Subgroups — C₂²×#16810400 #27536745

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133127	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810400	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂×#13133157	C₄	1	6	yes	complemented
8	C₂×#13133157	C₂²	1	6	yes	—
9	C₂²×#11719908	C₄	1	2	yes	complemented
10	C₂×#13133127	C₄	1	6	yes	—
11	C₂×#13133127	C₂²	1	6	yes	—
12	#16810400	C₂²	1	16	yes	direct product factor
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₅₃	C₄	1	6	yes	complemented
15	C₂×M₃₂×C₅₃	C₂²	1	6	yes	—
16	C₂²×C₁₆×C₅₃	C₄	1	2	yes	complemented
17	C₂³×Dic₅₃	—	2	1	no	—
18	C₂²×#88322	—	2	3	no	—
19	C₂²×#88322	—	2	1	no	—
20	C₂²×#88322	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₅₃	—	2	3	no	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
23	#13133157	C₂×C₄	1	12	yes	complemented
24	#13133157	C₂³	1	4	yes	—
25	C₂×#11719908	C₂×C₄	1	12	yes	complemented
26	#13133127	C₂×C₄	1	12	yes	—
27	#13133127	C₂³	1	4	yes	—
28	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
35	M₃₂×C₅₃	C₂×C₄	1	12	yes	complemented
36	M₃₂×C₅₃	C₂³	1	4	yes	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	12	yes	complemented
38	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₄²	1	3	yes	—
40	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	M₁₆	1	3	yes	—
42	C₂²×C₄×C₅₃	M₁₆	1	3	yes	—
43	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
45	C₂²×Dic₅₃	—	2	6	no	—
46	C₂²×Dic₅₃	—	4	4	no	—
47	C₂×C₈×C₅₃	C₄²	1	3	yes	—
48	C₂×C₈×C₅₃	C₄²	1	3	yes	—
49	C₂×C₈×C₅₃	C₄²	1	6	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
52	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₄×Dic₅₃	—	2	6	no	—
57	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
58	C₂×#88322	—	2	6	no	—
59	C₂×#88322	—	2	6	no	—
60	C₂×#88322	C₂²×C₄	1	4	yes	—
61	C₁₆×C₅₃	C₂²×C₄	1	8	yes	complemented
62	#11719908	C₂²×C₄	1	8	yes	complemented
63	C₂³×C₅₃	—	2	4	no	—
64	C₂³×C₅₃	#97	1	3	yes	—
65	C₂³×C₅₃	#97	1	1	yes	characteristic
66	C₂³×C₅₃	C₂×M₁₆	1	3	yes	—
67	C₂×C₄×C₅₃	#97	1	6	yes	—
68	C₂×C₄×C₅₃	#97	1	3	yes	—
69	C₂×C₄×C₅₃	#97	1	3	yes	—
70	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
72	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
73	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
74	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	4	no	—
77	C₂×Dic₅₃	—	4	12	no	—
78	C₈×C₅₃	C₂×C₄²	1	3	yes	—
79	C₈×C₅₃	C₂×C₄²	1	3	yes	—
80	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
82	#88322	—	2	4	no	—
83	C₂²×#347	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	—	2	6	no	—
85	C₂²×C₅₃	—	2	6	no	—
86	C₂²×C₅₃	#347	1	4	yes	complemented
87	C₂²×C₅₃	C₂×#97	1	3	yes	—
88	C₂²×C₅₃	C₂×#97	1	3	yes	—
89	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
90	C₄×C₅₃	C₂×#97	1	3	yes	—
91	C₄×C₅₃	C₂×#97	1	3	yes	—
92	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
93	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
94	Dic₅₃	—	4	4	no	—
95	C₂²×M₃₂	—	53	1	no	—
96	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
97	C₂²×#97	—	53	1	no	—
98	C₂×#347	—	53	12	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	3	no	—
101	C₂×C₅₃	C₂×#347	1	6	yes	complemented
102	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
103	C₂³×C₈	—	53	1	no	—
104	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
105	C₂²×C₄²	—	53	1	no	—
106	C₂×M₃₂	—	53	6	no	—
107	C₂×M₃₂	—	53	6	no	—
108	C₂×M₃₂	Dic₅₃	1	6	yes	complemented
109	C₂×M₃₂	C₂×D₅₃	1	6	yes	—
110	C₂²×C₁₆	—	53	2	no	—
111	C₂²×C₁₆	Dic₅₃	1	2	yes	complemented
112	C₂×#97	—	53	6	no	—
113	C₂×#97	—	53	6	no	—
114	#347	—	53	16	no	—
115	C₅₃	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₄	—	106	1	no	—
117	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
118	C₂²×C₈	—	53	6	no	—
119	C₂²×C₈	—	106	1	no	—
120	C₂²×C₈	—	106	3	no	—
121	C₂²×C₈	C₄×D₅₃	1	3	yes	—
122	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
123	C₂²×C₈	C₄×D₅₃	1	3	yes	—
124	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
125	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
127	C₂×C₄²	—	53	6	no	—
128	C₂×C₄²	—	106	3	no	—
129	M₃₂	—	53	12	no	—
130	M₃₂	—	53	4	no	—
131	M₃₂	C₂×Dic₅₃	1	12	yes	complemented
132	M₃₂	C₂²×D₅₃	1	4	yes	—
133	C₂×C₁₆	—	53	12	no	—
134	C₂×C₁₆	C₂×Dic₅₃	1	12	yes	complemented
135	#97	—	53	12	no	—
136	#97	—	53	4	no	—
137	C₂⁴	#11719912	1	1	yes	characteristic
138	C₂²×C₄	—	106	6	no	—
139	C₂²×C₄	—	212	4	no	—
140	C₂²×C₄	#11719912	1	3	yes	—
141	C₂²×C₄	#11719912	1	1	yes	characteristic
142	C₂²×C₄	#11719912	1	3	yes	—
143	C₂²×C₄	C₄×Dic₅₃	1	3	yes	—
144	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
146	C₂×C₈	—	53	4	no	—
147	C₂×C₈	—	106	6	no	—
148	C₂×C₈	—	106	6	no	—
149	C₂×C₈	C₄×Dic₅₃	1	6	yes	—
150	C₂×C₈	C₄×Dic₅₃	1	3	yes	—
151	C₂×C₈	C₄×Dic₅₃	1	3	yes	—
152	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
153	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
154	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
155	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
156	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
157	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
158	C₄²	—	53	4	no	—
159	C₄²	—	106	6	no	—
160	C₁₆	—	53	8	no	—
161	C₁₆	C₂²×Dic₅₃	1	8	yes	complemented
162	C₂³	—	2	4	no	—
163	C₂³	#13133127	1	3	yes	—
164	C₂³	#13133127	1	1	yes	characteristic
165	C₂³	C₂×#11719912	1	3	yes	—
166	C₂×C₄	—	106	4	no	—
167	C₂×C₄	—	212	12	no	—
168	C₂×C₄	#13133127	1	6	yes	—
169	C₂×C₄	#13133127	1	3	yes	—
170	C₂×C₄	#13133127	1	3	yes	—
171	C₂×C₄	C₂×C₄×Dic₅₃	1	3	yes	—
172	C₂×C₄	C₂×C₄×Dic₅₃	1	3	yes	—
173	C₂×C₄	C₂×#11719912	1	3	yes	—
174	C₂×C₄	C₂×#11719912	1	3	yes	—
175	C₂×C₄	C₂×#11719912	1	3	yes	—
176	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
177	C₈	—	106	4	no	—
178	C₈	C₂×C₄×Dic₅₃	1	3	yes	—
179	C₈	C₂×C₄×Dic₅₃	1	3	yes	—
180	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
181	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
182	C₂²	—	2	6	no	—
183	C₂²	—	2	6	no	—
184	C₂²	#16810400	1	4	yes	direct product factor
185	C₂²	C₂×#13133127	1	3	yes	—
186	C₂²	C₂×#13133127	1	3	yes	—
187	C₂²	C₂²×#11719912	1	1	yes	characteristic
188	C₄	—	212	4	no	—
189	C₄	C₂×#13133127	1	3	yes	—
190	C₄	C₂×#13133127	1	3	yes	—
191	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
192	C₄	C₂²×#11719912	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	3	no	—
195	C₂	C₂×#16810400	1	6	yes	direct product factor
196	C₂	C₂²×#13133127	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	#97
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂²×#347
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×#16810400	C₂	direct (N × H)
7. C₂×#13133157	C₄	semidirect
9. C₂²×#11719908	C₄	semidirect
12. #16810400	C₂²	direct (N × H)
14. C₂×M₃₂×C₅₃	C₄	semidirect
16. C₂²×C₁₆×C₅₃	C₄	semidirect
23. #13133157	C₂×C₄	semidirect
25. C₂×#11719908	C₂×C₄	semidirect
35. M₃₂×C₅₃	C₂×C₄	semidirect
37. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
61. C₁₆×C₅₃	C₂²×C₄	semidirect
62. #11719908	C₂²×C₄	semidirect
86. C₂²×C₅₃	#347	semidirect
101. C₂×C₅₃	C₂×#347	semidirect
108. C₂×M₃₂	Dic₅₃	semidirect
111. C₂²×C₁₆	Dic₅₃	semidirect
115. C₅₃	C₂²×#347	semidirect
131. M₃₂	C₂×Dic₅₃	semidirect
134. C₂×C₁₆	C₂×Dic₅₃	semidirect
161. C₁₆	C₂²×Dic₅₃	semidirect
184. C₂²	#16810400	direct (N × H)
195. C₂	C₂×#16810400	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×#13133157	2
C₂²×#13133127	2
C₂×#16810400	2
C₂²×M₃₂×C₅₃	2
C₂²×#347	53

Subgroups — C22×#16810400 #27536745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810400 #27536745