Subgroups — C₂×#22550293 #30651829

Order 16256.

368 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550293	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16861149	C₂	1	3	yes	maximal, complemented
3	C₂²×C₁₂₇⋊C₁₆	C₂	1	3	yes	maximal, complemented
4	C₂×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#16861149	C₂²	1	12	yes	complemented
6	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	3	yes	—
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	12	yes	complemented
9	C₂×#41×C₁₂₇	C₄	1	1	yes	characteristic
10	#131×C₁₂₇	C₄	1	4	yes	—
11	#131×C₁₂₇	C₂²	1	4	yes	—
12	C₂×M₁₆×C₁₂₇	C₄	1	3	yes	—
13	C₂²×C₈×C₁₂₇	C₂²	1	3	yes	—
14	C₁₂₇⋊C₁₆	C₂³	1	2	yes	—
15	C₁₂₇⋊C₁₆	C₂³	1	6	yes	—
16	#41×C₁₂₇	C₈	1	4	yes	—
17	#41×C₁₂₇	C₂×C₄	1	4	yes	—
18	C₂×Q₈×C₁₂₇	C₈	1	1	yes	characteristic
19	C₂×D₄×C₁₂₇	C₈	1	3	yes	—
20	C₂²×C₄×C₁₂₇	C₂×C₄	1	3	yes	—
21	M₁₆×C₁₂₇	C₂×C₄	1	12	yes	—
22	C₂×C₈×C₁₂₇	C₂×C₄	1	3	yes	—
23	C₂×C₈×C₁₂₇	C₂×C₄	1	6	yes	—
24	C₂×C₈×C₁₂₇	C₂³	1	6	yes	—
25	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂³×C₁₂₇	C₂×C₈	1	3	yes	—
27	Q₈×C₁₂₇	C₂×C₈	1	4	yes	—
28	D₄×C₁₂₇	C₂×C₈	1	12	yes	—
29	C₂×C₄×C₁₂₇	C₂×C₈	1	3	yes	—
30	C₂×C₄×C₁₂₇	C₂×C₈	1	6	yes	—
31	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
33	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₂₇	C₂²×C₄	1	6	yes	—
35	C₈×C₁₂₇	C₂⁴	1	1	yes	characteristic
36	C₂²×C₁₂₇	—	2	6	no	—
37	C₂²×C₁₂₇	C₂²×C₈	1	6	yes	—
38	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₁₂₇	C₂²×C₈	1	6	yes	—
41	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
42	C₂×C₁₂₇	—	2	6	no	—
43	C₂×C₁₂₇	#504	1	2	yes	complemented
44	C₂×C₁₂₇	C₂³×C₈	1	1	yes	characteristic
45	C₂×#504	—	127	1	no	maximal, Sylow
46	C₁₂₇	C₂×#504	1	1	yes	characteristic, complemented, Sylow
47	C₂×#131	D₁₂₇	1	1	yes	characteristic
48	#504	—	127	8	no	—
49	C₂×M₃₂	—	127	3	no	—
50	C₂²×C₁₆	—	127	3	no	—
51	C₂×#41	Dic₁₂₇	1	1	yes	characteristic
52	#131	Dic₁₂₇	1	4	yes	—
53	#131	C₂×D₁₂₇	1	4	yes	—
54	C₂×M₁₆	Dic₁₂₇	1	3	yes	—
55	C₂²×C₈	C₂×D₁₂₇	1	3	yes	—
56	M₃₂	—	127	12	no	—
57	C₂×C₁₆	—	127	12	no	—
58	C₂×C₁₆	—	127	1	no	—
59	C₂×C₁₆	—	127	3	no	—
60	#41	C₁₂₇⋊C₈	1	4	yes	—
61	#41	C₂×Dic₁₂₇	1	4	yes	—
62	C₂×Q₈	C₁₂₇⋊C₈	1	1	yes	characteristic
63	C₂×D₄	C₁₂₇⋊C₈	1	3	yes	—
64	C₂²×C₄	C₂×Dic₁₂₇	1	3	yes	—
65	M₁₆	C₂×Dic₁₂₇	1	12	yes	—
66	C₂×C₈	C₂×Dic₁₂₇	1	3	yes	—
67	C₂×C₈	C₂×Dic₁₂₇	1	6	yes	—
68	C₂×C₈	C₂²×D₁₂₇	1	6	yes	—
69	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
70	C₁₆	—	127	2	no	—
71	C₁₆	—	127	6	no	—
72	C₂³	C₂×C₁₂₇⋊C₈	1	3	yes	—
73	Q₈	C₂×C₁₂₇⋊C₈	1	4	yes	—
74	D₄	C₂×C₁₂₇⋊C₈	1	12	yes	—
75	C₂×C₄	C₂×C₁₂₇⋊C₈	1	3	yes	—
76	C₂×C₄	C₂×C₁₂₇⋊C₈	1	6	yes	—
77	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
78	C₂×C₄	C₂²×Dic₁₂₇	1	6	yes	—
79	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
80	C₈	C₂²×Dic₁₂₇	1	6	yes	—
81	C₈	C₂³×D₁₂₇	1	1	yes	characteristic
82	C₂²	—	2	6	no	—
83	C₂²	C₂²×C₁₂₇⋊C₈	1	6	yes	—
84	C₂²	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
85	C₄	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
86	C₄	C₂²×C₁₂₇⋊C₈	1	6	yes	—
87	C₄	C₂³×Dic₁₂₇	1	1	yes	characteristic
88	C₂	—	2	6	no	—
89	C₂	#22550293	1	2	yes	direct product factor
90	C₂	C₂³×C₁₂₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₈
Frattini	C₈	C₂³×D₁₂₇
Fitting	C₂×#131×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×C₈
prime radical	C₂×#41×C₁₂₇	C₄
cosocle	C₈×C₁₂₇	C₂⁴
hypercenter	C₂×#131	D₁₂₇
hypocenter	C₁₂₇	C₂×#504
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #22550293	C₂	direct (N × H)
2. C₂×#16861149	C₂	semidirect
3. C₂²×C₁₂₇⋊C₁₆	C₂	semidirect
5. #16861149	C₂²	semidirect
8. C₂×C₁₂₇⋊C₁₆	C₂²	semidirect
43. C₂×C₁₂₇	#504	semidirect
46. C₁₂₇	C₂×#504	semidirect
89. C₂	#22550293	direct (N × H)

Maximal subgroups

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

subgroup	index
#22550293	2
C₂×#16861149	2
C₂²×C₁₂₇⋊C₁₆	2
C₂×#131×C₁₂₇	2
C₂×#504	127

Subgroups — C2×#22550293 #30651829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550293 #30651829