Subgroups — #30638636

Order 16256.

220 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	C₄×#14523232	C₂	1	2	yes	maximal
3	#22549641	C₂	1	2	yes	maximal
4	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	C₂²	1	4	yes	—
6	#16861145	C₄	1	8	yes	complemented
7	#16861120	—	2	2	no	—
8	C₄×C₁₂₇⋊C₈	—	2	2	no	—
9	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	#97×C₁₂₇	C₄	1	4	yes	—
13	#14523232	—	2	4	no	—
14	#14523232	C₂×C₄	1	8	yes	—
15	C₂×C₁₂₇⋊C₈	—	2	4	no	—
16	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
17	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	—	2	2	no	—
19	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₂₇	D₄	1	2	yes	—
24	C₁₂₇⋊C₈	—	2	8	no	—
25	C₂³×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	—	4	2	no	—
27	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
29	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	#41	1	2	yes	—
32	C₈×C₁₂₇	—	2	2	no	—
33	C₈×C₁₂₇	C₄²	1	4	yes	—
34	C₂²×C₁₂₇	—	2	2	no	—
35	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₁₂₇	C₄×Q₈	1	2	yes	—
37	C₄×C₁₂₇	—	2	4	no	—
38	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₁₂₇	#117	1	1	yes	characteristic
40	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
41	C₂×C₁₂₇	—	2	2	no	—
42	C₂×C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₁₂₇	#430	1	2	yes	—
44	#1778	—	127	1	no	maximal, Sylow
45	C₁₂₇	#1778	1	1	yes	characteristic, complemented, Sylow
46	C₂×#108	—	127	2	no	—
47	C₄×M₁₆	—	127	2	no	—
48	C₂×#97	D₁₂₇	1	1	yes	characteristic
49	#341	—	127	2	no	—
50	C₂×M₁₆	—	127	4	no	—
51	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
53	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
54	#108	—	127	8	no	—
55	#97	—	254	2	no	—
56	#97	Dic₁₂₇	1	4	yes	—
57	C₄×C₈	—	254	2	no	—
58	C₂²×C₄	Dic₂₅₄	1	2	yes	—
59	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
60	M₁₆	—	127	8	no	—
61	M₁₆	—	254	4	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	254	4	no	—
64	C₂×C₈	C₄×D₁₂₇	1	4	yes	—
65	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
66	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
67	C₄²	C₂×Dic₁₂₇	1	2	yes	—
68	C₄²	D₅₀₈	1	2	yes	—
69	C₂³	C₂×Dic₂₅₄	1	1	yes	characteristic
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
72	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
73	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
74	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
75	C₂×C₄	#14523248	1	2	yes	—
76	C₈	—	2	2	no	—
77	C₈	—	254	8	no	—
78	C₈	C₄×Dic₁₂₇	1	4	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
81	C₂²	C₄×Dic₂₅₄	1	2	yes	—
82	C₄	—	2	4	no	—
83	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
84	C₄	#16861168	1	1	yes	characteristic
85	C₄	C₄×D₅₀₈	1	2	yes	—
86	C₂	—	2	2	no	—
87	C₂	C₄×C₄⋊Dic₁₂₇	1	1	yes	characteristic
88	C₂	#22549925	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₀₈
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#97×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₂₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#97	D₁₂₇
hypocenter	C₁₂₇	#1778
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
6. #16861145	C₄	semidirect
45. C₁₂₇	#1778	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861145	2
C₄×#14523232	2
#22549641	2
C₂×#97×C₁₂₇	2
#1778	127