Subgroups — #27282802

Order 13568.

290 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275491	C₂	1	2	yes	maximal
2	#20273471	C₂	1	1	yes	maximal, characteristic
3	#20273004	C₂	1	1	yes	maximal, characteristic
4	#20272612	C₂	1	2	yes	maximal
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	1	yes	characteristic
7	C₂×#13133153	—	2	2	no	—
8	#16810647	C₂²	1	2	yes	—
9	C₄×#11719917	C₂²	1	1	yes	characteristic
10	#16810388	—	2	2	no	—
11	#16810374	—	2	1	no	—
12	#16810374	—	2	1	no	—
13	#16810345	—	2	1	no	—
14	#16810344	—	2	1	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	C₄	1	2	yes	—
18	C₅₃×#335	C₄	1	2	yes	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	D₄	1	2	yes	—
21	#13133153	—	4	4	no	—
22	C₄⋊#88322	—	2	1	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₂²⋊#88322	Q₈	1	2	yes	—
26	#13133128	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	—	4	1	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅₃	#41	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	1	yes	characteristic
55	#11719917	—	2	2	no	—
56	#11719917	—	2	2	no	—
57	#11719917	—	4	2	no	—
58	#11719917	—	4	2	no	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	4	2	no	—
62	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	4	1	no	—
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
72	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₅₃	#137	1	2	yes	—
74	C₈×C₅₃	—	2	1	no	—
75	C₈×C₅₃	—	2	1	no	—
76	C₈×C₅₃	—	2	1	no	—
77	C₈×C₅₃	—	4	4	no	—
78	C₈×C₅₃	#108	1	2	yes	complemented
79	#88322	—	4	2	no	—
80	#88322	—	4	2	no	—
81	#88322	—	8	2	no	—
82	#9625	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	#389	1	1	yes	characteristic
86	C₂²×C₅₃	#501	1	2	yes	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
91	C₄×C₅₃	#430	1	1	yes	characteristic
92	C₄×C₅₃	#441	1	1	yes	characteristic
93	C₄×C₅₃	#442	1	1	yes	characteristic
94	C₄×C₅₃	#468	1	2	yes	—
95	#1950	—	53	2	no	—
96	#1566	—	53	1	no	—
97	C₂×#335	D₅₃	1	1	yes	characteristic
98	#1451	—	53	1	no	—
99	#1301	—	53	2	no	—
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	#1943	1	1	yes	characteristic
102	C₂×C₅₃	#1950	1	1	yes	characteristic
103	C₂×C₅₃	#1953	1	1	yes	characteristic
104	#431	—	53	1	no	—
105	C₂×#108	—	106	2	no	—
106	#423	—	53	2	no	—
107	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
108	C₄×M₁₆	—	53	1	no	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	#341	—	106	2	no	—
111	#335	Dic₅₃	1	2	yes	—
112	#335	Dic₅₃	1	2	yes	—
113	#334	—	106	1	no	—
114	#334	—	106	1	no	—
115	#322	—	106	1	no	—
116	#322	—	106	1	no	—
117	C₅₃	#9625	1	1	yes	characteristic, complemented, Sylow
118	C₂×M₁₆	—	53	2	no	—
119	C₂×M₁₆	—	106	2	no	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
122	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	#108	—	212	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	106	2	no	—
127	C₄⋊C₈	—	106	1	no	—
128	C₄⋊C₈	—	106	2	no	—
129	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
130	C₂²⋊C₈	—	53	2	no	—
131	#97	—	106	1	no	—
132	C₄×C₈	—	106	1	no	—
133	C₄×C₈	—	106	1	no	—
134	C₄×C₈	D₂₁₂	1	1	yes	characteristic
135	C₄×C₈	D₂₁₂	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
139	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
141	M₁₆	—	106	2	no	—
142	M₁₆	—	106	2	no	—
143	M₁₆	—	212	2	no	—
144	M₁₆	—	212	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	106	2	no	—
151	C₂×C₈	—	106	2	no	—
152	C₂×C₈	—	212	2	no	—
153	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
154	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
155	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
156	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
157	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
158	C₄²	#11719947	1	1	yes	characteristic
159	C₄²	#11719949	1	1	yes	characteristic
160	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
161	C₂³	#13133216	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
166	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
167	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
168	C₂×C₄	#13133192	1	1	yes	characteristic
169	C₂×C₄	#13133197	1	1	yes	characteristic
170	C₂×C₄	#13133208	1	1	yes	characteristic
171	C₂×C₄	#13133241	1	2	yes	—
172	C₂×C₄	#13133249	1	2	yes	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	4	4	no	—
177	C₈	—	424	2	no	—
178	C₈	—	212	2	no	—
179	C₈	—	212	2	no	—
180	C₈	#13133153	1	2	yes	complemented
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	#16810522	1	1	yes	characteristic
184	C₂²	#16811031	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	C₂×#13133153	1	1	yes	characteristic
189	C₄	#16810672	1	1	yes	characteristic
190	C₄	#16810730	1	1	yes	characteristic
191	C₄	#16810735	1	1	yes	characteristic
192	C₄	#16810870	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#20275454	1	1	yes	characteristic
195	C₂	#20275491	1	1	yes	characteristic
196	C₂	#20275506	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#9625
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
78. C₈×C₅₃	#108	semidirect
117. C₅₃	#9625	semidirect
180. C₈	#13133153	semidirect

Maximal subgroups

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

subgroup	index
#20275491	2
#20273471	2
#20273004	2
#20272612	2
C₂×C₅₃×#335	2
#9625	53