Subgroups — C₂×#20272724 #27294311

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810386	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272724	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#466	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
6	C₂²×#11719908	—	2	1	no	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₂²⋊#11719908	C₂²	1	2	yes	complemented
9	#16810386	—	2	1	no	—
10	#16810386	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#121×C₅₃	—	2	1	no	—
14	C₅₃×#466	—	2	1	no	—
15	C₅₃×#466	—	2	1	no	—
16	C₅₃×#466	C₄	1	2	yes	complemented
17	C₅₃×#466	C₂²	1	2	yes	—
18	C₂×#107×C₅₃	C₄	1	1	yes	characteristic, complemented
19	C₂×#102×C₅₃	—	2	1	no	—
20	C₂³×Dic₅₃	—	4	1	no	—
21	C₂×C₄⋊Dic₅₃	—	2	4	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×C₄⋊Dic₅₃	—	4	2	no	—
24	C₂×C₄⋊Dic₅₃	—	4	1	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂×#11719908	—	4	2	no	—
27	C₂²×D₄×C₅₃	—	2	1	no	—
28	C₂²×D₄×C₅₃	—	4	1	no	—
29	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂×D₈×C₅₃	—	2	1	no	—
31	C₂×D₈×C₅₃	—	4	2	no	—
32	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	complemented
33	C₂²×C₈×C₅₃	—	4	2	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	#121×C₅₃	—	2	2	no	—
41	#121×C₅₃	—	2	2	no	—
42	#121×C₅₃	—	4	1	no	—
43	#121×C₅₃	—	4	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
46	#107×C₅₃	—	2	1	no	—
47	#107×C₅₃	C₂×C₄	1	2	yes	complemented
48	#102×C₅₃	—	2	2	no	—
49	#102×C₅₃	—	2	2	no	—
50	C₂⁴×C₅₃	—	4	1	no	—
51	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	4	1	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	8	2	no	—
59	C₂×D₄×C₅₃	—	8	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₅₃	—	4	4	no	—
70	C₂²×Dic₅₃	—	4	2	no	—
71	C₂²×Dic₅₃	—	8	2	no	—
72	C₂²×Dic₅₃	—	8	2	no	—
73	D₈×C₅₃	—	4	2	no	—
74	D₈×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	4	1	no	—
80	C₂×C₈×C₅₃	—	4	1	no	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄⋊C₄×C₅₃	—	4	1	no	—
87	C₂²⋊C₄×C₅₃	—	4	2	no	—
88	C₂²⋊C₄×C₅₃	—	4	2	no	—
89	C₄⋊Dic₅₃	—	2	4	no	—
90	C₄⋊Dic₅₃	—	4	4	no	—
91	C₄⋊Dic₅₃	—	4	2	no	—
92	#11719908	—	4	2	no	—
93	C₂³×C₅₃	—	2	2	no	—
94	C₂³×C₅₃	—	2	2	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	—	8	2	no	—
97	C₂³×C₅₃	—	8	2	no	—
98	C₂³×C₅₃	—	8	2	no	—
99	C₂³×C₅₃	#102	1	1	yes	characteristic
100	C₂³×C₅₃	#102	1	1	yes	characteristic
101	C₂³×C₅₃	D₁₆	1	2	yes	—
102	C₂³×C₅₃	SD₃₂	1	2	yes	—
103	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
104	D₄×C₅₃	—	4	2	no	—
105	D₄×C₅₃	—	4	2	no	—
106	D₄×C₅₃	—	8	4	no	—
107	D₄×C₅₃	—	8	2	no	—
108	D₄×C₅₃	—	8	2	no	—
109	D₄×C₅₃	—	8	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	8	2	no	—
119	C₂×C₄×C₅₃	#99	1	2	yes	—
120	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#104	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×Dic₅₃	—	4	4	no	—
126	C₂×Dic₅₃	—	8	8	no	—
127	C₂×Dic₅₃	—	8	2	no	—
128	C₂×Dic₅₃	—	8	2	no	—
129	C₈×C₅₃	—	2	1	no	—
130	C₈×C₅₃	—	2	1	no	—
131	C₈×C₅₃	—	2	1	no	—
132	C₈×C₅₃	—	2	1	no	—
133	C₂×#1347	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	4	no	—
135	C₂²×C₅₃	—	2	4	no	—
136	C₂²×C₅₃	—	2	1	no	—
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	—	2	1	no	—
140	C₂²×C₅₃	—	8	2	no	—
141	C₂²×C₅₃	—	8	2	no	—
142	C₂²×C₅₃	—	8	2	no	—
143	C₂²×C₅₃	—	16	4	no	—
144	C₂²×C₅₃	#327	1	2	yes	—
145	C₂²×C₅₃	#357	1	2	yes	—
146	C₂²×C₅₃	#357	1	2	yes	—
147	C₂²×C₅₃	#360	1	2	yes	—
148	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
149	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
150	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	8	2	no	—
154	C₄×C₅₃	#327	1	2	yes	—
155	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
157	Dic₅₃	—	8	4	no	—
158	C₂×#466	D₅₃	1	1	yes	characteristic
159	C₂×C₂²⋊C₁₆	—	53	1	no	—
160	C₂×#340	—	53	1	no	—
161	#1347	—	53	4	no	—
162	C₂×C₅₃	—	2	1	no	—
163	C₂×C₅₃	—	2	2	no	—
164	C₂×C₅₃	—	2	1	no	—
165	C₂×C₅₃	—	16	2	no	—
166	C₂×C₅₃	#1347	1	4	yes	complemented
167	C₂×C₅₃	C₂×#327	1	1	yes	characteristic
168	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#360	1	1	yes	characteristic
170	C₂²×D₈	Dic₅₃	1	1	yes	characteristic, complemented
171	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
172	C₂×#121	—	2	1	no	—
173	C₂²×C₄⋊C₄	—	106	1	no	—
174	C₂²×C₁₆	—	106	1	no	—
175	#466	—	2	1	no	—
176	#466	—	2	1	no	—
177	#466	Dic₅₃	1	2	yes	complemented
178	#466	C₂×D₅₃	1	2	yes	—
179	C₂×#107	Dic₅₃	1	1	yes	characteristic, complemented
180	C₂×#102	—	2	1	no	—
181	C₂²⋊C₁₆	—	53	2	no	—
182	C₂²⋊C₁₆	—	106	1	no	—
183	#340	—	53	2	no	—
184	#340	—	106	1	no	—
185	C₅₃	C₂×#1347	1	1	yes	characteristic, complemented, Sylow
186	C₂²×D₄	—	2	1	no	—
187	C₂²×D₄	—	4	1	no	—
188	C₂³×C₄	—	212	1	no	—
189	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	4	2	no	—
192	C₂×D₈	C₂×Dic₅₃	1	2	yes	complemented
193	C₂²×C₈	—	4	2	no	—
194	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
195	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
200	#121	—	2	2	no	—
201	#121	—	2	2	no	—
202	#121	—	4	1	no	—
203	#121	—	4	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	106	4	no	—
206	C₂×C₄⋊C₄	—	106	2	no	—
207	C₂×C₄⋊C₄	—	212	2	no	—
208	C₂×C₄⋊C₄	—	212	1	no	—
209	C₂×C₂²⋊C₄	—	4	1	no	—
210	C₂×C₁₆	—	106	2	no	—
211	C₂×C₁₆	—	212	2	no	—
212	#107	—	2	1	no	—
213	#107	C₂×Dic₅₃	1	2	yes	complemented
214	#102	—	2	2	no	—
215	#102	—	2	2	no	—
216	C₂⁴	—	4	1	no	—
217	C₂⁴	#11719922	1	1	yes	characteristic
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	8	2	no	—
225	C₂×D₄	—	8	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	424	2	no	—
230	C₂²×C₄	—	424	2	no	—
231	C₂²×C₄	—	212	4	no	—
232	C₂²×C₄	—	212	2	no	—
233	C₂²×C₄	#11719922	1	1	yes	characteristic
234	C₂²×C₄	#11719923	1	1	yes	characteristic
235	C₂²×C₄	#11719923	1	1	yes	characteristic
236	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
239	D₈	—	4	2	no	—
240	D₈	—	4	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
249	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	106	4	no	—
254	C₄⋊C₄	—	212	2	no	—
255	C₄⋊C₄	—	212	4	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₁₆	—	212	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	8	2	no	—
263	C₂³	—	8	2	no	—
264	C₂³	—	8	2	no	—
265	C₂³	#13133138	1	1	yes	characteristic
266	C₂³	#13133138	1	1	yes	characteristic
267	C₂³	#13133158	1	2	yes	—
268	C₂³	#13133161	1	2	yes	—
269	C₂³	C₂×#11719922	1	1	yes	characteristic
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	8	2	no	—
273	D₄	—	8	2	no	—
274	D₄	—	8	4	no	—
275	D₄	—	8	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	424	8	no	—
286	C₂×C₄	—	424	2	no	—
287	C₂×C₄	—	424	2	no	—
288	C₂×C₄	—	212	4	no	—
289	C₂×C₄	#13133132	1	2	yes	—
290	C₂×C₄	#13133138	1	1	yes	characteristic
291	C₂×C₄	#13133138	1	1	yes	characteristic
292	C₂×C₄	#13133144	1	2	yes	—
293	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	16	4	no	—
309	C₂²	#16810357	1	2	yes	—
310	C₂²	#16810424	1	2	yes	—
311	C₂²	#16810424	1	2	yes	—
312	C₂²	#16810433	1	2	yes	—
313	C₂²	C₂×#13133138	1	1	yes	characteristic
314	C₂²	C₂×#13133158	1	1	yes	characteristic
315	C₂²	C₂×#13133161	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	424	4	no	—
320	C₄	#16810357	1	2	yes	—
321	C₄	C₂×#13133132	1	1	yes	characteristic
322	C₄	C₂×#13133144	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	16	2	no	—
327	C₂	#20272724	1	4	yes	direct product factor
328	C₂	C₂×#16810357	1	1	yes	characteristic
329	C₂	C₂×#16810424	1	1	yes	characteristic
330	C₂	C₂×#16810433	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#466	C₂
socle	C₂³×C₅₃	#102
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#466	D₅₃
hypocenter	C₅₃	C₂×#1347
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊#11719908	C₂	semidirect
2. C₂×#16810386	C₂	semidirect
3. #20272724	C₂	direct (N × H)
8. C₂²⋊#11719908	C₂²	semidirect
10. #16810386	C₂²	semidirect
11. C₂²×D₈×C₅₃	C₄	semidirect
16. C₅₃×#466	C₄	semidirect
18. C₂×#107×C₅₃	C₄	semidirect
32. C₂×D₈×C₅₃	C₂×C₄	semidirect
47. #107×C₅₃	C₂×C₄	semidirect
166. C₂×C₅₃	#1347	semidirect
170. C₂²×D₈	Dic₅₃	semidirect
177. #466	Dic₅₃	semidirect
179. C₂×#107	Dic₅₃	semidirect
185. C₅₃	C₂×#1347	semidirect
192. C₂×D₈	C₂×Dic₅₃	semidirect
213. #107	C₂×Dic₅₃	semidirect
327. C₂	#20272724	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×C₂²⋊#11719908	2
C₂×#16810386	2
#20272724	2
C₂×C₅₃×#466	2
C₂×#1347	53

Subgroups — C2×#20272724 #27294311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272724 #27294311