Subgroups — #27275393

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133152	C₂	1	1	yes	maximal, characteristic
2	#20273461	C₂	1	1	yes	maximal, characteristic
3	#20272996	C₂	1	1	yes	maximal, characteristic
4	M₁₆×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
6	#20272552	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1278	C₂	1	1	yes	maximal, characteristic
8	#16810676	C₂²	1	1	yes	characteristic
9	C₂×#13133152	C₄	1	2	yes	—
10	C₄×#11719917	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810387	C₄	1	2	yes	—
14	#16810375	C₄	1	2	yes	—
15	#16810373	C₄	1	2	yes	—
16	#16810345	—	2	1	no	—
17	#16810345	C₄	1	2	yes	—
18	#16810345	C₄	1	2	yes	—
19	#16810344	—	2	1	no	—
20	C₈×#88322	—	2	1	no	—
21	C₈×#88322	—	2	1	no	—
22	C₅₃×#431	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#11719917	C₂×C₄	1	2	yes	—
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	Q₈	1	2	yes	—
29	#13133152	C₈	1	8	yes	complemented
30	C₄⋊#88322	C₂×C₄	1	2	yes	—
31	C₂²⋊#88322	—	2	1	no	—
32	#13133128	—	2	1	no	—
33	C₄×#88322	—	2	1	no	—
34	C₄×#88322	—	2	1	no	—
35	C₄×#88322	C₂×C₄	1	1	yes	characteristic
36	C₄×#88322	C₂×C₄	1	1	yes	characteristic
37	C₄×#88322	C₂×C₄	1	1	yes	characteristic
38	C₄×#88322	C₂×C₄	1	1	yes	characteristic
39	C₄×#88322	C₂×C₄	1	2	yes	—
40	C₄×#88322	D₄	1	1	yes	characteristic
41	C₄×#88322	D₄	1	1	yes	characteristic
42	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	—	2	1	no	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₅₃	—	2	1	no	—
47	#97×C₅₃	—	2	1	no	—
48	C₄×C₈×C₅₃	—	2	1	no	—
49	C₄×C₈×C₅₃	—	2	1	no	—
50	C₄×C₈×C₅₃	—	2	1	no	—
51	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
55	M₁₆×C₅₃	—	2	2	no	—
56	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	#41	1	1	yes	characteristic
65	#11719917	—	2	2	no	—
66	#11719917	C₂×C₈	1	4	yes	—
67	C₂×#88322	—	2	2	no	—
68	C₂×#88322	—	2	2	no	—
69	C₂×#88322	—	2	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	C₄²	1	1	yes	characteristic
72	C₂×#88322	C₄²	1	1	yes	characteristic
73	C₂×#88322	C₄²	1	1	yes	characteristic
74	C₂×#88322	C₄²	1	1	yes	characteristic
75	C₂×#88322	C₄⋊C₄	1	2	yes	—
76	C₂×#88322	C₄⋊C₄	1	2	yes	—
77	C₂×#88322	C₂×C₈	1	4	yes	—
78	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	#131	1	2	yes	—
89	C₈×C₅₃	—	2	4	no	—
90	C₈×C₅₃	—	4	2	no	—
91	C₈×C₅₃	—	4	2	no	—
92	#88322	—	2	1	no	—
93	#88322	—	2	1	no	—
94	#88322	—	2	2	no	—
95	#88322	—	2	2	no	—
96	#88322	—	2	4	no	—
97	#88322	—	4	2	no	—
98	#88322	C₄×C₈	1	2	yes	—
99	#88322	C₄×C₈	1	2	yes	—
100	#8294	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
108	C₄×C₅₃	#405	1	1	yes	characteristic
109	C₄×C₅₃	#430	1	2	yes	—
110	C₄×C₅₃	C₈×D₄	1	2	yes	—
111	C₄×C₅₃	#444	1	2	yes	—
112	C₄×#108	—	53	1	no	—
113	#1564	—	53	1	no	—
114	C₈×M₁₆	—	53	1	no	—
115	C₈×M₁₆	—	53	1	no	—
116	C₂×#322	—	53	1	no	—
117	#1278	—	53	1	no	—
118	#1278	D₅₃	1	1	yes	characteristic
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₅₃	#1773	1	1	yes	characteristic
122	C₂×C₅₃	#1778	1	1	yes	characteristic
123	#431	—	53	1	no	—
124	#431	C₂×D₅₃	1	1	yes	characteristic
125	C₂×#108	—	53	2	no	—
126	C₄×M₁₆	—	53	1	no	—
127	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
128	C₂×C₄×C₈	—	53	1	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#341	—	53	2	no	—
132	#335	—	53	2	no	—
133	#334	—	53	2	no	—
134	#322	—	53	2	no	—
135	#322	—	53	2	no	—
136	#322	—	106	1	no	—
137	#322	—	106	1	no	—
138	C₈²	—	106	1	no	—
139	C₈²	—	106	1	no	—
140	C₅₃	#8294	1	1	yes	characteristic, complemented, Sylow
141	C₂×M₁₆	—	53	2	no	—
142	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	53	1	no	—
145	C₂²×C₈	—	53	1	no	—
146	C₂²×C₈	—	53	2	no	—
147	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
148	#108	—	53	8	no	—
149	C₄⋊C₈	—	53	2	no	—
150	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
151	C₂²⋊C₈	—	2	1	no	—
152	C₂²⋊C₈	—	106	1	no	—
153	#97	—	2	1	no	—
154	#97	—	106	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	—	53	1	no	—
159	C₄×C₈	—	53	1	no	—
160	C₄×C₈	—	53	2	no	—
161	C₄×C₈	—	53	1	no	—
162	C₄×C₈	—	53	1	no	—
163	C₄×C₈	—	53	1	no	—
164	C₄×C₈	—	53	1	no	—
165	C₄×C₈	—	106	1	no	—
166	C₄×C₈	—	106	1	no	—
167	C₄×C₈	C₄×D₅₃	1	2	yes	—
168	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
169	C₂²×C₄	#11719947	1	1	yes	characteristic
170	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
171	M₁₆	—	2	2	no	—
172	M₁₆	—	53	4	no	—
173	M₁₆	—	106	2	no	—
174	M₁₆	C₈×D₅₃	1	4	yes	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	53	1	no	—
180	C₂×C₈	—	53	1	no	—
181	C₂×C₈	—	53	4	no	—
182	C₂×C₈	—	53	1	no	—
183	C₂×C₈	—	53	1	no	—
184	C₂×C₈	—	53	2	no	—
185	C₂×C₈	—	53	2	no	—
186	C₂×C₈	—	106	2	no	—
187	C₂×C₈	—	106	2	no	—
188	C₂×C₈	—	106	2	no	—
189	C₂×C₈	—	212	2	no	—
190	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
191	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
192	C₄²	D₄×D₅₃	1	1	yes	characteristic
193	C₄²	#11719947	1	1	yes	characteristic
194	C₂³	#13133182	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
198	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄	#13133173	1	1	yes	characteristic
200	C₂×C₄	#13133177	1	1	yes	characteristic
201	C₂×C₄	#13133177	1	1	yes	characteristic
202	C₂×C₄	#13133182	1	1	yes	characteristic
203	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
204	C₂×C₄	#13133225	1	2	yes	—
205	C₈	—	2	4	no	—
206	C₈	—	4	2	no	—
207	C₈	—	4	2	no	—
208	C₈	—	53	2	no	—
209	C₈	—	53	2	no	—
210	C₈	—	106	1	no	—
211	C₈	—	106	1	no	—
212	C₈	—	106	2	no	—
213	C₈	—	106	2	no	—
214	C₈	—	106	4	no	—
215	C₈	—	212	2	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	#16810474	1	1	yes	characteristic
219	C₂²	#16810747	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
223	C₄	#16810580	1	1	yes	characteristic
224	C₄	#16810670	1	2	yes	—
225	C₄	#16810686	1	2	yes	—
226	C₄	#16810742	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	#20274576	1	1	yes	characteristic
229	C₂	#20274598	1	1	yes	characteristic
230	C₂	#20274620	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1278	C₂
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1278	D₅₃
hypocenter	C₅₃	#8294
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
29. #13133152	C₈	semidirect
140. C₅₃	#8294	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133152	2
#20273461	2
#20272996	2
M₁₆×#88322	2
C₂×#16810345	2
#20272552	2
C₅₃×#1278	2
#8294	53