Subgroups — #27277231

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274853	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	#20273144	C₂	1	2	yes	maximal
4	#20272612	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	#16810647	C₂²	1	2	yes	—
8	#16810388	—	2	1	no	—
9	#16810388	—	2	1	no	—
10	#16810376	—	2	1	no	—
11	#16810374	—	2	1	no	—
12	Q₈⋊#88322	—	2	4	no	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₅₃	—	2	1	no	—
15	C₈×Q₈×C₅₃	—	2	1	no	—
16	C₈×Q₈×C₅₃	C₄	1	2	yes	—
17	C₈×Q₈×C₅₃	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#11719917	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₂²⋊#88322	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₅₃	—	2	1	no	—
33	C₄×Q₈×C₅₃	—	2	1	no	—
34	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×Q₈×C₅₃	D₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	Q₈	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	C₄⋊C₈×C₅₃	—	2	1	no	—
43	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂×Q₈×C₅₃	—	2	4	no	—
51	C₂×Q₈×C₅₃	—	2	1	no	—
52	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	—	2	1	no	—
69	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	#11719917	—	4	2	no	—
77	#11719917	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
81	Q₈×C₅₃	—	2	4	no	—
82	Q₈×C₅₃	—	2	2	no	—
83	Q₈×C₅₃	#108	1	4	yes	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#135	1	2	yes	—
97	C₂×C₄×C₅₃	#137	1	2	yes	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	1	no	—
101	C₈×C₅₃	—	2	1	no	—
102	C₈×C₅₃	—	4	2	no	—
103	#88322	—	8	2	no	—
104	#88322	—	8	2	no	—
105	#8661	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	#380	1	1	yes	characteristic
109	C₂²×C₅₃	#479	1	2	yes	—
110	C₄×C₅₃	—	2	4	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	4	2	no	—
115	C₄×C₅₃	#416	1	1	yes	characteristic
116	C₄×C₅₃	#419	1	1	yes	characteristic
117	C₄×C₅₃	C₂×#108	1	2	yes	—
118	C₄×C₅₃	#464	1	2	yes	—
119	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
120	#1837	—	53	1	no	—
121	#1564	—	53	1	no	—
122	#1487	—	53	2	no	—
123	#1301	—	53	2	no	—
124	C₂×C₅₃	—	2	2	no	—
125	C₂×C₅₃	#1805	1	1	yes	characteristic
126	C₂×C₅₃	#1810	1	1	yes	characteristic
127	C₂×C₅₃	#1814	1	1	yes	characteristic
128	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
129	C₈×Q₈	—	2	1	no	—
130	C₈×Q₈	—	2	1	no	—
131	C₈×Q₈	Dic₅₃	1	2	yes	—
132	C₈×Q₈	Dic₅₃	1	2	yes	—
133	#431	—	53	2	no	—
134	#423	—	53	2	no	—
135	C₂×C₄⋊C₈	—	2	1	no	—
136	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
137	C₂×C₄×C₈	—	2	1	no	—
138	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
139	#341	—	106	1	no	—
140	#341	—	106	1	no	—
141	#335	—	106	1	no	—
142	#334	—	106	1	no	—
143	Q₈⋊C₈	—	106	4	no	—
144	C₅₃	#8661	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
146	C₂×M₁₆	—	106	2	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	D₂₁₂	1	2	yes	—
153	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
156	C₂×C₄²	—	2	1	no	—
157	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	106	2	no	—
162	C₄⋊C₈	—	106	2	no	—
163	C₄⋊C₈	—	106	2	no	—
164	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
165	C₂²⋊C₈	—	106	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	106	2	no	—
169	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
170	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
179	C₂²×C₄	#11719945	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	M₁₆	—	212	2	no	—
182	M₁₆	—	212	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
200	C₄²	#11719945	1	1	yes	characteristic
201	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂³	#13133196	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	2	no	—
206	Q₈	#13133153	1	4	yes	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	#13133176	1	1	yes	characteristic
216	C₂×C₄	#13133191	1	1	yes	characteristic
217	C₂×C₄	#13133191	1	1	yes	characteristic
218	C₂×C₄	#13133196	1	1	yes	characteristic
219	C₂×C₄	#13133237	1	2	yes	—
220	C₂×C₄	#13133248	1	2	yes	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	4	2	no	—
226	C₈	—	424	2	no	—
227	C₈	—	424	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#16810483	1	1	yes	characteristic
231	C₂²	#16810930	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	#16810619	1	1	yes	characteristic
238	C₄	#16810632	1	1	yes	characteristic
239	C₄	C₂×#13133153	1	2	yes	—
240	C₄	#16810845	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#20274730	1	1	yes	characteristic
243	C₂	#20274753	1	1	yes	characteristic
244	C₂	#20274769	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#8661
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
144. C₅₃	#8661	semidirect

Maximal subgroups

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

subgroup	index
#20274853	2
#20273463	2
#20273144	2
#20272612	2
C₂×C₈×Q₈×C₅₃	2
#8661	53