Subgroups — #27300303

Order 13568.

542 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11719908	C₂	1	2	yes	maximal
2	#20272793	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	C₂²	1	4	yes	—
5	C₂²⋊#11719908	—	2	4	no	—
6	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	—	2	4	no	—
10	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#11719908	—	2	4	no	—
13	C₂×#11719908	—	2	8	no	—
14	C₂³×C₄×C₅₃	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₅₃	C₈	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	—	2	4	no	—
18	C₂²×C₈×C₅₃	—	2	4	no	—
19	C₂²×C₈×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	D₄	1	4	yes	—
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	—	2	2	no	—
25	C₂×C₄²×C₅₃	—	2	2	no	—
26	C₂×C₄²×C₅₃	C₈	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₅₃	—	2	4	no	—
29	C₄×C₈×C₅₃	—	2	4	no	—
30	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	C₁₆	1	4	yes	complemented
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	—	2	8	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	8	no	—
49	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
51	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₅₃	#41	1	4	yes	—
53	C₄²×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	2	2	no	—
56	#11719908	—	4	4	no	—
57	C₂³×C₅₃	—	2	2	no	—
58	C₂³×C₅₃	—	2	2	no	—
59	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
60	C₂³×C₅₃	C₂×C₁₆	1	2	yes	—
61	C₂³×C₅₃	M₃₂	1	2	yes	—
62	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	8	no	—
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	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
75	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#117	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
80	C₈×C₅₃	—	2	2	no	—
81	C₈×C₅₃	—	2	4	no	—
82	C₈×C₅₃	#123	1	4	yes	—
83	#12844	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	—	2	2	no	—
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	2	4	no	—
87	C₂²×C₅₃	—	2	4	no	—
88	C₂²×C₅₃	C₂²⋊C₁₆	1	4	yes	—
89	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
90	C₂²×C₅₃	#431	1	2	yes	—
91	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
92	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	2	4	no	—
97	C₄×C₅₃	#381	1	1	yes	characteristic
98	C₄×C₅₃	#407	1	1	yes	characteristic
99	C₄×C₅₃	#432	1	2	yes	—
100	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
101	C₂×C₂²⋊C₁₆	—	53	2	no	—
102	#1374	—	53	4	no	—
103	C₂×C₅₃	—	2	4	no	—
104	C₂×C₅₃	#1817	1	1	yes	characteristic
105	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
106	C₂×C₅₃	#2112	1	1	yes	characteristic
107	C₂×C₅₃	#2162	1	2	yes	—
108	C₂×C₅₃	#2163	1	2	yes	—
109	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
110	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
111	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
112	C₂²×C₁₆	—	53	4	no	—
113	C₂×C₄×C₈	—	2	4	no	—
114	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
116	C₂²⋊C₁₆	—	106	4	no	—
117	C₅₃	#12844	1	1	yes	characteristic, complemented, Sylow
118	C₂³×C₄	#88322	1	1	yes	characteristic
119	C₂³×C₄	#88322	1	1	yes	characteristic
120	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	2	4	no	—
123	C₂²×C₈	—	2	4	no	—
124	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
125	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
127	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₄²	—	2	2	no	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	#88322	1	2	yes	—
131	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
132	C₂×C₁₆	—	106	8	no	—
133	C₂×C₁₆	—	106	4	no	—
134	C₄×C₈	—	2	4	no	—
135	C₄×C₈	—	2	4	no	—
136	C₂⁴	C₂×#88322	1	1	yes	characteristic
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	#11719908	1	4	yes	complemented
144	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
145	C₂²×C₄	C₂×#88322	1	2	yes	—
146	C₂²×C₄	C₂×#88322	1	2	yes	—
147	C₂²×C₄	#11719917	1	2	yes	—
148	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	#11719917	1	4	yes	—
156	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
157	C₂×C₈	#11719945	1	4	yes	—
158	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₁₆	—	212	4	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₂²⋊#88322	1	2	yes	—
166	C₂³	C₂×#11719908	1	2	yes	—
167	C₂³	#13133157	1	2	yes	—
168	C₂³	C₂²×#88322	1	1	yes	characteristic
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	C₂²⋊#88322	1	2	yes	—
180	C₂×C₄	C₂×#11719908	1	4	yes	—
181	C₂×C₄	#13133157	1	4	yes	—
182	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	#13133176	1	2	yes	—
184	C₂×C₄	C₂×#11719917	1	2	yes	—
185	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
186	C₈	—	2	4	no	—
187	C₈	—	2	2	no	—
188	C₈	#13133200	1	4	yes	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	C₂²⋊#11719908	1	4	yes	—
194	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
195	C₂²	#16810676	1	2	yes	—
196	C₂²	C₂²×#11719908	1	1	yes	characteristic
197	C₂²	C₂×#13133157	1	2	yes	—
198	C₂²	C₂×#13133157	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	4	no	—
202	C₄	#16810486	1	1	yes	characteristic
203	C₄	#16810587	1	1	yes	characteristic
204	C₄	#16810680	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	#20274783	1	1	yes	characteristic
207	C₂	C₂×C₂²⋊#11719908	1	1	yes	characteristic
208	C₂	#20276326	1	1	yes	characteristic
209	C₂	#20276514	1	2	yes	—
210	C₂	#20276518	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#12844
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20272793	C₂	semidirect
37. C₂²×C₄×C₅₃	C₁₆	semidirect
117. C₅₃	#12844	semidirect
143. C₂²×C₄	#11719908	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#11719908	2
#20272793	2
C₂²×C₄×C₈×C₅₃	2
#12844	53