Subgroups — C₂×C₅₃×#2819 #27244192

Order 13568.

3342 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2819	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅₃×#394	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#393	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₃×#392	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
9	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
10	C₂×#124×C₅₃	—	2	16	no	—
11	C₂×#124×C₅₃	C₂²	1	16	yes	complemented
12	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
14	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#394	C₂²	1	32	yes	complemented
18	C₅₃×#393	C₂²	1	32	yes	complemented
19	C₅₃×#392	C₂²	1	32	yes	complemented
20	C₅₃×#382	C₂²	1	8	yes	complemented
21	C₂×#95×C₅₃	C₂²	1	4	yes	—
22	C₂×#95×C₅₃	C₂²	1	4	yes	—
23	C₂⁵×C₅₃	C₂³	1	4	yes	—
24	C₂²×Q₈×C₅₃	—	2	8	no	—
25	C₂²×Q₈×C₅₃	—	4	8	no	—
26	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
27	C₂²×D₄×C₅₃	—	2	8	no	—
28	C₂²×D₄×C₅₃	—	4	8	no	—
29	C₂²×D₄×C₅₃	—	4	16	no	—
30	C₂²×D₄×C₅₃	—	4	8	no	—
31	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
32	C₂²×D₄×C₅₃	C₂³	1	16	yes	—
33	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
37	#124×C₅₃	—	2	64	no	—
38	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
40	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
41	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
42	C₂×C₂²⋊C₄×C₅₃	—	4	16	no	—
43	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
44	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
45	C₂×C₂²⋊C₄×C₅₃	C₂³	1	32	yes	—
46	C₂×C₄²×C₅₃	D₄	1	8	yes	complemented
47	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
48	#95×C₅₃	C₂³	1	16	yes	—
49	#95×C₅₃	C₂³	1	16	yes	—
50	C₂⁴×C₅₃	—	2	8	no	—
51	C₂⁴×C₅₃	—	2	16	no	—
52	C₂⁴×C₅₃	—	2	16	no	—
53	C₂⁴×C₅₃	—	2	8	no	—
54	C₂⁴×C₅₃	—	2	8	no	—
55	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
56	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂×Q₈×C₅₃	—	2	16	no	—
58	C₂×Q₈×C₅₃	—	4	16	no	—
59	C₂×Q₈×C₅₃	—	4	32	no	—
60	C₂×D₄×C₅₃	—	2	16	no	—
61	C₂×D₄×C₅₃	—	4	16	no	—
62	C₂×D₄×C₅₃	—	4	32	no	—
63	C₂×D₄×C₅₃	—	4	16	no	—
64	C₂×D₄×C₅₃	—	4	32	no	—
65	C₂×D₄×C₅₃	—	4	64	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	8	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	8	no	—
70	C₂²×C₄×C₅₃	—	2	16	no	—
71	C₂²×C₄×C₅₃	—	2	16	no	—
72	C₂²×C₄×C₅₃	—	2	8	no	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
77	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
78	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
79	C₄⋊C₄×C₅₃	—	4	8	no	—
80	C₂²⋊C₄×C₅₃	—	2	32	no	—
81	C₂²⋊C₄×C₅₃	—	2	32	no	—
82	C₂²⋊C₄×C₅₃	—	4	32	no	—
83	C₄²×C₅₃	C₂×D₄	1	16	yes	complemented
84	C₂³×C₅₃	—	2	8	no	—
85	C₂³×C₅₃	—	2	16	no	—
86	C₂³×C₅₃	—	2	8	no	—
87	C₂³×C₅₃	—	2	16	no	—
88	C₂³×C₅₃	—	2	8	no	—
89	C₂³×C₅₃	—	4	16	no	—
90	C₂³×C₅₃	—	4	16	no	—
91	C₂³×C₅₃	—	4	16	no	—
92	C₂³×C₅₃	—	4	32	no	—
93	C₂³×C₅₃	—	4	32	no	—
94	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
95	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
96	C₂³×C₅₃	H(5,𝔽₂)	1	2	yes	—
97	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
98	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
99	Q₈×C₅₃	—	4	32	no	—
100	D₄×C₅₃	—	4	32	no	—
101	D₄×C₅₃	—	4	64	no	—
102	C₂×C₄×C₅₃	—	2	16	no	—
103	C₂×C₄×C₅₃	—	2	8	no	—
104	C₂×C₄×C₅₃	—	2	8	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	8	no	—
108	C₂×C₄×C₅₃	—	2	16	no	—
109	C₂×C₄×C₅₃	—	2	8	no	—
110	C₂×C₄×C₅₃	—	2	16	no	—
111	C₂×C₄×C₅₃	—	4	16	no	—
112	C₂×C₄×C₅₃	—	4	16	no	—
113	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
114	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
115	C₂×#2819	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₅₃	—	4	8	no	—
117	C₂²×C₅₃	—	4	8	no	—
118	C₂²×C₅₃	—	4	8	no	—
119	C₂²×C₅₃	—	4	16	no	—
120	C₂²×C₅₃	—	4	16	no	—
121	C₂²×C₅₃	—	8	32	no	—
122	C₂²×C₅₃	#535	1	8	yes	—
123	C₂²×C₅₃	#537	1	4	yes	—
124	C₂²×C₅₃	H(3,𝔽₄)	1	8	yes	—
125	C₂²×C₅₃	#562	1	8	yes	—
126	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
128	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
129	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
130	C₄×C₅₃	—	2	8	no	—
131	C₄×C₅₃	—	2	4	no	—
132	C₄×C₅₃	—	4	8	no	—
133	C₄×C₅₃	—	4	8	no	—
134	C₂²×#124	C₂×C₅₃	1	2	yes	complemented
135	#2819	C₂×C₅₃	1	16	yes	direct product factor
136	C₂×#394	C₂×C₅₃	1	4	yes	complemented
137	C₂×#393	C₂×C₅₃	1	4	yes	complemented
138	C₂×#392	C₂×C₅₃	1	4	yes	complemented
139	C₂×#382	C₂×C₅₃	1	1	yes	characteristic, complemented
140	C₂×C₅₃	—	8	8	no	—
141	C₂×C₅₃	#2819	1	8	yes	direct product factor
142	C₂×C₅₃	C₂×#535	1	2	yes	—
143	C₂×C₅₃	C₂×#537	1	1	yes	characteristic
144	C₂×C₅₃	C₂×H(3,𝔽₄)	1	2	yes	—
145	C₂×C₅₃	C₂×#562	1	2	yes	—
146	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
147	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
148	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
149	C₂×#124	—	2	16	no	—
150	C₂×#124	C₂²×C₅₃	1	16	yes	complemented
151	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
152	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
153	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
154	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
155	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
156	#394	C₂²×C₅₃	1	32	yes	complemented
157	#393	C₂²×C₅₃	1	32	yes	complemented
158	#392	C₂²×C₅₃	1	32	yes	complemented
159	#382	C₂²×C₅₃	1	8	yes	complemented
160	C₂×#95	C₂²×C₅₃	1	4	yes	—
161	C₂×#95	C₂²×C₅₃	1	4	yes	—
162	C₅₃	C₂×#2819	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁵	C₂³×C₅₃	1	4	yes	—
164	C₂²×Q₈	—	2	8	no	—
165	C₂²×Q₈	—	4	8	no	—
166	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
167	C₂²×D₄	—	2	8	no	—
168	C₂²×D₄	—	4	8	no	—
169	C₂²×D₄	—	4	16	no	—
170	C₂²×D₄	—	4	8	no	—
171	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
172	C₂²×D₄	C₂³×C₅₃	1	16	yes	—
173	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
174	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
175	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
176	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
177	#124	—	2	64	no	—
178	C₂×C₄⋊C₄	—	4	4	no	—
179	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
180	C₂×C₂²⋊C₄	—	2	16	no	—
181	C₂×C₂²⋊C₄	—	2	16	no	—
182	C₂×C₂²⋊C₄	—	4	16	no	—
183	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
184	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
185	C₂×C₂²⋊C₄	C₂³×C₅₃	1	32	yes	—
186	C₂×C₄²	D₄×C₅₃	1	8	yes	complemented
187	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
188	#95	C₂³×C₅₃	1	16	yes	—
189	#95	C₂³×C₅₃	1	16	yes	—
190	C₂⁴	—	2	8	no	—
191	C₂⁴	—	2	16	no	—
192	C₂⁴	—	2	16	no	—
193	C₂⁴	—	2	8	no	—
194	C₂⁴	—	2	8	no	—
195	C₂⁴	C₂⁴×C₅₃	1	8	yes	—
196	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
197	C₂×Q₈	—	2	16	no	—
198	C₂×Q₈	—	4	16	no	—
199	C₂×Q₈	—	4	32	no	—
200	C₂×D₄	—	2	16	no	—
201	C₂×D₄	—	4	16	no	—
202	C₂×D₄	—	4	32	no	—
203	C₂×D₄	—	4	16	no	—
204	C₂×D₄	—	4	32	no	—
205	C₂×D₄	—	4	64	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	16	no	—
211	C₂²×C₄	—	2	16	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
214	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
215	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
216	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
217	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
218	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
219	C₄⋊C₄	—	4	8	no	—
220	C₂²⋊C₄	—	2	32	no	—
221	C₂²⋊C₄	—	2	32	no	—
222	C₂²⋊C₄	—	4	32	no	—
223	C₄²	C₂×D₄×C₅₃	1	16	yes	complemented
224	C₂³	—	2	8	no	—
225	C₂³	—	2	16	no	—
226	C₂³	—	2	8	no	—
227	C₂³	—	2	16	no	—
228	C₂³	—	2	8	no	—
229	C₂³	—	4	16	no	—
230	C₂³	—	4	16	no	—
231	C₂³	—	4	16	no	—
232	C₂³	—	4	32	no	—
233	C₂³	—	4	32	no	—
234	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
235	C₂³	H(5,𝔽₂)×C₅₃	1	8	yes	—
236	C₂³	H(5,𝔽₂)×C₅₃	1	2	yes	—
237	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
238	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
239	Q₈	—	4	32	no	—
240	D₄	—	4	32	no	—
241	D₄	—	4	64	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	16	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	16	no	—
252	C₂×C₄	—	4	16	no	—
253	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
254	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
255	C₂²	—	4	8	no	—
256	C₂²	—	4	16	no	—
257	C₂²	—	4	16	no	—
258	C₂²	—	4	8	no	—
259	C₂²	—	4	8	no	—
260	C₂²	—	8	32	no	—
261	C₂²	C₅₃×#535	1	8	yes	—
262	C₂²	C₅₃×#537	1	4	yes	—
263	C₂²	C₅₃×H(3,𝔽₄)	1	8	yes	—
264	C₂²	C₅₃×#562	1	8	yes	—
265	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
266	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
267	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	4	yes	—
268	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	2	8	no	—
271	C₄	—	4	8	no	—
272	C₄	—	4	8	no	—
273	C₂	—	8	8	no	—
274	C₂	C₅₃×#2819	1	8	yes	direct product factor
275	C₂	C₂×C₅₃×#535	1	2	yes	—
276	C₂	C₂×C₅₃×#537	1	1	yes	characteristic
277	C₂	C₂×C₅₃×H(3,𝔽₄)	1	2	yes	—
278	C₂	C₂×C₅₃×#562	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#124×C₅₃	C₂	semidirect
2. C₅₃×#2819	C₂	direct (N × H)
3. C₂×C₅₃×#394	C₂	semidirect
4. C₂×C₅₃×#393	C₂	semidirect
5. C₂×C₅₃×#392	C₂	semidirect
6. C₂×C₅₃×#382	C₂	semidirect
11. C₂×#124×C₅₃	C₂²	semidirect
17. C₅₃×#394	C₂²	semidirect
18. C₅₃×#393	C₂²	semidirect
19. C₅₃×#392	C₂²	semidirect
20. C₅₃×#382	C₂²	semidirect
46. C₂×C₄²×C₅₃	D₄	semidirect
83. C₄²×C₅₃	C₂×D₄	semidirect
115. C₂×#2819	C₅₃	direct (N × H)
134. C₂²×#124	C₂×C₅₃	semidirect
135. #2819	C₂×C₅₃	direct (N × H)
136. C₂×#394	C₂×C₅₃	semidirect
137. C₂×#393	C₂×C₅₃	semidirect
138. C₂×#392	C₂×C₅₃	semidirect
139. C₂×#382	C₂×C₅₃	semidirect
141. C₂×C₅₃	#2819	direct (N × H)
150. C₂×#124	C₂²×C₅₃	semidirect
156. #394	C₂²×C₅₃	semidirect
157. #393	C₂²×C₅₃	semidirect
158. #392	C₂²×C₅₃	semidirect
159. #382	C₂²×C₅₃	semidirect
162. C₅₃	C₂×#2819	direct (N × H)
186. C₂×C₄²	D₄×C₅₃	semidirect
223. C₄²	C₂×D₄×C₅₃	semidirect
274. C₂	C₅₃×#2819	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#124×C₅₃	2
C₅₃×#2819	2
C₂×C₅₃×#394	2
C₂×C₅₃×#393	2
C₂×C₅₃×#392	2
C₂×C₅₃×#382	2
C₂×#2819	53

Subgroups — C2×C53×#2819 #27244192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2819 #27244192