Subgroups — #27282806

Order 13568.

290 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275490	C₂	1	2	yes	maximal
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	#20272611	C₂	1	2	yes	maximal
5	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133152	—	2	2	no	—
7	#16810647	C₂²	1	2	yes	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	#16810387	—	2	2	no	—
10	#16810376	—	2	1	no	—
11	#16810376	—	2	1	no	—
12	#16810374	C₄	1	2	yes	—
13	#16810374	C₄	1	2	yes	—
14	C₅₃×#431	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#322	—	2	1	no	—
18	C₅₃×#322	—	2	1	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂²×#88322	—	2	2	no	—
21	#13133152	—	4	4	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
27	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	1	no	—
32	C₂²⋊C₈×C₅₃	Q₈	1	2	yes	—
33	#97×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	—	2	1	no	—
35	C₄×C₈×C₅₃	—	2	1	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	M₁₆×C₅₃	—	2	2	no	—
44	M₁₆×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	4	1	no	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₃	#41	1	1	yes	characteristic
57	C₄²×C₅₃	#41	1	1	yes	characteristic
58	#11719917	—	4	2	no	—
59	#11719917	—	4	2	no	—
60	C₂×#88322	—	2	4	no	—
61	C₂×#88322	—	4	2	no	—
62	C₂×#88322	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	4	1	no	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
74	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₅₃	#137	1	2	yes	—
76	C₈×C₅₃	—	2	1	no	—
77	C₈×C₅₃	—	2	1	no	—
78	C₈×C₅₃	—	2	1	no	—
79	C₈×C₅₃	—	4	2	no	—
80	C₈×C₅₃	—	4	1	no	—
81	C₈×C₅₃	—	4	1	no	—
82	C₈×C₅₃	#108	1	2	yes	complemented
83	#88322	—	4	4	no	—
84	#88322	—	8	2	no	—
85	#9626	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	#389	1	1	yes	characteristic
89	C₂²×C₅₃	#501	1	2	yes	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
94	C₄×C₅₃	#430	1	1	yes	characteristic
95	C₄×C₅₃	#440	1	2	yes	—
96	C₄×C₅₃	#469	1	1	yes	characteristic
97	C₄×C₅₃	#470	1	1	yes	characteristic
98	#1950	—	53	2	no	—
99	#1567	—	53	1	no	—
100	C₂×#334	—	53	1	no	—
101	#1451	D₅₃	1	1	yes	characteristic
102	#1301	—	53	2	no	—
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	#1944	1	1	yes	characteristic
105	C₂×C₅₃	#1950	1	1	yes	characteristic
106	C₂×C₅₃	#1953	1	1	yes	characteristic
107	#431	C₂×D₅₃	1	1	yes	characteristic
108	C₂×#108	—	106	2	no	—
109	#423	—	53	2	no	—
110	C₂×C₄⋊C₈	—	53	2	no	—
111	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
113	#341	—	106	2	no	—
114	#335	—	106	1	no	—
115	#335	—	106	1	no	—
116	#334	—	53	2	no	—
117	#334	—	53	2	no	—
118	#322	—	2	1	no	—
119	#322	—	2	1	no	—
120	C₅₃	#9626	1	1	yes	characteristic, complemented, Sylow
121	C₂×M₁₆	—	106	2	no	—
122	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
123	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
124	C₂²×C₈	—	106	2	no	—
125	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
126	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	#108	—	212	4	no	—
129	C₄⋊C₈	—	2	1	no	—
130	C₄⋊C₈	—	53	4	no	—
131	C₄⋊C₈	—	106	2	no	—
132	C₄⋊C₈	—	106	2	no	—
133	C₄⋊C₈	—	106	2	no	—
134	C₂²⋊C₈	Dic₁₀₆	1	2	yes	—
135	#97	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
139	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
142	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	212	2	no	—
148	M₁₆	—	212	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	106	4	no	—
155	C₂×C₈	—	212	2	no	—
156	C₂×C₈	—	212	2	no	—
157	C₂×C₈	—	212	2	no	—
158	C₂×C₈	#11719919	1	1	yes	characteristic
159	C₂×C₈	#11719919	1	1	yes	characteristic
160	C₂×C₈	#11719919	1	1	yes	characteristic
161	C₂×C₈	#11719919	1	1	yes	characteristic
162	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
163	C₄²	#11719945	1	1	yes	characteristic
164	C₄²	#11719945	1	1	yes	characteristic
165	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂³	#13133217	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
171	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
172	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
173	C₂×C₄	#13133191	1	1	yes	characteristic
174	C₂×C₄	#13133196	1	1	yes	characteristic
175	C₂×C₄	#13133209	1	1	yes	characteristic
176	C₂×C₄	#13133245	1	2	yes	—
177	C₂×C₄	#13133250	1	2	yes	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	4	2	no	—
182	C₈	—	4	1	no	—
183	C₈	—	4	1	no	—
184	C₈	—	424	2	no	—
185	C₈	—	212	4	no	—
186	C₈	#13133152	1	2	yes	complemented
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	#16810521	1	1	yes	characteristic
190	C₂²	#16811032	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#13133152	1	1	yes	characteristic
195	C₄	#16810671	1	1	yes	characteristic
196	C₄	#16810725	1	2	yes	—
197	C₄	#16810876	1	1	yes	characteristic
198	C₄	#16810881	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	#20275458	1	1	yes	characteristic
201	C₂	#20275490	1	1	yes	characteristic
202	C₂	#20275505	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#9626
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1451

Complements

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

N	H ≅ G/N	type
82. C₈×C₅₃	#108	semidirect
120. C₅₃	#9626	semidirect
186. C₈	#13133152	semidirect

Maximal subgroups

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

subgroup	index
#20275490	2
#20273474	2
C₂×#16810374	2
#20272611	2
C₅₃×#1451	2
#9626	53