Subgroups — #27289921

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	2	yes	maximal
2	C₂×#16810388	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₂²	1	4	yes	—
5	#16810647	—	2	4	no	—
6	#16810647	C₄	1	8	yes	—
7	C₂×C₄⋊#88322	—	2	4	no	—
8	#16810388	—	2	8	no	—
9	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	—	2	4	no	—
12	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
14	C₂×#11719917	—	2	4	no	—
15	C₂×#11719917	—	2	8	no	—
16	C₂×#11719917	—	2	8	no	—
17	C₂×#11719917	C₂×C₄	1	8	yes	—
18	C₂²×#88322	—	2	4	no	—
19	C₄⋊#88322	—	2	16	no	—
20	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	—	2	4	no	—
23	C₂²×C₈×C₅₃	—	2	4	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	D₄	1	4	yes	—
26	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	2	yes	—
31	C₂×C₄²×C₅₃	D₄	1	4	yes	—
32	C₄×C₈×C₅₃	—	2	4	no	—
33	C₄×C₈×C₅₃	—	2	4	no	—
34	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	—	2	8	no	—
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₅₃	#41	1	4	yes	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	—	2	8	no	—
49	C₂×C₈×C₅₃	—	2	8	no	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
51	C₄²×C₅₃	C₄²	1	2	yes	—
52	C₄²×C₅₃	C₄²	1	2	yes	—
53	C₄²×C₅₃	C₂²⋊C₄	1	8	yes	—
54	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
56	#11719917	—	2	16	no	—
57	#11719917	—	4	8	no	—
58	C₂×#88322	—	2	8	no	—
59	C₂×#88322	—	4	4	no	—
60	C₂×#88322	—	4	4	no	—
61	C₂³×C₅₃	—	2	4	no	—
62	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
63	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂³×C₅₃	#125	1	2	yes	—
65	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	16	no	—
68	C₂×C₄×C₅₃	#95	1	4	yes	—
69	C₂×C₄×C₅₃	#95	1	4	yes	—
70	C₂×C₄×C₅₃	#108	1	8	yes	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₅₃	#117	1	2	yes	—
76	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
77	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
78	C₂×C₄×C₅₃	#121	1	4	yes	—
79	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
80	C₂×C₄×C₅₃	#124	1	2	yes	—
81	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₈×C₅₃	—	2	8	no	—
84	#88322	—	4	8	no	—
85	#10844	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	4	no	—
87	C₂²×C₅₃	—	2	4	no	—
88	C₂²×C₅₃	—	2	4	no	—
89	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
90	C₂²×C₅₃	#382	1	1	yes	characteristic
91	C₂²×C₅₃	#384	1	1	yes	characteristic
92	C₂²×C₅₃	#389	1	4	yes	—
93	C₂²×C₅₃	C₂×#108	1	4	yes	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	#341	1	8	yes	complemented
96	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
97	C₄×C₅₃	#380	1	2	yes	—
98	C₄×C₅₃	#390	1	2	yes	—
99	C₄×C₅₃	#391	1	2	yes	—
100	C₂×#423	—	53	2	no	—
101	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
102	C₂×#341	—	53	4	no	—
103	C₂×C₅₃	—	2	4	no	—
104	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
105	C₂×C₅₃	C₂×#341	1	2	yes	—
106	C₂×C₅₃	#1837	1	2	yes	—
107	C₂×C₅₃	#1848	1	2	yes	—
108	C₂²×M₁₆	—	53	4	no	—
109	C₂³×C₈	C₂×D₅₃	1	2	yes	—
110	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
111	#423	—	53	8	no	—
112	#423	—	106	4	no	—
113	C₂×C₄⋊C₈	—	106	4	no	—
114	C₂×C₄×C₈	—	2	4	no	—
115	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
116	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
117	#341	—	106	8	no	—
118	C₅₃	#10844	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₄	Dic₁₀₆	1	2	yes	—
120	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
121	C₂×M₁₆	—	53	8	no	—
122	C₂×M₁₆	—	106	4	no	—
123	C₂×M₁₆	—	106	8	no	—
124	C₂×M₁₆	—	106	8	no	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	106	4	no	—
128	C₂²×C₈	Dic₁₀₆	1	4	yes	—
129	C₂²×C₈	D₂₁₂	1	4	yes	—
130	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
131	C₂×C₄²	C₄×D₅₃	1	4	yes	—
132	C₂×C₄²	D₂₁₂	1	2	yes	—
133	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
136	C₄⋊C₈	—	106	16	no	—
137	C₄×C₈	—	2	4	no	—
138	C₄×C₈	—	2	4	no	—
139	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	#11719919	1	8	yes	—
142	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
143	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
144	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
146	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
147	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
148	C₂²×C₄	#11719945	1	4	yes	—
149	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
151	M₁₆	—	106	16	no	—
152	M₁₆	—	212	8	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	2	8	no	—
157	C₂×C₈	—	106	8	no	—
158	C₂×C₈	—	212	4	no	—
159	C₂×C₈	—	212	4	no	—
160	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
161	C₄²	C₄×Dic₅₃	1	2	yes	—
162	C₄²	C₄×Dic₅₃	1	2	yes	—
163	C₄²	#11719921	1	8	yes	—
164	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
165	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
166	C₂³	—	2	4	no	—
167	C₂³	C₂×#11719919	1	2	yes	—
168	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
169	C₂³	#13133209	1	2	yes	—
170	C₂³	#13133217	1	2	yes	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	16	no	—
173	C₂×C₄	#13133124	1	4	yes	—
174	C₂×C₄	#13133124	1	4	yes	—
175	C₂×C₄	#13133153	1	8	yes	—
176	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂×#11719921	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
180	C₂×C₄	#13133176	1	2	yes	—
181	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
182	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
183	C₂×C₄	#13133191	1	4	yes	—
184	C₂×C₄	#13133196	1	4	yes	—
185	C₂×C₄	#13133205	1	2	yes	—
186	C₂×C₄	#13133214	1	1	yes	characteristic
187	C₂×C₄	#13133217	1	1	yes	characteristic
188	C₈	—	2	8	no	—
189	C₈	—	212	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₂×#13133124	1	1	yes	characteristic
194	C₂²	#16810489	1	1	yes	characteristic
195	C₂²	#16810493	1	1	yes	characteristic
196	C₂²	#16810521	1	4	yes	—
197	C₂²	C₂×#13133153	1	4	yes	—
198	C₄	—	2	4	no	—
199	C₄	#16810388	1	8	yes	complemented
200	C₄	C₄×C₄⋊Dic₅₃	1	2	yes	—
201	C₄	#16810483	1	2	yes	—
202	C₄	#16810525	1	2	yes	—
203	C₄	#16810529	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	#20272945	1	1	yes	characteristic
206	C₂	C₂×#16810388	1	2	yes	—
207	C₂	#20274853	1	2	yes	—
208	C₂	#20274887	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#10844
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
95. C₄×C₅₃	#341	semidirect
118. C₅₃	#10844	semidirect
199. C₄	#16810388	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
C₂×#16810388	2
C₂²×C₄×C₈×C₅₃	2
#10844	53