Subgroups — #27271276

Order 13568.

208 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
2	#20274887	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	—	2	1	no	—
5	C₂×#13133153	—	2	1	no	—
6	C₂×#13133153	—	2	1	no	—
7	#16810647	C₄	1	2	yes	—
8	#16810444	—	2	1	no	—
9	#16810444	C₄	1	2	yes	—
10	C₂×M₃₂×C₅₃	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#364	—	2	1	no	—
13	C₅₃×#364	C₄	1	2	yes	—
14	C₂×#11719917	—	2	2	no	—
15	#13133157	—	2	2	no	—
16	#13133157	—	4	1	no	—
17	C₂×#11719908	—	4	1	no	—
18	#13133153	—	4	1	no	—
19	#13133153	—	4	1	no	—
20	#13133153	—	4	1	no	—
21	#13133153	—	4	1	no	—
22	C₄⋊#88322	—	4	2	no	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₅₃	—	2	2	no	—
27	M₃₂×C₅₃	—	4	1	no	—
28	C₂×C₁₆×C₅₃	—	4	1	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	—	2	1	no	—
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
42	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	—	2	1	no	—
48	C₄²×C₅₃	C₄²	1	1	yes	characteristic
49	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	#11719917	—	4	2	no	—
51	#11719917	—	4	2	no	—
52	C₂×#88322	—	4	2	no	—
53	C₁₆×C₅₃	—	4	2	no	—
54	#11719908	—	4	2	no	—
55	C₂³×C₅₃	#101	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	4	2	no	—
59	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
65	C₈×C₅₃	#102	1	1	yes	characteristic
66	C₈×C₅₃	#102	1	1	yes	characteristic
67	C₈×C₅₃	#103	1	1	yes	characteristic
68	C₈×C₅₃	#103	1	1	yes	characteristic
69	C₈×C₅₃	#106	1	1	yes	characteristic
70	C₈×C₅₃	#106	1	1	yes	characteristic
71	C₈×C₅₃	#107	1	1	yes	characteristic
72	C₈×C₅₃	#107	1	1	yes	characteristic
73	#88322	—	8	2	no	—
74	#7273	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	4	1	no	—
76	C₂²×C₅₃	#332	1	1	yes	characteristic
77	C₂²×C₅₃	#332	1	1	yes	characteristic
78	C₂²×C₅₃	#343	1	1	yes	characteristic
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	4	1	no	—
81	C₄×C₅₃	#331	1	1	yes	characteristic
82	C₄×C₅₃	#333	1	1	yes	characteristic
83	C₄×C₅₃	#340	1	1	yes	characteristic
84	C₄×C₅₃	#340	1	1	yes	characteristic
85	C₂×#364	—	53	1	no	—
86	C₂×#364	D₅₃	1	1	yes	characteristic
87	#1848	—	53	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	—	2	1	no	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	#1296	1	1	yes	characteristic
92	C₂×M₃₂	—	2	1	no	—
93	C₂×M₃₂	—	106	1	no	—
94	C₂×#108	—	106	1	no	—
95	C₂×#108	—	106	1	no	—
96	#423	—	53	2	no	—
97	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
98	#364	—	2	1	no	—
99	#364	—	53	2	no	—
100	#364	—	106	1	no	—
101	#364	Dic₅₃	1	2	yes	—
102	C₅₃	#7273	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	106	2	no	—
104	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	4	1	no	—
109	M₃₂	—	106	2	no	—
110	M₃₂	—	212	1	no	—
111	C₂×C₁₆	—	4	1	no	—
112	C₂×C₁₆	—	212	1	no	—
113	#108	—	212	1	no	—
114	#108	—	212	1	no	—
115	#108	—	212	1	no	—
116	#108	—	212	1	no	—
117	C₄⋊C₈	—	212	2	no	—
118	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
119	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	#11719921	1	1	yes	characteristic
124	M₁₆	—	212	2	no	—
125	M₁₆	—	212	2	no	—
126	C₂×C₈	—	212	2	no	—
127	C₂×C₈	#11719913	1	1	yes	characteristic
128	C₂×C₈	#11719913	1	1	yes	characteristic
129	C₂×C₈	D₄₂₄	1	1	yes	characteristic
130	C₂×C₈	Dic₂₁₂	1	1	yes	characteristic
131	C₂×C₈	#11719919	1	1	yes	characteristic
132	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
133	C₂×C₈	#11719921	1	1	yes	characteristic
134	C₂×C₈	#11719922	1	1	yes	characteristic
135	C₂×C₈	#11719923	1	1	yes	characteristic
136	C₂×C₈	#11719924	1	1	yes	characteristic
137	C₂×C₈	#11719925	1	1	yes	characteristic
138	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
139	C₄²	—	2	1	no	—
140	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
141	C₄²	#11719919	1	1	yes	characteristic
142	C₁₆	—	4	2	no	—
143	C₁₆	—	212	2	no	—
144	C₂³	#13133135	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	#13133124	1	1	yes	characteristic
149	C₂×C₄	#13133133	1	1	yes	characteristic
150	C₂×C₄	#13133137	1	1	yes	characteristic
151	C₂×C₄	#13133139	1	1	yes	characteristic
152	C₂×C₄	#13133140	1	1	yes	characteristic
153	C₂×C₄	#13133142	1	1	yes	characteristic
154	C₈	—	424	2	no	—
155	C₈	#13133138	1	1	yes	characteristic
156	C₈	#13133139	1	1	yes	characteristic
157	C₈	#13133141	1	1	yes	characteristic
158	C₈	#13133142	1	1	yes	characteristic
159	C₈	#13133148	1	1	yes	characteristic
160	C₈	#13133149	1	1	yes	characteristic
161	C₈	#13133150	1	1	yes	characteristic
162	C₈	#13133151	1	1	yes	characteristic
163	C₂²	—	4	1	no	—
164	C₂²	#16810368	1	1	yes	characteristic
165	C₂²	#16810370	1	1	yes	characteristic
166	C₂²	#16810391	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	#16810366	1	1	yes	characteristic
170	C₄	#16810371	1	1	yes	characteristic
171	C₄	#16810385	1	1	yes	characteristic
172	C₄	#16810386	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#20272599	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810371
commutator	C₄²×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂×C₅₃	#1296
prime radical	C₂³×C₅₃	#101
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#7273
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
102. C₅₃	#7273	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
#20274887	2
C₂×C₅₃×#364	2
#7273	53