Subgroups — C₁₂₇×#3162 #30636114

Order 16256.

348 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533×C₁₂₇	C₂	1	2	yes	maximal
2	#501×C₁₂₇	C₂	1	8	yes	maximal, complemented
3	#427×C₁₂₇	C₂	1	4	yes	maximal
4	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
6	C₄⋊Q₈×C₁₂₇	C₂²	1	4	yes	—
7	#125×C₁₂₇	C₂²	1	4	yes	—
8	C₂²⋊Q₈×C₁₂₇	—	2	4	no	—
9	C₄×Q₈×C₁₂₇	—	2	4	no	—
10	#117×C₁₂₇	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	#107×C₁₂₇	C₂²	1	8	yes	—
13	#106×C₁₂₇	C₂²	1	8	yes	—
14	#97×C₁₂₇	C₂²	1	4	yes	—
15	C₂×Q₈×C₁₂₇	—	2	4	no	—
16	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
17	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	Q₈	1	8	yes	complemented
19	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
20	C₄⋊C₄×C₁₂₇	—	2	8	no	—
21	C₄⋊C₄×C₁₂₇	—	2	4	no	—
22	C₄⋊C₄×C₁₂₇	C₂³	1	8	yes	—
23	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
24	C₄²×C₁₂₇	—	2	4	no	—
25	C₄²×C₁₂₇	D₄	1	2	yes	—
26	C₄²×C₁₂₇	C₂³	1	2	yes	—
27	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
28	Q₈×C₁₂₇	—	4	8	no	—
29	C₂×C₄×C₁₂₇	—	2	8	no	—
30	C₂×C₄×C₁₂₇	—	4	2	no	—
31	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
32	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
35	C₈×C₁₂₇	C₂×Q₈	1	8	yes	—
36	C₂²×C₁₂₇	—	2	2	no	—
37	C₂²×C₁₂₇	C₄⋊Q₈	1	2	yes	—
38	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
39	C₄×C₁₂₇	—	2	4	no	—
40	C₄×C₁₂₇	—	4	8	no	—
41	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
42	C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
43	C₂×C₁₂₇	—	2	2	no	—
44	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₁₂₇	#575	1	2	yes	—
46	#3162	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₁₂₇	#3162	1	1	yes	characteristic, direct product factor, Sylow
48	#533	C₂×C₁₂₇	1	2	yes	—
49	#501	C₂×C₁₂₇	1	8	yes	complemented
50	#427	C₂×C₁₂₇	1	4	yes	—
51	C₂×#97	C₂×C₁₂₇	1	1	yes	characteristic
52	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
53	C₄⋊Q₈	C₂²×C₁₂₇	1	4	yes	—
54	#125	C₂²×C₁₂₇	1	4	yes	—
55	C₂²⋊Q₈	—	2	4	no	—
56	C₄×Q₈	—	2	4	no	—
57	#117	C₂²×C₁₂₇	1	4	yes	—
58	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
59	#107	C₂²×C₁₂₇	1	8	yes	—
60	#106	C₂²×C₁₂₇	1	8	yes	—
61	#97	C₂²×C₁₂₇	1	4	yes	—
62	C₂×Q₈	—	2	4	no	—
63	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
64	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
65	C₂×C₈	Q₈×C₁₂₇	1	8	yes	complemented
66	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
67	C₄⋊C₄	—	2	8	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	C₂³×C₁₂₇	1	8	yes	—
70	C₂²⋊C₄	—	2	4	no	—
71	C₄²	—	2	4	no	—
72	C₄²	D₄×C₁₂₇	1	2	yes	—
73	C₄²	C₂³×C₁₂₇	1	2	yes	—
74	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
75	Q₈	—	4	8	no	—
76	C₂×C₄	—	2	8	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
79	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
80	C₂×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
81	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
82	C₈	C₂×Q₈×C₁₂₇	1	8	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊Q₈×C₁₂₇	1	2	yes	—
85	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	4	8	no	—
88	C₄	C₄⋊Q₈×C₁₂₇	1	2	yes	—
89	C₄	C₂²×Q₈×C₁₂₇	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
92	C₂	#575×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×C₁₂₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #501×C₁₂₇	C₂	semidirect
18. C₂×C₈×C₁₂₇	Q₈	semidirect
46. #3162	C₁₂₇	direct (N × H)
47. C₁₂₇	#3162	direct (N × H)
49. #501	C₂×C₁₂₇	semidirect
65. C₂×C₈	Q₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#533×C₁₂₇	2
#501×C₁₂₇	2
#427×C₁₂₇	2
C₂×#97×C₁₂₇	2
#3162	127

Subgroups — C127×#3162 #30636114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#3162 #30636114