Subgroups — #27384563

Order 13568.

954 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	2	yes	maximal
2	C₂²×#13133153	C₂	1	2	yes	maximal
3	#20283279	C₂	1	2	yes	maximal
4	#20274890	C₂	1	4	yes	maximal, complemented
5	#20274887	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	4	yes	—
8	#16810672	—	2	8	no	—
9	C₂×#13133153	—	2	8	no	—
10	C₂×#13133153	C₂²	1	4	yes	—
11	C₂×#13133153	C₂²	1	4	yes	—
12	C₂×#13133153	C₂²	1	8	yes	—
13	#16810650	—	2	8	no	—
14	#16810647	C₂²	1	8	yes	—
15	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#405	C₄	1	8	yes	—
19	C₂×#97×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×#11719917	—	2	4	no	—
22	C₂×#11719917	—	2	8	no	—
23	C₂×#11719917	—	2	8	no	—
24	C₂×#11719917	C₂³	1	8	yes	—
25	C₂²×#88322	—	2	4	no	—
26	#13133153	—	2	8	no	—
27	#13133153	—	2	8	no	—
28	#13133153	—	2	16	no	—
29	C₄⋊#88322	—	2	16	no	—
30	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
33	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₅₃	D₄	1	4	yes	—
36	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	#117×C₅₃	C₂×C₄	1	4	yes	—
41	#117×C₅₃	D₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
43	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
45	#97×C₅₃	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
47	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₅₃	C₄⋊C₄	1	16	yes	—
58	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
68	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
69	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
70	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
71	C₄²×C₅₃	C₂×D₄	1	4	yes	—
72	#11719917	—	2	8	no	—
73	#11719917	—	2	8	no	—
74	#11719917	—	4	8	no	—
75	C₂×#88322	—	2	8	no	—
76	C₂×#88322	—	4	8	no	—
77	C₂³×C₅₃	—	4	2	no	—
78	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
80	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
81	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	—	4	4	no	—
83	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₅₃	#127	1	4	yes	—
88	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
89	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
97	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
98	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
99	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
100	#88322	—	4	8	no	—
101	#20807	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	4	no	—
104	C₂²×C₅₃	—	4	2	no	—
105	C₂²×C₅₃	—	4	2	no	—
106	C₂²×C₅₃	#384	1	4	yes	—
107	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
110	C₄×C₅₃	—	2	8	no	—
111	C₄×C₅₃	#384	1	4	yes	—
112	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
115	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂×#430	—	53	2	no	—
117	C₂²×#108	—	53	2	no	—
118	#2905	—	53	2	no	—
119	C₂×#405	D₅₃	1	1	yes	characteristic
120	#1849	—	53	4	no	—
121	#1848	—	53	4	no	—
122	C₂×C₅₃	—	2	1	no	—
123	C₂×C₅₃	—	2	1	no	—
124	C₂×C₅₃	—	2	4	no	—
125	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
126	C₂×C₅₃	#2916	1	2	yes	—
127	C₂²×M₁₆	—	53	4	no	—
128	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
129	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
130	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
131	#430	—	106	8	no	—
132	C₂×#108	—	53	4	no	—
133	C₂×#108	—	53	8	no	—
134	C₂×#108	—	53	4	no	—
135	C₂×#108	—	106	8	no	—
136	#424	—	106	8	no	—
137	#423	—	53	8	no	—
138	#405	Dic₅₃	1	8	yes	—
139	C₂×#97	C₂×D₅₃	1	2	yes	—
140	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
141	C₅₃	#20807	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
143	C₂×M₁₆	—	53	8	no	—
144	C₂×M₁₆	—	106	4	no	—
145	C₂×M₁₆	—	106	8	no	—
146	C₂×M₁₆	—	106	8	no	—
147	C₂×M₁₆	Dic₁₀₆	1	4	yes	—
148	C₂×M₁₆	D₂₁₂	1	4	yes	—
149	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
150	C₂²×C₈	—	106	4	no	—
151	C₂²×C₈	Dic₁₀₆	1	4	yes	—
152	C₂²×C₈	D₂₁₂	1	4	yes	—
153	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
154	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
157	#117	D₂₁₂	1	4	yes	—
158	#117	C₂×Dic₅₃	1	4	yes	—
159	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
160	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
161	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
162	#108	—	106	8	no	—
163	#108	—	106	8	no	—
164	#108	—	106	16	no	—
165	C₄⋊C₈	—	106	16	no	—
166	#97	C₂×Dic₅₃	1	8	yes	—
167	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
168	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
169	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
170	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
171	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
172	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
173	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
174	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
175	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
177	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
178	M₁₆	—	106	8	no	—
179	M₁₆	—	106	8	no	—
180	M₁₆	—	212	8	no	—
181	M₁₆	C₄⋊Dic₅₃	1	16	yes	—
182	C₂×C₈	—	106	8	no	—
183	C₂×C₈	—	212	8	no	—
184	C₂×C₈	C₄⋊Dic₅₃	1	16	yes	—
185	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
186	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
187	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
188	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
189	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
190	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
191	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
192	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
193	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
194	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
195	C₂²⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
196	C₄²	C₂×D₂₁₂	1	4	yes	—
197	C₄²	C₂²×Dic₅₃	1	4	yes	—
198	C₂³	—	4	2	no	—
199	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
200	C₂³	#13133217	1	2	yes	—
201	C₂³	#13133217	1	2	yes	—
202	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
208	C₂×C₄	#13133214	1	4	yes	—
209	C₂×C₄	#13133217	1	4	yes	—
210	C₂×C₄	#13133217	1	2	yes	—
211	C₂×C₄	#13133217	1	2	yes	—
212	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
214	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
216	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
217	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
218	C₈	—	212	8	no	—
219	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
220	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
221	C₈	C₂×C₄⋊Dic₅₃	1	8	yes	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	#16810493	1	4	yes	—
227	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
228	C₂²	C₂×#13133217	1	1	yes	characteristic
229	C₂²	C₂×#13133217	1	1	yes	characteristic
230	C₄	—	2	8	no	—
231	C₄	#16810493	1	4	yes	—
232	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
233	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
234	C₄	C₂×#13133214	1	1	yes	characteristic
235	C₄	C₂×#13133217	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	4	no	—
239	C₂	C₂×#16810493	1	1	yes	characteristic
240	C₂	#20283326	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#20807
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #20274890	C₂	semidirect
5. #20274887	C₂	semidirect
141. C₅₃	#20807	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
C₂²×#13133153	2
#20283279	2
#20274890	2
#20274887	2
C₂×C₅₃×#405	2
#20807	53