Subgroups — #27277831

Order 13568.

462 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	#20274866	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810375	C₂	1	2	yes	maximal
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133146	—	2	4	no	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810376	—	2	2	no	—
15	#16810375	—	2	4	no	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#335	—	2	2	no	—
19	C₂²×#88322	—	2	2	no	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
25	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	#13133146	—	4	8	no	—
29	C₄×#88322	D₄	1	2	yes	—
30	C₄×#88322	Q₈	1	2	yes	—
31	C₈×Dic₅₃	—	2	2	no	—
32	C₈×Dic₅₃	—	2	2	no	—
33	C₈×Dic₅₃	—	2	2	no	—
34	C₈×Dic₅₃	C₈	1	4	yes	complemented
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	D₄	1	2	yes	—
42	C₄×C₈×C₅₃	Q₈	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	—	4	4	no	—
52	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₄×Dic₅₃	—	2	1	no	—
61	C₄×Dic₅₃	—	2	1	no	—
62	C₄×Dic₅₃	—	2	4	no	—
63	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
64	C₄×Dic₅₃	M₁₆	1	2	yes	—
65	C₄×Dic₅₃	D₈	1	4	yes	—
66	C₄×Dic₅₃	Q₁₆	1	4	yes	—
67	C₂×#88322	—	2	1	no	—
68	C₂×#88322	—	2	1	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂×#88322	—	4	4	no	—
71	C₂×#88322	C₄⋊C₄	1	2	yes	—
72	C₂×#88322	C₄⋊C₄	1	2	yes	—
73	C₂×#88322	C₂×C₈	1	2	yes	—
74	C₂×#88322	M₁₆	1	2	yes	—
75	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
89	C₂×Dic₅₃	—	2	2	no	—
90	C₂×Dic₅₃	—	2	2	no	—
91	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
92	C₂×Dic₅₃	#107	1	4	yes	—
93	C₂×Dic₅₃	#107	1	4	yes	—
94	C₂×Dic₅₃	#108	1	4	yes	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	—	8	2	no	—
97	C₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
98	#88322	—	2	2	no	—
99	#88322	—	8	2	no	—
100	#88322	C₄⋊C₈	1	4	yes	complemented
101	#8806	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	#384	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₂²×C₅₃	#423	1	1	yes	characteristic
105	C₂²×C₅₃	C₂×#107	1	2	yes	—
106	C₂²×C₅₃	C₂×#108	1	2	yes	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₄×C₅₃	#423	1	1	yes	characteristic
112	C₄×C₅₃	#423	1	1	yes	characteristic
113	C₄×C₅₃	#493	1	1	yes	characteristic
114	C₄×C₅₃	#494	1	1	yes	characteristic
115	C₄×C₅₃	#500	1	2	yes	—
116	Dic₅₃	—	2	4	no	—
117	Dic₅₃	#335	1	8	yes	complemented
118	#1842	—	53	2	no	—
119	C₄²×C₈	—	53	1	no	—
120	C₂×#335	—	53	2	no	—
121	C₂×#335	—	53	1	no	—
122	C₂×#335	D₅₃	1	1	yes	characteristic
123	C₂×C₅₃	C₂×#335	1	2	yes	—
124	C₂×C₅₃	#1567	1	2	yes	—
125	C₂×C₅₃	#1842	1	1	yes	characteristic
126	C₂×C₅₃	#1845	1	1	yes	characteristic
127	C₂×C₅₃	#1848	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	53	2	no	—
129	C₂×C₄⋊C₈	—	106	4	no	—
130	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
131	C₂×C₄×C₈	—	53	1	no	—
132	C₂×C₄×C₈	—	53	1	no	—
133	C₂×C₄×C₈	—	53	1	no	—
134	C₂×C₄×C₈	—	53	1	no	—
135	C₂×C₄×C₈	—	53	1	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
137	C₄³	—	53	1	no	—
138	#335	—	2	2	no	—
139	#335	—	106	4	no	—
140	#335	—	106	2	no	—
141	C₅₃	#8806	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	53	1	no	—
144	C₂²×C₈	—	53	1	no	—
145	C₂²×C₈	—	106	2	no	—
146	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₄²	—	53	1	no	—
149	C₂×C₄²	—	53	2	no	—
150	C₂×C₄²	—	53	1	no	—
151	C₂×C₄²	—	53	2	no	—
152	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	106	2	no	—
156	C₄⋊C₈	—	106	2	no	—
157	C₄⋊C₈	—	212	8	no	—
158	C₄×C₈	—	53	4	no	—
159	C₄×C₈	—	53	2	no	—
160	C₄×C₈	—	53	2	no	—
161	C₄×C₈	—	106	2	no	—
162	C₄×C₈	—	106	2	no	—
163	C₄×C₈	—	106	2	no	—
164	C₄×C₈	Dic₁₀₆	1	2	yes	—
165	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
166	C₂²×C₄	—	53	2	no	—
167	C₂²×C₄	—	53	2	no	—
168	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
169	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	—	53	2	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	53	2	no	—
178	C₂×C₈	—	53	2	no	—
179	C₂×C₈	—	106	1	no	—
180	C₂×C₈	—	106	1	no	—
181	C₂×C₈	—	212	2	no	—
182	C₂×C₈	—	212	4	no	—
183	C₂×C₈	C₈×D₅₃	1	2	yes	—
184	C₂×C₈	#11719912	1	2	yes	—
185	C₂×C₈	#11719919	1	2	yes	—
186	C₂×C₈	#11719919	1	2	yes	—
187	C₄²	—	53	2	no	—
188	C₄²	—	53	2	no	—
189	C₄²	—	53	4	no	—
190	C₄²	—	53	4	no	—
191	C₄²	—	106	4	no	—
192	C₄²	—	106	1	no	—
193	C₄²	—	106	1	no	—
194	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
195	C₄²	D₄×D₅₃	1	1	yes	characteristic
196	C₄²	Q₈×D₅₃	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	53	4	no	—
201	C₂×C₄	—	53	4	no	—
202	C₂×C₄	—	53	4	no	—
203	C₂×C₄	—	53	4	no	—
204	C₂×C₄	—	106	2	no	—
205	C₂×C₄	—	106	2	no	—
206	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
208	C₂×C₄	#13133213	1	1	yes	characteristic
209	C₂×C₄	#13133215	1	1	yes	characteristic
210	C₂×C₄	#13133216	1	1	yes	characteristic
211	C₂×C₄	#13133216	1	1	yes	characteristic
212	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
213	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
214	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
215	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
216	C₂×C₄	D₈×D₅₃	1	2	yes	—
217	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
218	C₈	—	2	2	no	—
219	C₈	—	8	2	no	—
220	C₈	—	53	4	no	—
221	C₈	—	106	2	no	—
222	C₈	—	424	2	no	—
223	C₈	#13133146	1	4	yes	complemented
224	C₂²	#16810492	1	1	yes	characteristic
225	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
226	C₂²	#16810645	1	1	yes	characteristic
227	C₂²	#107×D₅₃	1	2	yes	—
228	C₂²	#108×D₅₃	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	53	8	no	—
232	C₄	—	106	4	no	—
233	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
234	C₄	C₂×#13133146	1	1	yes	characteristic
235	C₄	#16810645	1	1	yes	characteristic
236	C₄	#16810646	1	1	yes	characteristic
237	C₄	#16810998	1	1	yes	characteristic
238	C₄	#16811001	1	1	yes	characteristic
239	C₄	#16811027	1	2	yes	—
240	C₂	#335×D₅₃	1	2	yes	—
241	C₂	#20273472	1	2	yes	—
242	C₂	#20274866	1	1	yes	characteristic
243	C₂	#20274873	1	1	yes	characteristic
244	C₂	#20274885	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#8806
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
34. C₈×Dic₅₃	C₈	semidirect
97. C₈×C₅₃	C₄⋊C₈	semidirect
100. #88322	C₄⋊C₈	semidirect
117. Dic₅₃	#335	semidirect
141. C₅₃	#8806	semidirect
223. C₈	#13133146	semidirect

Maximal subgroups

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

subgroup	index
#20274866	2
C₄×C₈×Dic₅₃	2
C₂×#16810376	2
C₂×#16810375	2
C₂×C₅₃×#335	2
#8806	53