Subgroups — C₂×C₅₃×#1361 #27219439

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1361	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₈×C₅₃	—	2	1	no	—
6	C₂²×D₈×C₅₃	C₄	1	2	yes	—
7	C₂×#127×C₅₃	—	2	1	no	—
8	C₂²×C₁₆×C₅₃	—	2	1	no	—
9	C₅₃×#493	—	2	1	no	—
10	C₅₃×#493	—	2	1	no	—
11	C₅₃×#493	C₄	1	2	yes	—
12	C₅₃×#493	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×C₄⋊C₁₆	—	2	1	no	—
16	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
17	C₅₃×#335	—	2	1	no	—
18	C₅₃×#335	C₂²	1	2	yes	complemented
19	C₂²×D₄×C₅₃	—	2	2	no	—
20	C₂²×D₄×C₅₃	—	4	1	no	—
21	C₂²×D₄×C₅₃	—	4	1	no	—
22	C₂×D₈×C₅₃	—	2	2	no	—
23	C₂×D₈×C₅₃	—	2	2	no	—
24	C₂×D₈×C₅₃	—	2	2	no	—
25	C₂×D₈×C₅₃	—	4	2	no	—
26	C₂×D₈×C₅₃	—	4	2	no	—
27	C₂×D₈×C₅₃	—	4	4	no	—
28	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₅₃	—	4	1	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	#127×C₅₃	—	2	2	no	—
33	#127×C₅₃	—	2	2	no	—
34	#127×C₅₃	—	4	1	no	—
35	#127×C₅₃	—	4	1	no	—
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	—	4	2	no	—
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	—	4	1	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	C₂⁴×C₅₃	—	4	2	no	—
46	C₂×D₄×C₅₃	—	2	4	no	—
47	C₂×D₄×C₅₃	—	4	4	no	—
48	C₂×D₄×C₅₃	—	4	4	no	—
49	C₂×D₄×C₅₃	—	4	2	no	—
50	C₂×D₄×C₅₃	—	4	2	no	—
51	C₂×D₄×C₅₃	—	4	2	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	8	4	no	—
55	C₂×D₄×C₅₃	—	8	2	no	—
56	C₂×D₄×C₅₃	—	8	2	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
60	D₈×C₅₃	—	4	4	no	—
61	D₈×C₅₃	—	4	4	no	—
62	D₈×C₅₃	—	4	4	no	—
63	D₈×C₅₃	—	4	2	no	—
64	D₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	8	2	no	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	D₈	1	2	yes	—
71	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₅₃	D₈	1	1	yes	characteristic
76	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₅₃	—	4	2	no	—
79	C₂³×C₅₃	—	4	4	no	—
80	C₂³×C₅₃	—	8	4	no	—
81	C₂³×C₅₃	—	8	4	no	—
82	C₂³×C₅₃	—	8	4	no	—
83	C₂³×C₅₃	#102	1	1	yes	characteristic
84	D₄×C₅₃	—	4	4	no	—
85	D₄×C₅₃	—	4	4	no	—
86	D₄×C₅₃	—	8	4	no	—
87	D₄×C₅₃	—	8	4	no	—
88	D₄×C₅₃	—	8	4	no	—
89	D₄×C₅₃	—	8	4	no	—
90	D₄×C₅₃	—	8	2	no	—
91	D₄×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	#100	1	2	yes	—
94	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#102	1	2	yes	—
97	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
103	C₈×C₅₃	—	2	1	no	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	8	2	no	—
106	C₈×C₅₃	#102	1	2	yes	—
107	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
108	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×#1361	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₅₃	—	8	4	no	—
111	C₂²×C₅₃	—	8	4	no	—
112	C₂²×C₅₃	—	8	4	no	—
113	C₂²×C₅₃	—	16	8	no	—
114	C₂²×C₅₃	#331	1	2	yes	—
115	C₂²×C₅₃	#357	1	2	yes	—
116	C₂²×C₅₃	#361	1	2	yes	—
117	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	#331	1	2	yes	—
121	C₄×C₅₃	#357	1	2	yes	—
122	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
123	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
124	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
125	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
126	C₂×#493	C₂×C₅₃	1	1	yes	characteristic
127	C₂×C₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
128	C₂×#335	C₂×C₅₃	1	1	yes	characteristic, complemented
129	#1361	C₂×C₅₃	1	4	yes	direct product factor
130	C₂×C₅₃	—	16	4	no	—
131	C₂×C₅₃	#1361	1	4	yes	direct product factor
132	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
133	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
135	C₂²×D₈	—	2	1	no	—
136	C₂²×D₈	C₄×C₅₃	1	2	yes	—
137	C₂×#127	—	2	1	no	—
138	C₂²×C₁₆	—	2	1	no	—
139	#493	—	2	1	no	—
140	#493	—	2	1	no	—
141	#493	C₄×C₅₃	1	2	yes	—
142	#493	C₂²×C₅₃	1	2	yes	—
143	C₂×C₄⋊C₈	—	2	1	no	—
144	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
145	C₄⋊C₁₆	—	2	1	no	—
146	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	complemented
147	#335	—	2	1	no	—
148	#335	C₂²×C₅₃	1	2	yes	complemented
149	C₅₃	C₂×#1361	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×D₄	—	2	2	no	—
151	C₂²×D₄	—	4	1	no	—
152	C₂²×D₄	—	4	1	no	—
153	C₂×D₈	—	2	2	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂×D₈	—	4	4	no	—
157	C₂×D₈	—	4	2	no	—
158	C₂×D₈	—	4	2	no	—
159	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
160	C₂²×C₈	—	4	1	no	—
161	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
163	#127	—	2	2	no	—
164	#127	—	2	2	no	—
165	#127	—	4	1	no	—
166	#127	—	4	1	no	—
167	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	4	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	4	1	no	—
172	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
173	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
174	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
175	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
176	C₂⁴	—	4	2	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	4	4	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	8	4	no	—
186	C₂×D₄	—	8	2	no	—
187	C₂×D₄	—	8	2	no	—
188	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
190	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
191	D₈	—	4	4	no	—
192	D₈	—	4	4	no	—
193	D₈	—	4	4	no	—
194	D₈	—	4	2	no	—
195	D₈	—	4	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	8	2	no	—
199	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
201	C₂×C₈	D₈×C₅₃	1	2	yes	—
202	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
203	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
205	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
206	C₄²	D₈×C₅₃	1	1	yes	characteristic
207	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
209	C₁₆	—	4	2	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	8	4	no	—
212	C₂³	—	8	4	no	—
213	C₂³	—	8	4	no	—
214	C₂³	#102×C₅₃	1	1	yes	characteristic
215	D₄	—	4	4	no	—
216	D₄	—	4	4	no	—
217	D₄	—	8	4	no	—
218	D₄	—	8	4	no	—
219	D₄	—	8	2	no	—
220	D₄	—	8	2	no	—
221	D₄	—	8	4	no	—
222	D₄	—	8	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	#100×C₅₃	1	2	yes	—
225	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
226	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
227	C₂×C₄	#102×C₅₃	1	2	yes	—
228	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
229	C₂×C₄	D₁₆×C₅₃	1	2	yes	—
230	C₂×C₄	SD₃₂×C₅₃	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	8	2	no	—
237	C₈	#102×C₅₃	1	2	yes	—
238	C₈	C₂×D₈×C₅₃	1	1	yes	characteristic
239	C₈	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
240	C₂²	—	8	4	no	—
241	C₂²	—	8	4	no	—
242	C₂²	—	8	4	no	—
243	C₂²	—	16	8	no	—
244	C₂²	C₅₃×#331	1	2	yes	—
245	C₂²	C₅₃×#357	1	2	yes	—
246	C₂²	C₅₃×#361	1	2	yes	—
247	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	C₅₃×#331	1	2	yes	—
251	C₄	C₅₃×#357	1	2	yes	—
252	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
253	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
254	C₄	C₂×D₁₆×C₅₃	1	1	yes	characteristic
255	C₄	C₂×SD₃₂×C₅₃	1	1	yes	characteristic
256	C₂	—	16	4	no	—
257	C₂	C₅₃×#1361	1	4	yes	direct product factor
258	C₂	C₂×C₅₃×#331	1	1	yes	characteristic
259	C₂	C₂×C₅₃×#357	1	1	yes	characteristic
260	C₂	C₂×C₅₃×#361	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₂×C₅₃×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₅₃×#335	C₂	semidirect
4. C₅₃×#1361	C₂	direct (N × H)
16. C₅₃×C₄⋊C₁₆	C₂²	semidirect
18. C₅₃×#335	C₂²	semidirect
109. C₂×#1361	C₅₃	direct (N × H)
127. C₂×C₄⋊C₁₆	C₂×C₅₃	semidirect
128. C₂×#335	C₂×C₅₃	semidirect
129. #1361	C₂×C₅₃	direct (N × H)
131. C₂×C₅₃	#1361	direct (N × H)
146. C₄⋊C₁₆	C₂²×C₅₃	semidirect
148. #335	C₂²×C₅₃	semidirect
149. C₅₃	C₂×#1361	direct (N × H)
257. C₂	C₅₃×#1361	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#493	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×C₅₃×#335	2
C₅₃×#1361	2
C₂×#1361	53

Subgroups — C2×C53×#1361 #27219439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1361 #27219439