Subgroups — C₂×#20274228 #27366621

Order 13568.

758 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810828	C₂	1	2	yes	maximal
3	#20274228	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810368	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719923	—	2	2	no	—
8	C₂×#13133217	C₂²	1	2	yes	—
9	#16811028	—	2	2	no	—
10	#16811028	C₂²	1	4	yes	complemented
11	#16810828	—	2	4	no	—
12	#16810828	C₂²	1	8	yes	—
13	C₂×#13133151	—	2	1	no	—
14	C₂×#13133151	—	2	1	no	—
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×#13133140	—	2	2	no	—
17	#16810374	C₂²	1	4	yes	complemented
18	#16810368	C₂²	1	8	yes	complemented
19	C₂²×M₁₆×C₅₃	—	2	1	no	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#436	—	2	2	no	—
22	C₅₃×#436	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₀₆	—	2	2	no	—
27	C₂×#11719923	—	2	8	no	—
28	C₂×#11719923	—	4	8	no	—
29	C₂²×#88322	—	2	2	no	—
30	#13133217	—	2	2	no	—
31	#13133217	—	2	2	no	—
32	#13133217	—	2	2	no	—
33	#13133217	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₄⋊Dic₅₃	—	2	2	no	—
36	#13133151	—	2	4	no	—
37	#13133151	—	2	4	no	—
38	C₄⋊#88322	—	2	2	no	—
39	C₄⋊#88322	C₂³	1	4	yes	—
40	#13133140	—	2	8	no	—
41	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	—	2	1	no	—
43	C₂×M₁₆×C₅₃	—	2	4	no	—
44	C₂×M₁₆×C₅₃	—	4	4	no	—
45	C₂²×C₈×C₅₃	—	2	1	no	—
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×D₄×C₅₃	—	2	1	no	—
49	C₄×D₄×C₅₃	—	2	1	no	—
50	C₄×D₄×C₅₃	D₄	1	2	yes	—
51	C₄×D₄×C₅₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	—	2	1	no	—
56	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
57	C₂²⋊C₈×C₅₃	—	2	4	no	—
58	C₄×C₈×C₅₃	D₄	1	2	yes	—
59	C₄×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂⁴×C₅₃	—	2	1	no	—
61	C₂×D₄×C₅₃	—	2	1	no	—
62	C₂×D₄×C₅₃	—	2	4	no	—
63	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₃	—	2	1	no	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	4	2	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
72	C₂²×Dic₅₃	—	4	2	no	—
73	C₂×Dic₁₀₆	—	2	4	no	—
74	C₂×Dic₁₀₆	—	4	4	no	—
75	C₂×Dic₁₀₆	—	4	4	no	—
76	C₂×Dic₁₀₆	—	4	2	no	—
77	M₁₆×C₅₃	—	4	8	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
82	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₅₃	—	2	1	no	—
85	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
86	C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₂²⋊C₄×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	#41	1	1	yes	characteristic
91	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
92	#11719923	—	4	16	no	—
93	C₄⋊Dic₅₃	—	2	4	no	—
94	C₄⋊Dic₅₃	—	2	4	no	—
95	C₄⋊Dic₅₃	—	4	2	no	—
96	C₄⋊Dic₅₃	—	4	2	no	—
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	—	4	4	no	—
99	C₂³×C₅₃	—	2	2	no	—
100	C₂³×C₅₃	—	4	2	no	—
101	C₂³×C₅₃	—	4	2	no	—
102	C₂³×C₅₃	—	4	2	no	—
103	C₂³×C₅₃	#121	1	1	yes	characteristic
104	D₄×C₅₃	—	2	4	no	—
105	D₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
117	C₂×C₄×C₅₃	#137	1	2	yes	—
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₅₃	—	4	4	no	—
122	C₂×Dic₅₃	—	8	4	no	—
123	Dic₁₀₆	—	4	4	no	—
124	Dic₁₀₆	—	4	4	no	—
125	Dic₁₀₆	—	8	4	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	4	2	no	—
128	C₈×C₅₃	C₂×SD₁₆	1	4	yes	complemented
129	#88322	—	4	4	no	—
130	C₂×#1693	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	—	4	2	no	—
134	C₂²×C₅₃	—	8	4	no	—
135	C₂²×C₅₃	#461	1	2	yes	—
136	C₂²×C₅₃	#469	1	2	yes	—
137	C₂²×C₅₃	#473	1	2	yes	—
138	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	4	2	no	—
142	C₄×C₅₃	#461	1	2	yes	—
143	C₄×C₅₃	#469	1	2	yes	—
144	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
145	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
146	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
147	Dic₅₃	—	8	4	no	—
148	C₂×#500	—	53	1	no	—
149	C₂×#461	—	53	2	no	—
150	C₂×#436	D₅₃	1	1	yes	characteristic
151	#1693	—	53	8	no	—
152	C₂×#334	—	53	1	no	—
153	C₂×#332	—	53	2	no	—
154	C₂×C₅₃	—	8	2	no	—
155	C₂×C₅₃	#1693	1	4	yes	complemented
156	C₂×C₅₃	C₂×#461	1	1	yes	characteristic
157	C₂×C₅₃	C₂×#469	1	1	yes	characteristic
158	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
159	C₂²×SD₁₆	—	106	2	no	—
160	C₂²×M₁₆	—	2	1	no	—
161	C₂×C₄⋊Q₈	—	53	2	no	—
162	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
163	#500	—	53	4	no	—
164	#500	—	106	2	no	—
165	#461	—	53	8	no	—
166	#461	—	106	4	no	—
167	#436	—	2	2	no	—
168	#436	C₂×D₅₃	1	4	yes	—
169	C₂×#107	—	106	1	no	—
170	C₂×#107	—	106	1	no	—
171	C₂×C₄⋊C₈	—	53	2	no	—
172	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
173	C₂×#103	—	106	2	no	—
174	C₂×C₂²⋊C₈	—	2	1	no	—
175	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
176	#334	—	53	4	no	—
177	#332	—	53	8	no	—
178	C₅₃	C₂×#1693	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	—	106	2	no	—
180	C₂²×D₄	D₂₁₂	1	1	yes	characteristic
181	C₂³×C₄	—	2	1	no	—
182	C₂×SD₁₆	—	106	8	no	—
183	C₂×SD₁₆	—	212	8	no	—
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	4	4	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	106	2	no	—
188	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₄⋊Q₈	—	53	4	no	—
191	C₄⋊Q₈	—	106	2	no	—
192	C₄⋊Q₈	—	106	2	no	—
193	C₄⋊Q₈	—	106	2	no	—
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	D₂₁₂	1	2	yes	—
197	C₄×D₄	C₂²×D₅₃	1	2	yes	—
198	C₂×C₄⋊C₄	—	106	2	no	—
199	C₂×C₄⋊C₄	—	106	2	no	—
200	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	—	2	1	no	—
202	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
203	#107	—	106	4	no	—
204	#107	—	106	4	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	53	4	no	—
207	C₄⋊C₈	—	106	2	no	—
208	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
209	#103	—	106	8	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
212	C₄×C₈	C₂²×D₅₃	1	2	yes	—
213	C₂⁴	—	2	1	no	—
214	C₂×Q₈	—	106	4	no	—
215	C₂×Q₈	—	212	4	no	—
216	C₂×Q₈	—	212	4	no	—
217	C₂×Q₈	—	212	2	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	C₂×D₂₁₂	1	2	yes	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	212	2	no	—
227	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
228	C₂²×C₄	#11719945	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
230	SD₁₆	—	212	16	no	—
231	M₁₆	—	4	8	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	106	4	no	—
236	C₂×C₈	—	212	4	no	—
237	C₂×C₈	#11719923	1	4	yes	complemented
238	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	106	4	no	—
242	C₄⋊C₄	—	106	4	no	—
243	C₄⋊C₄	—	212	2	no	—
244	C₄⋊C₄	—	212	2	no	—
245	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
249	C₄²	#11719945	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
251	C₄²	C₂³×D₅₃	1	1	yes	characteristic
252	C₂³	—	2	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	#13133191	1	1	yes	characteristic
257	Q₈	—	424	4	no	—
258	Q₈	—	212	4	no	—
259	Q₈	—	212	4	no	—
260	D₄	—	2	4	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	424	4	no	—
269	C₂×C₄	—	212	4	no	—
270	C₂×C₄	#13133191	1	1	yes	characteristic
271	C₂×C₄	#13133191	1	1	yes	characteristic
272	C₂×C₄	#13133191	1	1	yes	characteristic
273	C₂×C₄	C₂×#11719923	1	2	yes	—
274	C₂×C₄	#13133245	1	4	yes	—
275	C₂×C₄	#13133247	1	2	yes	—
276	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
278	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	4	2	no	—
281	C₈	—	212	4	no	—
282	C₈	C₂×#11719923	1	4	yes	complemented
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	8	4	no	—
287	C₂²	#16810828	1	2	yes	—
288	C₂²	#16810876	1	2	yes	—
289	C₂²	#16810898	1	2	yes	—
290	C₂²	C₂×#13133191	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	2	no	—
294	C₄	—	424	4	no	—
295	C₄	#16810828	1	2	yes	—
296	C₄	#16810876	1	2	yes	—
297	C₄	C₂²×#11719923	1	1	yes	characteristic
298	C₄	C₂×#13133245	1	2	yes	—
299	C₄	C₂×#13133247	1	1	yes	characteristic
300	C₂	—	8	2	no	—
301	C₂	#20274228	1	4	yes	direct product factor
302	C₂	C₂×#16810828	1	1	yes	characteristic
303	C₂	C₂×#16810876	1	1	yes	characteristic
304	C₂	C₂×#16810898	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#436	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂²×D₄×C₅₃	D₄
hypercenter	C₂×#436	D₅₃
hypocenter	C₅₃	C₂×#1693
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. C₂×#16811028	C₂	semidirect
3. #20274228	C₂	direct (N × H)
4. C₂×#16810374	C₂	semidirect
5. C₂×#16810368	C₂	semidirect
10. #16811028	C₂²	semidirect
17. #16810374	C₂²	semidirect
18. #16810368	C₂²	semidirect
81. C₂×C₈×C₅₃	SD₁₆	semidirect
128. C₈×C₅₃	C₂×SD₁₆	semidirect
155. C₂×C₅₃	#1693	semidirect
178. C₅₃	C₂×#1693	semidirect
237. C₂×C₈	#11719923	semidirect
282. C₈	C₂×#11719923	semidirect
301. C₂	#20274228	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×#16811028	2
C₂×#16810828	2
#20274228	2
C₂×#16810374	2
C₂×#16810368	2
C₂×C₅₃×#436	2
C₂×#1693	53

Subgroups — C2×#20274228 #27366621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274228 #27366621