Subgroups — C₂×C₅₃×#1526 #27225149

Order 13568.

1122 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	C₂×C₅₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#1526	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#330	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₅₃	—	2	2	no	—
6	C₅₃×#533	—	2	4	no	—
7	C₅₃×#533	C₄	1	4	yes	—
8	C₅₃×#533	C₂²	1	4	yes	—
9	C₂×C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
10	C₂×#125×C₅₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
12	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
13	C₂×#117×C₅₃	C₄	1	2	yes	—
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₅₃	—	2	4	no	—
16	C₄×M₁₆×C₅₃	C₂²	1	8	yes	—
17	C₂×#97×C₅₃	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₅₃	—	2	2	no	—
19	C₅₃×#330	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₅₃	—	2	2	no	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	D₄	1	2	yes	—
23	C₂×M₁₆×C₅₃	—	2	16	no	—
24	C₂×M₁₆×C₅₃	—	2	8	no	—
25	C₂²×C₈×C₅₃	—	2	4	no	—
26	C₄⋊Q₈×C₅₃	—	2	4	no	—
27	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
28	#125×C₅₃	—	2	4	no	—
29	#125×C₅₃	C₂×C₄	1	4	yes	—
30	#125×C₅₃	C₂³	1	4	yes	—
31	C₂²⋊Q₈×C₅₃	—	2	8	no	—
32	C₂²⋊Q₈×C₅₃	—	2	8	no	—
33	C₄×Q₈×C₅₃	—	2	8	no	—
34	C₄×Q₈×C₅₃	—	2	8	no	—
35	#117×C₅₃	—	2	4	no	—
36	#117×C₅₃	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	D₄	1	4	yes	—
45	C₂×C₄²×C₅₃	D₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
48	#97×C₅₃	—	2	4	no	—
49	#97×C₅₃	C₂³	1	8	yes	—
50	C₄×C₈×C₅₃	—	2	8	no	—
51	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Q₈×C₅₃	—	2	8	no	—
53	C₂×Q₈×C₅₃	—	4	8	no	—
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	8	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
65	M₁₆×C₅₃	—	2	32	no	—
66	C₂×C₈×C₅₃	—	2	16	no	—
67	C₂×C₈×C₅₃	—	2	8	no	—
68	C₄⋊C₄×C₅₃	—	2	8	no	—
69	C₄⋊C₄×C₅₃	—	2	8	no	—
70	C₄⋊C₄×C₅₃	—	2	4	no	—
71	C₄⋊C₄×C₅₃	—	2	8	no	—
72	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
73	C₂²⋊C₄×C₅₃	—	2	8	no	—
74	C₄²×C₅₃	—	2	8	no	—
75	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
76	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₅₃	C₂×D₄	1	4	yes	—
80	C₄²×C₅₃	C₂×D₄	1	2	yes	—
81	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	—	2	4	no	—
84	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₅₃	—	4	16	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	8	no	—
92	C₂×C₄×C₅₃	—	2	8	no	—
93	C₂×C₄×C₅₃	—	2	8	no	—
94	C₂×C₄×C₅₃	—	4	8	no	—
95	C₂×C₄×C₅₃	#101	1	4	yes	—
96	C₂×C₄×C₅₃	#104	1	8	yes	—
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
103	C₈×C₅₃	—	2	16	no	—
104	C₂×#1526	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	4	no	—
107	C₂²×C₅₃	—	4	4	no	—
108	C₂²×C₅₃	C₂×#101	1	2	yes	—
109	C₂²×C₅₃	C₂×#104	1	4	yes	—
110	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	4	8	no	—
116	C₄×C₅₃	C₂×#101	1	4	yes	—
117	C₄×C₅₃	C₂×#104	1	8	yes	—
118	C₂×#533	C₂×C₅₃	1	1	yes	characteristic
119	C₂×C₄×M₁₆	C₂×C₅₃	1	2	yes	—
120	#1526	C₂×C₅₃	1	8	yes	direct product factor
121	C₂×#330	C₂×C₅₃	1	4	yes	complemented
122	C₂×C₅₃	—	4	2	no	—
123	C₂×C₅₃	#1526	1	4	yes	direct product factor
124	C₂×C₅₃	C₂²×#101	1	1	yes	characteristic
125	C₂×C₅₃	C₂²×#104	1	2	yes	—
126	C₂²×M₁₆	—	2	2	no	—
127	#533	—	2	4	no	—
128	#533	C₄×C₅₃	1	4	yes	—
129	#533	C₂²×C₅₃	1	4	yes	—
130	C₂×C₄⋊Q₈	C₄×C₅₃	1	2	yes	—
131	C₂×#125	C₂²×C₅₃	1	2	yes	—
132	C₂×C₂²⋊Q₈	—	2	2	no	—
133	C₂×C₄×Q₈	—	2	2	no	—
134	C₂×#117	C₄×C₅₃	1	2	yes	—
135	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
136	C₄×M₁₆	—	2	4	no	—
137	C₄×M₁₆	C₂²×C₅₃	1	8	yes	—
138	C₂×#97	C₂²×C₅₃	1	4	yes	—
139	C₂×C₄×C₈	—	2	2	no	—
140	#330	C₂²×C₅₃	1	16	yes	complemented
141	C₅₃	C₂×#1526	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×Q₈	—	2	2	no	—
143	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
144	C₂³×C₄	D₄×C₅₃	1	2	yes	—
145	C₂×M₁₆	—	2	16	no	—
146	C₂×M₁₆	—	2	8	no	—
147	C₂²×C₈	—	2	4	no	—
148	C₄⋊Q₈	—	2	4	no	—
149	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
150	#125	—	2	4	no	—
151	#125	C₂×C₄×C₅₃	1	4	yes	—
152	#125	C₂³×C₅₃	1	4	yes	—
153	C₂²⋊Q₈	—	2	8	no	—
154	C₂²⋊Q₈	—	2	8	no	—
155	C₄×Q₈	—	2	8	no	—
156	C₄×Q₈	—	2	8	no	—
157	#117	—	2	4	no	—
158	#117	C₂×C₄×C₅₃	1	8	yes	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
162	C₂×C₂²⋊C₄	—	2	2	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
166	C₂×C₄²	D₄×C₅₃	1	2	yes	—
167	C₂×C₄²	D₄×C₅₃	1	4	yes	—
168	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
170	#97	—	2	4	no	—
171	#97	C₂³×C₅₃	1	8	yes	—
172	C₄×C₈	—	2	8	no	—
173	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
174	C₂×Q₈	—	2	8	no	—
175	C₂×Q₈	—	4	8	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
180	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
181	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
185	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
187	M₁₆	—	2	32	no	—
188	C₂×C₈	—	2	16	no	—
189	C₂×C₈	—	2	8	no	—
190	C₄⋊C₄	—	2	8	no	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
195	C₂²⋊C₄	—	2	8	no	—
196	C₄²	—	2	8	no	—
197	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
198	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
199	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
200	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
201	C₄²	C₂×D₄×C₅₃	1	4	yes	—
202	C₄²	C₂×D₄×C₅₃	1	2	yes	—
203	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
204	C₂³	—	2	2	no	—
205	C₂³	—	2	4	no	—
206	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
207	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
208	Q₈	—	4	16	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	#101×C₅₃	1	4	yes	—
218	C₂×C₄	#104×C₅₃	1	8	yes	—
219	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
223	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
225	C₈	—	2	16	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	4	4	no	—
229	C₂²	C₂×#101×C₅₃	1	2	yes	—
230	C₂²	C₂×#104×C₅₃	1	4	yes	—
231	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	4	8	no	—
237	C₄	C₂×#101×C₅₃	1	4	yes	—
238	C₄	C₂×#104×C₅₃	1	8	yes	—
239	C₂	—	4	2	no	—
240	C₂	C₅₃×#1526	1	4	yes	direct product factor
241	C₂	C₂²×#101×C₅₃	1	1	yes	characteristic
242	C₂	C₂²×#104×C₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1526	C₂	direct (N × H)
4. C₂×C₅₃×#330	C₂	semidirect
19. C₅₃×#330	C₂²	semidirect
104. C₂×#1526	C₅₃	direct (N × H)
120. #1526	C₂×C₅₃	direct (N × H)
121. C₂×#330	C₂×C₅₃	semidirect
123. C₂×C₅₃	#1526	direct (N × H)
140. #330	C₂²×C₅₃	semidirect
141. C₅₃	C₂×#1526	direct (N × H)
240. C₂	C₅₃×#1526	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×#533	2
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#1526	2
C₂×C₅₃×#330	2
C₂×#1526	53

Subgroups — C2×C53×#1526 #27225149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1526 #27225149