Subgroups — #27273561

Order 13568.

670 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	C₂×#16810680	C₂	1	2	yes	maximal
2	#20272550	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810680	—	2	4	no	—
5	#16810680	C₄	1	8	yes	—
6	C₂×C₄⋊#88322	C₂²	1	4	yes	—
7	C₂×C₂²⋊#88322	—	2	2	no	—
8	C₂×#13133128	—	2	2	no	—
9	C₂³×C₈×C₅₃	C₄	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	C₂²	1	2	yes	—
14	C₂²×#88322	—	2	4	no	—
15	C₄⋊#88322	—	2	4	no	—
16	C₄⋊#88322	C₂×C₄	1	8	yes	—
17	C₂²⋊#88322	—	2	8	no	—
18	#13133128	—	2	8	no	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	—	2	8	no	—
23	C₂²×C₈×C₅₃	—	2	4	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	4	yes	—
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₃	—	2	4	no	—
32	C₄×C₈×C₅₃	—	2	4	no	—
33	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	2	4	no	—
36	C₂²×C₄×C₅₃	—	2	4	no	—
37	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	—	2	16	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
51	C₄²×C₅₃	C₄²	1	2	yes	—
52	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
53	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
56	C₄²×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×#88322	—	2	8	no	—
58	C₂×#88322	—	4	8	no	—
59	C₂³×C₅₃	—	2	2	no	—
60	C₂³×C₅₃	—	2	2	no	—
61	C₂³×C₅₃	#95	1	2	yes	—
62	C₂³×C₅₃	#108	1	4	yes	—
63	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	8	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	4	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	8	no	—
72	C₂×C₄×C₅₃	#95	1	2	yes	—
73	C₂×C₄×C₅₃	#95	1	2	yes	—
74	C₂×C₄×C₅₃	#95	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
80	C₂×C₄×C₅₃	#135	1	4	yes	—
81	C₈×C₅₃	—	2	8	no	—
82	#88322	—	4	8	no	—
83	#7875	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	—	2	8	no	—
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	2	2	no	—
87	C₂²×C₅₃	#341	1	4	yes	—
88	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
89	C₂²×C₅₃	#423	1	2	yes	—
90	C₂²×C₅₃	#427	1	2	yes	—
91	C₂²×C₅₃	C₂×#108	1	2	yes	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
97	C₄×C₅₃	#407	1	2	yes	—
98	C₄×C₅₃	#416	1	4	yes	—
99	C₂×#432	—	53	2	no	—
100	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
101	#1277	—	53	4	no	—
102	C₂×C₅₃	—	2	4	no	—
103	C₂×C₅₃	#1568	1	4	yes	—
104	C₂×C₅₃	#1728	1	1	yes	characteristic
105	C₂×C₅₃	#1735	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
107	C₂³×C₈	Dic₅₃	1	2	yes	—
108	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
109	#432	—	53	8	no	—
110	#432	—	106	4	no	—
111	C₂×C₄⋊C₈	—	53	4	no	—
112	C₂×C₂²⋊C₈	—	106	2	no	—
113	C₂×#97	—	106	2	no	—
114	C₂×C₄×C₈	—	2	4	no	—
115	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
117	C₅₃	#7875	1	1	yes	characteristic, complemented, Sylow
118	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
119	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
120	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
121	C₂²×C₈	—	2	4	no	—
122	C₂²×C₈	—	2	8	no	—
123	C₂²×C₈	—	106	4	no	—
124	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
125	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
126	C₂×C₄²	C₄×D₅₃	1	4	yes	—
127	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
128	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
130	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
131	C₄⋊C₈	—	53	8	no	—
132	C₄⋊C₈	—	106	4	no	—
133	C₂²⋊C₈	—	106	8	no	—
134	#97	—	106	8	no	—
135	C₄×C₈	—	2	4	no	—
136	C₄×C₈	—	2	4	no	—
137	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
142	C₂²×C₄	#11719919	1	4	yes	—
143	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
146	C₂²×C₄	#11719921	1	4	yes	—
147	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
148	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
151	C₂×C₈	—	2	16	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	106	8	no	—
155	C₂×C₈	—	212	8	no	—
156	C₂×C₈	#11719917	1	8	yes	—
157	C₄²	C₄×Dic₅₃	1	2	yes	—
158	C₄²	#11719919	1	4	yes	—
159	C₄²	#11719921	1	4	yes	—
160	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
161	C₄²	C₂×C₄×D₅₃	1	2	yes	—
162	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	#13133124	1	2	yes	—
166	C₂³	#13133153	1	4	yes	—
167	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
168	C₂×C₄	—	2	8	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	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	#13133124	1	2	yes	—
177	C₂×C₄	#13133124	1	2	yes	—
178	C₂×C₄	#13133124	1	2	yes	—
179	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×#11719921	1	2	yes	—
181	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂×#11719919	1	2	yes	—
183	C₂×C₄	C₂×#11719917	1	4	yes	—
184	C₂×C₄	#13133239	1	4	yes	—
185	C₈	—	2	8	no	—
186	C₈	—	212	8	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	#16810388	1	4	yes	—
191	C₂²	C₂×#13133124	1	1	yes	characteristic
192	C₂²	#16810647	1	2	yes	—
193	C₂²	#16810661	1	2	yes	—
194	C₂²	C₂×#13133153	1	2	yes	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	C₄×#11719917	1	2	yes	—
200	C₄	#16810587	1	2	yes	—
201	C₄	#16810620	1	4	yes	—
202	C₂	—	2	4	no	—
203	C₂	#20273478	1	4	yes	—
204	C₂	#20274409	1	1	yes	characteristic
205	C₂	#20274432	1	1	yes	characteristic
206	C₂	C₂×#16810388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#7875
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
2. #20272550	C₂	semidirect
117. C₅₃	#7875	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810680	2
#20272550	2
C₂²×C₄×C₈×C₅₃	2
#7875	53