Subgroups — #27403103

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11719917	C₂	1	2	yes	maximal
2	#20275491	C₂	1	2	yes	maximal
3	#20275491	C₂	1	4	yes	maximal
4	#20274856	C₂	1	2	yes	maximal
5	#20274853	C₂	1	1	yes	maximal, characteristic
6	#20274621	C₂	1	2	yes	maximal
7	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
9	#16810754	—	2	4	no	—
10	#16810754	—	2	2	no	—
11	Q₈×#88322	—	2	2	no	—
12	C₂×#13133153	C₂²	1	2	yes	—
13	C₂×#13133153	C₂²	1	4	yes	—
14	#16810647	C₂²	1	2	yes	—
15	#16810647	C₂²	1	4	yes	—
16	C₄×#11719917	C₂²	1	2	yes	—
17	C₄×#11719917	C₂²	1	4	yes	—
18	#16810388	C₂²	1	2	yes	—
19	#16810388	C₂²	1	4	yes	—
20	#16810388	C₂²	1	2	yes	—
21	#16810388	C₂²	1	1	yes	characteristic
22	#16810388	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#446	C₄	1	4	yes	—
25	C₅₃×#446	C₄	1	4	yes	—
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
28	C₂×#97×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#11719917	C₂³	1	2	yes	—
31	C₂×#11719917	C₂³	1	4	yes	—
32	C₂×#11719917	C₂³	1	2	yes	—
33	#13133153	—	2	4	no	—
34	#13133153	—	2	8	no	—
35	C₄⋊#88322	—	2	4	no	—
36	C₄⋊#88322	—	2	2	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	—	2	4	no	—
39	#13133128	—	2	4	no	—
40	#13133128	—	2	2	no	—
41	C₄×#88322	—	2	2	no	—
42	C₄×#88322	—	2	4	no	—
43	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×Q₈×C₅₃	D₄	1	2	yes	—
50	C₄×Q₈×C₅₃	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
55	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
56	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
58	#97×C₅₃	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×Q₈×C₅₃	—	4	2	no	—
62	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂×Q₈×C₅₃	C₄⋊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	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
75	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
76	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
77	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
78	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
79	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	4	yes	—
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	#11719917	—	2	2	no	—
86	#11719917	—	2	2	no	—
87	#11719917	—	2	4	no	—
88	#11719917	—	2	2	no	—
89	#11719917	—	2	4	no	—
90	#11719917	#41	1	4	yes	—
91	C₂×#88322	—	2	2	no	—
92	C₂×#88322	—	2	2	no	—
93	C₂×#88322	—	2	4	no	—
94	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
95	Q₈×C₅₃	—	2	4	no	—
96	Q₈×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	4	1	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
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	4	yes	—
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	C₂×#41	1	2	yes	—
110	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	—	2	4	no	—
113	C₈×C₅₃	—	2	2	no	—
114	#88322	—	2	4	no	—
115	#88322	—	4	2	no	—
116	#88322	—	4	4	no	—
117	#22747	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₅₃	#557	1	2	yes	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	4	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	4	no	—
127	C₄×C₅₃	#430	1	2	yes	—
128	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
129	C₄×C₅₃	#519	1	1	yes	characteristic
130	C₄×C₅₃	#548	1	2	yes	—
131	Q₈×M₁₆	—	53	2	no	—
132	C₂×#446	D₅₃	1	1	yes	characteristic
133	#1950	—	53	4	no	—
134	#1950	—	53	2	no	—
135	#1838	—	53	2	no	—
136	#1837	—	53	1	no	—
137	#1778	—	53	2	no	—
138	C₄×#108	—	53	1	no	—
139	C₂×C₅₃	—	2	2	no	—
140	C₂×C₅₃	#2351	1	1	yes	characteristic
141	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
142	C₂×C₅₃	#2916	1	1	yes	characteristic
143	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
144	#446	—	106	2	no	—
145	#446	—	106	4	no	—
146	#446	Dic₅₃	1	4	yes	—
147	#446	Dic₅₃	1	4	yes	—
148	C₈×Q₈	—	106	2	no	—
149	C₂×#108	—	53	2	no	—
150	C₂×#108	—	53	4	no	—
151	#423	—	53	2	no	—
152	#423	—	53	4	no	—
153	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
154	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
155	C₄×M₁₆	—	53	2	no	—
156	C₄×M₁₆	—	53	4	no	—
157	C₂×#97	C₂×D₅₃	1	2	yes	—
158	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
159	#341	—	53	2	no	—
160	#341	—	53	2	no	—
161	#341	—	53	1	no	—
162	#341	—	53	4	no	—
163	#341	—	53	1	no	—
164	C₅₃	#22747	1	1	yes	characteristic, complemented, Sylow
165	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
166	C₂×M₁₆	—	53	2	no	—
167	C₂×M₁₆	—	53	4	no	—
168	C₂×M₁₆	—	53	2	no	—
169	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
172	C₄×Q₈	D₂₁₂	1	2	yes	—
173	C₄×Q₈	D₂₁₂	1	2	yes	—
174	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
175	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
176	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
177	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
178	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
180	#108	—	106	4	no	—
181	#108	—	106	8	no	—
182	C₄⋊C₈	—	106	2	no	—
183	C₄⋊C₈	—	106	4	no	—
184	C₄⋊C₈	—	106	4	no	—
185	C₄⋊C₈	—	106	2	no	—
186	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
187	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
188	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
189	#97	—	106	4	no	—
190	#97	—	106	2	no	—
191	#97	C₂×Dic₅₃	1	8	yes	—
192	C₄×C₈	—	106	2	no	—
193	C₄×C₈	—	106	4	no	—
194	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
195	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
198	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
199	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
200	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
201	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
202	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
203	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
204	M₁₆	—	53	4	no	—
205	M₁₆	—	106	2	no	—
206	M₁₆	—	106	2	no	—
207	M₁₆	—	106	4	no	—
208	M₁₆	—	106	4	no	—
209	M₁₆	—	106	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	106	2	no	—
214	C₂×C₈	—	106	2	no	—
215	C₂×C₈	—	106	4	no	—
216	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
217	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
218	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
219	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
220	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
221	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
222	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
223	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
224	C₄²	C₂×D₂₁₂	1	4	yes	—
225	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
226	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
227	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
228	C₄²	C₂²×Dic₅₃	1	4	yes	—
229	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
244	C₂×C₄	#41×D₅₃	1	2	yes	—
245	C₂×C₄	#13133265	1	2	yes	—
246	C₈	—	2	2	no	—
247	C₈	—	2	4	no	—
248	C₈	—	2	2	no	—
249	C₈	—	106	4	no	—
250	C₈	—	212	2	no	—
251	C₈	—	212	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
255	C₂²	#16811349	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	1	no	—
261	C₄	#16810672	1	2	yes	—
262	C₄	#41×Dic₅₃	1	1	yes	characteristic
263	C₄	#16811101	1	1	yes	characteristic
264	C₄	#16811285	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	#20277636	1	1	yes	characteristic
267	C₂	C₂×#16810672	1	1	yes	characteristic
268	C₂	#20283326	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#446	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#446	D₅₃
hypocenter	C₅₃	#22747
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
164. C₅₃	#22747	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11719917	2
#20275491	2
#20274856	2
#20274853	2
#20274621	2
C₄×#13133153	2
C₂×C₅₃×#446	2
#22747	53