Subgroups — #27294071

Order 13568.

314 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	#20276514	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	#20272679	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₄⋊#11719908	—	2	1	no	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	#16810376	—	2	1	no	—
10	#16810376	C₂²	1	2	yes	—
11	C₅₃×#533	—	2	1	no	—
12	C₅₃×#500	C₄	1	1	yes	characteristic
13	C₅₃×#494	C₄	1	1	yes	characteristic
14	C₅₃×#487	—	2	1	no	—
15	C₅₃×#467	—	2	1	no	—
16	C₅₃×#467	C₄	1	2	yes	—
17	C₄×Q₁₆×C₅₃	—	2	1	no	—
18	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	4	1	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	4	1	no	—
24	C₂×Q₁₆×C₅₃	—	2	1	no	—
25	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₄⋊Q₈×C₅₃	—	2	1	no	—
29	C₄⋊Q₈×C₅₃	—	2	1	no	—
30	#125×C₅₃	—	4	1	no	—
31	C₂²⋊Q₈×C₅₃	—	2	2	no	—
32	C₂²⋊Q₈×C₅₃	—	4	1	no	—
33	C₄×Q₈×C₅₃	—	2	2	no	—
34	C₄×Q₈×C₅₃	—	4	1	no	—
35	#117×C₅₃	—	4	1	no	—
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	#107×C₅₃	—	2	1	no	—
38	#107×C₅₃	C₂×C₄	1	2	yes	—
39	#103×C₅₃	—	2	2	no	—
40	#103×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₅₃	—	2	2	no	—
46	C₂×Q₈×C₅₃	—	4	1	no	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
50	Q₁₆×C₅₃	—	2	4	no	—
51	Q₁₆×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	4	1	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₄⋊C₄×C₅₃	—	4	1	no	—
60	C₄⋊C₄×C₅₃	—	4	1	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	4	1	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₂²⋊C₄×C₅₃	—	4	2	no	—
65	C₄²×C₅₃	—	4	2	no	—
66	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	D₈	1	1	yes	characteristic
68	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	—	8	2	no	—
72	#11719908	—	4	2	no	—
73	C₂³×C₅₃	#102	1	1	yes	characteristic
74	Q₈×C₅₃	—	4	2	no	—
75	Q₈×C₅₃	—	4	2	no	—
76	Q₈×C₅₃	—	8	2	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
86	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	#131	1	2	yes	—
94	#88322	—	8	2	no	—
95	#11646	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	#358	1	2	yes	—
99	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	8	2	no	—
104	C₄×C₅₃	—	8	2	no	—
105	C₄×C₅₃	#358	1	2	yes	—
106	C₄×C₅₃	#408	1	1	yes	characteristic
107	C₄×C₅₃	#421	1	1	yes	characteristic
108	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
109	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
110	#3170	D₅₃	1	1	yes	characteristic
111	#2162	—	53	1	no	—
112	C₂×#335	—	53	1	no	—
113	#1332	—	53	4	no	—
114	C₂×C₅₃	—	2	2	no	—
115	C₂×C₅₃	#1480	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
117	C₂×C₅₃	#2143	1	1	yes	characteristic
118	#533	—	2	1	no	—
119	#500	Dic₅₃	1	1	yes	characteristic
120	#494	Dic₅₃	1	1	yes	characteristic
121	#487	—	2	1	no	—
122	#467	—	2	1	no	—
123	#467	Dic₅₃	1	2	yes	—
124	C₄×Q₁₆	—	2	1	no	—
125	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
126	C₂×C₄⋊C₈	—	106	1	no	—
127	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
128	C₄⋊C₁₆	—	106	1	no	—
129	C₂²⋊C₁₆	—	106	1	no	—
130	C₄×C₁₆	—	53	2	no	—
131	#335	—	53	2	no	—
132	#335	—	106	1	no	—
133	C₅₃	#11646	1	1	yes	characteristic, complemented, Sylow
134	C₂×Q₁₆	—	2	1	no	—
135	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
136	C₂²×C₈	—	212	1	no	—
137	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	—	2	1	no	—
141	#125	—	4	1	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	—	4	1	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	4	1	no	—
146	#117	—	4	1	no	—
147	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂×C₁₆	—	106	2	no	—
149	#107	—	2	1	no	—
150	#107	C₂×Dic₅₃	1	2	yes	—
151	C₄⋊C₈	—	106	2	no	—
152	C₄⋊C₈	—	212	1	no	—
153	#103	—	2	2	no	—
154	#103	—	2	2	no	—
155	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₅₃	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	—	2	2	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂²×C₄	#11719922	1	1	yes	characteristic
162	C₂²×C₄	#11719923	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
164	Q₁₆	—	2	4	no	—
165	Q₁₆	—	4	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	424	2	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
173	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	#11719922	1	1	yes	characteristic
183	C₄²	#11719923	1	1	yes	characteristic
184	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
185	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
186	C₁₆	—	212	2	no	—
187	C₂³	#13133138	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	4	2	no	—
190	Q₈	—	8	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	#13133138	1	1	yes	characteristic
197	C₂×C₄	#13133138	1	1	yes	characteristic
198	C₂×C₄	#13133138	1	1	yes	characteristic
199	C₂×C₄	#13133160	1	2	yes	—
200	C₂×C₄	#13133163	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
202	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
203	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	424	2	no	—
208	C₈	#13133224	1	2	yes	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	#16810427	1	2	yes	—
212	C₂²	C₂×#13133138	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	8	2	no	—
217	C₄	—	8	2	no	—
218	C₄	#16810427	1	2	yes	—
219	C₄	#16810590	1	1	yes	characteristic
220	C₄	#16810640	1	1	yes	characteristic
221	C₄	C₂×#13133160	1	1	yes	characteristic
222	C₄	C₂×#13133163	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	#20273114	1	1	yes	characteristic
225	C₂	C₂×#16810427	1	1	yes	characteristic
226	C₂	#20276443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#11646
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #20272679	C₂	semidirect
133. C₅₃	#11646	semidirect

Maximal subgroups

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

subgroup	index
#20276514	2
C₂×#16810376	2
#20272679	2
C₅₃×#3170	2
#11646	53