Subgroups — C₂×#20276503 #27537126

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
3	#20276503	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810451	C₂	1	2	yes	maximal, complemented
5	C₂×#16810449	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133176	—	2	1	no	—
8	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
9	#16810661	—	2	2	no	—
10	#16810661	C₂²	1	4	yes	—
11	C₂×#13133151	—	2	1	no	—
12	C₂×#13133151	—	2	2	no	—
13	C₂×#13133151	C₂²	1	1	yes	characteristic
14	C₂×#13133151	C₂²	1	2	yes	—
15	C₂×#13133151	C₂²	1	1	yes	characteristic
16	C₂×#13133151	C₂²	1	4	yes	—
17	C₂×#13133149	—	2	1	no	—
18	#16810451	C₂²	1	8	yes	complemented
19	#16810449	C₂²	1	8	yes	complemented
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₅₃	C₄	1	8	yes	complemented
22	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
23	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
24	C₂³×Dic₅₃	—	4	1	no	—
25	#13133176	—	2	4	no	—
26	#13133176	—	4	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	4	no	—
29	C₂×C₄⋊Dic₅₃	—	2	1	no	—
30	C₂×C₄⋊Dic₅₃	—	2	1	no	—
31	C₂×C₄⋊Dic₅₃	—	4	1	no	—
32	C₂×C₄⋊Dic₅₃	—	4	2	no	—
33	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
34	C₂×C₄×Dic₅₃	—	4	1	no	—
35	#13133151	—	2	4	no	—
36	#13133151	—	2	2	no	—
37	#13133151	—	2	1	no	—
38	#13133151	—	2	1	no	—
39	#13133151	C₂³	1	4	yes	—
40	#13133151	C₂³	1	2	yes	—
41	#13133151	C₂³	1	2	yes	—
42	#13133149	—	2	4	no	—
43	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	D₄	1	2	yes	—
47	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	M₃₂×C₅₃	C₂×C₄	1	16	yes	complemented
51	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
52	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
53	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	D₈	1	2	yes	—
57	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×Dic₅₃	—	4	1	no	—
61	C₂²×Dic₅₃	—	4	1	no	—
62	C₂²×Dic₅₃	—	4	4	no	—
63	C₂²×Dic₅₃	—	4	2	no	—
64	C₂²×Dic₅₃	—	8	4	no	—
65	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂²⋊Dic₅₃	—	4	4	no	—
75	C₄⋊Dic₅₃	—	2	2	no	—
76	C₄⋊Dic₅₃	—	2	2	no	—
77	C₄⋊Dic₅₃	—	2	4	no	—
78	C₄⋊Dic₅₃	—	4	4	no	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₄⋊Dic₅₃	—	4	1	no	—
81	C₄⋊Dic₅₃	—	4	1	no	—
82	C₄×Dic₅₃	—	4	4	no	—
83	C₁₆×C₅₃	C₂²×C₄	1	8	yes	complemented
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	#107	1	4	yes	—
87	C₂³×C₅₃	C₂×D₈	1	2	yes	—
88	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#107	1	8	yes	—
90	C₂×C₄×C₅₃	#107	1	4	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₅₃	—	4	4	no	—
99	C₂×Dic₅₃	—	4	2	no	—
100	C₂×Dic₅₃	—	4	2	no	—
101	C₂×Dic₅₃	—	8	8	no	—
102	C₂×Dic₅₃	—	8	2	no	—
103	C₂×Dic₅₃	—	8	4	no	—
104	C₂×Dic₅₃	—	8	2	no	—
105	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×#2159	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	8	no	—
113	C₂²×C₅₃	C₂×#107	1	4	yes	—
114	C₂²×C₅₃	C₂×#107	1	2	yes	—
115	C₂²×C₅₃	#509	1	2	yes	—
116	C₂²×C₅₃	#510	1	2	yes	—
117	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
118	C₄×C₅₃	C₂×#107	1	4	yes	—
119	C₄×C₅₃	C₂×#107	1	2	yes	—
120	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
122	Dic₅₃	—	8	2	no	—
123	Dic₅₃	—	8	4	no	—
124	Dic₅₃	—	8	2	no	—
125	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
126	C₂×#427	—	53	1	no	—
127	C₂²×#107	—	53	1	no	—
128	#2159	—	53	8	no	—
129	C₂×#367	—	53	2	no	—
130	C₂×#366	—	53	2	no	—
131	C₂×C₅₃	—	2	4	no	—
132	C₂×C₅₃	#2159	1	4	yes	complemented
133	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
136	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
137	C₂×#117	—	106	1	no	—
138	C₂²×C₄⋊C₄	—	106	1	no	—
139	C₂×M₃₂	Dic₅₃	1	8	yes	complemented
140	C₂×M₃₂	C₂×D₅₃	1	4	yes	—
141	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
142	#427	—	53	4	no	—
143	#427	—	106	2	no	—
144	C₂×#107	—	53	2	no	—
145	C₂×#107	—	53	4	no	—
146	C₂×#107	—	53	1	no	—
147	C₂×#107	—	53	1	no	—
148	C₂×#107	—	106	1	no	—
149	C₂×#107	—	106	2	no	—
150	C₂×#106	—	106	1	no	—
151	#367	—	53	8	no	—
152	#366	—	53	8	no	—
153	C₅₃	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	—	212	1	no	—
155	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
156	C₂²×C₈	Dic₁₀₆	1	4	yes	—
157	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
158	C₂²×C₈	D₂₁₂	1	2	yes	—
159	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
160	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
162	#117	—	106	4	no	—
163	#117	—	212	2	no	—
164	C₂×C₄⋊C₄	—	106	4	no	—
165	C₂×C₄⋊C₄	—	106	2	no	—
166	C₂×C₄⋊C₄	—	106	1	no	—
167	C₂×C₄⋊C₄	—	106	1	no	—
168	C₂×C₄⋊C₄	—	212	2	no	—
169	C₂×C₄⋊C₄	—	212	1	no	—
170	C₂×C₂²⋊C₄	—	212	1	no	—
171	C₂×C₄²	—	212	1	no	—
172	M₃₂	C₂×Dic₅₃	1	16	yes	complemented
173	C₂×C₁₆	C₂×Dic₅₃	1	8	yes	complemented
174	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
175	#107	—	53	2	no	—
176	#107	—	53	4	no	—
177	#107	—	53	2	no	—
178	#107	—	106	4	no	—
179	#107	—	106	2	no	—
180	#107	—	106	1	no	—
181	#107	—	106	1	no	—
182	#106	—	106	4	no	—
183	C₂⁴	D₄₂₄	1	1	yes	characteristic
184	C₂²×C₄	—	424	4	no	—
185	C₂²×C₄	—	212	4	no	—
186	C₂²×C₄	—	212	1	no	—
187	C₂²×C₄	—	212	1	no	—
188	C₂²×C₄	—	212	2	no	—
189	C₂²×C₄	D₄₂₄	1	2	yes	—
190	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
191	C₂²×C₄	Dic₂₁₂	1	4	yes	—
192	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
193	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
194	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
195	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
196	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
197	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
198	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
199	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
200	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
202	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
203	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
204	C₄⋊C₄	—	106	2	no	—
205	C₄⋊C₄	—	106	2	no	—
206	C₄⋊C₄	—	106	4	no	—
207	C₄⋊C₄	—	212	4	no	—
208	C₄⋊C₄	—	212	2	no	—
209	C₄⋊C₄	—	212	1	no	—
210	C₄⋊C₄	—	212	1	no	—
211	C₂²⋊C₄	—	212	4	no	—
212	C₄²	—	212	4	no	—
213	C₁₆	C₂²×Dic₅₃	1	8	yes	complemented
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	#13133151	1	4	yes	—
217	C₂³	C₂×D₄₂₄	1	2	yes	—
218	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
219	C₂×C₄	—	424	8	no	—
220	C₂×C₄	—	424	4	no	—
221	C₂×C₄	—	424	2	no	—
222	C₂×C₄	—	424	2	no	—
223	C₂×C₄	—	212	2	no	—
224	C₂×C₄	—	212	2	no	—
225	C₂×C₄	—	212	4	no	—
226	C₂×C₄	#13133151	1	8	yes	—
227	C₂×C₄	#13133151	1	4	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
230	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
231	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
232	C₂×C₄	C₂×Dic₂₁₂	1	4	yes	—
233	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
234	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
235	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
236	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
237	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
238	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	C₂×#13133151	1	4	yes	—
243	C₂²	C₂×#13133151	1	2	yes	—
244	C₂²	#16811061	1	2	yes	—
245	C₂²	#16811066	1	2	yes	—
246	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
247	C₄	—	424	4	no	—
248	C₄	—	424	2	no	—
249	C₄	—	424	2	no	—
250	C₄	C₂×#13133151	1	4	yes	—
251	C₄	C₂×#13133151	1	2	yes	—
252	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
253	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#20276503	1	4	yes	direct product factor
256	C₂	C₂²×#13133151	1	1	yes	characteristic
257	C₂	C₂×#16811061	1	1	yes	characteristic
258	C₂	C₂×#16811066	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#2159
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20276503	C₂	direct (N × H)
4. C₂×#16810451	C₂	semidirect
5. C₂×#16810449	C₂	semidirect
18. #16810451	C₂²	semidirect
19. #16810449	C₂²	semidirect
21. C₂×M₃₂×C₅₃	C₄	semidirect
50. M₃₂×C₅₃	C₂×C₄	semidirect
51. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
83. C₁₆×C₅₃	C₂²×C₄	semidirect
132. C₂×C₅₃	#2159	semidirect
139. C₂×M₃₂	Dic₅₃	semidirect
153. C₅₃	C₂×#2159	semidirect
172. M₃₂	C₂×Dic₅₃	semidirect
173. C₂×C₁₆	C₂×Dic₅₃	semidirect
213. C₁₆	C₂²×Dic₅₃	semidirect
255. C₂	#20276503	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×#16810661	2
C₂²×#13133151	2
#20276503	2
C₂×#16810451	2
C₂×#16810449	2
C₂²×M₃₂×C₅₃	2
C₂×#2159	53

Subgroups — C2×#20276503 #27537126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276503 #27537126