Subgroups — C₂×#20272726 #27294314

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810864	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272726	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#340	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic, complemented
6	C₂×#13133196	—	2	1	no	—
7	C₂²×#11719908	—	2	1	no	—
8	#16810864	—	2	1	no	—
9	#16810864	—	2	1	no	—
10	#16810864	C₄	1	2	yes	complemented
11	#16810864	C₂²	1	2	yes	—
12	C₂×#13133151	C₄	1	1	yes	characteristic, complemented
13	C₂×#13133142	—	2	1	no	—
14	C₂²⋊#11719908	—	2	1	no	—
15	C₂²⋊#11719908	C₂²	1	2	yes	—
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₅₃×#340	—	2	1	no	—
19	C₅₃×#340	C₂²	1	2	yes	—
20	C₂²×Dic₁₀₆	—	2	1	no	—
21	C₂×Dic₂₁₂	—	2	1	no	—
22	C₂×Dic₂₁₂	—	4	2	no	—
23	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	complemented
24	#13133196	—	2	2	no	—
25	#13133196	—	2	2	no	—
26	#13133196	—	4	1	no	—
27	#13133196	—	4	1	no	—
28	C₂×C₄⋊Dic₅₃	—	2	1	no	—
29	C₂×#11719919	—	4	1	no	—
30	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	—	4	2	no	—
33	#13133151	—	2	1	no	—
34	#13133151	C₂×C₄	1	2	yes	complemented
35	#13133142	—	2	2	no	—
36	#13133142	—	2	2	no	—
37	C₂³×C₄×C₅₃	—	4	1	no	—
38	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	—	4	2	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
50	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	—	4	4	no	—
54	C₂²×C₄×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	—	8	2	no	—
56	C₂²×C₄×C₅₃	—	8	2	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₅₃	—	4	1	no	—
64	C₂²×Dic₅₃	—	4	1	no	—
65	C₂×Dic₁₀₆	—	2	2	no	—
66	C₂×Dic₁₀₆	—	4	2	no	—
67	C₂×Dic₁₀₆	—	4	2	no	—
68	C₂×Dic₁₀₆	—	4	1	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	—	4	4	no	—
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₂²⋊Dic₅₃	—	4	2	no	—
83	C₂²⋊Dic₅₃	—	4	2	no	—
84	C₄⋊Dic₅₃	—	2	2	no	—
85	C₄⋊Dic₅₃	—	4	1	no	—
86	#11719919	—	4	2	no	—
87	#11719919	—	4	2	no	—
88	Dic₂₁₂	—	4	2	no	—
89	Dic₂₁₂	—	4	2	no	—
90	#11719908	—	4	2	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	#102	1	1	yes	characteristic
94	C₂³×C₅₃	#102	1	1	yes	characteristic
95	C₂³×C₅₃	SD₃₂	1	2	yes	—
96	C₂³×C₅₃	Q₃₂	1	2	yes	—
97	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	8	8	no	—
107	C₂×C₄×C₅₃	—	8	2	no	—
108	C₂×C₄×C₅₃	—	8	2	no	—
109	C₂×C₄×C₅₃	#99	1	2	yes	—
110	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#104	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×Dic₅₃	—	4	2	no	—
116	C₂×Dic₅₃	—	4	2	no	—
117	C₂×Dic₅₃	—	8	2	no	—
118	C₂×Dic₅₃	—	8	2	no	—
119	Dic₁₀₆	—	4	2	no	—
120	Dic₁₀₆	—	4	2	no	—
121	Dic₁₀₆	—	8	2	no	—
122	C₈×C₅₃	—	2	1	no	—
123	C₈×C₅₃	—	2	1	no	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₂×#1348	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	2	4	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	#327	1	2	yes	—
134	C₂²×C₅₃	#358	1	2	yes	—
135	C₂²×C₅₃	#358	1	2	yes	—
136	C₂²×C₅₃	#360	1	2	yes	—
137	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
139	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	8	4	no	—
143	C₄×C₅₃	#327	1	2	yes	—
144	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
145	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
146	Dic₅₃	—	8	2	no	—
147	Dic₅₃	—	8	2	no	—
148	C₂×#467	—	53	1	no	—
149	C₂×C₂²⋊C₁₆	—	53	1	no	—
150	C₂×#340	D₅₃	1	1	yes	characteristic
151	#1348	—	53	4	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	#1348	1	4	yes	complemented
156	C₂×C₅₃	C₂×#327	1	1	yes	characteristic
157	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
158	C₂×C₅₃	C₂×#360	1	1	yes	characteristic
159	C₂²×Q₁₆	—	53	1	no	—
160	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
161	C₂×C₂²⋊Q₈	—	106	1	no	—
162	C₂²×C₄⋊C₄	—	2	1	no	—
163	C₂²×C₁₆	—	106	1	no	—
164	#467	—	53	2	no	—
165	#467	—	53	2	no	—
166	#467	—	106	1	no	—
167	#467	—	106	1	no	—
168	C₂×#107	—	53	1	no	—
169	C₂×#103	—	106	1	no	—
170	C₂²⋊C₁₆	—	53	2	no	—
171	C₂²⋊C₁₆	—	106	1	no	—
172	#340	—	2	1	no	—
173	#340	C₂×D₅₃	1	2	yes	—
174	C₅₃	C₂×#1348	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₈	—	106	1	no	—
176	C₂³×C₄	—	4	1	no	—
177	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂×Q₁₆	—	53	2	no	—
179	C₂×Q₁₆	—	106	1	no	—
180	C₂×Q₁₆	—	212	2	no	—
181	C₂²×C₈	—	4	2	no	—
182	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
183	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
185	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
186	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	C₂²⋊Q₈	—	106	2	no	—
189	C₂²⋊Q₈	—	106	2	no	—
190	C₂²⋊Q₈	—	212	1	no	—
191	C₂²⋊Q₈	—	212	1	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	4	2	no	—
195	C₂×C₄⋊C₄	—	4	1	no	—
196	C₂×C₄⋊C₄	—	106	1	no	—
197	C₂×C₄⋊C₄	—	212	1	no	—
198	C₂×C₂²⋊C₄	—	212	1	no	—
199	C₂×C₁₆	—	106	2	no	—
200	C₂×C₁₆	—	212	2	no	—
201	#107	—	53	2	no	—
202	#107	—	106	1	no	—
203	#103	—	106	2	no	—
204	#103	—	106	2	no	—
205	C₂⁴	#11719922	1	1	yes	characteristic
206	C₂×Q₈	—	106	2	no	—
207	C₂×Q₈	—	212	2	no	—
208	C₂×Q₈	—	212	2	no	—
209	C₂×Q₈	—	212	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	8	2	no	—
215	C₂²×C₄	—	8	2	no	—
216	C₂²×C₄	—	212	1	no	—
217	C₂²×C₄	—	212	1	no	—
218	C₂²×C₄	#11719921	1	1	yes	characteristic
219	C₂²×C₄	#11719921	1	1	yes	characteristic
220	C₂²×C₄	#11719922	1	1	yes	characteristic
221	C₂²×C₄	#11719924	1	1	yes	characteristic
222	C₂²×C₄	#11719924	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
224	Q₁₆	—	212	2	no	—
225	Q₁₆	—	212	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	#11719921	1	1	yes	characteristic
233	C₂×C₈	#11719921	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	106	2	no	—
240	C₄⋊C₄	—	212	1	no	—
241	C₄⋊C₄	—	212	2	no	—
242	C₄⋊C₄	—	212	2	no	—
243	C₂²⋊C₄	—	212	2	no	—
244	C₂²⋊C₄	—	212	2	no	—
245	C₁₆	—	212	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	#13133137	1	1	yes	characteristic
249	C₂³	#13133137	1	1	yes	characteristic
250	C₂³	#13133161	1	2	yes	—
251	C₂³	#13133163	1	2	yes	—
252	C₂³	C₂×#11719922	1	1	yes	characteristic
253	Q₈	—	424	2	no	—
254	Q₈	—	212	2	no	—
255	Q₈	—	212	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	8	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	424	2	no	—
268	C₂×C₄	—	424	2	no	—
269	C₂×C₄	—	212	2	no	—
270	C₂×C₄	—	212	2	no	—
271	C₂×C₄	#13133131	1	2	yes	—
272	C₂×C₄	#13133137	1	1	yes	characteristic
273	C₂×C₄	#13133137	1	1	yes	characteristic
274	C₂×C₄	#13133145	1	2	yes	—
275	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
276	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	#16810356	1	2	yes	—
288	C₂²	#16810426	1	2	yes	—
289	C₂²	#16810426	1	2	yes	—
290	C₂²	#16810432	1	2	yes	—
291	C₂²	C₂×#13133137	1	1	yes	characteristic
292	C₂²	C₂×#13133161	1	1	yes	characteristic
293	C₂²	C₂×#13133163	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	8	4	no	—
297	C₄	—	424	2	no	—
298	C₄	—	424	2	no	—
299	C₄	#16810356	1	2	yes	—
300	C₄	C₂×#13133131	1	1	yes	characteristic
301	C₄	C₂×#13133145	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	2	no	—
305	C₂	#20272726	1	4	yes	direct product factor
306	C₂	C₂×#16810356	1	1	yes	characteristic
307	C₂	C₂×#16810426	1	1	yes	characteristic
308	C₂	C₂×#16810432	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133137
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#340	C₂
socle	C₂³×C₅₃	#102
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#340	D₅₃
hypocenter	C₅₃	C₂×#1348
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
3. #20272726	C₂	direct (N × H)
5. C₂²×Dic₂₁₂	C₄	semidirect
10. #16810864	C₄	semidirect
12. C₂×#13133151	C₄	semidirect
23. C₂×Dic₂₁₂	C₂×C₄	semidirect
34. #13133151	C₂×C₄	semidirect
155. C₂×C₅₃	#1348	semidirect
174. C₅₃	C₂×#1348	semidirect
305. C₂	#20272726	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×#16810864	2
C₂×C₂²⋊#11719908	2
#20272726	2
C₂×C₅₃×#340	2
C₂×#1348	53

Subgroups — C2×#20272726 #27294314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272726 #27294314