Subgroups — #27295359

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272768	C₂	1	4	yes	maximal, complemented
4	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
5	#16811180	—	2	1	no	—
6	C₂²×#11719908	—	2	1	no	—
7	#16811028	C₄	1	1	yes	characteristic
8	#16811028	C₄	1	1	yes	characteristic
9	#16811025	—	2	1	no	—
10	#16811025	C₂²	1	2	yes	—
11	#16811021	—	2	1	no	—
12	#16810661	—	2	1	no	—
13	#16810661	C₄	1	2	yes	—
14	C₄⋊#11719908	—	2	1	no	—
15	C₄⋊#11719908	C₂²	1	2	yes	—
16	C₅₃×#431	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#335	—	2	1	no	—
19	C₅₃×#334	C₂²	1	2	yes	—
20	#13133217	—	2	1	no	—
21	#13133217	—	2	1	no	—
22	#13133209	—	2	2	no	—
23	#13133196	—	4	1	no	—
24	#13133196	—	4	1	no	—
25	C₄×Dic₁₀₆	—	4	1	no	—
26	C₄×Dic₁₀₆	—	4	1	no	—
27	#13133176	—	2	2	no	—
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	—	4	2	no	—
30	#13133151	—	2	1	no	—
31	#13133151	—	2	1	no	—
32	#13133151	C₂×C₄	1	2	yes	—
33	#13133151	C₂×C₄	1	2	yes	—
34	#13133149	—	2	2	no	—
35	#13133149	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₂²⋊C₈×C₅₃	—	4	1	no	—
41	C₄×C₈×C₅₃	—	4	1	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂×Dic₁₀₆	—	4	1	no	—
50	C₂×Dic₁₀₆	—	4	1	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₃	D₈	1	1	yes	characteristic
61	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²⋊Dic₅₃	—	4	2	no	—
64	C₄⋊Dic₅₃	—	2	2	no	—
65	C₄⋊Dic₅₃	—	2	2	no	—
66	C₄⋊Dic₅₃	—	4	1	no	—
67	C₄⋊Dic₅₃	—	4	1	no	—
68	#11719919	—	4	2	no	—
69	#11719919	—	4	2	no	—
70	C₄×Dic₅₃	—	4	2	no	—
71	#11719908	—	4	2	no	—
72	C₂³×C₅₃	#102	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	4	1	no	—
75	C₂×C₄×C₅₃	#101	1	2	yes	—
76	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#103	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	Dic₁₀₆	—	8	2	no	—
86	Dic₁₀₆	—	8	2	no	—
87	C₈×C₅₃	—	2	1	no	—
88	C₈×C₅₃	—	2	1	no	—
89	C₈×C₅₃	—	8	2	no	—
90	C₈×C₅₃	#103	1	2	yes	—
91	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
93	#11934	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	#332	1	2	yes	—
97	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
98	C₄×C₅₃	—	2	1	no	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	#332	1	2	yes	—
102	C₄×C₅₃	#359	1	2	yes	—
103	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
104	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
105	Dic₅₃	—	8	2	no	—
106	Dic₅₃	—	8	2	no	—
107	#3160	—	53	1	no	—
108	C₂×C₄⋊C₁₆	—	53	1	no	—
109	#1564	D₅₃	1	1	yes	characteristic
110	#1363	—	53	4	no	—
111	C₂×C₅₃	—	2	2	no	—
112	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
113	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
114	C₂×C₅₃	#2145	1	1	yes	characteristic
115	#533	—	106	1	no	—
116	C₂²×C₁₆	—	106	1	no	—
117	#500	—	53	1	no	—
118	#500	—	53	1	no	—
119	#499	—	53	2	no	—
120	#499	—	106	1	no	—
121	#498	—	106	1	no	—
122	#431	—	2	1	no	—
123	#427	—	53	2	no	—
124	#427	—	106	1	no	—
125	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
126	C₄⋊C₁₆	—	53	2	no	—
127	C₄⋊C₁₆	—	106	1	no	—
128	#335	—	2	1	no	—
129	#334	C₂×D₅₃	1	2	yes	—
130	C₅₃	#11934	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
132	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
133	C₄⋊Q₈	—	106	1	no	—
134	C₄⋊Q₈	—	106	1	no	—
135	#125	—	106	2	no	—
136	C₂²⋊Q₈	—	212	1	no	—
137	C₂²⋊Q₈	—	212	1	no	—
138	C₄×Q₈	—	212	1	no	—
139	C₄×Q₈	—	212	1	no	—
140	#117	—	106	2	no	—
141	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
142	C₂×C₁₆	—	106	2	no	—
143	C₂×C₁₆	—	212	2	no	—
144	#107	—	53	2	no	—
145	#107	—	53	2	no	—
146	#107	—	106	1	no	—
147	#107	—	106	1	no	—
148	#106	—	106	2	no	—
149	#106	—	106	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₂²⋊C₈	—	4	1	no	—
152	C₄×C₈	—	4	1	no	—
153	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
154	C₄×C₈	D₂₁₂	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
156	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
157	C₂×Q₈	—	212	1	no	—
158	C₂×Q₈	—	212	1	no	—
159	C₂²×C₄	#11719921	1	1	yes	characteristic
160	C₂²×C₄	#11719922	1	1	yes	characteristic
161	C₂²×C₄	#11719924	1	1	yes	characteristic
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	#11719913	1	2	yes	—
165	C₂×C₈	Dic₂₁₂	1	2	yes	—
166	C₂×C₈	#11719921	1	1	yes	characteristic
167	C₂×C₈	#11719921	1	1	yes	characteristic
168	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
170	C₄⋊C₄	—	106	2	no	—
171	C₄⋊C₄	—	106	2	no	—
172	C₄⋊C₄	—	212	1	no	—
173	C₄⋊C₄	—	212	1	no	—
174	C₄⋊C₄	—	212	2	no	—
175	C₄⋊C₄	—	212	2	no	—
176	C₂²⋊C₄	—	212	2	no	—
177	C₄²	—	212	2	no	—
178	C₄²	#11719921	1	1	yes	characteristic
179	C₄²	#11719922	1	1	yes	characteristic
180	C₄²	#11719924	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
182	C₁₆	—	212	2	no	—
183	C₂³	#13133137	1	1	yes	characteristic
184	Q₈	—	424	2	no	—
185	Q₈	—	424	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	212	2	no	—
189	C₂×C₄	—	212	2	no	—
190	C₂×C₄	#13133135	1	2	yes	—
191	C₂×C₄	#13133137	1	1	yes	characteristic
192	C₂×C₄	#13133137	1	1	yes	characteristic
193	C₂×C₄	#13133137	1	1	yes	characteristic
194	C₂×C₄	#13133142	1	2	yes	—
195	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
196	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
197	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	8	2	no	—
201	C₈	#13133142	1	2	yes	—
202	C₈	C₂×#11719913	1	1	yes	characteristic
203	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	#16810370	1	2	yes	—
207	C₂²	C₂×#13133137	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	424	2	no	—
212	C₄	—	424	2	no	—
213	C₄	#16810370	1	2	yes	—
214	C₄	#16810429	1	2	yes	—
215	C₄	C₂×#13133135	1	1	yes	characteristic
216	C₄	C₂×#13133142	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	C₂×#16810370	1	1	yes	characteristic
219	C₂	C₂×#16810429	1	1	yes	characteristic
220	C₂	#20276450	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133137
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#11934
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #20272768	C₂	semidirect
130. C₅₃	#11934	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
C₂×C₄⋊#11719908	2
#20272768	2
C₅₃×#1564	2
#11934	53