Subgroups — C₂×C₅₃×#2812 #27244158

Order 13568.

4438 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#120×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2812	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅₃×#397	C₂	1	3	yes	maximal, complemented
4	C₂×C₅₃×#394	C₂	1	6	yes	maximal, complemented
5	C₂×C₅₃×#392	C₂	1	3	yes	maximal, complemented
6	C₂×C₅₃×#379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₅₃	C₂²	1	6	yes	—
9	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
10	C₂×#120×C₅₃	—	2	16	no	—
11	C₂×#120×C₅₃	C₂²	1	16	yes	complemented
12	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	complemented
13	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
14	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
15	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
16	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	complemented
17	C₅₃×#397	C₂²	1	24	yes	complemented
18	C₅₃×#394	C₂²	1	48	yes	complemented
19	C₅₃×#392	C₂²	1	24	yes	complemented
20	C₅₃×#379	C₂²	1	8	yes	complemented
21	C₂×#95×C₅₃	C₂²	1	3	yes	complemented
22	C₂×#95×C₅₃	C₂²	1	3	yes	complemented
23	C₂⁵×C₅₃	—	4	6	no	—
24	C₂⁵×C₅₃	—	4	6	no	—
25	C₂⁵×C₅₃	D₄	1	8	yes	complemented
26	C₂⁵×C₅₃	C₂³	1	2	yes	—
27	C₂⁵×C₅₃	C₂³	1	3	yes	—
28	C₂⁵×C₅₃	C₂³	1	4	yes	—
29	C₂²×D₄×C₅₃	—	2	24	no	—
30	C₂²×D₄×C₅₃	—	4	12	no	—
31	C₂²×D₄×C₅₃	—	4	12	no	—
32	C₂²×D₄×C₅₃	—	4	6	no	—
33	C₂²×D₄×C₅₃	C₂³	1	24	yes	—
34	C₂²×D₄×C₅₃	C₂³	1	4	yes	complemented
35	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
38	#120×C₅₃	—	2	64	no	—
39	C₂×C₄⋊C₄×C₅₃	—	4	6	no	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
41	C₂×C₂²⋊C₄×C₅₃	—	2	24	no	—
42	C₂×C₂²⋊C₄×C₅₃	—	4	6	no	—
43	C₂×C₂²⋊C₄×C₅₃	—	4	6	no	—
44	C₂×C₂²⋊C₄×C₅₃	—	4	12	no	—
45	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
46	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
47	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
48	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
49	#95×C₅₃	C₂³	1	12	yes	complemented
50	#95×C₅₃	C₂³	1	12	yes	complemented
51	C₂⁴×C₅₃	—	2	6	no	—
52	C₂⁴×C₅₃	—	2	6	no	—
53	C₂⁴×C₅₃	—	2	12	no	—
54	C₂⁴×C₅₃	—	2	12	no	—
55	C₂⁴×C₅₃	—	2	4	no	—
56	C₂⁴×C₅₃	—	2	12	no	—
57	C₂⁴×C₅₃	—	2	12	no	—
58	C₂⁴×C₅₃	—	4	48	no	—
59	C₂⁴×C₅₃	—	4	12	no	—
60	C₂⁴×C₅₃	—	4	12	no	—
61	C₂⁴×C₅₃	—	4	48	no	—
62	C₂⁴×C₅₃	—	4	16	no	—
63	C₂⁴×C₅₃	C₂×D₄	1	12	yes	—
64	C₂⁴×C₅₃	C₂×D₄	1	16	yes	complemented
65	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
66	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
67	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂×D₄×C₅₃	—	2	48	no	—
69	C₂×D₄×C₅₃	—	4	24	no	—
70	C₂×D₄×C₅₃	—	4	24	no	—
71	C₂×D₄×C₅₃	—	4	12	no	—
72	C₂×D₄×C₅₃	—	4	96	no	—
73	C₂×D₄×C₅₃	—	4	16	no	—
74	C₂²×C₄×C₅₃	—	2	12	no	—
75	C₂²×C₄×C₅₃	—	2	12	no	—
76	C₂²×C₄×C₅₃	—	2	6	no	—
77	C₂²×C₄×C₅₃	—	2	12	no	—
78	C₂²×C₄×C₅₃	—	2	6	no	—
79	C₂²×C₄×C₅₃	—	2	12	no	—
80	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
82	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
83	C₄⋊C₄×C₅₃	—	4	12	no	—
84	C₂²⋊C₄×C₅₃	—	2	48	no	—
85	C₂²⋊C₄×C₅₃	—	4	12	no	—
86	C₂²⋊C₄×C₅₃	—	4	12	no	—
87	C₂²⋊C₄×C₅₃	—	4	24	no	—
88	C₂³×C₅₃	—	2	48	no	—
89	C₂³×C₅₃	—	2	24	no	—
90	C₂³×C₅₃	—	2	12	no	—
91	C₂³×C₅₃	—	2	12	no	—
92	C₂³×C₅₃	—	2	6	no	—
93	C₂³×C₅₃	—	2	6	no	—
94	C₂³×C₅₃	—	2	4	no	—
95	C₂³×C₅₃	—	2	12	no	—
96	C₂³×C₅₃	—	2	12	no	—
97	C₂³×C₅₃	—	4	48	no	—
98	C₂³×C₅₃	—	4	48	no	—
99	C₂³×C₅₃	—	4	16	no	—
100	C₂³×C₅₃	—	4	128	no	—
101	C₂³×C₅₃	—	4	24	no	—
102	C₂³×C₅₃	—	4	12	no	—
103	C₂³×C₅₃	—	4	12	no	—
104	C₂³×C₅₃	—	4	24	no	—
105	C₂³×C₅₃	—	4	24	no	—
106	C₂³×C₅₃	—	4	8	no	—
107	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
108	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
109	C₂³×C₅₃	H(5,𝔽₂)	1	6	yes	—
110	C₂³×C₅₃	H(5,𝔽₂)	1	6	yes	—
111	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
112	D₄×C₅₃	—	4	96	no	—
113	D₄×C₅₃	—	4	16	no	—
114	C₂×C₄×C₅₃	—	2	12	no	—
115	C₂×C₄×C₅₃	—	2	12	no	—
116	C₂×C₄×C₅₃	—	2	6	no	—
117	C₂×C₄×C₅₃	—	2	12	no	—
118	C₂×C₄×C₅₃	—	2	12	no	—
119	C₂×C₄×C₅₃	—	2	6	no	—
120	C₂×C₄×C₅₃	—	4	24	no	—
121	C₂×C₄×C₅₃	—	4	12	no	—
122	C₂×C₄×C₅₃	—	4	4	no	—
123	C₂×#2812	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₅₃	—	2	24	no	—
125	C₂²×C₅₃	—	2	12	no	—
126	C₂²×C₅₃	—	4	64	no	—
127	C₂²×C₅₃	—	4	6	no	—
128	C₂²×C₅₃	—	4	12	no	—
129	C₂²×C₅₃	—	4	6	no	—
130	C₂²×C₅₃	—	4	48	no	—
131	C₂²×C₅₃	—	4	12	no	—
132	C₂²×C₅₃	—	4	12	no	—
133	C₂²×C₅₃	—	4	4	no	—
134	C₂²×C₅₃	—	8	16	no	—
135	C₂²×C₅₃	#534	1	12	yes	—
136	C₂²×C₅₃	#560	1	12	yes	—
137	C₂²×C₅₃	H(3,𝔽₄)	1	4	yes	—
138	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
139	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	3	yes	—
140	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	3	yes	—
141	C₄×C₅₃	—	4	12	no	—
142	C₄×C₅₃	—	4	6	no	—
143	C₄×C₅₃	—	4	2	no	—
144	C₂²×#120	C₂×C₅₃	1	2	yes	complemented
145	#2812	C₂×C₅₃	1	16	yes	direct product factor
146	C₂×#397	C₂×C₅₃	1	3	yes	complemented
147	C₂×#394	C₂×C₅₃	1	6	yes	complemented
148	C₂×#392	C₂×C₅₃	1	3	yes	complemented
149	C₂×#379	C₂×C₅₃	1	1	yes	characteristic, complemented
150	C₂×C₅₃	—	4	12	no	—
151	C₂×C₅₃	—	8	4	no	—
152	C₂×C₅₃	#2812	1	8	yes	direct product factor
153	C₂×C₅₃	C₂×#534	1	3	yes	—
154	C₂×C₅₃	C₂×#560	1	3	yes	—
155	C₂×C₅₃	C₂×H(3,𝔽₄)	1	1	yes	characteristic
156	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
157	C₂³×D₄	C₂²×C₅₃	1	6	yes	—
158	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
159	C₂×#120	—	2	16	no	—
160	C₂×#120	C₂²×C₅₃	1	16	yes	complemented
161	C₂²×C₄⋊C₄	C₂²×C₅₃	1	3	yes	complemented
162	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	—
163	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
164	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
165	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	complemented
166	#397	C₂²×C₅₃	1	24	yes	complemented
167	#394	C₂²×C₅₃	1	48	yes	complemented
168	#392	C₂²×C₅₃	1	24	yes	complemented
169	#379	C₂²×C₅₃	1	8	yes	complemented
170	C₂×#95	C₂²×C₅₃	1	3	yes	complemented
171	C₂×#95	C₂²×C₅₃	1	3	yes	complemented
172	C₅₃	C₂×#2812	1	1	yes	characteristic, direct product factor, Sylow
173	C₂⁵	—	4	6	no	—
174	C₂⁵	—	4	6	no	—
175	C₂⁵	D₄×C₅₃	1	8	yes	complemented
176	C₂⁵	C₂³×C₅₃	1	2	yes	—
177	C₂⁵	C₂³×C₅₃	1	3	yes	—
178	C₂⁵	C₂³×C₅₃	1	4	yes	—
179	C₂²×D₄	—	2	24	no	—
180	C₂²×D₄	—	4	12	no	—
181	C₂²×D₄	—	4	12	no	—
182	C₂²×D₄	—	4	6	no	—
183	C₂²×D₄	C₂³×C₅₃	1	24	yes	—
184	C₂²×D₄	C₂³×C₅₃	1	4	yes	complemented
185	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
186	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
188	#120	—	2	64	no	—
189	C₂×C₄⋊C₄	—	4	6	no	—
190	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	complemented
191	C₂×C₂²⋊C₄	—	2	24	no	—
192	C₂×C₂²⋊C₄	—	4	6	no	—
193	C₂×C₂²⋊C₄	—	4	6	no	—
194	C₂×C₂²⋊C₄	—	4	12	no	—
195	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	—
196	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
197	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
198	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	complemented
199	#95	C₂³×C₅₃	1	12	yes	complemented
200	#95	C₂³×C₅₃	1	12	yes	complemented
201	C₂⁴	—	2	6	no	—
202	C₂⁴	—	2	6	no	—
203	C₂⁴	—	2	12	no	—
204	C₂⁴	—	2	12	no	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	—	2	12	no	—
207	C₂⁴	—	2	12	no	—
208	C₂⁴	—	4	48	no	—
209	C₂⁴	—	4	12	no	—
210	C₂⁴	—	4	12	no	—
211	C₂⁴	—	4	48	no	—
212	C₂⁴	—	4	16	no	—
213	C₂⁴	C₂×D₄×C₅₃	1	12	yes	—
214	C₂⁴	C₂×D₄×C₅₃	1	16	yes	complemented
215	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
216	C₂⁴	C₂⁴×C₅₃	1	6	yes	—
217	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
218	C₂×D₄	—	2	48	no	—
219	C₂×D₄	—	4	24	no	—
220	C₂×D₄	—	4	24	no	—
221	C₂×D₄	—	4	12	no	—
222	C₂×D₄	—	4	96	no	—
223	C₂×D₄	—	4	16	no	—
224	C₂²×C₄	—	2	12	no	—
225	C₂²×C₄	—	2	12	no	—
226	C₂²×C₄	—	2	6	no	—
227	C₂²×C₄	—	2	12	no	—
228	C₂²×C₄	—	2	6	no	—
229	C₂²×C₄	—	2	12	no	—
230	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
231	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
232	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
233	C₄⋊C₄	—	4	12	no	—
234	C₂²⋊C₄	—	2	48	no	—
235	C₂²⋊C₄	—	4	12	no	—
236	C₂²⋊C₄	—	4	12	no	—
237	C₂²⋊C₄	—	4	24	no	—
238	C₂³	—	2	48	no	—
239	C₂³	—	2	24	no	—
240	C₂³	—	2	6	no	—
241	C₂³	—	2	6	no	—
242	C₂³	—	2	12	no	—
243	C₂³	—	2	12	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	2	12	no	—
246	C₂³	—	2	12	no	—
247	C₂³	—	4	48	no	—
248	C₂³	—	4	48	no	—
249	C₂³	—	4	16	no	—
250	C₂³	—	4	128	no	—
251	C₂³	—	4	24	no	—
252	C₂³	—	4	12	no	—
253	C₂³	—	4	12	no	—
254	C₂³	—	4	24	no	—
255	C₂³	—	4	24	no	—
256	C₂³	—	4	8	no	—
257	C₂³	C₂²×D₄×C₅₃	1	12	yes	—
258	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
259	C₂³	H(5,𝔽₂)×C₅₃	1	6	yes	—
260	C₂³	H(5,𝔽₂)×C₅₃	1	6	yes	—
261	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
262	D₄	—	4	96	no	—
263	D₄	—	4	16	no	—
264	C₂×C₄	—	2	12	no	—
265	C₂×C₄	—	2	12	no	—
266	C₂×C₄	—	2	6	no	—
267	C₂×C₄	—	2	6	no	—
268	C₂×C₄	—	2	12	no	—
269	C₂×C₄	—	2	12	no	—
270	C₂×C₄	—	4	24	no	—
271	C₂×C₄	—	4	12	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂²	—	2	24	no	—
274	C₂²	—	2	12	no	—
275	C₂²	—	4	64	no	—
276	C₂²	—	4	12	no	—
277	C₂²	—	4	6	no	—
278	C₂²	—	4	6	no	—
279	C₂²	—	4	48	no	—
280	C₂²	—	4	12	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	12	no	—
283	C₂²	—	8	16	no	—
284	C₂²	C₅₃×#534	1	12	yes	—
285	C₂²	C₅₃×#560	1	12	yes	—
286	C₂²	C₅₃×H(3,𝔽₄)	1	4	yes	—
287	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
288	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	3	yes	—
289	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	3	yes	—
290	C₄	—	4	6	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	12	no	—
293	C₂	—	4	12	no	—
294	C₂	—	8	4	no	—
295	C₂	C₅₃×#2812	1	8	yes	direct product factor
296	C₂	C₂×C₅₃×#534	1	3	yes	—
297	C₂	C₂×C₅₃×#560	1	3	yes	—
298	C₂	C₂×C₅₃×H(3,𝔽₄)	1	1	yes	characteristic

	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×#120×C₅₃	C₂	semidirect
2. C₅₃×#2812	C₂	direct (N × H)
3. C₂×C₅₃×#397	C₂	semidirect
4. C₂×C₅₃×#394	C₂	semidirect
5. C₂×C₅₃×#392	C₂	semidirect
6. C₂×C₅₃×#379	C₂	semidirect
9. C₂³×D₄×C₅₃	C₂²	semidirect
11. C₂×#120×C₅₃	C₂²	semidirect
12. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
16. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
17. C₅₃×#397	C₂²	semidirect
18. C₅₃×#394	C₂²	semidirect
19. C₅₃×#392	C₂²	semidirect
20. C₅₃×#379	C₂²	semidirect
21. C₂×#95×C₅₃	C₂²	semidirect
22. C₂×#95×C₅₃	C₂²	semidirect
25. C₂⁵×C₅₃	D₄	semidirect
34. C₂²×D₄×C₅₃	C₂³	semidirect
40. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
48. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
49. #95×C₅₃	C₂³	semidirect
50. #95×C₅₃	C₂³	semidirect
64. C₂⁴×C₅₃	C₂×D₄	semidirect
123. C₂×#2812	C₅₃	direct (N × H)
144. C₂²×#120	C₂×C₅₃	semidirect
145. #2812	C₂×C₅₃	direct (N × H)
146. C₂×#397	C₂×C₅₃	semidirect
147. C₂×#394	C₂×C₅₃	semidirect
148. C₂×#392	C₂×C₅₃	semidirect
149. C₂×#379	C₂×C₅₃	semidirect
152. C₂×C₅₃	#2812	direct (N × H)
158. C₂³×D₄	C₂²×C₅₃	semidirect
160. C₂×#120	C₂²×C₅₃	semidirect
161. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
165. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
166. #397	C₂²×C₅₃	semidirect
167. #394	C₂²×C₅₃	semidirect
168. #392	C₂²×C₅₃	semidirect
169. #379	C₂²×C₅₃	semidirect
170. C₂×#95	C₂²×C₅₃	semidirect
171. C₂×#95	C₂²×C₅₃	semidirect
172. C₅₃	C₂×#2812	direct (N × H)
175. C₂⁵	D₄×C₅₃	semidirect
184. C₂²×D₄	C₂³×C₅₃	semidirect
190. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
198. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
199. #95	C₂³×C₅₃	semidirect
200. #95	C₂³×C₅₃	semidirect
214. C₂⁴	C₂×D₄×C₅₃	semidirect
295. C₂	C₅₃×#2812	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×#120×C₅₃	2
C₅₃×#2812	2
C₂×C₅₃×#397	2
C₂×C₅₃×#394	2
C₂×C₅₃×#392	2
C₂×C₅₃×#379	2
C₂×#2812	53

Subgroups — C2×C53×#2812 #27244158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2812 #27244158