Subgroups — #30638482

Order 16256.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861120	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22549631	C₂	1	4	yes	maximal
4	#376×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	#16861120	C₄	1	4	yes	complemented
8	C₄×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
9	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	#95×C₁₂₇	C₄	1	4	yes	—
12	C₂×C₁₂₇⋊C₈	—	2	4	no	—
13	C₂×C₁₂₇⋊C₈	—	2	4	no	—
14	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
15	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₄²×C₁₂₇	—	2	4	no	—
20	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₁₂₇⋊C₈	—	2	4	no	—
23	C₁₂₇⋊C₈	—	2	4	no	—
24	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₂₇	—	2	4	no	—
26	C₂×C₄×C₁₂₇	—	2	4	no	—
27	C₂×C₄×C₁₂₇	—	2	8	no	—
28	C₂×C₄×C₁₂₇	C₄²	1	4	yes	—
29	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	#41	1	4	yes	—
32	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₁₂₇	#117	1	2	yes	—
34	C₂²×C₁₂₇	#131	1	4	yes	—
35	C₄×C₁₂₇	—	2	4	no	—
36	C₄×C₁₂₇	—	4	4	no	—
37	C₄×C₁₂₇	#117	1	2	yes	—
38	C₄×C₁₂₇	#117	1	2	yes	—
39	C₂×C₁₂₇	#376	1	1	yes	characteristic
40	C₂×C₁₂₇	#405	1	2	yes	—
41	C₂×C₁₂₇	#433	1	4	yes	—
42	#1746	—	127	1	no	maximal, Sylow
43	C₁₂₇	#1746	1	1	yes	characteristic, complemented, Sylow
44	C₂×#97	—	127	1	no	—
45	C₂×C₄×C₈	—	127	1	no	—
46	#376	D₁₂₇	1	1	yes	characteristic
47	#336	—	127	4	no	—
48	C₂²×C₈	—	127	2	no	—
49	C₂²×C₈	—	127	2	no	—
50	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
51	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
52	#97	—	127	4	no	—
53	C₄×C₈	—	127	4	no	—
54	#95	Dic₁₂₇	1	4	yes	—
55	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
56	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
57	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
58	C₂×C₈	—	127	4	no	—
59	C₂×C₈	—	127	4	no	—
60	C₂×C₈	—	254	4	no	—
61	C₂×C₈	—	254	4	no	—
62	C₄²	—	2	4	no	—
63	C₄²	C₄×D₁₂₇	1	2	yes	—
64	C₄²	C₄×D₁₂₇	1	2	yes	—
65	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	8	no	—
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	C₄×Dic₁₂₇	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
71	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
72	C₂×C₄	#14523248	1	4	yes	—
73	C₈	—	254	4	no	—
74	C₈	—	254	4	no	—
75	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
76	C₂²	#16861168	1	2	yes	—
77	C₂²	#16861216	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	4	4	no	—
80	C₄	#16861166	1	2	yes	—
81	C₄	#16861166	1	2	yes	—
82	C₂	#22549722	1	1	yes	characteristic
83	C₂	#22549832	1	2	yes	—
84	C₂	#22549937	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂×C₄²
Fitting	#376×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#376	D₁₂₇
hypocenter	C₁₂₇	#1746
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
7. #16861120	C₄	semidirect
8. C₄×C₁₂₇⋊C₈	C₄	semidirect
43. C₁₂₇	#1746	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861120	2
C₂×C₄×C₁₂₇⋊C₈	2
#22549631	2
#376×C₁₂₇	2
#1746	127