Subgroups — C₂×#20274115 #27366356

Order 13568.

726 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×#16810914	C₂	1	2	yes	maximal
3	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
4	#20274115	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810371	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₅₃	—	2	1	no	—
8	C₂×#13133217	C₂²	1	2	yes	—
9	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
10	#16811028	—	2	2	no	—
11	#16811028	C₂²	1	4	yes	complemented
12	#16810914	—	2	4	no	—
13	#16810914	C₂²	1	8	yes	—
14	C₈×D₂₁₂	—	2	2	no	—
15	C₈×D₂₁₂	C₂²	1	4	yes	—
16	C₂×#13133151	—	2	1	no	—
17	C₂×#13133151	—	2	1	no	—
18	C₂×#13133149	—	2	2	no	—
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×#13133139	—	2	2	no	—
21	C₂×#13133129	—	2	1	no	—
22	#16810371	C₂²	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#335	C₂²	1	4	yes	complemented
26	C₂²×Dic₁₀₆	—	2	2	no	—
27	C₂²×D₂₁₂	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₅₃	—	2	1	no	—
29	C₂²×#88322	—	2	1	no	—
30	C₂×C₈×D₅₃	—	2	4	no	—
31	C₂×C₈×D₅₃	—	4	4	no	—
32	#13133217	—	2	2	no	—
33	#13133217	—	2	2	no	—
34	#13133217	—	2	2	no	—
35	#13133217	C₂³	1	4	yes	—
36	C₄×D₂₁₂	—	2	1	no	—
37	C₄×D₂₁₂	—	2	1	no	—
38	C₄×D₂₁₂	D₄	1	2	yes	—
39	C₄×D₂₁₂	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₅₃	—	2	2	no	—
41	C₂×C₄⋊Dic₅₃	—	2	2	no	—
42	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
43	C₂×#11719921	—	2	1	no	—
44	#13133151	—	2	4	no	—
45	#13133151	—	2	4	no	—
46	#13133149	—	2	8	no	—
47	C₄⋊#88322	—	2	1	no	—
48	C₄⋊#88322	C₂³	1	2	yes	—
49	#13133139	—	2	8	no	—
50	#13133129	—	2	4	no	—
51	C₂²×C₈×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	—	2	2	no	—
56	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
57	C₄×C₈×C₅₃	D₄	1	2	yes	—
58	C₄×C₈×C₅₃	C₂³	1	2	yes	—
59	C₂³×D₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂²×Dic₅₃	—	2	1	no	—
64	C₂²×Dic₅₃	—	4	2	no	—
65	C₂×D₂₁₂	—	2	1	no	—
66	C₂×D₂₁₂	—	2	2	no	—
67	C₂×D₂₁₂	Q₁₆	1	4	yes	—
68	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
69	C₂×C₄×D₅₃	—	2	2	no	—
70	C₂×C₄×D₅₃	—	2	2	no	—
71	C₂×C₄×D₅₃	—	4	2	no	—
72	C₂×C₄×D₅₃	—	4	2	no	—
73	C₂×Dic₁₀₆	—	2	4	no	—
74	C₂×Dic₁₀₆	—	4	2	no	—
75	C₂×Dic₁₀₆	—	8	4	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	4	4	no	—
79	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
80	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	#41	1	1	yes	characteristic
85	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
86	#11719921	—	2	2	no	—
87	#11719921	—	2	2	no	—
88	C₄⋊Dic₅₃	—	2	1	no	—
89	C₄⋊Dic₅₃	—	2	4	no	—
90	C₄⋊Dic₅₃	—	2	4	no	—
91	C₄⋊Dic₅₃	—	4	2	no	—
92	C₄⋊Dic₅₃	—	4	2	no	—
93	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	4	2	no	—
96	C₈×D₅₃	—	4	8	no	—
97	C₂³×C₅₃	#121	1	1	yes	characteristic
98	C₂²×D₅₃	—	2	2	no	—
99	C₂²×D₅₃	—	2	4	no	—
100	C₂²×D₅₃	—	4	2	no	—
101	C₂²×D₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₅₃	#135	1	2	yes	—
109	C₂×C₄×C₅₃	#137	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
113	C₂×Dic₅₃	—	2	2	no	—
114	C₂×Dic₅₃	—	4	2	no	—
115	C₂×Dic₅₃	—	4	4	no	—
116	C₂×Dic₅₃	—	8	4	no	—
117	D₂₁₂	—	2	2	no	—
118	D₂₁₂	—	2	4	no	—
119	D₂₁₂	C₂×Q₁₆	1	4	yes	—
120	C₄×D₅₃	—	4	2	no	—
121	C₄×D₅₃	—	4	2	no	—
122	C₄×D₅₃	—	4	4	no	—
123	Dic₁₀₆	—	8	4	no	—
124	Dic₁₀₆	—	8	4	no	—
125	C₈×C₅₃	—	2	2	no	—
126	C₈×C₅₃	—	4	4	no	—
127	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
128	#88322	—	4	2	no	—
129	C₂×#1675	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	#461	1	2	yes	—
131	C₂²×C₅₃	#467	1	2	yes	—
132	C₂²×C₅₃	#472	1	2	yes	—
133	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
134	C₂×D₅₃	—	2	4	no	—
135	C₂×D₅₃	—	4	8	no	—
136	C₂×D₅₃	—	4	2	no	—
137	C₂×D₅₃	—	4	2	no	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	#461	1	2	yes	—
141	C₄×C₅₃	#467	1	2	yes	—
142	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
143	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
144	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
145	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
146	Dic₅₃	—	4	2	no	—
147	Dic₅₃	—	8	4	no	—
148	C₂×#500	—	53	1	no	—
149	C₂×#476	—	53	2	no	—
150	C₂×C₈×D₄	—	53	1	no	—
151	#1675	—	53	8	no	—
152	C₂×#335	D₅₃	1	1	yes	characteristic, complemented
153	C₂×#333	—	53	2	no	—
154	C₂×C₅₃	#1675	1	4	yes	complemented
155	C₂×C₅₃	C₂×#461	1	1	yes	characteristic
156	C₂×C₅₃	C₂×#467	1	1	yes	characteristic
157	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
158	D₅₃	—	4	4	no	—
159	C₂³×C₈	—	106	1	no	—
160	C₂×C₄⋊Q₈	—	53	2	no	—
161	C₂×C₄×D₄	—	53	1	no	—
162	#500	—	53	4	no	—
163	#500	—	106	2	no	—
164	#476	—	53	8	no	—
165	#476	—	106	4	no	—
166	C₈×D₄	—	53	4	no	—
167	C₈×D₄	—	106	2	no	—
168	C₂×#107	—	106	1	no	—
169	C₂×#107	—	106	1	no	—
170	C₂×#106	—	106	2	no	—
171	C₂×C₄⋊C₈	—	53	1	no	—
172	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
173	C₂×#102	—	106	2	no	—
174	C₂×C₂²⋊C₈	—	106	1	no	—
175	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
176	#335	C₂×D₅₃	1	4	yes	complemented
177	#333	—	53	8	no	—
178	C₅₃	C₂×#1675	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	—	106	2	no	—
180	C₂²×D₄	—	53	1	no	—
181	C₂³×C₄	—	106	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	106	1	no	—
184	C₂²×C₈	—	106	4	no	—
185	C₂²×C₈	—	212	4	no	—
186	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
187	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
188	C₄⋊Q₈	—	53	4	no	—
189	C₄⋊Q₈	—	106	2	no	—
190	C₄⋊Q₈	—	106	2	no	—
191	C₄⋊Q₈	—	106	2	no	—
192	C₄×D₄	—	53	2	no	—
193	C₄×D₄	—	53	2	no	—
194	C₄×D₄	—	106	1	no	—
195	C₄×D₄	—	106	1	no	—
196	C₂×C₄⋊C₄	—	53	1	no	—
197	C₂×C₄⋊C₄	—	106	2	no	—
198	C₂×C₄⋊C₄	—	106	2	no	—
199	C₂×C₂²⋊C₄	—	106	1	no	—
200	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
201	#107	—	106	4	no	—
202	#107	—	106	4	no	—
203	#106	—	106	8	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	53	2	no	—
206	C₄⋊C₈	—	106	1	no	—
207	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
208	#102	—	106	8	no	—
209	C₂²⋊C₈	—	106	4	no	—
210	C₄×C₈	D₂₁₂	1	2	yes	—
211	C₄×C₈	C₂²×D₅₃	1	2	yes	—
212	C₂⁴	—	106	1	no	—
213	C₂×Q₈	—	106	4	no	—
214	C₂×Q₈	—	424	4	no	—
215	C₂×Q₈	—	212	2	no	—
216	C₂×D₄	—	53	4	no	—
217	C₂×D₄	—	53	2	no	—
218	C₂×D₄	—	106	1	no	—
219	C₂×D₄	—	106	2	no	—
220	C₂²×C₄	—	106	1	no	—
221	C₂²×C₄	—	106	2	no	—
222	C₂²×C₄	—	106	2	no	—
223	C₂²×C₄	—	212	2	no	—
224	C₂²×C₄	—	212	2	no	—
225	C₂²×C₄	—	212	2	no	—
226	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
228	C₂²×C₄	#11719949	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	4	no	—
232	C₂×C₈	—	106	2	no	—
233	C₂×C₈	—	212	2	no	—
234	C₂×C₈	—	212	8	no	—
235	C₂×C₈	#11719913	1	4	yes	—
236	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
237	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
238	C₄⋊C₄	—	53	2	no	—
239	C₄⋊C₄	—	106	1	no	—
240	C₄⋊C₄	—	106	4	no	—
241	C₄⋊C₄	—	106	4	no	—
242	C₄⋊C₄	—	212	2	no	—
243	C₄⋊C₄	—	212	2	no	—
244	C₂²⋊C₄	—	106	2	no	—
245	C₂²⋊C₄	—	106	2	no	—
246	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
247	C₄²	D₄×D₅₃	1	1	yes	characteristic
248	C₄²	#11719949	1	1	yes	characteristic
249	C₄²	C₂³×D₅₃	1	1	yes	characteristic
250	C₂³	—	106	2	no	—
251	C₂³	—	106	4	no	—
252	C₂³	—	212	2	no	—
253	C₂³	—	212	2	no	—
254	C₂³	#13133192	1	1	yes	characteristic
255	Q₈	—	424	4	no	—
256	Q₈	—	424	4	no	—
257	D₄	—	53	4	no	—
258	D₄	—	106	4	no	—
259	D₄	—	106	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	106	2	no	—
262	C₂×C₄	—	424	4	no	—
263	C₂×C₄	—	212	4	no	—
264	C₂×C₄	—	212	2	no	—
265	C₂×C₄	—	212	4	no	—
266	C₂×C₄	—	212	2	no	—
267	C₂×C₄	—	212	2	no	—
268	C₂×C₄	#13133192	1	1	yes	characteristic
269	C₂×C₄	#13133192	1	1	yes	characteristic
270	C₂×C₄	#13133192	1	1	yes	characteristic
271	C₂×C₄	C₂×#11719913	1	2	yes	—
272	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
273	C₂×C₄	#13133236	1	2	yes	—
274	C₂×C₄	#13133250	1	2	yes	—
275	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
277	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	4	no	—
280	C₈	—	212	2	no	—
281	C₈	C₂×#11719913	1	4	yes	—
282	C₂²	—	106	4	no	—
283	C₂²	—	212	2	no	—
284	C₂²	—	212	8	no	—
285	C₂²	—	212	2	no	—
286	C₂²	#16810829	1	2	yes	—
287	C₂²	#16810865	1	2	yes	—
288	C₂²	#16810894	1	2	yes	—
289	C₂²	C₂×#13133192	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	424	4	no	—
293	C₄	—	212	2	no	—
294	C₄	#16810829	1	2	yes	—
295	C₄	#16810865	1	2	yes	—
296	C₄	C₂²×#11719913	1	1	yes	characteristic
297	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
298	C₄	C₂×#13133236	1	1	yes	characteristic
299	C₄	C₂×#13133250	1	1	yes	characteristic
300	C₂	—	212	4	no	—
301	C₂	#20274115	1	4	yes	direct product factor
302	C₂	C₂×#16810829	1	1	yes	characteristic
303	C₂	C₂×#16810865	1	1	yes	characteristic
304	C₂	C₂×#16810894	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂²×D₂₁₂	D₄
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	C₂×#1675
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
1. C₂×#16811028	C₂	semidirect
4. #20274115	C₂	direct (N × H)
5. C₂×#16810371	C₂	semidirect
6. C₂×C₅₃×#335	C₂	semidirect
11. #16811028	C₂²	semidirect
22. #16810371	C₂²	semidirect
25. C₅₃×#335	C₂²	semidirect
152. C₂×#335	D₅₃	semidirect
154. C₂×C₅₃	#1675	semidirect
176. #335	C₂×D₅₃	semidirect
178. C₅₃	C₂×#1675	semidirect
301. C₂	#20274115	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×#16810914	2
C₂×C₈×D₂₁₂	2
#20274115	2
C₂×#16810371	2
C₂×C₅₃×#335	2
C₂×#1675	53

Subgroups — C2×#20274115 #27366356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274115 #27366356