Subgroups — C₂×#20276521 #27537166

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
2	#20276521	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810402	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810402	C₂	1	1	yes	maximal, characteristic
6	C₂×#16810398	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719912	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×#13133174	C₄	1	1	yes	characteristic
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	#16810577	C₄	1	4	yes	—
15	#16810577	C₂²	1	4	yes	—
16	C₂×#13133128	C₄	1	1	yes	characteristic
17	C₂×#13133127	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊#11719908	C₂²	1	4	yes	complemented
20	#16810402	C₂²	1	4	yes	—
21	#16810402	C₂²	1	4	yes	—
22	#16810398	C₂²	1	4	yes	complemented
23	C₁₆×Dic₅₃	C₂²	1	4	yes	complemented
24	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
28	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×#11719912	—	2	4	no	—
30	C₂×#11719912	C₂×C₄	1	4	yes	—
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₅₃	—	2	2	no	—
34	C₂×C₈×D₅₃	—	2	2	no	—
35	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₅₃	C₂³	1	2	yes	—
37	#13133174	C₈	1	4	yes	—
38	#13133174	C₂×C₄	1	4	yes	—
39	C₂×#11719919	C₈	1	1	yes	characteristic
40	C₂×#11719919	C₈	1	1	yes	characteristic
41	C₂×#11719921	C₈	1	1	yes	characteristic
42	C₂×#11719921	C₈	1	1	yes	characteristic
43	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×#11719908	—	2	2	no	—
45	C₂×#11719908	—	2	2	no	—
46	C₂×#11719908	C₂³	1	2	yes	—
47	C₂×#11719908	C₂³	1	2	yes	—
48	#13133128	C₂×C₄	1	4	yes	—
49	#13133127	C₂×C₄	1	4	yes	—
50	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
51	C₈×Dic₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₅₃	—	2	2	no	—
56	C₂×C₁₆×C₅₃	—	2	2	no	—
57	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
58	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₅₃	—	2	2	no	—
69	C₂×C₄×D₅₃	—	2	2	no	—
70	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	#41	1	2	yes	—
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×C₈	1	2	yes	—
79	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
80	#11719921	C₂×C₈	1	4	yes	—
81	#11719921	C₂×C₈	1	4	yes	—
82	#11719919	C₂×C₈	1	4	yes	—
83	#11719919	C₂×C₈	1	4	yes	—
84	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
85	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
86	C₂×#88322	—	2	2	no	—
87	C₂×#88322	M₁₆	1	4	yes	—
88	C₂×#88322	C₂²×C₄	1	2	yes	—
89	C₂×#88322	C₂²×C₄	1	2	yes	—
90	#11719912	—	2	8	no	—
91	C₈×D₅₃	—	2	4	no	—
92	C₈×D₅₃	—	2	2	no	—
93	C₈×D₅₃	—	2	2	no	—
94	C₁₆×C₅₃	—	2	2	no	—
95	C₁₆×C₅₃	—	2	2	no	—
96	#11719908	—	2	2	no	—
97	#11719908	—	2	2	no	—
98	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
99	C₂²×D₅₃	—	2	2	no	—
100	C₂²×D₅₃	—	2	2	no	—
101	C₂²×D₅₃	—	2	2	no	—
102	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	#117	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₅₃	—	2	2	no	—
112	C₂×Dic₅₃	—	2	2	no	—
113	C₂×Dic₅₃	—	2	2	no	—
114	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
115	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
116	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
117	C₄×D₅₃	—	2	2	no	—
118	C₄×D₅₃	—	2	2	no	—
119	C₄×D₅₃	—	2	4	no	—
120	C₈×C₅₃	—	2	2	no	—
121	C₈×C₅₃	#117	1	2	yes	—
122	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
123	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
124	#88322	—	2	2	no	—
125	#88322	C₂×M₁₆	1	4	yes	—
126	C₂×#2164	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	#431	1	2	yes	—
128	C₂²×C₅₃	#504	1	2	yes	—
129	C₂²×C₅₃	#504	1	2	yes	—
130	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
131	C₂×D₅₃	—	2	2	no	—
132	C₂×D₅₃	—	2	2	no	—
133	C₂×D₅₃	—	2	2	no	—
134	C₂×D₅₃	—	4	4	no	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	#431	1	2	yes	—
138	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
139	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
140	Dic₅₃	—	2	2	no	—
141	Dic₅₃	—	2	2	no	—
142	Dic₅₃	—	2	2	no	—
143	C₂×#405	—	53	1	no	—
144	#2164	—	53	8	no	—
145	C₂×C₄⋊C₁₆	—	53	1	no	—
146	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
147	C₂×C₂²⋊C₁₆	—	53	1	no	—
148	C₂×C₂²⋊C₁₆	—	53	1	no	—
149	C₂×#346	—	53	1	no	—
150	C₂×C₄×C₁₆	—	53	1	no	—
151	C₂×C₅₃	#2164	1	4	yes	complemented
152	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
153	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
154	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
155	D₅₃	—	4	2	no	—
156	C₂²×M₁₆	—	53	1	no	—
157	C₂³×C₈	—	53	1	no	—
158	C₂×#117	—	53	1	no	—
159	C₂²×C₁₆	—	53	1	no	—
160	C₂²×C₁₆	—	53	1	no	—
161	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
162	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
163	#405	—	53	4	no	—
164	#405	—	53	4	no	—
165	C₂×#97	—	53	1	no	—
166	C₂×#97	—	53	1	no	—
167	C₂×C₄×C₈	—	53	1	no	—
168	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
169	C₄⋊C₁₆	—	53	4	no	—
170	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	complemented
171	C₂²⋊C₁₆	—	53	4	no	—
172	C₂²⋊C₁₆	—	53	4	no	—
173	#346	—	53	4	no	—
174	C₄×C₁₆	—	53	4	no	—
175	C₅₃	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	—	53	1	no	—
177	C₂×M₁₆	—	53	4	no	—
178	C₂×M₁₆	—	106	4	no	—
179	C₂²×C₈	—	53	1	no	—
180	C₂²×C₈	—	53	1	no	—
181	C₂²×C₈	—	53	2	no	—
182	C₂²×C₈	—	53	2	no	—
183	C₂²×C₈	—	106	2	no	—
184	C₂²×C₈	—	106	2	no	—
185	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
187	#117	—	53	4	no	—
188	#117	—	53	4	no	—
189	C₂×C₄⋊C₄	—	53	1	no	—
190	C₂×C₄⋊C₄	—	53	1	no	—
191	C₂×C₂²⋊C₄	—	53	1	no	—
192	C₂×C₂²⋊C₄	—	53	1	no	—
193	C₂×C₄²	—	53	1	no	—
194	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	—	53	2	no	—
198	C₂×C₁₆	—	53	2	no	—
199	C₂×C₁₆	—	106	2	no	—
200	C₂×C₁₆	—	106	2	no	—
201	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
202	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
203	#97	—	53	4	no	—
204	#97	—	53	4	no	—
205	C₄×C₈	—	53	2	no	—
206	C₄×C₈	—	53	2	no	—
207	C₄×C₈	C₄×D₅₃	1	2	yes	—
208	C₄×C₈	C₂²×D₅₃	1	2	yes	—
209	C₂⁴	—	53	1	no	—
210	C₂²×C₄	—	53	1	no	—
211	C₂²×C₄	—	53	1	no	—
212	C₂²×C₄	—	53	1	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	—	106	2	no	—
216	C₂²×C₄	—	106	2	no	—
217	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
220	M₁₆	—	106	8	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	53	4	no	—
223	C₂×C₈	—	53	2	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	106	2	no	—
226	C₂×C₈	—	106	4	no	—
227	C₂×C₈	—	106	2	no	—
228	C₂×C₈	—	106	2	no	—
229	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
230	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₂×C₈	#11719947	1	2	yes	—
232	C₂×C₈	#11719949	1	2	yes	—
233	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
234	C₄⋊C₄	—	53	4	no	—
235	C₄⋊C₄	—	53	4	no	—
236	C₂²⋊C₄	—	53	4	no	—
237	C₂²⋊C₄	—	53	4	no	—
238	C₄²	—	53	2	no	—
239	C₄²	—	53	2	no	—
240	C₄²	C₈×D₅₃	1	2	yes	—
241	C₄²	C₂×C₄×D₅₃	1	2	yes	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	106	2	no	—
245	C₁₆	—	106	2	no	—
246	C₂³	—	53	2	no	—
247	C₂³	—	106	2	no	—
248	C₂³	—	106	2	no	—
249	C₂³	—	106	2	no	—
250	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	53	2	no	—
254	C₂×C₄	—	53	2	no	—
255	C₂×C₄	—	53	2	no	—
256	C₂×C₄	—	106	2	no	—
257	C₂×C₄	—	106	2	no	—
258	C₂×C₄	—	106	2	no	—
259	C₂×C₄	—	106	2	no	—
260	C₂×C₄	—	106	2	no	—
261	C₂×C₄	—	106	4	no	—
262	C₂×C₄	#13133175	1	2	yes	—
263	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
264	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
265	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
266	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
267	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	53	4	no	—
270	C₈	—	106	2	no	—
271	C₈	#13133175	1	2	yes	—
272	C₈	C₂×#11719947	1	1	yes	characteristic
273	C₈	C₂×#11719949	1	1	yes	characteristic
274	C₂²	—	106	2	no	—
275	C₂²	—	106	2	no	—
276	C₂²	—	106	2	no	—
277	C₂²	—	212	4	no	—
278	C₂²	#16810675	1	2	yes	—
279	C₂²	#16811039	1	2	yes	—
280	C₂²	#16811041	1	2	yes	—
281	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	106	2	no	—
285	C₄	—	106	2	no	—
286	C₄	—	106	2	no	—
287	C₄	#16810675	1	2	yes	—
288	C₄	C₂×#13133175	1	1	yes	characteristic
289	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
290	C₂	—	212	2	no	—
291	C₂	#20276521	1	4	yes	direct product factor
292	C₂	C₂×#16810675	1	1	yes	characteristic
293	C₂	C₂×#16811039	1	1	yes	characteristic
294	C₂	C₂×#16811041	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	C₂×#2164
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276521	C₂	direct (N × H)
3. C₂×C₄⋊#11719908	C₂	semidirect
6. C₂×#16810398	C₂	semidirect
7. C₂×C₁₆×Dic₅₃	C₂	semidirect
8. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
19. C₄⋊#11719908	C₂²	semidirect
22. #16810398	C₂²	semidirect
23. C₁₆×Dic₅₃	C₂²	semidirect
27. C₅₃×C₄⋊C₁₆	C₂²	semidirect
146. C₂×C₄⋊C₁₆	D₅₃	semidirect
151. C₂×C₅₃	#2164	semidirect
170. C₄⋊C₁₆	C₂×D₅₃	semidirect
175. C₅₃	C₂×#2164	semidirect
291. C₂	#20276521	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×#16810577	2
#20276521	2
C₂×C₄⋊#11719908	2
C₂×#16810402	2
C₂×#16810398	2
C₂×C₁₆×Dic₅₃	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×#2164	53

Subgroups — C2×#20276521 #27537166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276521 #27537166