Subgroups — #27402989

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₀₆	C₂	1	2	yes	maximal
2	C₂×#16810753	C₂	1	1	yes	maximal, characteristic
3	#20275490	C₂	1	2	yes	maximal
4	#20275489	C₂	1	4	yes	maximal
5	#20274888	C₂	1	2	yes	maximal
6	#20274887	C₂	1	1	yes	maximal, characteristic
7	#20274623	C₂	1	2	yes	maximal
8	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#16810753	—	2	2	no	—
11	#16810753	C₄	1	4	yes	—
12	#16810753	C₄	1	4	yes	—
13	#16810751	—	2	4	no	—
14	C₈×Dic₁₀₆	—	2	2	no	—
15	C₂×#13133153	C₂²	1	2	yes	—
16	C₂×#13133153	C₂²	1	2	yes	—
17	C₂×#13133152	C₂²	1	4	yes	—
18	#16810647	C₂²	1	2	yes	—
19	#16810646	C₂²	1	4	yes	—
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×#13133146	C₂²	1	2	yes	—
22	M₁₆×Dic₅₃	C₂²	1	4	yes	—
23	C₂×#13133127	C₂²	1	2	yes	—
24	#16810387	C₂²	1	4	yes	—
25	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#341	C₂²	1	2	yes	—
30	C₂²×Dic₁₀₆	Q₈	1	1	yes	characteristic
31	C₂×#11719917	C₂³	1	4	yes	—
32	C₂²×#88322	C₂³	1	2	yes	—
33	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₀₆	D₄	1	2	yes	—
36	C₄×Dic₁₀₆	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
38	C₂×#11719919	Q₈	1	2	yes	—
39	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
40	#13133153	—	2	4	no	—
41	#13133153	—	4	2	no	—
42	#13133152	—	4	4	no	—
43	C₄⋊#88322	—	2	2	no	—
44	C₄⋊#88322	—	2	2	no	—
45	C₄⋊#88322	C₂×C₄	1	2	yes	—
46	C₄⋊#88322	C₂×C₄	1	2	yes	—
47	#13133146	—	2	4	no	—
48	#13133146	—	2	4	no	—
49	#13133146	C₂×C₄	1	8	yes	—
50	#13133127	—	2	4	no	—
51	#13133127	C₂×C₄	1	8	yes	—
52	C₈×Dic₅₃	—	2	4	no	—
53	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
54	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	#108×C₅₃	—	2	2	no	—
59	#108×C₅₃	Q₈	1	4	yes	complemented
60	#97×C₅₃	—	2	2	no	—
61	C₄×C₈×C₅₃	—	2	2	no	—
62	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
68	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
69	C₂×Dic₁₀₆	—	4	2	no	—
70	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
71	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
72	M₁₆×C₅₃	—	2	2	no	—
73	M₁₆×C₅₃	—	2	2	no	—
74	M₁₆×C₅₃	—	2	2	no	—
75	M₁₆×C₅₃	C₂×Q₈	1	4	yes	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₅₃	#41	1	2	yes	—
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
89	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
90	#11719919	C₄⋊C₄	1	4	yes	—
91	#11719919	C₄⋊C₄	1	4	yes	—
92	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
93	C₄×Dic₅₃	C₂×D₄	1	4	yes	—
94	#11719917	—	2	4	no	—
95	#11719917	—	2	4	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	4	no	—
99	C₂×#88322	C₂²×C₄	1	4	yes	—
100	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
113	C₂×Dic₅₃	—	4	2	no	—
114	C₂×Dic₅₃	—	4	2	no	—
115	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
117	Dic₁₀₆	—	2	4	no	—
118	Dic₁₀₆	—	4	2	no	—
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	2	4	no	—
121	C₈×C₅₃	—	4	2	no	—
122	C₈×C₅₃	C₄×Q₈	1	4	yes	—
123	#88322	—	2	4	no	—
124	#88322	—	4	4	no	—
125	#22733	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₅₃	#557	1	1	yes	characteristic
130	C₂²×C₅₃	Q₈²	1	1	yes	characteristic
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	#430	1	2	yes	—
135	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
136	C₄×C₅₃	#520	1	1	yes	characteristic
137	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
138	C₄×C₅₃	#550	1	1	yes	characteristic
139	Dic₅₃	—	2	4	no	—
140	Dic₅₃	—	2	4	no	—
141	Q₈×M₁₆	—	53	2	no	—
142	C₂×#446	—	53	1	no	—
143	#1950	—	53	2	no	—
144	#1950	—	53	4	no	—
145	#1849	—	53	2	no	—
146	#1848	—	53	1	no	—
147	#1778	—	53	2	no	—
148	C₄×#108	D₅₃	1	1	yes	characteristic
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
153	C₂×C₅₃	#2916	1	1	yes	characteristic
154	C₂×C₄×Q₈	—	53	1	no	—
155	#446	—	53	4	no	—
156	#446	—	53	4	no	—
157	#446	—	106	2	no	—
158	#446	—	106	4	no	—
159	C₈×Q₈	—	106	2	no	—
160	C₂×#108	—	53	2	no	—
161	C₂×#108	—	53	4	no	—
162	C₂×#108	—	53	2	no	—
163	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
164	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
165	#423	—	53	4	no	—
166	#423	—	53	2	no	—
167	C₂×C₄⋊C₈	—	53	2	no	—
168	C₂×C₄⋊C₈	—	53	1	no	—
169	C₄×M₁₆	—	53	4	no	—
170	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
171	C₂×#97	—	53	2	no	—
172	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
173	#341	—	53	4	no	—
174	#341	C₂×D₅₃	1	2	yes	—
175	C₅₃	#22733	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	53	1	no	—
177	C₂×M₁₆	—	53	4	no	—
178	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
179	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
180	C₂²×C₈	—	53	2	no	—
181	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	C₄×Q₈	—	53	2	no	—
184	C₄×Q₈	—	53	2	no	—
185	C₄×Q₈	—	53	2	no	—
186	C₄×Q₈	—	53	2	no	—
187	C₂×C₄⋊C₄	—	53	2	no	—
188	C₂×C₄⋊C₄	—	53	1	no	—
189	C₂×C₄²	—	53	2	no	—
190	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
191	#108	—	2	2	no	—
192	#108	—	106	4	no	—
193	#108	—	212	2	no	—
194	#108	—	212	4	no	—
195	#108	Dic₁₀₆	1	4	yes	complemented
196	C₄⋊C₈	—	53	8	no	—
197	C₄⋊C₈	—	53	2	no	—
198	C₄⋊C₈	—	53	2	no	—
199	C₄⋊C₈	—	106	4	no	—
200	C₄⋊C₈	—	106	4	no	—
201	C₄⋊C₈	—	106	2	no	—
202	C₄⋊C₈	—	106	2	no	—
203	#97	—	2	2	no	—
204	#97	—	53	8	no	—
205	#97	—	106	4	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	—	106	4	no	—
208	C₄×C₈	C₄×D₅₃	1	2	yes	—
209	C₄×C₈	C₄×D₅₃	1	2	yes	—
210	C₂×Q₈	—	53	2	no	—
211	C₂×Q₈	—	53	2	no	—
212	C₂×Q₈	—	212	2	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
216	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	106	4	no	—
222	M₁₆	—	106	4	no	—
223	M₁₆	C₂×Dic₁₀₆	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	53	4	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
235	C₂×C₈	#11719945	1	2	yes	—
236	C₄⋊C₄	—	53	4	no	—
237	C₄⋊C₄	—	53	4	no	—
238	C₄⋊C₄	—	53	2	no	—
239	C₄⋊C₄	—	53	2	no	—
240	C₄²	—	53	4	no	—
241	C₄²	—	53	4	no	—
242	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
244	C₄²	D₄×D₅₃	1	1	yes	characteristic
245	C₄²	D₄×D₅₃	1	1	yes	characteristic
246	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
247	Q₈	—	106	4	no	—
248	Q₈	—	212	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	53	4	no	—
252	C₂×C₄	—	53	4	no	—
253	C₂×C₄	—	212	2	no	—
254	C₂×C₄	—	212	2	no	—
255	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
257	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
262	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
263	C₂×C₄	#13133263	1	1	yes	characteristic
264	C₂×C₄	#13133264	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	106	4	no	—
269	C₈	—	212	4	no	—
270	C₈	C₄×Dic₁₀₆	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
274	C₂²	#16811351	1	1	yes	characteristic
275	C₂²	Q₈×Dic₁₀₆	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	106	4	no	—
280	C₄	—	106	4	no	—
281	C₄	#16810670	1	2	yes	—
282	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
283	C₄	#16811103	1	1	yes	characteristic
284	C₄	D₄×Dic₁₀₆	1	1	yes	characteristic
285	C₄	#16811294	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₄⋊C₄×Dic₁₀₆	1	1	yes	characteristic
289	C₂	C₂×#16810670	1	1	yes	characteristic
290	C₂	#20283323	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	#22733
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
59. #108×C₅₃	Q₈	semidirect
175. C₅₃	#22733	semidirect
195. #108	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₀₆	2
C₂×#16810753	2
#20275490	2
#20275489	2
#20274888	2
#20274887	2
#20274623	2
C₄×#108×C₅₃	2
#22733	53