Subgroups — #28328552

Order 13568.

460 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291132	C₂	1	4	yes	maximal, complemented
2	#20291129	C₂	1	4	yes	maximal, complemented
3	#20291081	C₂	1	2	yes	maximal
4	#20291080	C₂	1	2	yes	maximal
5	#20291073	C₂	1	2	yes	maximal
6	#20283648	C₂	1	1	yes	maximal, characteristic
7	#16811544	C₂²	1	2	yes	—
8	#16811544	C₂²	1	2	yes	—
9	#16811544	C₂²	1	4	yes	—
10	C₄⋊#11719910	C₂²	1	4	yes	—
11	C₂²⋊#11719910	C₂²	1	4	yes	—
12	#16811512	C₂²	1	2	yes	—
13	#16811511	C₂²	1	2	yes	—
14	#16811509	C₂²	1	4	yes	—
15	#16811509	C₂²	1	2	yes	—
16	C₄×#11719910	C₂²	1	2	yes	—
17	#16811176	C₄	1	1	yes	characteristic
18	#16810752	C₄	1	4	yes	—
19	#16810747	C₂²	1	4	yes	—
20	#16810649	C₄	1	1	yes	characteristic
21	#16810646	C₄	1	1	yes	characteristic
22	#16810581	C₂²	1	1	yes	characteristic
23	#16810579	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#424	C₄	1	1	yes	characteristic
26	C₂×#11719910	C₂³	1	4	yes	—
27	C₂×#11719910	C₂³	1	4	yes	—
28	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂³	1	1	yes	characteristic
30	#13133216	C₈	1	2	yes	—
31	#13133208	C₈	1	2	yes	—
32	#13133194	C₈	1	4	yes	—
33	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
34	#13133176	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	#13133146	C₂×C₄	1	4	yes	—
37	#13133146	C₂×C₄	1	4	yes	—
38	#13133128	C₂×C₄	1	2	yes	—
39	#13133127	C₂×C₄	1	2	yes	—
40	C₄×#88322	C₂³	1	2	yes	—
41	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
42	C₈×Dic₅₃	C₂³	1	2	yes	—
43	C₈×Dic₅₃	C₂³	1	2	yes	—
44	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
50	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
51	M₁₆×C₅₃	—	4	1	no	—
52	C₂×C₈×C₅₃	—	4	1	no	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
55	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
56	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
57	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
59	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
60	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
61	#11719919	C₂×C₈	1	4	yes	—
62	#11719919	C₂×C₈	1	4	yes	—
63	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
64	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
66	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
67	#11719917	—	4	1	no	—
68	C₂×#88322	—	2	2	no	—
69	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
70	C₂×#88322	C₂²×C₄	1	2	yes	—
71	C₂×#88322	C₂⁴	1	1	yes	characteristic
72	#11719910	—	2	4	no	—
73	#11719910	—	2	4	no	—
74	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
78	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
80	C₂×Dic₅₃	—	2	4	no	—
81	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
82	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
83	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
84	Dic₁₀₆	—	2	8	no	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	—	2	2	no	—
87	#88322	—	2	1	no	—
88	#88322	—	2	2	no	—
89	#88322	2₋¹⁺⁴	1	2	yes	—
90	#33500	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	2	no	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	#519	1	1	yes	characteristic
97	C₄×C₅₃	#568	1	1	yes	characteristic
98	Dic₅₃	—	2	4	no	—
99	Dic₅₃	—	2	2	no	—
100	Dic₅₃	#504	1	4	yes	—
101	#2960	—	53	1	no	—
102	#2183	—	53	4	no	—
103	Q₈×C₁₆	—	53	4	no	—
104	#2164	—	53	2	no	—
105	#2164	—	53	2	no	—
106	#2162	—	53	2	no	—
107	C₂×C₅₃	—	4	1	no	—
108	C₂×C₅₃	#2982	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#504	1	2	yes	—
110	#533	—	53	1	no	—
111	#446	—	53	4	no	—
112	C₈×Q₈	—	53	4	no	—
113	#424	—	53	1	no	—
114	#424	#6157	1	1	yes	characteristic
115	#423	—	53	1	no	—
116	#405	—	53	1	no	—
117	#405	—	53	1	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₄⋊C₁₆	—	53	4	no	—
120	C₄⋊C₁₆	—	53	2	no	—
121	C₄⋊C₁₆	—	53	2	no	—
122	C₄⋊C₁₆	—	53	4	no	—
123	C₂²⋊C₁₆	—	53	4	no	—
124	#346	—	53	2	no	—
125	#346	—	53	2	no	—
126	C₄×C₁₆	—	53	4	no	—
127	C₄×C₁₆	—	53	2	no	—
128	C₄×C₁₆	—	53	2	no	—
129	C₅₃	#33500	1	1	yes	characteristic, complemented, Sylow
130	C₂×M₁₆	—	53	1	no	—
131	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
134	C₄⋊Q₈	—	53	2	no	—
135	#125	—	53	2	no	—
136	C₂²⋊Q₈	—	53	4	no	—
137	C₄×Q₈	—	53	4	no	—
138	#117	—	53	1	no	—
139	#117	C₂×#6157	1	1	yes	characteristic
140	C₂×C₄²	—	53	1	no	—
141	C₂×C₁₆	—	53	4	no	—
142	C₂×C₁₆	—	53	4	no	—
143	C₄⋊C₈	—	53	4	no	—
144	C₄⋊C₈	—	53	4	no	—
145	C₄⋊C₈	C₂×#6157	1	4	yes	—
146	#97	—	53	2	no	—
147	#97	—	53	2	no	—
148	C₄×C₈	—	53	2	no	—
149	C₄×C₈	—	53	2	no	—
150	C₄×C₈	—	53	2	no	—
151	C₄×C₈	—	53	2	no	—
152	C₂×Q₈	—	53	4	no	—
153	C₂²×C₄	—	53	2	no	—
154	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
155	M₁₆	—	4	1	no	—
156	M₁₆	—	212	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	53	1	no	—
159	C₂×C₈	—	53	1	no	—
160	C₂×C₈	—	53	2	no	—
161	C₂×C₈	—	106	2	no	—
162	C₂×C₈	C₂²×#6157	1	2	yes	—
163	C₂×C₈	C₂²×#6157	1	2	yes	—
164	C₄⋊C₄	—	53	2	no	—
165	C₄⋊C₄	—	53	4	no	—
166	C₄⋊C₄	—	53	4	no	—
167	C₄⋊C₄	—	53	4	no	—
168	C₄⋊C₄	D₅₃⋊C₈	1	2	yes	—
169	C₂²⋊C₄	—	53	2	no	—
170	C₂²⋊C₄	D₅₃⋊C₈	1	2	yes	—
171	C₄²	—	53	1	no	—
172	C₄²	—	53	1	no	—
173	C₄²	—	53	2	no	—
174	C₄²	—	53	2	no	—
175	C₄²	C₂²×#6157	1	2	yes	—
176	C₁₆	—	106	4	no	—
177	C₁₆	—	106	4	no	—
178	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
179	Q₈	—	106	8	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	53	2	no	—
183	C₂×C₄	—	53	2	no	—
184	C₂×C₄	—	53	4	no	—
185	C₂×C₄	—	106	4	no	—
186	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
187	C₂×C₄	C₂×D₅₃⋊C₈	1	4	yes	—
188	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	2	2	no	—
191	C₈	—	53	2	no	—
192	C₈	—	106	1	no	—
193	C₈	—	106	2	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	2	4	no	—
199	C₄	—	53	4	no	—
200	C₄	—	106	2	no	—
201	C₄	—	106	4	no	—
202	C₄	#16811691	1	1	yes	characteristic
203	C₄	#16811701	1	1	yes	characteristic
204	C₂	—	4	1	no	—
205	C₂	#20291610	1	1	yes	characteristic
206	C₂	D₅₃⋊#504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#424	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#424	#6157
hypocenter	C₅₃	#33500
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
1. #20291132	C₂	semidirect
2. #20291129	C₂	semidirect
129. C₅₃	#33500	semidirect

Maximal subgroups

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

subgroup	index
#20291132	2
#20291129	2
#20291081	2
#20291080	2
#20291073	2
#20283648	2
#33500	53