Subgroups — #27382182

Order 13568.

1126 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
3	#20277071	C₂	1	2	yes	maximal
4	#20274612	C₂	1	4	yes	maximal, complemented
5	C₄×#13133151	C₂	1	4	yes	maximal, complemented
6	#20274435	C₂	1	2	yes	maximal
7	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133176	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#16810661	C₄	1	8	yes	complemented
11	C₂×#13133151	C₄	1	8	yes	complemented
12	C₂×#13133151	C₂²	1	2	yes	—
13	C₂×#13133151	C₂²	1	2	yes	—
14	C₂×#13133151	C₂²	1	2	yes	—
15	C₂×#13133149	C₂²	1	2	yes	—
16	C₄⋊C₄×Dic₅₃	—	2	4	no	—
17	C₄×C₄⋊Dic₅₃	C₂²	1	8	yes	—
18	C₂²⋊C₄×Dic₅₃	—	2	4	no	—
19	#16810386	C₂²	1	8	yes	—
20	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#405	C₄	1	8	yes	complemented
24	C₂×#97×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
26	C₂³×Dic₅₃	—	2	2	no	—
27	#13133176	—	2	4	no	—
28	#13133176	C₂×C₄	1	8	yes	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	4	no	—
31	C₂×C₄⋊Dic₅₃	C₂×C₄	1	8	yes	—
32	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
34	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
35	C₂×C₄×Dic₅₃	—	2	2	no	—
36	C₂×C₄×Dic₅₃	—	2	8	no	—
37	#13133151	C₂×C₄	1	8	yes	—
38	#13133151	C₂×C₄	1	8	yes	—
39	#13133151	C₂×C₄	1	8	yes	complemented
40	#13133149	C₂×C₄	1	8	yes	complemented
41	#13133124	—	2	8	no	—
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	#117×C₅₃	C₂×C₄	1	4	yes	—
53	#117×C₅₃	Q₈	1	4	yes	—
54	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
57	#97×C₅₃	C₂×C₄	1	8	yes	complemented
58	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
59	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₅₃	#41	1	2	yes	—
67	C₂²×C₄×C₅₃	#41	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₅₃	—	2	4	no	—
70	C₂²×Dic₅₃	—	2	4	no	—
71	C₂²×Dic₅₃	—	2	8	no	—
72	C₂²×Dic₅₃	—	8	4	no	—
73	M₁₆×C₅₃	—	2	8	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	C₄²	1	16	yes	complemented
77	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
84	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
85	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
86	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
87	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
88	C₂²⋊Dic₅₃	—	2	8	no	—
89	C₄⋊Dic₅₃	—	2	8	no	—
90	C₄⋊Dic₅₃	—	2	4	no	—
91	C₄⋊Dic₅₃	—	2	4	no	—
92	C₄⋊Dic₅₃	C₂²×C₄	1	8	yes	—
93	C₄⋊Dic₅₃	C₂²×C₄	1	8	yes	—
94	C₄×Dic₅₃	—	2	8	no	—
95	C₄×Dic₅₃	—	8	8	no	—
96	C₂³×C₅₃	—	2	2	no	—
97	C₂³×C₅₃	—	2	2	no	—
98	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₅₃	C₄×D₄	1	2	yes	—
100	C₂³×C₅₃	C₄×D₄	1	2	yes	—
101	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
106	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₅₃	#117	1	4	yes	—
108	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
118	C₂×Dic₅₃	—	2	8	no	—
119	C₂×Dic₅₃	—	2	8	no	—
120	C₂×Dic₅₃	—	4	8	no	—
121	C₂×Dic₅₃	—	4	4	no	—
122	C₂×Dic₅₃	—	4	4	no	—
123	C₂×Dic₅₃	—	8	8	no	—
124	C₈×C₅₃	—	2	4	no	—
125	C₈×C₅₃	C₂×C₄²	1	4	yes	—
126	C₈×C₅₃	C₂×C₄²	1	4	yes	—
127	#20455	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	2	4	no	—
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	2	4	no	—
132	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
133	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₅₃	#576	1	2	yes	—
137	C₂²×C₅₃	#578	1	2	yes	—
138	C₄×C₅₃	—	2	8	no	—
139	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
140	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
141	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
142	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
144	Dic₅₃	—	4	8	no	—
145	Dic₅₃	—	4	8	no	—
146	C₂×#427	—	53	1	no	—
147	C₂²×#107	—	53	1	no	—
148	C₂×#405	D₅₃	1	1	yes	characteristic
149	#2274	—	53	2	no	—
150	#1776	—	53	4	no	—
151	C₄×#107	—	53	4	no	—
152	#1736	—	53	2	no	—
153	C₂×C₅₃	—	2	4	no	—
154	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₅₃	#2913	1	2	yes	—
156	C₂×C₅₃	#2948	1	1	yes	characteristic
157	C₂×C₅₃	#2950	1	1	yes	characteristic
158	C₂×C₅₃	#2951	1	2	yes	—
159	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
160	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
161	C₂×#117	—	53	2	no	—
162	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
163	C₂²×C₄⋊C₄	—	53	2	no	—
164	#427	—	53	8	no	—
165	C₂×#107	—	53	8	no	—
166	C₂×#107	—	53	2	no	—
167	C₂×#107	—	53	2	no	—
168	C₂×#107	—	53	2	no	—
169	C₂×#106	—	53	2	no	—
170	#405	Dic₅₃	1	8	yes	complemented
171	C₂×#97	C₂×D₅₃	1	2	yes	—
172	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
173	C₄×C₄⋊C₄	—	53	8	no	—
174	C₄×C₄⋊C₄	—	106	4	no	—
175	C₄×C₂²⋊C₄	—	106	4	no	—
176	#340	—	53	8	no	—
177	C₅₃	#20455	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	106	2	no	—
179	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
183	C₂²×C₈	C₄×D₅₃	1	4	yes	—
184	C₂²×C₈	C₄×D₅₃	1	4	yes	—
185	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
186	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
187	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
189	#117	—	53	8	no	—
190	#117	—	106	4	no	—
191	#117	Dic₁₀₆	1	4	yes	—
192	#117	C₂×Dic₅₃	1	4	yes	—
193	C₂×C₄⋊C₄	—	53	8	no	—
194	C₂×C₄⋊C₄	—	53	4	no	—
195	C₂×C₄⋊C₄	—	53	4	no	—
196	C₂×C₄⋊C₄	—	106	2	no	—
197	C₂×C₄⋊C₄	—	106	4	no	—
198	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
199	C₂×C₂²⋊C₄	—	106	2	no	—
200	C₂×C₂²⋊C₄	D₂₁₂	1	2	yes	—
201	C₂×C₄²	—	106	2	no	—
202	C₂×C₄²	—	106	8	no	—
203	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
204	#107	—	53	8	no	—
205	#107	—	53	8	no	—
206	#107	—	53	8	no	—
207	#106	—	53	8	no	—
208	#97	C₂×Dic₅₃	1	8	yes	complemented
209	C₄×C₈	C₂×Dic₅₃	1	8	yes	complemented
210	#95	—	106	8	no	—
211	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
212	C₂²×C₄	—	106	8	no	—
213	C₂²×C₄	—	106	4	no	—
214	C₂²×C₄	—	106	4	no	—
215	C₂²×C₄	—	424	4	no	—
216	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
219	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
220	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
221	C₂²×C₄	#11719945	1	2	yes	—
222	C₂²×C₄	#11719945	1	2	yes	—
223	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
224	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
225	M₁₆	—	2	8	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₄×Dic₅₃	1	16	yes	complemented
229	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
230	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
231	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
233	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
234	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
235	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
236	C₄⋊C₄	—	53	8	no	—
237	C₄⋊C₄	—	53	8	no	—
238	C₄⋊C₄	—	106	8	no	—
239	C₄⋊C₄	—	106	4	no	—
240	C₄⋊C₄	—	106	4	no	—
241	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
242	C₂²⋊C₄	—	106	8	no	—
243	C₂²⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
244	C₄²	—	106	8	no	—
245	C₄²	—	424	8	no	—
246	C₄²	C₂×Dic₁₀₆	1	4	yes	—
247	C₄²	C₂²×Dic₅₃	1	4	yes	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
251	C₂³	C₄×D₂₁₂	1	2	yes	—
252	C₂³	C₄×D₂₁₂	1	2	yes	—
253	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	106	8	no	—
257	C₂×C₄	—	106	8	no	—
258	C₂×C₄	—	424	8	no	—
259	C₂×C₄	—	212	4	no	—
260	C₂×C₄	—	212	8	no	—
261	C₂×C₄	—	212	4	no	—
262	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
263	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
264	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
265	C₂×C₄	#13133176	1	4	yes	—
266	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
267	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
268	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
269	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
270	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
274	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
275	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
278	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	C₄×C₄⋊Dic₅₃	1	4	yes	—
284	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
285	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
286	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
287	C₂²	#16811428	1	2	yes	—
288	C₂²	#16811440	1	2	yes	—
289	C₄	—	2	8	no	—
290	C₄	—	212	8	no	—
291	C₄	—	212	8	no	—
292	C₄	C₄×C₄⋊Dic₅₃	1	4	yes	—
293	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
294	C₄	C₂×#13133176	1	1	yes	characteristic
295	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
296	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
297	C₂	—	2	4	no	—
298	C₂	C₂×C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
299	C₂	#20283315	1	2	yes	—
300	C₂	#20283568	1	1	yes	characteristic
301	C₂	#20283584	1	1	yes	characteristic
302	C₂	#20283592	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#20455
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #20274612	C₂	semidirect
5. C₄×#13133151	C₂	semidirect
10. #16810661	C₄	semidirect
11. C₂×#13133151	C₄	semidirect
23. C₅₃×#405	C₄	semidirect
39. #13133151	C₂×C₄	semidirect
40. #13133149	C₂×C₄	semidirect
57. #97×C₅₃	C₂×C₄	semidirect
58. C₄×C₈×C₅₃	C₂×C₄	semidirect
76. C₂×C₈×C₅₃	C₄²	semidirect
170. #405	Dic₅₃	semidirect
177. C₅₃	#20455	semidirect
208. #97	C₂×Dic₅₃	semidirect
209. C₄×C₈	C₂×Dic₅₃	semidirect
228. C₂×C₈	C₄×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810661	2
C₂²×#13133151	2
#20277071	2
#20274612	2
C₄×#13133151	2
#20274435	2
C₂×C₅₃×#405	2
#20455	53