Subgroups — C₂×#20273008 #27349696

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#20273008	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810345	C₂	1	2	yes	maximal, complemented
5	C₂×#16810344	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₄	1	2	yes	—
8	#16810676	C₄	1	4	yes	—
9	#16810676	C₂²	1	4	yes	—
10	C₂×C₄⋊#88322	C₄	1	2	yes	—
11	C₂×C₂²⋊#88322	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₄	1	4	yes	—
13	C₂×C₄×#88322	C₄	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄×#88322	C₂²	1	2	yes	—
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	C₂×C₄×#88322	C₂²	1	4	yes	—
18	#16810345	C₂²	1	8	yes	complemented
19	#16810344	C₂²	1	8	yes	complemented
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#431	C₄	1	4	yes	—
22	C₅₃×#431	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
24	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
26	C₂²×#88322	—	2	8	no	—
27	C₂²×#88322	C₂×C₄	1	2	yes	—
28	C₂²×#88322	C₂×C₄	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	8	yes	—
30	C₂²×#88322	C₂×C₄	1	4	yes	—
31	C₄⋊#88322	C₂×C₄	1	8	yes	—
32	C₂²⋊#88322	C₂×C₄	1	8	yes	—
33	C₄×#88322	C₂×C₄	1	2	yes	—
34	C₄×#88322	C₂×C₄	1	2	yes	—
35	C₄×#88322	C₂×C₄	1	8	yes	—
36	C₄×#88322	C₂×C₄	1	4	yes	—
37	C₄×#88322	C₂³	1	2	yes	—
38	C₄×#88322	C₂³	1	2	yes	—
39	C₄×#88322	C₂³	1	4	yes	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
53	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₅₃	C₂³	1	4	yes	—
56	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	—	2	8	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
74	C₄²×C₅₃	C₄²	1	2	yes	—
75	C₄²×C₅₃	C₄²	1	2	yes	—
76	C₄²×C₅₃	C₄²	1	2	yes	—
77	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
84	C₂×#88322	—	2	16	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	2	8	no	—
88	C₂×#88322	C₂²×C₄	1	4	yes	—
89	C₂×#88322	C₂²×C₄	1	4	yes	—
90	C₂×#88322	C₂²×C₄	1	8	yes	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	#97	1	4	yes	—
94	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
95	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	#97	1	8	yes	—
103	C₂×C₄×C₅₃	#97	1	4	yes	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
111	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
112	C₂×C₄×C₅₃	#131	1	4	yes	—
113	C₂×C₄×C₅₃	#131	1	4	yes	—
114	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
117	C₈×C₅₃	—	2	8	no	—
118	#88322	—	2	4	no	—
119	#88322	—	2	4	no	—
120	#88322	—	2	8	no	—
121	C₂×#1452	—	53	1	no	maximal, Sylow
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	2	8	no	—
125	C₂²×C₅₃	C₂×#97	1	4	yes	—
126	C₂²×C₅₃	C₂×#97	1	2	yes	—
127	C₂²×C₅₃	#405	1	2	yes	—
128	C₂²×C₅₃	#405	1	2	yes	—
129	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	C₂×#97	1	2	yes	—
133	C₄×C₅₃	C₂×#97	1	4	yes	—
134	C₄×C₅₃	#405	1	2	yes	—
135	C₄×C₅₃	#405	1	2	yes	—
136	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
137	C₄×C₅₃	C₂×#131	1	2	yes	—
138	C₄×C₅₃	C₂×#131	1	2	yes	—
139	C₂×#431	—	53	1	no	—
140	C₂×#431	D₅₃	1	1	yes	characteristic
141	C₂²×C₄×C₈	—	53	1	no	—
142	#1452	—	53	8	no	—
143	C₂×#322	—	53	2	no	—
144	C₂×#322	—	53	2	no	—
145	C₂×C₅₃	—	2	4	no	—
146	C₂×C₅₃	#1452	1	4	yes	complemented
147	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
149	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
150	C₂³×C₈	—	53	2	no	—
151	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
152	#431	—	53	4	no	—
153	#431	—	53	4	no	—
154	#431	Dic₅₃	1	4	yes	—
155	#431	C₂×D₅₃	1	4	yes	—
156	C₂×C₄⋊C₈	—	53	2	no	—
157	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
158	C₂×C₂²⋊C₈	—	53	2	no	—
159	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
160	C₂×C₄×C₈	—	53	2	no	—
161	C₂×C₄×C₈	—	53	4	no	—
162	C₂×C₄×C₈	—	53	4	no	—
163	C₂×C₄×C₈	—	53	1	no	—
164	C₂×C₄×C₈	—	53	1	no	—
165	C₂×C₄×C₈	—	53	2	no	—
166	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
167	#322	—	53	8	no	—
168	#322	—	53	8	no	—
169	C₅₃	C₂×#1452	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
171	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
172	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
173	C₂²×C₈	—	53	8	no	—
174	C₂²×C₈	—	53	2	no	—
175	C₂²×C₈	—	53	2	no	—
176	C₂²×C₈	—	53	4	no	—
177	C₂²×C₈	—	106	8	no	—
178	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
179	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄²	C₄×D₅₃	1	2	yes	—
181	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₅₃	1	2	yes	—
183	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
185	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
186	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
187	C₄⋊C₈	—	53	8	no	—
188	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
189	C₂²⋊C₈	—	53	8	no	—
190	C₂²⋊C₈	C₂×Dic₅₃	1	8	yes	—
191	C₄×C₈	—	53	8	no	—
192	C₄×C₈	—	53	2	no	—
193	C₄×C₈	—	53	2	no	—
194	C₄×C₈	—	53	4	no	—
195	C₄×C₈	—	53	2	no	—
196	C₄×C₈	—	53	2	no	—
197	C₄×C₈	—	53	4	no	—
198	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
199	C₄×C₈	C₂²×D₅₃	1	4	yes	—
200	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	#11719912	1	8	yes	—
204	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
206	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
208	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
209	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
216	C₂×C₈	—	2	8	no	—
217	C₂×C₈	—	53	4	no	—
218	C₂×C₈	—	53	4	no	—
219	C₂×C₈	—	53	8	no	—
220	C₂×C₈	—	106	16	no	—
221	C₂×C₈	—	106	4	no	—
222	C₂×C₈	—	106	4	no	—
223	C₂×C₈	—	106	8	no	—
224	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
225	C₄²	C₄×Dic₅₃	1	2	yes	—
226	C₄²	C₄×Dic₅₃	1	2	yes	—
227	C₄²	C₄×Dic₅₃	1	2	yes	—
228	C₄²	C₂×C₄×D₅₃	1	2	yes	—
229	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₄²	C₂×C₄×D₅₃	1	2	yes	—
232	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
233	C₄²	C₂²×Dic₅₃	1	2	yes	—
234	C₄²	C₂³×D₅₃	1	1	yes	characteristic
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	#13133127	1	4	yes	—
238	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
239	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	#13133127	1	8	yes	—
247	C₂×C₄	#13133127	1	4	yes	—
248	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
249	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
250	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
252	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#11719912	1	8	yes	—
255	C₂×C₄	C₂×#11719912	1	4	yes	—
256	C₂×C₄	#13133224	1	4	yes	—
257	C₂×C₄	#13133225	1	4	yes	—
258	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
261	C₈	—	2	8	no	—
262	C₈	—	106	4	no	—
263	C₈	—	106	4	no	—
264	C₈	—	106	8	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	C₂×#13133127	1	4	yes	—
269	C₂²	C₂×#13133127	1	2	yes	—
270	C₂²	#16810579	1	2	yes	—
271	C₂²	#16810581	1	2	yes	—
272	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	C₂×#13133127	1	2	yes	—
276	C₄	C₂×#13133127	1	4	yes	—
277	C₄	#16810579	1	2	yes	—
278	C₄	#16810581	1	2	yes	—
279	C₄	C₂²×#11719912	1	2	yes	—
280	C₄	C₂×#13133224	1	2	yes	—
281	C₄	C₂×#13133225	1	2	yes	—
282	C₂	—	2	4	no	—
283	C₂	#20273008	1	4	yes	direct product factor
284	C₂	C₂²×#13133127	1	1	yes	characteristic
285	C₂	C₂×#16810579	1	1	yes	characteristic
286	C₂	C₂×#16810581	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	C₂×#1452
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20273008	C₂	direct (N × H)
4. C₂×#16810345	C₂	semidirect
5. C₂×#16810344	C₂	semidirect
18. #16810345	C₂²	semidirect
19. #16810344	C₂²	semidirect
146. C₂×C₅₃	#1452	semidirect
169. C₅₃	C₂×#1452	semidirect
283. C₂	#20273008	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×#16810676	2
C₂²×C₄×#88322	2
#20273008	2
C₂×#16810345	2
C₂×#16810344	2
C₂×C₅₃×#431	2
C₂×#1452	53

Subgroups — C2×#20273008 #27349696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273008 #27349696