Subgroups — #27537151

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811041	C₂	1	1	yes	maximal, characteristic
2	#20283323	C₂	1	2	yes	maximal
3	#20276510	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276508	C₂	1	4	yes	maximal, complemented
5	C₂×#16810452	C₂	1	2	yes	maximal, complemented
6	#368×D₅₃	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811389	—	2	2	no	—
9	C₂×#13133223	C₂²	1	1	yes	characteristic
10	#16811041	C₄	1	8	yes	—
11	C₂×#13133156	C₂²	1	2	yes	—
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
14	#16810671	—	2	2	no	—
15	#16810670	—	2	4	no	—
16	C₂×#13133153	C₂²	1	2	yes	—
17	#108×D₅₃	C₂²	1	8	yes	—
18	#16810453	C₂²	1	4	yes	—
19	#16810452	C₂²	1	4	yes	—
20	#16810452	C₂²	1	4	yes	—
21	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#108×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#368	C₂²	1	4	yes	—
24	C₂×#11719945	D₄	1	1	yes	characteristic
25	#13133225	—	4	4	no	—
26	#13133223	—	4	4	no	—
27	#13133223	C₂×C₄	1	4	yes	—
28	#13133223	Q₈	1	4	yes	—
29	C₂×#11719917	—	2	2	no	—
30	C₂×#11719917	D₄	1	1	yes	characteristic
31	C₂×#11719912	D₄	1	2	yes	—
32	M₁₆×D₅₃	—	2	8	no	—
33	C₂×C₈×D₅₃	C₂³	1	2	yes	—
34	#13133156	C₂×C₄	1	8	yes	—
35	C₂×#11719908	C₂×C₄	1	4	yes	—
36	C₂×#11719908	C₂³	1	1	yes	characteristic
37	C₂×#11719908	C₂³	1	1	yes	characteristic
38	C₁₆×D₅₃	C₂×C₄	1	8	yes	—
39	#13133153	—	2	2	no	—
40	#13133153	C₂³	1	4	yes	—
41	#13133152	—	2	8	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
45	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
46	#108×C₅₃	—	2	2	no	—
47	#108×C₅₃	C₂³	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₂²⋊D₅₃	Q₁₆	1	2	yes	—
50	#11719945	C₄⋊C₄	1	4	yes	—
51	#11719945	D₈	1	4	yes	—
52	C₂×D₂₁₂	Q₁₆	1	1	yes	characteristic
53	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
54	C₂×Dic₁₀₆	Q₁₆	1	1	yes	characteristic
55	M₁₆×C₅₃	—	2	4	no	—
56	M₁₆×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
62	#11719917	—	2	4	no	—
63	#11719917	—	4	2	no	—
64	#11719917	C₄⋊C₄	1	4	yes	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	C₂×D₄	1	2	yes	—
67	#11719912	—	4	8	no	—
68	#11719912	C₄⋊C₄	1	8	yes	—
69	C₈×D₅₃	—	4	8	no	—
70	C₈×D₅₃	C₂²×C₄	1	4	yes	—
71	C₈×D₅₃	C₂×Q₈	1	4	yes	—
72	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
73	#11719908	C₂²×C₄	1	2	yes	—
74	#11719908	C₂²×C₄	1	2	yes	—
75	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×D₅₃	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
81	C₂²⋊D₅₃	#107	1	8	yes	—
82	C₂×Dic₅₃	C₂×Q₁₆	1	2	yes	—
83	D₂₁₂	#107	1	4	yes	—
84	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
85	C₄×D₅₃	C₂×D₈	1	4	yes	—
86	Dic₁₀₆	#107	1	4	yes	—
87	C₈×C₅₃	—	4	4	no	—
88	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
91	#88322	—	4	4	no	—
92	#88322	C₂×C₄⋊C₄	1	4	yes	—
93	#33414	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	4	1	no	—
95	C₂²×C₅₃	C₂×#107	1	2	yes	—
96	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
97	C₂×D₅₃	—	4	2	no	—
98	C₂×D₅₃	C₂×#107	1	4	yes	—
99	C₄×C₅₃	C₂×#107	1	2	yes	—
100	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
102	Dic₅₃	C₂×#107	1	4	yes	—
103	C₂×#504	—	53	1	no	—
104	#2916	—	53	2	no	—
105	#2161	—	53	1	no	—
106	#2161	—	53	4	no	—
107	#2161	D₅₃	1	1	yes	characteristic, complemented
108	C₂×#368	—	53	2	no	—
109	C₂×#368	—	53	4	no	—
110	C₂×C₅₃	—	2	1	no	—
111	C₂×C₅₃	—	2	2	no	—
112	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
113	D₅₃	—	2	4	no	—
114	#568	—	106	2	no	—
115	C₂×#131	—	53	1	no	—
116	#504	—	53	8	no	—
117	C₂×M₃₂	—	53	2	no	—
118	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
119	C₂²×C₁₆	—	53	1	no	—
120	C₂²×C₁₆	—	53	2	no	—
121	#430	—	106	4	no	—
122	#430	—	106	2	no	—
123	C₂×#108	—	53	2	no	—
124	C₂×#108	—	53	8	no	—
125	C₂×#108	C₂×D₅₃	1	2	yes	—
126	#368	—	53	4	no	—
127	#368	—	53	4	no	—
128	#368	—	53	4	no	—
129	#368	C₂×D₅₃	1	4	yes	—
130	C₅₃	#33414	1	1	yes	characteristic, complemented, Sylow
131	C₂×#41	—	53	1	no	—
132	#131	—	53	4	no	—
133	#131	—	53	4	no	—
134	#131	—	212	4	no	—
135	#131	—	212	4	no	—
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	53	1	no	—
138	C₂×M₁₆	—	53	2	no	—
139	C₂×M₁₆	—	106	8	no	—
140	C₂×M₁₆	—	106	2	no	—
141	C₂²×C₈	—	53	2	no	—
142	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
143	M₃₂	—	53	8	no	—
144	M₃₂	C₄×D₅₃	1	4	yes	—
145	C₂×C₁₆	—	53	4	no	—
146	C₂×C₁₆	—	53	1	no	—
147	C₂×C₁₆	—	53	1	no	—
148	C₂×C₁₆	—	53	8	no	—
149	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
150	#108	—	2	2	no	—
151	#108	—	53	4	no	—
152	#108	—	106	8	no	—
153	#108	—	106	2	no	—
154	#108	C₂²×D₅₃	1	4	yes	—
155	#41	—	53	4	no	—
156	#41	—	53	4	no	—
157	C₂×Q₈	—	53	1	no	—
158	C₂×D₄	—	53	2	no	—
159	C₂×D₄	—	53	1	no	—
160	C₂²×C₄	—	53	2	no	—
161	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
162	M₁₆	—	2	4	no	—
163	M₁₆	—	4	2	no	—
164	M₁₆	—	53	4	no	—
165	M₁₆	—	53	8	no	—
166	M₁₆	—	106	4	no	—
167	M₁₆	—	212	8	no	—
168	M₁₆	—	212	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	53	2	no	—
171	C₂×C₈	—	53	4	no	—
172	C₂×C₈	—	53	4	no	—
173	C₂×C₈	—	212	8	no	—
174	C₂×C₈	—	212	2	no	—
175	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
176	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
177	C₂×C₈	Q₈×D₅₃	1	2	yes	—
178	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
179	C₁₆	—	53	2	no	—
180	C₁₆	—	53	2	no	—
181	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
182	C₂³	—	53	2	no	—
183	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
184	Q₈	—	53	4	no	—
185	D₄	—	53	8	no	—
186	D₄	—	53	4	no	—
187	C₂×C₄	—	53	2	no	—
188	C₂×C₄	—	53	4	no	—
189	C₂×C₄	—	53	4	no	—
190	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
191	C₂×C₄	D₈×D₅₃	1	2	yes	—
192	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
193	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
194	C₈	—	4	4	no	—
195	C₈	—	53	4	no	—
196	C₈	—	212	4	no	—
197	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
198	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
199	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
200	C₂²	—	4	1	no	—
201	C₂²	—	53	4	no	—
202	C₂²	—	212	2	no	—
203	C₂²	#107×D₅₃	1	2	yes	—
204	C₂²	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
205	C₄	—	53	4	no	—
206	C₄	#107×D₅₃	1	2	yes	—
207	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
208	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	2	no	—
211	C₂	—	106	4	no	—
212	C₂	C₂×#107×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2161	C₂
socle	C₂×C₅₃	C₂²×#107
prime radical	C₂×#11719945	D₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2161	D₅₃
hypocenter	C₅₃	#33414
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

Complements

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

N	H ≅ G/N	type
3. #20276510	C₂	semidirect
4. #20276508	C₂	semidirect
5. C₂×#16810452	C₂	semidirect
6. #368×D₅₃	C₂	semidirect
7. C₅₃×#2161	C₂	semidirect
107. #2161	D₅₃	semidirect
130. C₅₃	#33414	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811041	2
#20283323	2
#20276510	2
#20276508	2
C₂×#16810452	2
#368×D₅₃	2
C₅₃×#2161	2
#33414	53