Subgroups — #27300268

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810451	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810449	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274744	C₂	1	2	yes	maximal
4	#20272807	C₂	1	2	yes	maximal, complemented
5	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	C₂²	1	2	yes	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	#16810483	—	2	2	no	—
9	#16810451	—	2	2	no	—
10	#16810449	—	2	2	no	—
11	#16810386	—	2	2	no	—
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₅₃	—	2	2	no	—
14	C₂×M₃₂×C₅₃	—	2	2	no	—
15	C₂×M₃₂×C₅₃	C₄	1	4	yes	complemented
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₂²⋊Dic₅₃	—	4	2	no	—
21	#13133151	—	2	2	no	—
22	#13133151	—	2	2	no	—
23	#13133151	—	2	2	no	—
24	#13133151	—	2	2	no	—
25	#13133124	—	4	2	no	—
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	M₃₂×C₅₃	—	2	8	no	—
36	C₂×C₁₆×C₅₃	—	2	1	no	—
37	C₂×C₁₆×C₅₃	—	2	1	no	—
38	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
41	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
42	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	4	2	no	—
52	C₂²×Dic₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
66	C₂²⋊Dic₅₃	—	8	4	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	C₄⋊Dic₅₃	—	4	2	no	—
69	C₄⋊Dic₅₃	—	4	2	no	—
70	C₄⋊Dic₅₃	—	4	2	no	—
71	C₁₆×C₅₃	—	2	2	no	—
72	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	complemented
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	—	2	1	no	—
75	C₂³×C₅₃	—	2	2	no	—
76	C₂³×C₅₃	—	2	2	no	—
77	C₂³×C₅₃	#107	1	2	yes	—
78	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	#107	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
92	C₂×Dic₅₃	—	8	2	no	—
93	C₂×Dic₅₃	—	8	2	no	—
94	C₂×Dic₅₃	—	8	2	no	—
95	C₂×Dic₅₃	—	8	2	no	—
96	C₂×Dic₅₃	—	8	2	no	—
97	C₂×Dic₅₃	—	8	2	no	—
98	C₈×C₅₃	—	2	2	no	—
99	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₈×C₅₃	#117	1	1	yes	characteristic
101	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
102	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
103	#12836	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	4	no	—
106	C₂²×C₅₃	—	4	1	no	—
107	C₂²×C₅₃	—	4	1	no	—
108	C₂²×C₅₃	—	4	2	no	—
109	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
110	C₂²×C₅₃	#466	1	1	yes	characteristic
111	C₂²×C₅₃	#466	1	1	yes	characteristic
112	C₂²×C₅₃	#509	1	2	yes	—
113	C₂²×C₅₃	#510	1	2	yes	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	#380	1	1	yes	characteristic
116	C₄×C₅₃	#427	1	1	yes	characteristic
117	C₄×C₅₃	#467	1	1	yes	characteristic
118	C₄×C₅₃	#467	1	1	yes	characteristic
119	Dic₅₃	—	16	2	no	—
120	Dic₅₃	—	16	2	no	—
121	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
122	C₂×#367	—	53	1	no	—
123	C₂×#366	—	53	1	no	—
124	#1808	—	53	2	no	—
125	#1380	—	53	2	no	—
126	C₂×C₅₃	—	4	2	no	—
127	C₂×C₅₃	#1808	1	1	yes	characteristic
128	C₂×C₅₃	#2159	1	2	yes	—
129	C₂×C₅₃	#2218	1	1	yes	characteristic
130	C₂×C₅₃	#2219	1	1	yes	characteristic
131	C₂×C₅₃	#2220	1	2	yes	—
132	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
133	C₂×M₃₂	—	2	2	no	—
134	C₂×M₃₂	—	2	2	no	—
135	C₂×M₃₂	Dic₅₃	1	4	yes	complemented
136	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
137	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
138	C₂×#107	—	53	2	no	—
139	C₂×#107	—	53	2	no	—
140	#380	—	106	2	no	—
141	#367	—	106	2	no	—
142	#366	—	106	2	no	—
143	#340	—	106	2	no	—
144	C₅₃	#12836	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	Dic₁₀₆	1	2	yes	—
151	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
152	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
153	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	106	2	no	—
155	C₂×C₄⋊C₄	—	106	2	no	—
156	C₂×C₂²⋊C₄	—	212	2	no	—
157	M₃₂	—	2	8	no	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	—	2	1	no	—
160	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
161	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
162	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
163	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
164	#107	—	106	2	no	—
165	#107	—	106	2	no	—
166	#107	—	106	2	no	—
167	#107	—	106	2	no	—
168	#95	—	212	2	no	—
169	C₂⁴	D₄₂₄	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	212	2	no	—
175	C₂²×C₄	—	212	2	no	—
176	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
177	C₂²×C₄	Dic₂₁₂	1	2	yes	—
178	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
179	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
187	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
188	C₂×C₈	#11719945	1	1	yes	characteristic
189	C₂×C₈	#11719945	1	1	yes	characteristic
190	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
193	C₄⋊C₄	—	212	2	no	—
194	C₄⋊C₄	—	212	2	no	—
195	C₄⋊C₄	—	212	2	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₂²⋊C₄	—	424	4	no	—
198	C₁₆	—	2	2	no	—
199	C₁₆	C₂²⋊Dic₅₃	1	4	yes	complemented
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	#13133151	1	2	yes	—
205	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	424	2	no	—
213	C₂×C₄	—	424	2	no	—
214	C₂×C₄	—	424	2	no	—
215	C₂×C₄	—	424	2	no	—
216	C₂×C₄	—	424	2	no	—
217	C₂×C₄	—	424	2	no	—
218	C₂×C₄	#13133151	1	2	yes	—
219	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
220	C₂×C₄	#13133191	1	1	yes	characteristic
221	C₂×C₄	#13133191	1	1	yes	characteristic
222	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
223	C₂×C₄	#13133239	1	1	yes	characteristic
224	C₂×C₄	#13133239	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
227	C₈	#13133176	1	1	yes	characteristic
228	C₈	#13133196	1	1	yes	characteristic
229	C₈	#13133196	1	1	yes	characteristic
230	C₂²	—	2	4	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	C₂×#13133151	1	1	yes	characteristic
236	C₂²	#16810859	1	1	yes	characteristic
237	C₂²	#16810859	1	1	yes	characteristic
238	C₂²	#16811061	1	2	yes	—
239	C₂²	#16811066	1	2	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	848	2	no	—
242	C₄	—	848	2	no	—
243	C₄	#16810483	1	1	yes	characteristic
244	C₄	#16810661	1	1	yes	characteristic
245	C₄	#16810864	1	1	yes	characteristic
246	C₄	#16810864	1	1	yes	characteristic
247	C₂	—	4	2	no	—
248	C₂	#20274744	1	1	yes	characteristic
249	C₂	#20276503	1	2	yes	—
250	C₂	#20276818	1	1	yes	characteristic
251	C₂	#20276823	1	1	yes	characteristic
252	C₂	#20276828	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#12836
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#16810451	C₂	semidirect
2. C₂×#16810449	C₂	semidirect
4. #20272807	C₂	semidirect
15. C₂×M₃₂×C₅₃	C₄	semidirect
72. C₁₆×C₅₃	C₂²⋊C₄	semidirect
135. C₂×M₃₂	Dic₅₃	semidirect
144. C₅₃	#12836	semidirect
199. C₁₆	C₂²⋊Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810451	2
C₂×#16810449	2
#20274744	2
#20272807	2
C₂²×M₃₂×C₅₃	2
#12836	53