Subgroups — M₆₄×#6157 #28312313

Order 13568.

250 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×#6157	C₂	1	1	yes	maximal, characteristic
3	#20289821	C₂	1	2	yes	maximal, complemented
4	C₃₂×#6157	C₂	1	2	yes	maximal, complemented
5	M₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₈×#6157	C₄	1	1	yes	characteristic
8	#16811556	C₄	1	2	yes	complemented
9	#16811555	C₄	1	2	yes	complemented
10	D₅₃⋊C₃₂	C₂²	1	2	yes	—
11	C₁₆×#6157	C₄	1	1	yes	characteristic
12	C₁₆×#6157	C₄	1	1	yes	characteristic
13	C₁₆×#6157	C₂²	1	1	yes	characteristic
14	C₁₆×#6157	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
16	#16810457	C₄	1	1	yes	characteristic, complemented
17	#16810456	C₄	1	2	yes	complemented
18	C₃₂×D₅₃	C₂²	1	2	yes	—
19	C₅₃×M₆₄	C₄	1	1	yes	characteristic, complemented
20	C₂×D₅₃⋊C₈	C₈	1	1	yes	characteristic
21	C₂×C₄×#6157	C₈	1	1	yes	characteristic
22	D₅₃⋊C₁₆	C₈	1	2	yes	—
23	D₅₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
24	D₅₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×#11719910	C₈	1	2	yes	—
26	C₈×#6157	C₈	1	1	yes	characteristic
27	C₈×#6157	C₈	1	1	yes	characteristic
28	C₈×#6157	C₂×C₄	1	1	yes	characteristic
29	C₈×#6157	C₂×C₄	1	1	yes	characteristic
30	#13133267	C₂×C₄	1	4	yes	complemented
31	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
33	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
36	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
37	#13133123	C₂×C₄	1	2	yes	complemented
38	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
40	C₂²×#6157	C₁₆	1	2	yes	—
41	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×#88324	C₁₆	1	2	yes	—
43	C₄×#6157	C₁₆	1	2	yes	—
44	C₄×#6157	C₂×C₈	1	1	yes	characteristic
45	C₄×#6157	C₂×C₈	1	1	yes	characteristic
46	D₅₃⋊C₈	C₁₆	1	2	yes	—
47	D₅₃⋊C₈	C₂×C₈	1	1	yes	characteristic
48	D₅₃⋊C₈	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
50	C₂×#88322	C₂×C₈	1	1	yes	characteristic
51	C₈×D₅₃	C₄²	1	1	yes	characteristic
52	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
53	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
54	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
55	#11719910	C₂×C₈	1	2	yes	—
56	#11719910	C₂×C₈	1	2	yes	—
57	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
58	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	#11719908	C₄²	1	1	yes	characteristic
60	#11719908	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₅₃	C₂×C₁₆	1	1	yes	characteristic
62	C₂×#6157	—	2	4	no	—
63	C₂×#6157	C₂×C₁₆	1	2	yes	—
64	C₂×#6157	C₂×C₁₆	1	2	yes	—
65	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
66	C₂×Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
67	C₄×D₅₃	C₄×C₈	1	1	yes	characteristic
68	C₄×D₅₃	C₂×C₁₆	1	1	yes	characteristic
69	C₄×D₅₃	C₂×C₁₆	1	1	yes	characteristic
70	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
71	#88324	C₂×C₁₆	1	2	yes	—
72	#88324	C₂×C₁₆	1	2	yes	—
73	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
74	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
75	#88322	C₄×C₈	1	1	yes	characteristic
76	#88322	C₂²×C₈	1	1	yes	characteristic
77	C₄×M₆₄	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
79	C₂×D₅₃	—	2	2	no	—
80	C₂×D₅₃	C₄×C₁₆	1	1	yes	characteristic
81	C₂×D₅₃	C₂²×C₁₆	1	1	yes	characteristic
82	#6157	—	2	2	no	—
83	#6157	M₆₄	1	4	yes	direct product factor
84	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
85	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
86	Dic₅₃	C₄×C₁₆	1	1	yes	characteristic
87	Dic₅₃	C₂²×C₁₆	1	1	yes	characteristic
88	C₂×M₆₄	—	53	1	no	—
89	C₂×M₆₄	—	53	1	no	—
90	C₂×C₄×C₁₆	—	53	1	no	—
91	#1397	—	53	2	no	—
92	C₄×C₃₂	—	53	2	no	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
95	D₅₃	—	2	1	no	—
96	D₅₃	C₂×M₆₄	1	2	yes	—
97	C₂²×C₁₆	—	53	1	no	—
98	C₂²×C₁₆	—	53	1	no	—
99	C₂×C₄×C₈	—	53	1	no	—
100	M₆₄	—	53	2	no	—
101	M₆₄	—	53	2	no	—
102	M₆₄	—	53	1	no	—
103	M₆₄	—	53	2	no	—
104	M₆₄	#6157	1	1	yes	characteristic, direct product factor
105	C₂×C₃₂	—	53	2	no	—
106	C₂×C₃₂	—	53	2	no	—
107	C₄×C₁₆	—	53	1	no	—
108	C₄×C₁₆	—	53	1	no	—
109	C₄×C₁₆	—	53	1	no	—
110	C₄×C₁₆	—	53	1	no	—
111	C₅₃	C₄×M₆₄	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	53	1	no	—
113	C₂²×C₈	—	53	1	no	—
114	C₂×C₄²	—	53	1	no	—
115	C₂×C₁₆	—	53	1	no	—
116	C₂×C₁₆	—	53	1	no	—
117	C₂×C₁₆	—	53	2	no	—
118	C₂×C₁₆	—	53	2	no	—
119	C₂×C₁₆	—	53	1	no	—
120	C₂×C₁₆	—	53	1	no	—
121	C₂×C₁₆	—	53	1	no	—
122	C₂×C₁₆	—	53	1	no	—
123	C₂×C₁₆	—	53	1	no	—
124	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
125	C₄×C₈	—	53	1	no	—
126	C₄×C₈	—	53	1	no	—
127	C₄×C₈	—	53	1	no	—
128	C₄×C₈	—	53	1	no	—
129	C₃₂	—	53	4	no	—
130	C₃₂	—	53	2	no	—
131	C₃₂	C₂×#6157	1	2	yes	complemented
132	C₂²×C₄	—	53	1	no	—
133	C₂²×C₄	—	53	2	no	—
134	C₂×C₈	—	53	2	no	—
135	C₂×C₈	—	53	1	no	—
136	C₂×C₈	—	53	1	no	—
137	C₂×C₈	—	53	2	no	—
138	C₂×C₈	—	53	1	no	—
139	C₂×C₈	—	53	1	no	—
140	C₂×C₈	—	53	1	no	—
141	C₂×C₈	—	53	1	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	C₄×#6157	1	1	yes	characteristic
144	C₄²	—	53	2	no	—
145	C₄²	—	53	1	no	—
146	C₄²	—	53	1	no	—
147	C₁₆	—	53	2	no	—
148	C₁₆	—	53	2	no	—
149	C₁₆	—	53	1	no	—
150	C₁₆	—	53	1	no	—
151	C₁₆	C₄×#6157	1	1	yes	characteristic
152	C₁₆	C₂²×#6157	1	1	yes	characteristic
153	C₂³	—	53	1	no	—
154	C₂×C₄	—	53	2	no	—
155	C₂×C₄	—	53	1	no	—
156	C₂×C₄	—	53	1	no	—
157	C₂×C₄	—	53	1	no	—
158	C₂×C₄	—	53	1	no	—
159	C₂×C₄	—	53	1	no	—
160	C₂×C₄	—	53	2	no	—
161	C₂×C₄	—	106	4	no	—
162	C₂×C₄	C₈×#6157	1	1	yes	characteristic
163	C₈	—	53	2	no	—
164	C₈	—	53	2	no	—
165	C₈	—	53	1	no	—
166	C₈	—	53	1	no	—
167	C₈	C₈×#6157	1	1	yes	characteristic
168	C₈	C₂×C₄×#6157	1	1	yes	characteristic
169	C₂²	—	53	1	no	—
170	C₂²	—	53	1	no	—
171	C₂²	—	106	2	no	—
172	C₂²	C₁₆×#6157	1	1	yes	characteristic
173	C₄	—	53	4	no	—
174	C₄	—	53	1	no	—
175	C₄	—	53	1	no	—
176	C₄	—	106	2	no	—
177	C₄	C₁₆×#6157	1	1	yes	characteristic
178	C₄	C₂×C₈×#6157	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	53	2	no	—
181	C₂	—	106	1	no	—
182	C₂	C₂×C₁₆×#6157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×#6157
commutator, socle	C₂×C₅₃	C₂×C₄×C₁₆
Fitting	C₅₃×M₆₄	C₄
prime radical	C₂²×D₅₃	C₂×C₁₆
cosocle	C₁₆×D₅₃	C₂³
hypercenter	M₆₄	#6157
hypocenter	C₅₃	C₄×M₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
3. #20289821	C₂	semidirect
4. C₃₂×#6157	C₂	semidirect
8. #16811556	C₄	semidirect
9. #16811555	C₄	semidirect
16. #16810457	C₄	semidirect
17. #16810456	C₄	semidirect
19. C₅₃×M₆₄	C₄	semidirect
30. #13133267	C₂×C₄	semidirect
37. #13133123	C₂×C₄	semidirect
39. C₃₂×C₅₃	C₂×C₄	semidirect
83. #6157	M₆₄	direct (N × H)
104. M₆₄	#6157	direct (N × H)
111. C₅₃	C₄×M₆₄	semidirect
131. C₃₂	C₂×#6157	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊M₆₄	2
C₂×C₁₆×#6157	2
#20289821	2
C₃₂×#6157	2
M₆₄×D₅₃	2
C₄×M₆₄	53

Subgroups — M64×#6157 #28312313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₆₄×#6157 #28312313