Subgroups — #27302659

Order 13568.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 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	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
3	#20272811	C₂	1	4	yes	maximal
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#16811180	—	2	1	no	—
6	#16811028	C₄	1	1	yes	characteristic
7	#16811028	C₄	1	1	yes	characteristic
8	#16811025	—	2	1	no	—
9	#16811025	C₄	1	2	yes	—
10	#16811021	—	2	1	no	—
11	C₂×#13133153	C₂²	1	2	yes	—
12	#16810661	—	2	1	no	—
13	#16810661	C₂²	1	2	yes	—
14	C₄×#11719917	—	2	1	no	—
15	#16810388	—	2	1	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×C₄⋊C₁₆	—	2	1	no	—
19	C₅₃×C₄⋊C₁₆	—	2	1	no	—
20	C₂×#11719917	—	2	2	no	—
21	#13133217	—	2	1	no	—
22	#13133217	—	2	1	no	—
23	#13133209	—	2	2	no	—
24	#13133196	—	4	1	no	—
25	#13133196	—	4	1	no	—
26	C₄×Dic₁₀₆	—	4	1	no	—
27	C₄×Dic₁₀₆	—	4	1	no	—
28	#13133176	—	2	2	no	—
29	#13133153	—	2	2	no	—
30	#13133153	Q₈	1	4	yes	complemented
31	#13133151	—	2	1	no	—
32	#13133151	—	2	1	no	—
33	#13133151	C₂×C₄	1	2	yes	—
34	#13133151	C₂×C₄	1	2	yes	—
35	#13133149	—	2	2	no	—
36	#13133149	—	2	2	no	—
37	#13133128	—	4	1	no	—
38	C₄×#88322	—	4	1	no	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₃	—	2	2	no	—
43	C₂×C₁₆×C₅₃	—	2	4	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₁₀₆	—	4	1	no	—
52	C₂×Dic₁₀₆	—	4	1	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₅₃	#41	1	2	yes	—
60	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅₃	D₈	1	1	yes	characteristic
63	C₄²×C₅₃	D₈	1	1	yes	characteristic
64	C₂²⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	2	2	no	—
66	C₄⋊Dic₅₃	—	2	2	no	—
67	C₄⋊Dic₅₃	—	4	1	no	—
68	C₄⋊Dic₅₃	—	4	1	no	—
69	#11719919	—	4	2	no	—
70	#11719919	—	4	2	no	—
71	C₄×Dic₅₃	—	4	2	no	—
72	#11719917	—	2	4	no	—
73	#11719917	—	4	2	no	—
74	C₂×#88322	—	4	2	no	—
75	C₁₆×C₅₃	—	4	2	no	—
76	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	4	1	no	—
79	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#137	1	2	yes	—
87	C₂×Dic₅₃	—	4	2	no	—
88	C₂×Dic₅₃	—	4	2	no	—
89	Dic₁₀₆	—	8	2	no	—
90	Dic₁₀₆	—	8	2	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	C₄×Q₈	1	2	yes	—
94	C₈×C₅₃	#124	1	1	yes	characteristic
95	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
96	#88322	—	4	4	no	—
97	#13274	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
101	C₂²×C₅₃	#475	1	2	yes	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	#359	1	2	yes	—
106	C₄×C₅₃	#391	1	1	yes	characteristic
107	C₄×C₅₃	#419	1	1	yes	characteristic
108	C₄×C₅₃	#474	1	2	yes	—
109	Dic₅₃	—	8	2	no	—
110	Dic₅₃	—	8	2	no	—
111	#3160	—	53	1	no	—
112	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
113	C₄×#108	—	53	1	no	—
114	#1382	—	53	4	no	—
115	C₂×C₅₃	—	2	2	no	—
116	C₂×C₅₃	#1988	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
118	C₂×C₅₃	#2143	1	1	yes	characteristic
119	#533	—	106	1	no	—
120	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
121	#500	—	53	1	no	—
122	#500	—	53	1	no	—
123	#499	—	53	2	no	—
124	#499	—	106	1	no	—
125	#498	—	106	1	no	—
126	C₂×#108	—	53	2	no	—
127	#427	—	53	2	no	—
128	#427	—	106	1	no	—
129	C₄×M₁₆	—	106	1	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	C₄⋊C₁₆	—	2	1	no	—
132	C₄⋊C₁₆	—	2	1	no	—
133	#341	—	106	1	no	—
134	C₅₃	#13274	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	106	2	no	—
136	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
138	C₄⋊Q₈	—	106	1	no	—
139	C₄⋊Q₈	—	106	1	no	—
140	#125	—	106	2	no	—
141	C₂²⋊Q₈	—	212	1	no	—
142	C₂²⋊Q₈	—	212	1	no	—
143	C₄×Q₈	—	212	1	no	—
144	C₄×Q₈	—	212	1	no	—
145	#117	—	106	2	no	—
146	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
147	C₂×C₁₆	—	2	4	no	—
148	C₂×C₁₆	—	2	2	no	—
149	#108	—	53	4	no	—
150	#108	—	106	2	no	—
151	#107	—	53	2	no	—
152	#107	—	53	2	no	—
153	#107	—	106	1	no	—
154	#107	—	106	1	no	—
155	#106	—	106	2	no	—
156	#106	—	106	2	no	—
157	#97	—	212	1	no	—
158	C₄×C₈	—	212	1	no	—
159	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
160	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
163	C₂×Q₈	—	212	1	no	—
164	C₂×Q₈	—	212	1	no	—
165	C₂²×C₄	#11719913	1	1	yes	characteristic
166	C₂²×C₄	#11719913	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
168	M₁₆	—	106	4	no	—
169	M₁₆	—	212	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	212	2	no	—
172	C₂×C₈	#11719921	1	1	yes	characteristic
173	C₂×C₈	#11719921	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₈	#11719947	1	2	yes	—
176	C₂×C₈	Q₈×D₅₃	1	2	yes	—
177	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	106	2	no	—
180	C₄⋊C₄	—	212	2	no	—
181	C₄⋊C₄	—	212	2	no	—
182	C₄⋊C₄	—	212	1	no	—
183	C₄⋊C₄	—	212	1	no	—
184	C₂²⋊C₄	—	212	2	no	—
185	C₄²	—	212	2	no	—
186	C₄²	D₄₂₄	1	1	yes	characteristic
187	C₄²	D₄₂₄	1	1	yes	characteristic
188	C₄²	#11719921	1	1	yes	characteristic
189	C₄²	#11719921	1	1	yes	characteristic
190	C₁₆	—	4	2	no	—
191	C₂³	C₂×#11719913	1	1	yes	characteristic
192	Q₈	—	424	2	no	—
193	Q₈	—	424	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	212	2	no	—
197	C₂×C₄	—	212	2	no	—
198	C₂×C₄	#13133139	1	1	yes	characteristic
199	C₂×C₄	#13133139	1	1	yes	characteristic
200	C₂×C₄	#13133139	1	1	yes	characteristic
201	C₂×C₄	#13133139	1	1	yes	characteristic
202	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
203	C₂×C₄	#13133197	1	2	yes	—
204	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
205	C₂×C₄	#13133250	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	212	4	no	—
209	C₈	#13133182	1	2	yes	—
210	C₈	#13133204	1	1	yes	characteristic
211	C₈	#13133215	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#13133139	1	1	yes	characteristic
215	C₂²	#16810909	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	424	2	no	—
220	C₄	—	424	2	no	—
221	C₄	#16810431	1	2	yes	—
222	C₄	#16810528	1	1	yes	characteristic
223	C₄	#16810633	1	1	yes	characteristic
224	C₄	#16810904	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#20275666	1	1	yes	characteristic
227	C₂	C₂×#16810431	1	1	yes	characteristic
228	C₂	#20276444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#13274
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
30. #13133153	Q₈	semidirect
134. C₅₃	#13274	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
C₄×#13133153	2
#20272811	2
C₂×C₅₃×C₄⋊C₁₆	2
#13274	53