Subgroups — C₂×#22550274 #30649229

Order 16256.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861209	C₂	1	2	yes	maximal
2	#22550274	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16861141	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₄	—	2	2	no	—
6	#16861209	—	2	4	no	—
7	#16861209	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
10	#16861141	C₂²	1	16	yes	—
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	8	no	—
15	C₂×Dic₂₅₄	—	4	8	no	—
16	C₂²×Dic₁₂₇	—	2	4	no	—
17	C₄⋊Dic₁₂₇	—	2	4	no	—
18	C₄⋊Dic₁₂₇	—	2	8	no	—
19	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
20	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	Q₈	1	8	yes	complemented
23	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
24	C₄²×C₁₂₇	D₄	1	2	yes	—
25	C₄²×C₁₂₇	C₂³	1	2	yes	—
26	Dic₂₅₄	—	4	16	no	—
27	C₂×Dic₁₂₇	—	2	8	no	—
28	C₂×Dic₁₂₇	—	4	8	no	—
29	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	SD₁₆	1	8	yes	—
31	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
35	C₈×C₁₂₇	C₂×Q₈	1	8	yes	complemented
36	C₂²×C₁₂₇	C₄⋊Q₈	1	2	yes	—
37	C₂²×C₁₂₇	C₂×SD₁₆	1	4	yes	—
38	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
39	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
40	C₄×C₁₂₇	C₂×SD₁₆	1	8	yes	—
41	C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
42	Dic₁₂₇	—	4	8	no	—
43	C₂×C₁₂₇	#498	1	4	yes	complemented
44	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₁₂₇	C₂²×SD₁₆	1	2	yes	—
46	C₂×#498	—	127	1	no	maximal, Sylow
47	C₁₂₇	C₂×#498	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₄⋊Q₈	—	127	2	no	—
49	#498	—	127	8	no	—
50	C₂×#106	—	127	4	no	—
51	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
52	C₂²×Q₈	—	254	2	no	—
53	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
54	C₄⋊Q₈	—	127	8	no	—
55	C₄⋊Q₈	—	254	4	no	—
56	C₂×C₄⋊C₄	—	127	4	no	—
57	C₂×C₄⋊C₄	—	254	2	no	—
58	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
59	#106	—	127	16	no	—
60	C₄×C₈	C₂×D₁₂₇	1	4	yes	—
61	C₂×Q₈	—	508	8	no	—
62	C₂×Q₈	—	254	8	no	—
63	C₂²×C₄	—	254	4	no	—
64	C₂²×C₄	D₅₀₈	1	2	yes	—
65	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
66	C₂×C₈	Dic₂₅₄	1	8	yes	complemented
67	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
68	C₄⋊C₄	—	127	8	no	—
69	C₄⋊C₄	—	254	4	no	—
70	C₄⋊C₄	—	254	8	no	—
71	C₄²	D₅₀₈	1	2	yes	—
72	C₄²	C₂²×D₁₂₇	1	2	yes	—
73	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
74	Q₈	—	508	16	no	—
75	C₂×C₄	—	508	8	no	—
76	C₂×C₄	—	254	8	no	—
77	C₂×C₄	#14523237	1	8	yes	—
78	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
79	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
80	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
81	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
82	C₈	C₂×Dic₂₅₄	1	8	yes	complemented
83	C₂²	#16861209	1	2	yes	—
84	C₂²	C₂×#14523237	1	4	yes	—
85	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
86	C₄	—	508	8	no	—
87	C₄	#16861209	1	2	yes	—
88	C₄	C₂×#14523237	1	8	yes	—
89	C₄	C₂²×Dic₂₅₄	1	2	yes	—
90	C₂	#22550274	1	4	yes	direct product factor
91	C₂	C₂×#16861209	1	1	yes	characteristic
92	C₂	C₂²×#14523237	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×#498
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
2. #22550274	C₂	direct (N × H)
22. C₂×C₈×C₁₂₇	Q₈	semidirect
35. C₈×C₁₂₇	C₂×Q₈	semidirect
43. C₂×C₁₂₇	#498	semidirect
47. C₁₂₇	C₂×#498	semidirect
66. C₂×C₈	Dic₂₅₄	semidirect
82. C₈	C₂×Dic₂₅₄	semidirect
90. C₂	#22550274	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×#16861209	2
#22550274	2
C₂×#16861141	2
C₂×C₄×C₈×C₁₂₇	2
C₂×#498	127

Subgroups — C2×#22550274 #30649229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550274 #30649229