Subgroups — #27302656

Order 13568.

334 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811025	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
3	#20272820	C₂	1	2	yes	maximal
4	#20272815	C₂	1	2	yes	maximal
5	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133209	—	2	1	no	—
7	C₂×#13133157	C₂²	1	2	yes	—
8	#16811025	—	2	2	no	—
9	#16811025	C₄	1	4	yes	complemented
10	C₂×#13133151	—	2	1	no	—
11	C₂×#13133149	C₂²	1	2	yes	—
12	#16810444	—	2	2	no	—
13	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#427	C₂²	1	1	yes	characteristic
15	C₅₃×#405	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#339	—	2	1	no	—
18	#13133209	—	2	4	no	—
19	#13133209	—	4	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×#11719919	—	4	2	no	—
22	#13133157	—	2	2	no	—
23	#13133157	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	#13133151	—	2	4	no	—
26	#13133149	—	2	2	no	—
27	#13133149	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	#117×C₅₃	—	2	1	no	—
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	#108×C₅₃	—	2	1	no	—
34	#108×C₅₃	Q₈	1	2	yes	—
35	#107×C₅₃	Q₈	1	2	yes	—
36	#106×C₅₃	—	2	1	no	—
37	#97×C₅₃	—	4	1	no	—
38	C₄×C₈×C₅₃	—	4	1	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	4	2	no	—
44	M₁₆×C₅₃	—	2	2	no	—
45	M₁₆×C₅₃	—	4	1	no	—
46	C₂×C₈×C₅₃	—	4	1	no	—
47	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₂²⋊C₄×C₅₃	—	4	1	no	—
59	C₄²×C₅₃	—	2	1	no	—
60	C₄²×C₅₃	—	4	1	no	—
61	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₄⋊Dic₅₃	—	2	4	no	—
63	C₄⋊Dic₅₃	—	4	2	no	—
64	#11719919	—	4	8	no	—
65	#11719908	—	4	2	no	—
66	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×Dic₅₃	—	4	4	no	—
77	C₂×Dic₅₃	—	8	4	no	—
78	C₈×C₅₃	—	4	2	no	—
79	C₈×C₅₃	#103	1	2	yes	—
80	C₈×C₅₃	#103	1	2	yes	—
81	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
83	C₈×C₅₃	#124	1	1	yes	characteristic
84	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
85	#13273	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	2	no	—
87	C₂²×C₅₃	#418	1	1	yes	characteristic
88	C₂²×C₅₃	#476	1	1	yes	characteristic
89	C₂²×C₅₃	#477	1	1	yes	characteristic
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	—	8	2	no	—
92	C₄×C₅₃	#391	1	1	yes	characteristic
93	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
94	C₄×C₅₃	#476	1	1	yes	characteristic
95	C₄×C₅₃	#477	1	1	yes	characteristic
96	Dic₅₃	—	8	4	no	—
97	C₂×#499	—	53	1	no	—
98	C₂×#364	—	53	1	no	—
99	#1772	D₅₃	1	1	yes	characteristic
100	#1386	—	53	2	no	—
101	#1383	—	53	2	no	—
102	C₂×C₅₃	—	2	1	no	—
103	C₂×C₅₃	—	2	1	no	—
104	C₂×C₅₃	#1987	1	1	yes	characteristic
105	C₂×C₅₃	#2231	1	2	yes	—
106	C₂×#125	—	106	1	no	—
107	C₂×M₃₂	—	53	2	no	—
108	#499	—	53	4	no	—
109	#499	—	106	2	no	—
110	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
111	#427	C₂×D₅₃	1	1	yes	characteristic
112	C₂×#107	—	106	1	no	—
113	C₂×#106	—	53	2	no	—
114	#405	—	2	1	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#364	—	106	2	no	—
117	#339	—	2	1	no	—
118	C₅₃	#13273	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	2	1	no	—
120	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
122	#125	—	106	4	no	—
123	#125	—	212	2	no	—
124	#117	—	2	1	no	—
125	C₂×C₄⋊C₄	—	106	2	no	—
126	C₂×C₄⋊C₄	—	212	2	no	—
127	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
128	M₃₂	—	106	2	no	—
129	M₃₂	—	106	2	no	—
130	C₂×C₁₆	—	106	2	no	—
131	#108	—	2	1	no	—
132	#108	Dic₁₀₆	1	2	yes	—
133	#107	—	106	4	no	—
134	#107	Dic₁₀₆	1	2	yes	—
135	#106	—	2	1	no	—
136	#106	—	53	4	no	—
137	#106	—	106	2	no	—
138	#97	—	4	1	no	—
139	C₄×C₈	—	4	1	no	—
140	C₄×C₈	C₄×D₅₃	1	2	yes	—
141	C₄×C₈	D₂₁₂	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	212	2	no	—
144	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	#11719921	1	2	yes	—
149	C₂×C₈	#11719923	1	1	yes	characteristic
150	C₂×C₈	#11719923	1	1	yes	characteristic
151	C₂×C₈	#11719925	1	2	yes	—
152	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
153	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
154	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
155	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
156	C₂×C₈	#11719945	1	1	yes	characteristic
157	C₂×C₈	#11719945	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	106	4	no	—
160	C₄⋊C₄	—	212	8	no	—
161	C₄⋊C₄	—	212	2	no	—
162	C₂²⋊C₄	—	4	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	#11719921	1	2	yes	—
166	C₁₆	—	212	2	no	—
167	C₂³	#13133242	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	424	4	no	—
172	C₂×C₄	—	212	4	no	—
173	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
174	C₂×C₄	#13133196	1	1	yes	characteristic
175	C₂×C₄	#13133196	1	1	yes	characteristic
176	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
177	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
178	C₂×C₄	#13133242	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	#13133140	1	2	yes	—
181	C₈	#13133140	1	2	yes	—
182	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
183	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
184	C₈	#13133205	1	1	yes	characteristic
185	C₈	#13133217	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#16810627	1	1	yes	characteristic
188	C₂²	#16810913	1	1	yes	characteristic
189	C₂²	#16810918	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	8	2	no	—
192	C₄	—	424	4	no	—
193	C₄	#16810529	1	1	yes	characteristic
194	C₄	C₂×#13133140	1	1	yes	characteristic
195	C₄	#16810913	1	1	yes	characteristic
196	C₄	#16810918	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#20275663	1	1	yes	characteristic
200	C₂	#20276891	1	2	yes	—

	subgroup	G/H
center	C₂²	#16810913
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#13273
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
9. #16811025	C₄	semidirect
118. C₅₃	#13273	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811025	2
C₂×#16810444	2
#20272820	2
#20272815	2
C₅₃×#1772	2
#13273	53