Subgroups — C₂×#20273500 #27353172

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810680	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20273500	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810376	C₂	1	2	yes	maximal, complemented
5	C₂×#16810374	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	—	2	1	no	—
8	#16810680	—	2	2	no	—
9	#16810680	C₂²	1	4	yes	—
10	C₂×C₄⋊#88322	—	2	1	no	—
11	C₂×C₄⋊#88322	—	2	2	no	—
12	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	4	yes	—
16	C₂×C₂²⋊#88322	—	2	1	no	—
17	C₂×#13133128	—	2	1	no	—
18	#16810376	C₂²	1	8	yes	complemented
19	#16810374	C₂²	1	8	yes	complemented
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#431	C₄	1	4	yes	—
22	C₅₃×#431	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
24	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
26	C₂²×#88322	—	2	1	no	—
27	C₂²×#88322	—	2	1	no	—
28	C₂²×#88322	—	2	4	no	—
29	C₂²×#88322	—	2	2	no	—
30	C₂²×#88322	—	4	4	no	—
31	C₄⋊#88322	—	2	4	no	—
32	C₄⋊#88322	—	2	2	no	—
33	C₄⋊#88322	—	2	1	no	—
34	C₄⋊#88322	—	2	1	no	—
35	C₄⋊#88322	C₂³	1	2	yes	—
36	C₄⋊#88322	C₂³	1	4	yes	—
37	C₄⋊#88322	C₂³	1	2	yes	—
38	C₂²⋊#88322	—	2	4	no	—
39	#13133128	—	2	4	no	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
52	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
53	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₅₃	C₂³	1	4	yes	—
56	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₅₃	—	2	8	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
75	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
76	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₅₃	C₂×D₄	1	2	yes	—
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
83	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂×#88322	—	2	2	no	—
86	C₂×#88322	—	2	2	no	—
87	C₂×#88322	—	2	4	no	—
88	C₂×#88322	—	4	8	no	—
89	C₂×#88322	—	4	2	no	—
90	C₂×#88322	—	4	4	no	—
91	C₂×#88322	—	4	2	no	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	—	2	2	no	—
94	C₂³×C₅₃	#108	1	4	yes	—
95	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#131	1	4	yes	—
115	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₅₃	#137	1	2	yes	—
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₅₃	—	2	8	no	—
121	#88322	—	4	2	no	—
122	#88322	—	4	4	no	—
123	#88322	—	4	2	no	—
124	C₂×#1573	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	2	8	no	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	#424	1	2	yes	—
129	C₂²×C₅₃	#428	1	2	yes	—
130	C₂²×C₅₃	C₂×#108	1	4	yes	—
131	C₂²×C₅₃	C₂×#108	1	2	yes	—
132	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	#424	1	2	yes	—
139	C₄×C₅₃	#428	1	2	yes	—
140	C₄×C₅₃	C₂×#131	1	2	yes	—
141	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
143	C₂×#432	—	53	1	no	—
144	C₂×#431	D₅₃	1	1	yes	characteristic
145	C₂²×C₄⋊C₈	—	53	1	no	—
146	#1573	—	53	8	no	—
147	C₂×#335	—	53	2	no	—
148	C₂×#334	—	53	2	no	—
149	C₂×C₅₃	—	2	4	no	—
150	C₂×C₅₃	#1573	1	4	yes	complemented
151	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
153	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
154	C₂³×C₈	—	106	1	no	—
155	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
156	#432	—	53	4	no	—
157	#432	—	106	2	no	—
158	#431	Dic₅₃	1	4	yes	—
159	#431	C₂×D₅₃	1	4	yes	—
160	C₂×C₄⋊C₈	—	53	1	no	—
161	C₂×C₄⋊C₈	—	53	2	no	—
162	C₂×C₄⋊C₈	—	53	1	no	—
163	C₂×C₄⋊C₈	—	53	4	no	—
164	C₂×C₄⋊C₈	—	106	2	no	—
165	C₂×C₄⋊C₈	—	106	1	no	—
166	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
167	C₂×C₂²⋊C₈	—	106	1	no	—
168	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
169	C₂×#97	—	106	1	no	—
170	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
171	#335	—	53	8	no	—
172	#334	—	53	8	no	—
173	C₅₃	C₂×#1573	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
175	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
176	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₈	—	106	4	no	—
178	C₂²×C₈	—	106	2	no	—
179	C₂²×C₈	—	106	1	no	—
180	C₂²×C₈	—	106	1	no	—
181	C₂²×C₈	—	212	4	no	—
182	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
183	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
184	C₂×C₄²	Dic₁₀₆	1	2	yes	—
185	C₂×C₄²	D₂₁₂	1	2	yes	—
186	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
189	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
190	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
191	C₄⋊C₈	—	53	2	no	—
192	C₄⋊C₈	—	53	4	no	—
193	C₄⋊C₈	—	53	2	no	—
194	C₄⋊C₈	—	106	4	no	—
195	C₄⋊C₈	—	106	2	no	—
196	C₄⋊C₈	—	106	1	no	—
197	C₄⋊C₈	—	106	1	no	—
198	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
199	C₂²⋊C₈	—	106	4	no	—
200	C₂²⋊C₈	C₂×Dic₅₃	1	8	yes	—
201	#97	—	106	4	no	—
202	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
203	C₄×C₈	C₂²×D₅₃	1	4	yes	—
204	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
211	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
213	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
214	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
216	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
218	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	106	2	no	—
224	C₂×C₈	—	106	4	no	—
225	C₂×C₈	—	212	8	no	—
226	C₂×C₈	—	212	4	no	—
227	C₂×C₈	—	212	2	no	—
228	C₂×C₈	—	212	2	no	—
229	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
230	C₄²	C₄⋊Dic₅₃	1	2	yes	—
231	C₄²	C₄⋊Dic₅₃	1	2	yes	—
232	C₄²	C₄⋊Dic₅₃	1	2	yes	—
233	C₄²	C₂×Dic₁₀₆	1	2	yes	—
234	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
235	C₄²	C₂×D₂₁₂	1	2	yes	—
236	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
237	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
238	C₄²	C₂²×Dic₅₃	1	2	yes	—
239	C₄²	C₂³×D₅₃	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	#13133153	1	4	yes	—
243	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
244	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
259	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
262	C₂×C₄	#13133224	1	4	yes	—
263	C₂×C₄	#13133245	1	2	yes	—
264	C₂×C₄	#13133250	1	2	yes	—
265	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
267	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	212	2	no	—
270	C₈	—	212	4	no	—
271	C₈	—	212	2	no	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	#16810650	1	2	yes	—
276	C₂²	#16810665	1	2	yes	—
277	C₂²	C₂×#13133153	1	4	yes	—
278	C₂²	C₂×#13133153	1	2	yes	—
279	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	#16810650	1	2	yes	—
286	C₄	#16810665	1	2	yes	—
287	C₄	C₂×#13133224	1	2	yes	—
288	C₄	C₂×#13133245	1	1	yes	characteristic
289	C₄	C₂×#13133250	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	#20273500	1	4	yes	direct product factor
292	C₂	C₂×#16810650	1	1	yes	characteristic
293	C₂	C₂×#16810665	1	1	yes	characteristic
294	C₂	C₂²×#13133153	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	C₂×#1573
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20273500	C₂	direct (N × H)
4. C₂×#16810376	C₂	semidirect
5. C₂×#16810374	C₂	semidirect
18. #16810376	C₂²	semidirect
19. #16810374	C₂²	semidirect
150. C₂×C₅₃	#1573	semidirect
173. C₅₃	C₂×#1573	semidirect
291. C₂	#20273500	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×#16810680	2
C₂²×C₄⋊#88322	2
#20273500	2
C₂×#16810376	2
C₂×#16810374	2
C₂×C₅₃×#431	2
C₂×#1573	53

Subgroups — C2×#20273500 #27353172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273500 #27353172