Subgroups — #27537747

Order 13568.

452 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	#20276624	C₂	1	2	yes	maximal, complemented
2	#20276624	C₂	1	4	yes	maximal, complemented
3	Q₈×#11719908	C₂	1	2	yes	maximal, complemented
4	#20276522	C₂	1	4	yes	maximal
5	#20276518	C₂	1	1	yes	maximal, characteristic
6	#20276514	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#11719908	C₂²	1	2	yes	—
9	C₄⋊#11719908	C₂²	1	2	yes	—
10	C₄⋊#11719908	C₂²	1	4	yes	—
11	C₄⋊#11719908	C₂²	1	4	yes	—
12	C₂²⋊#11719908	C₂²	1	2	yes	—
13	C₂²⋊#11719908	C₂²	1	2	yes	—
14	#16810399	C₂²	1	2	yes	—
15	#16810399	C₂²	1	4	yes	—
16	C₄×#11719908	C₂²	1	1	yes	characteristic
17	C₄×#11719908	C₂²	1	1	yes	characteristic
18	C₄×#11719908	C₂²	1	4	yes	—
19	C₅₃×#533	C₄	1	1	yes	characteristic
20	C₅₃×#446	C₄	1	2	yes	—
21	C₅₃×#446	C₄	1	2	yes	—
22	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
23	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
24	C₅₃×#424	C₄	1	2	yes	—
25	C₅₃×#423	C₄	1	1	yes	characteristic
26	C₅₃×#405	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#11719908	C₂³	1	2	yes	—
29	C₂×#11719908	C₂³	1	2	yes	—
30	C₂×#11719908	C₂³	1	4	yes	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
36	#125×C₅₃	C₈	1	2	yes	—
37	C₂²⋊Q₈×C₅₃	C₈	1	2	yes	—
38	C₂²⋊Q₈×C₅₃	C₈	1	2	yes	—
39	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
41	#117×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
46	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
47	#97×C₅₃	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂³	1	4	yes	—
52	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
54	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	M₁₆×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	4	1	no	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
67	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
68	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
69	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
70	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
71	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
72	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
73	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
78	#11719908	—	2	2	no	—
79	#11719908	—	2	2	no	—
80	#11719908	—	2	4	no	—
81	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	Q₈×C₅₃	—	2	4	no	—
83	Q₈×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
96	C₈×C₅₃	—	2	1	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	4	no	—
100	C₈×C₅₃	2₋¹⁺⁴	1	2	yes	—
101	#33499	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	4	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	4	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	#504	1	2	yes	—
113	C₄×C₅₃	#519	1	1	yes	characteristic
114	C₄×C₅₃	#568	1	1	yes	characteristic
115	#2960	D₅₃	1	1	yes	characteristic
116	#2183	—	53	2	no	—
117	#2183	—	53	4	no	—
118	Q₈×C₁₆	—	53	2	no	—
119	#2164	—	53	4	no	—
120	#2163	—	53	1	no	—
121	#2162	—	53	1	no	—
122	C₂×C₅₃	—	4	1	no	—
123	C₂×C₅₃	#2982	1	1	yes	characteristic
124	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
125	C₂×C₅₃	#3407	1	1	yes	characteristic
126	#533	Dic₅₃	1	1	yes	characteristic
127	#446	Dic₅₃	1	2	yes	—
128	#446	Dic₅₃	1	2	yes	—
129	C₈×Q₈	C₂×D₅₃	1	2	yes	—
130	C₈×Q₈	C₂×D₅₃	1	2	yes	—
131	#424	Dic₅₃	1	2	yes	—
132	#423	Dic₅₃	1	1	yes	characteristic
133	#405	C₂×D₅₃	1	2	yes	—
134	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₄⋊C₁₆	—	53	2	no	—
136	C₄⋊C₁₆	—	53	2	no	—
137	C₄⋊C₁₆	—	53	4	no	—
138	C₄⋊C₁₆	—	53	4	no	—
139	C₂²⋊C₁₆	—	53	2	no	—
140	C₂²⋊C₁₆	—	53	2	no	—
141	#346	—	53	2	no	—
142	#346	—	53	4	no	—
143	C₄×C₁₆	—	53	1	no	—
144	C₄×C₁₆	—	53	1	no	—
145	C₄×C₁₆	—	53	4	no	—
146	C₅₃	#33499	1	1	yes	characteristic, complemented, Sylow
147	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
148	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
149	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
150	C₄⋊Q₈	#88322	1	1	yes	characteristic
151	C₄⋊Q₈	#88322	1	1	yes	characteristic
152	#125	#88322	1	2	yes	—
153	C₂²⋊Q₈	#88322	1	2	yes	—
154	C₂²⋊Q₈	#88322	1	2	yes	—
155	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
156	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
157	#117	C₂×Dic₅₃	1	2	yes	—
158	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
159	C₂×C₁₆	—	53	2	no	—
160	C₂×C₁₆	—	53	2	no	—
161	C₂×C₁₆	—	53	4	no	—
162	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
163	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
164	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
165	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
166	#97	C₂×Dic₅₃	1	4	yes	—
167	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
169	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₅₃	1	4	yes	—
171	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₂×Q₈	C₂×#88322	1	2	yes	—
173	C₂×Q₈	C₂×#88322	1	2	yes	—
174	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
175	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
177	M₁₆	—	4	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
181	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
182	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
184	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
185	C₄⋊C₄	C₂×#88322	1	2	yes	—
186	C₄⋊C₄	C₂×#88322	1	2	yes	—
187	C₄⋊C₄	C₂×#88322	1	4	yes	—
188	C₄⋊C₄	C₂×#88322	1	4	yes	—
189	C₄⋊C₄	C₂×#88322	1	2	yes	—
190	C₄⋊C₄	C₂×#88322	1	2	yes	—
191	C₂²⋊C₄	C₂×#88322	1	4	yes	—
192	C₄²	C₂×#88322	1	1	yes	characteristic
193	C₄²	C₂×#88322	1	1	yes	characteristic
194	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
195	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
196	C₄²	C₂²×Dic₅₃	1	4	yes	—
197	C₁₆	—	106	2	no	—
198	C₁₆	—	106	2	no	—
199	C₁₆	—	106	4	no	—
200	C₂³	C₂²×#88322	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
208	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
209	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
210	C₂×C₄	C₂²×#88322	1	4	yes	—
211	C₂×C₄	C₂²×#88322	1	4	yes	—
212	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
213	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
214	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	4	no	—
219	C₈	#13133265	1	2	yes	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	C₂³×#88322	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	4	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	4	no	—
229	C₄	—	2	1	no	—
230	C₄	#16811040	1	2	yes	—
231	C₄	#16811101	1	1	yes	characteristic
232	C₄	#16811390	1	1	yes	characteristic
233	C₂	—	4	1	no	—
234	C₂	#20283796	1	1	yes	characteristic
235	C₂	C₂×#16811040	1	1	yes	characteristic
236	C₂	#20287818	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2960	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#33499
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #20276624	C₂	semidirect
2. #20276624	C₂	semidirect
3. Q₈×#11719908	C₂	semidirect
146. C₅₃	#33499	semidirect

Maximal subgroups

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

subgroup	index
#20276624	2
Q₈×#11719908	2
#20276522	2
#20276518	2
#20276514	2
C₅₃×#2960	2
#33499	53