Subgroups — #27299461

Order 13568.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810400	C₂	1	2	yes	maximal
3	#20274582	C₂	1	1	yes	maximal, characteristic
4	#20272794	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
8	C₂×#13133127	C₂²	1	2	yes	—
9	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
10	C₄⋊#11719908	C₄	1	2	yes	complemented
11	C₄⋊#11719908	C₄	1	2	yes	complemented
12	#16810400	—	2	4	no	—
13	#16810377	C₄	1	2	yes	—
14	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#346	C₄	1	2	yes	complemented
18	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
19	C₂²×#88322	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	—	2	1	no	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
22	#13133157	—	2	2	no	—
23	#13133157	—	2	2	no	—
24	C₂×#11719908	C₂×C₄	1	4	yes	—
25	C₄⋊#88322	—	2	1	no	—
26	C₄⋊#88322	—	2	1	no	—
27	#13133127	—	2	2	no	—
28	#13133127	—	2	2	no	—
29	#13133124	—	2	1	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₅₃	—	2	1	no	—
34	M₃₂×C₅₃	—	2	1	no	—
35	M₃₂×C₅₃	D₄	1	2	yes	—
36	M₃₂×C₅₃	Q₈	1	2	yes	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
57	C₄²×C₅₃	C₄²	1	1	yes	characteristic
58	C₄²×C₅₃	C₄²	1	1	yes	characteristic
59	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
60	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
61	C₄⋊Dic₅₃	—	4	2	no	—
62	C₄×Dic₅₃	—	2	2	no	—
63	C₄×Dic₅₃	—	2	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	2	2	no	—
67	C₁₆×C₅₃	—	2	2	no	—
68	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
69	#11719908	—	2	4	no	—
70	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	4	1	no	—
73	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	4	2	no	—
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	—	2	1	no	—
88	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₈×C₅₃	#117	1	1	yes	characteristic
90	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
91	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
92	#88322	—	4	2	no	—
93	#12666	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
97	C₂²×C₅₃	#436	1	1	yes	characteristic
98	C₂²×C₅₃	#446	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	#347	1	2	yes	complemented
103	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₄×C₅₃	#405	1	1	yes	characteristic
105	C₄×C₅₃	#436	1	1	yes	characteristic
106	C₄×C₅₃	#446	1	1	yes	characteristic
107	Dic₅₃	—	8	2	no	—
108	#2150	—	53	1	no	—
109	C₂×#347	—	53	2	no	—
110	C₄×M₃₂	D₅₃	1	1	yes	characteristic
111	#1770	—	53	1	no	—
112	#1375	—	53	2	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	#1770	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#347	1	1	yes	characteristic
117	C₂×C₅₃	#2110	1	1	yes	characteristic
118	C₂×M₃₂	—	53	2	no	—
119	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
120	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
121	C₂×C₄⋊C₈	—	53	1	no	—
122	C₂×#97	—	53	2	no	—
123	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
124	C₄×C₄⋊C₄	—	53	1	no	—
125	C₄⋊C₁₆	—	53	2	no	—
126	C₄⋊C₁₆	—	53	2	no	—
127	#347	—	106	4	no	—
128	#346	Dic₅₃	1	2	yes	complemented
129	C₄×C₁₆	Dic₅₃	1	2	yes	complemented
130	#336	—	53	2	no	—
131	C₅₃	#12666	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	53	2	no	—
133	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	106	1	no	—
136	C₂×C₄²	—	53	2	no	—
137	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
138	M₃₂	—	2	1	no	—
139	M₃₂	—	2	1	no	—
140	M₃₂	—	106	2	no	—
141	M₃₂	—	106	2	no	—
142	M₃₂	Dic₁₀₆	1	2	yes	—
143	M₃₂	D₂₁₂	1	2	yes	—
144	C₂×C₁₆	—	53	4	no	—
145	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
146	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
147	C₄⋊C₈	—	106	1	no	—
148	C₄⋊C₈	—	106	1	no	—
149	#97	—	106	2	no	—
150	#97	—	106	2	no	—
151	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
152	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
153	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
154	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
155	#95	—	106	1	no	—
156	C₂²×C₄	—	106	2	no	—
157	C₂²×C₄	#11719912	1	1	yes	characteristic
158	C₂²×C₄	#11719912	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	2	no	—
164	C₂×C₈	—	106	2	no	—
165	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
166	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
167	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
168	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
170	C₂×C₈	#11719945	1	1	yes	characteristic
171	C₂×C₈	#11719945	1	1	yes	characteristic
172	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
173	C₄⋊C₄	—	212	2	no	—
174	C₄²	—	106	2	no	—
175	C₄²	—	106	2	no	—
176	C₄²	#11719912	1	1	yes	characteristic
177	C₄²	#11719912	1	1	yes	characteristic
178	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
179	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
180	C₁₆	—	2	2	no	—
181	C₁₆	—	106	4	no	—
182	C₁₆	C₄⋊Dic₅₃	1	4	yes	complemented
183	C₂³	C₂×#11719912	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	212	2	no	—
187	C₂×C₄	—	212	2	no	—
188	C₂×C₄	—	212	2	no	—
189	C₂×C₄	#13133127	1	1	yes	characteristic
190	C₂×C₄	#13133127	1	1	yes	characteristic
191	C₂×C₄	#13133127	1	1	yes	characteristic
192	C₂×C₄	#13133127	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
194	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
195	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
196	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
197	C₂×C₄	#13133223	1	1	yes	characteristic
198	C₂×C₄	#13133223	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	212	2	no	—
202	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
203	C₈	#13133176	1	1	yes	characteristic
204	C₈	C₄×D₂₁₂	1	1	yes	characteristic
205	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#13133127	1	1	yes	characteristic
209	C₂²	#16810702	1	1	yes	characteristic
210	C₂²	#16810753	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	424	2	no	—
215	C₄	#16810400	1	2	yes	complemented
216	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
217	C₄	#16810578	1	1	yes	characteristic
218	C₄	#16810702	1	1	yes	characteristic
219	C₄	#16810753	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#20274582	1	1	yes	characteristic
223	C₂	C₂×#16810400	1	1	yes	characteristic
224	C₂	#20276319	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12666
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. C₄⋊#11719908	C₄	semidirect
11. C₄⋊#11719908	C₄	semidirect
17. C₅₃×#346	C₄	semidirect
18. C₄×C₁₆×C₅₃	C₄	semidirect
68. C₁₆×C₅₃	C₄⋊C₄	semidirect
102. C₄×C₅₃	#347	semidirect
128. #346	Dic₅₃	semidirect
129. C₄×C₁₆	Dic₅₃	semidirect
131. C₅₃	#12666	semidirect
182. C₁₆	C₄⋊Dic₅₃	semidirect
215. C₄	#16810400	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₂×#16810400	2
#20274582	2
#20272794	2
C₄×M₃₂×C₅₃	2
#12666	53