Subgroups — #27349675

Order 13568.

914 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	2	yes	maximal
2	#20272998	C₂	1	4	yes	maximal, complemented
3	M₁₆×#88322	C₂	1	8	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₄	1	4	yes	—
6	#16810676	C₄	1	4	yes	—
7	#16810676	C₂²	1	4	yes	—
8	#16810676	C₂²	1	4	yes	—
9	C₂×C₄⋊#88322	C₄	1	4	yes	—
10	C₂×C₂²⋊#88322	C₄	1	4	yes	—
11	C₂×C₄×#88322	C₂²	1	4	yes	—
12	#16810344	C₂²	1	8	yes	complemented
13	C₈×#88322	C₂²	1	8	yes	complemented
14	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₅₃	C₄	1	4	yes	—
17	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
18	C₂×#97×C₅₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
20	C₂²×#88322	C₂×C₄	1	8	yes	—
21	C₄⋊#88322	C₂×C₄	1	8	yes	—
22	C₄⋊#88322	C₂×C₄	1	8	yes	—
23	C₂²⋊#88322	C₂×C₄	1	8	yes	—
24	C₂²⋊#88322	C₂×C₄	1	8	yes	—
25	C₄×#88322	C₂×C₄	1	8	yes	—
26	C₄×#88322	C₂³	1	8	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₅₃	C₈	1	16	yes	complemented
30	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#97×C₅₃	C₂×C₄	1	8	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₅₃	C₂³	1	4	yes	—
42	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	4	4	no	—
44	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
45	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
48	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	M₁₆×C₅₃	C₂×C₈	1	32	yes	complemented
56	C₂×C₈×C₅₃	C₂×C₈	1	16	yes	complemented
57	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
58	C₄²×C₅₃	C₄²	1	1	yes	characteristic
59	C₄²×C₅₃	C₄²	1	2	yes	—
60	C₄²×C₅₃	C₄²	1	1	yes	characteristic
61	C₄²×C₅₃	C₄²	1	2	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
63	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
64	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
65	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂×#88322	—	4	8	no	—
69	C₂×#88322	C₂²×C₄	1	16	yes	—
70	C₂³×C₅₃	—	4	2	no	—
71	C₂³×C₅₃	C₄×C₈	1	4	yes	—
72	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
81	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
82	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
88	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
89	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₅₃	C₂²×C₈	1	16	yes	—
92	#88322	—	2	16	no	—
93	#17314	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	—	2	2	no	—
97	C₂²×C₅₃	—	4	4	no	—
98	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
99	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
100	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
105	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
106	C₄×C₅₃	C₂³×C₈	1	2	yes	—
107	C₄×C₅₃	#568	1	4	yes	—
108	C₂×#431	—	53	2	no	—
109	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
110	#1450	—	53	4	no	—
111	C₈×M₁₆	—	53	8	no	—
112	C₂×C₅₃	—	2	2	no	—
113	C₂×C₅₃	—	2	4	no	—
114	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
115	C₂×C₅₃	#2873	1	2	yes	—
116	C₂²×M₁₆	Dic₅₃	1	2	yes	—
117	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
118	#431	—	53	4	no	—
119	#431	—	53	4	no	—
120	#431	—	53	4	no	—
121	#431	—	53	4	no	—
122	C₂×C₄⋊C₈	—	53	4	no	—
123	C₂×C₂²⋊C₈	—	53	4	no	—
124	C₄×M₁₆	Dic₅₃	1	4	yes	—
125	C₄×M₁₆	C₂×D₅₃	1	4	yes	—
126	C₂×#97	Dic₅₃	1	2	yes	—
127	C₂×C₄×C₈	—	53	4	no	—
128	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
129	#322	—	53	8	no	—
130	C₈²	—	53	8	no	—
131	C₅₃	#17314	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₄	C₄×D₅₃	1	2	yes	—
133	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
134	C₂×M₁₆	#88322	1	16	yes	complemented
135	C₂×M₁₆	C₂×Dic₅₃	1	8	yes	—
136	C₂²×C₈	—	53	8	no	—
137	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
138	C₂×C₄²	C₄×D₅₃	1	2	yes	—
139	C₂×C₄²	C₄×D₅₃	1	2	yes	—
140	C₂×C₄²	C₄×D₅₃	1	2	yes	—
141	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
142	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
143	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
144	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
145	C₄⋊C₈	—	53	8	no	—
146	C₄⋊C₈	—	53	8	no	—
147	C₂²⋊C₈	—	53	8	no	—
148	C₂²⋊C₈	—	53	8	no	—
149	#97	C₂×Dic₅₃	1	8	yes	—
150	C₄×C₈	—	53	8	no	—
151	C₄×C₈	—	53	8	no	—
152	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
153	C₄×C₈	C₂²×D₅₃	1	4	yes	—
154	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	C₂×#88322	1	8	yes	—
157	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
158	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
159	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
160	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
161	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
163	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
164	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
165	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
167	M₁₆	C₂×#88322	1	32	yes	complemented
168	C₂×C₈	—	53	16	no	—
169	C₂×C₈	—	212	8	no	—
170	C₂×C₈	C₂×#88322	1	16	yes	complemented
171	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
172	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
173	C₄²	C₄×Dic₅₃	1	2	yes	—
174	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
175	C₄²	C₄×Dic₅₃	1	2	yes	—
176	C₄²	C₂×C₄×D₅₃	1	2	yes	—
177	C₄²	C₂×C₄×D₅₃	1	2	yes	—
178	C₄²	C₂×C₄×D₅₃	1	2	yes	—
179	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
180	C₄²	C₂²×Dic₅₃	1	2	yes	—
181	C₄²	C₂³×D₅₃	1	1	yes	characteristic
182	C₂³	—	4	2	no	—
183	C₂³	C₄×#88322	1	4	yes	—
184	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
185	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	C₄×#88322	1	8	yes	—
193	C₂×C₄	C₄×#88322	1	4	yes	—
194	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
195	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
196	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
197	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
198	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
199	C₂×C₄	C₂²×#88322	1	4	yes	—
200	C₂×C₄	C₂²×#88322	1	8	yes	—
201	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
202	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
203	C₈	—	106	16	no	—
204	C₈	C₂²×#88322	1	16	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	C₂×C₄×#88322	1	2	yes	—
210	C₂²	C₂×C₄×#88322	1	4	yes	—
211	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	C₂×C₄×#88322	1	2	yes	—
216	C₄	C₂×C₄×#88322	1	4	yes	—
217	C₄	C₂³×#88322	1	2	yes	—
218	C₄	#16811389	1	4	yes	—
219	C₂	—	2	4	no	—
220	C₂	—	2	2	no	—
221	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
222	C₂	#20283133	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#17314
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20272998	C₂	semidirect
3. M₁₆×#88322	C₂	semidirect
12. #16810344	C₂²	semidirect
13. C₈×#88322	C₂²	semidirect
29. C₂×M₁₆×C₅₃	C₈	semidirect
55. M₁₆×C₅₃	C₂×C₈	semidirect
56. C₂×C₈×C₅₃	C₂×C₈	semidirect
131. C₅₃	#17314	semidirect
134. C₂×M₁₆	#88322	semidirect
167. M₁₆	C₂×#88322	semidirect
170. C₂×C₈	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
#20272998	2
M₁₆×#88322	2
C₂×C₄×M₁₆×C₅₃	2
#17314	53