Subgroups — C₂×#20276885 #27538821

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811020	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810739	C₂	1	1	yes	maximal, characteristic
3	#20276885	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810447	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810443	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810434	C₂	1	1	yes	maximal, characteristic
7	C₂×#16810432	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	C₂×#13133225	—	2	1	no	—
11	C₂×#13133216	—	2	1	no	—
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	#16811020	—	2	2	no	—
14	#16811020	C₂²	1	4	yes	complemented
15	#16810739	—	2	2	no	—
16	#16810739	C₂²	1	4	yes	—
17	C₂×#13133149	C₂²	1	1	yes	characteristic
18	C₂×#13133148	—	2	1	no	—
19	C₂×#13133143	—	2	1	no	—
20	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	#16810447	C₂²	1	4	yes	complemented
22	#16810443	C₂²	1	4	yes	complemented
23	#16810434	C₂²	1	4	yes	—
24	#16810432	C₂²	1	4	yes	—
25	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#365	C₂²	1	4	yes	complemented
29	C₂×#11719945	—	2	1	no	—
30	C₂²×Dic₁₀₆	—	4	1	no	—
31	#13133239	—	2	1	no	—
32	#13133239	—	2	1	no	—
33	#13133239	Q₈	1	2	yes	complemented
34	#13133239	C₂³	1	2	yes	—
35	C₂×Dic₂₁₂	Q₈	1	1	yes	characteristic, complemented
36	C₂×#11719913	—	2	1	no	—
37	C₂×D₄₂₄	Q₈	1	1	yes	characteristic, complemented
38	#13133225	—	2	4	no	—
39	#13133225	—	4	2	no	—
40	C₂×#11719912	—	4	1	no	—
41	C₂²×#88322	D₄	1	1	yes	characteristic
42	C₂×C₈×D₅₃	—	4	1	no	—
43	#13133216	—	2	4	no	—
44	#13133216	—	4	2	no	—
45	C₂×C₄⋊Dic₅₃	—	2	1	no	—
46	C₂×#11719919	—	4	1	no	—
47	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
48	#13133157	—	2	1	no	—
49	#13133157	C₂³	1	2	yes	—
50	C₂×#11719908	—	2	1	no	—
51	#13133149	—	2	1	no	—
52	#13133149	C₂³	1	2	yes	—
53	#13133148	—	2	4	no	—
54	#13133143	—	2	4	no	—
55	C₈×Dic₅₃	D₄	1	2	yes	—
56	C₈×Dic₅₃	C₂³	1	2	yes	—
57	C₂×M₁₆×C₅₃	—	2	1	no	—
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
60	M₃₂×C₅₃	—	2	1	no	—
61	M₃₂×C₅₃	C₂³	1	2	yes	—
62	C₂×C₁₆×C₅₃	—	2	1	no	—
63	#108×C₅₃	—	2	1	no	—
64	#108×C₅₃	C₂³	1	2	yes	—
65	#106×C₅₃	—	2	1	no	—
66	#106×C₅₃	C₂³	1	2	yes	—
67	C₂²×C₄×C₅₃	—	4	1	no	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₃	—	4	1	no	—
70	C₂²×Dic₅₃	—	2	1	no	—
71	C₂²×Dic₅₃	—	4	1	no	—
72	#11719945	—	2	2	no	—
73	#11719945	—	2	2	no	—
74	#11719945	—	4	1	no	—
75	#11719945	—	4	1	no	—
76	C₂×D₂₁₂	—	2	1	no	—
77	C₂×C₄×D₅₃	—	4	1	no	—
78	C₂×Dic₁₀₆	—	2	1	no	—
79	C₂×Dic₁₀₆	—	4	4	no	—
80	C₂×Dic₁₀₆	—	8	4	no	—
81	M₁₆×C₅₃	—	2	2	no	—
82	M₁₆×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	—	4	1	no	—
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₄⋊C₄×C₅₃	—	4	1	no	—
92	C₄⋊Dic₅₃	—	2	2	no	—
93	C₄⋊Dic₅₃	—	4	1	no	—
94	#11719919	—	4	4	no	—
95	C₄×Dic₅₃	—	2	1	no	—
96	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
97	C₂×#88322	SD₁₆	1	1	yes	characteristic
98	C₂×#88322	SD₁₆	1	1	yes	characteristic
99	C₂×#88322	SD₁₆	1	2	yes	—
100	C₂×#88322	C₂×D₄	1	2	yes	—
101	Dic₂₁₂	—	2	1	no	—
102	Dic₂₁₂	C₂×Q₈	1	2	yes	complemented
103	D₄₂₄	—	2	1	no	—
104	D₄₂₄	C₂×Q₈	1	2	yes	complemented
105	#11719913	—	2	2	no	—
106	#11719913	—	2	2	no	—
107	#11719912	—	4	4	no	—
108	C₈×D₅₃	—	4	4	no	—
109	C₁₆×C₅₃	—	2	2	no	—
110	#11719908	—	2	2	no	—
111	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂²×D₅₃	—	4	1	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	8	2	no	—
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₂²⋊D₅₃	—	4	2	no	—
122	C₂²⋊D₅₃	—	4	2	no	—
123	C₂×Dic₅₃	—	2	4	no	—
124	C₂×Dic₅₃	—	2	2	no	—
125	C₂×Dic₅₃	—	4	1	no	—
126	C₂×Dic₅₃	—	4	2	no	—
127	C₂×Dic₅₃	—	8	2	no	—
128	D₂₁₂	—	2	2	no	—
129	D₂₁₂	—	4	1	no	—
130	C₄×D₅₃	—	4	2	no	—
131	C₄×D₅₃	—	4	2	no	—
132	Dic₁₀₆	—	2	2	no	—
133	Dic₁₀₆	—	4	1	no	—
134	Dic₁₀₆	—	8	8	no	—
135	C₈×C₅₃	—	4	2	no	—
136	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
139	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
140	#88322	C₂×SD₁₆	1	2	yes	—
141	#88322	C₂×SD₁₆	1	2	yes	—
142	C₂×#2230	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	#476	1	1	yes	characteristic
145	C₂²×C₅₃	#476	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×D₅₃	—	4	2	no	—
148	C₂×D₅₃	—	8	2	no	—
149	C₄×C₅₃	—	8	2	no	—
150	C₄×C₅₃	#476	1	1	yes	characteristic
151	C₄×C₅₃	#476	1	1	yes	characteristic
152	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
153	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
154	Dic₅₃	—	2	4	no	—
155	Dic₅₃	—	4	2	no	—
156	Dic₅₃	—	8	2	no	—
157	C₂×#498	—	53	1	no	—
158	C₂×#443	—	53	1	no	—
159	#2230	—	53	8	no	—
160	C₂×#365	—	53	1	no	—
161	C₂×#365	D₅₃	1	1	yes	characteristic, complemented
162	C₂×#364	—	53	1	no	—
163	C₂×#360	—	53	1	no	—
164	C₂×#360	—	53	1	no	—
165	C₂×C₅₃	—	2	1	no	—
166	C₂×C₅₃	—	2	1	no	—
167	C₂×C₅₃	#2230	1	2	yes	complemented
168	C₂×C₅₃	C₂×#476	1	1	yes	characteristic
169	D₅₃	—	8	2	no	—
170	C₂×#135	—	53	1	no	—
171	C₂×#131	—	106	1	no	—
172	C₂×C₄⋊Q₈	—	106	1	no	—
173	C₂×M₃₂	—	53	1	no	—
174	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
175	#498	—	53	4	no	—
176	#498	—	106	2	no	—
177	#443	—	53	4	no	—
178	#443	—	106	2	no	—
179	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
180	C₂×#106	—	53	1	no	—
181	C₂×#106	—	106	1	no	—
182	C₂×#106	C₂×D₅₃	1	1	yes	characteristic
183	C₂×#104	—	106	1	no	—
184	C₂×C₄×C₈	—	53	1	no	—
185	#365	—	53	4	no	—
186	#365	C₂×D₅₃	1	4	yes	complemented
187	#364	—	53	4	no	—
188	#360	—	53	4	no	—
189	#360	—	53	4	no	—
190	C₅₃	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	—	106	1	no	—
192	C₂²×Q₈	—	212	1	no	—
193	#135	—	53	2	no	—
194	#135	—	53	2	no	—
195	#135	—	106	1	no	—
196	#135	—	106	1	no	—
197	C₂×Q₁₆	—	53	1	no	—
198	C₂×SD₁₆	—	106	1	no	—
199	C₂×D₈	—	53	1	no	—
200	#131	—	106	4	no	—
201	#131	—	212	2	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	212	1	no	—
204	C₂²×C₈	—	53	1	no	—
205	C₂²×C₈	—	212	1	no	—
206	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	C₄⋊Q₈	—	106	4	no	—
208	C₄⋊Q₈	—	212	2	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	106	1	no	—
211	C₂×C₄⋊C₄	—	212	1	no	—
212	C₂×C₄²	—	53	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	53	2	no	—
215	M₃₂	—	106	1	no	—
216	M₃₂	C₂²×D₅₃	1	2	yes	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	106	1	no	—
219	#108	—	2	1	no	—
220	#108	C₂²×D₅₃	1	2	yes	—
221	#106	—	2	1	no	—
222	#106	—	53	2	no	—
223	#106	—	106	4	no	—
224	#106	—	106	1	no	—
225	#106	C₂²×D₅₃	1	2	yes	—
226	#104	—	106	4	no	—
227	C₄×C₈	—	53	2	no	—
228	C₄×C₈	—	53	2	no	—
229	#41	—	106	2	no	—
230	#41	—	106	2	no	—
231	#41	—	212	1	no	—
232	#41	—	212	1	no	—
233	C₂×Q₈	—	106	1	no	—
234	C₂×Q₈	—	424	4	no	—
235	C₂×Q₈	—	212	4	no	—
236	C₂×D₄	—	106	1	no	—
237	C₂×D₄	—	212	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	106	1	no	—
240	C₂²×C₄	—	212	1	no	—
241	C₂²×C₄	—	212	1	no	—
242	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
243	Q₁₆	—	53	2	no	—
244	Q₁₆	—	106	1	no	—
245	SD₁₆	—	106	2	no	—
246	SD₁₆	—	106	2	no	—
247	D₈	—	53	2	no	—
248	D₈	—	106	1	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	212	4	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	53	1	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	212	4	no	—
258	C₂×C₈	#11719945	1	1	yes	characteristic
259	C₂×C₈	#11719945	1	1	yes	characteristic
260	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
261	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
262	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	106	2	no	—
267	C₄⋊C₄	—	212	4	no	—
268	C₄⋊C₄	—	212	1	no	—
269	C₄²	—	53	2	no	—
270	C₄²	—	106	1	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	106	2	no	—
273	C₂³	—	212	1	no	—
274	C₂³	#13133244	1	1	yes	characteristic
275	Q₈	—	106	2	no	—
276	Q₈	—	424	8	no	—
277	Q₈	—	212	1	no	—
278	D₄	—	106	2	no	—
279	D₄	—	212	2	no	—
280	D₄	—	212	2	no	—
281	D₄	—	212	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	106	4	no	—
285	C₂×C₄	—	106	2	no	—
286	C₂×C₄	—	424	2	no	—
287	C₂×C₄	—	212	2	no	—
288	C₂×C₄	—	212	1	no	—
289	C₂×C₄	—	212	2	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	#13133195	1	1	yes	characteristic
292	C₂×C₄	#13133195	1	1	yes	characteristic
293	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
295	C₂×C₄	#13133244	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
297	C₈	—	4	2	no	—
298	C₈	—	53	2	no	—
299	C₈	—	53	2	no	—
300	C₈	#13133195	1	1	yes	characteristic
301	C₈	#13133195	1	1	yes	characteristic
302	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
303	C₈	C₂×#11719945	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	424	2	no	—
306	C₂²	—	212	2	no	—
307	C₂²	#16810912	1	1	yes	characteristic
308	C₂²	#16810912	1	1	yes	characteristic
309	C₂²	C₂×#13133244	1	1	yes	characteristic
310	C₄	—	8	2	no	—
311	C₄	—	106	4	no	—
312	C₄	—	424	2	no	—
313	C₄	—	212	2	no	—
314	C₄	#16810912	1	1	yes	characteristic
315	C₄	#16810912	1	1	yes	characteristic
316	C₄	C₂×#13133195	1	1	yes	characteristic
317	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	424	2	no	—
321	C₂	#20276885	1	2	yes	direct product factor
322	C₂	C₂×#16810912	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810912
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#365	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂×#13133239	C₂²
hypercenter	C₂×#365	D₅₃
hypocenter	C₅₃	C₂×#2230
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
1. C₂×#16811020	C₂	semidirect
3. #20276885	C₂	direct (N × H)
4. C₂×#16810447	C₂	semidirect
5. C₂×#16810443	C₂	semidirect
8. C₂×C₅₃×#365	C₂	semidirect
14. #16811020	C₂²	semidirect
21. #16810447	C₂²	semidirect
22. #16810443	C₂²	semidirect
28. C₅₃×#365	C₂²	semidirect
33. #13133239	Q₈	semidirect
35. C₂×Dic₂₁₂	Q₈	semidirect
37. C₂×D₄₂₄	Q₈	semidirect
102. Dic₂₁₂	C₂×Q₈	semidirect
104. D₄₂₄	C₂×Q₈	semidirect
161. C₂×#365	D₅₃	semidirect
167. C₂×C₅₃	#2230	semidirect
186. #365	C₂×D₅₃	semidirect
190. C₅₃	C₂×#2230	semidirect
321. C₂	#20276885	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×#16811020	2
C₂×#16810739	2
#20276885	2
C₂×#16810447	2
C₂×#16810443	2
C₂×#16810434	2
C₂×#16810432	2
C₂×C₅₃×#365	2
C₂×#2230	53

Subgroups — C2×#20276885 #27538821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276885 #27538821