Subgroups — C₂×#20276757 #27538319

Order 13568.

766 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133160	C₂	1	2	yes	maximal
2	C₂×C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
3	#20276757	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810427	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₁₂	—	2	1	no	—
9	C₂×#13133160	—	2	8	no	—
10	C₂×#13133160	C₂²	1	8	yes	—
11	C₂²×#11719908	C₂²	1	2	yes	—
12	C₄×D₄₂₄	—	2	2	no	—
13	C₄×D₄₂₄	C₂²	1	4	yes	—
14	C₂×#13133151	C₂²	1	1	yes	characteristic
15	C₂×#13133139	—	2	1	no	—
16	C₄⋊#11719908	C₂²	1	4	yes	complemented
17	#16810427	C₂²	1	8	yes	complemented
18	C₂²×Q₁₆×C₅₃	—	2	1	no	—
19	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
21	C₅₃×#494	—	2	2	no	—
22	C₅₃×#494	C₂²	1	4	yes	complemented
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×D₂₁₂	—	2	1	no	—
25	C₂²×C₄×D₅₃	—	4	1	no	—
26	C₂×D₄₂₄	—	2	1	no	—
27	C₂×D₄₂₄	—	2	2	no	—
28	C₂×D₄₂₄	D₄	1	4	yes	—
29	C₂×D₄₂₄	C₂³	1	2	yes	—
30	C₄×D₂₁₂	—	2	4	no	—
31	C₄×D₂₁₂	—	4	2	no	—
32	C₂×C₄⋊Dic₅₃	—	2	1	no	—
33	C₂×#11719921	—	4	1	no	—
34	#13133160	—	2	16	no	—
35	C₂×#11719908	—	2	4	no	—
36	C₂×#11719908	C₂³	1	4	yes	—
37	#13133151	—	2	1	no	—
38	#13133151	C₂³	1	2	yes	—
39	#13133139	—	2	4	no	—
40	C₂²×Q₈×C₅₃	—	2	2	no	—
41	C₂×Q₁₆×C₅₃	—	2	2	no	—
42	C₂×Q₁₆×C₅₃	—	2	4	no	—
43	C₂×Q₁₆×C₅₃	—	2	4	no	—
44	C₂×Q₁₆×C₅₃	—	2	4	no	—
45	C₂×Q₁₆×C₅₃	—	4	4	no	—
46	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
47	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₃	—	2	4	no	—
50	C₄⋊Q₈×C₅₃	—	4	2	no	—
51	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
52	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	2	yes	—
55	C₄×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂³×D₅₃	—	4	1	no	—
57	C₂×Q₈×C₅₃	—	2	4	no	—
58	C₂×Q₈×C₅₃	—	4	2	no	—
59	C₂×Q₈×C₅₃	—	4	4	no	—
60	C₂×Q₈×C₅₃	—	4	4	no	—
61	C₂²×C₄×C₅₃	—	4	2	no	—
62	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₅₃	—	4	1	no	—
66	C₂×D₂₁₂	—	2	4	no	—
67	C₂×D₂₁₂	—	2	2	no	—
68	C₂×D₂₁₂	—	4	1	no	—
69	C₂×D₂₁₂	—	4	2	no	—
70	C₂×C₄×D₅₃	—	4	4	no	—
71	C₂×C₄×D₅₃	—	8	4	no	—
72	Q₁₆×C₅₃	—	2	4	no	—
73	Q₁₆×C₅₃	—	2	4	no	—
74	Q₁₆×C₅₃	—	4	8	no	—
75	Q₁₆×C₅₃	—	4	4	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₅₃	#41	1	2	yes	—
81	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	—	4	4	no	—
83	C₄⋊C₄×C₅₃	—	4	4	no	—
84	C₄²×C₅₃	D₈	1	2	yes	—
85	C₄²×C₅₃	C₂×D₄	1	2	yes	—
86	#11719921	—	4	4	no	—
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	—	4	1	no	—
89	D₄₂₄	—	2	4	no	—
90	D₄₂₄	—	2	2	no	—
91	D₄₂₄	C₂×D₄	1	4	yes	—
92	#11719908	—	2	4	no	—
93	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
94	C₂²×D₅₃	—	4	2	no	—
95	C₂²×D₅₃	—	4	4	no	—
96	C₂²×D₅₃	—	8	2	no	—
97	C₂²×D₅₃	—	8	2	no	—
98	Q₈×C₅₃	—	4	4	no	—
99	Q₈×C₅₃	—	4	4	no	—
100	Q₈×C₅₃	—	8	4	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	4	no	—
103	C₂×C₄×C₅₃	—	8	4	no	—
104	C₂×C₄×C₅₃	SD₃₂	1	4	yes	complemented
105	C₂×C₄×C₅₃	#121	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₅₃	—	4	2	no	—
112	C₂×Dic₅₃	—	8	2	no	—
113	D₂₁₂	—	2	4	no	—
114	D₂₁₂	—	4	2	no	—
115	D₂₁₂	—	4	4	no	—
116	C₄×D₅₃	—	8	8	no	—
117	C₈×C₅₃	—	2	2	no	—
118	C₈×C₅₃	#121	1	2	yes	—
119	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
121	C₂×#2208	—	53	1	no	maximal, Sylow
122	C₂²×C₅₃	#459	1	2	yes	—
123	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
124	C₂²×C₅₃	#510	1	2	yes	—
125	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
126	C₂×D₅₃	—	4	4	no	—
127	C₂×D₅₃	—	8	2	no	—
128	C₂×D₅₃	—	8	2	no	—
129	C₂×D₅₃	—	8	8	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	8	4	no	—
132	C₄×C₅₃	#459	1	2	yes	—
133	C₄×C₅₃	C₂×SD₃₂	1	4	yes	complemented
134	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
135	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
136	Dic₅₃	—	8	2	no	—
137	C₂²×SD₃₂	—	53	2	no	—
138	C₂×#494	D₅₃	1	1	yes	characteristic, complemented
139	C₂×C₄×D₈	—	53	1	no	—
140	#2208	—	53	8	no	—
141	C₂×C₄⋊C₁₆	—	53	1	no	—
142	C₂×#358	—	53	2	no	—
143	C₂×C₅₃	#2208	1	4	yes	complemented
144	C₂×C₅₃	C₂×#459	1	1	yes	characteristic
145	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
146	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
147	D₅₃	—	8	4	no	—
148	C₂²×Q₁₆	—	2	1	no	—
149	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
150	C₂²×D₈	—	53	1	no	—
151	C₂×C₄⋊Q₈	—	2	1	no	—
152	C₂×C₄×D₄	—	106	1	no	—
153	C₂×SD₃₂	—	53	8	no	—
154	C₂×SD₃₂	—	106	8	no	—
155	C₂²×C₁₆	—	53	2	no	—
156	#494	—	2	2	no	—
157	#494	C₂×D₅₃	1	4	yes	complemented
158	C₄×D₈	—	53	4	no	—
159	C₄×D₈	—	106	2	no	—
160	C₂×#107	—	53	1	no	—
161	C₂×#102	—	106	1	no	—
162	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
163	C₄⋊C₁₆	—	53	4	no	—
164	#358	—	53	8	no	—
165	C₅₃	C₂×#2208	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×D₄	—	106	1	no	—
168	C₂³×C₄	—	212	1	no	—
169	C₂×Q₁₆	—	2	4	no	—
170	C₂×Q₁₆	—	2	4	no	—
171	C₂×Q₁₆	—	2	4	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	—	4	4	no	—
174	C₂×Q₁₆	C₂²×D₅₃	1	4	yes	—
175	C₂×D₈	—	53	4	no	—
176	C₂×D₈	—	53	2	no	—
177	C₂×D₈	—	106	2	no	—
178	C₂×D₈	—	106	1	no	—
179	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	C₄⋊Q₈	—	2	4	no	—
182	C₄⋊Q₈	—	4	2	no	—
183	C₄×D₄	—	106	4	no	—
184	C₄×D₄	—	212	2	no	—
185	C₂×C₄⋊C₄	—	4	1	no	—
186	C₂×C₄⋊C₄	—	4	1	no	—
187	C₂×C₄⋊C₄	—	106	1	no	—
188	C₂×C₂²⋊C₄	—	212	1	no	—
189	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
190	SD₃₂	—	106	16	no	—
191	C₂×C₁₆	—	53	4	no	—
192	C₂×C₁₆	—	106	4	no	—
193	#107	—	53	2	no	—
194	#107	—	106	1	no	—
195	#102	—	106	4	no	—
196	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
197	C₄×C₈	C₂²×D₅₃	1	2	yes	—
198	C₂⁴	—	212	1	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×D₄	—	106	4	no	—
204	C₂×D₄	—	106	2	no	—
205	C₂×D₄	—	212	1	no	—
206	C₂×D₄	—	212	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	424	4	no	—
209	C₂²×C₄	—	212	1	no	—
210	C₂²×C₄	—	212	4	no	—
211	C₂²×C₄	#11719922	1	1	yes	characteristic
212	C₂²×C₄	#11719922	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	2	4	no	—
216	Q₁₆	—	4	8	no	—
217	Q₁₆	—	4	4	no	—
218	D₈	—	53	4	no	—
219	D₈	—	106	4	no	—
220	D₈	—	106	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	D₄×D₅₃	1	2	yes	—
223	C₂×C₈	#11719947	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	106	2	no	—
230	C₄⋊C₄	—	212	1	no	—
231	C₂²⋊C₄	—	212	4	no	—
232	C₄²	#11719922	1	2	yes	—
233	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
234	C₁₆	—	106	4	no	—
235	C₂³	—	424	2	no	—
236	C₂³	—	424	2	no	—
237	C₂³	—	212	4	no	—
238	C₂³	—	212	2	no	—
239	C₂³	C₂×#11719922	1	1	yes	characteristic
240	Q₈	—	4	4	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	8	4	no	—
243	D₄	—	106	4	no	—
244	D₄	—	212	4	no	—
245	D₄	—	212	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	—	424	2	no	—
250	C₂×C₄	—	424	8	no	—
251	C₂×C₄	—	212	2	no	—
252	C₂×C₄	#13133160	1	4	yes	complemented
253	C₂×C₄	#13133188	1	2	yes	—
254	C₂×C₄	C₂×#11719922	1	2	yes	—
255	C₂×C₄	C₂×#11719922	1	2	yes	—
256	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
257	C₂×C₄	#13133243	1	2	yes	—
258	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	#13133188	1	2	yes	—
261	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
262	C₈	C₂×#11719947	1	1	yes	characteristic
263	C₂²	—	424	8	no	—
264	C₂²	—	424	2	no	—
265	C₂²	—	424	2	no	—
266	C₂²	—	212	4	no	—
267	C₂²	#16810815	1	2	yes	—
268	C₂²	C₂×#13133160	1	2	yes	—
269	C₂²	#16811064	1	2	yes	—
270	C₂²	C₂²×#11719922	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	8	4	no	—
273	C₄	—	424	2	no	—
274	C₄	#16810815	1	2	yes	—
275	C₄	C₂×#13133160	1	4	yes	complemented
276	C₄	C₂×#13133188	1	1	yes	characteristic
277	C₄	C₂×#13133243	1	1	yes	characteristic
278	C₂	—	424	4	no	—
279	C₂	#20276757	1	4	yes	direct product factor
280	C₂	C₂×#16810815	1	1	yes	characteristic
281	C₂	C₂²×#13133160	1	1	yes	characteristic
282	C₂	C₂×#16811064	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	C₂×#2208
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #20276757	C₂	direct (N × H)
4. C₂×C₄⋊#11719908	C₂	semidirect
5. C₂×#16810427	C₂	semidirect
6. C₂×C₅₃×#494	C₂	semidirect
16. C₄⋊#11719908	C₂²	semidirect
17. #16810427	C₂²	semidirect
22. C₅₃×#494	C₂²	semidirect
104. C₂×C₄×C₅₃	SD₃₂	semidirect
133. C₄×C₅₃	C₂×SD₃₂	semidirect
138. C₂×#494	D₅₃	semidirect
143. C₂×C₅₃	#2208	semidirect
157. #494	C₂×D₅₃	semidirect
165. C₅₃	C₂×#2208	semidirect
252. C₂×C₄	#13133160	semidirect
275. C₄	C₂×#13133160	semidirect
279. C₂	#20276757	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13133160	2
C₂×C₄×D₄₂₄	2
#20276757	2
C₂×C₄⋊#11719908	2
C₂×#16810427	2
C₂×C₅₃×#494	2
C₂×#2208	53

Subgroups — C2×#20276757 #27538319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276757 #27538319