Subgroups — C₂×#22549706 #30640522

Order 16256.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861145	C₂	1	2	yes	maximal
2	#22549706	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523232	—	2	2	no	—
5	#16861145	—	2	2	no	—
6	#16861145	C₂²	1	4	yes	—
7	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₁₂₇	C₄	1	2	yes	—
9	C₂×C₁₆×C₁₂₇	C₄	1	2	yes	—
10	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
11	#14523232	—	2	4	no	—
12	#14523232	—	4	2	no	—
13	C₂×C₁₂₇⋊C₈	—	4	2	no	—
14	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
22	C₁₂₇⋊C₈	—	4	4	no	—
23	C₂³×C₁₂₇	Q₁₆	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
30	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
31	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
32	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₁₂₇	—	2	2	no	—
35	C₂²×C₁₂₇	#107	1	1	yes	characteristic
36	C₂²×C₁₂₇	#107	1	1	yes	characteristic
37	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
38	C₄×C₁₂₇	#107	1	1	yes	characteristic
39	C₄×C₁₂₇	#107	1	1	yes	characteristic
40	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
42	C₂×C₁₂₇	—	2	1	no	—
43	C₂×C₁₂₇	—	2	1	no	—
44	C₂×C₁₂₇	#368	1	2	yes	complemented
45	C₂×C₁₂₇	C₂×#107	1	1	yes	characteristic
46	C₂×#368	—	127	1	no	maximal, Sylow
47	C₁₂₇	C₂×#368	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₁₆	D₁₂₇	1	1	yes	characteristic
49	C₂×#108	—	127	2	no	—
50	#368	—	127	4	no	—
51	C₂×M₁₆	—	254	2	no	—
52	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
53	C₂×C₁₆	Dic₁₂₇	1	2	yes	—
54	C₂×C₁₆	Dic₁₂₇	1	2	yes	—
55	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
56	#108	—	127	4	no	—
57	#108	—	254	2	no	—
58	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
59	M₁₆	—	508	2	no	—
60	M₁₆	—	254	4	no	—
61	C₂×C₈	—	508	2	no	—
62	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
63	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
64	C₂×C₈	D₅₀₈	1	1	yes	characteristic
65	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
66	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
67	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
68	C₁₆	C₂×Dic₁₂₇	1	4	yes	—
69	C₂³	Dic₅₀₈	1	1	yes	characteristic
70	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
71	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
72	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
73	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
74	C₂×C₄	Dic₅₀₈	1	1	yes	characteristic
75	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
76	C₈	—	508	4	no	—
77	C₈	C₄⋊Dic₁₂₇	1	1	yes	characteristic
78	C₈	C₄⋊Dic₁₂₇	1	1	yes	characteristic
79	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
80	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
81	C₂²	—	2	2	no	—
82	C₂²	#16861143	1	1	yes	characteristic
83	C₂²	#16861143	1	1	yes	characteristic
84	C₂²	C₂×Dic₅₀₈	1	1	yes	characteristic
85	C₄	#16861143	1	1	yes	characteristic
86	C₄	#16861143	1	1	yes	characteristic
87	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
88	C₄	C₂×D₁₀₁₆	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	2	1	no	—
91	C₂	#22549706	1	2	yes	direct product factor
92	C₂	C₂×#16861143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₀₁₆
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	C₂²×C₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#107
prime radical	C₂³×C₁₂₇	Q₁₆
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂²×C₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×#368
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22549706	C₂	direct (N × H)
44. C₂×C₁₂₇	#368	semidirect
47. C₁₂₇	C₂×#368	semidirect
91. C₂	#22549706	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16861145	2
#22549706	2
C₂²×C₁₆×C₁₂₇	2
C₂×#368	127

Subgroups — C2×#22549706 #30640522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549706 #30640522