Subgroups — #27300401

Order 13568.

442 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276326	C₂	1	1	yes	maximal, characteristic
2	#20274753	C₂	1	1	yes	maximal, characteristic
3	#20272823	C₂	1	4	yes	maximal, complemented
4	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133153	C₂²	1	2	yes	—
7	#16810587	—	2	1	no	—
8	C₂²⋊#11719908	—	2	1	no	—
9	C₂²⋊#11719908	—	2	1	no	—
10	#16810388	—	2	1	no	—
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
13	C₂×#107×C₅₃	—	2	1	no	—
14	C₂×#107×C₅₃	—	2	1	no	—
15	C₂×#107×C₅₃	—	2	1	no	—
16	C₂×#107×C₅₃	C₄	1	2	yes	—
17	C₂×#107×C₅₃	C₄	1	2	yes	—
18	C₂×#107×C₅₃	C₂²	1	2	yes	—
19	C₂×#11719917	—	2	2	no	—
20	#13133157	—	2	4	no	—
21	C₂×#11719908	—	2	2	no	—
22	#13133153	—	2	4	no	—
23	C₂²⋊#88322	—	4	1	no	—
24	C₂²⋊#88322	—	4	1	no	—
25	C₂³×C₄×C₅₃	—	4	1	no	—
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	4	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
39	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
40	#107×C₅₃	—	2	2	no	—
41	#107×C₅₃	—	2	2	no	—
42	#107×C₅₃	—	2	2	no	—
43	#107×C₅₃	C₂×C₄	1	4	yes	—
44	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	—	4	2	no	—
48	C₂²×C₄×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	8	4	no	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
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	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	#41	1	4	yes	—
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
71	#11719917	—	4	2	no	—
72	#11719917	—	4	2	no	—
73	C₂×#88322	—	4	2	no	—
74	#11719908	—	4	2	no	—
75	C₂³×C₅₃	—	2	2	no	—
76	C₂³×C₅₃	—	2	2	no	—
77	C₂³×C₅₃	—	2	2	no	—
78	C₂³×C₅₃	#102	1	1	yes	characteristic
79	C₂³×C₅₃	#102	1	1	yes	characteristic
80	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
88	C₂×C₄×C₅₃	—	4	4	no	—
89	C₂×C₄×C₅₃	—	8	4	no	—
90	C₂×C₄×C₅₃	—	8	4	no	—
91	C₂×C₄×C₅₃	—	8	4	no	—
92	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#123	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#137	1	2	yes	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	#117	1	2	yes	—
101	C₈×C₅₃	#123	1	2	yes	—
102	#88322	—	8	2	no	—
103	#12871	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	8	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	#361	1	2	yes	—
112	C₂²×C₅₃	#362	1	2	yes	—
113	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
114	C₂²×C₅₃	#483	1	2	yes	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	8	4	no	—
118	C₄×C₅₃	#381	1	1	yes	characteristic
119	C₄×C₅₃	#419	1	1	yes	characteristic
120	C₄×C₅₃	#480	1	2	yes	—
121	C₂²×#107	D₅₃	1	1	yes	characteristic
122	#2112	—	53	1	no	—
123	#1810	—	53	1	no	—
124	#1387	—	53	4	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	#1823	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
131	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
132	C₂²×C₄⋊C₄	—	2	1	no	—
133	C₂×M₃₂	—	53	2	no	—
134	C₂×#108	—	53	2	no	—
135	C₂×#107	—	2	1	no	—
136	C₂×#107	—	2	1	no	—
137	C₂×#107	—	2	1	no	—
138	C₂×#107	Dic₅₃	1	2	yes	—
139	C₂×#107	Dic₅₃	1	2	yes	—
140	C₂×#107	C₂×D₅₃	1	2	yes	—
141	#407	—	106	1	no	—
142	C₂²⋊C₁₆	—	106	1	no	—
143	C₂²⋊C₁₆	—	106	1	no	—
144	#341	—	106	1	no	—
145	C₅₃	#12871	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	—	4	1	no	—
147	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂×M₁₆	—	106	2	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
154	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	4	1	no	—
160	C₂×C₄⋊C₄	—	4	1	no	—
161	C₂×C₄⋊C₄	—	4	1	no	—
162	M₃₂	—	106	4	no	—
163	C₂×C₁₆	—	106	2	no	—
164	#108	—	106	4	no	—
165	#107	—	2	2	no	—
166	#107	—	2	2	no	—
167	#107	—	2	2	no	—
168	#107	C₂×Dic₅₃	1	4	yes	—
169	C₂²⋊C₈	—	212	1	no	—
170	C₂²⋊C₈	—	212	1	no	—
171	C₂⁴	#11719923	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	8	4	no	—
178	C₂²×C₄	#11719922	1	1	yes	characteristic
179	C₂²×C₄	#11719922	1	1	yes	characteristic
180	C₂²×C₄	#11719923	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	M₁₆	—	212	2	no	—
185	M₁₆	—	212	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	212	2	no	—
192	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
193	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
194	C₂×C₈	#11719945	1	4	yes	—
195	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₁₆	—	212	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	#13133138	1	1	yes	characteristic
206	C₂³	#13133138	1	1	yes	characteristic
207	C₂³	C₂×#11719923	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	#13133138	1	1	yes	characteristic
220	C₂×C₄	#13133138	1	1	yes	characteristic
221	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
222	C₂×C₄	#13133200	1	2	yes	—
223	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
224	C₂×C₄	#13133248	1	2	yes	—
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	424	2	no	—
228	C₈	#13133176	1	2	yes	—
229	C₈	#13133200	1	2	yes	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	#16810436	1	2	yes	—
238	C₂²	#16810439	1	2	yes	—
239	C₂²	C₂×#13133138	1	1	yes	characteristic
240	C₂²	#16810953	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	8	4	no	—
244	C₄	#16810486	1	1	yes	characteristic
245	C₄	#16810632	1	1	yes	characteristic
246	C₄	#16810936	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	2	2	no	—
250	C₂	#20274801	1	1	yes	characteristic
251	C₂	C₂×#16810436	1	1	yes	characteristic
252	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	C₂²×#107×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×#107	D₅₃
hypocenter	C₅₃	#12871
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #20272823	C₂	semidirect
145. C₅₃	#12871	semidirect

Maximal subgroups

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

subgroup	index
#20276326	2
#20274753	2
#20272823	2
C₂²×#107×C₅₃	2
#12871	53