Subgroups — #27304295

Order 13568.

290 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811039	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810457	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272854	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133223	C₄	1	1	yes	characteristic
6	#16811039	—	2	1	no	—
7	#16811039	—	2	1	no	—
8	#16811039	—	2	1	no	—
9	#16811039	—	2	1	no	—
10	C₂×#13133157	C₄	1	1	yes	characteristic
11	C₂×#13133156	C₂²	1	2	yes	—
12	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
13	#16810457	—	2	1	no	—
14	#16810457	C₂²	1	2	yes	—
15	C₂×#13133123	—	2	1	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×M₆₄	—	2	1	no	—
18	C₅₃×M₆₄	C₂²	1	2	yes	—
19	C₂×C₃₂×C₅₃	—	2	1	no	—
20	C₂×#11719945	C₈	1	1	yes	characteristic
21	#13133223	—	2	1	no	—
22	#13133223	—	2	1	no	—
23	#13133223	—	2	1	no	—
24	#13133223	—	2	1	no	—
25	C₂×#11719917	C₈	1	1	yes	characteristic
26	C₂×#11719912	C₈	1	2	yes	—
27	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
28	#13133157	—	2	1	no	—
29	#13133157	—	2	1	no	—
30	#13133156	—	2	2	no	—
31	#13133156	—	2	2	no	—
32	C₂×#11719908	C₂×C₄	1	2	yes	—
33	C₁₆×D₅₃	—	2	2	no	—
34	C₁₆×D₅₃	—	2	2	no	—
35	#13133123	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
43	C₃₂×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₂²⋊D₅₃	C₁₆	1	2	yes	—
46	#11719945	—	2	1	no	—
47	#11719945	—	2	1	no	—
48	#11719945	—	2	1	no	—
49	#11719945	—	2	1	no	—
50	C₂×D₂₁₂	C₁₆	1	1	yes	characteristic
51	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
52	C₂×Dic₁₀₆	C₁₆	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
57	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	#11719917	—	2	1	no	—
60	#11719917	—	2	1	no	—
61	C₂×#88322	C₂×C₈	1	2	yes	—
62	#11719912	—	2	2	no	—
63	#11719912	—	2	2	no	—
64	C₈×D₅₃	—	2	2	no	—
65	C₈×D₅₃	—	2	2	no	—
66	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
70	#11719908	—	2	2	no	—
71	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
72	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
77	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
79	C₂²⋊D₅₃	—	2	2	no	—
80	C₂²⋊D₅₃	—	2	2	no	—
81	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
82	D₂₁₂	—	2	1	no	—
83	D₂₁₂	—	2	1	no	—
84	C₄×D₅₃	—	2	2	no	—
85	C₄×D₅₃	—	2	2	no	—
86	Dic₁₀₆	—	2	1	no	—
87	Dic₁₀₆	—	2	1	no	—
88	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
89	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
90	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	#88322	—	2	2	no	—
93	#13556	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	4	1	no	—
95	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
96	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
97	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
98	C₂×D₅₃	—	2	2	no	—
99	C₂×D₅₃	—	4	2	no	—
100	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
101	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
104	Dic₅₃	—	2	2	no	—
105	C₂×#504	—	53	1	no	—
106	C₂×M₆₄	—	53	1	no	—
107	C₂×M₆₄	D₅₃	1	1	yes	characteristic, complemented
108	#1400	—	53	4	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	—	2	1	no	—
111	C₂×C₅₃	—	2	1	no	—
112	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
113	D₅₃	—	4	2	no	—
114	C₂×#131	—	53	1	no	—
115	#504	—	106	1	no	—
116	#504	—	106	1	no	—
117	#504	—	106	1	no	—
118	#504	—	106	1	no	—
119	C₂×M₃₂	—	53	1	no	—
120	C₂×M₃₂	—	53	2	no	—
121	C₂²×C₁₆	—	53	2	no	—
122	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
123	M₆₄	—	2	1	no	—
124	M₆₄	—	53	2	no	—
125	M₆₄	—	106	1	no	—
126	M₆₄	C₂×D₅₃	1	2	yes	—
127	C₂×C₃₂	—	2	1	no	—
128	C₂×C₃₂	—	106	1	no	—
129	C₅₃	#13556	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	—	53	1	no	—
131	#131	—	106	1	no	—
132	#131	—	106	1	no	—
133	#131	—	106	1	no	—
134	#131	—	106	1	no	—
135	C₂×M₁₆	—	53	1	no	—
136	C₂×M₁₆	—	53	2	no	—
137	C₂²×C₈	—	53	2	no	—
138	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
139	M₃₂	—	106	1	no	—
140	M₃₂	—	106	1	no	—
141	M₃₂	—	106	2	no	—
142	M₃₂	—	106	2	no	—
143	C₂×C₁₆	—	53	2	no	—
144	C₂×C₁₆	—	106	2	no	—
145	C₂×C₁₆	—	106	2	no	—
146	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
147	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
148	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
149	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
152	C₃₂	—	2	2	no	—
153	C₃₂	—	106	2	no	—
154	#41	—	106	1	no	—
155	#41	—	106	1	no	—
156	#41	—	106	1	no	—
157	#41	—	106	1	no	—
158	C₂×Q₈	—	53	1	no	—
159	C₂×D₄	—	53	2	no	—
160	C₂×D₄	—	53	1	no	—
161	C₂²×C₄	—	53	2	no	—
162	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
163	M₁₆	—	106	1	no	—
164	M₁₆	—	106	1	no	—
165	M₁₆	—	106	2	no	—
166	M₁₆	—	106	2	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
171	C₂×C₈	#11719912	1	1	yes	characteristic
172	C₂×C₈	#11719912	1	1	yes	characteristic
173	C₂×C₈	#11719921	1	1	yes	characteristic
174	C₂×C₈	#11719921	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
176	C₁₆	—	106	2	no	—
177	C₁₆	#11719921	1	1	yes	characteristic
178	C₁₆	#11719921	1	1	yes	characteristic
179	C₁₆	C₂×D₂₁₂	1	1	yes	characteristic
180	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂³	—	53	2	no	—
182	C₂³	C₁₆×D₅₃	1	1	yes	characteristic
183	Q₈	—	106	1	no	—
184	Q₈	—	106	1	no	—
185	D₄	—	106	2	no	—
186	D₄	—	106	2	no	—
187	D₄	—	106	1	no	—
188	D₄	—	106	1	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	106	2	no	—
191	C₂×C₄	—	106	2	no	—
192	C₂×C₄	#13133129	1	1	yes	characteristic
193	C₂×C₄	#13133129	1	1	yes	characteristic
194	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
195	C₂×C₄	#13133156	1	1	yes	characteristic
196	C₂×C₄	#13133156	1	1	yes	characteristic
197	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
198	C₈	—	106	2	no	—
199	C₈	#13133129	1	1	yes	characteristic
200	C₈	#13133129	1	1	yes	characteristic
201	C₈	C₂×#11719921	1	1	yes	characteristic
202	C₈	C₂×#11719912	1	1	yes	characteristic
203	C₂²	—	4	1	no	—
204	C₂²	—	106	2	no	—
205	C₂²	—	212	2	no	—
206	C₂²	#16810402	1	1	yes	characteristic
207	C₂²	#16810402	1	1	yes	characteristic
208	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
209	C₄	—	106	2	no	—
210	C₄	#16810402	1	1	yes	characteristic
211	C₄	#16810402	1	1	yes	characteristic
212	C₄	C₂×#13133129	1	1	yes	characteristic
213	C₄	C₂×#13133156	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	212	2	no	—
218	C₂	C₂×#16810402	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#11719921
commutator	C₂²×C₅₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂×C₅₃	C₂×C₂²⋊C₁₆
prime radical	C₂×#11719945	C₈
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	#13556
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#16810457	C₂	semidirect
3. #20272854	C₂	semidirect
4. C₂×C₅₃×M₆₄	C₂	semidirect
107. C₂×M₆₄	D₅₃	semidirect
129. C₅₃	#13556	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16811039	2
C₂×#16810457	2
#20272854	2
C₂×C₅₃×M₆₄	2
#13556	53