Subgroups — #27299416

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
2	M₃₂×Dic₅₃	C₂	1	2	yes	maximal
3	#20274401	C₂	1	1	yes	maximal, characteristic
4	#20272796	C₂	1	2	yes	maximal
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133128	C₄	1	1	yes	characteristic
8	C₂×#13133127	C₄	1	1	yes	characteristic
9	C₂×#13133127	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810469	C₄	1	1	yes	characteristic
13	#16810444	C₄	1	4	yes	complemented
14	#16810398	—	2	2	no	—
15	C₁₆×Dic₅₃	—	2	2	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#364	C₄	1	4	yes	complemented
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	#13133157	—	2	2	no	—
24	#13133157	C₂×C₄	1	4	yes	—
25	C₂×#11719908	—	2	2	no	—
26	#13133128	—	2	2	no	—
27	#13133127	C₈	1	4	yes	—
28	#13133127	C₈	1	4	yes	—
29	C₈×Dic₅₃	—	2	1	no	—
30	C₈×Dic₅₃	—	2	1	no	—
31	C₈×Dic₅₃	D₄	1	2	yes	—
32	C₈×Dic₅₃	Q₈	1	2	yes	—
33	#13133124	—	2	2	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₅₃	—	2	2	no	—
38	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₁₆×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	—	2	1	no	—
45	C₂²×Dic₅₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₅₃	C₄²	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
55	C₄²×C₅₃	—	2	1	no	—
56	C₄²×C₅₃	C₄²	1	2	yes	—
57	C₄×Dic₅₃	—	2	1	no	—
58	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
59	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
60	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	—	2	1	no	—
63	C₂×#88322	C₄⋊C₄	1	2	yes	—
64	C₂×#88322	C₂×C₈	1	2	yes	—
65	C₂×#88322	C₂×C₈	1	2	yes	—
66	C₂×#88322	M₁₆	1	2	yes	—
67	C₁₆×C₅₃	—	2	4	no	—
68	#11719908	—	2	4	no	—
69	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₅₃	—	2	2	no	—
79	C₂×Dic₅₃	—	4	2	no	—
80	C₂×Dic₅₃	—	4	2	no	—
81	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
82	C₈×C₅₃	C₄×C₈	1	4	yes	—
83	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
84	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
85	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
87	#88322	—	2	2	no	—
88	#88322	C₄⋊C₈	1	4	yes	—
89	#12656	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	2	no	—
91	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
92	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₅₃	#431	1	1	yes	characteristic
94	C₄×C₅₃	—	4	2	no	—
95	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₅₃	#431	1	1	yes	characteristic
99	Dic₅₃	—	2	4	no	—
100	Dic₅₃	—	4	2	no	—
101	C₂×#364	—	53	1	no	—
102	C₂×#364	D₅₃	1	1	yes	characteristic
103	C₄×M₃₂	—	53	2	no	—
104	#1726	—	53	1	no	—
105	#1376	—	53	2	no	—
106	C₂×C₅₃	—	2	1	no	—
107	C₂×C₅₃	—	2	1	no	—
108	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
109	C₂×C₅₃	#2153	1	2	yes	—
110	C₂×M₃₂	—	53	2	no	—
111	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
112	C₂×#97	—	53	1	no	—
113	C₂×#97	—	53	1	no	—
114	C₂×#97	—	53	1	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
118	#376	—	53	1	no	—
119	#364	—	53	4	no	—
120	#364	Dic₅₃	1	4	yes	complemented
121	#346	—	106	2	no	—
122	C₄×C₁₆	—	106	2	no	—
123	C₅₃	#12656	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	—	53	1	no	—
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₄²	—	53	1	no	—
129	C₂×C₄²	—	53	1	no	—
130	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
131	M₃₂	—	2	2	no	—
132	M₃₂	—	53	4	no	—
133	M₃₂	—	106	2	no	—
134	M₃₂	C₂×Dic₅₃	1	4	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	106	2	no	—
137	#97	—	53	4	no	—
138	#97	—	53	4	no	—
139	#97	—	106	2	no	—
140	C₄×C₈	—	53	2	no	—
141	C₄×C₈	—	53	2	no	—
142	C₄×C₈	—	106	1	no	—
143	C₄×C₈	—	106	1	no	—
144	C₄×C₈	C₄×D₅₃	1	2	yes	—
145	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
146	#95	—	106	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	53	2	no	—
149	C₂²×C₄	—	106	1	no	—
150	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	106	1	no	—
156	C₂×C₈	—	106	1	no	—
157	C₂×C₈	C₈×D₅₃	1	2	yes	—
158	C₂×C₈	C₈×D₅₃	1	2	yes	—
159	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
161	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
162	C₂×C₈	#11719947	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
164	C₂×C₈	#11719949	1	1	yes	characteristic
165	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
166	C₄²	—	2	1	no	—
167	C₄²	—	53	2	no	—
168	C₄²	—	53	2	no	—
169	C₄²	—	53	2	no	—
170	C₄²	—	106	1	no	—
171	C₄²	C₄×Dic₅₃	1	2	yes	—
172	C₁₆	—	2	4	no	—
173	C₁₆	—	106	4	no	—
174	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	53	4	no	—
178	C₂×C₄	—	106	2	no	—
179	C₂×C₄	—	212	2	no	—
180	C₂×C₄	—	212	2	no	—
181	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
182	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
183	C₂×C₄	#13133175	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
186	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
187	C₈	—	53	4	no	—
188	C₈	—	106	2	no	—
189	C₈	C₈×Dic₅₃	1	4	yes	—
190	C₈	#13133179	1	1	yes	characteristic
191	C₈	D₄×Dic₅₃	1	1	yes	characteristic
192	C₈	#13133182	1	1	yes	characteristic
193	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	M₁₆×Dic₅₃	1	1	yes	characteristic
196	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
197	C₂²	#16810675	1	1	yes	characteristic
198	C₄	—	4	2	no	—
199	C₄	—	106	4	no	—
200	C₄	—	212	2	no	—
201	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
202	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
203	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
204	C₄	#16810675	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	C₄⋊C₈×Dic₅₃	1	1	yes	characteristic
208	C₂	#20276479	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#12656
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
13. #16810444	C₄	semidirect
18. C₅₃×#364	C₄	semidirect
120. #364	Dic₅₃	semidirect
123. C₅₃	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
M₃₂×Dic₅₃	2
#20274401	2
#20272796	2
C₂×C₅₃×#364	2
#12656	53