Subgroups — #27300044

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810443	C₂	1	2	yes	maximal, complemented
3	C₂×#16810405	C₂	1	1	yes	maximal, characteristic
4	#20272794	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#349	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂×#13133157	C₂²	1	2	yes	—
9	M₁₆×Dic₅₃	—	2	4	no	—
10	C₂×#13133127	C₄	1	1	yes	characteristic
11	C₂×#13133127	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810443	—	2	4	no	—
15	#16810405	—	2	2	no	—
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#349	—	2	2	no	—
19	C₂³×Dic₅₃	—	2	1	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	C₈	1	4	yes	—
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	—	2	2	no	—
26	C₂×C₄×Dic₅₃	—	2	2	no	—
27	C₂×C₄×Dic₅₃	—	2	1	no	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	#13133157	—	2	2	no	—
31	#13133157	—	2	2	no	—
32	C₂×#11719908	—	2	2	no	—
33	#13133127	—	2	2	no	—
34	#13133127	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₈×Dic₅₃	D₄	1	4	yes	—
37	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
41	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	M₃₂×C₅₃	—	2	2	no	—
45	M₃₂×C₅₃	—	2	2	no	—
46	C₂×C₁₆×C₅₃	—	2	2	no	—
47	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	2	1	no	—
57	C₂²×Dic₅₃	—	2	4	no	—
58	C₂²×Dic₅₃	—	2	1	no	—
59	C₂²×Dic₅₃	—	4	4	no	—
60	M₁₆×C₅₃	—	2	4	no	—
61	M₁₆×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	2	yes	—
69	C₄×Dic₅₃	—	2	4	no	—
70	C₄×Dic₅₃	—	2	1	no	—
71	C₄×Dic₅₃	—	2	1	no	—
72	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
73	C₄×Dic₅₃	M₁₆	1	2	yes	—
74	#11719917	—	2	8	no	—
75	C₂×#88322	—	2	1	no	—
76	C₂×#88322	—	2	1	no	—
77	C₂×#88322	C₂²⋊C₄	1	2	yes	—
78	C₂×#88322	C₂×C₈	1	2	yes	—
79	C₂×#88322	C₂×C₈	1	2	yes	—
80	C₂×#88322	M₁₆	1	2	yes	—
81	C₁₆×C₅₃	—	4	2	no	—
82	#11719908	—	4	2	no	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂³×C₅₃	—	2	1	no	—
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
87	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	4	no	—
102	C₂×Dic₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	—	4	8	no	—
105	C₂×Dic₅₃	—	4	2	no	—
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	#121	1	2	yes	—
108	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
109	#88322	—	2	2	no	—
110	#88322	C₂²⋊C₈	1	4	yes	—
111	#12791	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	#407	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₅₃	#432	1	1	yes	characteristic
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	#380	1	1	yes	characteristic
122	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₄×C₅₃	#423	1	1	yes	characteristic
124	C₄×C₅₃	#431	1	1	yes	characteristic
125	Dic₅₃	—	2	4	no	—
126	Dic₅₃	—	4	2	no	—
127	C₂×C₄×M₁₆	—	53	1	no	—
128	C₂×#364	—	53	2	no	—
129	C₂×#349	—	53	1	no	—
130	C₂×#349	D₅₃	1	1	yes	characteristic
131	#1375	—	53	2	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	4	2	no	—
135	C₂×C₅₃	#1797	1	1	yes	characteristic
136	C₂×C₅₃	#2165	1	2	yes	—
137	C₂²×M₁₆	—	53	1	no	—
138	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
139	C₂²×C₄²	—	53	1	no	—
140	C₂×M₃₂	—	53	2	no	—
141	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
142	C₄×M₁₆	—	106	4	no	—
143	C₂×#97	—	53	1	no	—
144	C₂×#97	—	53	1	no	—
145	C₂×C₄×C₈	—	53	1	no	—
146	C₂×C₄×C₈	—	53	1	no	—
147	#364	—	106	4	no	—
148	#349	—	2	2	no	—
149	#349	—	106	2	no	—
150	C₅₃	#12791	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	—	106	1	no	—
152	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	53	4	no	—
156	C₂×M₁₆	—	106	2	no	—
157	C₂×M₁₆	—	106	2	no	—
158	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
159	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
160	C₂²×C₈	—	53	1	no	—
161	C₂²×C₈	—	53	1	no	—
162	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
164	C₂×C₄²	—	53	1	no	—
165	C₂×C₄²	—	53	1	no	—
166	C₂×C₄²	—	106	2	no	—
167	C₂×C₄²	—	106	2	no	—
168	C₂×C₄²	—	106	1	no	—
169	M₃₂	—	2	2	no	—
170	M₃₂	—	2	2	no	—
171	M₃₂	—	106	2	no	—
172	M₃₂	—	106	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	106	2	no	—
175	#97	—	106	2	no	—
176	#97	—	106	2	no	—
177	C₄×C₈	—	53	4	no	—
178	C₄×C₈	—	106	2	no	—
179	C₂⁴	#11719912	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	106	1	no	—
185	C₂²×C₄	—	106	4	no	—
186	C₂²×C₄	—	106	1	no	—
187	C₂²×C₄	—	212	4	no	—
188	C₂²×C₄	C₈×D₅₃	1	2	yes	—
189	C₂²×C₄	#11719912	1	1	yes	characteristic
190	C₂²×C₄	#11719919	1	2	yes	—
191	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	106	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	53	2	no	—
198	C₂×C₈	—	53	2	no	—
199	C₂×C₈	—	53	2	no	—
200	C₂×C₈	—	53	2	no	—
201	C₂×C₈	—	106	1	no	—
202	C₂×C₈	—	106	1	no	—
203	C₂×C₈	#11719919	1	2	yes	—
204	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
205	C₂×C₈	D₄×D₅₃	1	2	yes	—
206	C₂×C₈	#11719947	1	2	yes	—
207	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
208	C₄²	—	53	2	no	—
209	C₄²	—	53	2	no	—
210	C₄²	—	106	4	no	—
211	C₄²	—	106	1	no	—
212	C₄²	—	106	1	no	—
213	C₁₆	—	4	2	no	—
214	C₁₆	—	212	2	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	4	2	no	—
218	C₂³	#13133146	1	2	yes	—
219	C₂³	C₂×#11719912	1	1	yes	characteristic
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	106	4	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	212	8	no	—
230	C₂×C₄	—	212	2	no	—
231	C₂×C₄	#13133146	1	2	yes	—
232	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
234	C₂×C₄	#13133173	1	1	yes	characteristic
235	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
236	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
237	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	53	4	no	—
240	C₈	—	106	2	no	—
241	C₈	#13133189	1	2	yes	—
242	C₈	#13133194	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#16810585	1	1	yes	characteristic
249	C₂²	C₂×#13133146	1	1	yes	characteristic
250	C₂²	#16810677	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	106	4	no	—
253	C₄	—	212	2	no	—
254	C₄	#16810481	1	1	yes	characteristic
255	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
256	C₄	#16810646	1	1	yes	characteristic
257	C₄	#16810673	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	#20274693	1	1	yes	characteristic
262	C₂	#20276524	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133173
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#349	C₂
socle	C₂²×C₅₃	#432
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#349	D₅₃
hypocenter	C₅₃	#12791
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
2. C₂×#16810443	C₂	semidirect
4. #20272794	C₂	semidirect
150. C₅₃	#12791	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₅₃	2
C₂×#16810443	2
C₂×#16810405	2
#20272794	2
C₂×C₅₃×#349	2
#12791	53