Subgroups — #30637325

Order 16256.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523232	C₂	1	2	yes	maximal
2	#22549618	C₂	1	4	yes	maximal, complemented
3	#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523232	—	2	2	no	—
5	#16861120	C₂²	1	4	yes	—
6	C₄×C₁₂₇⋊C₈	—	2	2	no	—
7	C₄⋊Q₈×C₁₂₇	C₄	1	2	yes	—
8	#125×C₁₂₇	C₂²	1	2	yes	—
9	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
10	C₄×Q₈×C₁₂₇	—	2	2	no	—
11	#117×C₁₂₇	C₄	1	2	yes	—
12	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	#14523232	—	2	8	no	—
14	C₂×C₁₂₇⋊C₈	—	2	4	no	—
15	C₂×Q₈×C₁₂₇	—	2	2	no	—
16	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
18	C₄⋊C₄×C₁₂₇	—	2	2	no	—
19	C₄⋊C₄×C₁₂₇	—	2	4	no	—
20	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
21	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
22	C₄²×C₁₂₇	—	2	2	no	—
23	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₂₇	D₄	1	2	yes	—
25	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₁₂₇⋊C₈	—	2	8	no	—
27	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
28	Q₈×C₁₂₇	—	4	4	no	—
29	C₂×C₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄×C₁₂₇	—	2	4	no	—
31	C₂×C₄×C₁₂₇	—	2	4	no	—
32	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
36	C₂²×C₁₂₇	—	2	1	no	—
37	C₂²×C₁₂₇	—	2	2	no	—
38	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₄×C₁₂₇	—	2	1	no	—
40	C₄×C₁₂₇	—	2	2	no	—
41	C₄×C₁₂₇	—	4	4	no	—
42	C₄×C₁₂₇	#101	1	2	yes	—
43	C₄×C₁₂₇	#104	1	4	yes	—
44	C₂×C₁₂₇	—	4	1	no	—
45	C₂×C₁₂₇	C₂×#101	1	1	yes	characteristic
46	C₂×C₁₂₇	C₂×#104	1	2	yes	—
47	#1526	—	127	1	no	maximal, Sylow
48	C₁₂₇	#1526	1	1	yes	characteristic, complemented, Sylow
49	#533	D₁₂₇	1	1	yes	characteristic
50	C₄×M₁₆	—	127	2	no	—
51	#330	—	127	4	no	—
52	C₂×M₁₆	—	254	2	no	—
53	C₄⋊Q₈	Dic₁₂₇	1	2	yes	—
54	#125	C₂×D₁₂₇	1	2	yes	—
55	C₂²⋊Q₈	—	2	2	no	—
56	C₄×Q₈	—	2	2	no	—
57	#117	Dic₁₂₇	1	2	yes	—
58	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
59	#97	—	127	4	no	—
60	C₄×C₈	—	254	2	no	—
61	C₂×Q₈	—	2	2	no	—
62	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
63	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
64	M₁₆	—	254	8	no	—
65	C₂×C₈	—	254	4	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
69	C₂²⋊C₄	—	2	2	no	—
70	C₄²	—	2	2	no	—
71	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
72	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
73	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
75	Q₈	—	4	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
80	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
81	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
82	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
83	C₈	—	254	8	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	2	no	—
86	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#16861128	1	2	yes	—
91	C₄	#16861136	1	4	yes	—
92	C₂	—	4	1	no	—
93	C₂	C₂×#16861128	1	1	yes	characteristic
94	C₂	C₂×#16861136	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#533×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₂₇	C₂²⋊C₄
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	#533	D₁₂₇
hypocenter	C₁₂₇	#1526
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #22549618	C₂	semidirect
48. C₁₂₇	#1526	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523232	2
#22549618	2
#533×C₁₂₇	2
#1526	127