Subgroups — #27276976

Order 13568.

372 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273102	C₂	1	2	yes	maximal
4	#20272566	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2922	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	#16810376	—	2	2	no	—
9	#16810376	—	2	1	no	—
10	#16810374	—	2	1	no	—
11	Q₈⋊#88322	—	2	2	no	—
12	D₄⋊#88322	—	2	2	no	—
13	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#433	—	2	1	no	—
15	C₅₃×#433	—	2	1	no	—
16	C₅₃×#431	—	2	2	no	—
17	C₅₃×#424	C₄	1	2	yes	—
18	C₅₃×#408	C₄	1	2	yes	—
19	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₂²⋊#88322	—	2	2	no	—
26	C₄×#88322	—	2	2	no	—
27	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₄×D₄×C₅₃	—	2	1	no	—
33	C₄×D₄×C₅₃	—	2	1	no	—
34	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
35	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
36	#117×C₅₃	—	2	1	no	—
37	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₂²⋊C₈×C₅₃	—	2	2	no	—
43	C₂²⋊C₈×C₅₃	—	2	2	no	—
44	#97×C₅₃	—	2	1	no	—
45	C₄×C₈×C₅₃	—	2	1	no	—
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	#41×C₅₃	—	2	4	no	—
50	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₄×C₅₃	—	2	1	no	—
52	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
59	M₁₆×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂²⋊C₄×C₅₃	—	2	1	no	—
69	C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₂²⋊C₄×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	—	2	1	no	—
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	8	2	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	#117	1	1	yes	characteristic
83	Q₈×C₅₃	—	2	2	no	—
84	D₄×C₅₃	—	2	2	no	—
85	D₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	1	no	—
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#131	1	4	yes	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	2	2	no	—
105	C₈×C₅₃	—	4	2	no	—
106	#88322	—	8	2	no	—
107	#88322	—	8	2	no	—
108	#8606	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	4	1	no	—
112	C₂²×C₅₃	—	4	1	no	—
113	C₂²×C₅₃	—	4	1	no	—
114	C₂²×C₅₃	#380	1	1	yes	characteristic
115	C₂²×C₅₃	#433	1	2	yes	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	4	1	no	—
120	C₄×C₅₃	—	4	1	no	—
121	C₄×C₅₃	—	4	1	no	—
122	C₄×C₅₃	#408	1	2	yes	—
123	C₄×C₅₃	#424	1	2	yes	—
124	C₄×C₅₃	#472	1	1	yes	characteristic
125	C₄×C₅₃	#473	1	1	yes	characteristic
126	#2922	D₅₃	1	1	yes	characteristic
127	#1564	—	53	1	no	—
128	C₂×#335	—	53	1	no	—
129	#1477	—	53	2	no	—
130	#1283	—	53	2	no	—
131	C₂×C₅₃	—	2	2	no	—
132	C₂×C₅₃	—	8	1	no	—
133	C₂×C₅₃	#1798	1	1	yes	characteristic
134	C₂×C₅₃	#1814	1	1	yes	characteristic
135	C₂×C₅₃	#1816	1	1	yes	characteristic
136	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
137	#433	—	2	1	no	—
138	#433	—	2	1	no	—
139	#431	—	2	2	no	—
140	#431	—	53	2	no	—
141	#424	Dic₅₃	1	2	yes	—
142	C₂×C₄⋊C₈	—	53	2	no	—
143	#408	Dic₅₃	1	2	yes	—
144	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
146	#335	—	106	2	no	—
147	#335	—	106	1	no	—
148	#334	—	106	1	no	—
149	Q₈⋊C₈	—	106	2	no	—
150	D₄⋊C₈	—	106	2	no	—
151	C₅₃	#8606	1	1	yes	characteristic, complemented, Sylow
152	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
153	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
154	C₂²×C₈	—	106	2	no	—
155	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
156	C₄×Q₈	Dic₁₀₆	1	1	yes	characteristic
157	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	Dic₁₀₆	1	1	yes	characteristic
161	C₄×D₄	D₂₁₂	1	1	yes	characteristic
162	#117	—	2	1	no	—
163	#117	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	106	2	no	—
169	C₄⋊C₈	—	106	2	no	—
170	C₄⋊C₈	—	106	2	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	C₂²⋊C₈	—	2	2	no	—
173	C₂²⋊C₈	—	106	2	no	—
174	#97	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	106	2	no	—
177	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
179	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	#41	—	2	4	no	—
181	C₂×Q₈	C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	C₄⋊Dic₅₃	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	#11719945	1	1	yes	characteristic
188	C₂²×C₄	#11719945	1	1	yes	characteristic
189	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
190	M₁₆	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	424	2	no	—
194	C₂×C₈	—	212	2	no	—
195	C₂×C₈	—	212	2	no	—
196	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
197	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
201	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
209	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
210	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
211	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	#13133176	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	D₄	—	2	2	no	—
216	D₄	—	4	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
229	C₂×C₄	#13133191	1	1	yes	characteristic
230	C₂×C₄	#13133191	1	1	yes	characteristic
231	C₂×C₄	#13133196	1	1	yes	characteristic
232	C₂×C₄	#13133196	1	1	yes	characteristic
233	C₂×C₄	#13133224	1	4	yes	—
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	4	2	no	—
237	C₈	—	424	2	no	—
238	C₈	—	424	2	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	#16810483	1	1	yes	characteristic
245	C₂²	#16810684	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	#16810590	1	2	yes	—
253	C₄	#16810650	1	2	yes	—
254	C₄	#16810893	1	1	yes	characteristic
255	C₄	#16810898	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	8	1	no	—
258	C₂	#20274699	1	1	yes	characteristic
259	C₂	#20274769	1	1	yes	characteristic
260	C₂	#20274779	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2922	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2922	D₅₃
hypocenter	C₅₃	#8606
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2922

Complements

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

N	H ≅ G/N	type
1. #20273463	C₂	semidirect
2. C₂×#16810376	C₂	semidirect
4. #20272566	C₂	semidirect
151. C₅₃	#8606	semidirect

Maximal subgroups

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

subgroup	index
#20273463	2
C₂×#16810376	2
#20273102	2
#20272566	2
C₅₃×#2922	2
#8606	53