Subgroups — C₂×#20272544 #27273488

Order 13568.

758 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×#16810676	C₂	1	2	yes	maximal
2	#20272544	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
4	#16810676	—	2	2	no	—
5	#16810676	—	2	2	no	—
6	#16810676	C₄	1	4	yes	complemented
7	#16810676	C₂²	1	4	yes	—
8	C₂×C₄⋊#88322	—	2	2	no	—
9	C₂×C₂²⋊#88322	—	2	2	no	—
10	C₂×C₄×#88322	C₄	1	4	yes	complemented
11	C₂²×C₄²×C₅₃	C₄	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#376	—	2	2	no	—
14	C₅₃×#376	C₄	1	2	yes	—
15	C₅₃×#376	C₂²	1	2	yes	—
16	C₂×#95×C₅₃	—	2	2	no	—
17	C₂²×#88322	—	2	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₂²⋊#88322	—	2	4	no	—
21	C₂²⋊#88322	—	2	4	no	—
22	C₄×#88322	—	2	4	no	—
23	C₄×#88322	C₂×C₄	1	8	yes	complemented
24	C₂³×C₄×C₅₃	—	2	2	no	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	—	2	2	no	—
28	C₂×C₄²×C₅₃	—	2	4	no	—
29	C₂×C₄²×C₅₃	C₈	1	8	yes	complemented
30	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	#95×C₅₃	—	2	4	no	—
41	#95×C₅₃	—	2	4	no	—
42	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	8	no	—
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	—	4	4	no	—
46	C₂²×C₄×C₅₃	—	4	8	no	—
47	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₄²×C₅₃	—	2	8	no	—
62	C₄²×C₅₃	—	2	4	no	—
63	C₄²×C₅₃	C₄²	1	2	yes	—
64	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×C₈	1	8	yes	complemented
71	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×#88322	—	2	8	no	—
75	C₂×#88322	—	4	8	no	—
76	C₂³×C₅₃	—	2	4	no	—
77	C₂³×C₅₃	#95	1	1	yes	characteristic
78	C₂³×C₅₃	#95	1	1	yes	characteristic
79	C₂³×C₅₃	C₂²⋊C₈	1	4	yes	—
80	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	8	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	8	no	—
89	C₂×C₄×C₅₃	—	4	16	no	—
90	C₂×C₄×C₅₃	#95	1	2	yes	—
91	C₂×C₄×C₅₃	#95	1	2	yes	—
92	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
96	C₂×C₄×C₅₃	#97	1	2	yes	—
97	C₂×C₄×C₅₃	#97	1	2	yes	—
98	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
99	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
100	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
107	#88322	—	4	8	no	—
108	C₂×#1274	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	2	8	no	—
112	C₂²×C₅₃	#336	1	2	yes	—
113	C₂²×C₅₃	#336	1	2	yes	—
114	C₂²×C₅₃	H(3,ℤ/4)	1	4	yes	—
115	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	4	8	no	—
122	C₄×C₅₃	#336	1	2	yes	—
123	C₄×C₅₃	#336	1	2	yes	—
124	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
125	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
126	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
127	C₂×#431	—	53	2	no	—
128	C₂×#376	D₅₃	1	1	yes	characteristic
129	#1274	—	53	4	no	—
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	—	2	2	no	—
132	C₂×C₅₃	#1274	1	4	yes	complemented
133	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
134	C₂×C₅₃	C₂×H(3,ℤ/4)	1	2	yes	—
135	C₂²×C₄²	Dic₅₃	1	2	yes	—
136	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
137	#431	—	53	4	no	—
138	#431	—	53	4	no	—
139	#431	—	106	2	no	—
140	#431	—	106	2	no	—
141	C₂×C₄⋊C₈	—	106	2	no	—
142	C₂×C₂²⋊C₈	—	106	2	no	—
143	C₂×C₄×C₈	—	53	4	no	—
144	#376	—	2	2	no	—
145	#376	Dic₅₃	1	2	yes	—
146	#376	C₂×D₅₃	1	2	yes	—
147	C₂×#95	—	2	2	no	—
148	C₅₃	C₂×#1274	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	—	2	2	no	—
150	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
151	C₂³×C₄	C₂²⋊D₅₃	1	2	yes	—
152	C₂²×C₈	—	106	4	no	—
153	C₂×C₄²	—	2	4	no	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	#88322	1	8	yes	complemented
156	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
157	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
158	C₂×C₄²	C₄×D₅₃	1	2	yes	—
159	C₂×C₄²	C₄×D₅₃	1	2	yes	—
160	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
161	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
163	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
165	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
166	C₄⋊C₈	—	106	4	no	—
167	C₄⋊C₈	—	106	4	no	—
168	C₂²⋊C₈	—	106	4	no	—
169	C₂²⋊C₈	—	106	4	no	—
170	C₄×C₈	—	53	8	no	—
171	C₄×C₈	—	106	4	no	—
172	#95	—	2	4	no	—
173	#95	—	2	4	no	—
174	C₂⁴	C₂²⋊Dic₅₃	1	1	yes	characteristic
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	—	4	8	no	—
179	C₂²×C₄	C₂×#88322	1	4	yes	—
180	C₂²×C₄	#11719917	1	4	yes	—
181	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
183	C₂²×C₄	#11719919	1	2	yes	—
184	C₂²×C₄	#11719919	1	2	yes	—
185	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
187	C₂²×C₄	#11719921	1	2	yes	—
188	C₂²×C₄	#11719921	1	2	yes	—
189	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
190	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
193	C₂×C₈	—	106	8	no	—
194	C₂×C₈	—	212	8	no	—
195	C₄²	—	2	8	no	—
196	C₄²	—	2	4	no	—
197	C₄²	C₂×#88322	1	8	yes	complemented
198	C₄²	C₄×Dic₅₃	1	2	yes	—
199	C₄²	#11719919	1	2	yes	—
200	C₄²	#11719919	1	2	yes	—
201	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
202	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
203	C₄²	#11719921	1	2	yes	—
204	C₄²	#11719921	1	2	yes	—
205	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
206	C₄²	C₂×C₄×D₅₃	1	2	yes	—
207	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
208	C₂³	—	2	4	no	—
209	C₂³	#13133124	1	1	yes	characteristic
210	C₂³	#13133124	1	1	yes	characteristic
211	C₂³	C₂²⋊#88322	1	4	yes	—
212	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	16	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	#13133124	1	2	yes	—
223	C₂×C₄	#13133124	1	2	yes	—
224	C₂×C₄	#13133124	1	1	yes	characteristic
225	C₂×C₄	#13133124	1	1	yes	characteristic
226	C₂×C₄	C₄×#88322	1	2	yes	—
227	C₂×C₄	C₄×#88322	1	2	yes	—
228	C₂×C₄	#13133128	1	2	yes	—
229	C₂×C₄	#13133128	1	2	yes	—
230	C₂×C₄	C₂²⋊#88322	1	4	yes	—
231	C₂×C₄	C₄⋊#88322	1	4	yes	—
232	C₂×C₄	C₄⋊#88322	1	4	yes	—
233	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×#11719921	1	2	yes	—
235	C₂×C₄	C₂×#11719919	1	2	yes	—
236	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂²×#88322	1	2	yes	—
238	C₂×C₄	C₂×#11719917	1	2	yes	—
239	C₈	—	212	8	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	#16810378	1	2	yes	—
244	C₂²	#16810378	1	2	yes	—
245	C₂²	#16810379	1	4	yes	—
246	C₂²	C₂×#13133124	1	1	yes	characteristic
247	C₂²	C₂×C₂²⋊#88322	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	8	no	—
253	C₄	#16810378	1	2	yes	—
254	C₄	#16810378	1	2	yes	—
255	C₄	C₂×C₄×#88322	1	1	yes	characteristic
256	C₄	C₂×#13133128	1	1	yes	characteristic
257	C₄	C₂×C₄⋊#88322	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	2	2	no	—
260	C₂	#20272544	1	4	yes	direct product factor
261	C₂	C₂×#16810378	1	1	yes	characteristic
262	C₂	C₂×#16810379	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#376	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#376	D₅₃
hypocenter	C₅₃	C₂×#1274
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #20272544	C₂	direct (N × H)
6. #16810676	C₄	semidirect
10. C₂×C₄×#88322	C₄	semidirect
23. C₄×#88322	C₂×C₄	semidirect
29. C₂×C₄²×C₅₃	C₈	semidirect
70. C₄²×C₅₃	C₂×C₈	semidirect
132. C₂×C₅₃	#1274	semidirect
148. C₅₃	C₂×#1274	semidirect
155. C₂×C₄²	#88322	semidirect
197. C₄²	C₂×#88322	semidirect
260. C₂	#20272544	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
#20272544	2
C₂×C₅₃×#376	2
C₂×#1274	53

Subgroups — C2×#20272544 #27273488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272544 #27273488