Subgroups — C₂×C₆₇×M₁₂₈ #30692987

Order 17152.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×M₁₂₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₆₄×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₆×C₆₇	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₆₇	C₄	1	2	yes	—
6	C₂×C₃₂×C₆₇	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₆₇	C₂²	1	2	yes	—
8	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
9	C₆₄×C₆₇	C₂²	1	4	yes	complemented
10	C₂²×C₈×C₆₇	C₈	1	1	yes	characteristic
11	C₂×C₁₆×C₆₇	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₆₇	C₈	1	2	yes	—
13	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
14	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₆₇	C₂×C₄	1	2	yes	—
16	C₃₂×C₆₇	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₆₇	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₆₇	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₆₇	C₁₆	1	2	yes	—
20	C₂×C₈×C₆₇	C₁₆	1	1	yes	characteristic
21	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
22	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₆₇	C₂×C₈	1	2	yes	—
24	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
25	C₁₆×C₆₇	C₂²×C₄	1	1	yes	characteristic
26	C₂³×C₆₇	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₆₇	C₃₂	1	2	yes	—
28	C₂×C₄×C₆₇	C₃₂	1	1	yes	characteristic
29	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
30	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₆₇	C₂×C₁₆	1	2	yes	—
32	C₈×C₆₇	C₂×C₁₆	1	1	yes	characteristic
33	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
34	C₂²×C₆₇	—	2	2	no	—
35	C₂²×C₆₇	C₂×C₃₂	1	2	yes	—
36	C₂²×C₆₇	C₂×C₃₂	1	1	yes	characteristic
37	C₄×C₆₇	C₂×C₃₂	1	1	yes	characteristic
38	C₄×C₆₇	C₂×C₃₂	1	2	yes	—
39	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
40	C₂×M₁₂₈	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×C₆₇	—	2	2	no	—
42	C₂×C₆₇	M₁₂₈	1	2	yes	direct product factor
43	C₂×C₆₇	C₂²×C₃₂	1	1	yes	characteristic
44	C₂²×C₃₂	C₂×C₆₇	1	1	yes	characteristic
45	M₁₂₈	C₂×C₆₇	1	4	yes	direct product factor
46	C₂×C₆₄	C₂×C₆₇	1	2	yes	complemented
47	C₆₇	C₂×M₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₁₆	C₄×C₆₇	1	1	yes	characteristic
49	C₂×C₃₂	C₄×C₆₇	1	2	yes	—
50	C₂×C₃₂	C₄×C₆₇	1	1	yes	characteristic
51	C₂×C₃₂	C₂²×C₆₇	1	1	yes	characteristic
52	C₂×C₃₂	C₂²×C₆₇	1	2	yes	—
53	C₆₄	C₂²×C₆₇	1	4	yes	complemented
54	C₂²×C₈	C₈×C₆₇	1	1	yes	characteristic
55	C₂×C₁₆	C₈×C₆₇	1	1	yes	characteristic
56	C₂×C₁₆	C₈×C₆₇	1	2	yes	—
57	C₂×C₁₆	C₂×C₄×C₆₇	1	2	yes	—
58	C₂×C₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
59	C₃₂	C₂×C₄×C₆₇	1	1	yes	characteristic
60	C₃₂	C₂×C₄×C₆₇	1	2	yes	—
61	C₃₂	C₂³×C₆₇	1	1	yes	characteristic
62	C₂²×C₄	C₁₆×C₆₇	1	1	yes	characteristic
63	C₂×C₈	C₁₆×C₆₇	1	2	yes	—
64	C₂×C₈	C₁₆×C₆₇	1	1	yes	characteristic
65	C₂×C₈	C₂×C₈×C₆₇	1	2	yes	—
66	C₂×C₈	C₂×C₈×C₆₇	1	1	yes	characteristic
67	C₁₆	C₂×C₈×C₆₇	1	2	yes	—
68	C₁₆	C₂×C₈×C₆₇	1	1	yes	characteristic
69	C₁₆	C₂²×C₄×C₆₇	1	1	yes	characteristic
70	C₂³	C₃₂×C₆₇	1	1	yes	characteristic
71	C₂×C₄	C₃₂×C₆₇	1	2	yes	—
72	C₂×C₄	C₃₂×C₆₇	1	1	yes	characteristic
73	C₂×C₄	C₂×C₁₆×C₆₇	1	2	yes	—
74	C₂×C₄	C₂×C₁₆×C₆₇	1	1	yes	characteristic
75	C₈	C₂×C₁₆×C₆₇	1	1	yes	characteristic
76	C₈	C₂×C₁₆×C₆₇	1	2	yes	—
77	C₈	C₂²×C₈×C₆₇	1	1	yes	characteristic
78	C₂²	—	2	2	no	—
79	C₂²	C₂×C₃₂×C₆₇	1	2	yes	—
80	C₂²	C₂×C₃₂×C₆₇	1	1	yes	characteristic
81	C₄	C₂×C₃₂×C₆₇	1	2	yes	—
82	C₄	C₂×C₃₂×C₆₇	1	1	yes	characteristic
83	C₄	C₂²×C₁₆×C₆₇	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	C₆₇×M₁₂₈	1	2	yes	direct product factor
86	C₂	C₂²×C₃₂×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂×C₆₇	C₂²
commutator	C₂	C₂²×C₃₂×C₆₇
Frattini, cosocle	C₃₂	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×C₃₂
prime radical	C₂³×C₆₇	C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×M₁₂₈	C₂	direct (N × H)
3. C₂×C₆₄×C₆₇	C₂	semidirect
9. C₆₄×C₆₇	C₂²	semidirect
40. C₂×M₁₂₈	C₆₇	direct (N × H)
42. C₂×C₆₇	M₁₂₈	direct (N × H)
45. M₁₂₈	C₂×C₆₇	direct (N × H)
46. C₂×C₆₄	C₂×C₆₇	semidirect
47. C₆₇	C₂×M₁₂₈	direct (N × H)
53. C₆₄	C₂²×C₆₇	semidirect
85. C₂	C₆₇×M₁₂₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₂×C₆₇	2
C₆₇×M₁₂₈	2
C₂×C₆₄×C₆₇	2
C₂×M₁₂₈	67

Subgroups — C2×C67×M128 #30692987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×M₁₂₈ #30692987