Subgroups — #27275331

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133149	C₂	1	1	yes	maximal, characteristic
2	#20274582	C₂	1	2	yes	maximal
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	#20272555	C₂	1	2	yes	maximal
5	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	C₄	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133127	—	2	2	no	—
9	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#16810386	C₄	1	2	yes	—
11	#16810377	—	2	2	no	—
12	#16810374	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₈²×C₅₃	C₄	1	4	yes	complemented
17	C₂²×#88322	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₅₃	—	2	2	no	—
20	#13133149	—	2	4	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	C₂×C₄	1	4	yes	—
23	#13133127	—	4	4	no	—
24	#13133124	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	2	4	no	—
41	C₂×C₈×C₅₃	—	2	4	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	C₄²	1	2	yes	—
46	C₂×C₈×C₅₃	C₄²	1	2	yes	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
50	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	1	yes	characteristic
55	C₄⋊Dic₅₃	—	2	4	no	—
56	C₄⋊Dic₅₃	—	2	4	no	—
57	C₄×Dic₅₃	—	4	2	no	—
58	C₄×Dic₅₃	—	4	2	no	—
59	C₂×#88322	—	2	4	no	—
60	C₂×#88322	—	4	2	no	—
61	C₂×#88322	—	4	2	no	—
62	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₅₃	#131	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₅₃	#135	1	2	yes	—
74	C₂×Dic₅₃	—	4	4	no	—
75	C₂×Dic₅₃	—	4	4	no	—
76	C₂×Dic₅₃	—	4	4	no	—
77	C₈×C₅₃	—	2	4	no	—
78	C₈×C₅₃	—	2	2	no	—
79	C₈×C₅₃	—	2	2	no	—
80	C₈×C₅₃	#97	1	4	yes	complemented
81	C₈×C₅₃	#106	1	4	yes	complemented
82	#88322	—	4	4	no	—
83	#8283	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₅₃	#436	1	2	yes	—
86	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
87	C₂²×C₅₃	#443	1	2	yes	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
91	C₄×C₅₃	#405	1	1	yes	characteristic
92	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
93	C₄×C₅₃	#427	1	1	yes	characteristic
94	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
95	C₄×C₅₃	#446	1	2	yes	—
96	Dic₅₃	—	8	4	no	—
97	C₄×#106	—	53	1	no	—
98	#1770	—	53	2	no	—
99	C₂×#334	—	53	1	no	—
100	C₂×C₈²	D₅₃	1	1	yes	characteristic
101	#1279	—	53	2	no	—
102	C₂×C₅₃	#1590	1	4	yes	—
103	C₂×C₅₃	#1770	1	1	yes	characteristic
104	C₂×C₅₃	#1772	1	1	yes	characteristic
105	C₂×C₅₃	C₄×#106	1	1	yes	characteristic
106	C₂×#106	—	53	2	no	—
107	C₂×C₄⋊C₈	—	53	2	no	—
108	C₂×#97	—	106	2	no	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
112	C₄×C₄⋊C₄	—	53	2	no	—
113	#340	—	53	2	no	—
114	#336	—	106	2	no	—
115	#334	—	53	4	no	—
116	C₈²	Dic₅₃	1	4	yes	complemented
117	C₅₃	#8283	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	106	2	no	—
120	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
122	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
123	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
124	C₂×C₄⋊C₄	—	53	4	no	—
125	C₂×C₄²	—	106	2	no	—
126	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
127	#106	—	106	4	no	—
128	C₄⋊C₈	—	53	4	no	—
129	C₄⋊C₈	—	106	2	no	—
130	#97	—	212	4	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	Dic₁₀₆	1	2	yes	—
133	C₄×C₈	C₄×D₅₃	1	2	yes	—
134	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
135	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
136	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
137	#95	—	106	2	no	—
138	C₂²×C₄	—	106	4	no	—
139	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
141	C₂²×C₄	#11719945	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	106	4	no	—
147	C₂×C₈	—	212	2	no	—
148	C₂×C₈	—	212	2	no	—
149	C₂×C₈	#11719912	1	4	yes	—
150	C₂×C₈	#11719913	1	4	yes	—
151	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
152	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
153	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
154	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
155	C₄⋊C₄	—	106	4	no	—
156	C₄⋊C₄	—	106	4	no	—
157	C₄²	—	212	2	no	—
158	C₄²	—	212	2	no	—
159	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
161	C₄²	#11719945	1	1	yes	characteristic
162	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
163	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	212	4	no	—
166	C₂×C₄	—	212	4	no	—
167	C₂×C₄	—	212	4	no	—
168	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
170	C₂×C₄	#13133176	1	1	yes	characteristic
171	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
172	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
173	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
174	C₂×C₄	C₂×#11719912	1	2	yes	—
175	C₂×C₄	#13133223	1	2	yes	—
176	C₂×C₄	C₂×#11719913	1	2	yes	—
177	C₂×C₄	#13133239	1	2	yes	—
178	C₈	—	2	4	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	—	212	4	no	—
182	C₈	#13133127	1	4	yes	complemented
183	C₈	#13133149	1	4	yes	complemented
184	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
185	C₂²	#16810702	1	2	yes	—
186	C₂²	C₄×#11719913	1	2	yes	—
187	C₂²	#16810741	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	424	4	no	—
191	C₄	C₂×#13133127	1	1	yes	characteristic
192	C₄	#16810578	1	1	yes	characteristic
193	C₄	C₂×#13133149	1	1	yes	characteristic
194	C₄	#16810661	1	1	yes	characteristic
195	C₄	C₄×#11719913	1	2	yes	—
196	C₄	#16810753	1	2	yes	—
197	C₂	#20273607	1	4	yes	—
198	C₂	#20274582	1	1	yes	characteristic
199	C₂	#20274594	1	1	yes	characteristic
200	C₂	C₄×#13133149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#8283
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
12. #16810374	C₄	semidirect
16. C₈²×C₅₃	C₄	semidirect
80. C₈×C₅₃	#97	semidirect
81. C₈×C₅₃	#106	semidirect
116. C₈²	Dic₅₃	semidirect
117. C₅₃	#8283	semidirect
182. C₈	#13133127	semidirect
183. C₈	#13133149	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133149	2
#20274582	2
C₂×#16810374	2
#20272555	2
C₂×C₈²×C₅₃	2
#8283	53