Subgroups — #27271043

Order 13568.

386 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11719908	C₂	1	2	yes	maximal
2	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719908	—	2	2	no	—
4	C₂²⋊#11719908	—	2	2	no	—
5	C₂²⋊#11719908	C₄	1	4	yes	complemented
6	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
9	C₅₃×#336	—	2	2	no	—
10	C₂×#11719908	—	2	4	no	—
11	C₂×#11719908	—	4	2	no	—
12	C₂×#11719908	—	4	2	no	—
13	C₂³×C₄×C₅₃	C₈	1	2	yes	—
14	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	—	2	1	no	—
16	C₂²×C₈×C₅₃	—	2	1	no	—
17	C₂²×C₈×C₅₃	—	4	1	no	—
18	C₂²×C₈×C₅₃	—	4	1	no	—
19	C₂²×C₈×C₅₃	—	4	1	no	—
20	C₂²×C₈×C₅₃	—	4	1	no	—
21	C₂²×C₈×C₅₃	C₈	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	D₄	1	2	yes	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	—	2	1	no	—
27	C₂×C₄²×C₅₃	—	2	1	no	—
28	C₂×C₄²×C₅₃	—	4	1	no	—
29	C₂×C₄²×C₅₃	—	4	1	no	—
30	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	—	2	1	no	—
32	C₂²×C₄×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	—	2	1	no	—
34	C₂²×C₄×C₅₃	—	2	1	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	4	2	no	—
38	C₂²×C₄×C₅₃	—	4	2	no	—
39	C₂²×C₄×C₅₃	C₁₆	1	4	yes	complemented
40	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	—	4	1	no	—
51	C₂×C₈×C₅₃	—	4	1	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	—	4	1	no	—
54	C₂×C₈×C₅₃	—	4	1	no	—
55	C₂×C₈×C₅₃	—	4	2	no	—
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₄²×C₅₃	—	4	2	no	—
59	C₄²×C₅₃	—	4	1	no	—
60	C₄²×C₅₃	—	4	1	no	—
61	#11719908	—	4	4	no	—
62	C₂³×C₅₃	—	2	1	no	—
63	C₂³×C₅₃	—	2	1	no	—
64	C₂³×C₅₃	—	2	2	no	—
65	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
66	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
67	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
68	C₂³×C₅₃	C₂×C₁₆	1	2	yes	—
69	C₂³×C₅₃	M₃₂	1	2	yes	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
93	C₈×C₅₃	—	2	2	no	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	—	4	2	no	—
96	#7183	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	2	2	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	#323	1	1	yes	characteristic
106	C₂²×C₅₃	#324	1	1	yes	characteristic
107	C₂²×C₅₃	#336	1	1	yes	characteristic
108	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
109	C₂²×C₅₃	#346	1	1	yes	characteristic
110	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
111	C₂²×C₅₃	#349	1	2	yes	—
112	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	4	4	no	—
116	C₄×C₅₃	#323	1	1	yes	characteristic
117	C₄×C₅₃	#324	1	1	yes	characteristic
118	C₄×C₅₃	#342	1	1	yes	characteristic
119	C₄×C₅₃	#343	1	1	yes	characteristic
120	C₂×C₂²⋊C₁₆	—	53	2	no	—
121	C₂×#336	D₅₃	1	1	yes	characteristic
122	C₂×C₅₃	—	2	1	no	—
123	C₂×C₅₃	—	2	1	no	—
124	C₂×C₅₃	—	2	2	no	—
125	C₂×C₅₃	#1280	1	1	yes	characteristic
126	C₂×C₅₃	#1314	1	2	yes	—
127	C₂×C₅₃	#1322	1	2	yes	—
128	C₂×C₅₃	#1374	1	1	yes	characteristic
129	C₂×C₅₃	#1377	1	1	yes	characteristic
130	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
131	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
132	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
133	C₂²×C₁₆	—	106	2	no	—
134	C₂²⋊C₁₆	—	53	4	no	—
135	C₂²⋊C₁₆	—	106	2	no	—
136	#336	—	2	2	no	—
137	C₅₃	#7183	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	#88322	1	2	yes	—
139	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	4	1	no	—
143	C₂²×C₈	—	4	1	no	—
144	C₂²×C₈	—	4	1	no	—
145	C₂²×C₈	—	4	1	no	—
146	C₂²×C₈	#88322	1	2	yes	—
147	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
148	C₂²×C₈	C₄×D₅₃	1	2	yes	—
149	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
150	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	—	4	1	no	—
154	C₂×C₄²	—	4	1	no	—
155	C₂×C₁₆	—	106	4	no	—
156	C₂×C₁₆	—	212	2	no	—
157	C₂×C₁₆	—	212	2	no	—
158	C₂⁴	C₂×#88322	1	1	yes	characteristic
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	#11719908	1	4	yes	complemented
168	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
169	C₂²×C₄	#11719917	1	1	yes	characteristic
170	C₂²×C₄	#11719917	1	1	yes	characteristic
171	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	#11719919	1	2	yes	—
185	C₂×C₈	#11719921	1	2	yes	—
186	C₄²	—	4	2	no	—
187	C₄²	—	4	1	no	—
188	C₄²	—	4	1	no	—
189	C₁₆	—	212	4	no	—
190	C₂³	—	2	1	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	2	no	—
193	C₂³	C₄×#88322	1	1	yes	characteristic
194	C₂³	C₂²⋊#88322	1	1	yes	characteristic
195	C₂³	C₄⋊#88322	1	1	yes	characteristic
196	C₂³	C₂×#11719908	1	2	yes	—
197	C₂³	#13133157	1	2	yes	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	#13133124	1	1	yes	characteristic
214	C₂×C₄	#13133128	1	1	yes	characteristic
215	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
216	C₂×C₄	#13133132	1	1	yes	characteristic
217	C₂×C₄	#13133132	1	1	yes	characteristic
218	C₂×C₄	#13133134	1	1	yes	characteristic
219	C₂×C₄	#13133136	1	1	yes	characteristic
220	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	4	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	#16810347	1	1	yes	characteristic
233	C₂²	#16810349	1	1	yes	characteristic
234	C₂²	#16810378	1	1	yes	characteristic
235	C₂²	C₄×#11719908	1	1	yes	characteristic
236	C₂²	#16810399	1	1	yes	characteristic
237	C₂²	C₂²⋊#11719908	1	2	yes	—
238	C₂²	#16810405	1	2	yes	—
239	C₂²	C₄⋊#11719908	1	2	yes	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	4	no	—
243	C₄	#16810347	1	1	yes	characteristic
244	C₄	#16810349	1	1	yes	characteristic
245	C₄	#16810390	1	1	yes	characteristic
246	C₄	#16810392	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	—	2	2	no	—
250	C₂	#20272558	1	1	yes	characteristic
251	C₂	#20272639	1	2	yes	—
252	C₂	#20272655	1	2	yes	—
253	C₂	#20272793	1	1	yes	characteristic
254	C₂	#20272799	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₄×C₁₆
Frattini	C₂³×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂³×C₈×C₅₃	C₂²
hypercenter	C₂×#336	D₅₃
hypocenter	C₅₃	#7183
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
5. C₂²⋊#11719908	C₄	semidirect
39. C₂²×C₄×C₅₃	C₁₆	semidirect
137. C₅₃	#7183	semidirect
167. C₂²×C₄	#11719908	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#11719908	2
C₂×C₅₃×#336	2
#7183	53