	2	8	no	—
29	C₂×Dic₁₂₇	—	4	8	no	—
30	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	Q₁₆	1	8	yes	complemented
32	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
34	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
35	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
36	C₈×C₁₂₇	C₂×D₄	1	8	yes	—
37	C₂²×C₁₂₇	#127	1	2	yes	—
38	C₂²×C₁₂₇	C₂×Q₁₆	1	4	yes	—
39	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
40	C₄×C₁₂₇	#127	1	2	yes	—
41	C₄×C₁₂₇	C₂×Q₁₆	1	8	yes	complemented
42	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
43	Dic₁₂₇	—	4	8	no	—
44	C₂×C₁₂₇	#494	1	4	yes	complemented
45	C₂×C₁₂₇	C₂×#127	1	1	yes	characteristic
46	C₂×C₁₂₇	C₂²×Q₁₆	1	2	yes	—
47	C₂×#494	—	127	1	no	maximal, Sylow
48	C₁₂₇	C₂×#494	1	1	yes	characteristic, complemented, Sylow
49	C₂²×Q₁₆	—	127	4	no	—
50	C₂×C₄⋊Q₈	—	127	2	no	—
51	#494	—	127	8	no	—
52	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
53	C₂²×Q₈	—	127	4	no	—
54	C₂×Q₁₆	—	127	16	no	—
55	C₂×Q₁₆	—	254	16	no	—
56	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
57	C₄⋊Q₈	—	127	8	no	—
58	C₄⋊Q₈	—	254	4	no	—
59	C₂×C₄⋊C₄	—	254	4	no	—
60	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
61	C₄×C₈	C₂×D₁₂₇	1	4	yes	—
62	C₂×Q₈	—	127	8	no	—
63	C₂×Q₈	—	254	16	no	—
64	C₂×Q₈	—	254	4	no	—
65	C₂²×C₄	—	254	4	no	—
66	C₂²×C₄	D₅₀₈	1	2	yes	—
67	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
68	Q₁₆	—	254	32	no	—
69	C₂×C₈	D₅₀₈	1	8	yes	—
70	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
71	C₄⋊C₄	—	254	16	no	—
72	C₄²	D₅₀₈	1	2	yes	—
73	C₄²	C₂²×D₁₂₇	1	2	yes	—
74	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
75	Q₈	—	508	8	no	—
76	Q₈	—	254	16	no	—
77	C₂×C₄	—	508	8	no	—
78	C₂×C₄	—	254	8	no	—
79	C₂×C₄	Dic₅₀₈	1	8	yes	complemented
80	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
81	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
82	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
83	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
84	C₈	C₂×D₅₀₈	1	8	yes	—
85	C₂²	#16861206	1	2	yes	—
86	C₂²	C₂×Dic₅₀₈	1	4	yes	—
87	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
88	C₄	—	508	8	no	—
89	C₄	#16861206	1	2	yes	—
90	C₄	C₂×Dic₅₀₈	1	8	yes	complemented
91	C₄	C₂²×D₅₀₈	1	2	yes	—
92	C₂	#22550256	1	4	yes	direct product factor
93	C₂	C₂×#16861206	1	1	yes	characteristic
94	C₂	C₂²×Dic₅₀₈	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#494
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #22550256	C₂	direct (N × H)
31. C₂×C₄×C₁₂₇	Q₁₆	semidirect
41. C₄×C₁₂₇	C₂×Q₁₆	semidirect
44. C₂×C₁₂₇	#494	semidirect
48. C₁₂₇	C₂×#494	semidirect
79. C₂×C₄	Dic₅₀₈	semidirect
90. C₄	C₂×Dic₅₀₈	semidirect
92. C₂	#22550256	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₀₈	2
C₂×#16861209	2
#22550256	2
C₂×C₄×C₈×C₁₂₇	2
C₂×#494	127

Subgroups — C2×#22550256 #30649146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550256 #30649146