Subgroups — #27353264

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810680	C₂	1	2	yes	maximal
2	#20273500	C₂	1	8	yes	maximal, complemented
3	#20273478	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	#16810680	—	2	2	no	—
6	#16810680	—	2	2	no	—
7	#16810680	C₂²	1	4	yes	—
8	#16810680	C₂²	1	4	yes	—
9	C₂×C₄⋊#88322	C₂²	1	4	yes	—
10	C₂×C₂²⋊#88322	—	2	2	no	—
11	C₂×#13133128	—	2	2	no	—
12	#16810376	C₂²	1	8	yes	complemented
13	#16810374	C₂²	1	8	yes	complemented
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#431	C₄	1	4	yes	—
16	C₅₃×#431	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
18	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
20	C₂²×#88322	—	2	4	no	—
21	C₄⋊#88322	—	2	4	no	—
22	C₄⋊#88322	C₂³	1	8	yes	—
23	C₂²⋊#88322	—	2	4	no	—
24	C₂²⋊#88322	—	2	4	no	—
25	#13133128	—	2	4	no	—
26	#13133128	—	2	4	no	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
40	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂³	1	4	yes	—
43	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₅₃	—	4	4	no	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
61	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	2	yes	—
68	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
69	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₂×#88322	—	2	8	no	—
72	C₂×#88322	—	4	8	no	—
73	C₂³×C₅₃	—	4	1	no	—
74	C₂³×C₅₃	—	4	1	no	—
75	C₂³×C₅₃	#108	1	4	yes	—
76	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	4	1	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
99	C₈×C₅₃	—	2	8	no	—
100	#88322	—	4	8	no	—
101	#17818	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	4	4	no	—
107	C₂²×C₅₃	C₂×#108	1	4	yes	—
108	C₂²×C₅₃	C₂×#108	1	2	yes	—
109	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	#568	1	2	yes	—
118	C₄×C₅₃	#575	1	2	yes	—
119	C₂×#432	—	53	2	no	—
120	C₂×#431	D₅₃	1	1	yes	characteristic
121	#1573	—	53	8	no	—
122	#1568	—	53	4	no	—
123	C₂×C₅₃	—	2	4	no	—
124	C₂×C₅₃	—	2	2	no	—
125	C₂×C₅₃	#2905	1	1	yes	characteristic
126	C₂×C₅₃	#2911	1	1	yes	characteristic
127	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
128	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
129	#432	—	53	4	no	—
130	#432	—	53	4	no	—
131	#432	—	106	2	no	—
132	#432	—	106	2	no	—
133	#431	Dic₅₃	1	4	yes	—
134	#431	C₂×D₅₃	1	4	yes	—
135	C₂×C₄⋊C₈	—	53	4	no	—
136	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
137	C₂×C₂²⋊C₈	—	106	2	no	—
138	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
139	C₂×#97	—	106	2	no	—
140	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
141	#335	—	53	8	no	—
142	#334	—	53	8	no	—
143	C₅₃	#17818	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
145	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
146	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
147	C₂²×C₈	—	106	4	no	—
148	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
149	C₂×C₄²	Dic₁₀₆	1	2	yes	—
150	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
151	C₂×C₄²	D₂₁₂	1	2	yes	—
152	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
155	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
156	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
157	C₄⋊C₈	—	53	8	no	—
158	C₄⋊C₈	—	106	4	no	—
159	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
160	C₂²⋊C₈	—	106	4	no	—
161	C₂²⋊C₈	—	106	4	no	—
162	C₂²⋊C₈	C₂×Dic₅₃	1	8	yes	—
163	#97	—	106	4	no	—
164	#97	—	106	4	no	—
165	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
166	C₄×C₈	C₂²×D₅₃	1	4	yes	—
167	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
172	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
174	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
177	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
178	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
179	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	—	106	8	no	—
185	C₂×C₈	—	212	8	no	—
186	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
187	C₄²	C₄⋊Dic₅₃	1	2	yes	—
188	C₄²	C₄⋊Dic₅₃	1	2	yes	—
189	C₄²	C₄⋊Dic₅₃	1	2	yes	—
190	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
191	C₄²	C₂×Dic₁₀₆	1	2	yes	—
192	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
193	C₄²	C₂×D₂₁₂	1	2	yes	—
194	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
195	C₄²	C₂²×Dic₅₃	1	2	yes	—
196	C₄²	C₂³×D₅₃	1	1	yes	characteristic
197	C₂³	—	4	1	no	—
198	C₂³	—	4	1	no	—
199	C₂³	#13133153	1	4	yes	—
200	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
201	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
216	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
220	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
222	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
223	C₈	—	2	8	no	—
224	C₈	—	212	8	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	4	4	no	—
230	C₂²	C₂×#13133153	1	2	yes	—
231	C₂²	C₂×#13133153	1	4	yes	—
232	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	1	no	—
240	C₄	#16811390	1	2	yes	—
241	C₄	#16811423	1	2	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	2	4	no	—
244	C₂	#20283279	1	1	yes	characteristic
245	C₂	#20283305	1	1	yes	characteristic
246	C₂	C₂²×#13133153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#17818
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #20273500	C₂	semidirect
3. #20273478	C₂	semidirect
12. #16810376	C₂²	semidirect
13. #16810374	C₂²	semidirect
143. C₅₃	#17818	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810680	2
#20273500	2
#20273478	2
C₂×C₅₃×#431	2
#17818	53