Subgroups — #30638870

Order 16256.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549900	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861120	C₂	1	1	yes	maximal, characteristic
3	#22549640	C₂	1	4	yes	maximal
4	#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	C₂²	1	2	yes	—
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
8	#16861120	C₄	1	4	yes	—
9	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₂₇	—	2	2	no	—
12	#14523232	—	2	4	no	—
13	C₂×C₁₂₇⋊C₈	—	2	2	no	—
14	C₂×C₁₂₇⋊C₈	—	2	2	no	—
15	C₂×C₁₂₇⋊C₈	—	2	2	no	—
16	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
18	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
19	M₁₆×C₁₂₇	—	2	4	no	—
20	C₂×C₈×C₁₂₇	—	2	2	no	—
21	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₂₇	D₄	1	2	yes	—
23	C₁₂₇⋊C₈	—	2	4	no	—
24	C₁₂₇⋊C₈	—	4	2	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	1	yes	characteristic
29	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	#41	1	2	yes	—
31	C₂×C₄×C₁₂₇	#41	1	2	yes	—
32	C₈×C₁₂₇	—	4	2	no	—
33	C₂²×C₁₂₇	—	2	2	no	—
34	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₁₂₇	#125	1	2	yes	—
36	C₄×C₁₂₇	—	2	4	no	—
37	C₄×C₁₂₇	#117	1	2	yes	—
38	C₄×C₁₂₇	#124	1	1	yes	characteristic
39	C₄×C₁₂₇	#124	1	1	yes	characteristic
40	C₂×C₁₂₇	—	2	2	no	—
41	C₂×C₁₂₇	#382	1	1	yes	characteristic
42	C₂×C₁₂₇	#430	1	2	yes	—
43	#1838	—	127	1	no	maximal, Sylow
44	C₁₂₇	#1838	1	1	yes	characteristic, complemented, Sylow
45	#423	—	127	1	no	—
46	#423	D₁₂₇	1	1	yes	characteristic
47	C₂×#97	—	127	1	no	—
48	#341	—	127	4	no	—
49	C₂×M₁₆	—	127	2	no	—
50	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
51	C₂²×C₈	—	127	2	no	—
52	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	—	254	2	no	—
55	#97	—	127	4	no	—
56	C₂²×C₄	Dic₂₅₄	1	2	yes	—
57	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
58	M₁₆	—	2	4	no	—
59	M₁₆	—	254	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	127	4	no	—
62	C₂×C₈	—	254	2	no	—
63	C₂×C₈	—	254	2	no	—
64	C₂×C₈	—	254	2	no	—
65	C₄²	C₄×D₁₂₇	1	2	yes	—
66	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
67	C₂³	C₂×Dic₂₅₄	1	1	yes	characteristic
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
71	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
72	C₂×C₄	#14523247	1	2	yes	—
73	C₂×C₄	#14523249	1	2	yes	—
74	C₈	—	4	2	no	—
75	C₈	—	508	2	no	—
76	C₈	—	254	4	no	—
77	C₂²	—	2	2	no	—
78	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
79	C₂²	#16861200	1	2	yes	—
80	C₄	—	2	4	no	—
81	C₄	#16861167	1	2	yes	—
82	C₄	#16861196	1	1	yes	characteristic
83	C₄	#16861198	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#22549741	1	1	yes	characteristic
86	C₂	#22549924	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#423×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₂₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#423	D₁₂₇
hypocenter	C₁₂₇	#1838
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
44. C₁₂₇	#1838	semidirect

Maximal subgroups

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

subgroup	index
#22549900	2
C₂×#16861120	2
#22549640	2
#423×C₁₂₇	2
#1838	127