Subgroups — C₅₃×#12824 #27220336

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#360	C₂	1	2	yes	maximal
4	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1383	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#131×C₅₃	—	2	1	no	—
8	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#443	—	2	2	no	—
10	C₅₃×#443	C₄	1	4	yes	complemented
11	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#107×C₅₃	—	2	1	no	—
13	C₂×#106×C₅₃	C₂²	1	2	yes	—
14	C₂×#104×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#382	—	2	1	no	—
17	C₅₃×#364	—	2	2	no	—
18	C₅₃×#360	—	2	4	no	—
19	C₂×#41×C₅₃	—	2	1	no	—
20	#135×C₅₃	—	2	1	no	—
21	#135×C₅₃	—	2	1	no	—
22	#135×C₅₃	C₂×C₄	1	2	yes	—
23	#135×C₅₃	D₄	1	2	yes	—
24	C₂×Q₁₆×C₅₃	Q₈	1	1	yes	characteristic
25	C₂×SD₁₆×C₅₃	—	2	1	no	—
26	C₂×D₈×C₅₃	Q₈	1	1	yes	characteristic
27	#131×C₅₃	—	2	4	no	—
28	#131×C₅₃	—	4	2	no	—
29	C₂×M₁₆×C₅₃	—	2	1	no	—
30	C₂×M₁₆×C₅₃	—	4	1	no	—
31	C₂²×C₈×C₅₃	—	4	1	no	—
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×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	M₃₂×C₅₃	—	2	2	no	—
37	M₃₂×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	#108×C₅₃	—	2	1	no	—
40	#108×C₅₃	C₂×C₄	1	2	yes	—
41	#107×C₅₃	—	4	2	no	—
42	#106×C₅₃	—	2	2	no	—
43	#106×C₅₃	—	2	2	no	—
44	#104×C₅₃	—	2	4	no	—
45	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	#95×C₅₃	—	4	2	no	—
48	#41×C₅₃	—	2	2	no	—
49	#41×C₅₃	—	2	2	no	—
50	#41×C₅₃	—	4	1	no	—
51	#41×C₅₃	—	4	1	no	—
52	C₂×Q₈×C₅₃	—	2	1	no	—
53	C₂×D₄×C₅₃	—	2	1	no	—
54	C₂×D₄×C₅₃	—	4	1	no	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	—	4	1	no	—
57	C₂²×C₄×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	Q₁₆×C₅₃	—	2	1	no	—
60	Q₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
61	SD₁₆×C₅₃	—	2	2	no	—
62	SD₁₆×C₅₃	—	2	2	no	—
63	D₈×C₅₃	—	2	1	no	—
64	D₈×C₅₃	C₄⋊C₄	1	2	yes	—
65	M₁₆×C₅₃	—	2	2	no	—
66	M₁₆×C₅₃	—	4	1	no	—
67	M₁₆×C₅₃	—	4	4	no	—
68	C₂×C₈×C₅₃	—	4	1	no	—
69	C₂×C₈×C₅₃	—	4	4	no	—
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₅₃	D₈	1	2	yes	—
72	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
73	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
80	C₄⋊C₄×C₅₃	—	4	2	no	—
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₄²×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
84	C₁₆×C₅₃	—	4	2	no	—
85	C₂³×C₅₃	—	4	1	no	—
86	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	Q₈×C₅₃	—	2	2	no	—
88	Q₈×C₅₃	—	4	1	no	—
89	D₄×C₅₃	—	2	2	no	—
90	D₄×C₅₃	—	4	2	no	—
91	D₄×C₅₃	—	4	2	no	—
92	D₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	1	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	8	4	no	—
99	C₂×C₄×C₅₃	—	8	2	no	—
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
106	C₈×C₅₃	—	4	2	no	—
107	C₈×C₅₃	#102	1	2	yes	—
108	C₈×C₅₃	#102	1	2	yes	—
109	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₈×C₅₃	#117	1	1	yes	characteristic
111	C₈×C₅₃	#121	1	1	yes	characteristic
112	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
113	#12824	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	8	2	no	—
117	C₂²×C₅₃	#417	1	1	yes	characteristic
118	C₂²×C₅₃	#459	1	1	yes	characteristic
119	C₂²×C₅₃	#476	1	1	yes	characteristic
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	—	4	2	no	—
122	C₄×C₅₃	—	16	2	no	—
123	C₄×C₅₃	#380	1	1	yes	characteristic
124	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
125	C₄×C₅₃	#461	1	1	yes	characteristic
126	C₄×C₅₃	#474	1	1	yes	characteristic
127	C₂×#443	C₂×C₅₃	1	1	yes	characteristic
128	C₂×#364	C₂×C₅₃	1	1	yes	characteristic, complemented
129	C₂×#360	C₂×C₅₃	1	2	yes	—
130	#1846	C₂×C₅₃	1	1	yes	characteristic, complemented
131	#1383	C₂×C₅₃	1	2	yes	complemented
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	8	2	no	—
135	C₂×C₅₃	#1802	1	1	yes	characteristic
136	C₂×C₅₃	#2231	1	2	yes	—
137	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
138	C₂×#131	—	2	1	no	—
139	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
140	#443	—	2	2	no	—
141	#443	C₄×C₅₃	1	4	yes	complemented
142	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
143	C₂×#107	—	2	1	no	—
144	C₂×#106	C₂²×C₅₃	1	2	yes	—
145	C₂×#104	—	2	1	no	—
146	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
147	#382	—	2	1	no	—
148	#364	—	2	2	no	—
149	#360	—	2	4	no	—
150	C₅₃	#12824	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#41	—	2	1	no	—
152	#135	—	2	1	no	—
153	#135	—	2	1	no	—
154	#135	C₂×C₄×C₅₃	1	2	yes	—
155	#135	D₄×C₅₃	1	2	yes	—
156	C₂×Q₁₆	Q₈×C₅₃	1	1	yes	characteristic
157	C₂×SD₁₆	—	2	1	no	—
158	C₂×D₈	Q₈×C₅₃	1	1	yes	characteristic
159	#131	—	2	4	no	—
160	#131	—	4	2	no	—
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	—	4	1	no	—
163	C₂²×C₈	—	4	1	no	—
164	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
168	M₃₂	—	2	2	no	—
169	M₃₂	—	2	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	#108	—	2	1	no	—
172	#108	C₂×C₄×C₅₃	1	2	yes	—
173	#107	—	4	2	no	—
174	#106	—	2	2	no	—
175	#106	—	2	2	no	—
176	#104	—	2	4	no	—
177	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
178	C₄×C₈	D₄×C₅₃	1	2	yes	—
179	#95	—	4	2	no	—
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	#41	—	4	1	no	—
183	#41	—	4	1	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
191	Q₁₆	—	2	1	no	—
192	Q₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
193	SD₁₆	—	2	2	no	—
194	SD₁₆	—	2	2	no	—
195	D₈	—	2	1	no	—
196	D₈	C₄⋊C₄×C₅₃	1	2	yes	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	4	1	no	—
199	M₁₆	—	4	4	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
203	C₂×C₈	D₈×C₅₃	1	2	yes	—
204	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
205	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
206	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
209	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
210	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
211	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄²	—	2	1	no	—
215	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
216	C₁₆	—	4	2	no	—
217	C₂³	—	4	1	no	—
218	C₂³	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
219	Q₈	—	2	2	no	—
220	Q₈	—	4	1	no	—
221	D₄	—	2	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
237	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
238	C₈	—	4	2	no	—
239	C₈	#102×C₅₃	1	2	yes	—
240	C₈	#102×C₅₃	1	2	yes	—
241	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
242	C₈	#117×C₅₃	1	1	yes	characteristic
243	C₈	#121×C₅₃	1	1	yes	characteristic
244	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	C₅₃×#417	1	1	yes	characteristic
249	C₂²	C₅₃×#459	1	1	yes	characteristic
250	C₂²	C₅₃×#476	1	1	yes	characteristic
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	16	2	no	—
254	C₄	C₅₃×#380	1	1	yes	characteristic
255	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
256	C₄	C₅₃×#461	1	1	yes	characteristic
257	C₄	C₅₃×#474	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	8	2	no	—
261	C₂	C₅₃×#1802	1	1	yes	characteristic
262	C₂	C₅₃×#2231	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	#476
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×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×#364	C₂	semidirect
4. C₅₃×#1846	C₂	semidirect
5. C₅₃×#1383	C₂	semidirect
10. C₅₃×#443	C₄	semidirect
113. #12824	C₅₃	direct (N × H)
128. C₂×#364	C₂×C₅₃	semidirect
130. #1846	C₂×C₅₃	semidirect
131. #1383	C₂×C₅₃	semidirect
141. #443	C₄×C₅₃	semidirect
150. C₅₃	#12824	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×#443	2
C₂×C₅₃×#364	2
C₂×C₅₃×#360	2
C₅₃×#1846	2
C₅₃×#1383	2
#12824	53

Subgroups — C53×#12824 #27220336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12824 #27220336