Subgroups — C₂²×#16810446 #27537099

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133149	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810446	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	1	no	—
6	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
7	C₂×#13133153	—	2	3	no	—
8	C₂×#13133153	C₂²	1	6	yes	—
9	C₂×#13133149	—	2	3	no	—
10	C₂×#13133149	C₂²	1	6	yes	—
11	#16810446	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₅₃	C₄	1	6	yes	complemented
14	C₂×M₃₂×C₅₃	C₂²	1	6	yes	—
15	C₂²×C₁₆×C₅₃	C₄	1	2	yes	complemented
16	C₂³×Dic₅₃	—	4	1	no	—
17	C₂×#11719917	—	2	6	no	—
18	C₂×#11719917	—	4	3	no	—
19	C₂²×#88322	—	4	1	no	—
20	C₂×C₄⋊Dic₅₃	—	2	6	no	—
21	C₂×C₄⋊Dic₅₃	—	4	3	no	—
22	#13133153	—	2	6	no	—
23	#13133153	C₂³	1	4	yes	—
24	#13133149	—	2	6	no	—
25	#13133149	C₂³	1	4	yes	—
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
29	C₂²×C₈×C₅₃	D₄	1	3	yes	—
30	C₂²×C₈×C₅₃	Q₈	1	3	yes	—
31	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
33	M₃₂×C₅₃	C₂×C₄	1	12	yes	complemented
34	M₃₂×C₅₃	C₂³	1	4	yes	—
35	C₂×C₁₆×C₅₃	C₂×C₄	1	12	yes	complemented
36	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	SD₁₆	1	3	yes	—
40	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	SD₁₆	1	3	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
43	C₂²×Dic₅₃	—	4	6	no	—
44	C₂²×Dic₅₃	—	8	4	no	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	3	yes	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	6	yes	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	3	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
51	C₂×C₈×C₅₃	C₂×Q₈	1	3	yes	—
52	C₂×C₈×C₅₃	C₂×Q₈	1	3	yes	—
53	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₄⋊Dic₅₃	—	2	4	no	—
55	C₄⋊Dic₅₃	—	4	6	no	—
56	#11719917	—	2	4	no	—
57	#11719917	—	4	6	no	—
58	C₂×#88322	—	4	6	no	—
59	C₁₆×C₅₃	C₂²×C₄	1	8	yes	complemented
60	C₂³×C₅₃	—	2	4	no	—
61	C₂³×C₅₃	#106	1	3	yes	—
62	C₂³×C₅₃	#106	1	1	yes	characteristic
63	C₂³×C₅₃	C₂×SD₁₆	1	3	yes	—
64	C₂×C₄×C₅₃	#106	1	6	yes	—
65	C₂×C₄×C₅₃	#106	1	3	yes	—
66	C₂×C₄×C₅₃	#106	1	3	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₅₃	C₂×SD₁₆	1	3	yes	—
70	C₂×C₄×C₅₃	C₂×SD₁₆	1	3	yes	—
71	C₂×C₄×C₅₃	C₂×SD₁₆	1	3	yes	—
72	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×Dic₅₃	—	4	4	no	—
74	C₂×Dic₅₃	—	8	12	no	—
75	C₈×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
76	C₈×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
78	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
79	#88322	—	4	4	no	—
80	C₂²×#365	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	—	2	6	no	—
82	C₂²×C₅₃	—	2	6	no	—
83	C₂²×C₅₃	#365	1	4	yes	complemented
84	C₂²×C₅₃	C₂×#106	1	3	yes	—
85	C₂²×C₅₃	C₂×#106	1	3	yes	—
86	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
87	C₄×C₅₃	C₂×#106	1	3	yes	—
88	C₄×C₅₃	C₂×#106	1	3	yes	—
89	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
91	Dic₅₃	—	8	4	no	—
92	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
93	C₂²×#108	—	53	1	no	—
94	C₂²×#106	—	53	1	no	—
95	C₂×#365	—	53	12	no	—
96	C₂×C₅₃	—	2	1	no	—
97	C₂×C₅₃	—	2	3	no	—
98	C₂×C₅₃	C₂×#365	1	6	yes	complemented
99	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
100	C₂²×M₁₆	—	106	1	no	—
101	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₂²×C₄⋊C₄	—	106	1	no	—
103	C₂×M₃₂	Dic₅₃	1	6	yes	complemented
104	C₂×M₃₂	C₂×D₅₃	1	6	yes	—
105	C₂²×C₁₆	Dic₅₃	1	2	yes	complemented
106	C₂×#108	—	53	6	no	—
107	C₂×#108	—	106	3	no	—
108	C₂×#106	—	53	6	no	—
109	C₂×#106	—	106	3	no	—
110	#365	—	53	16	no	—
111	C₅₃	C₂²×#365	1	1	yes	characteristic, complemented, Sylow
112	C₂³×C₄	—	212	1	no	—
113	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
114	C₂×M₁₆	—	106	6	no	—
115	C₂×M₁₆	—	212	3	no	—
116	C₂²×C₈	—	212	1	no	—
117	C₂²×C₈	Dic₁₀₆	1	3	yes	—
118	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
119	C₂²×C₈	D₂₁₂	1	3	yes	—
120	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
122	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
123	C₂×C₄⋊C₄	—	106	6	no	—
124	C₂×C₄⋊C₄	—	212	3	no	—
125	M₃₂	C₂×Dic₅₃	1	12	yes	complemented
126	M₃₂	C₂²×D₅₃	1	4	yes	—
127	C₂×C₁₆	C₂×Dic₅₃	1	12	yes	complemented
128	#108	—	53	4	no	—
129	#108	—	106	6	no	—
130	#106	—	53	4	no	—
131	#106	—	106	6	no	—
132	C₂⁴	#11719913	1	1	yes	characteristic
133	C₂²×C₄	—	424	4	no	—
134	C₂²×C₄	—	212	6	no	—
135	C₂²×C₄	#11719913	1	3	yes	—
136	C₂²×C₄	#11719913	1	3	yes	—
137	C₂²×C₄	#11719913	1	1	yes	characteristic
138	C₂²×C₄	C₄⋊Dic₅₃	1	3	yes	—
139	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
141	M₁₆	—	106	4	no	—
142	M₁₆	—	212	6	no	—
143	C₂×C₈	—	212	6	no	—
144	C₂×C₈	C₄⋊Dic₅₃	1	3	yes	—
145	C₂×C₈	C₄⋊Dic₅₃	1	6	yes	—
146	C₂×C₈	C₄⋊Dic₅₃	1	3	yes	—
147	C₂×C₈	C₂×Dic₁₀₆	1	3	yes	—
148	C₂×C₈	C₂×Dic₁₀₆	1	3	yes	—
149	C₂×C₈	C₂×D₂₁₂	1	3	yes	—
150	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
151	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
152	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
153	C₄⋊C₄	—	106	4	no	—
154	C₄⋊C₄	—	212	6	no	—
155	C₁₆	C₂²×Dic₅₃	1	8	yes	complemented
156	C₂³	—	2	4	no	—
157	C₂³	#13133149	1	3	yes	—
158	C₂³	#13133149	1	1	yes	characteristic
159	C₂³	C₂×#11719913	1	3	yes	—
160	C₂×C₄	—	424	12	no	—
161	C₂×C₄	—	212	4	no	—
162	C₂×C₄	#13133149	1	6	yes	—
163	C₂×C₄	#13133149	1	3	yes	—
164	C₂×C₄	#13133149	1	3	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
167	C₂×C₄	C₂×#11719913	1	3	yes	—
168	C₂×C₄	C₂×#11719913	1	3	yes	—
169	C₂×C₄	C₂×#11719913	1	3	yes	—
170	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
171	C₈	—	212	4	no	—
172	C₈	C₂×C₄⋊Dic₅₃	1	3	yes	—
173	C₈	C₂×C₄⋊Dic₅₃	1	3	yes	—
174	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
175	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
176	C₂²	—	2	6	no	—
177	C₂²	—	2	6	no	—
178	C₂²	#16810446	1	4	yes	direct product factor
179	C₂²	C₂×#13133149	1	3	yes	—
180	C₂²	C₂×#13133149	1	3	yes	—
181	C₂²	C₂²×#11719913	1	1	yes	characteristic
182	C₄	—	424	4	no	—
183	C₄	C₂×#13133149	1	3	yes	—
184	C₄	C₂×#13133149	1	3	yes	—
185	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
186	C₄	C₂²×#11719913	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	3	no	—
189	C₂	C₂×#16810446	1	6	yes	direct product factor
190	C₂	C₂²×#13133149	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133149
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	#106
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂²×#365
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×#16810446	C₂	direct (N × H)
11. #16810446	C₂²	direct (N × H)
13. C₂×M₃₂×C₅₃	C₄	semidirect
15. C₂²×C₁₆×C₅₃	C₄	semidirect
33. M₃₂×C₅₃	C₂×C₄	semidirect
35. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
59. C₁₆×C₅₃	C₂²×C₄	semidirect
83. C₂²×C₅₃	#365	semidirect
98. C₂×C₅₃	C₂×#365	semidirect
103. C₂×M₃₂	Dic₅₃	semidirect
105. C₂²×C₁₆	Dic₅₃	semidirect
111. C₅₃	C₂²×#365	semidirect
125. M₃₂	C₂×Dic₅₃	semidirect
127. C₂×C₁₆	C₂×Dic₅₃	semidirect
155. C₁₆	C₂²×Dic₅₃	semidirect
178. C₂²	#16810446	direct (N × H)
189. C₂	C₂×#16810446	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×#13133153	2
C₂²×#13133149	2
C₂×#16810446	2
C₂²×M₃₂×C₅₃	2
C₂²×#365	53

Subgroups — C22×#16810446 #27537099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810446 #27537099