Subgroups — #27537078

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₅₃	C₂	1	2	yes	maximal
2	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
3	#20276472	C₂	1	8	yes	maximal, complemented
4	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276470	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11719912	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×#13133174	C₄	1	1	yes	characteristic
10	C₂×#13133157	C₂²	1	2	yes	—
11	C₂×#13133156	C₂²	1	4	yes	—
12	M₃₂×D₅₃	—	2	8	no	—
13	C₂×C₁₆×D₅₃	C₂²	1	4	yes	—
14	#16810577	C₄	1	4	yes	—
15	#16810577	C₂²	1	4	yes	—
16	C₂×#13133128	C₄	1	1	yes	characteristic
17	C₂×#13133127	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊#11719908	C₂²	1	4	yes	complemented
20	#16810441	C₂²	1	8	yes	complemented
21	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
24	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×#11719912	—	2	4	no	—
26	C₂×#11719912	C₂×C₄	1	4	yes	—
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₅₃	D₄	1	4	yes	—
31	C₂×C₈×D₅₃	Q₈	1	4	yes	—
32	C₂×C₈×D₅₃	C₂³	1	2	yes	—
33	#13133174	C₈	1	4	yes	—
34	#13133174	C₂×C₄	1	4	yes	—
35	C₂×#11719919	C₈	1	2	yes	—
36	C₂×#11719921	C₈	1	2	yes	—
37	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	#13133157	—	2	4	no	—
39	#13133156	—	2	8	no	—
40	C₂×#11719908	C₂³	1	4	yes	—
41	C₁₆×D₅₃	—	2	8	no	—
42	#13133128	C₂×C₄	1	4	yes	—
43	#13133127	C₂×C₄	1	4	yes	—
44	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
45	C₈×Dic₅₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	M₃₂×C₅₃	—	2	4	no	—
50	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
51	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
57	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄×D₅₃	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
60	C₂×C₄×D₅₃	M₁₆	1	4	yes	—
61	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×C₈	1	2	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	#11719921	C₂×C₈	1	8	yes	—
71	#11719919	C₂×C₈	1	8	yes	—
72	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
73	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	C₄⋊C₄	1	4	yes	—
76	C₂×#88322	C₂²×C₄	1	2	yes	—
77	C₂×#88322	C₂²×C₄	1	2	yes	—
78	#11719912	—	2	8	no	—
79	C₈×D₅₃	C₄⋊C₄	1	8	yes	—
80	C₈×D₅₃	C₂×D₄	1	4	yes	—
81	C₈×D₅₃	C₂×Q₈	1	4	yes	—
82	C₁₆×C₅₃	—	2	4	no	—
83	#11719908	—	2	4	no	—
84	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂²×D₅₃	—	4	2	no	—
86	C₂²×D₅₃	C₄⋊C₈	1	4	yes	—
87	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×Dic₅₃	—	4	2	no	—
95	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
96	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
97	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
98	C₄×D₅₃	C₄⋊C₈	1	8	yes	—
99	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
105	#88322	—	2	2	no	—
106	#88322	C₂×C₄⋊C₄	1	4	yes	—
107	#33397	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
110	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
111	C₂×D₅₃	—	2	4	no	—
112	C₂×D₅₃	—	4	4	no	—
113	C₂×D₅₃	C₂×C₄⋊C₈	1	4	yes	—
114	C₄×C₅₃	—	2	4	no	—
115	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
116	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
118	Dic₅₃	—	2	4	no	—
119	Dic₅₃	C₂×C₄⋊C₈	1	4	yes	—
120	C₂²×M₃₂	—	53	2	no	—
121	C₂×#405	—	53	1	no	—
122	#2151	—	53	8	no	—
123	#2150	—	53	2	no	—
124	#2150	—	53	1	no	—
125	#2150	D₅₃	1	1	yes	characteristic, complemented
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₅₃	#3407	1	2	yes	—
129	D₅₃	—	2	4	no	—
130	C₂²×M₁₆	—	53	1	no	—
131	C₂³×C₈	—	53	1	no	—
132	C₂×#117	—	53	1	no	—
133	C₂×M₃₂	—	53	2	no	—
134	C₂×M₃₂	—	53	4	no	—
135	C₂×M₃₂	—	106	8	no	—
136	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
137	C₂²×C₁₆	—	53	4	no	—
138	#405	—	53	4	no	—
139	#405	—	53	4	no	—
140	C₂×#97	—	53	1	no	—
141	C₂×#97	—	53	1	no	—
142	C₂×C₄×C₈	—	53	1	no	—
143	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
144	C₄⋊C₁₆	—	53	8	no	—
145	C₄⋊C₁₆	—	53	4	no	—
146	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	complemented
147	C₅₃	#33397	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	—	53	1	no	—
149	C₂×M₁₆	—	53	4	no	—
150	C₂×M₁₆	—	106	4	no	—
151	C₂²×C₈	—	53	1	no	—
152	C₂²×C₈	—	53	2	no	—
153	C₂²×C₈	—	53	4	no	—
154	C₂²×C₈	—	53	4	no	—
155	C₂²×C₈	—	53	2	no	—
156	C₂²×C₈	—	53	1	no	—
157	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	#117	—	53	4	no	—
160	#117	—	53	4	no	—
161	C₂×C₄⋊C₄	—	53	2	no	—
162	C₂×C₂²⋊C₄	—	53	2	no	—
163	C₂×C₄²	—	53	1	no	—
164	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
165	M₃₂	—	2	4	no	—
166	M₃₂	—	106	8	no	—
167	M₃₂	—	106	4	no	—
168	C₂×C₁₆	—	53	4	no	—
169	C₂×C₁₆	—	106	8	no	—
170	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
171	#97	—	53	4	no	—
172	#97	—	53	4	no	—
173	C₄×C₈	—	53	2	no	—
174	C₄×C₈	—	53	2	no	—
175	C₄×C₈	C₄×D₅₃	1	2	yes	—
176	C₄×C₈	C₂²×D₅₃	1	2	yes	—
177	C₂⁴	—	53	1	no	—
178	C₂²×C₄	—	53	1	no	—
179	C₂²×C₄	—	53	4	no	—
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	53	4	no	—
182	C₂²×C₄	—	53	2	no	—
183	C₂²×C₄	—	53	2	no	—
184	C₂²×C₄	C₈×D₅₃	1	2	yes	—
185	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
186	M₁₆	—	106	8	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	53	4	no	—
189	C₂×C₈	—	53	2	no	—
190	C₂×C₈	—	53	8	no	—
191	C₂×C₈	—	53	4	no	—
192	C₂×C₈	—	53	4	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	106	2	no	—
195	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
196	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
197	C₂×C₈	D₄×D₅₃	1	2	yes	—
198	C₂×C₈	Q₈×D₅₃	1	2	yes	—
199	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
200	C₄⋊C₄	—	53	8	no	—
201	C₂²⋊C₄	—	53	8	no	—
202	C₄²	—	53	2	no	—
203	C₄²	—	53	2	no	—
204	C₄²	C₈×D₅₃	1	2	yes	—
205	C₄²	C₂×C₄×D₅₃	1	2	yes	—
206	C₁₆	—	2	4	no	—
207	C₁₆	—	106	4	no	—
208	C₂³	—	53	2	no	—
209	C₂³	—	53	4	no	—
210	C₂³	—	212	2	no	—
211	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	53	2	no	—
214	C₂×C₄	—	53	4	no	—
215	C₂×C₄	—	53	4	no	—
216	C₂×C₄	—	53	4	no	—
217	C₂×C₄	—	53	8	no	—
218	C₂×C₄	—	53	4	no	—
219	C₂×C₄	—	212	2	no	—
220	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
221	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
222	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
223	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
224	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	53	4	no	—
227	C₈	—	106	2	no	—
228	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
229	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
230	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	53	4	no	—
233	C₂²	—	106	4	no	—
234	C₂²	—	212	4	no	—
235	C₂²	C₄⋊C₈×D₅₃	1	2	yes	—
236	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	53	4	no	—
239	C₄	—	106	4	no	—
240	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
241	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
242	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	106	4	no	—
245	C₂	C₂×C₄⋊C₈×D₅₃	1	1	yes	characteristic
246	C₂	#20287817	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2150	C₂
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#33397
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #20276472	C₂	semidirect
4. #20276471	C₂	semidirect
5. #20276470	C₂	semidirect
6. C₅₃×#2150	C₂	semidirect
19. C₄⋊#11719908	C₂²	semidirect
20. #16810441	C₂²	semidirect
23. C₅₃×C₄⋊C₁₆	C₂²	semidirect
125. #2150	D₅₃	semidirect
146. C₄⋊C₁₆	C₂×D₅₃	semidirect
147. C₅₃	#33397	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₅₃	2
C₂×#16810577	2
#20276472	2
#20276471	2
#20276470	2
C₅₃×#2150	2
#33397	53