Subgroups — C₂²×C₄×Q₈×C₁₂₇ #30636375

Order 16256.

1664 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₂₇	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	3	yes	maximal
4	C₂²×C₄²×C₁₂₇	C₂	1	3	yes	maximal
5	C₂²×Q₈×C₁₂₇	C₄	1	16	yes	direct product factor
6	C₂²×Q₈×C₁₂₇	C₂²	1	12	yes	—
7	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
8	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
9	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₁₂₇	C₂²	1	64	yes	direct product factor
11	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	36	yes	—
12	C₂×C₄²×C₁₂₇	C₂²	1	36	yes	—
13	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
14	C₂×Q₈×C₁₂₇	C₂×C₄	1	96	yes	direct product factor
15	C₂×Q₈×C₁₂₇	C₂³	1	16	yes	—
16	C₂²×C₄×C₁₂₇	—	2	12	no	—
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	24	yes	—
18	C₂²×C₄×C₁₂₇	Q₈	1	8	yes	direct product factor
19	C₂²×C₄×C₁₂₇	C₂³	1	18	yes	—
20	C₂²×C₄×C₁₂₇	C₂³	1	18	yes	—
21	C₂²×C₄×C₁₂₇	C₂³	1	6	yes	—
22	C₄⋊C₄×C₁₂₇	C₂³	1	48	yes	—
23	C₄²×C₁₂₇	C₂³	1	48	yes	—
24	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
25	C₂³×C₁₂₇	C₂×Q₈	1	4	yes	—
26	C₂³×C₁₂₇	#41	1	4	yes	—
27	C₂³×C₁₂₇	C₂⁴	1	3	yes	—
28	Q₈×C₁₂₇	C₂²×C₄	1	64	yes	direct product factor
29	C₂×C₄×C₁₂₇	—	2	36	no	—
30	C₂×C₄×C₁₂₇	C₂²×C₄	1	72	yes	—
31	C₂×C₄×C₁₂₇	C₂×Q₈	1	24	yes	direct product factor
32	C₂×C₄×C₁₂₇	C₂⁴	1	12	yes	—
33	C₂×C₄×C₁₂₇	C₂⁴	1	12	yes	—
34	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
35	C₂²×C₁₂₇	C₄×Q₈	1	16	yes	direct product factor
36	C₂²×C₁₂₇	C₂³×C₄	1	6	yes	—
37	C₂²×C₁₂₇	C₂²×Q₈	1	6	yes	—
38	C₂²×C₁₂₇	C₂×#41	1	6	yes	—
39	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
40	C₄×C₁₂₇	—	2	12	no	—
41	C₄×C₁₂₇	C₂³×C₄	1	24	yes	—
42	C₄×C₁₂₇	C₂²×Q₈	1	8	yes	direct product factor
43	C₂×C₁₂₇	C₂×C₄×Q₈	1	12	yes	direct product factor
44	C₂×C₁₂₇	C₂⁴×C₄	1	1	yes	characteristic
45	C₂×C₁₂₇	C₂³×Q₈	1	1	yes	characteristic
46	C₂×C₁₂₇	C₂²×#41	1	1	yes	characteristic
47	C₂²×C₄×Q₈	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₁₂₇	C₂²×C₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
49	C₂³×Q₈	C₂×C₁₂₇	1	1	yes	characteristic
50	C₂×C₄×Q₈	C₂×C₁₂₇	1	24	yes	direct product factor
51	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	3	yes	—
52	C₂²×C₄²	C₂×C₁₂₇	1	3	yes	—
53	C₂²×Q₈	C₄×C₁₂₇	1	16	yes	direct product factor
54	C₂²×Q₈	C₂²×C₁₂₇	1	12	yes	—
55	C₂³×C₄	C₂²×C₁₂₇	1	3	yes	—
56	C₂³×C₄	C₂²×C₁₂₇	1	3	yes	—
57	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
58	C₄×Q₈	C₂²×C₁₂₇	1	64	yes	direct product factor
59	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	36	yes	—
60	C₂×C₄²	C₂²×C₁₂₇	1	36	yes	—
61	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
62	C₂×Q₈	C₂×C₄×C₁₂₇	1	96	yes	direct product factor
63	C₂×Q₈	C₂³×C₁₂₇	1	16	yes	—
64	C₂²×C₄	—	2	12	no	—
65	C₂²×C₄	C₂×C₄×C₁₂₇	1	24	yes	—
66	C₂²×C₄	Q₈×C₁₂₇	1	8	yes	direct product factor
67	C₂²×C₄	C₂³×C₁₂₇	1	18	yes	—
68	C₂²×C₄	C₂³×C₁₂₇	1	18	yes	—
69	C₂²×C₄	C₂³×C₁₂₇	1	6	yes	—
70	C₄⋊C₄	C₂³×C₁₂₇	1	48	yes	—
71	C₄²	C₂³×C₁₂₇	1	48	yes	—
72	C₂³	C₂²×C₄×C₁₂₇	1	4	yes	—
73	C₂³	C₂×Q₈×C₁₂₇	1	4	yes	—
74	C₂³	#41×C₁₂₇	1	4	yes	—
75	C₂³	C₂⁴×C₁₂₇	1	3	yes	—
76	Q₈	C₂²×C₄×C₁₂₇	1	64	yes	direct product factor
77	C₂×C₄	—	2	36	no	—
78	C₂×C₄	C₂²×C₄×C₁₂₇	1	72	yes	—
79	C₂×C₄	C₂×Q₈×C₁₂₇	1	24	yes	direct product factor
80	C₂×C₄	C₂⁴×C₁₂₇	1	12	yes	—
81	C₂×C₄	C₂⁴×C₁₂₇	1	12	yes	—
82	C₂×C₄	C₂⁴×C₁₂₇	1	4	yes	—
83	C₂²	C₄×Q₈×C₁₂₇	1	16	yes	direct product factor
84	C₂²	C₂³×C₄×C₁₂₇	1	6	yes	—
85	C₂²	C₂²×Q₈×C₁₂₇	1	6	yes	—
86	C₂²	C₂×#41×C₁₂₇	1	6	yes	—
87	C₂²	C₂⁵×C₁₂₇	1	1	yes	characteristic
88	C₄	—	2	12	no	—
89	C₄	C₂³×C₄×C₁₂₇	1	24	yes	—
90	C₄	C₂²×Q₈×C₁₂₇	1	8	yes	direct product factor
91	C₂	C₂×C₄×Q₈×C₁₂₇	1	12	yes	direct product factor
92	C₂	C₂⁴×C₄×C₁₂₇	1	1	yes	characteristic
93	C₂	C₂³×Q₈×C₁₂₇	1	1	yes	characteristic
94	C₂	C₂²×#41×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₁₂₇	C₂²
commutator	C₂	C₂⁴×C₄×C₁₂₇
Frattini, cosocle	C₂²	C₂⁵×C₁₂₇
socle, prime radical	C₂⁴×C₁₂₇	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₁₂₇	C₂	direct (N × H)
5. C₂²×Q₈×C₁₂₇	C₄	direct (N × H)
10. C₄×Q₈×C₁₂₇	C₂²	direct (N × H)
14. C₂×Q₈×C₁₂₇	C₂×C₄	direct (N × H)
18. C₂²×C₄×C₁₂₇	Q₈	direct (N × H)
28. Q₈×C₁₂₇	C₂²×C₄	direct (N × H)
31. C₂×C₄×C₁₂₇	C₂×Q₈	direct (N × H)
35. C₂²×C₁₂₇	C₄×Q₈	direct (N × H)
42. C₄×C₁₂₇	C₂²×Q₈	direct (N × H)
43. C₂×C₁₂₇	C₂×C₄×Q₈	direct (N × H)
47. C₂²×C₄×Q₈	C₁₂₇	direct (N × H)
48. C₁₂₇	C₂²×C₄×Q₈	direct (N × H)
50. C₂×C₄×Q₈	C₂×C₁₂₇	direct (N × H)
53. C₂²×Q₈	C₄×C₁₂₇	direct (N × H)
58. C₄×Q₈	C₂²×C₁₂₇	direct (N × H)
62. C₂×Q₈	C₂×C₄×C₁₂₇	direct (N × H)
66. C₂²×C₄	Q₈×C₁₂₇	direct (N × H)
76. Q₈	C₂²×C₄×C₁₂₇	direct (N × H)
79. C₂×C₄	C₂×Q₈×C₁₂₇	direct (N × H)
83. C₂²	C₄×Q₈×C₁₂₇	direct (N × H)
90. C₄	C₂²×Q₈×C₁₂₇	direct (N × H)
91. C₂	C₂×C₄×Q₈×C₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈×C₁₂₇	2
C₂×C₄×Q₈×C₁₂₇	2
C₂²×C₄⋊C₄×C₁₂₇	2
C₂²×C₄²×C₁₂₇	2
C₂²×C₄×Q₈	127

Subgroups — C22×C4×Q8×C127 #30636375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×Q₈×C₁₂₇ #30636375