Subgroups — #28310277

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊#108	C₂	1	2	yes	maximal
2	#20289775	C₂	1	4	yes	maximal, complemented
3	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊M₁₆	—	2	2	no	—
5	C₂×#13133288	C₂²	1	4	yes	—
6	D₅₃⋊#108	—	2	4	no	—
7	D₅₃⋊#108	C₄	1	8	yes	—
8	C₂×#13133287	—	2	2	no	—
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₄	1	1	yes	characteristic
11	C₂×C₁₆×D₅₃	—	2	4	no	—
12	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
13	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#11719936	—	2	4	no	—
16	D₅₃⋊M₁₆	—	2	8	no	—
17	D₅₃⋊M₁₆	—	4	4	no	—
18	C₂×#11719940	—	4	2	no	—
19	C₂×D₅₃⋊C₈	—	4	2	no	—
20	#13133288	—	2	4	no	—
21	#13133288	C₂×C₄	1	8	yes	—
22	#13133287	—	2	8	no	—
23	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
24	C₂²×#88322	Q₈	1	1	yes	characteristic
25	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₅₃	D₄	1	4	yes	—
28	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
30	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	C₂×C₄	1	2	yes	—
33	C₁₆×D₅₃	—	2	4	no	—
34	C₁₆×D₅₃	—	2	4	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂³×D₅₃	Q₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
41	C₂×C₄×D₅₃	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
44	C₂×C₄×D₅₃	SD₁₆	1	4	yes	—
45	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
47	#11719940	—	4	8	no	—
48	C₂×#88324	—	4	4	no	—
49	#11719936	—	2	8	no	—
50	#11719936	—	4	4	no	—
51	D₅₃⋊C₈	—	4	8	no	—
52	C₂×C₈×C₅₃	C₄²	1	2	yes	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
57	C₂×#88322	C₄⋊C₄	1	4	yes	—
58	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
59	C₈×D₅₃	C₄²	1	2	yes	—
60	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
61	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
62	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
63	C₈×D₅₃	C₂²×C₄	1	2	yes	—
64	C₈×D₅₃	C₂×D₄	1	2	yes	—
65	C₁₆×C₅₃	—	2	2	no	—
66	#11719908	—	2	2	no	—
67	C₂³×C₅₃	#107	1	1	yes	characteristic
68	C₂²×D₅₃	—	2	4	no	—
69	C₂²×D₅₃	#103	1	4	yes	—
70	C₂²×D₅₃	#107	1	1	yes	characteristic
71	C₂²×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#95	1	2	yes	—
73	C₂×C₄×C₅₃	#106	1	2	yes	—
74	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×Dic₅₃	#102	1	4	yes	—
77	C₂×Dic₅₃	#107	1	1	yes	characteristic
78	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
79	C₄×D₅₃	#95	1	2	yes	—
80	C₄×D₅₃	#102	1	4	yes	—
81	C₄×D₅₃	#103	1	4	yes	—
82	C₄×D₅₃	#106	1	2	yes	—
83	C₄×D₅₃	C₂×C₂²⋊C₄	1	2	yes	—
84	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
85	#88324	—	4	8	no	—
86	C₈×C₅₃	#95	1	2	yes	—
87	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
88	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	#88322	#95	1	2	yes	—
90	#88322	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×#1381	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	#340	1	2	yes	—
94	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
95	C₂×D₅₃	—	2	2	no	—
96	C₂×D₅₃	—	2	8	no	—
97	C₂×D₅₃	#340	1	2	yes	—
98	C₂×D₅₃	#368	1	4	yes	—
99	C₂×D₅₃	C₂×#103	1	2	yes	—
100	C₄×C₅₃	#340	1	2	yes	—
101	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
102	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
103	Dic₅₃	#340	1	2	yes	—
104	Dic₅₃	C₂×#102	1	2	yes	—
105	C₂³×C₁₆	—	53	1	no	—
106	C₂²×#108	—	53	2	no	—
107	#1381	—	53	4	no	—
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
110	C₂×C₅₃	C₂×#368	1	2	yes	—
111	D₅₃	—	2	2	no	—
112	D₅₃	#1381	1	4	yes	complemented
113	C₂²×M₁₆	—	106	2	no	—
114	C₂³×C₈	—	53	1	no	—
115	C₂²×C₁₆	—	53	1	no	—
116	C₂²×C₁₆	—	53	2	no	—
117	C₂²×C₁₆	—	53	2	no	—
118	C₂²×C₁₆	—	106	4	no	—
119	C₂²×C₁₆	#6157	1	1	yes	characteristic
120	C₂×#108	—	53	4	no	—
121	C₂×#108	—	53	8	no	—
122	C₂×#108	—	106	2	no	—
123	C₂×#108	—	106	4	no	—
124	C₅₃	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₄	—	53	1	no	—
126	C₂×M₁₆	—	106	8	no	—
127	C₂×M₁₆	—	106	4	no	—
128	C₂×M₁₆	—	212	2	no	—
129	C₂×M₁₆	—	212	4	no	—
130	C₂²×C₈	—	53	1	no	—
131	C₂²×C₈	—	53	4	no	—
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	53	4	no	—
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	212	2	no	—
138	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	53	2	no	—
141	C₂×C₁₆	—	106	4	no	—
142	C₂×C₁₆	—	106	4	no	—
143	C₂×C₁₆	—	106	2	no	—
144	C₂×C₁₆	C₂×#6157	1	2	yes	—
145	#108	—	53	8	no	—
146	#108	—	106	8	no	—
147	#108	—	106	4	no	—
148	C₂⁴	—	53	1	no	—
149	C₂²×C₄	—	53	1	no	—
150	C₂²×C₄	—	53	4	no	—
151	C₂²×C₄	—	53	4	no	—
152	C₂²×C₄	—	53	1	no	—
153	C₂²×C₄	—	53	1	no	—
154	C₂²×C₄	—	53	1	no	—
155	C₂²×C₄	—	53	1	no	—
156	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
157	M₁₆	—	106	8	no	—
158	M₁₆	—	212	4	no	—
159	M₁₆	—	212	8	no	—
160	C₂×C₈	—	53	4	no	—
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	53	1	no	—
163	C₂×C₈	—	53	4	no	—
164	C₂×C₈	—	53	4	no	—
165	C₂×C₈	—	53	2	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	212	4	no	—
170	C₂×C₈	—	212	8	no	—
171	C₂×C₈	C₄×#6157	1	2	yes	—
172	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊#6157	1	2	yes	—
174	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
175	C₁₆	—	2	2	no	—
176	C₁₆	—	106	2	no	—
177	C₂³	—	53	1	no	—
178	C₂³	—	53	1	no	—
179	C₂³	—	53	4	no	—
180	C₂³	—	106	4	no	—
181	C₂³	#13133286	1	1	yes	characteristic
182	C₂×C₄	—	53	1	no	—
183	C₂×C₄	—	53	1	no	—
184	C₂×C₄	—	53	4	no	—
185	C₂×C₄	—	53	2	no	—
186	C₂×C₄	—	53	2	no	—
187	C₂×C₄	—	53	2	no	—
188	C₂×C₄	—	53	2	no	—
189	C₂×C₄	—	53	4	no	—
190	C₂×C₄	—	53	4	no	—
191	C₂×C₄	(C₂×#6157)⋊C₄	1	2	yes	—
192	C₂×C₄	#13133285	1	2	yes	—
193	C₂×C₄	#13133286	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
195	C₈	—	53	2	no	—
196	C₈	—	53	2	no	—
197	C₈	—	212	8	no	—
198	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
199	C₈	C₂×C₄×#6157	1	1	yes	characteristic
200	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
201	C₂²	—	2	2	no	—
202	C₂²	—	53	2	no	—
203	C₂²	—	53	4	no	—
204	C₂²	—	53	2	no	—
205	C₂²	—	106	8	no	—
206	C₂²	—	106	2	no	—
207	C₂²	#16811497	1	2	yes	—
208	C₂²	C₂×#13133286	1	1	yes	characteristic
209	C₄	—	53	2	no	—
210	C₄	—	53	2	no	—
211	C₄	#16811497	1	2	yes	—
212	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
213	C₄	C₂×#13133285	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	—	53	4	no	—
216	C₂	—	106	2	no	—
217	C₂	C₂×#16811497	1	1	yes	characteristic
218	C₂	D₅₃⋊#368	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133286
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₂×#1381
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20289775	C₂	semidirect
112. D₅₃	#1381	semidirect
124. C₅₃	C₂×#1381	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₃⋊#108	2
#20289775	2
C₂²×C₁₆×D₅₃	2
C₂×#1381	53