Subgroups — C₂×D₅₃⋊M₆₄ #28328605

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811556	C₂	1	2	yes	maximal, complemented
2	D₅₃⋊M₆₄	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16811555	C₂	1	2	yes	maximal, complemented
4	C₂×D₅₃⋊C₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811556	C₂²	1	8	yes	complemented
7	#16811555	C₂²	1	8	yes	complemented
8	D₅₃⋊C₃₂	C₂²	1	8	yes	complemented
9	C₂×#13133267	C₂²	1	4	yes	complemented
10	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
11	C₂²×#11719908	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₅₃	C₄	1	4	yes	—
13	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
14	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
15	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₅₃	C₂²	1	4	yes	—
17	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
19	#13133267	C₂³	1	8	yes	complemented
20	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
21	C₂²×#88322	C₈	1	1	yes	characteristic
22	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
23	C₂×C₈×D₅₃	C₈	1	4	yes	—
24	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
25	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719908	C₂×C₄	1	2	yes	—
29	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719908	C₂³	1	1	yes	characteristic
31	C₂×#11719908	C₂³	1	2	yes	—
32	C₁₆×D₅₃	C₂×C₄	1	2	yes	—
33	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
34	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
35	C₁₆×D₅₃	C₂×C₄	1	2	yes	—
36	C₁₆×D₅₃	C₂³	1	2	yes	—
37	C₁₆×D₅₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×D₅₃	C₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₁₆	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₁₆	1	4	yes	—
47	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
48	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
50	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×#88322	C₂×C₈	1	4	yes	—
56	C₂×#88322	C₂×C₈	1	1	yes	characteristic
57	C₂×#88322	C₂×C₈	1	1	yes	characteristic
58	C₈×D₅₃	C₂×C₈	1	4	yes	—
59	C₈×D₅₃	C₂×C₈	1	2	yes	—
60	C₈×D₅₃	C₂×C₈	1	2	yes	—
61	C₈×D₅₃	C₂×C₈	1	4	yes	—
62	C₈×D₅₃	C₂²×C₄	1	2	yes	—
63	C₈×D₅₃	C₂²×C₄	1	2	yes	—
64	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
65	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
66	#11719908	C₂²×C₄	1	1	yes	characteristic
67	#11719908	C₂²×C₄	1	2	yes	—
68	#11719908	C₂⁴	1	1	yes	characteristic
69	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
70	C₂²×D₅₃	—	2	4	no	—
71	C₂²×D₅₃	C₂×C₁₆	1	1	yes	characteristic
72	C₂²×D₅₃	C₂×C₁₆	1	4	yes	—
73	C₂²×D₅₃	C₂×C₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
79	C₂×Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
80	C₂×Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
81	C₄×D₅₃	C₂×C₁₆	1	4	yes	—
82	C₄×D₅₃	C₂×C₁₆	1	2	yes	—
83	C₄×D₅₃	C₂×C₁₆	1	2	yes	—
84	C₄×D₅₃	C₂×C₁₆	1	4	yes	—
85	C₄×D₅₃	C₂²×C₈	1	2	yes	—
86	C₄×D₅₃	C₂²×C₈	1	2	yes	—
87	C₈×C₅₃	C₂²×C₈	1	2	yes	—
88	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	#88322	C₂²×C₈	1	2	yes	—
91	#88322	C₂²×C₈	1	2	yes	—
92	C₂²×M₆₄	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
95	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
96	C₂×D₅₃	—	2	2	no	—
97	C₂×D₅₃	—	2	8	no	—
98	C₂×D₅₃	M₆₄	1	4	yes	complemented
99	C₂×D₅₃	C₂²×C₁₆	1	2	yes	—
100	C₂×D₅₃	C₂²×C₁₆	1	2	yes	—
101	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
102	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
103	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
104	Dic₅₃	C₂²×C₁₆	1	2	yes	—
105	Dic₅₃	C₂²×C₁₆	1	2	yes	—
106	C₂³×C₁₆	—	53	1	no	—
107	C₂×M₆₄	—	53	2	no	—
108	C₂×M₆₄	—	53	8	no	—
109	C₂×M₆₄	—	53	2	no	—
110	C₂²×C₃₂	—	53	2	no	—
111	C₂×C₅₃	—	2	2	no	—
112	C₂×C₅₃	C₂×M₆₄	1	2	yes	complemented
113	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
114	D₅₃	—	2	2	no	—
115	D₅₃	C₂×M₆₄	1	4	yes	complemented
116	C₂³×C₈	—	53	1	no	—
117	C₂²×C₁₆	—	53	1	no	—
118	C₂²×C₁₆	—	53	4	no	—
119	C₂²×C₁₆	—	53	1	no	—
120	C₂²×C₁₆	—	53	1	no	—
121	C₂²×C₁₆	—	53	4	no	—
122	C₂²×C₁₆	—	53	1	no	—
123	C₂²×C₁₆	—	53	1	no	—
124	C₂²×C₁₆	#6157	1	1	yes	characteristic
125	M₆₄	—	53	8	no	—
126	M₆₄	—	53	8	no	—
127	C₂×C₃₂	—	53	4	no	—
128	C₂×C₃₂	—	53	8	no	—
129	C₅₃	C₂²×M₆₄	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₄	—	53	1	no	—
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	4	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	53	1	no	—
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	53	4	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
139	C₂×C₁₆	—	53	2	no	—
140	C₂×C₁₆	—	53	1	no	—
141	C₂×C₁₆	—	53	1	no	—
142	C₂×C₁₆	—	53	2	no	—
143	C₂×C₁₆	—	53	2	no	—
144	C₂×C₁₆	—	53	4	no	—
145	C₂×C₁₆	—	53	2	no	—
146	C₂×C₁₆	—	53	4	no	—
147	C₂×C₁₆	—	53	2	no	—
148	C₂×C₁₆	—	53	2	no	—
149	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
150	C₂×C₁₆	C₂×#6157	1	4	yes	—
151	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
152	C₃₂	—	53	8	no	—
153	C₂⁴	—	53	1	no	—
154	C₂²×C₄	—	53	1	no	—
155	C₂²×C₄	—	53	1	no	—
156	C₂²×C₄	—	53	4	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	53	4	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
162	C₂×C₈	—	53	4	no	—
163	C₂×C₈	—	53	1	no	—
164	C₂×C₈	—	53	1	no	—
165	C₂×C₈	—	53	4	no	—
166	C₂×C₈	—	53	4	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	—	53	2	no	—
171	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
172	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
173	C₂×C₈	C₂²×#6157	1	2	yes	—
174	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
175	C₁₆	—	53	1	no	—
176	C₁₆	—	53	1	no	—
177	C₁₆	—	53	2	no	—
178	C₁₆	C₂²×#6157	1	2	yes	—
179	C₁₆	C₂²×#6157	1	2	yes	—
180	C₂³	—	53	1	no	—
181	C₂³	—	53	4	no	—
182	C₂³	—	53	1	no	—
183	C₂³	—	106	4	no	—
184	C₂³	D₅₃⋊C₁₆	1	1	yes	characteristic
185	C₂×C₄	—	53	4	no	—
186	C₂×C₄	—	53	1	no	—
187	C₂×C₄	—	53	1	no	—
188	C₂×C₄	—	53	4	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	2	no	—
191	C₂×C₄	—	53	4	no	—
192	C₂×C₄	—	53	2	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
195	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
196	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
197	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
198	C₈	—	53	2	no	—
199	C₈	—	53	2	no	—
200	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
201	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
202	C₈	C₂³×#6157	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	—	53	4	no	—
205	C₂²	—	53	2	no	—
206	C₂²	—	53	2	no	—
207	C₂²	—	106	8	no	—
208	C₂²	—	106	2	no	—
209	C₂²	C₂×D₅₃⋊C₁₆	1	2	yes	—
210	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
211	C₄	—	53	2	no	—
212	C₄	—	53	2	no	—
213	C₄	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
214	C₄	C₂×D₅₃⋊C₁₆	1	2	yes	—
215	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
216	C₂	—	2	2	no	—
217	C₂	—	53	4	no	—
218	C₂	—	106	2	no	—
219	C₂	D₅₃⋊M₆₄	1	2	yes	direct product factor
220	C₂	C₂²×D₅₃⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×#6157
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₈	C₂³×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂³×D₅₃	C₁₆
cosocle	#11719908	C₂⁴
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₂²×M₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16811556	C₂	semidirect
2. D₅₃⋊M₆₄	C₂	direct (N × H)
3. C₂×#16811555	C₂	semidirect
4. C₂×D₅₃⋊C₃₂	C₂	semidirect
6. #16811556	C₂²	semidirect
7. #16811555	C₂²	semidirect
8. D₅₃⋊C₃₂	C₂²	semidirect
9. C₂×#13133267	C₂²	semidirect
19. #13133267	C₂³	semidirect
98. C₂×D₅₃	M₆₄	semidirect
112. C₂×C₅₃	C₂×M₆₄	semidirect
115. D₅₃	C₂×M₆₄	semidirect
129. C₅₃	C₂²×M₆₄	semidirect
219. C₂	D₅₃⋊M₆₄	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×#16811556	2
D₅₃⋊M₆₄	2
C₂×#16811555	2
C₂×D₅₃⋊C₃₂	2
C₂²×C₁₆×D₅₃	2
C₂²×M₆₄	53

Subgroups — C2×D53⋊M64 #28328605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅₃⋊M₆₄ #28328605