Subgroups — C₂×M₁₆×Dic₁₂₇ #30647137

Order 16256.

656 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523232	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	M₁₆×Dic₁₂₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16861119	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₂₇	C₄	1	2	yes	—
8	C₂×#14523232	C₄	1	8	yes	complemented
9	C₂×#14523232	C₂²	1	4	yes	—
10	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₂₇	C₄	1	4	yes	—
12	C₂×C₄×Dic₁₂₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₂₇	C₂²	1	4	yes	—
16	#16861119	C₂²	1	8	yes	complemented
17	C₈×Dic₁₂₇	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₁₂₇	C₄	1	8	yes	complemented
20	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
21	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
22	C₂²×Dic₁₂₇	—	2	8	no	—
23	C₂²×Dic₁₂₇	C₂×C₄	1	8	yes	—
24	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
25	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
26	#14523232	C₂×C₄	1	16	yes	complemented
27	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	complemented
28	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
29	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
30	C₄×Dic₁₂₇	C₂×C₄	1	8	yes	—
31	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₂₇	C₂³	1	2	yes	—
33	C₄×Dic₁₂₇	C₂³	1	2	yes	—
34	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	M₁₆×C₁₂₇	C₂×C₄	1	16	yes	complemented
42	C₂×C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
43	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
44	C₂×Dic₁₂₇	—	2	4	no	—
45	C₂×Dic₁₂₇	—	2	16	no	—
46	C₂×Dic₁₂₇	M₁₆	1	8	yes	direct product factor
47	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
48	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
49	C₁₂₇⋊C₈	C₂²×C₄	1	8	yes	complemented
50	C₂³×C₁₂₇	—	2	4	no	—
51	C₂³×C₁₂₇	C₄²	1	4	yes	—
52	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
53	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	C₄²	1	8	yes	—
55	C₂×C₄×C₁₂₇	C₄²	1	4	yes	—
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
63	C₈×C₁₂₇	C₂²×C₄	1	8	yes	complemented
64	C₂²×C₁₂₇	—	2	4	no	—
65	C₂²×C₁₂₇	—	2	8	no	—
66	C₂²×C₁₂₇	C₂×C₄²	1	4	yes	—
67	C₂²×C₁₂₇	C₂×C₄²	1	2	yes	—
68	C₂²×C₁₂₇	C₂×M₁₆	1	4	yes	—
69	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
70	C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
71	C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
72	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
74	Dic₁₂₇	—	2	4	no	—
75	Dic₁₂₇	C₂×M₁₆	1	8	yes	direct product factor
76	C₂×C₁₂₇	—	2	4	no	—
77	C₂×C₁₂₇	C₄×M₁₆	1	4	yes	complemented
78	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
79	C₂×C₁₂₇	C₂²×M₁₆	1	2	yes	—
80	C₂×C₄×M₁₆	—	127	1	no	maximal, Sylow
81	C₁₂₇	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂²×M₁₆	—	127	1	no	—
83	C₂²×M₁₆	D₁₂₇	1	1	yes	characteristic
84	C₂²×C₄²	—	127	1	no	—
85	C₄×M₁₆	—	127	8	no	—
86	C₂×#97	—	127	2	no	—
87	C₂×C₄×C₈	—	127	2	no	—
88	C₂³×C₄	—	127	2	no	—
89	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
90	C₂×M₁₆	—	127	8	no	—
91	C₂×M₁₆	—	127	4	no	—
92	C₂×M₁₆	Dic₁₂₇	1	8	yes	direct product factor
93	C₂×M₁₆	C₂×D₁₂₇	1	4	yes	—
94	C₂²×C₈	—	127	2	no	—
95	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
96	C₂×C₄²	—	127	2	no	—
97	C₂×C₄²	—	127	4	no	—
98	C₂×C₄²	—	127	4	no	—
99	C₂×C₄²	—	127	1	no	—
100	C₂×C₄²	—	127	1	no	—
101	#97	—	127	8	no	—
102	C₄×C₈	—	127	8	no	—
103	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
104	C₂²×C₄	—	127	2	no	—
105	C₂²×C₄	—	127	8	no	—
106	C₂²×C₄	—	127	2	no	—
107	C₂²×C₄	—	254	8	no	—
108	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
110	C₂²×C₄	C₄×D₁₂₇	1	4	yes	—
111	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
112	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
113	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
114	M₁₆	—	127	16	no	—
115	M₁₆	C₂×Dic₁₂₇	1	16	yes	direct product factor
116	C₂×C₈	—	127	8	no	—
117	C₂×C₈	—	127	4	no	—
118	C₂×C₈	C₂×Dic₁₂₇	1	8	yes	complemented
119	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
120	C₄²	—	127	2	no	—
121	C₄²	—	127	8	no	—
122	C₄²	—	127	2	no	—
123	C₄²	—	127	2	no	—
124	C₄²	—	127	2	no	—
125	C₂³	—	2	4	no	—
126	C₂³	C₄×Dic₁₂₇	1	4	yes	—
127	C₂³	C₂×C₄×D₁₂₇	1	2	yes	—
128	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
129	C₂×C₄	—	127	8	no	—
130	C₂×C₄	—	127	4	no	—
131	C₂×C₄	—	127	4	no	—
132	C₂×C₄	—	254	16	no	—
133	C₂×C₄	—	254	4	no	—
134	C₂×C₄	C₄×Dic₁₂₇	1	8	yes	—
135	C₂×C₄	C₄×Dic₁₂₇	1	4	yes	—
136	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
138	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
139	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
140	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
141	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
142	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
143	C₈	—	127	8	no	—
144	C₈	C₂²×Dic₁₂₇	1	8	yes	complemented
145	C₂²	—	2	8	no	—
146	C₂²	—	2	4	no	—
147	C₂²	C₂×C₄×Dic₁₂₇	1	4	yes	—
148	C₂²	C₂×C₄×Dic₁₂₇	1	2	yes	—
149	C₂²	M₁₆×D₁₂₇	1	4	yes	—
150	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
151	C₄	—	127	8	no	—
152	C₄	—	254	4	no	—
153	C₄	C₂×C₄×Dic₁₂₇	1	2	yes	—
154	C₄	C₂×C₄×Dic₁₂₇	1	4	yes	—
155	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
156	C₄	C₂³×Dic₁₂₇	1	1	yes	characteristic
157	C₂	—	2	4	no	—
158	C₂	M₁₆×Dic₁₂₇	1	4	yes	direct product factor
159	C₂	C₂²×C₄×Dic₁₂₇	1	1	yes	characteristic
160	C₂	C₂×M₁₆×D₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂²×M₁₆×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂²×M₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. M₁₆×Dic₁₂₇	C₂	direct (N × H)
4. C₂×#16861119	C₂	semidirect
5. C₂×C₈×Dic₁₂₇	C₂	semidirect
8. C₂×#14523232	C₄	semidirect
16. #16861119	C₂²	semidirect
17. C₈×Dic₁₂₇	C₂²	semidirect
19. C₂×M₁₆×C₁₂₇	C₄	semidirect
26. #14523232	C₂×C₄	semidirect
27. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
41. M₁₆×C₁₂₇	C₂×C₄	semidirect
42. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
46. C₂×Dic₁₂₇	M₁₆	direct (N × H)
49. C₁₂₇⋊C₈	C₂²×C₄	semidirect
63. C₈×C₁₂₇	C₂²×C₄	semidirect
75. Dic₁₂₇	C₂×M₁₆	direct (N × H)
77. C₂×C₁₂₇	C₄×M₁₆	semidirect
81. C₁₂₇	C₂×C₄×M₁₆	semidirect
92. C₂×M₁₆	Dic₁₂₇	direct (N × H)
115. M₁₆	C₂×Dic₁₂₇	direct (N × H)
118. C₂×C₈	C₂×Dic₁₂₇	semidirect
144. C₈	C₂²×Dic₁₂₇	semidirect
158. C₂	M₁₆×Dic₁₂₇	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×#14523232	2
C₂²×C₄×Dic₁₂₇	2
M₁₆×Dic₁₂₇	2
C₂×#16861119	2
C₂×C₈×Dic₁₂₇	2
C₂²×M₁₆×C₁₂₇	2
C₂×C₄×M₁₆	127

Subgroups — C2×M16×Dic127 #30647137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₆×Dic₁₂₇ #30647137