Subgroups — C₂×C₅₃×#1307 #27215629

Order 13568.

570 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×#432	C₂	1	2	yes	maximal
2	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1307	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#432	—	2	2	no	—
7	C₅₃×#432	—	2	2	no	—
8	C₅₃×#432	C₄	1	4	yes	—
9	C₅₃×#432	C₂²	1	4	yes	—
10	C₅₃×#423	—	2	2	no	—
11	C₅₃×#423	C₄	1	2	yes	—
12	C₅₃×#423	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
15	C₂×C₂²⋊C₈×C₅₃	C₄	1	4	yes	—
16	C₂×#97×C₅₃	—	2	2	no	—
17	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₅₃	—	2	2	no	—
20	C₂×M₁₆×C₅₃	—	4	4	no	—
21	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	4	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	4	no	—
34	C₄⋊C₈×C₅₃	—	2	4	no	—
35	C₄⋊C₈×C₅₃	—	2	4	no	—
36	C₄⋊C₈×C₅₃	—	2	4	no	—
37	C₂²⋊C₈×C₅₃	—	2	4	no	—
38	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
39	#97×C₅₃	—	2	4	no	—
40	#97×C₅₃	—	2	4	no	—
41	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	4	4	no	—
45	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
57	M₁₆×C₅₃	—	4	4	no	—
58	M₁₆×C₅₃	—	4	4	no	—
59	C₂×C₈×C₅₃	—	2	8	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	4	8	no	—
62	C₂×C₈×C₅₃	—	4	4	no	—
63	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×D₄	1	2	yes	—
72	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₂³×C₅₃	—	2	2	no	—
74	C₂³×C₅₃	—	2	4	no	—
75	C₂³×C₅₃	#95	1	1	yes	characteristic
76	C₂³×C₅₃	#95	1	1	yes	characteristic
77	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#95	1	2	yes	—
93	C₂×C₄×C₅₃	#95	1	2	yes	—
94	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#100	1	2	yes	—
96	C₂×C₄×C₅₃	#101	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₈×C₅₃	—	4	8	no	—
102	C₈×C₅₃	—	4	4	no	—
103	C₂×#1307	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	4	4	no	—
109	C₂²×C₅₃	#343	1	2	yes	—
110	C₂²×C₅₃	#344	1	4	yes	—
111	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	#343	1	2	yes	—
119	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
120	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
121	C₂×#432	C₂×C₅₃	1	2	yes	—
122	C₂×#423	C₂×C₅₃	1	1	yes	characteristic
123	#1307	C₂×C₅₃	1	4	yes	direct product factor
124	C₂×C₅₃	—	4	1	no	—
125	C₂×C₅₃	—	4	1	no	—
126	C₂×C₅₃	#1307	1	4	yes	direct product factor
127	C₂×C₅₃	C₂×#343	1	1	yes	characteristic
128	C₂×C₅₃	C₂×#344	1	2	yes	—
129	C₂²×M₁₆	C₄×C₅₃	1	2	yes	—
130	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
131	#432	—	2	2	no	—
132	#432	—	2	2	no	—
133	#432	C₄×C₅₃	1	4	yes	—
134	#432	C₂²×C₅₃	1	4	yes	—
135	#423	—	2	2	no	—
136	#423	C₄×C₅₃	1	2	yes	—
137	#423	C₂²×C₅₃	1	2	yes	—
138	C₂×C₄⋊C₈	—	2	2	no	—
139	C₂×C₄⋊C₈	—	2	2	no	—
140	C₂×C₂²⋊C₈	C₄×C₅₃	1	4	yes	—
141	C₂×#97	—	2	2	no	—
142	C₅₃	C₂×#1307	1	1	yes	characteristic, direct product factor, Sylow
143	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
144	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	4	4	no	—
147	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
152	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
153	C₂×C₄²	D₄×C₅₃	1	2	yes	—
154	C₂×C₄²	D₄×C₅₃	1	2	yes	—
155	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
156	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
157	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
158	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
159	C₄⋊C₈	—	2	4	no	—
160	C₄⋊C₈	—	2	4	no	—
161	C₄⋊C₈	—	2	4	no	—
162	C₄⋊C₈	—	2	4	no	—
163	C₂²⋊C₈	—	2	4	no	—
164	C₂²⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
165	#97	—	2	4	no	—
166	#97	—	2	4	no	—
167	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
173	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
174	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
176	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
179	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊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	M₁₆	—	4	4	no	—
184	M₁₆	—	4	4	no	—
185	C₂×C₈	—	2	8	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	8	no	—
188	C₂×C₈	—	4	4	no	—
189	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
190	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
191	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
192	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
193	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
194	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
195	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
196	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
197	C₄²	C₂×D₄×C₅₃	1	2	yes	—
198	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
199	C₂³	—	2	2	no	—
200	C₂³	—	2	4	no	—
201	C₂³	#95×C₅₃	1	1	yes	characteristic
202	C₂³	#95×C₅₃	1	1	yes	characteristic
203	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
218	C₂×C₄	#95×C₅₃	1	2	yes	—
219	C₂×C₄	#95×C₅₃	1	2	yes	—
220	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
221	C₂×C₄	#100×C₅₃	1	2	yes	—
222	C₂×C₄	#101×C₅₃	1	2	yes	—
223	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
227	C₈	—	4	8	no	—
228	C₈	—	4	4	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	4	no	—
234	C₂²	C₅₃×#343	1	2	yes	—
235	C₂²	C₅₃×#344	1	4	yes	—
236	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	C₅₃×#343	1	2	yes	—
244	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
245	C₄	C₂×#101×C₅₃	1	1	yes	characteristic
246	C₂	—	4	1	no	—
247	C₂	—	4	1	no	—
248	C₂	C₅₃×#1307	1	4	yes	direct product factor
249	C₂	C₂×C₅₃×#343	1	1	yes	characteristic
250	C₂	C₂×C₅₃×#344	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	#95
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1307	C₂	direct (N × H)
103. C₂×#1307	C₅₃	direct (N × H)
123. #1307	C₂×C₅₃	direct (N × H)
126. C₂×C₅₃	#1307	direct (N × H)
142. C₅₃	C₂×#1307	direct (N × H)
248. C₂	C₅₃×#1307	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×#432	2
C₂×C₅₃×#423	2
C₅₃×#1307	2
C₂×#1307	53

Subgroups — C2×C53×#1307 #27215629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1307 #27215629