Subgroups — #27270954

Order 13568.

262 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	—	2	1	no	—
5	C₂×C₄×#88322	C₄	1	1	yes	characteristic
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	C₄	1	2	yes	—
9	C₈×#88322	—	2	1	no	—
10	C₈×#88322	—	2	1	no	—
11	C₂²×C₁₆×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×C₄⋊C₁₆	—	2	1	no	—
14	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	—
15	C₂²×#88322	—	2	1	no	—
16	C₂²×#88322	C₈	1	2	yes	—
17	C₂×#11719908	—	2	2	no	—
18	C₂×#11719908	—	4	1	no	—
19	C₂×#11719908	—	4	1	no	—
20	C₄×#88322	—	2	1	no	—
21	C₄×#88322	—	2	1	no	—
22	C₄×#88322	—	2	1	no	—
23	C₄×#88322	C₈	1	2	yes	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	—	2	2	no	—
28	C₂×C₁₆×C₅₃	—	4	1	no	—
29	C₂×C₁₆×C₅₃	—	4	1	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	C₄²	1	1	yes	characteristic
48	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂×#88322	—	2	4	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	2	no	—
54	C₂×#88322	—	2	2	no	—
55	C₂×#88322	C₁₆	1	4	yes	complemented
56	C₁₆×C₅₃	—	4	2	no	—
57	#11719908	—	4	2	no	—
58	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
71	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
72	C₈×C₅₃	—	2	1	no	—
73	C₈×C₅₃	—	2	1	no	—
74	C₈×C₅₃	#102	1	1	yes	characteristic
75	C₈×C₅₃	#102	1	1	yes	characteristic
76	C₈×C₅₃	#103	1	1	yes	characteristic
77	C₈×C₅₃	#103	1	1	yes	characteristic
78	#88322	—	2	4	no	—
79	#88322	—	2	4	no	—
80	#7151	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	#334	1	1	yes	characteristic
82	C₂²×C₅₃	#335	1	1	yes	characteristic
83	C₂²×C₅₃	#336	1	1	yes	characteristic
84	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
85	C₂²×C₅₃	#364	1	2	yes	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	#334	1	1	yes	characteristic
89	C₄×C₅₃	#335	1	1	yes	characteristic
90	C₄×C₅₃	#340	1	1	yes	characteristic
91	C₄×C₅₃	#341	1	1	yes	characteristic
92	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
93	C₄×C₅₃	#346	1	1	yes	characteristic
94	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
95	C₂×C₄⋊C₁₆	—	53	1	no	—
96	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
97	C₂×C₈²	—	53	1	no	—
98	C₂×C₅₃	#1277	1	1	yes	characteristic
99	C₂×C₅₃	#1367	1	2	yes	—
100	C₂×C₅₃	#1370	1	2	yes	—
101	C₂×C₅₃	#1374	1	1	yes	characteristic
102	C₂×C₅₃	#1376	1	1	yes	characteristic
103	C₂²×C₁₆	—	2	1	no	—
104	C₂²×C₁₆	—	106	1	no	—
105	C₂×C₄×C₈	—	53	1	no	—
106	C₂×C₄×C₈	—	53	1	no	—
107	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
108	C₄⋊C₁₆	—	2	1	no	—
109	C₄⋊C₁₆	—	53	2	no	—
110	C₄⋊C₁₆	—	106	1	no	—
111	C₄⋊C₁₆	Dic₅₃	1	2	yes	—
112	C₈²	—	106	1	no	—
113	C₈²	—	106	1	no	—
114	C₅₃	#7151	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	53	2	no	—
116	C₂²×C₈	—	106	1	no	—
117	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
118	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
119	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	4	1	no	—
122	C₂×C₁₆	—	4	1	no	—
123	C₂×C₁₆	—	106	2	no	—
124	C₂×C₁₆	—	212	1	no	—
125	C₂×C₁₆	—	212	1	no	—
126	C₄×C₈	—	53	2	no	—
127	C₄×C₈	—	106	1	no	—
128	C₄×C₈	—	106	1	no	—
129	C₄×C₈	—	106	1	no	—
130	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
131	C₄×C₈	D₂₁₂	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
135	C₂²×C₄	#11719912	1	1	yes	characteristic
136	C₂²×C₄	#11719919	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	53	4	no	—
140	C₂×C₈	—	106	4	no	—
141	C₂×C₈	—	106	2	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	—	106	2	no	—
144	C₂×C₈	#11719919	1	1	yes	characteristic
145	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
146	C₂×C₈	#11719921	1	1	yes	characteristic
147	C₂×C₈	#11719922	1	1	yes	characteristic
148	C₂×C₈	#11719923	1	1	yes	characteristic
149	C₂×C₈	#11719924	1	1	yes	characteristic
150	C₂×C₈	#11719925	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
152	C₄²	C₈×D₅₃	1	1	yes	characteristic
153	C₄²	#11719912	1	1	yes	characteristic
154	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
155	C₄²	#11719921	1	1	yes	characteristic
156	C₁₆	—	4	2	no	—
157	C₁₆	—	212	2	no	—
158	C₂³	#13133146	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	#13133124	1	1	yes	characteristic
161	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
162	C₂×C₄	#13133127	1	1	yes	characteristic
163	C₂×C₄	#13133129	1	1	yes	characteristic
164	C₂×C₄	#13133129	1	1	yes	characteristic
165	C₂×C₄	#13133146	1	1	yes	characteristic
166	C₂×C₄	#13133148	1	1	yes	characteristic
167	C₂×C₄	#13133150	1	1	yes	characteristic
168	C₂×C₄	#13133152	1	1	yes	characteristic
169	C₂×C₄	#13133152	1	1	yes	characteristic
170	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
171	C₂×C₄	#13133156	1	2	yes	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	106	4	no	—
175	C₈	—	106	4	no	—
176	C₈	#13133137	1	1	yes	characteristic
177	C₈	#13133138	1	1	yes	characteristic
178	C₈	#13133140	1	1	yes	characteristic
179	C₈	#13133141	1	1	yes	characteristic
180	C₂²	#16810373	1	1	yes	characteristic
181	C₂²	#16810375	1	1	yes	characteristic
182	C₂²	#16810377	1	1	yes	characteristic
183	C₂²	#16810441	1	2	yes	—
184	C₂²	#16810443	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	#16810373	1	1	yes	characteristic
188	C₄	#16810375	1	1	yes	characteristic
189	C₄	#16810385	1	1	yes	characteristic
190	C₄	#16810387	1	1	yes	characteristic
191	C₄	C₁₆×Dic₅₃	1	1	yes	characteristic
192	C₄	#16810398	1	1	yes	characteristic
193	C₄	#16810402	1	2	yes	—
194	C₂	#20272549	1	1	yes	characteristic
195	C₂	#20272778	1	2	yes	—
196	C₂	#20272784	1	2	yes	—
197	C₂	#20272792	1	1	yes	characteristic
198	C₂	#20272796	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#7151
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
55. C₂×#88322	C₁₆	semidirect
114. C₅₃	#7151	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×C₈×#88322	2
C₂×C₅₃×C₄⋊C₁₆	2
#7151	53