Subgroups — #27299127

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 266 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×#16810399	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272793	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂²×#11719908	C₂²	1	2	yes	—
8	C₂²⋊#11719908	C₄	1	8	yes	complemented
9	#16810399	—	2	2	no	—
10	C₄×#11719908	—	2	2	no	—
11	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
14	C₅₃×#405	—	2	2	no	—
15	C₅₃×#405	—	2	2	no	—
16	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	C₂×C₄	1	4	yes	—
25	C₂×#11719908	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₅₃	—	2	2	no	—
28	C₂×M₁₆×C₅₃	—	2	2	no	—
29	C₂×M₁₆×C₅₃	C₈	1	4	yes	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	—	2	1	no	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	#117×C₅₃	—	2	2	no	—
39	#117×C₅₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
41	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	#97×C₅₃	—	2	2	no	—
45	#97×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	—	2	1	no	—
47	C₄×C₈×C₅₃	—	2	1	no	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₅₃	—	2	8	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	C₄²	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₂²⋊C₄×C₅₃	—	2	4	no	—
79	C₄²×C₅₃	—	2	1	no	—
80	C₄²×C₅₃	—	2	1	no	—
81	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅₃	M₁₆	1	2	yes	—
83	#11719908	—	2	4	no	—
84	#11719908	—	2	4	no	—
85	C₂³×C₅₃	—	2	1	no	—
86	C₂³×C₅₃	—	2	1	no	—
87	C₂³×C₅₃	—	2	4	no	—
88	C₂³×C₅₃	C₄×C₈	1	2	yes	—
89	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
110	C₈×C₅₃	C₄×D₄	1	2	yes	—
111	C₈×C₅₃	C₄×D₄	1	2	yes	—
112	#12600	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₅₃	#431	1	1	yes	characteristic
120	C₂²×C₅₃	#504	1	4	yes	—
121	C₄×C₅₃	—	2	4	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	4	2	no	—
124	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
126	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₄×C₅₃	#431	1	1	yes	characteristic
128	C₂×#405	D₅₃	1	1	yes	characteristic
129	C₂×C₂²⋊C₁₆	—	53	2	no	—
130	C₂×#346	—	53	1	no	—
131	C₂×C₄×C₁₆	—	53	1	no	—
132	#1374	—	53	2	no	—
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₅₃	#2108	1	2	yes	—
136	C₂×C₅₃	#2113	1	2	yes	—
137	C₂×C₅₃	#2164	1	2	yes	—
138	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
139	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
140	C₂×#117	Dic₅₃	1	1	yes	characteristic
141	C₂²×C₁₆	—	53	2	no	—
142	C₂²×C₁₆	—	53	2	no	—
143	#405	—	2	2	no	—
144	#405	—	2	2	no	—
145	C₂×#97	Dic₅₃	1	1	yes	characteristic
146	C₂×#97	Dic₅₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
149	C₂²⋊C₁₆	—	53	8	no	—
150	#346	—	106	2	no	—
151	C₄×C₁₆	—	106	2	no	—
152	C₅₃	#12600	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	#88322	1	4	yes	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	C₄×D₅₃	1	4	yes	—
161	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
165	#117	—	2	2	no	—
166	#117	—	2	2	no	—
167	C₂×C₄⋊C₄	#88322	1	2	yes	—
168	C₂×C₂²⋊C₄	#88322	1	2	yes	—
169	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂×C₁₆	—	53	4	no	—
172	C₂×C₁₆	—	53	4	no	—
173	C₂×C₁₆	—	106	2	no	—
174	C₂×C₁₆	—	106	2	no	—
175	C₂×C₁₆	—	106	2	no	—
176	C₂×C₁₆	—	106	2	no	—
177	#97	—	2	2	no	—
178	#97	—	2	2	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
182	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
183	C₂⁴	C₂×#88322	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂×#88322	1	2	yes	—
189	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
190	C₂²×C₄	C₂×#88322	1	2	yes	—
191	C₂²×C₄	C₂×#88322	1	2	yes	—
192	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
194	M₁₆	—	2	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	C₂×#88322	1	2	yes	—
202	C₂×C₈	C₂×#88322	1	2	yes	—
203	C₂×C₈	#11719917	1	2	yes	—
204	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
205	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
207	C₂×C₈	#11719945	1	2	yes	—
208	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
209	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	#11719917	1	2	yes	—
215	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
216	C₁₆	—	106	4	no	—
217	C₁₆	—	106	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	C₄×#88322	1	2	yes	—
222	C₂³	C₂²×#88322	1	1	yes	characteristic
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	C₄×#88322	1	2	yes	—
233	C₂×C₄	C₂²⋊#88322	1	4	yes	—
234	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
236	C₂×C₄	#13133176	1	1	yes	characteristic
237	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
238	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
239	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	C₂²⋊#88322	1	4	yes	—
243	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
244	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
250	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
251	C₂²	#16810676	1	1	yes	characteristic
252	C₂²	#16811040	1	4	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	—	4	2	no	—
256	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
257	C₄	C₄×#11719917	1	1	yes	characteristic
258	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
259	C₄	#16810676	1	1	yes	characteristic
260	C₂	—	4	2	no	—
261	C₂	C₄×C₂²⋊#88322	1	1	yes	characteristic
262	C₂	#20276312	1	2	yes	—
263	C₂	#20276329	1	2	yes	—
264	C₂	#20276522	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₅₃×#405	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#12600
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. C₂×#16810399	C₂	semidirect
3. C₂×C₄×#11719908	C₂	semidirect
4. #20272793	C₂	semidirect
8. C₂²⋊#11719908	C₄	semidirect
152. C₅₃	#12600	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×#16810399	2
C₂×C₄×#11719908	2
#20272793	2
C₂×C₅₃×#405	2
#12600	53