Subgroups — #27295382

Order 13568.

334 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	#20272769	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133217	—	2	1	no	—
6	#16811028	—	2	1	no	—
7	#16811028	—	2	1	no	—
8	#16811028	C₄	1	2	yes	—
9	#16811028	C₂²	1	2	yes	—
10	#16810676	—	2	1	no	—
11	C₂×#13133151	—	2	1	no	—
12	C₂×#13133151	C₄	1	2	yes	—
13	#16810376	C₂²	1	2	yes	—
14	#16810374	—	2	1	no	—
15	C₂×M₃₂×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×C₄⋊C₁₆	—	2	1	no	—
18	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂²×Dic₁₀₆	—	4	1	no	—
20	#13133217	—	2	2	no	—
21	#13133217	—	2	2	no	—
22	#13133217	—	4	1	no	—
23	#13133217	—	4	1	no	—
24	C₂×C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	—	4	1	no	—
26	#13133151	—	2	2	no	—
27	#13133151	—	2	2	no	—
28	#13133151	—	2	2	no	—
29	#13133151	C₂×C₄	1	4	yes	—
30	C₄⋊#88322	—	2	2	no	—
31	C₂²⋊#88322	—	4	1	no	—
32	C₄×#88322	—	4	1	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	M₃₂×C₅₃	—	4	2	no	—
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂×Dic₁₀₆	—	4	2	no	—
47	C₂×Dic₁₀₆	—	4	2	no	—
48	C₂×Dic₁₀₆	—	8	4	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
53	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄²×C₅₃	D₈	1	1	yes	characteristic
58	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄⋊Dic₅₃	—	2	4	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₄⋊Dic₅₃	—	4	2	no	—
64	C₂×#88322	—	4	2	no	—
65	C₁₆×C₅₃	—	4	2	no	—
66	C₂³×C₅₃	#102	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	#101	1	2	yes	—
70	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#103	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×Dic₅₃	—	4	4	no	—
78	C₂×Dic₅₃	—	8	4	no	—
79	Dic₁₀₆	—	8	4	no	—
80	Dic₁₀₆	—	8	4	no	—
81	C₈×C₅₃	—	2	1	no	—
82	C₈×C₅₃	—	2	1	no	—
83	C₈×C₅₃	#103	1	2	yes	—
84	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
86	#88322	—	8	2	no	—
87	#11940	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	—	2	1	no	—
89	C₂²×C₅₃	—	2	1	no	—
90	C₂²×C₅₃	#332	1	2	yes	—
91	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	#332	1	2	yes	—
95	C₄×C₅₃	#362	1	2	yes	—
96	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
97	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
98	C₄×C₅₃	#509	1	2	yes	—
99	Dic₅₃	—	8	4	no	—
100	C₂×#500	—	53	1	no	—
101	#2150	D₅₃	1	1	yes	characteristic
102	#1564	—	53	1	no	—
103	#1364	—	53	4	no	—
104	C₂×C₅₃	—	2	2	no	—
105	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
106	C₂×C₅₃	#2140	1	1	yes	characteristic
107	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	—	106	1	no	—
109	C₂×M₃₂	—	2	1	no	—
110	#500	—	53	2	no	—
111	#500	—	53	2	no	—
112	#500	—	106	1	no	—
113	#500	—	106	1	no	—
114	#431	—	106	1	no	—
115	C₂×#107	—	53	2	no	—
116	C₂×#107	—	106	1	no	—
117	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
118	C₄⋊C₁₆	—	2	1	no	—
119	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
120	#335	—	53	2	no	—
121	#334	—	106	1	no	—
122	C₅₃	#11940	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	—	212	1	no	—
124	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₄⋊Q₈	—	106	2	no	—
127	C₄⋊Q₈	—	106	2	no	—
128	C₄⋊Q₈	—	212	1	no	—
129	C₄⋊Q₈	—	212	1	no	—
130	C₂×C₄⋊C₄	—	106	2	no	—
131	C₂×C₄⋊C₄	—	212	1	no	—
132	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
133	M₃₂	—	4	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	#107	—	53	4	no	—
136	#107	—	106	2	no	—
137	#107	—	106	2	no	—
138	#107	—	106	2	no	—
139	C₄⋊C₈	—	106	2	no	—
140	C₂²⋊C₈	—	212	1	no	—
141	C₄×C₈	—	212	1	no	—
142	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
143	C₄×C₈	D₂₁₂	1	1	yes	characteristic
144	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
145	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
146	C₂×Q₈	—	424	4	no	—
147	C₂×Q₈	—	212	2	no	—
148	C₂×Q₈	—	212	2	no	—
149	C₂²×C₄	—	212	2	no	—
150	C₂²×C₄	#11719913	1	1	yes	characteristic
151	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
152	C₂²×C₄	#11719921	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	212	2	no	—
155	C₂×C₈	#11719921	1	1	yes	characteristic
156	C₂×C₈	#11719921	1	1	yes	characteristic
157	C₂×C₈	#11719923	1	2	yes	—
158	C₂×C₈	#11719925	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
161	C₄⋊C₄	—	106	4	no	—
162	C₄⋊C₄	—	212	2	no	—
163	C₄⋊C₄	—	212	2	no	—
164	C₄⋊C₄	—	212	2	no	—
165	C₄²	#11719913	1	1	yes	characteristic
166	C₄²	D₄₂₄	1	1	yes	characteristic
167	C₄²	#11719921	1	1	yes	characteristic
168	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
169	C₁₆	—	4	2	no	—
170	C₂³	#13133139	1	1	yes	characteristic
171	Q₈	—	424	4	no	—
172	Q₈	—	424	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	424	4	no	—
176	C₂×C₄	—	212	4	no	—
177	C₂×C₄	#13133135	1	2	yes	—
178	C₂×C₄	#13133139	1	1	yes	characteristic
179	C₂×C₄	#13133139	1	1	yes	characteristic
180	C₂×C₄	#13133139	1	1	yes	characteristic
181	C₂×C₄	#13133140	1	2	yes	—
182	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	424	2	no	—
188	C₈	#13133140	1	2	yes	—
189	C₈	C₂×#11719923	1	1	yes	characteristic
190	C₈	C₂×#11719925	1	1	yes	characteristic
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#16810368	1	2	yes	—
194	C₂²	C₂×#13133139	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	424	4	no	—
198	C₄	#16810368	1	2	yes	—
199	C₄	#16810440	1	2	yes	—
200	C₄	C₂×#13133135	1	1	yes	characteristic
201	C₄	C₂×#13133140	1	1	yes	characteristic
202	C₄	#16811061	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	C₂×#16810368	1	1	yes	characteristic
205	C₂	#20276432	1	1	yes	characteristic
206	C₂	C₂×#16810440	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133139
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#11940
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #20272769	C₂	semidirect
122. C₅₃	#11940	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811028	2
#20273463	2
#20272769	2
C₅₃×#2150	2
#11940	53