Subgroups — C₂²×#16861155 #30651842

Order 16256.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861155	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	—	2	1	no	—
6	C₂×Dic₅₀₈	—	2	3	no	—
7	C₂×Dic₅₀₈	C₂²	1	6	yes	—
8	#16861155	C₂²	1	16	yes	direct product factor
9	C₂×C₁₂₇⋊C₁₆	C₂²	1	6	yes	—
10	C₂²×Q₈×C₁₂₇	—	2	1	no	—
11	C₂×Q₁₆×C₁₂₇	—	2	3	no	—
12	C₂×Q₁₆×C₁₂₇	C₂²	1	6	yes	—
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₅₄	—	2	6	no	—
15	C₂×Dic₂₅₄	—	4	3	no	—
16	C₂²×Dic₁₂₇	—	4	1	no	—
17	Dic₅₀₈	—	2	6	no	—
18	Dic₅₀₈	C₂³	1	4	yes	—
19	C₁₂₇⋊C₁₆	C₂³	1	4	yes	—
20	C₂×Q₈×C₁₂₇	—	2	6	no	—
21	C₂×Q₈×C₁₂₇	—	4	3	no	—
22	C₂²×C₄×C₁₂₇	—	4	1	no	—
23	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
24	Q₁₆×C₁₂₇	—	2	6	no	—
25	Q₁₆×C₁₂₇	C₂³	1	4	yes	—
26	C₂×C₈×C₁₂₇	D₄	1	3	yes	—
27	C₂×C₈×C₁₂₇	C₂³	1	3	yes	—
28	Dic₂₅₄	—	2	4	no	—
29	Dic₂₅₄	—	4	6	no	—
30	C₂×Dic₁₂₇	—	4	6	no	—
31	C₂³×C₁₂₇	D₈	1	1	yes	characteristic
32	Q₈×C₁₂₇	—	2	4	no	—
33	Q₈×C₁₂₇	—	4	6	no	—
34	C₂×C₄×C₁₂₇	—	4	6	no	—
35	C₂×C₄×C₁₂₇	D₈	1	3	yes	—
36	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
37	C₈×C₁₂₇	C₂×D₄	1	3	yes	—
38	C₈×C₁₂₇	C₂⁴	1	1	yes	characteristic
39	C₂²×C₁₂₇	Q₃₂	1	4	yes	complemented
40	C₂²×C₁₂₇	C₂×D₈	1	3	yes	—
41	C₄×C₁₂₇	—	4	4	no	—
42	C₄×C₁₂₇	C₂×D₈	1	3	yes	—
43	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
44	Dic₁₂₇	—	4	4	no	—
45	C₂×C₁₂₇	C₂×Q₃₂	1	6	yes	complemented
46	C₂×C₁₂₇	C₂²×D₈	1	1	yes	characteristic
47	C₂²×Q₃₂	—	127	1	no	maximal, Sylow
48	C₁₂₇	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
49	C₂²×Q₁₆	—	127	1	no	—
50	C₂²×Q₁₆	D₁₂₇	1	1	yes	characteristic
51	C₂×Q₃₂	—	127	12	no	—
52	C₂²×C₁₆	—	127	1	no	—
53	C₂²×Q₈	—	2	1	no	—
54	C₂²×Q₈	—	254	1	no	—
55	C₂×Q₁₆	—	2	3	no	—
56	C₂×Q₁₆	—	127	6	no	—
57	C₂×Q₁₆	—	254	3	no	—
58	C₂×Q₁₆	C₂×D₁₂₇	1	6	yes	—
59	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
60	Q₃₂	—	127	16	no	—
61	C₂×C₁₆	—	127	6	no	—
62	C₂×Q₈	—	2	6	no	—
63	C₂×Q₈	—	4	3	no	—
64	C₂×Q₈	—	508	3	no	—
65	C₂×Q₈	—	254	6	no	—
66	C₂²×C₄	—	4	1	no	—
67	C₂²×C₄	—	508	1	no	—
68	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
69	Q₁₆	—	2	6	no	—
70	Q₁₆	—	127	4	no	—
71	Q₁₆	—	254	6	no	—
72	Q₁₆	C₂²×D₁₂₇	1	4	yes	—
73	C₂×C₈	C₂²⋊D₁₂₇	1	3	yes	—
74	C₂×C₈	C₂²×D₁₂₇	1	3	yes	—
75	C₁₆	—	127	4	no	—
76	C₂³	#14523233	1	1	yes	characteristic
77	Q₈	—	2	4	no	—
78	Q₈	—	4	6	no	—
79	Q₈	—	508	6	no	—
80	Q₈	—	254	4	no	—
81	C₂×C₄	—	4	6	no	—
82	C₂×C₄	—	508	6	no	—
83	C₂×C₄	#14523233	1	3	yes	—
84	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
85	C₈	C₂×C₂²⋊D₁₂₇	1	3	yes	—
86	C₈	C₂³×D₁₂₇	1	1	yes	characteristic
87	C₂²	#16861155	1	4	yes	direct product factor
88	C₂²	C₂×#14523233	1	3	yes	—
89	C₄	—	4	4	no	—
90	C₄	—	508	4	no	—
91	C₄	C₂×#14523233	1	3	yes	—
92	C₄	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
93	C₂	C₂×#16861155	1	6	yes	direct product factor
94	C₂	C₂²×#14523233	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14523233
commutator, cosocle	C₈×C₁₂₇	C₂⁴
Frattini	C₈	C₂³×D₁₂₇
Fitting	C₂²×Q₁₆×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	D₈
hypercenter	C₂²×Q₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16861155	C₂	direct (N × H)
8. #16861155	C₂²	direct (N × H)
39. C₂²×C₁₂₇	Q₃₂	semidirect
45. C₂×C₁₂₇	C₂×Q₃₂	semidirect
48. C₁₂₇	C₂²×Q₃₂	semidirect
87. C₂²	#16861155	direct (N × H)
93. C₂	C₂×#16861155	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×#16861155	2
C₂²×C₁₂₇⋊C₁₆	2
C₂²×Q₁₆×C₁₂₇	2
C₂²×Q₃₂	127

Subgroups — C22×#16861155 #30651842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16861155 #30651842