Subgroups — C₂×C₄×C₈×D₁₂₇ #30647129

Order 16256.

760 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₂₇	C₂	1	2	yes	maximal
2	C₂×C₄²×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×D₁₂₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₂₇	C₄	1	2	yes	—
8	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
9	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₈×D₁₂₇	C₄	1	16	yes	direct product factor
11	C₂×C₈×D₁₂₇	C₂²	1	8	yes	—
12	C₂×C₄×Dic₁₂₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₄²×D₁₂₇	C₄	1	4	yes	—
15	C₄²×D₁₂₇	C₂²	1	4	yes	—
16	C₄×C₁₂₇⋊C₈	C₂²	1	4	yes	complemented
17	C₈×Dic₁₂₇	C₂²	1	8	yes	complemented
18	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₂₇	C₂²	1	4	yes	complemented
21	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×D₁₂₇	C₈	1	16	yes	direct product factor
25	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₁₂₇	C₂×C₄	1	8	yes	—
27	C₂×C₄×D₁₂₇	C₂×C₄	1	8	yes	—
28	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
29	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	complemented
30	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
31	C₈×D₁₂₇	C₂×C₄	1	32	yes	direct product factor
32	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
33	C₄×Dic₁₂₇	C₂×C₄	1	8	yes	—
34	C₄×Dic₁₂₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
39	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₄²×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×D₁₂₇	C₄²	1	4	yes	—
42	C₂²×D₁₂₇	C₂×C₈	1	8	yes	—
43	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
44	C₂×Dic₁₂₇	C₄²	1	4	yes	—
45	C₂×Dic₁₂₇	C₂×C₈	1	8	yes	complemented
46	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
47	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
48	C₄×D₁₂₇	C₄²	1	8	yes	—
49	C₄×D₁₂₇	C₂×C₈	1	32	yes	direct product factor
50	C₄×D₁₂₇	C₂²×C₄	1	8	yes	—
51	C₁₂₇⋊C₈	C₂²×C₄	1	8	yes	complemented
52	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	C₂×C₈	1	8	yes	complemented
54	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
58	C₈×C₁₂₇	C₂²×C₄	1	8	yes	complemented
59	C₂²×C₁₂₇	C₂×C₄²	1	2	yes	—
60	C₂²×C₁₂₇	C₂²×C₈	1	4	yes	—
61	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
62	C₂×D₁₂₇	C₄×C₈	1	16	yes	direct product factor
63	C₂×D₁₂₇	C₂×C₄²	1	4	yes	—
64	C₂×D₁₂₇	C₂²×C₈	1	8	yes	—
65	C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
66	C₄×C₁₂₇	C₂²×C₈	1	8	yes	complemented
67	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
68	Dic₁₂₇	C₂×C₄²	1	4	yes	—
69	Dic₁₂₇	C₂²×C₈	1	8	yes	complemented
70	C₂×C₁₂₇	C₂×C₄×C₈	1	4	yes	complemented
71	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
72	C₂×C₁₂₇	C₂³×C₈	1	2	yes	—
73	D₁₂₇	C₂×C₄×C₈	1	8	yes	direct product factor
74	C₂²×C₄×C₈	—	127	1	no	maximal, Sylow
75	C₁₂₇	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₈	—	127	2	no	—
77	C₂²×C₄²	—	127	1	no	—
78	C₂×C₄×C₈	—	127	2	no	—
79	C₂×C₄×C₈	—	127	1	no	—
80	C₂×C₄×C₈	—	127	8	no	—
81	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic, direct product factor
82	C₂³×C₄	—	127	2	no	—
83	C₂³×C₄	—	127	1	no	—
84	C₂²×C₈	—	127	2	no	—
85	C₂²×C₈	—	127	16	no	—
86	C₂²×C₈	—	127	8	no	—
87	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
88	C₂×C₄²	—	127	2	no	—
89	C₂×C₄²	—	127	1	no	—
90	C₂×C₄²	—	127	4	no	—
91	C₂×C₄²	—	127	4	no	—
92	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
93	C₄×C₈	—	127	8	no	—
94	C₄×C₈	—	127	4	no	—
95	C₄×C₈	C₂×D₁₂₇	1	4	yes	direct product factor
96	C₂⁴	—	127	1	no	—
97	C₂²×C₄	—	127	2	no	—
98	C₂²×C₄	—	127	1	no	—
99	C₂²×C₄	—	127	16	no	—
100	C₂²×C₄	—	127	8	no	—
101	C₂²×C₄	—	127	2	no	—
102	C₂²×C₄	—	127	8	no	—
103	C₂²×C₄	—	127	2	no	—
104	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
105	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
106	C₂×C₈	—	127	8	no	—
107	C₂×C₈	—	127	4	no	—
108	C₂×C₈	—	127	32	no	—
109	C₂×C₈	C₄×D₁₂₇	1	8	yes	direct product factor
110	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
111	C₄²	—	127	8	no	—
112	C₄²	—	127	2	no	—
113	C₄²	—	127	2	no	—
114	C₄²	C₄×D₁₂₇	1	2	yes	—
115	C₄²	C₂²×D₁₂₇	1	2	yes	—
116	C₂³	—	127	8	no	—
117	C₂³	—	127	2	no	—
118	C₂³	—	127	4	no	—
119	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
120	C₂×C₄	—	127	8	no	—
121	C₂×C₄	—	127	2	no	—
122	C₂×C₄	—	127	4	no	—
123	C₂×C₄	—	127	4	no	—
124	C₂×C₄	—	127	32	no	—
125	C₂×C₄	—	127	8	no	—
126	C₂×C₄	—	127	8	no	—
127	C₂×C₄	C₈×D₁₂₇	1	8	yes	direct product factor
128	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
129	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
130	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
131	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
132	C₈	—	127	8	no	—
133	C₈	C₂×C₄×D₁₂₇	1	8	yes	direct product factor
134	C₂²	—	127	4	no	—
135	C₂²	—	127	16	no	—
136	C₂²	—	127	8	no	—
137	C₂²	C₄²×D₁₂₇	1	2	yes	—
138	C₂²	C₂×C₈×D₁₂₇	1	4	yes	—
139	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
140	C₄	—	127	4	no	—
141	C₄	—	127	8	no	—
142	C₄	C₄²×D₁₂₇	1	2	yes	—
143	C₄	C₂×C₈×D₁₂₇	1	8	yes	direct product factor
144	C₄	C₂²×C₄×D₁₂₇	1	2	yes	—
145	C₂	—	127	8	no	—
146	C₂	C₄×C₈×D₁₂₇	1	4	yes	direct product factor
147	C₂	C₂×C₄²×D₁₂₇	1	1	yes	characteristic
148	C₂	C₂²×C₈×D₁₂₇	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	D₁₂₇
commutator, hypocenter	C₁₂₇	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂³×D₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
LCS	G ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₁₂₇⋊C₈	C₂	semidirect
4. C₂×C₈×Dic₁₂₇	C₂	semidirect
5. C₄×C₈×D₁₂₇	C₂	direct (N × H)
6. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
10. C₂×C₈×D₁₂₇	C₄	direct (N × H)
16. C₄×C₁₂₇⋊C₈	C₂²	semidirect
17. C₈×Dic₁₂₇	C₂²	semidirect
20. C₄×C₈×C₁₂₇	C₂²	semidirect
24. C₂×C₄×D₁₂₇	C₈	direct (N × H)
29. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
31. C₈×D₁₂₇	C₂×C₄	direct (N × H)
37. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
45. C₂×Dic₁₂₇	C₂×C₈	semidirect
49. C₄×D₁₂₇	C₂×C₈	direct (N × H)
51. C₁₂₇⋊C₈	C₂²×C₄	semidirect
53. C₂×C₄×C₁₂₇	C₂×C₈	semidirect
58. C₈×C₁₂₇	C₂²×C₄	semidirect
62. C₂×D₁₂₇	C₄×C₈	direct (N × H)
66. C₄×C₁₂₇	C₂²×C₈	semidirect
69. Dic₁₂₇	C₂²×C₈	semidirect
70. C₂×C₁₂₇	C₂×C₄×C₈	semidirect
73. D₁₂₇	C₂×C₄×C₈	direct (N × H)
75. C₁₂₇	C₂²×C₄×C₈	semidirect
81. C₂×C₄×C₈	D₁₂₇	direct (N × H)
95. C₄×C₈	C₂×D₁₂₇	direct (N × H)
109. C₂×C₈	C₄×D₁₂₇	direct (N × H)
127. C₂×C₄	C₈×D₁₂₇	direct (N × H)
133. C₈	C₂×C₄×D₁₂₇	direct (N × H)
143. C₄	C₂×C₈×D₁₂₇	direct (N × H)
146. C₂	C₄×C₈×D₁₂₇	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×C₈×D₁₂₇	2
C₂×C₄²×D₁₂₇	2
C₂×C₄×C₁₂₇⋊C₈	2
C₂×C₈×Dic₁₂₇	2
C₄×C₈×D₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
C₂²×C₄×C₈	127

Subgroups — C2×C4×C8×D127 #30647129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₈×D₁₂₇ #30647129