Subgroups — #27295702

Order 13568.

246 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276513	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₈×#11719912	C₂	1	1	yes	maximal, characteristic
4	#20272785	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272784	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272778	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1367	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₅₃	—	2	1	no	—
9	#16810674	C₄	1	1	yes	characteristic
10	C₄×#11719912	C₄	1	1	yes	characteristic
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊#11719908	C₂²	1	1	yes	characteristic
13	C₄⋊#11719908	C₂²	1	1	yes	characteristic
14	#16810441	—	2	1	no	—
15	#16810402	—	2	1	no	—
16	C₁₆×Dic₅₃	—	2	1	no	—
17	#16810345	C₄	1	1	yes	characteristic
18	#16810344	C₄	1	1	yes	characteristic
19	C₈×#88322	C₂²	1	1	yes	characteristic
20	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₈²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719912	C₈	1	1	yes	characteristic
24	C₂×#11719912	C₈	1	1	yes	characteristic
25	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
26	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
27	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719908	—	2	1	no	—
29	C₂×#11719908	—	2	1	no	—
30	C₁₆×D₅₃	—	4	2	no	—
31	#13133146	—	2	1	no	—
32	#13133129	—	2	1	no	—
33	#13133127	C₈	1	1	yes	characteristic
34	#13133127	C₈	1	1	yes	characteristic
35	C₄×#88322	C₂×C₄	1	1	yes	characteristic
36	C₄×#88322	C₂×C₄	1	1	yes	characteristic
37	C₄×#88322	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅₃	D₄	1	1	yes	characteristic
39	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
40	C₂×C₁₆×C₅₃	—	2	1	no	—
41	C₂×C₁₆×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
56	C₄×Dic₅₃	M₁₆	1	1	yes	characteristic
57	C₂×#88322	—	2	1	no	—
58	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
59	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
60	C₂×#88322	C₂×C₈	1	1	yes	characteristic
61	C₂×#88322	C₂×C₈	1	1	yes	characteristic
62	#11719912	—	2	2	no	—
63	#11719912	C₁₆	1	4	yes	complemented
64	C₈×D₅₃	—	2	1	no	—
65	C₈×D₅₃	—	2	1	no	—
66	C₈×D₅₃	—	4	1	no	—
67	C₁₆×C₅₃	—	4	1	no	—
68	C₁₆×C₅₃	—	4	1	no	—
69	#11719908	—	4	1	no	—
70	#11719908	—	4	1	no	—
71	C₂²×D₅₃	C₄⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
76	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
77	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
78	C₄×D₅₃	—	2	1	no	—
79	C₄×D₅₃	—	2	1	no	—
80	C₄×D₅₃	—	4	1	no	—
81	C₄×D₅₃	C₂×C₁₆	1	2	yes	—
82	C₄×D₅₃	M₃₂	1	2	yes	—
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	2	1	no	—
85	C₈×C₅₃	—	4	1	no	—
86	C₈×C₅₃	C₂×C₁₆	1	2	yes	complemented
87	C₈×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₈×C₅₃	#137	1	1	yes	characteristic
89	#88322	—	2	1	no	—
90	#88322	—	2	1	no	—
91	#88322	—	2	1	no	—
92	#88322	—	4	1	no	—
93	#88322	C₂×C₁₆	1	2	yes	complemented
94	#12015	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₂×D₅₃	—	2	1	no	—
97	C₂×D₅₃	—	2	1	no	—
98	C₂×D₅₃	C₄⋊C₁₆	1	2	yes	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	#428	1	1	yes	characteristic
101	C₄×C₅₃	#430	1	1	yes	characteristic
102	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
103	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
104	Dic₅₃	—	2	1	no	—
105	Dic₅₃	—	2	1	no	—
106	Dic₅₃	C₄⋊C₁₆	1	2	yes	—
107	#2162	—	53	1	no	—
108	C₂×C₄⋊C₁₆	—	53	1	no	—
109	C₈×M₁₆	—	53	1	no	—
110	#1370	—	53	1	no	—
111	#1370	—	53	1	no	—
112	#1367	—	53	1	no	—
113	#1367	D₅₃	1	1	yes	characteristic, complemented
114	C₂×C₅₃	#1565	1	1	yes	characteristic
115	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
116	C₂×C₅₃	#2153	1	1	yes	characteristic
117	D₅₃	—	2	2	no	—
118	C₂²×C₁₆	—	106	1	no	—
119	#431	—	53	1	no	—
120	C₄×M₁₆	—	53	1	no	—
121	C₂×C₄×C₈	—	53	1	no	—
122	C₄⋊C₁₆	—	53	1	no	—
123	C₄⋊C₁₆	—	53	1	no	—
124	C₄⋊C₁₆	—	106	1	no	—
125	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
126	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
127	C₂²⋊C₁₆	—	106	1	no	—
128	C₄×C₁₆	—	106	1	no	—
129	#322	—	53	1	no	—
130	#322	—	53	1	no	—
131	C₈²	—	53	1	no	—
132	C₈²	C₂×D₅₃	1	1	yes	characteristic
133	C₅₃	#12015	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	53	1	no	—
135	C₂×M₁₆	—	53	1	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂×C₄²	—	53	1	no	—
139	C₂×C₁₆	—	2	1	no	—
140	C₂×C₁₆	—	2	1	no	—
141	C₂×C₁₆	—	106	1	no	—
142	C₂×C₁₆	—	106	1	no	—
143	C₂×C₁₆	—	212	2	no	—
144	C₄⋊C₈	—	106	1	no	—
145	C₂²⋊C₈	—	106	1	no	—
146	#97	—	53	1	no	—
147	#97	—	53	1	no	—
148	C₄×C₈	—	53	1	no	—
149	C₄×C₈	—	53	1	no	—
150	C₄×C₈	—	53	1	no	—
151	C₄×C₈	—	53	1	no	—
152	C₄×C₈	—	53	1	no	—
153	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
154	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	M₁₆	—	53	4	no	—
160	M₁₆	—	106	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	53	1	no	—
163	C₂×C₈	—	53	1	no	—
164	C₂×C₈	—	53	1	no	—
165	C₂×C₈	—	53	1	no	—
166	C₂×C₈	—	106	1	no	—
167	C₂×C₈	—	106	1	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	212	1	no	—
170	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
171	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
172	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
173	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
174	C₄²	—	53	1	no	—
175	C₄²	—	53	1	no	—
176	C₄²	—	53	1	no	—
177	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
178	C₁₆	—	4	1	no	—
179	C₁₆	—	4	1	no	—
180	C₁₆	—	212	1	no	—
181	C₁₆	—	212	1	no	—
182	C₂³	—	53	1	no	—
183	C₂×C₄	—	53	1	no	—
184	C₂×C₄	—	53	2	no	—
185	C₂×C₄	—	53	2	no	—
186	C₂×C₄	—	53	1	no	—
187	C₂×C₄	—	53	1	no	—
188	C₂×C₄	—	106	1	no	—
189	C₂×C₄	—	106	1	no	—
190	C₂×C₄	—	212	1	no	—
191	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
192	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
193	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	4	1	no	—
197	C₈	—	53	2	no	—
198	C₈	—	106	1	no	—
199	C₈	—	106	1	no	—
200	C₈	—	106	1	no	—
201	C₈	—	212	1	no	—
202	C₈	C₁₆×D₅₃	1	2	yes	complemented
203	C₈	#13133244	1	1	yes	characteristic
204	C₈	#13133246	1	1	yes	characteristic
205	C₂²	—	53	2	no	—
206	C₂²	—	106	1	no	—
207	C₂²	—	106	1	no	—
208	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	53	2	no	—
211	C₄	—	106	1	no	—
212	C₄	—	106	1	no	—
213	C₄	#16810662	1	1	yes	characteristic
214	C₄	#16810670	1	1	yes	characteristic
215	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
216	C₄	M₃₂×D₅₃	1	1	yes	characteristic
217	C₂	—	106	2	no	—
218	C₂	#20273464	1	1	yes	characteristic
219	C₂	C₄⋊C₁₆×D₅₃	1	1	yes	characteristic
220	C₂	#20276479	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1367	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂²×D₅₃	C₄⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1367	D₅₃
hypocenter	C₅₃	#12015
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

Complements

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

N	H ≅ G/N	type
4. #20272785	C₂	semidirect
5. #20272784	C₂	semidirect
6. #20272778	C₂	semidirect
7. C₅₃×#1367	C₂	semidirect
63. #11719912	C₁₆	semidirect
86. C₈×C₅₃	C₂×C₁₆	semidirect
93. #88322	C₂×C₁₆	semidirect
113. #1367	D₅₃	semidirect
133. C₅₃	#12015	semidirect
202. C₈	C₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20276513	2
C₄⋊C₁₆×D₅₃	2
C₈×#11719912	2
#20272785	2
#20272784	2
#20272778	2
C₅₃×#1367	2
#12015	53