Subgroups — #27302229

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810446	C₂	1	2	yes	maximal
2	#20274887	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133149	C₂	1	1	yes	maximal, characteristic
4	#20272814	C₂	1	2	yes	maximal
5	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	—	2	1	no	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	C₂×#13133149	C₂²	1	1	yes	characteristic
9	C₂×#13133149	C₂²	1	1	yes	characteristic
10	#16810647	—	2	1	no	—
11	C₄×C₄⋊Dic₅₃	—	2	1	no	—
12	#16810446	—	2	4	no	—
13	#16810386	—	2	1	no	—
14	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
17	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
18	C₂×#11719917	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	1	no	—
20	C₂×C₄⋊Dic₅₃	—	2	1	no	—
21	C₂×C₄×Dic₅₃	—	4	1	no	—
22	#13133153	—	2	2	no	—
23	#13133153	—	2	2	no	—
24	#13133153	—	4	2	no	—
25	#13133149	—	2	1	no	—
26	#13133149	—	2	1	no	—
27	#13133149	D₄	1	2	yes	—
28	#13133149	Q₈	1	2	yes	—
29	C₄⋊#88322	—	4	1	no	—
30	C₄⋊#88322	—	4	1	no	—
31	#13133124	—	4	1	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	M₃₂×C₅₃	—	2	2	no	—
36	M₃₂×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	1	no	—
46	C₂²×Dic₅₃	—	4	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
57	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
60	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
61	C₄⋊Dic₅₃	—	2	2	no	—
62	C₄⋊Dic₅₃	—	2	2	no	—
63	C₄⋊Dic₅₃	—	4	1	no	—
64	C₄⋊Dic₅₃	—	4	1	no	—
65	C₄×Dic₅₃	—	8	2	no	—
66	#11719917	—	4	2	no	—
67	#11719917	—	4	2	no	—
68	C₂×#88322	—	4	2	no	—
69	C₁₆×C₅₃	—	2	4	no	—
70	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	4	1	no	—
73	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	8	1	no	—
86	C₂×Dic₅₃	—	8	1	no	—
87	C₂×Dic₅₃	—	8	1	no	—
88	C₂×Dic₅₃	—	8	1	no	—
89	C₈×C₅₃	—	2	1	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
93	C₈×C₅₃	#125	1	1	yes	characteristic
94	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
95	#88322	—	8	2	no	—
96	#13185	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
100	C₂²×C₅₃	#461	1	1	yes	characteristic
101	C₂²×C₅₃	#476	1	1	yes	characteristic
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	#365	1	2	yes	complemented
106	C₄×C₅₃	#389	1	1	yes	characteristic
107	C₄×C₅₃	#428	1	1	yes	characteristic
108	C₄×C₅₃	#460	1	1	yes	characteristic
109	C₄×C₅₃	#475	1	1	yes	characteristic
110	Dic₅₃	—	8	2	no	—
111	Dic₅₃	—	8	2	no	—
112	C₂×#365	—	53	2	no	—
113	#2150	D₅₃	1	1	yes	characteristic
114	#1848	—	53	1	no	—
115	C₄×#106	—	53	1	no	—
116	#1383	—	53	2	no	—
117	C₂×C₅₃	—	2	1	no	—
118	C₂×C₅₃	—	2	1	no	—
119	C₂×C₅₃	#1948	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#365	1	1	yes	characteristic
121	C₂×C₅₃	#2155	1	1	yes	characteristic
122	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
123	C₂×#108	—	53	2	no	—
124	C₂×#108	—	106	1	no	—
125	C₂×#106	—	53	1	no	—
126	C₂×#106	—	53	1	no	—
127	#423	—	106	1	no	—
128	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
129	C₄×C₄⋊C₄	—	106	1	no	—
130	#365	—	106	4	no	—
131	C₄⋊C₁₆	Dic₅₃	1	2	yes	complemented
132	C₄⋊C₁₆	Dic₅₃	1	2	yes	complemented
133	#340	—	106	1	no	—
134	C₅₃	#13185	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	106	2	no	—
136	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	106	1	no	—
139	C₂×C₄⋊C₄	—	106	1	no	—
140	C₂×C₄²	—	212	1	no	—
141	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
142	M₃₂	—	2	2	no	—
143	M₃₂	—	2	2	no	—
144	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
145	#108	—	106	2	no	—
146	#108	—	106	2	no	—
147	#108	—	212	2	no	—
148	#106	—	53	2	no	—
149	#106	—	53	2	no	—
150	#106	—	106	1	no	—
151	#106	—	106	1	no	—
152	C₄⋊C₈	—	212	1	no	—
153	C₄⋊C₈	—	212	1	no	—
154	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
155	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
156	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
157	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
158	#95	—	212	1	no	—
159	C₂²×C₄	—	212	1	no	—
160	C₂²×C₄	—	212	1	no	—
161	C₂²×C₄	#11719913	1	1	yes	characteristic
162	C₂²×C₄	#11719913	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
164	M₁₆	—	212	2	no	—
165	M₁₆	—	212	2	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	212	2	no	—
169	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
170	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
172	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
173	C₂×C₈	#11719947	1	1	yes	characteristic
174	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
175	C₂×C₈	#11719949	1	1	yes	characteristic
176	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
177	C₄⋊C₄	—	106	2	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	212	1	no	—
180	C₄⋊C₄	—	212	1	no	—
181	C₄²	—	424	2	no	—
182	C₄²	#11719913	1	1	yes	characteristic
183	C₄²	#11719913	1	1	yes	characteristic
184	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
185	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
186	C₁₆	—	2	4	no	—
187	C₂³	C₂×#11719913	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	424	1	no	—
191	C₂×C₄	—	424	1	no	—
192	C₂×C₄	—	424	1	no	—
193	C₂×C₄	—	424	1	no	—
194	C₂×C₄	—	212	2	no	—
195	C₂×C₄	—	212	2	no	—
196	C₂×C₄	#13133149	1	1	yes	characteristic
197	C₂×C₄	#13133149	1	1	yes	characteristic
198	C₂×C₄	#13133149	1	1	yes	characteristic
199	C₂×C₄	#13133149	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
201	C₂×C₄	#13133192	1	1	yes	characteristic
202	C₂×C₄	#13133197	1	1	yes	characteristic
203	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
204	C₂×C₄	#13133245	1	1	yes	characteristic
205	C₂×C₄	#13133250	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	424	2	no	—
209	C₈	D₄×Dic₅₃	1	1	yes	characteristic
210	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
211	C₈	#13133208	1	1	yes	characteristic
212	C₈	#13133216	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×#13133149	1	1	yes	characteristic
216	C₂²	#16810829	1	1	yes	characteristic
217	C₂²	#16810914	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	424	2	no	—
222	C₄	—	424	2	no	—
223	C₄	#16810446	1	2	yes	complemented
224	C₄	#16810522	1	1	yes	characteristic
225	C₄	#16810665	1	1	yes	characteristic
226	C₄	#16810824	1	1	yes	characteristic
227	C₄	#16810909	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	#20275479	1	1	yes	characteristic
231	C₂	C₂×#16810446	1	1	yes	characteristic
232	C₂	#20276488	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133149
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#13185
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
16. C₅₃×C₄⋊C₁₆	C₄	semidirect
17. C₅₃×C₄⋊C₁₆	C₄	semidirect
105. C₄×C₅₃	#365	semidirect
131. C₄⋊C₁₆	Dic₅₃	semidirect
132. C₄⋊C₁₆	Dic₅₃	semidirect
134. C₅₃	#13185	semidirect
223. C₄	#16810446	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810446	2
#20274887	2
C₄×#13133149	2
#20272814	2
C₅₃×#2150	2
#13185	53