Subgroups — #27282629

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275433	C₂	1	2	yes	maximal
2	C₄×#13133149	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	#20272555	C₂	1	2	yes	maximal
5	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	C₄	1	1	yes	characteristic
7	C₂×#13133149	C₄	1	1	yes	characteristic
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×#13133146	—	2	2	no	—
10	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	#16810386	C₄	1	2	yes	—
12	#16810377	—	2	2	no	—
13	#16810374	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₈²×C₅₃	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅₃	—	2	2	no	—
22	#13133149	—	2	2	no	—
23	#13133149	C₈	1	4	yes	complemented
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	#13133146	—	4	4	no	—
27	#13133124	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	—	2	1	no	—
35	C₄×C₈×C₅₃	—	2	1	no	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₄×C₈×C₅₃	Q₈	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	2	no	—
43	C₂²×Dic₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
54	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
55	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₅₃	#41	1	1	yes	characteristic
59	C₄⋊Dic₅₃	—	2	2	no	—
60	C₄⋊Dic₅₃	—	2	2	no	—
61	C₄⋊Dic₅₃	—	2	2	no	—
62	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
63	C₄×Dic₅₃	—	4	2	no	—
64	C₄×Dic₅₃	—	4	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₅₃	#135	1	2	yes	—
82	C₂×Dic₅₃	—	2	4	no	—
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	4	2	no	—
86	C₂×Dic₅₃	—	4	2	no	—
87	C₂×Dic₅₃	—	4	2	no	—
88	C₈×C₅₃	—	2	4	no	—
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	—	2	4	no	—
91	C₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
92	#88322	—	8	2	no	—
93	#9593	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#389	1	1	yes	characteristic
95	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
96	C₂²×C₅₃	#436	1	1	yes	characteristic
97	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
98	C₂²×C₅₃	#443	1	2	yes	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₅₃	#424	1	1	yes	characteristic
103	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
104	C₄×C₅₃	#446	1	1	yes	characteristic
105	C₄×C₅₃	#465	1	2	yes	—
106	C₄×C₅₃	#498	1	1	yes	characteristic
107	C₄×C₅₃	#499	1	1	yes	characteristic
108	Dic₅₃	—	4	4	no	—
109	Dic₅₃	—	8	2	no	—
110	#1939	—	53	2	no	—
111	C₄×#106	—	53	1	no	—
112	C₂×#334	—	53	1	no	—
113	C₂×C₈²	D₅₃	1	1	yes	characteristic
114	#1279	—	53	2	no	—
115	C₂×C₅₃	C₈×SD₁₆	1	2	yes	—
116	C₂×C₅₃	#1590	1	2	yes	—
117	C₂×C₅₃	#1939	1	1	yes	characteristic
118	C₂×C₅₃	#1941	1	1	yes	characteristic
119	C₂×C₅₃	#1945	1	1	yes	characteristic
120	C₂×#106	—	53	1	no	—
121	C₂×#106	—	53	1	no	—
122	C₂×C₄⋊C₈	—	53	2	no	—
123	C₂×C₄⋊C₈	—	106	2	no	—
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
127	C₄×C₄⋊C₄	—	53	2	no	—
128	#340	—	53	2	no	—
129	#336	—	106	2	no	—
130	#334	—	106	2	no	—
131	C₈²	—	2	2	no	—
132	C₅₃	#9593	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	—	106	2	no	—
135	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
137	C₂²×C₈	D₂₁₂	1	2	yes	—
138	C₂×C₄⋊C₄	—	53	2	no	—
139	C₂×C₄⋊C₄	—	53	2	no	—
140	C₂×C₄²	—	106	2	no	—
141	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
142	#106	—	53	4	no	—
143	#106	—	106	2	no	—
144	C₄⋊C₈	—	106	2	no	—
145	C₄⋊C₈	—	106	2	no	—
146	C₄⋊C₈	—	212	4	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	—	2	1	no	—
150	C₄×C₈	Dic₁₀₆	1	2	yes	—
151	C₄×C₈	Dic₁₀₆	1	2	yes	—
152	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
153	#95	—	106	2	no	—
154	C₂²×C₄	—	106	2	no	—
155	C₂²×C₄	—	106	2	no	—
156	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
158	C₂²×C₄	#11719945	1	1	yes	characteristic
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	212	2	no	—
166	C₂×C₈	—	212	2	no	—
167	C₂×C₈	—	212	2	no	—
168	C₂×C₈	C₈×D₅₃	1	2	yes	—
169	C₂×C₈	#11719912	1	2	yes	—
170	C₂×C₈	#11719913	1	4	yes	—
171	C₂×C₈	#11719919	1	2	yes	—
172	C₂×C₈	#11719919	1	2	yes	—
173	C₄⋊C₄	—	53	4	no	—
174	C₄⋊C₄	—	106	2	no	—
175	C₄⋊C₄	—	106	2	no	—
176	C₄⋊C₄	—	106	2	no	—
177	C₄²	—	212	2	no	—
178	C₄²	—	212	2	no	—
179	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
181	C₄²	#11719945	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
183	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	106	4	no	—
186	C₂×C₄	—	212	2	no	—
187	C₂×C₄	—	212	2	no	—
188	C₂×C₄	—	212	2	no	—
189	C₂×C₄	—	212	2	no	—
190	C₂×C₄	—	212	2	no	—
191	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
192	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
193	C₂×C₄	#13133191	1	1	yes	characteristic
194	C₂×C₄	#13133196	1	1	yes	characteristic
195	C₂×C₄	#13133209	1	1	yes	characteristic
196	C₂×C₄	#13133217	1	1	yes	characteristic
197	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
198	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
199	C₂×C₄	#13133223	1	1	yes	characteristic
200	C₂×C₄	#13133223	1	1	yes	characteristic
201	C₂×C₄	C₂×#11719913	1	2	yes	—
202	C₂×C₄	#13133239	1	2	yes	—
203	C₈	—	2	4	no	—
204	C₈	—	2	4	no	—
205	C₈	—	2	2	no	—
206	C₈	—	424	2	no	—
207	C₈	#13133146	1	4	yes	complemented
208	C₂²	#16810521	1	1	yes	characteristic
209	C₂²	C₈×D₂₁₂	1	1	yes	characteristic
210	C₂²	#16810702	1	1	yes	characteristic
211	C₂²	C₄×#11719913	1	2	yes	—
212	C₂²	#16810741	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	424	2	no	—
216	C₄	—	212	4	no	—
217	C₄	C₂×#13133146	1	1	yes	characteristic
218	C₄	#16810649	1	1	yes	characteristic
219	C₄	C₈×Dic₁₀₆	1	1	yes	characteristic
220	C₄	#16810753	1	1	yes	characteristic
221	C₄	#16810852	1	2	yes	—
222	C₄	#16811021	1	1	yes	characteristic
223	C₄	#16811025	1	1	yes	characteristic
224	C₂	C₈×#11719913	1	2	yes	—
225	C₂	#20273607	1	2	yes	—
226	C₂	#20275433	1	1	yes	characteristic
227	C₂	#20275443	1	1	yes	characteristic
228	C₂	#20275463	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#9593
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
23. #13133149	C₈	semidirect
91. C₈×C₅₃	C₄⋊C₈	semidirect
132. C₅₃	#9593	semidirect
207. C₈	#13133146	semidirect

Maximal subgroups

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

subgroup	index
#20275433	2
C₄×#13133149	2
C₂×#16810374	2
#20272555	2
C₂×C₈²×C₅₃	2
#9593	53