Subgroups — C₂×C₄×C₁₁×C₁₂₈ #72189754

Order 11264.

378 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₁×C₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₁×C₁₂₈	C₂	1	2	yes	maximal
3	C₄×C₁₁×C₁₂₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₂₈	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₁₁×C₆₄	C₄	1	1	yes	characteristic
6	C₂²×C₁₁×C₆₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₁×C₃₂	C₄	1	1	yes	characteristic
8	C₂×C₁₁×C₁₂₈	C₄	1	8	yes	direct product factor
9	C₂×C₁₁×C₁₂₈	C₂²	1	4	yes	—
10	C₄×C₁₁×C₆₄	C₄	1	2	yes	—
11	C₄×C₁₁×C₆₄	C₂²	1	2	yes	—
12	C₂²×C₁₁×C₃₂	C₈	1	1	yes	characteristic
13	C₂²×C₁₁×C₃₂	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₁×C₁₆	C₈	1	1	yes	characteristic
15	C₂×C₁₁×C₆₄	C₈	1	4	yes	—
16	C₂×C₁₁×C₆₄	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₁×C₆₄	C₂×C₄	1	2	yes	—
18	C₂×C₁₁×C₆₄	C₂×C₄	1	4	yes	—
19	C₂×C₁₁×C₆₄	C₂³	1	1	yes	characteristic
20	C₄×C₁₁×C₃₂	C₈	1	2	yes	—
21	C₄×C₁₁×C₃₂	C₂×C₄	1	2	yes	—
22	C₁₁×C₁₂₈	C₂×C₄	1	8	yes	direct product factor
23	C₂²×C₁₁×C₁₆	C₁₆	1	1	yes	characteristic
24	C₂²×C₁₁×C₁₆	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₁	C₁₆	1	1	yes	characteristic
26	C₂×C₁₁×C₃₂	C₁₆	1	4	yes	—
27	C₂×C₁₁×C₃₂	C₄²	1	2	yes	—
28	C₂×C₁₁×C₃₂	C₂×C₈	1	1	yes	characteristic
29	C₂×C₁₁×C₃₂	C₂×C₈	1	2	yes	—
30	C₂×C₁₁×C₃₂	C₂×C₈	1	2	yes	—
31	C₂×C₁₁×C₃₂	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₁₁×C₁₆	C₁₆	1	2	yes	—
33	C₄×C₁₁×C₁₆	C₂×C₈	1	2	yes	—
34	C₁₁×C₆₄	C₄²	1	2	yes	—
35	C₁₁×C₆₄	C₂×C₈	1	4	yes	—
36	C₁₁×C₆₄	C₂²×C₄	1	2	yes	—
37	C₂²×C₁₂₈	C₂×C₁₁	1	2	yes	—
38	C₂×C₄×C₆₄	C₂×C₁₁	1	1	yes	characteristic
39	C₄×C₁₂₈	C₂×C₁₁	1	4	yes	direct product factor
40	C₂²×C₈×C₁₁	C₃₂	1	1	yes	characteristic
41	C₂²×C₈×C₁₁	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄²×C₁₁	C₃₂	1	1	yes	characteristic
43	C₂×C₁₁×C₁₆	C₃₂	1	4	yes	—
44	C₂×C₁₁×C₁₆	C₄×C₈	1	2	yes	—
45	C₂×C₁₁×C₁₆	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₁₁×C₁₆	C₂×C₁₆	1	2	yes	—
47	C₂×C₁₁×C₁₆	C₂×C₁₆	1	2	yes	—
48	C₂×C₁₁×C₁₆	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₈×C₁₁	C₃₂	1	2	yes	—
50	C₄×C₈×C₁₁	C₂×C₁₆	1	2	yes	—
51	C₁₁×C₃₂	C₄×C₈	1	2	yes	—
52	C₁₁×C₃₂	C₂×C₁₆	1	4	yes	—
53	C₁₁×C₃₂	C₂×C₄²	1	1	yes	characteristic
54	C₁₁×C₃₂	C₂²×C₈	1	1	yes	characteristic
55	C₂²×C₆₄	C₄×C₁₁	1	1	yes	characteristic
56	C₂²×C₆₄	C₂²×C₁₁	1	1	yes	characteristic
57	C₂×C₄×C₃₂	C₄×C₁₁	1	1	yes	characteristic
58	C₂×C₁₂₈	C₄×C₁₁	1	8	yes	direct product factor
59	C₂×C₁₂₈	C₂²×C₁₁	1	4	yes	—
60	C₄×C₆₄	C₄×C₁₁	1	2	yes	—
61	C₄×C₆₄	C₂²×C₁₁	1	2	yes	—
62	C₂²×C₄×C₁₁	C₆₄	1	2	yes	—
63	C₂²×C₄×C₁₁	C₂×C₃₂	1	1	yes	characteristic
64	C₂×C₈×C₁₁	C₆₄	1	4	yes	—
65	C₂×C₈×C₁₁	C₄×C₁₆	1	2	yes	—
66	C₂×C₈×C₁₁	C₂×C₃₂	1	1	yes	characteristic
67	C₂×C₈×C₁₁	C₂×C₃₂	1	2	yes	—
68	C₂×C₈×C₁₁	C₂×C₃₂	1	2	yes	—
69	C₂×C₈×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
70	C₄²×C₁₁	C₆₄	1	2	yes	—
71	C₄²×C₁₁	C₂×C₃₂	1	2	yes	—
72	C₁₁×C₁₆	C₄×C₁₆	1	2	yes	—
73	C₁₁×C₁₆	C₂×C₃₂	1	4	yes	—
74	C₁₁×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
75	C₁₁×C₁₆	C₂²×C₁₆	1	1	yes	characteristic
76	C₂²×C₃₂	C₈×C₁₁	1	1	yes	characteristic
77	C₂²×C₃₂	C₂×C₄×C₁₁	1	1	yes	characteristic
78	C₂×C₄×C₁₆	C₈×C₁₁	1	1	yes	characteristic
79	C₂×C₆₄	C₈×C₁₁	1	4	yes	—
80	C₂×C₆₄	C₂×C₄×C₁₁	1	1	yes	characteristic
81	C₂×C₆₄	C₂×C₄×C₁₁	1	2	yes	—
82	C₂×C₆₄	C₂×C₄×C₁₁	1	4	yes	—
83	C₂×C₆₄	C₂³×C₁₁	1	1	yes	characteristic
84	C₄×C₃₂	C₈×C₁₁	1	2	yes	—
85	C₄×C₃₂	C₂×C₄×C₁₁	1	2	yes	—
86	C₁₂₈	C₂×C₄×C₁₁	1	8	yes	direct product factor
87	C₂³×C₁₁	C₂×C₆₄	1	1	yes	characteristic
88	C₂×C₄×C₁₁	C₁₂₈	1	8	yes	direct product factor
89	C₂×C₄×C₁₁	C₄×C₃₂	1	2	yes	—
90	C₂×C₄×C₁₁	C₂×C₆₄	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₂×C₆₄	1	2	yes	—
92	C₂×C₄×C₁₁	C₂×C₆₄	1	4	yes	—
93	C₂×C₄×C₁₁	C₂²×C₃₂	1	1	yes	characteristic
94	C₈×C₁₁	C₄×C₃₂	1	2	yes	—
95	C₈×C₁₁	C₂×C₆₄	1	4	yes	—
96	C₈×C₁₁	C₂×C₄×C₁₆	1	1	yes	characteristic
97	C₈×C₁₁	C₂²×C₃₂	1	1	yes	characteristic
98	C₂²×C₁₆	C₁₁×C₁₆	1	1	yes	characteristic
99	C₂²×C₁₆	C₂×C₈×C₁₁	1	1	yes	characteristic
100	C₂×C₄×C₈	C₁₁×C₁₆	1	1	yes	characteristic
101	C₂×C₃₂	C₁₁×C₁₆	1	4	yes	—
102	C₂×C₃₂	C₄²×C₁₁	1	2	yes	—
103	C₂×C₃₂	C₂×C₈×C₁₁	1	1	yes	characteristic
104	C₂×C₃₂	C₂×C₈×C₁₁	1	2	yes	—
105	C₂×C₃₂	C₂×C₈×C₁₁	1	2	yes	—
106	C₂×C₃₂	C₂²×C₄×C₁₁	1	1	yes	characteristic
107	C₄×C₁₆	C₁₁×C₁₆	1	2	yes	—
108	C₄×C₁₆	C₂×C₈×C₁₁	1	2	yes	—
109	C₆₄	C₄²×C₁₁	1	2	yes	—
110	C₆₄	C₂×C₈×C₁₁	1	4	yes	—
111	C₆₄	C₂²×C₄×C₁₁	1	2	yes	—
112	C₂²×C₁₁	C₄×C₆₄	1	2	yes	—
113	C₂²×C₁₁	C₂×C₁₂₈	1	4	yes	—
114	C₂²×C₁₁	C₂²×C₆₄	1	1	yes	characteristic
115	C₄×C₁₁	C₄×C₆₄	1	2	yes	—
116	C₄×C₁₁	C₂×C₁₂₈	1	8	yes	direct product factor
117	C₄×C₁₁	C₂×C₄×C₃₂	1	1	yes	characteristic
118	C₄×C₁₁	C₂²×C₆₄	1	1	yes	characteristic
119	C₂²×C₈	C₁₁×C₃₂	1	1	yes	characteristic
120	C₂²×C₈	C₂×C₁₁×C₁₆	1	1	yes	characteristic
121	C₂×C₄²	C₁₁×C₃₂	1	1	yes	characteristic
122	C₂×C₁₆	C₁₁×C₃₂	1	4	yes	—
123	C₂×C₁₆	C₄×C₈×C₁₁	1	2	yes	—
124	C₂×C₁₆	C₂×C₁₁×C₁₆	1	1	yes	characteristic
125	C₂×C₁₆	C₂×C₁₁×C₁₆	1	2	yes	—
126	C₂×C₁₆	C₂×C₁₁×C₁₆	1	2	yes	—
127	C₂×C₁₆	C₂²×C₈×C₁₁	1	1	yes	characteristic
128	C₄×C₈	C₁₁×C₃₂	1	2	yes	—
129	C₄×C₈	C₂×C₁₁×C₁₆	1	2	yes	—
130	C₃₂	C₄×C₈×C₁₁	1	2	yes	—
131	C₃₂	C₂×C₁₁×C₁₆	1	4	yes	—
132	C₃₂	C₂×C₄²×C₁₁	1	1	yes	characteristic
133	C₃₂	C₂²×C₈×C₁₁	1	1	yes	characteristic
134	C₂×C₁₁	C₄×C₁₂₈	1	4	yes	direct product factor
135	C₂×C₁₁	C₂×C₄×C₆₄	1	1	yes	characteristic
136	C₂×C₁₁	C₂²×C₁₂₈	1	2	yes	—
137	C₂²×C₄	C₁₁×C₆₄	1	2	yes	—
138	C₂²×C₄	C₂×C₁₁×C₃₂	1	1	yes	characteristic
139	C₂×C₈	C₁₁×C₆₄	1	4	yes	—
140	C₂×C₈	C₄×C₁₁×C₁₆	1	2	yes	—
141	C₂×C₈	C₂×C₁₁×C₃₂	1	1	yes	characteristic
142	C₂×C₈	C₂×C₁₁×C₃₂	1	2	yes	—
143	C₂×C₈	C₂×C₁₁×C₃₂	1	2	yes	—
144	C₂×C₈	C₂²×C₁₁×C₁₆	1	1	yes	characteristic
145	C₄²	C₁₁×C₆₄	1	2	yes	—
146	C₄²	C₂×C₁₁×C₃₂	1	2	yes	—
147	C₁₆	C₄×C₁₁×C₁₆	1	2	yes	—
148	C₁₆	C₂×C₁₁×C₃₂	1	4	yes	—
149	C₁₆	C₂×C₄×C₈×C₁₁	1	1	yes	characteristic
150	C₁₆	C₂²×C₁₁×C₁₆	1	1	yes	characteristic
151	C₁₁	C₂×C₄×C₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
152	C₂³	C₂×C₁₁×C₆₄	1	1	yes	characteristic
153	C₂×C₄	C₁₁×C₁₂₈	1	8	yes	direct product factor
154	C₂×C₄	C₄×C₁₁×C₃₂	1	2	yes	—
155	C₂×C₄	C₂×C₁₁×C₆₄	1	1	yes	characteristic
156	C₂×C₄	C₂×C₁₁×C₆₄	1	2	yes	—
157	C₂×C₄	C₂×C₁₁×C₆₄	1	4	yes	—
158	C₂×C₄	C₂²×C₁₁×C₃₂	1	1	yes	characteristic
159	C₈	C₄×C₁₁×C₃₂	1	2	yes	—
160	C₈	C₂×C₁₁×C₆₄	1	4	yes	—
161	C₈	C₂×C₄×C₁₁×C₁₆	1	1	yes	characteristic
162	C₈	C₂²×C₁₁×C₃₂	1	1	yes	characteristic
163	C₂²	C₄×C₁₁×C₆₄	1	2	yes	—
164	C₂²	C₂×C₁₁×C₁₂₈	1	4	yes	—
165	C₂²	C₂²×C₁₁×C₆₄	1	1	yes	characteristic
166	C₄	C₄×C₁₁×C₆₄	1	2	yes	—
167	C₄	C₂×C₁₁×C₁₂₈	1	8	yes	direct product factor
168	C₄	C₂×C₄×C₁₁×C₃₂	1	1	yes	characteristic
169	C₄	C₂²×C₁₁×C₆₄	1	1	yes	characteristic
170	C₂	C₄×C₁₁×C₁₂₈	1	4	yes	direct product factor
171	C₂	C₂²×C₁₁×C₁₂₈	1	2	yes	—
172	C₂	C₂×C₄×C₁₁×C₆₄	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₆₄	C₂³×C₁₁
socle, prime radical	C₂³×C₁₁	C₂×C₆₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₁₁×C₁₂₈	C₂	direct (N × H)
4. C₂×C₄×C₁₂₈	C₁₁	direct (N × H)
8. C₂×C₁₁×C₁₂₈	C₄	direct (N × H)
22. C₁₁×C₁₂₈	C₂×C₄	direct (N × H)
39. C₄×C₁₂₈	C₂×C₁₁	direct (N × H)
58. C₂×C₁₂₈	C₄×C₁₁	direct (N × H)
86. C₁₂₈	C₂×C₄×C₁₁	direct (N × H)
88. C₂×C₄×C₁₁	C₁₂₈	direct (N × H)
116. C₄×C₁₁	C₂×C₁₂₈	direct (N × H)
134. C₂×C₁₁	C₄×C₁₂₈	direct (N × H)
151. C₁₁	C₂×C₄×C₁₂₈	direct (N × H)
153. C₂×C₄	C₁₁×C₁₂₈	direct (N × H)
167. C₄	C₂×C₁₁×C₁₂₈	direct (N × H)
170. C₂	C₄×C₁₁×C₁₂₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₁×C₆₄	2
C₂²×C₁₁×C₁₂₈	2
C₄×C₁₁×C₁₂₈	2
C₂×C₄×C₁₂₈	11

Subgroups — C2×C4×C11×C128 #72189754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₁×C₁₂₈ #72189754