Subgroups — #27277842

Order 13568.

418 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	#20274867	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133128	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810375	C₂	1	2	yes	maximal
4	C₂×#16810373	C₂	1	2	yes	maximal
5	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133128	C₄	1	2	yes	—
9	C₂×#13133128	C₄	1	1	yes	characteristic
10	C₂×#13133128	C₄	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	2	yes	—
12	#16810375	—	2	4	no	—
13	#16810373	—	2	4	no	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂²×#88322	C₂×C₄	1	2	yes	—
20	C₂²×#88322	C₂×C₄	1	2	yes	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	4	4	no	—
24	#13133128	—	2	2	no	—
25	#13133128	—	2	4	no	—
26	#13133128	C₈	1	4	yes	complemented
27	C₄×#88322	D₄	1	4	yes	—
28	C₄×#88322	Q₈	1	4	yes	—
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	4	1	no	—
44	C₂×C₈×C₅₃	—	4	1	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	4	2	no	—
51	C₄²×C₅₃	C₂×C₈	1	2	yes	—
52	C₄²×C₅₃	M₁₆	1	2	yes	—
53	C₄²×C₅₃	C₂×D₄	1	2	yes	—
54	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	4	no	—
59	C₂×#88322	C₄⋊C₄	1	4	yes	—
60	C₂×#88322	C₄⋊C₄	1	4	yes	—
61	C₂×#88322	C₂×C₈	1	4	yes	—
62	C₂×#88322	M₁₆	1	4	yes	—
63	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₅₃	#137	1	2	yes	—
80	C₈×C₅₃	—	8	1	no	—
81	C₈×C₅₃	—	8	1	no	—
82	#88322	—	2	4	no	—
83	#88322	—	8	2	no	—
84	#88322	C₄⋊C₈	1	8	yes	complemented
85	#8808	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	#384	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₅₃	#423	1	1	yes	characteristic
89	C₂²×C₅₃	#428	1	2	yes	—
90	C₂²×C₅₃	#430	1	2	yes	—
91	C₄×C₅₃	—	2	4	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	4	2	no	—
94	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
95	C₄×C₅₃	#423	1	2	yes	—
96	C₄×C₅₃	#496	1	1	yes	characteristic
97	C₄×C₅₃	#497	1	1	yes	characteristic
98	C₄×C₅₃	#501	1	2	yes	—
99	#1842	—	53	1	no	—
100	#1831	D₅₃	1	1	yes	characteristic
101	C₄×#97	—	53	1	no	—
102	C₂×#335	—	53	2	no	—
103	C₂×#334	—	53	2	no	—
104	C₂×C₅₃	#1565	1	2	yes	—
105	C₂×C₅₃	#1569	1	2	yes	—
106	C₂×C₅₃	#1842	1	1	yes	characteristic
107	C₂×C₅₃	#1847	1	1	yes	characteristic
108	C₂×C₅₃	#1849	1	1	yes	characteristic
109	C₂×C₄⋊C₈	—	53	2	no	—
110	C₂×C₄⋊C₈	—	106	2	no	—
111	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
112	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
113	C₂×#97	—	53	1	no	—
114	C₂×#97	—	53	2	no	—
115	C₂×#97	—	53	1	no	—
116	C₂×C₄×C₈	—	53	2	no	—
117	C₄³	C₂×D₅₃	1	1	yes	characteristic
118	#336	—	2	2	no	—
119	#335	—	106	4	no	—
120	#334	—	106	4	no	—
121	C₅₃	#8808	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	—	53	2	no	—
125	C₂²×C₈	—	53	2	no	—
126	C₂²×C₈	—	106	2	no	—
127	C₂×C₄²	Dic₁₀₆	1	2	yes	—
128	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
129	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
131	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	106	2	no	—
135	C₄⋊C₈	—	106	2	no	—
136	C₄⋊C₈	—	212	4	no	—
137	#97	—	53	4	no	—
138	#97	—	106	4	no	—
139	#97	—	106	2	no	—
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	—	53	4	no	—
142	C₂²×C₄	#11719919	1	2	yes	—
143	C₂²×C₄	#11719919	1	2	yes	—
144	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	53	4	no	—
152	C₂×C₈	—	53	4	no	—
153	C₂×C₈	—	53	4	no	—
154	C₂×C₈	—	53	4	no	—
155	C₂×C₈	—	106	2	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	212	2	no	—
158	C₂×C₈	—	212	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₈×D₅₃	1	2	yes	—
164	C₄²	#11719912	1	2	yes	—
165	C₄²	D₄×D₅₃	1	2	yes	—
166	C₄²	Q₈×D₅₃	1	2	yes	—
167	C₂³	C₂×#11719919	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	#13133146	1	4	yes	—
175	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
176	C₂×C₄	#13133213	1	1	yes	characteristic
177	C₂×C₄	#13133215	1	1	yes	characteristic
178	C₂×C₄	#13133216	1	2	yes	—
179	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
180	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
181	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
182	C₂×C₄	#13133242	1	2	yes	—
183	C₂×C₄	#13133247	1	2	yes	—
184	C₈	—	8	1	no	—
185	C₈	—	8	1	no	—
186	C₈	—	53	8	no	—
187	C₈	—	106	4	no	—
188	C₈	—	424	2	no	—
189	C₂²	#16810492	1	1	yes	characteristic
190	C₂²	C₂×#13133146	1	1	yes	characteristic
191	C₂²	#16810646	1	1	yes	characteristic
192	C₂²	#16810666	1	2	yes	—
193	C₂²	#16810671	1	2	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
198	C₄	#16810645	1	2	yes	—
199	C₄	#16811013	1	1	yes	characteristic
200	C₄	#16811018	1	1	yes	characteristic
201	C₄	#16811033	1	2	yes	—
202	C₂	#20273468	1	2	yes	—
203	C₂	#20273483	1	2	yes	—
204	C₂	#20274866	1	1	yes	characteristic
205	C₂	#20274884	1	1	yes	characteristic
206	C₂	#20274889	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#8808
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
26. #13133128	C₈	semidirect
84. #88322	C₄⋊C₈	semidirect
121. C₅₃	#8808	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₄×#13133128	2
C₂×#16810375	2
C₂×#16810373	2
C₅₃×#1831	2
#8808	53