Subgroups — C₂×#20290937 #28327927

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊M₃₂	C₂	1	2	yes	maximal
2	#20290937	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16811516	C₂	1	2	yes	maximal, complemented
4	C₂×#16811515	C₂	1	2	yes	maximal, complemented
5	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133292	C₄	1	4	yes	complemented
7	D₅₃⋊M₃₂	C₄	1	8	yes	complemented
8	D₅₃⋊M₃₂	C₂²	1	8	yes	—
9	C₂×#13133291	C₂²	1	4	yes	—
10	C₂×D₅₃⋊C₁₆	C₄	1	4	yes	complemented
11	#16811516	C₂²	1	8	yes	complemented
12	#16811515	C₂²	1	8	yes	complemented
13	C₂²×#11719912	C₄	1	1	yes	characteristic
14	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₂×#13133174	C₄	1	1	yes	characteristic
16	#16810577	C₄	1	4	yes	—
17	#16810577	C₂²	1	4	yes	—
18	C₂×#13133128	C₂²	1	1	yes	characteristic
19	C₂×#13133127	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
22	#13133292	C₂×C₄	1	16	yes	complemented
23	#13133291	C₂×C₄	1	8	yes	—
24	#13133291	C₂³	1	8	yes	—
25	D₅₃⋊C₁₆	C₂×C₄	1	16	yes	complemented
26	C₂×#11719910	C₂×C₄	1	8	yes	—
27	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719912	—	2	2	no	—
29	C₂×#11719912	—	2	2	no	—
30	C₂×#11719912	C₂×C₄	1	4	yes	—
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂³	1	1	yes	characteristic
33	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₅₃	C₂³	1	2	yes	—
38	#13133174	—	2	2	no	—
39	#13133174	C₂×C₄	1	4	yes	—
40	C₂×#11719919	—	2	1	no	—
41	C₂×#11719921	—	2	1	no	—
42	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	#13133128	C₂×C₄	1	2	yes	—
44	#13133128	C₂³	1	2	yes	—
45	#13133127	C₂×C₄	1	2	yes	—
46	#13133127	C₂³	1	2	yes	—
47	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
52	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	1	no	—
56	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
57	C₂×C₄×D₅₃	C₄²	1	2	yes	—
58	C₂×C₄×D₅₃	C₄²	1	2	yes	—
59	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
60	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
61	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
62	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
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	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
72	#11719921	—	2	4	no	—
73	#11719919	—	2	4	no	—
74	C₄×Dic₅₃	—	2	1	no	—
75	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
76	C₂×#88322	—	2	1	no	—
77	C₂×#88322	—	2	1	no	—
78	C₂×#88322	C₂²×C₄	1	2	yes	—
79	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
80	C₂×#88322	C₂²×C₄	1	2	yes	—
81	C₂×#88322	C₂²×C₄	1	2	yes	—
82	C₂×#88322	C₂⁴	1	1	yes	characteristic
83	#11719912	—	2	8	no	—
84	C₈×D₅₃	C₄²	1	4	yes	—
85	C₈×D₅₃	C₄²	1	4	yes	—
86	C₈×D₅₃	C₂²×C₄	1	4	yes	—
87	C₈×D₅₃	C₂²×C₄	1	4	yes	—
88	#11719910	C₂²×C₄	1	16	yes	—
89	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂²×D₅₃	—	4	2	no	—
91	C₂²×D₅₃	#97	1	2	yes	—
92	C₂²×D₅₃	#97	1	2	yes	—
93	C₂²×D₅₃	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×Dic₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	4	2	no	—
104	C₂×Dic₅₃	#97	1	2	yes	—
105	C₂×Dic₅₃	#97	1	2	yes	—
106	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
107	C₄×D₅₃	#97	1	4	yes	—
108	C₄×D₅₃	#97	1	4	yes	—
109	C₄×D₅₃	C₂×C₄²	1	4	yes	—
110	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	C₂×C₄²	1	2	yes	—
113	C₈×C₅₃	C₂×C₄²	1	2	yes	—
114	#88322	—	2	2	no	—
115	#88322	C₂×C₄²	1	2	yes	—
116	#88322	C₂³×C₄	1	2	yes	—
117	C₂×#2110	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
121	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×D₅₃	—	2	4	no	—
123	C₂×D₅₃	—	4	2	no	—
124	C₂×D₅₃	—	4	2	no	—
125	C₂×D₅₃	C₂×#97	1	4	yes	—
126	C₄×C₅₃	—	4	2	no	—
127	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
128	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
129	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
130	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
131	Dic₅₃	—	4	2	no	—
132	Dic₅₃	C₂×#97	1	4	yes	—
133	C₂²×M₃₂	—	53	2	no	—
134	C₂×#405	—	53	1	no	—
135	#2110	—	53	8	no	—
136	C₂×#347	—	53	2	no	—
137	C₂×#347	—	53	2	no	—
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	#2110	1	2	yes	complemented
141	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
142	D₅₃	—	2	4	no	—
143	C₂²×M₁₆	—	53	1	no	—
144	C₂³×C₈	—	53	1	no	—
145	C₂×#117	—	53	1	no	—
146	C₂×M₃₂	—	53	4	no	—
147	C₂×M₃₂	—	53	8	no	—
148	C₂×M₃₂	—	53	8	no	—
149	C₂×M₃₂	—	53	4	no	—
150	C₂²×C₁₆	—	53	4	no	—
151	#405	—	53	4	no	—
152	#405	—	53	4	no	—
153	C₂×#97	—	53	1	no	—
154	C₂×#97	—	53	1	no	—
155	C₂×C₄×C₈	—	53	1	no	—
156	C₂×C₄×C₈	#6157	1	1	yes	characteristic
157	#347	—	53	8	no	—
158	#347	—	53	8	no	—
159	C₅₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	—	53	1	no	—
161	C₂×M₁₆	—	53	4	no	—
162	C₂×M₁₆	—	106	2	no	—
163	C₂×M₁₆	—	106	2	no	—
164	C₂²×C₈	—	53	1	no	—
165	C₂²×C₈	—	53	2	no	—
166	C₂²×C₈	—	53	4	no	—
167	C₂²×C₈	—	53	2	no	—
168	C₂²×C₈	—	53	2	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	—	53	1	no	—
171	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
172	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
173	#117	—	53	4	no	—
174	#117	—	106	2	no	—
175	C₂×C₄⋊C₄	—	106	1	no	—
176	C₂×C₂²⋊C₄	—	106	1	no	—
177	C₂×C₄²	—	53	1	no	—
178	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
179	M₃₂	—	53	8	no	—
180	M₃₂	—	53	8	no	—
181	M₃₂	—	53	16	no	—
182	C₂×C₁₆	—	53	8	no	—
183	C₂×C₁₆	—	53	16	no	—
184	#97	—	53	2	no	—
185	#97	—	53	2	no	—
186	#97	—	53	2	no	—
187	#97	—	53	2	no	—
188	C₄×C₈	—	53	4	no	—
189	C₄×C₈	C₂×#6157	1	4	yes	—
190	C₂⁴	—	53	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	53	2	no	—
194	C₂²×C₄	—	53	2	no	—
195	C₂²×C₄	—	53	2	no	—
196	C₂²×C₄	—	53	2	no	—
197	C₂²×C₄	—	53	2	no	—
198	C₂²×C₄	—	53	2	no	—
199	C₂²×C₄	—	106	1	no	—
200	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
201	M₁₆	—	106	8	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	53	2	no	—
205	C₂×C₈	—	53	2	no	—
206	C₂×C₈	—	53	1	no	—
207	C₂×C₈	—	53	1	no	—
208	C₂×C₈	—	53	4	no	—
209	C₂×C₈	—	53	4	no	—
210	C₂×C₈	—	53	4	no	—
211	C₂×C₈	—	53	4	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	106	1	no	—
214	C₂×C₈	—	106	1	no	—
215	C₂×C₈	C₄×#6157	1	1	yes	characteristic
216	C₂×C₈	C₄×#6157	1	2	yes	—
217	C₂×C₈	C₄×#6157	1	1	yes	characteristic
218	C₂×C₈	C₂²×#6157	1	2	yes	—
219	C₂×C₈	C₂²×#6157	1	2	yes	—
220	C₄⋊C₄	—	106	4	no	—
221	C₂²⋊C₄	—	106	4	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	53	2	no	—
224	C₄²	—	106	1	no	—
225	C₄²	C₂²×#6157	1	2	yes	—
226	C₁₆	—	53	16	no	—
227	C₂³	—	53	2	no	—
228	C₂³	—	53	2	no	—
229	C₂³	—	53	2	no	—
230	C₂³	—	212	2	no	—
231	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	53	2	no	—
235	C₂×C₄	—	53	2	no	—
236	C₂×C₄	—	53	2	no	—
237	C₂×C₄	—	53	4	no	—
238	C₂×C₄	—	53	4	no	—
239	C₂×C₄	—	53	4	no	—
240	C₂×C₄	—	53	4	no	—
241	C₂×C₄	—	106	2	no	—
242	C₂×C₄	—	212	2	no	—
243	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
245	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
246	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
247	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
248	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	53	2	no	—
251	C₈	—	53	2	no	—
252	C₈	—	106	2	no	—
253	C₈	C₂×C₄×#6157	1	2	yes	—
254	C₈	C₂×C₄×#6157	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	53	4	no	—
257	C₂²	—	106	4	no	—
258	C₂²	—	212	2	no	—
259	C₂²	—	212	2	no	—
260	C₂²	#16811582	1	1	yes	characteristic
261	C₂²	#16811582	1	1	yes	characteristic
262	C₂²	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	53	4	no	—
265	C₄	—	212	2	no	—
266	C₄	#16811582	1	1	yes	characteristic
267	C₄	#16811582	1	1	yes	characteristic
268	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
269	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	106	4	no	—
273	C₂	#20290937	1	2	yes	direct product factor
274	C₂	C₂×#16811582	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₂×#2110
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20290937	C₂	direct (N × H)
3. C₂×#16811516	C₂	semidirect
4. C₂×#16811515	C₂	semidirect
6. C₂×#13133292	C₄	semidirect
7. D₅₃⋊M₃₂	C₄	semidirect
10. C₂×D₅₃⋊C₁₆	C₄	semidirect
11. #16811516	C₂²	semidirect
12. #16811515	C₂²	semidirect
22. #13133292	C₂×C₄	semidirect
25. D₅₃⋊C₁₆	C₂×C₄	semidirect
140. C₂×C₅₃	#2110	semidirect
159. C₅₃	C₂×#2110	semidirect
273. C₂	#20290937	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×D₅₃⋊M₃₂	2
#20290937	2
C₂×#16811516	2
C₂×#16811515	2
C₂×#16810577	2
C₂×#2110	53

Subgroups — C2×#20290937 #28327927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20290937 #28327927