Subgroups — C₂×C₄×M₃₂×C₅₃ #27240834

Order 13568.

918 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₅₃	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₅₃	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#346	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₅₃	C₄	1	16	yes	direct product factor
10	C₂×M₃₂×C₅₃	C₂²	1	8	yes	—
11	C₂²×C₁₆×C₅₃	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#346	C₂²	1	8	yes	complemented
19	C₄×C₁₆×C₅₃	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₅₃	C₈	1	2	yes	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₈	1	8	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
31	C₂×C₄²×C₅₃	C₈	1	4	yes	—
32	C₂×C₄²×C₅₃	C₈	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₅₃	C₂×C₄	1	32	yes	direct product factor
37	C₂×C₁₆×C₅₃	C₂×C₄	1	16	yes	complemented
38	C₂×C₁₆×C₅₃	C₂³	1	8	yes	—
39	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	C₂³	1	2	yes	—
44	C₄×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	8	no	—
47	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄²	1	8	yes	—
57	C₂×C₈×C₅₃	C₄²	1	4	yes	—
58	C₂×C₈×C₅₃	C₂×C₈	1	16	yes	—
59	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×C₈	1	2	yes	—
69	C₄²×C₅₃	C₂×C₈	1	8	yes	—
70	C₄²×C₅₃	C₂×C₈	1	2	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₁₆×C₅₃	C₂²×C₄	1	16	yes	complemented
74	C₂³×C₅₃	—	2	4	no	—
75	C₂³×C₅₃	C₄×C₈	1	4	yes	—
76	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
77	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	16	no	—
80	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
81	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
82	C₂×C₄×C₅₃	M₃₂	1	8	yes	direct product factor
83	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
87	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₅₃	C₂×C₄²	1	2	yes	—
93	C₈×C₅₃	C₂×C₄²	1	4	yes	—
94	C₈×C₅₃	C₂²×C₈	1	4	yes	—
95	C₈×C₅₃	C₂²×C₈	1	4	yes	—
96	C₈×C₅₃	C₂³×C₄	1	2	yes	—
97	C₂×C₄×M₃₂	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₅₃	—	2	8	no	—
99	C₂²×C₅₃	—	2	4	no	—
100	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
101	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
102	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
103	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
106	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
107	C₄×C₅₃	C₂×M₃₂	1	8	yes	direct product factor
108	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
109	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
110	C₂²×M₃₂	C₂×C₅₃	1	2	yes	—
111	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
112	C₄×M₃₂	C₂×C₅₃	1	8	yes	direct product factor
113	C₂×#346	C₂×C₅₃	1	2	yes	complemented
114	C₂×C₄×C₁₆	C₂×C₅₃	1	2	yes	complemented
115	C₂×C₅₃	—	2	4	no	—
116	C₂×C₅₃	C₄×M₃₂	1	4	yes	direct product factor
117	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
118	C₂×C₅₃	C₂²×M₃₂	1	2	yes	—
119	C₂³×C₈	C₄×C₅₃	1	1	yes	characteristic
120	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
121	C₂²×C₄²	C₄×C₅₃	1	1	yes	characteristic
122	C₂×M₃₂	C₄×C₅₃	1	16	yes	direct product factor
123	C₂×M₃₂	C₂²×C₅₃	1	8	yes	—
124	C₂²×C₁₆	C₂²×C₅₃	1	4	yes	—
125	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	—
126	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
127	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
131	#346	C₂²×C₅₃	1	8	yes	complemented
132	C₄×C₁₆	C₂²×C₅₃	1	8	yes	complemented
133	C₅₃	C₂×C₄×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
134	C₂³×C₄	C₈×C₅₃	1	2	yes	—
135	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
136	C₂²×C₈	C₈×C₅₃	1	8	yes	—
137	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
138	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
139	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
141	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
142	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
145	C₂×C₄²	C₈×C₅₃	1	4	yes	—
146	C₂×C₄²	C₈×C₅₃	1	2	yes	—
147	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
148	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
149	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
150	M₃₂	C₂×C₄×C₅₃	1	32	yes	direct product factor
151	C₂×C₁₆	C₂×C₄×C₅₃	1	16	yes	complemented
152	C₂×C₁₆	C₂³×C₅₃	1	8	yes	—
153	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
154	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
155	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
156	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
157	C₄×C₈	C₂³×C₅₃	1	2	yes	—
158	C₄×C₈	C₂³×C₅₃	1	2	yes	—
159	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
162	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
163	C₂²×C₄	C₂×C₈×C₅₃	1	8	yes	—
164	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
165	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	—
167	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
168	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
169	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
170	C₂×C₈	C₄²×C₅₃	1	8	yes	—
171	C₂×C₈	C₄²×C₅₃	1	4	yes	—
172	C₂×C₈	C₂×C₈×C₅₃	1	16	yes	—
173	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
174	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
175	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
177	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
178	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
179	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
180	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
181	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
182	C₄²	C₂×C₈×C₅₃	1	2	yes	—
183	C₄²	C₂×C₈×C₅₃	1	8	yes	—
184	C₄²	C₂×C₈×C₅₃	1	2	yes	—
185	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
186	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
187	C₁₆	C₂²×C₄×C₅₃	1	16	yes	complemented
188	C₂³	—	2	4	no	—
189	C₂³	C₄×C₈×C₅₃	1	4	yes	—
190	C₂³	C₂²×C₈×C₅₃	1	2	yes	—
191	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	C₄×C₈×C₅₃	1	8	yes	—
195	C₂×C₄	C₄×C₈×C₅₃	1	4	yes	—
196	C₂×C₄	M₃₂×C₅₃	1	8	yes	direct product factor
197	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
198	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
199	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
200	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
201	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
202	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
203	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
205	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
206	C₈	C₂×C₄²×C₅₃	1	2	yes	—
207	C₈	C₂×C₄²×C₅₃	1	4	yes	—
208	C₈	C₂²×C₈×C₅₃	1	4	yes	—
209	C₈	C₂²×C₈×C₅₃	1	4	yes	—
210	C₈	C₂³×C₄×C₅₃	1	2	yes	—
211	C₂²	—	2	8	no	—
212	C₂²	—	2	4	no	—
213	C₂²	C₂×C₄×C₈×C₅₃	1	4	yes	—
214	C₂²	C₂×C₄×C₈×C₅₃	1	2	yes	—
215	C₂²	C₂×M₃₂×C₅₃	1	4	yes	—
216	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	C₂×C₄×C₈×C₅₃	1	2	yes	—
219	C₄	C₂×C₄×C₈×C₅₃	1	4	yes	—
220	C₄	C₂×M₃₂×C₅₃	1	8	yes	direct product factor
221	C₄	C₂²×C₄²×C₅₃	1	1	yes	characteristic
222	C₄	C₂³×C₈×C₅₃	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	C₄×M₃₂×C₅₃	1	4	yes	direct product factor
225	C₂	C₂²×C₄×C₈×C₅₃	1	1	yes	characteristic
226	C₂	C₂²×M₃₂×C₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈×C₅₃	C₂²
commutator	C₂	C₂²×C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×M₃₂×C₅₃	C₂	direct (N × H)
4. C₂×C₅₃×#346	C₂	semidirect
5. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
9. C₂×M₃₂×C₅₃	C₄	direct (N × H)
18. C₅₃×#346	C₂²	semidirect
19. C₄×C₁₆×C₅₃	C₂²	semidirect
36. M₃₂×C₅₃	C₂×C₄	direct (N × H)
37. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
73. C₁₆×C₅₃	C₂²×C₄	semidirect
82. C₂×C₄×C₅₃	M₃₂	direct (N × H)
97. C₂×C₄×M₃₂	C₅₃	direct (N × H)
107. C₄×C₅₃	C₂×M₃₂	direct (N × H)
112. C₄×M₃₂	C₂×C₅₃	direct (N × H)
113. C₂×#346	C₂×C₅₃	semidirect
114. C₂×C₄×C₁₆	C₂×C₅₃	semidirect
116. C₂×C₅₃	C₄×M₃₂	direct (N × H)
122. C₂×M₃₂	C₄×C₅₃	direct (N × H)
131. #346	C₂²×C₅₃	semidirect
132. C₄×C₁₆	C₂²×C₅₃	semidirect
133. C₅₃	C₂×C₄×M₃₂	direct (N × H)
150. M₃₂	C₂×C₄×C₅₃	direct (N × H)
151. C₂×C₁₆	C₂×C₄×C₅₃	semidirect
187. C₁₆	C₂²×C₄×C₅₃	semidirect
196. C₂×C₄	M₃₂×C₅₃	direct (N × H)
220. C₄	C₂×M₃₂×C₅₃	direct (N × H)
224. C₂	C₄×M₃₂×C₅₃	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×M₃₂×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₄×M₃₂×C₅₃	2
C₂×C₅₃×#346	2
C₂×C₄×C₁₆×C₅₃	2
C₂×C₄×M₃₂	53

Subgroups — C2×C4×M32×C53 #27240834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×M₃₂×C₅₃ #27240834