Subgroups — C₁₂₇×#2161 #30635113

Order 16256.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×C₁₂₇	C₂	1	2	yes	maximal
3	#368×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₁₂₇	—	2	2	no	—
5	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
6	M₃₂×C₁₂₇	C₄	1	4	yes	—
7	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
8	#108×C₁₂₇	—	2	2	no	—
9	#108×C₁₂₇	C₂²	1	4	yes	—
10	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
11	M₁₆×C₁₂₇	—	2	4	no	—
12	M₁₆×C₁₂₇	—	4	2	no	—
13	C₂×C₈×C₁₂₇	—	4	2	no	—
14	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
17	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
18	C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
19	C₂³×C₁₂₇	Q₁₆	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
21	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
22	C₂×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
24	C₈×C₁₂₇	—	4	4	no	—
25	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
26	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
27	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₁₂₇	—	4	1	no	—
29	C₂²×C₁₂₇	#107	1	2	yes	—
30	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
31	C₄×C₁₂₇	#107	1	2	yes	—
32	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
34	C₂×C₁₂₇	—	2	2	no	—
35	C₂×C₁₂₇	—	2	1	no	—
36	C₂×C₁₂₇	C₂×#107	1	1	yes	characteristic
37	#2161	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₁₂₇	#2161	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
40	C₂×#108	C₂×C₁₂₇	1	2	yes	—
41	#368	C₂×C₁₂₇	1	4	yes	complemented
42	C₂×M₁₆	—	2	2	no	—
43	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
44	M₃₂	C₄×C₁₂₇	1	4	yes	—
45	C₂×C₁₆	C₂²×C₁₂₇	1	2	yes	—
46	#108	—	2	2	no	—
47	#108	C₂²×C₁₂₇	1	4	yes	—
48	C₂²×C₄	D₄×C₁₂₇	1	1	yes	characteristic
49	M₁₆	—	2	4	no	—
50	M₁₆	—	4	2	no	—
51	C₂×C₈	—	4	2	no	—
52	C₂×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
53	C₂×C₈	D₄×C₁₂₇	1	1	yes	characteristic
54	C₂×C₈	Q₈×C₁₂₇	1	2	yes	—
55	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
56	C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
57	C₂³	Q₁₆×C₁₂₇	1	1	yes	characteristic
58	C₂×C₄	C₄⋊C₄×C₁₂₇	1	2	yes	—
59	C₂×C₄	D₈×C₁₂₇	1	2	yes	—
60	C₂×C₄	Q₁₆×C₁₂₇	1	1	yes	characteristic
61	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
62	C₈	—	4	4	no	—
63	C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
64	C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
65	C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
66	C₂²	—	4	1	no	—
67	C₂²	#107×C₁₂₇	1	2	yes	—
68	C₂²	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
69	C₄	#107×C₁₂₇	1	2	yes	—
70	C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
71	C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
72	C₂	—	2	2	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×#107×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₂₇	C₂×D₈
commutator	C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	C₂×#107
prime radical	C₂³×C₁₂₇	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #368×C₁₂₇	C₂	semidirect
37. #2161	C₁₂₇	direct (N × H)
38. C₁₂₇	#2161	direct (N × H)
41. #368	C₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×C₁₂₇	2
C₂×#108×C₁₂₇	2
#368×C₁₂₇	2
#2161	127

Subgroups — C127×#2161 #30635113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2161 #30635113