Subgroups — C₄×C₅₃×#349 #27220113

Order 13568.

522 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×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#349	C₂	1	2	yes	maximal
3	C₄×M₃₂×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1377	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
9	C₄×M₁₆×C₅₃	—	2	2	no	—
10	C₄×M₁₆×C₅₃	—	2	2	no	—
11	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#349	C₄	1	8	yes	direct product factor
16	C₅₃×#346	—	2	2	no	—
17	C₄×C₁₆×C₅₃	—	2	2	no	—
18	C₂³×C₄×C₅₃	C₈	1	2	yes	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	C₈	1	4	yes	—
25	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	—	2	1	no	—
31	C₂×C₄²×C₅₃	—	2	1	no	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₈	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₅₃	—	2	4	no	—
37	M₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
38	C₂×C₁₆×C₅₃	—	2	4	no	—
39	#97×C₅₃	—	2	2	no	—
40	#97×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	4	4	no	—
53	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₅₃	—	2	8	no	—
60	M₁₆×C₅₃	—	2	8	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	C₄²	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	#41	1	2	yes	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅₃	M₁₆	1	2	yes	—
80	C₁₆×C₅₃	—	2	8	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	—	4	2	no	—
84	C₂³×C₅₃	C₄×C₈	1	2	yes	—
85	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
109	C₈×C₅₃	C₄×D₄	1	4	yes	—
110	C₄×#349	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	4	2	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₅₃	#431	1	1	yes	characteristic
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	4	2	no	—
122	C₄×C₅₃	#349	1	4	yes	direct product factor
123	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
125	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₅₃	#431	1	1	yes	characteristic
127	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
128	C₂×#349	C₂×C₅₃	1	2	yes	—
129	C₄×M₃₂	C₂×C₅₃	1	2	yes	complemented
130	#1377	C₂×C₅₃	1	2	yes	complemented
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#349	1	2	yes	—
136	C₂²×M₁₆	C₄×C₅₃	1	1	yes	characteristic
137	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
138	C₂²×C₄²	C₄×C₅₃	1	1	yes	characteristic
139	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
140	C₄×M₁₆	—	2	2	no	—
141	C₄×M₁₆	—	2	2	no	—
142	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
143	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
146	#349	C₄×C₅₃	1	8	yes	direct product factor
147	#346	—	2	2	no	—
148	C₄×C₁₆	—	2	2	no	—
149	C₅₃	C₄×#349	1	1	yes	characteristic, direct product factor, Sylow
150	C₂³×C₄	C₈×C₅₃	1	2	yes	—
151	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	C₈×C₅₃	1	4	yes	—
157	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
158	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₈×C₅₃	1	2	yes	—
166	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
167	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
168	M₃₂	—	2	4	no	—
169	M₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
170	C₂×C₁₆	—	2	4	no	—
171	#97	—	2	2	no	—
172	#97	—	2	2	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	D₄×C₅₃	1	2	yes	—
176	C₄×C₈	D₄×C₅₃	1	2	yes	—
177	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
186	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
188	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
189	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
190	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
191	M₁₆	—	2	8	no	—
192	M₁₆	—	2	8	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	C₄²×C₅₃	1	2	yes	—
198	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
199	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
200	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
201	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
202	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
204	C₂×C₈	#41×C₅₃	1	2	yes	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
210	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
211	C₄²	M₁₆×C₅₃	1	2	yes	—
212	C₁₆	—	2	8	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	4	2	no	—
216	C₂³	C₄×C₈×C₅₃	1	2	yes	—
217	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
230	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
231	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
232	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	C₂²⋊C₈×C₅₃	1	4	yes	—
241	C₈	C₄×D₄×C₅₃	1	4	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
248	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
249	C₂²	C₅₃×#431	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	C₅₃×#349	1	4	yes	direct product factor
254	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
255	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
256	C₄	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
257	C₄	C₅₃×#431	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	C₄×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
262	C₂	C₂×C₅₃×#349	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅₃	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×M₃₂×C₅₃	C₂	semidirect
4. C₅₃×#1377	C₂	semidirect
15. C₅₃×#349	C₄	direct (N × H)
37. M₃₂×C₅₃	C₂×C₄	semidirect
110. C₄×#349	C₅₃	direct (N × H)
122. C₄×C₅₃	#349	direct (N × H)
129. C₄×M₃₂	C₂×C₅₃	semidirect
130. #1377	C₂×C₅₃	semidirect
146. #349	C₄×C₅₃	direct (N × H)
149. C₅₃	C₄×#349	direct (N × H)
169. M₃₂	C₂×C₄×C₅₃	semidirect
253. C₄	C₅₃×#349	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×C₄×M₁₆×C₅₃	2
C₂×C₅₃×#349	2
C₄×M₃₂×C₅₃	2
C₅₃×#1377	2
C₄×#349	53

Subgroups — C4×C53×#349 #27220113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#349 #27220113