Subgroups — #27537176

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
2	#20276519	C₂	1	8	yes	maximal, complemented
3	#20276470	C₂	1	2	yes	maximal, complemented
4	#20276326	C₂	1	1	yes	maximal, characteristic
5	M₃₂×Dic₅₃	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2112	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₄	1	1	yes	characteristic
8	C₂×#13133176	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16810578	C₄	1	4	yes	—
11	#16810578	C₂²	1	4	yes	—
12	C₂×#13133127	C₄	1	1	yes	characteristic
13	C₂×#13133127	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	#16810441	C₂²	1	8	yes	complemented
17	C₂²⋊#11719908	C₂²	1	4	yes	—
18	#16810398	C₂²	1	4	yes	complemented
19	C₁₆×Dic₅₃	C₂²	1	4	yes	complemented
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
22	C₅₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
23	C₂×#11719917	—	2	4	no	—
24	C₂×#11719917	C₂×C₄	1	4	yes	—
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	#13133176	C₈	1	4	yes	—
28	#13133176	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
30	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	#13133157	—	2	4	no	—
34	C₂×#11719908	C₂³	1	4	yes	—
35	#13133127	C₂×C₄	1	4	yes	—
36	#13133127	C₂×C₄	1	4	yes	—
37	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
38	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
39	C₈×Dic₅₃	C₂³	1	2	yes	—
40	C₈×Dic₅₃	C₂³	1	2	yes	—
41	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	—	2	4	no	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	M₃₂×C₅₃	—	2	4	no	—
48	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
49	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	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	2	yes	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	#41	1	4	yes	—
64	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂²⋊Dic₅₃	C₂×C₈	1	8	yes	—
66	C₄⋊Dic₅₃	C₂×C₈	1	8	yes	—
67	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
68	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
69	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
70	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
71	#11719917	—	2	8	no	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	M₁₆	1	4	yes	—
74	C₂×#88322	C₂²×C₄	1	2	yes	—
75	C₂×#88322	C₂²×C₄	1	2	yes	—
76	C₁₆×C₅₃	—	2	4	no	—
77	#11719908	—	2	4	no	—
78	C₂³×C₅₃	—	2	2	no	—
79	C₂³×C₅₃	—	4	2	no	—
80	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
81	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	#117	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
94	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	#117	1	2	yes	—
97	C₈×C₅₃	C₂×#41	1	2	yes	—
98	#88322	—	2	2	no	—
99	#88322	C₂×M₁₆	1	4	yes	—
100	#33419	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	2	4	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	4	4	no	—
105	C₂²×C₅₃	#431	1	2	yes	—
106	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	#431	1	2	yes	—
109	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
110	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
111	Dic₅₃	—	2	4	no	—
112	Dic₅₃	—	2	4	no	—
113	C₂×#405	—	53	1	no	—
114	#2164	—	53	8	no	—
115	#2150	—	53	2	no	—
116	#2112	—	53	1	no	—
117	#2112	D₅₃	1	1	yes	characteristic
118	C₄×M₃₂	—	53	2	no	—
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	—	4	2	no	—
121	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
122	C₂×C₅₃	#3407	1	2	yes	—
123	C₂²×M₁₆	—	53	1	no	—
124	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂×#117	—	53	1	no	—
126	C₂×M₃₂	—	53	2	no	—
127	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
128	#405	—	53	4	no	—
129	#405	—	53	4	no	—
130	C₂×#97	—	53	1	no	—
131	C₂×#97	—	53	1	no	—
132	C₂×C₄×C₈	—	53	1	no	—
133	C₂×C₄×C₈	—	53	1	no	—
134	C₄⋊C₁₆	—	53	8	no	—
135	C₂²⋊C₁₆	—	53	4	no	—
136	C₂²⋊C₁₆	C₂×D₅₃	1	4	yes	—
137	#346	—	53	4	no	—
138	C₄×C₁₆	—	53	4	no	—
139	C₅₃	#33419	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
141	C₂×M₁₆	—	53	4	no	—
142	C₂×M₁₆	—	106	4	no	—
143	C₂²×C₈	—	2	4	no	—
144	C₂²×C₈	—	53	1	no	—
145	C₂²×C₈	—	53	1	no	—
146	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₈	C₄×D₅₃	1	2	yes	—
148	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
149	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
150	#117	—	53	4	no	—
151	#117	—	53	4	no	—
152	C₂×C₄⋊C₄	—	53	2	no	—
153	C₂×C₂²⋊C₄	—	53	2	no	—
154	C₂×C₄²	—	53	1	no	—
155	C₂×C₄²	—	53	1	no	—
156	M₃₂	—	2	4	no	—
157	M₃₂	—	106	4	no	—
158	C₂×C₁₆	—	53	4	no	—
159	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
160	#97	—	53	4	no	—
161	#97	—	53	4	no	—
162	C₄×C₈	—	53	2	no	—
163	C₄×C₈	—	53	2	no	—
164	C₄×C₈	—	53	2	no	—
165	C₄×C₈	—	53	2	no	—
166	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	53	2	no	—
170	C₂²×C₄	—	53	2	no	—
171	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₈×D₅₃	1	2	yes	—
173	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
174	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
175	M₁₆	—	106	8	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	53	4	no	—
180	C₂×C₈	—	53	2	no	—
181	C₂×C₈	—	53	2	no	—
182	C₂×C₈	—	106	2	no	—
183	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
184	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
185	C₂×C₈	#11719947	1	4	yes	—
186	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
187	C₄⋊C₄	—	53	8	no	—
188	C₂²⋊C₄	—	53	8	no	—
189	C₄²	—	53	2	no	—
190	C₄²	—	53	2	no	—
191	C₄²	—	53	2	no	—
192	C₄²	—	53	2	no	—
193	C₁₆	—	2	4	no	—
194	C₁₆	—	106	4	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	C₂×C₈×D₅₃	1	2	yes	—
198	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	53	4	no	—
204	C₂×C₄	—	53	4	no	—
205	C₂×C₄	—	212	2	no	—
206	C₂×C₄	—	212	2	no	—
207	C₂×C₄	#13133173	1	2	yes	—
208	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
209	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
210	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
211	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	53	4	no	—
214	C₈	—	106	2	no	—
215	C₈	#13133173	1	2	yes	—
216	C₈	C₂×#11719947	1	2	yes	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	#16810673	1	2	yes	—
222	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	106	4	no	—
225	C₄	—	106	4	no	—
226	C₄	#16810673	1	2	yes	—
227	C₄	C₂×#13133173	1	1	yes	characteristic
228	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	4	2	no	—
231	C₂	C₂×#16810673	1	1	yes	characteristic
232	C₂	#20287817	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2112	C₂
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2112	D₅₃
hypocenter	C₅₃	#33419
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
2. #20276519	C₂	semidirect
3. #20276470	C₂	semidirect
5. M₃₂×Dic₅₃	C₂	semidirect
16. #16810441	C₂²	semidirect
18. #16810398	C₂²	semidirect
19. C₁₆×Dic₅₃	C₂²	semidirect
139. C₅₃	#33419	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810578	2
#20276519	2
#20276470	2
#20276326	2
M₃₂×Dic₅₃	2
C₅₃×#2112	2
#33419	53