Subgroups — C₂×#20276546 #27537255

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133156	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20276546	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810402	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
8	C₂×C₄×D₂₁₂	C₄	1	1	yes	characteristic
9	C₂×#13133156	—	2	8	no	—
10	C₂×#13133156	C₂²	1	8	yes	complemented
11	C₂²×#11719908	C₂²	1	1	yes	characteristic
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	C₈×D₂₁₂	C₄	1	4	yes	—
14	C₈×D₂₁₂	C₂²	1	4	yes	—
15	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
16	C₂×#13133129	C₄	1	2	yes	—
17	#16810402	C₂²	1	8	yes	complemented
18	C₄×#11719908	C₂²	1	4	yes	complemented
19	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
22	C₂²×D₂₁₂	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂×C₈×D₅₃	—	2	4	no	—
26	C₂×C₈×D₅₃	—	2	4	no	—
27	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₈×D₅₃	C₂³	1	4	yes	—
29	C₄×D₂₁₂	C₈	1	4	yes	—
30	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
32	C₂×#11719921	C₈	1	2	yes	—
33	#13133156	—	2	16	no	—
34	C₂×#11719908	—	2	2	no	—
35	C₂×#11719908	D₄	1	4	yes	complemented
36	C₂×#11719908	C₂³	1	2	yes	—
37	C₂×#11719908	C₂³	1	2	yes	—
38	C₄⋊#88322	C₂×C₄	1	4	yes	—
39	#13133129	C₂×C₄	1	8	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	—	2	4	no	—
44	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂³×D₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
52	C₂×D₂₁₂	—	2	4	no	—
53	C₂×D₂₁₂	C₂×C₈	1	4	yes	—
54	C₂×C₄×D₅₃	—	2	4	no	—
55	C₂×C₄×D₅₃	—	2	4	no	—
56	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
57	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₅₃	#41	1	2	yes	—
63	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×C₈	1	2	yes	—
65	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
66	#11719921	C₂×C₈	1	8	yes	—
67	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
68	C₂×#88322	—	2	4	no	—
69	C₂×#88322	C₂²×C₄	1	4	yes	—
70	C₈×D₅₃	—	2	8	no	—
71	C₈×D₅₃	—	2	4	no	—
72	C₈×D₅₃	—	2	4	no	—
73	C₁₆×C₅₃	—	2	4	no	—
74	#11719908	—	2	2	no	—
75	#11719908	C₂×D₄	1	4	yes	complemented
76	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂²×D₅₃	—	2	4	no	—
78	C₂²×D₅₃	—	2	4	no	—
79	C₂²×D₅₃	—	2	4	no	—
80	C₂²×D₅₃	C₂²×C₈	1	4	yes	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	M₃₂	1	4	yes	complemented
83	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₅₃	#131	1	2	yes	—
88	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×Dic₅₃	—	2	4	no	—
90	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
91	D₂₁₂	—	2	8	no	—
92	C₄×D₅₃	—	2	4	no	—
93	C₄×D₅₃	—	2	4	no	—
94	C₄×D₅₃	—	2	8	no	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	C₄×D₄	1	2	yes	—
97	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
99	#88322	—	2	4	no	—
100	C₂×#2169	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	C₈×D₄	1	2	yes	—
102	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
103	C₂²×C₅₃	#504	1	2	yes	—
104	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
105	C₂×D₅₃	—	2	4	no	—
106	C₂×D₅₃	—	2	4	no	—
107	C₂×D₅₃	—	2	4	no	—
108	C₂×D₅₃	—	4	8	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	C₈×D₄	1	2	yes	—
111	C₄×C₅₃	C₂×M₃₂	1	4	yes	complemented
112	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
113	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
114	Dic₅₃	—	2	4	no	—
115	C₂²×M₃₂	—	53	2	no	—
116	C₂×C₈×D₄	—	53	1	no	—
117	#2169	—	53	8	no	—
118	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
119	C₂×C₂²⋊C₁₆	—	53	2	no	—
120	C₂×C₄×C₁₆	—	53	1	no	—
121	C₂×C₅₃	#2169	1	4	yes	complemented
122	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
123	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
124	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
125	D₅₃	—	4	4	no	—
126	C₂³×C₈	—	53	2	no	—
127	C₂×C₄×D₄	—	53	1	no	—
128	C₂×M₃₂	—	53	8	no	—
129	C₂×M₃₂	—	106	8	no	—
130	C₂²×C₁₆	—	53	1	no	—
131	C₂²×C₁₆	—	53	1	no	—
132	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
133	C₈×D₄	—	53	4	no	—
134	C₈×D₄	—	53	4	no	—
135	C₂×C₄⋊C₈	—	53	1	no	—
136	C₂×C₂²⋊C₈	—	53	2	no	—
137	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
138	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	complemented
139	C₂²⋊C₁₆	—	53	8	no	—
140	C₄×C₁₆	—	53	4	no	—
141	C₅₃	C₂×#2169	1	1	yes	characteristic, complemented, Sylow
142	C₂²×D₄	—	53	1	no	—
143	C₂³×C₄	—	53	2	no	—
144	C₂²×C₈	—	53	2	no	—
145	C₂²×C₈	—	53	4	no	—
146	C₂²×C₈	—	53	4	no	—
147	C₂²×C₈	—	106	4	no	—
148	C₂²×C₈	—	106	4	no	—
149	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
151	C₄×D₄	—	53	4	no	—
152	C₄×D₄	—	53	4	no	—
153	C₂×C₄⋊C₄	—	53	1	no	—
154	C₂×C₂²⋊C₄	—	53	2	no	—
155	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
156	M₃₂	—	106	16	no	—
157	C₂×C₁₆	—	2	4	no	—
158	C₂×C₁₆	—	53	4	no	—
159	C₂×C₁₆	—	53	2	no	—
160	C₂×C₁₆	—	53	2	no	—
161	C₂×C₁₆	—	106	2	no	—
162	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
163	C₄⋊C₈	—	53	4	no	—
164	C₂²⋊C₈	—	53	8	no	—
165	C₄×C₈	C₄×D₅₃	1	2	yes	—
166	C₄×C₈	C₂²×D₅₃	1	2	yes	—
167	C₂⁴	—	53	2	no	—
168	C₂×D₄	—	53	4	no	—
169	C₂×D₄	—	106	4	no	—
170	C₂²×C₄	—	53	2	no	—
171	C₂²×C₄	—	53	4	no	—
172	C₂²×C₄	—	53	4	no	—
173	C₂²×C₄	—	106	4	no	—
174	C₂²×C₄	—	106	4	no	—
175	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	53	4	no	—
180	C₂×C₈	—	106	4	no	—
181	C₂×C₈	—	106	8	no	—
182	C₂×C₈	—	106	4	no	—
183	C₂×C₈	—	106	4	no	—
184	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
185	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
186	C₂×C₈	D₄×D₅₃	1	2	yes	—
187	C₂×C₈	#11719949	1	2	yes	—
188	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
189	C₄⋊C₄	—	53	4	no	—
190	C₂²⋊C₄	—	53	8	no	—
191	C₄²	C₈×D₅₃	1	2	yes	—
192	C₄²	C₂×C₄×D₅₃	1	2	yes	—
193	C₁₆	—	2	4	no	—
194	C₁₆	—	53	4	no	—
195	C₁₆	—	106	2	no	—
196	C₂³	—	53	4	no	—
197	C₂³	—	106	4	no	—
198	C₂³	—	106	4	no	—
199	C₂³	—	106	4	no	—
200	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
201	D₄	—	106	8	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	53	4	no	—
204	C₂×C₄	—	106	4	no	—
205	C₂×C₄	—	106	4	no	—
206	C₂×C₄	—	106	4	no	—
207	C₂×C₄	—	106	8	no	—
208	C₂×C₄	#13133156	1	4	yes	complemented
209	C₂×C₄	#13133179	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
212	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
213	C₂×C₄	#13133225	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	106	4	no	—
217	C₈	#13133179	1	2	yes	—
218	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
219	C₈	C₂×#11719949	1	1	yes	characteristic
220	C₂²	—	106	4	no	—
221	C₂²	—	106	4	no	—
222	C₂²	—	106	4	no	—
223	C₂²	—	212	8	no	—
224	C₂²	#16810688	1	2	yes	—
225	C₂²	C₂×#13133156	1	2	yes	—
226	C₂²	#16811041	1	2	yes	—
227	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	106	4	no	—
230	C₄	#16810688	1	2	yes	—
231	C₄	C₂×#13133156	1	4	yes	complemented
232	C₄	C₂×#13133179	1	1	yes	characteristic
233	C₄	C₂×#13133225	1	1	yes	characteristic
234	C₂	—	212	4	no	—
235	C₂	#20276546	1	4	yes	direct product factor
236	C₂	C₂×#16810688	1	1	yes	characteristic
237	C₂	C₂²×#13133156	1	1	yes	characteristic
238	C₂	C₂×#16811041	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂²×D₂₁₂	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	C₂×#2169
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133156	C₂	semidirect
3. #20276546	C₂	direct (N × H)
4. C₂×#16810402	C₂	semidirect
5. C₂×C₄×#11719908	C₂	semidirect
6. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
10. C₂×#13133156	C₂²	semidirect
17. #16810402	C₂²	semidirect
18. C₄×#11719908	C₂²	semidirect
21. C₅₃×C₄⋊C₁₆	C₂²	semidirect
35. C₂×#11719908	D₄	semidirect
75. #11719908	C₂×D₄	semidirect
82. C₂×C₄×C₅₃	M₃₂	semidirect
111. C₄×C₅₃	C₂×M₃₂	semidirect
118. C₂×C₄⋊C₁₆	D₅₃	semidirect
121. C₂×C₅₃	#2169	semidirect
138. C₄⋊C₁₆	C₂×D₅₃	semidirect
141. C₅₃	C₂×#2169	semidirect
208. C₂×C₄	#13133156	semidirect
231. C₄	C₂×#13133156	semidirect
235. C₂	#20276546	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×#13133156	2
C₂×C₈×D₂₁₂	2
#20276546	2
C₂×#16810402	2
C₂×C₄×#11719908	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×#2169	53

Subgroups — C2×#20276546 #27537255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276546 #27537255