Subgroups — #27394536

Order 13568.

946 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20275434	C₂	1	4	yes	maximal, complemented
4	#20274895	C₂	1	4	yes	maximal
5	#20274867	C₂	1	2	yes	maximal, complemented
6	#20274832	C₂	1	2	yes	maximal, complemented
7	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×#88322	C₂²	1	2	yes	—
9	#16810676	—	2	4	no	—
10	C₂×C₄⋊#88322	—	2	4	no	—
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	8	yes	—
16	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
17	C₂×C₄×#88322	C₂²	1	2	yes	—
18	#16810378	C₂²	1	8	yes	—
19	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
20	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
21	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#376	C₄	1	2	yes	—
27	C₄³×C₅₃	C₂²	1	2	yes	—
28	C₂²×#88322	—	2	4	no	—
29	C₂²×#88322	—	4	4	no	—
30	C₂²×#88322	C₂³	1	2	yes	—
31	C₂²×#88322	C₂³	1	2	yes	—
32	C₂²×#88322	C₂³	1	4	yes	—
33	C₄⋊#88322	—	2	4	no	—
34	C₄⋊#88322	—	2	4	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	—	2	2	no	—
37	C₄⋊#88322	—	4	8	no	—
38	C₂²⋊#88322	—	2	4	no	—
39	C₄×#88322	—	2	4	no	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	#117×C₅₃	—	2	4	no	—
44	#117×C₅₃	C₈	1	8	yes	complemented
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₅₃	D₄	1	4	yes	—
55	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	#95×C₅₃	C₂×C₄	1	4	yes	—
60	#95×C₅₃	C₂×C₄	1	4	yes	—
61	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
81	C₂²⋊C₄×C₅₃	—	2	4	no	—
82	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
83	C₄²×C₅₃	—	2	4	no	—
84	C₄²×C₅₃	—	2	4	no	—
85	C₄²×C₅₃	—	2	4	no	—
86	C₄²×C₅₃	—	4	4	no	—
87	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
88	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
89	C₄²×C₅₃	C₂×C₈	1	8	yes	complemented
90	C₄²×C₅₃	C₂×D₄	1	2	yes	—
91	C₄²×C₅₃	C₂×D₄	1	2	yes	—
92	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
93	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	—	2	4	no	—
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	—	2	8	no	—
99	C₂×#88322	—	2	4	no	—
100	C₂×#88322	—	4	8	no	—
101	C₂³×C₅₃	—	2	2	no	—
102	C₂³×C₅₃	—	2	2	no	—
103	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
105	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
106	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	2	4	no	—
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	—	4	4	no	—
119	C₂×C₄×C₅₃	C₄⋊C₈	1	16	yes	complemented
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
126	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
133	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
134	#88322	—	4	2	no	—
135	#88322	—	4	4	no	—
136	#88322	—	4	2	no	—
137	#21972	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	2	4	no	—
141	C₂²×C₅₃	—	2	4	no	—
142	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
143	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₅₃	#520	1	2	yes	—
145	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
146	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
147	C₂²×C₅₃	#568	1	2	yes	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	—	2	8	no	—
152	C₄×C₅₃	—	4	4	no	—
153	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
154	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
155	C₄×C₅₃	#535	1	1	yes	characteristic
156	C₄×C₅₃	#536	1	1	yes	characteristic
157	C₄×C₅₃	#544	1	2	yes	—
158	C₂×#431	—	53	1	no	—
159	C₂²×C₄⋊C₈	—	53	1	no	—
160	C₄×#117	D₅₃	1	1	yes	characteristic
161	#1939	—	53	4	no	—
162	#1850	—	53	4	no	—
163	#1842	—	53	2	no	—
164	#1831	—	53	2	no	—
165	C₂×C₅₃	—	2	4	no	—
166	C₂×C₅₃	#2317	1	1	yes	characteristic
167	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
168	C₂×C₅₃	#2905	1	1	yes	characteristic
169	C₂×C₅₃	#2988	1	2	yes	—
170	C₂×C₅₃	#2989	1	2	yes	—
171	C₂³×C₈	—	53	2	no	—
172	C₂×#117	Dic₅₃	1	1	yes	characteristic
173	C₂×#117	Dic₅₃	1	1	yes	characteristic
174	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
175	#431	—	106	4	no	—
176	C₂×C₄⋊C₈	—	53	1	no	—
177	C₂×C₄⋊C₈	—	53	2	no	—
178	C₂×C₄⋊C₈	—	53	1	no	—
179	C₂×C₄⋊C₈	—	53	2	no	—
180	C₂×C₄⋊C₈	—	53	8	no	—
181	C₂×C₄⋊C₈	—	106	4	no	—
182	C₂×C₂²⋊C₈	—	53	2	no	—
183	C₂×C₄×C₈	—	53	2	no	—
184	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
185	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
186	C₄×C₂²⋊C₄	Dic₅₃	1	2	yes	—
187	C₄×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
188	#376	Dic₅₃	1	2	yes	—
189	C₄³	C₂×D₅₃	1	2	yes	—
190	#336	—	53	8	no	—
191	C₅₃	#21972	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
193	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
194	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
195	C₂²×C₈	—	53	4	no	—
196	C₂²×C₈	—	53	2	no	—
197	C₂²×C₈	—	53	2	no	—
198	C₂²×C₈	—	106	4	no	—
199	C₂²×C₈	—	212	4	no	—
200	#117	—	2	4	no	—
201	#117	#88322	1	8	yes	complemented
202	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
204	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
205	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
206	C₂×C₄²	Dic₁₀₆	1	4	yes	—
207	C₂×C₄²	D₂₁₂	1	4	yes	—
208	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
209	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
210	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
211	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
212	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
214	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	C₄⋊C₈	—	106	4	no	—
217	C₄⋊C₈	—	106	4	no	—
218	C₄⋊C₈	—	106	2	no	—
219	C₄⋊C₈	—	106	2	no	—
220	C₄⋊C₈	—	212	8	no	—
221	C₂²⋊C₈	—	106	4	no	—
222	C₄×C₈	—	106	4	no	—
223	#95	C₂×Dic₅₃	1	4	yes	—
224	#95	C₂×Dic₅₃	1	4	yes	—
225	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₂×#88322	1	4	yes	—
230	C₂²×C₄	#11719917	1	4	yes	—
231	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
232	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
233	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
234	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
235	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
236	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
240	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
243	C₂×C₈	—	106	2	no	—
244	C₂×C₈	—	106	2	no	—
245	C₂×C₈	—	106	4	no	—
246	C₂×C₈	—	106	4	no	—
247	C₂×C₈	—	106	8	no	—
248	C₂×C₈	—	106	4	no	—
249	C₂×C₈	—	212	8	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	C₂×#88322	1	8	yes	complemented
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	C₂×#88322	1	8	yes	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	4	4	no	—
258	C₄²	C₂×#88322	1	8	yes	complemented
259	C₄²	C₄⋊Dic₅₃	1	4	yes	—
260	C₄²	C₄⋊Dic₅₃	1	4	yes	—
261	C₄²	C₂×Dic₁₀₆	1	2	yes	—
262	C₄²	C₂×Dic₁₀₆	1	2	yes	—
263	C₄²	C₂×D₂₁₂	1	2	yes	—
264	C₄²	C₂×D₂₁₂	1	2	yes	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
268	C₂³	C₂²×#88322	1	2	yes	—
269	C₂³	C₂×#11719917	1	2	yes	—
270	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	C₄⋊#88322	1	16	yes	complemented
284	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
285	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
286	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
287	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
289	C₂×C₄	C₂²×#88322	1	8	yes	—
290	C₂×C₄	C₂²×#88322	1	2	yes	—
291	C₂×C₄	C₂²×#88322	1	2	yes	—
292	C₂×C₄	C₂×#11719917	1	2	yes	—
293	C₂×C₄	C₂×#11719917	1	2	yes	—
294	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
295	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
296	C₂×C₄	#13133263	1	2	yes	—
297	C₂×C₄	#13133264	1	2	yes	—
298	C₈	—	212	2	no	—
299	C₈	—	212	4	no	—
300	C₈	—	212	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	4	no	—
305	C₂²	C₂×C₄⋊#88322	1	4	yes	—
306	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
307	C₂²	#16811106	1	2	yes	—
308	C₂²	C₂³×#88322	1	1	yes	characteristic
309	C₂²	C₂²×#11719917	1	1	yes	characteristic
310	C₂²	#16811390	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	8	no	—
315	C₄	—	4	4	no	—
316	C₄	C₂×C₄⋊#88322	1	4	yes	—
317	C₄	C₂×C₄⋊#88322	1	4	yes	—
318	C₄	#16811192	1	1	yes	characteristic
319	C₄	#16811199	1	1	yes	characteristic
320	C₄	#16811251	1	2	yes	—
321	C₂	—	2	4	no	—
322	C₂	#20277338	1	1	yes	characteristic
323	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
324	C₂	#20283279	1	1	yes	characteristic
325	C₂	#20283826	1	2	yes	—
326	C₂	#20283832	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₄×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	#21972
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20275434	C₂	semidirect
5. #20274867	C₂	semidirect
6. #20274832	C₂	semidirect
44. #117×C₅₃	C₈	semidirect
80. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
89. C₄²×C₅₃	C₂×C₈	semidirect
119. C₂×C₄×C₅₃	C₄⋊C₈	semidirect
191. C₅₃	#21972	semidirect
201. #117	#88322	semidirect
251. C₄⋊C₄	C₂×#88322	semidirect
258. C₄²	C₂×#88322	semidirect
283. C₂×C₄	C₄⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂²×C₄⋊#88322	2
#20275434	2
#20274895	2
#20274867	2
#20274832	2
C₄×#117×C₅₃	2
#21972	53