Subgroups — C₂×C₅₃×#2351 #27241625

Order 13568.

3158 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₅₃	C₂	1	2	yes	maximal
2	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2351	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#389	C₂	1	6	yes	maximal
5	C₂×C₅₃×#382	C₂	1	3	yes	maximal
6	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	3	yes	maximal
7	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₅₃	—	2	16	no	—
9	C₂×C₄×Q₈×C₅₃	C₄	1	16	yes	complemented
10	C₂×C₄×Q₈×C₅₃	C₂²	1	16	yes	—
11	C₂×C₄×Q₈×C₅₃	C₂²	1	8	yes	—
12	C₂²×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
13	C₂²×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
14	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	6	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
17	C₅₃×#389	C₂²	1	48	yes	—
18	C₅₃×#382	C₂²	1	24	yes	—
19	C₄×C₄⋊C₄×C₅₃	C₂²	1	24	yes	—
20	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#95×C₅₃	C₂²	1	3	yes	—
22	C₂×#95×C₅₃	C₂²	1	6	yes	—
23	C₂²×Q₈×C₅₃	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₅₃	D₄	1	8	yes	—
25	C₂²×Q₈×C₅₃	Q₈	1	8	yes	—
26	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
32	C₄×Q₈×C₅₃	—	2	64	no	—
33	C₄×Q₈×C₅₃	C₂×C₄	1	64	yes	complemented
34	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	24	yes	—
37	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
38	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
40	C₂×C₄²×C₅₃	—	2	24	no	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	24	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
44	#95×C₅₃	C₂³	1	4	yes	—
45	#95×C₅₃	C₂³	1	12	yes	—
46	#95×C₅₃	C₂³	1	24	yes	—
47	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂×Q₈×C₅₃	C₄⋊C₄	1	64	yes	complemented
49	C₂×Q₈×C₅₃	C₂²×C₄	1	16	yes	—
50	C₂×Q₈×C₅₃	C₂×D₄	1	16	yes	—
51	C₂×Q₈×C₅₃	C₂×Q₈	1	16	yes	—
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	2	12	no	—
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	12	no	—
57	C₂²×C₄×C₅₃	—	2	12	no	—
58	C₂²×C₄×C₅₃	—	2	12	no	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₅₃	C₂×Q₈	1	12	yes	—
64	C₂²×C₄×C₅₃	#41	1	8	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
70	C₄⋊C₄×C₅₃	—	2	48	no	—
71	C₄⋊C₄×C₅₃	—	2	48	no	—
72	C₄⋊C₄×C₅₃	C₂²×C₄	1	48	yes	—
73	C₄²×C₅₃	—	2	48	no	—
74	C₄²×C₅₃	C₂²×C₄	1	48	yes	—
75	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
76	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
77	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
78	C₂³×C₅₃	C₂×#41	1	4	yes	—
79	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
80	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
81	Q₈×C₅₃	C₂×C₄⋊C₄	1	64	yes	complemented
82	C₂×C₄×C₅₃	—	2	24	no	—
83	C₂×C₄×C₅₃	—	2	8	no	—
84	C₂×C₄×C₅₃	—	2	16	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	12	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	12	no	—
90	C₂×C₄×C₅₃	—	2	12	no	—
91	C₂×C₄×C₅₃	—	2	12	no	—
92	C₂×C₄×C₅₃	—	4	24	no	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	48	yes	—
94	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
97	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
98	C₂×C₄×C₅₃	C₂²×Q₈	1	12	yes	—
99	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
100	C₂×#2351	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₅₃	C₂²×C₄⋊C₄	1	4	yes	—
102	C₂²×C₅₃	C₄×#41	1	4	yes	—
103	C₂²×C₅₃	#519	1	4	yes	—
104	C₂²×C₅₃	#548	1	8	yes	—
105	C₂²×C₅₃	#557	1	8	yes	—
106	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
107	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
108	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
109	C₂²×C₅₃	C₂²×#41	1	2	yes	—
110	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
111	C₄×C₅₃	—	2	12	no	—
112	C₄×C₅₃	—	2	4	no	—
113	C₄×C₅₃	—	2	8	no	—
114	C₄×C₅₃	—	4	12	no	—
115	C₄×C₅₃	C₂²×C₄⋊C₄	1	24	yes	—
116	C₂²×C₄×Q₈	C₂×C₅₃	1	2	yes	—
117	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic
118	#2351	C₂×C₅₃	1	16	yes	direct product factor
119	C₂×#389	C₂×C₅₃	1	6	yes	—
120	C₂×#382	C₂×C₅₃	1	3	yes	—
121	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	3	yes	—
122	C₂×C₅₃	#2351	1	8	yes	direct product factor
123	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#519	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#548	1	2	yes	—
127	C₂×C₅₃	C₂×#557	1	2	yes	—
128	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
129	C₂×C₄×Q₈	—	2	16	no	—
130	C₂×C₄×Q₈	C₄×C₅₃	1	16	yes	complemented
131	C₂×C₄×Q₈	C₂²×C₅₃	1	16	yes	—
132	C₂×C₄×Q₈	C₂²×C₅₃	1	8	yes	—
133	C₂²×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
134	C₂²×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
135	C₂²×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
136	C₂²×C₄²	C₂²×C₅₃	1	6	yes	—
137	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
138	#389	C₂²×C₅₃	1	48	yes	—
139	#382	C₂²×C₅₃	1	24	yes	—
140	C₄×C₄⋊C₄	C₂²×C₅₃	1	24	yes	—
141	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
142	C₂×#95	C₂²×C₅₃	1	3	yes	—
143	C₂×#95	C₂²×C₅₃	1	6	yes	—
144	C₅₃	C₂×#2351	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×Q₈	C₂×C₄×C₅₃	1	8	yes	—
146	C₂²×Q₈	D₄×C₅₃	1	8	yes	—
147	C₂²×Q₈	Q₈×C₅₃	1	8	yes	—
148	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
149	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
150	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
151	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
152	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
153	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
154	C₄×Q₈	—	2	64	no	—
155	C₄×Q₈	C₂×C₄×C₅₃	1	64	yes	complemented
156	C₂×C₄⋊C₄	—	2	24	no	—
157	C₂×C₄⋊C₄	—	2	24	no	—
158	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	24	yes	—
159	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
160	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
161	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
162	C₂×C₄²	—	2	24	no	—
163	C₂×C₄²	C₂×C₄×C₅₃	1	24	yes	—
164	C₂×C₄²	C₂³×C₅₃	1	24	yes	—
165	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
166	#95	C₂³×C₅₃	1	4	yes	—
167	#95	C₂³×C₅₃	1	12	yes	—
168	#95	C₂³×C₅₃	1	24	yes	—
169	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
170	C₂×Q₈	C₄⋊C₄×C₅₃	1	64	yes	complemented
171	C₂×Q₈	C₂²×C₄×C₅₃	1	16	yes	—
172	C₂×Q₈	C₂×D₄×C₅₃	1	16	yes	—
173	C₂×Q₈	C₂×Q₈×C₅₃	1	16	yes	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	12	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	12	no	—
179	C₂²×C₄	—	2	12	no	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
182	C₂²×C₄	C₂²×C₄×C₅₃	1	12	yes	—
183	C₂²×C₄	C₂²×C₄×C₅₃	1	12	yes	—
184	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
185	C₂²×C₄	C₂×Q₈×C₅₃	1	12	yes	—
186	C₂²×C₄	#41×C₅₃	1	8	yes	—
187	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
188	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
189	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
190	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
191	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
192	C₄⋊C₄	—	2	48	no	—
193	C₄⋊C₄	—	2	48	no	—
194	C₄⋊C₄	C₂²×C₄×C₅₃	1	48	yes	—
195	C₄²	—	2	48	no	—
196	C₄²	C₂²×C₄×C₅₃	1	48	yes	—
197	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
198	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
199	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
200	C₂³	C₂×#41×C₅₃	1	4	yes	—
201	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
202	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
203	Q₈	C₂×C₄⋊C₄×C₅₃	1	64	yes	complemented
204	C₂×C₄	—	2	24	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	12	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	12	no	—
212	C₂×C₄	—	2	12	no	—
213	C₂×C₄	—	2	12	no	—
214	C₂×C₄	—	4	24	no	—
215	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	48	yes	—
216	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
217	C₂×C₄	C₂³×C₄×C₅₃	1	12	yes	—
218	C₂×C₄	C₂³×C₄×C₅₃	1	12	yes	—
219	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
220	C₂×C₄	C₂²×Q₈×C₅₃	1	12	yes	—
221	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
222	C₂²	C₂²×C₄⋊C₄×C₅₃	1	4	yes	—
223	C₂²	C₄×#41×C₅₃	1	4	yes	—
224	C₂²	C₅₃×#519	1	4	yes	—
225	C₂²	C₅₃×#548	1	8	yes	—
226	C₂²	C₅₃×#557	1	8	yes	—
227	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
228	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
229	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
230	C₂²	C₂²×#41×C₅₃	1	2	yes	—
231	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
232	C₄	—	2	8	no	—
233	C₄	—	2	12	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	12	no	—
236	C₄	C₂²×C₄⋊C₄×C₅₃	1	24	yes	—
237	C₂	C₅₃×#2351	1	8	yes	direct product factor
238	C₂	C₂³×C₄⋊C₄×C₅₃	1	1	yes	characteristic
239	C₂	C₂×C₄×#41×C₅₃	1	1	yes	characteristic
240	C₂	C₂×C₅₃×#519	1	1	yes	characteristic
241	C₂	C₂×C₅₃×#548	1	2	yes	—
242	C₂	C₂×C₅₃×#557	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2351	C₂	direct (N × H)
9. C₂×C₄×Q₈×C₅₃	C₄	semidirect
33. C₄×Q₈×C₅₃	C₂×C₄	semidirect
48. C₂×Q₈×C₅₃	C₄⋊C₄	semidirect
81. Q₈×C₅₃	C₂×C₄⋊C₄	semidirect
100. C₂×#2351	C₅₃	direct (N × H)
118. #2351	C₂×C₅₃	direct (N × H)
122. C₂×C₅₃	#2351	direct (N × H)
130. C₂×C₄×Q₈	C₄×C₅₃	semidirect
144. C₅₃	C₂×#2351	direct (N × H)
155. C₄×Q₈	C₂×C₄×C₅₃	semidirect
170. C₂×Q₈	C₄⋊C₄×C₅₃	semidirect
203. Q₈	C₂×C₄⋊C₄×C₅₃	semidirect
237. C₂	C₅₃×#2351	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×C₄×Q₈×C₅₃	2
C₅₃×#2351	2
C₂×C₅₃×#389	2
C₂×C₅₃×#382	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×#2351	53

Subgroups — C2×C53×#2351 #27241625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2351 #27241625