Subgroups — #28310428

Order 13568.

358 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×#16811512	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811509	C₂	1	1	yes	maximal, characteristic
3	#20289754	C₂	1	2	yes	maximal
4	#20289753	C₂	1	2	yes	maximal
5	#20274403	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719910	C₂²	1	2	yes	—
7	C₂²×#11719910	C₂²	1	2	yes	—
8	#16811512	C₄	1	4	yes	complemented
9	#16811509	C₄	1	4	yes	complemented
10	C₂×#13133127	C₄	1	1	yes	characteristic
11	C₂×#13133127	C₄	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	#16810469	C₄	1	1	yes	characteristic
16	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#11719910	—	2	2	no	—
18	C₂×#11719910	—	2	2	no	—
19	C₂×#11719910	—	2	2	no	—
20	C₂×#11719910	—	2	2	no	—
21	C₂×#11719910	C₂×C₄	1	4	yes	—
22	C₂×#11719910	C₂×C₄	1	4	yes	—
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂³	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	#13133127	—	2	2	no	—
28	#13133127	C₈	1	4	yes	—
29	C₄×#88322	—	2	2	no	—
30	C₈×Dic₅₃	—	2	2	no	—
31	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
32	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
33	#13133124	C₈	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	#97×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
41	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	C₄²	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
47	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
48	C₄²×C₅₃	—	2	1	no	—
49	C₄²×C₅₃	—	2	1	no	—
50	C₄×Dic₅₃	—	2	1	no	—
51	C₄×Dic₅₃	—	2	1	no	—
52	C₄×Dic₅₃	C₄²	1	2	yes	—
53	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
54	C₂×#88322	—	2	1	no	—
55	C₂×#88322	—	2	1	no	—
56	C₂×#88322	C₂×C₈	1	2	yes	—
57	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
58	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
59	C₂×#88322	#41	1	2	yes	—
60	C₂×#88322	#41	1	2	yes	—
61	#11719910	—	2	4	no	—
62	#11719910	—	2	4	no	—
63	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×Dic₅₃	—	2	4	no	—
73	C₂×Dic₅₃	—	2	2	no	—
74	C₂×Dic₅₃	—	2	4	no	—
75	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
76	C₈×C₅₃	—	2	2	no	—
77	C₈×C₅₃	#97	1	4	yes	—
78	#88322	—	2	2	no	—
79	#88322	#117	1	2	yes	—
80	#88322	#117	1	2	yes	—
81	#12592	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₅₃	#431	1	1	yes	characteristic
84	C₂²×C₅₃	#431	1	1	yes	characteristic
85	C₂²×C₅₃	#504	1	4	yes	—
86	C₄×C₅₃	—	4	2	no	—
87	C₄×C₅₃	#376	1	1	yes	characteristic
88	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
89	C₄×C₅₃	#431	1	1	yes	characteristic
90	C₄×C₅₃	#431	1	1	yes	characteristic
91	Dic₅₃	—	2	4	no	—
92	Dic₅₃	—	4	2	no	—
93	C₂×#346	—	53	1	no	—
94	C₂×C₄×C₁₆	—	53	1	no	—
95	#1726	—	53	1	no	—
96	#1374	—	53	2	no	—
97	#1374	—	53	2	no	—
98	C₂×C₅₃	#1744	1	1	yes	characteristic
99	C₂×C₅₃	#2108	1	2	yes	—
100	C₂×C₅₃	#2164	1	2	yes	—
101	C₂×C₅₃	#2164	1	2	yes	—
102	C₂²×C₁₆	—	53	2	no	—
103	C₂²×C₁₆	—	53	2	no	—
104	C₂×#97	—	53	1	no	—
105	C₂×#97	—	53	1	no	—
106	C₂×#97	#6157	1	1	yes	characteristic
107	C₂×C₄×C₈	—	53	1	no	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	#376	—	53	1	no	—
111	#346	—	53	4	no	—
112	C₄×C₁₆	—	53	4	no	—
113	C₅₃	#12592	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	—	53	1	no	—
115	C₂²×C₈	—	53	1	no	—
116	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
117	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
118	C₂×C₄²	—	53	1	no	—
119	C₂×C₄²	—	53	1	no	—
120	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
121	C₂×C₁₆	—	53	4	no	—
122	C₂×C₁₆	—	53	4	no	—
123	C₂×C₁₆	—	106	2	no	—
124	C₂×C₁₆	—	106	2	no	—
125	C₂×C₁₆	—	106	2	no	—
126	C₂×C₁₆	—	106	2	no	—
127	#97	—	2	2	no	—
128	#97	—	53	4	no	—
129	#97	—	106	2	no	—
130	C₄×C₈	—	53	2	no	—
131	C₄×C₈	—	53	2	no	—
132	C₄×C₈	—	106	2	no	—
133	C₄×C₈	—	106	2	no	—
134	#95	—	53	4	no	—
135	C₂²×C₄	—	53	2	no	—
136	C₂²×C₄	—	53	2	no	—
137	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
138	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	53	1	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	2	no	—
144	C₂×C₈	—	53	2	no	—
145	C₂×C₈	—	53	2	no	—
146	C₂×C₈	—	106	1	no	—
147	C₂×C₈	—	106	1	no	—
148	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
149	C₂×C₈	#11719936	1	2	yes	—
150	C₂×C₈	C₄×#6157	1	2	yes	—
151	C₂×C₈	#11719940	1	2	yes	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	53	2	no	—
155	C₄²	—	53	2	no	—
156	C₄²	—	106	1	no	—
157	C₄²	—	106	1	no	—
158	C₁₆	—	106	4	no	—
159	C₁₆	—	106	4	no	—
160	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	53	4	no	—
164	C₂×C₄	—	106	4	no	—
165	C₂×C₄	—	106	2	no	—
166	C₂×C₄	—	106	4	no	—
167	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
168	C₂×C₄	#13133296	1	1	yes	characteristic
169	C₂×C₄	#13133296	1	1	yes	characteristic
170	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
172	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	53	2	no	—
175	C₈	—	53	2	no	—
176	C₈	—	106	2	no	—
177	C₈	#13133272	1	4	yes	—
178	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
179	C₂²	#16811640	1	1	yes	characteristic
180	C₂²	#16811642	1	1	yes	characteristic
181	C₂²	#16811682	1	4	yes	—
182	C₄	—	4	2	no	—
183	C₄	—	106	4	no	—
184	C₄	—	212	2	no	—
185	C₄	#16811559	1	1	yes	characteristic
186	C₄	C₂×#13133272	1	1	yes	characteristic
187	C₄	#16811640	1	1	yes	characteristic
188	C₄	#16811642	1	1	yes	characteristic
189	C₂	#20290430	1	1	yes	characteristic
190	C₂	#20290925	1	2	yes	—
191	C₂	#20291081	1	2	yes	—
192	C₂	#20291083	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×#97×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	#12592
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
8. #16811512	C₄	semidirect
9. #16811509	C₄	semidirect
113. C₅₃	#12592	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811512	2
C₂×#16811509	2
#20289754	2
#20289753	2
#20274403	2
#12592	53