Subgroups — #27302726

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
2	#20274887	C₂	1	1	yes	maximal, characteristic
3	#20272823	C₂	1	4	yes	maximal
4	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133153	—	2	1	no	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810647	—	2	1	no	—
9	#16810399	—	2	1	no	—
10	C₄×#11719908	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
12	C₅₃×#500	—	2	1	no	—
13	C₅₃×#500	—	2	1	no	—
14	C₅₃×#500	C₄	1	2	yes	—
15	C₅₃×#500	C₄	1	2	yes	—
16	C₂×#107×C₅₃	—	2	1	no	—
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#11719917	—	2	2	no	—
20	#13133157	—	2	2	no	—
21	#13133157	Q₈	1	4	yes	complemented
22	C₂×#11719908	—	2	2	no	—
23	#13133153	—	2	4	no	—
24	#13133153	—	4	2	no	—
25	C₄⋊#88322	—	4	1	no	—
26	C₄⋊#88322	—	4	1	no	—
27	C₂²×Q₈×C₅₃	—	4	1	no	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	—	4	1	no	—
33	C₄⋊Q₈×C₅₃	—	4	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	#107×C₅₃	—	2	2	no	—
38	#107×C₅₃	—	2	2	no	—
39	#107×C₅₃	—	2	2	no	—
40	#107×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂×Q₈×C₅₃	—	4	2	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂×Q₈×C₅₃	—	8	4	no	—
48	C₂²×C₄×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₅₃	#41	1	2	yes	—
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅₃	D₈	1	1	yes	characteristic
67	C₄²×C₅₃	D₈	1	1	yes	characteristic
68	#11719917	—	4	2	no	—
69	#11719917	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	#11719908	—	2	4	no	—
72	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
73	Q₈×C₅₃	—	8	4	no	—
74	Q₈×C₅₃	—	8	4	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	4	4	no	—
78	C₂×C₄×C₅₃	—	8	4	no	—
79	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₈×C₅₃	#103	1	2	yes	—
90	C₈×C₅₃	#103	1	2	yes	—
91	C₈×C₅₃	C₄×Q₈	1	2	yes	—
92	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
93	#88322	—	8	2	no	—
94	#13290	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
98	C₂²×C₅₃	#487	1	1	yes	characteristic
99	C₂²×C₅₃	#498	1	1	yes	characteristic
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	8	4	no	—
103	C₄×C₅₃	#361	1	2	yes	—
104	C₄×C₅₃	#362	1	2	yes	—
105	C₄×C₅₃	#391	1	1	yes	characteristic
106	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
107	C₄×C₅₃	#486	1	1	yes	characteristic
108	C₄×C₅₃	#500	1	1	yes	characteristic
109	C₂×#500	D₅₃	1	1	yes	characteristic
110	C₄×M₃₂	—	53	1	no	—
111	#1848	—	53	1	no	—
112	#1387	—	53	4	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	#1990	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
118	C₂×C₄⋊Q₈	—	2	1	no	—
119	C₂×M₃₂	—	53	2	no	—
120	#500	—	2	1	no	—
121	#500	—	2	1	no	—
122	#500	Dic₅₃	1	2	yes	—
123	#500	Dic₅₃	1	2	yes	—
124	C₂×#108	—	53	2	no	—
125	C₂×#108	—	106	1	no	—
126	C₂×#107	—	2	1	no	—
127	C₂×#107	C₂×D₅₃	1	2	yes	—
128	#423	—	106	1	no	—
129	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
130	#346	—	106	1	no	—
131	C₄×C₁₆	—	106	1	no	—
132	C₅₃	#13290	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	4	1	no	—
134	C₂×M₁₆	—	106	2	no	—
135	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	4	1	no	—
140	C₄⋊Q₈	—	4	1	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
144	M₃₂	—	53	4	no	—
145	M₃₂	—	106	2	no	—
146	C₂×C₁₆	—	106	2	no	—
147	#108	—	106	4	no	—
148	#108	—	212	2	no	—
149	#107	—	2	2	no	—
150	#107	—	2	2	no	—
151	#107	—	2	2	no	—
152	#107	C₂×Dic₅₃	1	4	yes	—
153	C₄⋊C₈	—	212	1	no	—
154	C₄⋊C₈	—	212	1	no	—
155	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
156	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	8	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	#11719923	1	1	yes	characteristic
164	C₂²×C₄	#11719923	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
166	M₁₆	—	212	2	no	—
167	M₁₆	—	212	2	no	—
168	C₂×C₈	—	212	2	no	—
169	C₂×C₈	#11719924	1	2	yes	—
170	C₂×C₈	#11719925	1	2	yes	—
171	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
173	C₂×C₈	Q₈×D₅₃	1	2	yes	—
174	C₂×C₈	#11719949	1	2	yes	—
175	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄²	#11719922	1	1	yes	characteristic
182	C₄²	#11719922	1	1	yes	characteristic
183	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
184	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
185	C₁₆	—	106	4	no	—
186	C₂³	C₂×#11719923	1	1	yes	characteristic
187	Q₈	—	8	4	no	—
188	Q₈	—	8	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	#13133138	1	1	yes	characteristic
194	C₂×C₄	#13133138	1	1	yes	characteristic
195	C₂×C₄	#13133138	1	1	yes	characteristic
196	C₂×C₄	#13133138	1	1	yes	characteristic
197	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
198	C₂×C₄	#13133206	1	1	yes	characteristic
199	C₂×C₄	#13133215	1	1	yes	characteristic
200	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
201	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
202	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
203	C₈	—	424	2	no	—
204	C₈	#13133141	1	2	yes	—
205	C₈	#13133141	1	2	yes	—
206	C₈	Q₈×Dic₅₃	1	2	yes	—
207	C₈	#13133193	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₂×#13133138	1	1	yes	characteristic
211	C₂²	#16810975	1	1	yes	characteristic
212	C₂²	#16811019	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	4	no	—
216	C₄	#16810436	1	2	yes	—
217	C₄	#16810439	1	2	yes	—
218	C₄	#16810530	1	1	yes	characteristic
219	C₄	C₂×#13133141	1	1	yes	characteristic
220	C₄	#16810971	1	1	yes	characteristic
221	C₄	#16811026	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#20275677	1	1	yes	characteristic
225	C₂	C₂×#16810436	1	1	yes	characteristic
226	C₂	C₂×#16810439	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#13290
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
21. #13133157	Q₈	semidirect
132. C₅₃	#13290	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133157	2
#20274887	2
#20272823	2
C₂×C₅₃×#500	2
#13290	53