Subgroups — #27273558

Order 13568.

758 conjugacy classes of subgroups.

The subgroup lattice has 238 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	#20272550	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	—	2	2	no	—
5	#16810647	—	2	4	no	—
6	#16810647	C₄	1	8	yes	—
7	C₂×C₄⋊#88322	—	2	2	no	—
8	C₂×C₄⋊#88322	C₂²	1	4	yes	—
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	C₂×#11719917	—	2	8	no	—
15	C₂×#11719917	—	4	4	no	—
16	C₂×#11719917	—	4	4	no	—
17	C₂²×#88322	—	2	4	no	—
18	C₄⋊#88322	—	2	8	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₄⋊#88322	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	—	2	8	no	—
25	C₂²×C₈×C₅₃	—	2	4	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	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	D₄	1	4	yes	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₅₃	—	2	4	no	—
34	C₄×C₈×C₅₃	—	2	4	no	—
35	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	4	no	—
37	C₂²×C₄×C₅₃	—	2	4	no	—
38	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	D₈	1	2	yes	—
46	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	—	2	16	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
55	C₄²×C₅₃	C₄²	1	2	yes	—
56	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
59	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
60	C₄²×C₅₃	C₂×D₄	1	2	yes	—
61	#11719917	—	4	16	no	—
62	C₂×#88322	—	2	8	no	—
63	C₂×#88322	—	4	4	no	—
64	C₂×#88322	—	4	4	no	—
65	C₂³×C₅₃	—	2	2	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	#95	1	2	yes	—
68	C₂³×C₅₃	#106	1	2	yes	—
69	C₂³×C₅₃	#107	1	2	yes	—
70	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	8	no	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	#95	1	2	yes	—
77	C₂×C₄×C₅₃	#95	1	2	yes	—
78	C₂×C₄×C₅₃	#95	1	2	yes	—
79	C₂×C₄×C₅₃	#102	1	8	yes	—
80	C₂×C₄×C₅₃	#103	1	8	yes	—
81	C₂×C₄×C₅₃	#106	1	2	yes	—
82	C₂×C₄×C₅₃	#107	1	2	yes	—
83	C₂×C₄×C₅₃	#108	1	4	yes	—
84	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
89	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
92	C₈×C₅₃	—	2	8	no	—
93	#88322	—	4	8	no	—
94	#7874	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	4	no	—
96	C₂²×C₅₃	—	2	4	no	—
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	2	2	no	—
99	C₂²×C₅₃	#340	1	4	yes	—
100	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
101	C₂²×C₅₃	#423	1	2	yes	—
102	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×#108	1	2	yes	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	#340	1	4	yes	—
108	C₄×C₅₃	#341	1	4	yes	—
109	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
110	C₄×C₅₃	#407	1	2	yes	—
111	C₄×C₅₃	C₂×#102	1	2	yes	—
112	C₄×C₅₃	C₂×#103	1	2	yes	—
113	C₂×#423	—	53	2	no	—
114	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
115	#1277	—	53	4	no	—
116	C₂×C₅₃	—	2	4	no	—
117	C₂×C₅₃	#1566	1	2	yes	—
118	C₂×C₅₃	#1567	1	2	yes	—
119	C₂×C₅₃	#1728	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
121	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
122	C₂²×M₁₆	—	106	2	no	—
123	C₂³×C₈	Dic₅₃	1	2	yes	—
124	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
125	#423	—	53	8	no	—
126	#423	—	106	4	no	—
127	C₂×C₄⋊C₈	—	53	4	no	—
128	C₂×C₄⋊C₈	—	106	2	no	—
129	C₂×C₄×C₈	—	2	4	no	—
130	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
131	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
132	C₅₃	#7874	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
134	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
135	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
136	C₂×M₁₆	—	106	8	no	—
137	C₂×M₁₆	—	212	4	no	—
138	C₂×M₁₆	—	212	4	no	—
139	C₂²×C₈	—	2	4	no	—
140	C₂²×C₈	—	2	8	no	—
141	C₂²×C₈	—	106	4	no	—
142	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
143	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
144	C₂×C₄²	C₄×D₅₃	1	4	yes	—
145	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
148	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
149	C₄⋊C₈	—	53	8	no	—
150	C₄⋊C₈	—	106	4	no	—
151	C₄⋊C₈	—	106	8	no	—
152	C₄×C₈	—	2	4	no	—
153	C₄×C₈	—	2	4	no	—
154	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	#11719913	1	4	yes	—
158	C₂²×C₄	D₄₂₄	1	2	yes	—
159	C₂²×C₄	Dic₂₁₂	1	2	yes	—
160	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
161	C₂²×C₄	#11719919	1	4	yes	—
162	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
164	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
165	C₂²×C₄	#11719921	1	4	yes	—
166	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
167	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
169	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
170	M₁₆	—	212	16	no	—
171	C₂×C₈	—	2	16	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	106	8	no	—
175	C₂×C₈	—	212	4	no	—
176	C₂×C₈	—	212	4	no	—
177	C₂×C₈	#11719917	1	8	yes	complemented
178	C₄²	C₄×Dic₅₃	1	2	yes	—
179	C₄²	#11719919	1	4	yes	—
180	C₄²	#11719921	1	4	yes	—
181	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
182	C₄²	C₂×C₄×D₅₃	1	2	yes	—
183	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	#13133124	1	2	yes	—
187	C₂³	#13133149	1	2	yes	—
188	C₂³	#13133151	1	2	yes	—
189	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	#13133124	1	2	yes	—
196	C₂×C₄	#13133124	1	2	yes	—
197	C₂×C₄	#13133124	1	2	yes	—
198	C₂×C₄	#13133139	1	8	yes	—
199	C₂×C₄	#13133142	1	8	yes	—
200	C₂×C₄	#13133149	1	2	yes	—
201	C₂×C₄	#13133151	1	2	yes	—
202	C₂×C₄	#13133153	1	4	yes	—
203	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×#11719921	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂×C₄	C₂×#11719919	1	2	yes	—
207	C₂×C₄	C₂×#11719917	1	4	yes	—
208	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
209	C₂×C₄	C₂×#11719913	1	2	yes	—
210	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
211	C₈	—	2	8	no	—
212	C₈	—	212	8	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	#16810386	1	4	yes	—
218	C₂²	C₂×#13133124	1	1	yes	characteristic
219	C₂²	#16810647	1	2	yes	—
220	C₂²	C₂×#13133149	1	1	yes	characteristic
221	C₂²	C₂×#13133151	1	1	yes	characteristic
222	C₂²	C₂×#13133153	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	#16810386	1	4	yes	—
226	C₄	#16810388	1	4	yes	—
227	C₄	C₄×#11719917	1	2	yes	—
228	C₄	#16810587	1	2	yes	—
229	C₄	C₂×#13133139	1	2	yes	—
230	C₄	C₂×#13133142	1	2	yes	—
231	C₂	—	2	4	no	—
232	C₂	#20273471	1	2	yes	—
233	C₂	#20273474	1	2	yes	—
234	C₂	#20274409	1	1	yes	characteristic
235	C₂	C₂×#16810386	1	1	yes	characteristic
236	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	#7874
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
54. C₂×C₈×C₅₃	M₁₆	semidirect
132. C₅₃	#7874	semidirect
177. C₂×C₈	#11719917	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
#20272550	2
C₂²×C₄×C₈×C₅₃	2
#7874	53