Subgroups — #30647119

Order 16256.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549767	C₂	1	7	yes	maximal
2	#22549760	C₂	1	7	yes	maximal, complemented
3	C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₁₂₇⋊C₄	C₂²	1	7	yes	—
5	C₂×(C₂×D₁₂₇)⋊C₄	C₂²	1	7	yes	—
6	C₁₂₇⋊#95	C₂²	1	14	yes	—
7	C₂×C₄²×C₁₂₇	C₂²	1	7	yes	—
8	C₂³×D₁₂₇	C₂³	1	1	yes	characteristic
9	C₂²×Dic₁₂₇	C₂³	1	7	yes	—
10	Dic₁₂₇⋊C₄	—	2	14	no	—
11	(C₂×D₁₂₇)⋊C₄	—	2	14	no	—
12	C₂²×C₄×C₁₂₇	C₂³	1	7	yes	—
13	C₄²×C₁₂₇	—	2	14	no	—
14	C₂²×D₁₂₇	—	2	7	no	—
15	C₂×Dic₁₂₇	—	2	7	no	—
16	C₂×Dic₁₂₇	—	2	14	no	—
17	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	—	2	14	no	—
19	C₂×C₄×C₁₂₇	#41	1	14	yes	—
20	C₂²×C₁₂₇	C₂×#41	1	7	yes	—
21	C₂×D₁₂₇	—	4	7	no	—
22	C₄×C₁₂₇	—	2	14	no	—
23	Dic₁₂₇	—	4	7	no	—
24	C₂×C₁₂₇	#529	1	7	yes	—
25	D₁₂₇	—	8	1	no	—
26	#2865	—	127	1	no	maximal, Sylow
27	C₁₂₇	#2865	1	1	yes	characteristic, complemented, Sylow
28	#388	—	127	7	no	—
29	#387	—	127	7	no	—
30	C₄³	D₁₂₇	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄	—	127	7	no	—
32	C₂×C₂²⋊C₄	—	127	7	no	—
33	C₂×C₄²	C₂×D₁₂₇	1	7	yes	—
34	#95	—	127	14	no	—
35	C₂⁴	—	127	1	no	—
36	C₂²×C₄	—	127	7	no	—
37	C₂²×C₄	C₂²×D₁₂₇	1	7	yes	—
38	C₄⋊C₄	—	254	14	no	—
39	C₂²⋊C₄	—	254	14	no	—
40	C₄²	—	2	14	no	—
41	C₂³	—	254	7	no	—
42	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
43	C₂×C₄	—	2	14	no	—
44	C₂×C₄	—	254	7	no	—
45	C₂×C₄	—	254	14	no	—
46	C₂×C₄	#14523248	1	14	yes	—
47	C₂²	—	508	7	no	—
48	C₂²	C₂×#14523248	1	7	yes	—
49	C₄	—	2	14	no	—
50	C₄	—	508	7	no	—
51	C₂	—	1016	1	no	—
52	C₂	#22550410	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₂₇
commutator, socle, cosocle	C₂³×C₁₂₇	C₂⁴
Fitting	C₄³×C₁₂₇	C₂
prime radical	C₂³×D₁₂₇	C₂³
hypercenter	C₄³	D₁₂₇
hypocenter	C₁₂₇	#2865
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. #22549760	C₂	semidirect
3. C₄³×C₁₂₇	C₂	semidirect
27. C₁₂₇	#2865	semidirect
30. C₄³	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22549767	2
#22549760	2
C₄³×C₁₂₇	2
#2865	127