Subgroups — C₂×#20273478 #27353099

Order 13568.

766 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×#16810680	C₂	1	2	yes	maximal
2	#20273478	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810376	C₂	1	2	yes	maximal, complemented
4	C₂×#16810374	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810680	—	2	4	no	—
7	#16810680	C₂²	1	8	yes	—
8	C₂×C₄⋊#88322	C₂²	1	4	yes	—
9	C₂×C₂²⋊#88322	—	2	2	no	—
10	C₂×#13133128	—	2	2	no	—
11	#16810376	C₂²	1	8	yes	complemented
12	#16810374	C₂²	1	8	yes	complemented
13	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
18	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
22	C₂²×#88322	—	2	4	no	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	C₂³	1	8	yes	—
25	C₂²⋊#88322	—	2	8	no	—
26	#13133128	—	2	8	no	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	—	2	4	no	—
31	C₂²×C₈×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
46	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₅₃	C₂³	1	2	yes	—
51	C₄×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	8	no	—
72	C₂×C₈×C₅₃	—	2	8	no	—
73	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
74	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
80	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	2	yes	—
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
87	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₂×#88322	—	2	8	no	—
90	C₂×#88322	—	4	8	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	#108	1	4	yes	—
94	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₅₃	C₂×C₄⋊C₄	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	2	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₅₃	#135	1	4	yes	—
116	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	2	2	no	—
121	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
122	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
123	#88322	—	4	8	no	—
124	C₂×#1568	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	2	8	no	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	#423	1	2	yes	—
129	C₂²×C₅₃	#427	1	2	yes	—
130	C₂²×C₅₃	C₂×#108	1	4	yes	—
131	C₂²×C₅₃	C₂×#108	1	2	yes	—
132	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	#423	1	2	yes	—
139	C₄×C₅₃	#427	1	2	yes	—
140	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
141	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#135	1	2	yes	—
143	C₂×#432	—	53	2	no	—
144	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
145	#1568	—	53	8	no	—
146	C₂×#335	—	53	2	no	—
147	C₂×#334	—	53	2	no	—
148	C₂×C₅₃	—	2	4	no	—
149	C₂×C₅₃	#1568	1	4	yes	complemented
150	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
152	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
153	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
154	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
155	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
156	#432	—	53	8	no	—
157	#432	—	106	4	no	—
158	C₂×C₄⋊C₈	—	53	4	no	—
159	C₂×C₂²⋊C₈	—	106	2	no	—
160	C₂×#97	—	106	2	no	—
161	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
162	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
163	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×D₅₃	1	4	yes	—
167	#335	—	53	8	no	—
168	#334	—	53	8	no	—
169	C₅₃	C₂×#1568	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
171	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
172	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	106	4	no	—
176	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
178	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
179	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
180	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
182	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
183	C₂×C₄²	Dic₁₀₆	1	2	yes	—
184	C₂×C₄²	D₂₁₂	1	2	yes	—
185	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
187	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
189	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
190	C₄⋊C₈	—	53	8	no	—
191	C₄⋊C₈	—	106	4	no	—
192	C₂²⋊C₈	—	106	8	no	—
193	#97	—	106	8	no	—
194	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
195	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
196	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
197	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
198	C₄×C₈	C₂²×D₅₃	1	2	yes	—
199	C₄×C₈	C₂²×D₅₃	1	2	yes	—
200	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
207	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
209	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
210	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
212	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
214	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	106	8	no	—
222	C₂×C₈	—	212	8	no	—
223	C₂×C₈	#11719917	1	4	yes	—
224	C₂×C₈	#11719917	1	4	yes	—
225	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
226	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
227	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
228	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
229	C₄²	C₄⋊Dic₅₃	1	2	yes	—
230	C₄²	C₄⋊Dic₅₃	1	2	yes	—
231	C₄²	C₄⋊Dic₅₃	1	2	yes	—
232	C₄²	C₂×Dic₁₀₆	1	2	yes	—
233	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
234	C₄²	C₂×D₂₁₂	1	2	yes	—
235	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
236	C₄²	C₂²×Dic₅₃	1	2	yes	—
237	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
238	C₄²	C₂³×D₅₃	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	#13133153	1	4	yes	—
242	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
243	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
256	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#11719917	1	2	yes	—
262	C₂×C₄	C₂×#11719917	1	2	yes	—
263	C₂×C₄	#13133239	1	4	yes	—
264	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
266	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₈	—	212	8	no	—
270	C₈	C₂×#11719917	1	4	yes	—
271	C₈	C₂×#11719917	1	4	yes	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	#16810647	1	2	yes	—
276	C₂²	#16810661	1	2	yes	—
277	C₂²	C₂×#13133153	1	4	yes	—
278	C₂²	C₂×#13133153	1	2	yes	—
279	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	#16810647	1	2	yes	—
286	C₄	#16810661	1	2	yes	—
287	C₄	C₂²×#11719917	1	1	yes	characteristic
288	C₄	C₂²×#11719917	1	1	yes	characteristic
289	C₄	C₂×#13133239	1	2	yes	—
290	C₂	—	2	4	no	—
291	C₂	#20273478	1	4	yes	direct product factor
292	C₂	C₂×#16810647	1	1	yes	characteristic
293	C₂	C₂×#16810661	1	1	yes	characteristic
294	C₂	C₂²×#13133153	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#1568
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20273478	C₂	direct (N × H)
3. C₂×#16810376	C₂	semidirect
4. C₂×#16810374	C₂	semidirect
11. #16810376	C₂²	semidirect
12. #16810374	C₂²	semidirect
149. C₂×C₅₃	#1568	semidirect
169. C₅₃	C₂×#1568	semidirect
291. C₂	#20273478	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×#16810680	2
#20273478	2
C₂×#16810376	2
C₂×#16810374	2
C₂²×C₄×C₈×C₅₃	2
C₂×#1568	53

Subgroups — C2×#20273478 #27353099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273478 #27353099