Subgroups — C₂×C₅₃×#1708 #27227107

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#492	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#487	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#486	C₂	1	2	yes	maximal
5	C₅₃×#1708	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×SD₁₆×C₅₃	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#498	—	2	2	no	—
12	C₅₃×#498	C₂²	1	4	yes	complemented
13	C₅₃×#492	—	2	2	no	—
14	C₅₃×#492	C₂²	1	4	yes	—
15	C₅₃×#487	—	2	4	no	—
16	C₅₃×#487	C₂²	1	8	yes	complemented
17	C₅₃×#486	—	2	4	no	—
18	C₅₃×#486	C₂²	1	8	yes	—
19	C₂×#106×C₅₃	—	2	2	no	—
20	C₂×#103×C₅₃	—	2	4	no	—
21	C₂×#102×C₅₃	—	2	4	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₈²×C₅₃	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₅₃	—	2	1	no	—
26	C₂²×Q₈×C₅₃	—	2	2	no	—
27	C₂²×D₄×C₅₃	—	2	1	no	—
28	C₂²×D₄×C₅₃	—	2	2	no	—
29	C₂×SD₁₆×C₅₃	—	2	8	no	—
30	C₂×SD₁₆×C₅₃	—	4	8	no	—
31	C₂²×C₈×C₅₃	D₄	1	4	yes	—
32	C₂²×C₈×C₅₃	D₄	1	2	yes	—
33	C₄⋊Q₈×C₅₃	—	2	2	no	—
34	C₄⋊Q₈×C₅₃	—	2	1	no	—
35	C₄⋊Q₈×C₅₃	—	2	2	no	—
36	C₄⋊Q₈×C₅₃	—	2	2	no	—
37	C₄⋊Q₈×C₅₃	—	2	2	no	—
38	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
40	#127×C₅₃	—	2	2	no	—
41	#127×C₅₃	—	2	1	no	—
42	#127×C₅₃	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	#106×C₅₃	—	2	8	no	—
48	#103×C₅₃	—	2	16	no	—
49	#102×C₅₃	—	2	16	no	—
50	C₄×C₈×C₅₃	D₄	1	4	yes	—
51	C₄×C₈×C₅₃	D₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	4	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂⁴×C₅₃	—	4	1	no	—
55	C₂×Q₈×C₅₃	—	2	4	no	—
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	—	4	1	no	—
58	C₂×Q₈×C₅₃	—	4	2	no	—
59	C₂×Q₈×C₅₃	—	4	4	no	—
60	C₂×Q₈×C₅₃	—	4	8	no	—
61	C₂×D₄×C₅₃	—	2	4	no	—
62	C₂×D₄×C₅₃	—	2	2	no	—
63	C₂×D₄×C₅₃	—	4	1	no	—
64	C₂×D₄×C₅₃	—	4	2	no	—
65	C₂×D₄×C₅₃	—	4	4	no	—
66	C₂×D₄×C₅₃	—	4	8	no	—
67	C₂²×C₄×C₅₃	—	4	1	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	SD₁₆×C₅₃	—	4	16	no	—
72	C₂×C₈×C₅₃	—	2	8	no	—
73	C₂×C₈×C₅₃	SD₁₆	1	8	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	—	4	2	no	—
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₄²×C₅₃	C₂×D₄	1	2	yes	—
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	—	8	2	no	—
87	C₂³×C₅₃	—	8	4	no	—
88	C₂³×C₅₃	#127	1	1	yes	characteristic
89	Q₈×C₅₃	—	4	4	no	—
90	Q₈×C₅₃	—	4	8	no	—
91	Q₈×C₅₃	—	8	2	no	—
92	Q₈×C₅₃	—	8	4	no	—
93	D₄×C₅₃	—	4	4	no	—
94	D₄×C₅₃	—	4	8	no	—
95	D₄×C₅₃	—	8	2	no	—
96	D₄×C₅₃	—	8	4	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	4	no	—
99	C₂×C₄×C₅₃	—	8	2	no	—
100	C₂×C₄×C₅₃	—	8	4	no	—
101	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#127	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
104	C₂×C₄×C₅₃	#135	1	8	yes	—
105	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
107	C₈×C₅₃	—	2	8	no	—
108	C₈×C₅₃	C₂×SD₁₆	1	8	yes	—
109	C₂×#1708	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₅₃	—	8	2	no	—
111	C₂²×C₅₃	—	8	4	no	—
112	C₂²×C₅₃	—	16	4	no	—
113	C₂²×C₅₃	#492	1	2	yes	—
114	C₂²×C₅₃	#495	1	4	yes	—
115	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
116	C₄×C₅₃	—	8	2	no	—
117	C₄×C₅₃	—	8	4	no	—
118	C₄×C₅₃	#492	1	2	yes	—
119	C₄×C₅₃	#495	1	4	yes	—
120	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
121	C₄×C₅₃	C₂×#135	1	4	yes	—
122	C₂×#498	C₂×C₅₃	1	1	yes	characteristic, complemented
123	C₂×#492	C₂×C₅₃	1	1	yes	characteristic
124	C₂×#487	C₂×C₅₃	1	2	yes	complemented
125	C₂×#486	C₂×C₅₃	1	2	yes	—
126	#1708	C₂×C₅₃	1	8	yes	direct product factor
127	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic, complemented
128	C₂×C₅₃	—	16	2	no	—
129	C₂×C₅₃	#1708	1	4	yes	direct product factor
130	C₂×C₅₃	C₂×#492	1	1	yes	characteristic
131	C₂×C₅₃	C₂×#495	1	2	yes	—
132	C₂²×SD₁₆	—	2	2	no	—
133	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
134	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
135	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
136	#498	—	2	2	no	—
137	#498	C₂²×C₅₃	1	4	yes	complemented
138	#492	—	2	2	no	—
139	#492	C₂²×C₅₃	1	4	yes	—
140	#487	—	2	4	no	—
141	#487	C₂²×C₅₃	1	8	yes	complemented
142	#486	—	2	4	no	—
143	#486	C₂²×C₅₃	1	8	yes	—
144	C₂×#106	—	2	2	no	—
145	C₂×#103	—	2	4	no	—
146	C₂×#102	—	2	4	no	—
147	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
149	C₈²	C₂²×C₅₃	1	4	yes	complemented
150	C₅₃	C₂×#1708	1	1	yes	characteristic, direct product factor, Sylow
151	C₂²×Q₈	—	2	1	no	—
152	C₂²×Q₈	—	2	2	no	—
153	C₂²×D₄	—	2	1	no	—
154	C₂²×D₄	—	2	2	no	—
155	C₂×SD₁₆	—	2	8	no	—
156	C₂×SD₁₆	—	4	8	no	—
157	C₂²×C₈	D₄×C₅₃	1	4	yes	—
158	C₂²×C₈	D₄×C₅₃	1	2	yes	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	—	2	1	no	—
164	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
165	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
166	#127	—	2	2	no	—
167	#127	—	2	1	no	—
168	#127	C₂³×C₅₃	1	2	yes	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
173	#106	—	2	8	no	—
174	#103	—	2	16	no	—
175	#102	—	2	16	no	—
176	C₄×C₈	D₄×C₅₃	1	4	yes	—
177	C₄×C₈	D₄×C₅₃	1	2	yes	—
178	C₄×C₈	C₂³×C₅₃	1	4	yes	—
179	C₄×C₈	C₂³×C₅₃	1	2	yes	—
180	C₂⁴	—	4	1	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	4	1	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	4	8	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	8	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
197	SD₁₆	—	4	16	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	SD₁₆×C₅₃	1	8	yes	—
200	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
201	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄²	C₂×D₄×C₅₃	1	2	yes	—
209	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
210	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
211	C₂³	—	4	2	no	—
212	C₂³	—	8	2	no	—
213	C₂³	—	8	4	no	—
214	C₂³	#127×C₅₃	1	1	yes	characteristic
215	Q₈	—	4	4	no	—
216	Q₈	—	4	8	no	—
217	Q₈	—	8	2	no	—
218	Q₈	—	8	4	no	—
219	D₄	—	4	4	no	—
220	D₄	—	4	8	no	—
221	D₄	—	8	2	no	—
222	D₄	—	8	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	4	no	—
227	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
228	C₂×C₄	#127×C₅₃	1	2	yes	—
229	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
230	C₂×C₄	#135×C₅₃	1	8	yes	—
231	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
233	C₈	—	2	8	no	—
234	C₈	C₂×SD₁₆×C₅₃	1	8	yes	—
235	C₂²	—	8	2	no	—
236	C₂²	—	8	4	no	—
237	C₂²	—	16	4	no	—
238	C₂²	C₅₃×#492	1	2	yes	—
239	C₂²	C₅₃×#495	1	4	yes	—
240	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
241	C₄	—	8	2	no	—
242	C₄	—	8	4	no	—
243	C₄	C₅₃×#492	1	2	yes	—
244	C₄	C₅₃×#495	1	4	yes	—
245	C₄	C₂²×SD₁₆×C₅₃	1	2	yes	—
246	C₄	C₂×#135×C₅₃	1	4	yes	—
247	C₂	—	16	2	no	—
248	C₂	C₅₃×#1708	1	4	yes	direct product factor
249	C₂	C₂×C₅₃×#492	1	1	yes	characteristic
250	C₂	C₂×C₅₃×#495	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#498	C₂	semidirect
3. C₂×C₅₃×#487	C₂	semidirect
5. C₅₃×#1708	C₂	direct (N × H)
6. C₂×C₈²×C₅₃	C₂	semidirect
12. C₅₃×#498	C₂²	semidirect
16. C₅₃×#487	C₂²	semidirect
24. C₈²×C₅₃	C₂²	semidirect
109. C₂×#1708	C₅₃	direct (N × H)
122. C₂×#498	C₂×C₅₃	semidirect
124. C₂×#487	C₂×C₅₃	semidirect
126. #1708	C₂×C₅₃	direct (N × H)
127. C₂×C₈²	C₂×C₅₃	semidirect
129. C₂×C₅₃	#1708	direct (N × H)
137. #498	C₂²×C₅₃	semidirect
141. #487	C₂²×C₅₃	semidirect
149. C₈²	C₂²×C₅₃	semidirect
150. C₅₃	C₂×#1708	direct (N × H)
248. C₂	C₅₃×#1708	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×#498	2
C₂×C₅₃×#492	2
C₂×C₅₃×#487	2
C₂×C₅₃×#486	2
C₅₃×#1708	2
C₂×C₈²×C₅₃	2
C₂×#1708	53

Subgroups — C2×C53×#1708 #27227107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1708 #27227107