Subgroups — #27277758

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 190 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	#20272549	C₂	1	4	yes	maximal
4	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133128	C₄	1	2	yes	—
8	C₂×#13133128	C₄	1	1	yes	characteristic
9	C₂×#13133128	C₄	1	1	yes	characteristic
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#336	—	2	2	no	—
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂²×#88322	C₂×C₄	1	2	yes	—
17	C₄⋊#88322	—	2	4	no	—
18	C₄⋊#88322	—	4	4	no	—
19	#13133128	—	2	2	no	—
20	#13133128	—	2	4	no	—
21	#13133128	C₈	1	4	yes	complemented
22	C₄×#88322	—	2	2	no	—
23	C₄×#88322	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	4	no	—
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₄²×C₅₃	—	2	2	no	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	—	4	2	no	—
43	C₄²×C₅₃	C₂×C₈	1	2	yes	—
44	C₄²×C₅₃	M₁₆	1	2	yes	—
45	C₄²×C₅₃	#41	1	2	yes	—
46	C₄²×C₅₃	#41	1	2	yes	—
47	C₂×#88322	—	2	2	no	—
48	C₂×#88322	—	2	2	no	—
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	—	2	2	no	—
51	C₂×#88322	—	4	2	no	—
52	C₂×#88322	—	4	4	no	—
53	C₂×#88322	C₂×C₈	1	4	yes	—
54	C₂×#88322	M₁₆	1	4	yes	—
55	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
63	C₂×C₄×C₅₃	#117	1	2	yes	—
64	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#125	1	2	yes	—
67	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₅₃	#137	1	2	yes	—
72	C₈×C₅₃	—	8	2	no	—
73	#88322	—	2	4	no	—
74	#88322	—	2	4	no	—
75	#88322	—	8	2	no	—
76	#8786	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	#382	1	1	yes	characteristic
78	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₅₃	#423	1	1	yes	characteristic
80	C₂²×C₅₃	#428	1	2	yes	—
81	C₂²×C₅₃	#430	1	2	yes	—
82	C₄×C₅₃	—	2	4	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	4	2	no	—
85	C₄×C₅₃	#431	1	2	yes	—
86	C₄×C₅₃	#432	1	2	yes	—
87	C₄×C₅₃	#489	1	2	yes	—
88	C₄×C₅₃	#490	1	1	yes	characteristic
89	C₄×C₅₃	#491	1	1	yes	characteristic
90	#1842	—	53	1	no	—
91	#1831	D₅₃	1	1	yes	characteristic
92	C₄×#97	—	53	1	no	—
93	#1277	—	53	4	no	—
94	C₂×C₅₃	#1565	1	2	yes	—
95	C₂×C₅₃	#1569	1	2	yes	—
96	C₂×C₅₃	#1831	1	1	yes	characteristic
97	C₂×C₅₃	#1836	1	1	yes	characteristic
98	C₂×C₅₃	#1838	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	53	2	no	—
100	C₂×C₄⋊C₈	—	106	2	no	—
101	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
102	C₂×#97	—	53	1	no	—
103	C₂×#97	—	53	2	no	—
104	C₂×#97	—	53	1	no	—
105	C₂×C₄×C₈	—	53	2	no	—
106	C₄³	C₂×D₅₃	1	1	yes	characteristic
107	#336	—	2	2	no	—
108	C₅₃	#8786	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	53	2	no	—
111	C₂²×C₈	—	53	2	no	—
112	C₂²×C₈	—	106	2	no	—
113	C₂×C₄²	Dic₁₀₆	1	2	yes	—
114	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
117	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	106	4	no	—
120	C₄⋊C₈	—	212	4	no	—
121	#97	—	53	4	no	—
122	#97	—	106	4	no	—
123	#97	—	106	2	no	—
124	C₄×C₈	—	106	2	no	—
125	C₄×C₈	—	106	2	no	—
126	C₂²×C₄	#11719919	1	2	yes	—
127	C₂²×C₄	#11719919	1	2	yes	—
128	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	53	4	no	—
135	C₂×C₈	—	53	4	no	—
136	C₂×C₈	—	106	2	no	—
137	C₂×C₈	—	106	2	no	—
138	C₂×C₈	—	106	2	no	—
139	C₂×C₈	—	106	2	no	—
140	C₂×C₈	—	212	2	no	—
141	C₂×C₈	—	212	4	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₈×D₅₃	1	2	yes	—
147	C₄²	#11719912	1	2	yes	—
148	C₄²	#11719947	1	2	yes	—
149	C₄²	#11719949	1	2	yes	—
150	C₂³	C₂×#11719919	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#13133146	1	4	yes	—
158	C₂×C₄	#13133175	1	2	yes	—
159	C₂×C₄	#13133204	1	1	yes	characteristic
160	C₂×C₄	#13133206	1	1	yes	characteristic
161	C₂×C₄	#13133208	1	2	yes	—
162	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
163	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
164	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
165	C₂×C₄	#13133242	1	2	yes	—
166	C₂×C₄	#13133247	1	2	yes	—
167	C₈	—	8	2	no	—
168	C₈	—	106	4	no	—
169	C₈	—	106	4	no	—
170	C₈	—	424	2	no	—
171	C₂²	#16810488	1	1	yes	characteristic
172	C₂²	C₂×#13133146	1	1	yes	characteristic
173	C₂²	#16810646	1	1	yes	characteristic
174	C₂²	#16810666	1	2	yes	—
175	C₂²	#16810671	1	2	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	#16810675	1	2	yes	—
180	C₄	#16810679	1	2	yes	—
181	C₄	#16810982	1	2	yes	—
182	C₄	#16810989	1	1	yes	characteristic
183	C₄	#16810993	1	1	yes	characteristic
184	C₂	#20273468	1	2	yes	—
185	C₂	#20273483	1	2	yes	—
186	C₂	#20274831	1	1	yes	characteristic
187	C₂	#20274848	1	1	yes	characteristic
188	C₂	#20274855	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	#8786
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
21. #13133128	C₈	semidirect
108. C₅₃	#8786	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
#20272549	2
C₅₃×#1831	2
#8786	53