Subgroups — #27302709

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 196 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	#20274887	C₂	1	1	yes	maximal, characteristic
3	#20272811	C₂	1	4	yes	maximal
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
5	#16811180	—	2	1	no	—
6	#16811028	C₄	1	2	yes	—
7	#16811025	—	2	1	no	—
8	#16811025	C₄	1	2	yes	—
9	#16811021	—	2	1	no	—
10	C₂×#13133153	—	2	1	no	—
11	C₂×#13133153	C₂²	1	2	yes	—
12	#16810661	—	2	1	no	—
13	#16810661	C₂²	1	2	yes	—
14	#16810647	—	2	1	no	—
15	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#346	—	2	1	no	—
18	C₅₃×#346	—	2	1	no	—
19	C₂×#11719917	—	2	2	no	—
20	#13133217	—	2	2	no	—
21	#13133209	—	2	2	no	—
22	#13133196	—	4	2	no	—
23	C₄×Dic₁₀₆	—	4	2	no	—
24	#13133176	—	2	2	no	—
25	#13133153	—	2	4	no	—
26	#13133153	—	4	2	no	—
27	#13133151	—	2	2	no	—
28	#13133151	C₂×C₄	1	4	yes	—
29	#13133149	—	2	2	no	—
30	#13133149	—	2	2	no	—
31	C₄⋊#88322	—	4	2	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	Q₈	1	4	yes	complemented
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₁₀₆	—	4	2	no	—
45	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
48	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₅₃	#41	1	2	yes	—
53	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₄²×C₅₃	D₈	1	2	yes	—
56	C₂²⋊Dic₅₃	—	4	2	no	—
57	C₄⋊Dic₅₃	—	2	4	no	—
58	C₄⋊Dic₅₃	—	4	2	no	—
59	#11719919	—	4	4	no	—
60	C₄×Dic₅₃	—	4	2	no	—
61	#11719917	—	4	2	no	—
62	#11719917	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₁₆×C₅₃	—	2	4	no	—
65	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	4	2	no	—
67	C₂×C₄×C₅₃	#102	1	2	yes	—
68	C₂×C₄×C₅₃	#102	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×D₈	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×Dic₅₃	—	4	4	no	—
76	Dic₁₀₆	—	8	4	no	—
77	C₈×C₅₃	#103	1	2	yes	—
78	C₈×C₅₃	#103	1	2	yes	—
79	C₈×C₅₃	C₄×Q₈	1	2	yes	—
80	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
81	#88322	—	8	2	no	—
82	#13286	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
85	C₂²×C₅₃	#487	1	1	yes	characteristic
86	C₂²×C₅₃	#498	1	1	yes	characteristic
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	#391	1	1	yes	characteristic
90	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
91	C₄×C₅₃	#486	1	1	yes	characteristic
92	C₄×C₅₃	#500	1	1	yes	characteristic
93	Dic₅₃	—	8	4	no	—
94	#3160	—	53	1	no	—
95	C₂×#346	D₅₃	1	1	yes	characteristic
96	#1848	—	53	1	no	—
97	#1382	—	53	4	no	—
98	C₂×C₅₃	—	2	1	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	#1990	1	1	yes	characteristic
101	C₂×C₅₃	#2143	1	2	yes	—
102	#533	—	106	1	no	—
103	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
104	#500	—	53	2	no	—
105	#499	—	53	2	no	—
106	#499	—	106	1	no	—
107	#498	—	106	1	no	—
108	C₂×#108	—	53	2	no	—
109	C₂×#108	—	106	1	no	—
110	#427	—	53	2	no	—
111	#427	—	106	1	no	—
112	#423	—	106	1	no	—
113	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	#346	—	2	1	no	—
115	#346	—	2	1	no	—
116	C₅₃	#13286	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	106	2	no	—
118	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
120	C₄⋊Q₈	—	106	2	no	—
121	#125	—	106	2	no	—
122	C₂²⋊Q₈	—	212	2	no	—
123	C₄×Q₈	—	212	2	no	—
124	#117	—	106	2	no	—
125	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	Dic₁₀₆	1	4	yes	complemented
129	#108	—	106	4	no	—
130	#108	—	212	2	no	—
131	#107	—	53	4	no	—
132	#107	—	106	2	no	—
133	#106	—	106	2	no	—
134	#106	—	106	2	no	—
135	C₄⋊C₈	—	212	2	no	—
136	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
137	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂×Q₈	—	212	2	no	—
141	C₂²×C₄	#11719913	1	2	yes	—
142	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
143	M₁₆	—	212	2	no	—
144	M₁₆	—	212	2	no	—
145	C₂×C₈	—	212	2	no	—
146	C₂×C₈	#11719913	1	2	yes	—
147	C₂×C₈	Dic₂₁₂	1	2	yes	—
148	C₂×C₈	#11719921	1	1	yes	characteristic
149	C₂×C₈	#11719921	1	1	yes	characteristic
150	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
152	C₂×C₈	#11719945	1	2	yes	—
153	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
154	C₄⋊C₄	—	106	4	no	—
155	C₄⋊C₄	—	212	4	no	—
156	C₄⋊C₄	—	212	2	no	—
157	C₂²⋊C₄	—	212	2	no	—
158	C₄²	—	212	2	no	—
159	C₄²	D₄₂₄	1	2	yes	—
160	C₄²	#11719921	1	1	yes	characteristic
161	C₄²	#11719921	1	1	yes	characteristic
162	C₁₆	—	2	4	no	—
163	C₂³	C₂×#11719913	1	1	yes	characteristic
164	Q₈	—	424	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	212	4	no	—
167	C₂×C₄	#13133139	1	2	yes	—
168	C₂×C₄	#13133139	1	2	yes	—
169	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
170	C₂×C₄	#13133205	1	1	yes	characteristic
171	C₂×C₄	#13133217	1	1	yes	characteristic
172	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
173	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
174	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
175	C₈	—	424	2	no	—
176	C₈	#13133142	1	2	yes	—
177	C₈	#13133142	1	2	yes	—
178	C₈	C₄×Dic₁₀₆	1	2	yes	—
179	C₈	#13133196	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#13133139	1	1	yes	characteristic
182	C₂²	#16810974	1	1	yes	characteristic
183	C₂²	#16811021	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	424	4	no	—
187	C₄	#16810529	1	1	yes	characteristic
188	C₄	C₂×#13133142	1	1	yes	characteristic
189	C₄	#16810970	1	1	yes	characteristic
190	C₄	#16811028	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#20275676	1	1	yes	characteristic
194	C₂	#20276444	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#13286
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
37. C₂×C₁₆×C₅₃	Q₈	semidirect
116. C₅₃	#13286	semidirect
128. C₂×C₁₆	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
#20274887	2
#20272811	2
C₂×C₅₃×#346	2
#13286	53