Subgroups — #27302721

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811025	C₂	1	1	yes	maximal, characteristic
2	#20274887	C₂	1	1	yes	maximal, characteristic
3	#20272824	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133209	—	2	1	no	—
6	#16811025	—	2	1	no	—
7	#16811025	—	2	1	no	—
8	#16811025	C₄	1	2	yes	—
9	#16811025	C₄	1	2	yes	—
10	C₂×#13133153	—	2	1	no	—
11	C₂×#13133153	C₂²	1	2	yes	—
12	C₂×#13133151	C₂²	1	2	yes	—
13	C₂×#13133149	—	2	1	no	—
14	#16810647	—	2	1	no	—
15	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#346	—	2	1	no	—
18	C₄×C₁₆×C₅₃	—	2	1	no	—
19	C₂×#11719917	—	2	2	no	—
20	#13133209	—	2	2	no	—
21	#13133209	—	2	2	no	—
22	#13133209	—	4	2	no	—
23	C₂×C₄⋊Dic₅₃	—	2	2	no	—
24	C₂×#11719919	—	4	2	no	—
25	#13133153	—	2	4	no	—
26	#13133153	—	4	2	no	—
27	#13133151	—	2	2	no	—
28	#13133151	C₂×C₄	1	4	yes	—
29	#13133149	—	2	2	no	—
30	#13133149	—	2	2	no	—
31	C₄⋊#88322	—	4	2	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	M₃₂×C₅₃	—	2	2	no	—
36	M₃₂×C₅₃	Q₈	1	4	yes	—
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
48	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₅₃	#41	1	2	yes	—
53	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₄²×C₅₃	D₈	1	2	yes	—
56	C₄⋊Dic₅₃	—	2	4	no	—
57	C₄⋊Dic₅₃	—	4	2	no	—
58	#11719919	—	4	4	no	—
59	#11719919	—	4	4	no	—
60	#11719917	—	4	2	no	—
61	#11719917	—	4	2	no	—
62	C₂×#88322	—	4	2	no	—
63	C₁₆×C₅₃	—	2	4	no	—
64	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	4	2	no	—
66	C₂×C₄×C₅₃	#102	1	2	yes	—
67	C₂×C₄×C₅₃	#102	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×Dic₅₃	—	4	4	no	—
75	C₂×Dic₅₃	—	8	4	no	—
76	C₈×C₅₃	#103	1	2	yes	—
77	C₈×C₅₃	#103	1	2	yes	—
78	C₈×C₅₃	C₄×Q₈	1	2	yes	—
79	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
80	#88322	—	8	2	no	—
81	#13289	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	—	2	2	no	—
83	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
84	C₂²×C₅₃	#487	1	1	yes	characteristic
85	C₂²×C₅₃	#498	1	1	yes	characteristic
86	C₄×C₅₃	—	2	2	no	—
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	#391	1	1	yes	characteristic
89	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
90	C₄×C₅₃	#486	1	1	yes	characteristic
91	C₄×C₅₃	#500	1	1	yes	characteristic
92	Dic₅₃	—	8	4	no	—
93	C₂×#499	—	53	1	no	—
94	C₄×M₃₂	D₅₃	1	1	yes	characteristic
95	#1848	—	53	1	no	—
96	#1387	—	53	4	no	—
97	C₂×C₅₃	—	2	1	no	—
98	C₂×C₅₃	—	2	1	no	—
99	C₂×C₅₃	#1990	1	1	yes	characteristic
100	C₂×C₅₃	#2148	1	2	yes	—
101	C₂×#125	—	106	1	no	—
102	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
103	#499	—	53	2	no	—
104	#499	—	53	2	no	—
105	#499	—	106	1	no	—
106	#499	—	106	1	no	—
107	C₂×#108	—	53	2	no	—
108	C₂×#108	—	106	1	no	—
109	C₂×#107	—	53	2	no	—
110	C₂×#106	—	106	1	no	—
111	#423	—	106	1	no	—
112	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
113	#346	—	2	1	no	—
114	C₄×C₁₆	—	2	1	no	—
115	C₅₃	#13289	1	1	yes	characteristic, complemented, Sylow
116	C₂×M₁₆	—	106	2	no	—
117	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	#125	—	106	2	no	—
120	#125	—	106	2	no	—
121	#125	—	212	2	no	—
122	C₂×C₄⋊C₄	—	106	2	no	—
123	C₂×C₄⋊C₄	—	212	2	no	—
124	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	Dic₁₀₆	1	4	yes	—
127	C₂×C₁₆	—	2	2	no	—
128	#108	—	106	4	no	—
129	#108	—	212	2	no	—
130	#107	—	53	4	no	—
131	#107	—	106	2	no	—
132	#106	—	106	2	no	—
133	#106	—	106	2	no	—
134	C₄⋊C₈	—	212	2	no	—
135	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
136	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂²×C₄	—	212	2	no	—
140	C₂²×C₄	#11719913	1	2	yes	—
141	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
142	M₁₆	—	212	2	no	—
143	M₁₆	—	212	2	no	—
144	C₂×C₈	—	212	2	no	—
145	C₂×C₈	#11719913	1	2	yes	—
146	C₂×C₈	Dic₂₁₂	1	2	yes	—
147	C₂×C₈	#11719921	1	1	yes	characteristic
148	C₂×C₈	#11719921	1	1	yes	characteristic
149	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
151	C₂×C₈	#11719945	1	2	yes	—
152	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₄⋊C₄	—	106	4	no	—
154	C₄⋊C₄	—	212	4	no	—
155	C₄⋊C₄	—	212	4	no	—
156	C₄⋊C₄	—	212	2	no	—
157	C₄²	D₄₂₄	1	2	yes	—
158	C₄²	#11719921	1	1	yes	characteristic
159	C₄²	#11719921	1	1	yes	characteristic
160	C₁₆	—	2	4	no	—
161	C₂³	C₂×#11719913	1	1	yes	characteristic
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	424	4	no	—
164	C₂×C₄	—	212	4	no	—
165	C₂×C₄	#13133139	1	2	yes	—
166	C₂×C₄	#13133139	1	2	yes	—
167	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
168	C₂×C₄	#13133205	1	1	yes	characteristic
169	C₂×C₄	#13133217	1	1	yes	characteristic
170	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
173	C₈	—	424	2	no	—
174	C₈	#13133142	1	2	yes	—
175	C₈	#13133142	1	2	yes	—
176	C₈	C₄×Dic₁₀₆	1	2	yes	—
177	C₈	#13133196	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	C₂×#13133139	1	1	yes	characteristic
180	C₂²	#16810974	1	1	yes	characteristic
181	C₂²	#16811021	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	424	4	no	—
185	C₄	#16810529	1	1	yes	characteristic
186	C₄	C₂×#13133142	1	1	yes	characteristic
187	C₄	#16810970	1	1	yes	characteristic
188	C₄	#16811028	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#20275676	1	1	yes	characteristic
192	C₂	#20276465	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133139
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#13289
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
115. C₅₃	#13289	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811025	2
#20274887	2
#20272824	2
C₄×M₃₂×C₅₃	2
#13289	53