Subgroups — C₂×#22550354 #30651921

Order 16256.

1488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550354	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×Dic₁₂₇	C₂	1	8	yes	maximal, complemented
3	C₂×#16861168	C₂	1	6	yes	maximal
4	C₂³×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	Q₈×Dic₁₂₇	C₂²	1	64	yes	complemented
6	#16861168	C₂²	1	48	yes	—
7	C₂×C₄⋊Dic₁₂₇	C₂²	1	12	yes	—
8	C₂×C₂²⋊Dic₁₂₇	C₂²	1	4	yes	—
9	C₂×C₄×Dic₁₂₇	C₂²	1	12	yes	—
10	C₂²×Q₈×C₁₂₇	C₄	1	16	yes	complemented
11	C₂²×Q₈×C₁₂₇	C₂²	1	4	yes	—
12	C₂²×Q₈×C₁₂₇	C₂²	1	8	yes	—
13	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
14	C₂²×Dic₁₂₇	C₂³	1	8	yes	—
15	C₄⋊Dic₁₂₇	C₂³	1	48	yes	—
16	C₂²⋊Dic₁₂₇	C₂³	1	16	yes	—
17	C₄×Dic₁₂₇	C₂³	1	48	yes	—
18	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₂×Q₈×C₁₂₇	C₂×C₄	1	32	yes	—
20	C₂×Q₈×C₁₂₇	C₂×C₄	1	64	yes	complemented
21	C₂×Q₈×C₁₂₇	C₂³	1	16	yes	—
22	C₂²×C₄×C₁₂₇	C₂×C₄	1	24	yes	—
23	C₂²×C₄×C₁₂₇	C₂³	1	6	yes	—
24	C₂²×C₄×C₁₂₇	C₂³	1	12	yes	—
25	C₂×Dic₁₂₇	—	2	16	no	—
26	C₂×Dic₁₂₇	C₂⁴	1	16	yes	—
27	C₂³×C₁₂₇	—	2	4	no	—
28	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
29	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
30	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
31	Q₈×C₁₂₇	C₂²×C₄	1	64	yes	—
32	C₂×C₄×C₁₂₇	C₂²×C₄	1	24	yes	—
33	C₂×C₄×C₁₂₇	C₂²×C₄	1	48	yes	—
34	C₂×C₄×C₁₂₇	C₂⁴	1	12	yes	—
35	C₂²×C₁₂₇	—	2	8	no	—
36	C₂²×C₁₂₇	—	2	4	no	—
37	C₂²×C₁₂₇	C₂³×C₄	1	4	yes	—
38	C₂²×C₁₂₇	C₂³×C₄	1	2	yes	—
39	C₂²×C₁₂₇	2₋¹⁺⁴	1	4	yes	—
40	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
41	C₄×C₁₂₇	C₂³×C₄	1	24	yes	—
42	Dic₁₂₇	—	2	16	no	—
43	C₂×C₁₂₇	—	2	4	no	—
44	C₂×C₁₂₇	#519	1	4	yes	complemented
45	C₂×C₁₂₇	C₂⁴×C₄	1	1	yes	characteristic
46	C₂×C₁₂₇	C₂×2₋¹⁺⁴	1	2	yes	—
47	C₂×#519	—	127	1	no	maximal, Sylow
48	C₁₂₇	C₂×#519	1	1	yes	characteristic, complemented, Sylow
49	C₂³×Q₈	D₁₂₇	1	1	yes	characteristic
50	#519	—	127	16	no	—
51	C₂×C₄×Q₈	—	127	8	no	—
52	C₂×#117	—	127	6	no	—
53	C₂²×Q₈	Dic₁₂₇	1	16	yes	complemented
54	C₂²×Q₈	C₂×D₁₂₇	1	4	yes	—
55	C₂²×Q₈	C₂×D₁₂₇	1	8	yes	—
56	C₂³×C₄	C₂×D₁₂₇	1	3	yes	—
57	C₄×Q₈	—	127	64	no	—
58	#117	—	127	48	no	—
59	C₂×C₄⋊C₄	—	127	12	no	—
60	C₂×C₂²⋊C₄	—	127	4	no	—
61	C₂×C₄²	—	127	12	no	—
62	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
63	C₂×Q₈	C₂×Dic₁₂₇	1	32	yes	—
64	C₂×Q₈	C₂×Dic₁₂₇	1	64	yes	complemented
65	C₂×Q₈	C₂²×D₁₂₇	1	16	yes	—
66	C₂²×C₄	—	127	8	no	—
67	C₂²×C₄	C₂×Dic₁₂₇	1	24	yes	—
68	C₂²×C₄	C₂²×D₁₂₇	1	12	yes	—
69	C₂²×C₄	C₂²×D₁₂₇	1	6	yes	—
70	C₄⋊C₄	—	127	48	no	—
71	C₂²⋊C₄	—	127	16	no	—
72	C₄²	—	127	48	no	—
73	C₂³	—	2	4	no	—
74	C₂³	C₂²×Dic₁₂₇	1	4	yes	—
75	C₂³	C₂³×D₁₂₇	1	2	yes	—
76	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
77	Q₈	C₂²×Dic₁₂₇	1	64	yes	—
78	C₂×C₄	—	127	16	no	—
79	C₂×C₄	—	254	16	no	—
80	C₂×C₄	C₂²×Dic₁₂₇	1	24	yes	—
81	C₂×C₄	C₂²×Dic₁₂₇	1	48	yes	—
82	C₂×C₄	C₂³×D₁₂₇	1	12	yes	—
83	C₂²	—	2	8	no	—
84	C₂²	—	2	4	no	—
85	C₂²	C₂³×Dic₁₂₇	1	4	yes	—
86	C₂²	C₂³×Dic₁₂₇	1	2	yes	—
87	C₂²	#16861257	1	4	yes	—
88	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
89	C₄	—	254	16	no	—
90	C₄	C₂³×Dic₁₂₇	1	24	yes	—
91	C₂	—	2	4	no	—
92	C₂	#22550354	1	4	yes	direct product factor
93	C₂	C₂⁴×Dic₁₂₇	1	1	yes	characteristic
94	C₂	C₂×#16861257	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂³×Q₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂⁴
prime radical	C₂⁴×C₁₂₇	C₂³
cosocle	C₂²×C₁₂₇	C₂⁵
hypercenter	C₂³×Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#519
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. #22550354	C₂	direct (N × H)
2. C₂×Q₈×Dic₁₂₇	C₂	semidirect
5. Q₈×Dic₁₂₇	C₂²	semidirect
10. C₂²×Q₈×C₁₂₇	C₄	semidirect
20. C₂×Q₈×C₁₂₇	C₂×C₄	semidirect
44. C₂×C₁₂₇	#519	semidirect
48. C₁₂₇	C₂×#519	semidirect
53. C₂²×Q₈	Dic₁₂₇	semidirect
64. C₂×Q₈	C₂×Dic₁₂₇	semidirect
92. C₂	#22550354	direct (N × H)

Maximal subgroups

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

subgroup	index
#22550354	2
C₂×Q₈×Dic₁₂₇	2
C₂×#16861168	2
C₂³×Q₈×C₁₂₇	2
C₂×#519	127

Subgroups — C2×#22550354 #30651921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550354 #30651921