Subgroups — #27299212

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276336	C₂	1	1	yes	maximal, characteristic
2	#20276319	C₂	1	2	yes	maximal, complemented
3	#20274515	C₂	1	1	yes	maximal, characteristic
4	#20272803	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	#16810607	C₄	1	1	yes	characteristic
8	#16810597	C₄	1	1	yes	characteristic
9	#16810578	C₂²	1	2	yes	—
10	#16810405	C₄	1	4	yes	complemented
11	#16810400	—	2	4	no	—
12	#16810394	C₄	1	2	yes	—
13	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#504	—	2	4	no	—
15	C₂×M₃₂×C₅₃	C₄	1	2	yes	complemented
16	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₃	C₄	1	2	yes	complemented
18	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#11719917	C₂×C₄	1	2	yes	—
20	#13133176	C₂×C₄	1	2	yes	—
21	#13133157	—	2	2	no	—
22	#13133157	C₂×C₄	1	4	yes	—
23	C₂×#11719908	—	2	2	no	—
24	#13133136	—	2	1	no	—
25	#13133134	—	2	1	no	—
26	#13133132	—	2	2	no	—
27	#13133127	—	2	2	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
30	#131×C₅₃	—	2	2	no	—
31	#131×C₅₃	—	2	2	no	—
32	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	M₃₂×C₅₃	—	2	4	no	—
37	M₃₂×C₅₃	D₄	1	4	yes	—
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₂×C₁₆×C₅₃	—	2	4	no	—
40	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	#41×C₅₃	—	2	2	no	—
44	#41×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
46	C₂×D₄×C₅₃	M₁₆	1	2	yes	—
47	C₂×D₄×C₅₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₅₃	—	2	2	no	—
51	M₁₆×C₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₄²	1	2	yes	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₃	#41	1	2	yes	—
59	C₂²⋊Dic₅₃	—	2	2	no	—
60	C₄⋊Dic₅₃	—	2	2	no	—
61	C₄×Dic₅₃	—	2	2	no	—
62	#11719917	—	2	2	no	—
63	#11719917	—	2	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₁₆×C₅₃	—	2	1	no	—
66	C₁₆×C₅₃	—	2	1	no	—
67	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
68	#11719908	—	2	4	no	—
69	C₂³×C₅₃	#97	1	2	yes	—
70	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₅₃	—	2	2	no	—
72	D₄×C₅₃	—	2	4	no	—
73	D₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	#97	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#131	1	2	yes	—
81	C₂×Dic₅₃	—	4	2	no	—
82	C₈×C₅₃	—	2	2	no	—
83	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₅₃	#117	1	1	yes	characteristic
85	C₈×C₅₃	C₄×D₄	1	2	yes	—
86	#88322	—	4	2	no	—
87	#12617	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	4	1	no	—
90	C₂²×C₅₃	—	4	2	no	—
91	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
92	C₂²×C₅₃	#435	1	2	yes	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₅₃	#405	1	1	yes	characteristic
96	C₄×C₅₃	#435	1	2	yes	—
97	Dic₅₃	—	8	2	no	—
98	C₂×#504	D₅₃	1	1	yes	characteristic
99	#2115	—	53	1	no	—
100	#2110	—	53	2	no	—
101	#1757	—	53	1	no	—
102	#1379	—	53	2	no	—
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	—	2	1	no	—
105	C₂×C₅₃	—	4	2	no	—
106	C₂×C₅₃	#1752	1	1	yes	characteristic
107	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
108	#504	—	2	4	no	—
109	C₂×M₃₂	—	53	2	no	—
110	C₂×M₃₂	Dic₅₃	1	2	yes	complemented
111	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
112	C₂²×C₁₆	Dic₅₃	1	2	yes	complemented
113	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
114	#413	—	53	1	no	—
115	#410	—	53	1	no	—
116	#405	—	53	2	no	—
117	#349	—	53	4	no	—
118	#347	—	106	4	no	—
119	#344	—	53	2	no	—
120	C₅₃	#12617	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
122	#131	—	2	2	no	—
123	#131	—	2	2	no	—
124	C₂×M₁₆	—	53	2	no	—
125	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
126	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
127	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
128	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
129	#117	—	53	2	no	—
130	M₃₂	—	2	4	no	—
131	M₃₂	—	53	4	no	—
132	M₃₂	—	106	2	no	—
133	M₃₂	C₂²⋊D₅₃	1	4	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	4	no	—
136	C₂×C₁₆	—	106	2	no	—
137	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
138	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
140	#101	—	106	1	no	—
141	#100	—	106	1	no	—
142	#99	—	106	2	no	—
143	#97	—	106	2	no	—
144	C₄×C₈	—	106	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	#11719912	1	1	yes	characteristic
148	C₂×D₄	#11719912	1	2	yes	—
149	C₂×D₄	#11719912	1	1	yes	characteristic
150	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	106	2	no	—
155	M₁₆	—	106	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	106	2	no	—
159	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
161	C₂×C₈	#11719945	1	2	yes	—
162	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
163	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
164	C₄⋊C₄	—	106	2	no	—
165	C₂²⋊C₄	—	106	2	no	—
166	C₄²	—	106	2	no	—
167	C₁₆	—	2	1	no	—
168	C₁₆	—	2	1	no	—
169	C₁₆	—	106	4	no	—
170	C₁₆	C₂²⋊Dic₅₃	1	4	yes	—
171	C₂³	#13133127	1	2	yes	—
172	C₂³	C₂×#11719912	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	212	2	no	—
179	C₂×C₄	#13133127	1	2	yes	—
180	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
182	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
183	C₂×C₄	#13133223	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	212	2	no	—
186	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
187	C₈	#13133176	1	1	yes	characteristic
188	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#13133127	1	1	yes	characteristic
193	C₂²	#16810694	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	424	2	no	—
196	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
197	C₄	#16810578	1	1	yes	characteristic
198	C₄	#16810694	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	#20274491	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133176
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂×C₅₃	#1752
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#12617
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #20276319	C₂	semidirect
4. #20272803	C₂	semidirect
10. #16810405	C₄	semidirect
15. C₂×M₃₂×C₅₃	C₄	semidirect
17. C₂²×C₁₆×C₅₃	C₄	semidirect
110. C₂×M₃₂	Dic₅₃	semidirect
112. C₂²×C₁₆	Dic₅₃	semidirect
120. C₅₃	#12617	semidirect

Maximal subgroups

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

subgroup	index
#20276336	2
#20276319	2
#20274515	2
#20272803	2
C₂×C₅₃×#504	2
#12617	53