Subgroups — C₂×#20273630 #27353863

Order 13568.

806 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810702	C₂	1	2	yes	maximal, complemented
3	#20273630	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810352	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₂₄	C₄	1	1	yes	characteristic
8	C₂²×#11719912	—	2	2	no	—
9	C₂×C₄×D₂₁₂	C₂²	1	2	yes	—
10	C₄×D₄₂₄	C₄	1	4	yes	—
11	C₄×D₄₂₄	C₂²	1	4	yes	—
12	#16810702	—	2	4	no	—
13	#16810702	C₂²	1	8	yes	complemented
14	C₂×#13133151	C₄	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×#13133139	C₄	1	2	yes	—
17	C₂×#13133129	—	2	2	no	—
18	#16810374	C₂²	1	4	yes	complemented
19	#16810352	C₂²	1	8	yes	complemented
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#322	C₂²	1	4	yes	complemented
24	C₂²×D₂₁₂	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₅₃	—	2	2	no	—
26	C₂×D₄₂₄	—	2	4	no	—
27	C₂×D₄₂₄	C₂×C₄	1	4	yes	—
28	C₂×#11719912	—	2	8	no	—
29	C₂×#11719912	—	4	8	no	—
30	C₂²×#88322	—	2	2	no	—
31	C₄×D₂₁₂	—	2	2	no	—
32	C₄×D₂₁₂	—	2	2	no	—
33	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
34	C₄×D₂₁₂	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
36	C₂×#11719921	—	2	2	no	—
37	#13133151	C₂×C₄	1	4	yes	—
38	C₄⋊#88322	—	2	2	no	—
39	C₄⋊#88322	C₂³	1	4	yes	—
40	#13133139	C₂×C₄	1	8	yes	—
41	#13133129	—	2	8	no	—
42	C₂²×C₈×C₅₃	—	2	1	no	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	D₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₅₃	—	2	1	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₄×C₈×C₅₃	C₂³	1	2	yes	—
51	C₄×C₈×C₅₃	C₂³	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂³×D₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
57	C₂²×Dic₅₃	—	2	2	no	—
58	C₂×D₂₁₂	—	2	4	no	—
59	C₂×D₂₁₂	—	2	4	no	—
60	C₂×D₂₁₂	—	2	2	no	—
61	C₂×D₂₁₂	C₂²×C₄	1	4	yes	—
62	C₂×C₄×D₅₃	—	2	4	no	—
63	C₂×C₄×D₅₃	—	2	4	no	—
64	C₂×C₄×D₅₃	—	4	4	no	—
65	C₂×C₄×D₅₃	—	4	4	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
74	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	#41	1	1	yes	characteristic
77	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
78	#11719921	—	2	4	no	—
79	#11719921	—	2	4	no	—
80	C₄⋊Dic₅₃	—	2	2	no	—
81	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	4	4	no	—
84	D₄₂₄	—	2	8	no	—
85	#11719912	—	4	16	no	—
86	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
87	C₂²×D₅₃	—	2	4	no	—
88	C₂²×D₅₃	—	4	4	no	—
89	C₂²×D₅₃	—	4	4	no	—
90	C₂²×D₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₅₃	#131	1	2	yes	—
97	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
98	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	4	no	—
102	C₂×Dic₅₃	—	4	4	no	—
103	D₂₁₂	—	2	4	no	—
104	D₂₁₂	—	2	4	no	—
105	D₂₁₂	—	4	4	no	—
106	C₄×D₅₃	—	4	4	no	—
107	C₄×D₅₃	—	4	4	no	—
108	C₄×D₅₃	—	4	8	no	—
109	C₈×C₅₃	—	2	2	no	—
110	C₈×C₅₃	—	2	4	no	—
111	C₈×C₅₃	—	4	2	no	—
112	C₈×C₅₃	C₂×M₁₆	1	4	yes	complemented
113	#88322	—	4	4	no	—
114	C₂×#1594	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	#436	1	2	yes	—
116	C₂²×C₅₃	#442	1	2	yes	—
117	C₂²×C₅₃	#444	1	2	yes	—
118	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×D₅₃	—	4	4	no	—
120	C₂×D₅₃	—	4	4	no	—
121	C₂×D₅₃	—	4	4	no	—
122	C₂×D₅₃	—	8	8	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	#436	1	2	yes	—
126	C₄×C₅₃	#442	1	2	yes	—
127	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
128	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
129	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	Dic₅₃	—	4	4	no	—
131	C₂×C₄×D₈	—	53	1	no	—
132	C₂×#436	—	53	2	no	—
133	#1594	—	53	8	no	—
134	C₂×#334	—	53	1	no	—
135	C₂×D₄⋊C₈	—	53	2	no	—
136	C₂×#322	D₅₃	1	1	yes	characteristic, complemented
137	C₂×C₅₃	#1594	1	4	yes	complemented
138	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
139	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
140	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
141	D₅₃	—	8	4	no	—
142	C₂²×D₈	—	53	1	no	—
143	C₂²×M₁₆	—	106	2	no	—
144	C₂×C₄×D₄	—	53	2	no	—
145	C₄×D₈	—	53	4	no	—
146	C₄×D₈	—	53	4	no	—
147	#436	—	53	8	no	—
148	#436	—	106	4	no	—
149	C₂×#107	—	53	1	no	—
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×#102	—	53	2	no	—
152	C₂×C₂²⋊C₈	—	106	2	no	—
153	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
156	#334	—	53	4	no	—
157	D₄⋊C₈	—	53	8	no	—
158	#322	C₂×D₅₃	1	4	yes	complemented
159	C₅₃	C₂×#1594	1	1	yes	characteristic, complemented, Sylow
160	C₂²×D₄	—	53	2	no	—
161	C₂³×C₄	—	106	2	no	—
162	C₂×D₈	—	53	4	no	—
163	C₂×D₈	—	106	4	no	—
164	C₂×M₁₆	—	106	8	no	—
165	C₂×M₁₆	—	212	8	no	—
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	—	106	2	no	—
168	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
169	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
170	C₂²×C₈	D₂₁₂	1	2	yes	—
171	C₄×D₄	—	53	4	no	—
172	C₄×D₄	—	53	4	no	—
173	C₄×D₄	—	106	2	no	—
174	C₄×D₄	—	106	2	no	—
175	C₂×C₄⋊C₄	—	53	2	no	—
176	C₂×C₂²⋊C₄	—	106	2	no	—
177	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
178	#107	—	53	4	no	—
179	C₄⋊C₈	—	53	4	no	—
180	C₄⋊C₈	—	106	2	no	—
181	#102	—	53	8	no	—
182	C₂²⋊C₈	—	106	8	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	C₄×D₅₃	1	2	yes	—
185	C₄×C₈	D₂₁₂	1	2	yes	—
186	C₄×C₈	C₂²×D₅₃	1	2	yes	—
187	C₄×C₈	C₂²×D₅₃	1	2	yes	—
188	C₄×C₈	C₂²×D₅₃	1	2	yes	—
189	C₂⁴	—	106	2	no	—
190	C₂×D₄	—	53	4	no	—
191	C₂×D₄	—	106	2	no	—
192	C₂×D₄	—	106	4	no	—
193	C₂×D₄	—	106	4	no	—
194	C₂²×C₄	—	106	2	no	—
195	C₂²×C₄	—	106	4	no	—
196	C₂²×C₄	—	106	4	no	—
197	C₂²×C₄	—	212	4	no	—
198	C₂²×C₄	—	212	4	no	—
199	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
201	C₂²×C₄	#11719945	1	1	yes	characteristic
202	D₈	—	106	8	no	—
203	M₁₆	—	212	16	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	—	212	4	no	—
210	C₂×C₈	#11719912	1	4	yes	complemented
211	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
213	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
214	C₄⋊C₄	—	53	4	no	—
215	C₄⋊C₄	—	106	2	no	—
216	C₂²⋊C₄	—	106	4	no	—
217	C₂²⋊C₄	—	106	4	no	—
218	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
219	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
220	C₄²	#11719945	1	1	yes	characteristic
221	C₄²	C₂³×D₅₃	1	1	yes	characteristic
222	C₂³	—	106	4	no	—
223	C₂³	—	212	4	no	—
224	C₂³	—	212	4	no	—
225	C₂³	—	212	4	no	—
226	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
227	D₄	—	106	4	no	—
228	D₄	—	106	4	no	—
229	D₄	—	212	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	106	4	no	—
232	C₂×C₄	—	212	4	no	—
233	C₂×C₄	—	212	4	no	—
234	C₂×C₄	—	212	4	no	—
235	C₂×C₄	—	212	8	no	—
236	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
237	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
238	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
239	C₂×C₄	C₂×#11719912	1	2	yes	—
240	C₂×C₄	#13133223	1	2	yes	—
241	C₂×C₄	#13133245	1	4	yes	—
242	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
244	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	2	4	no	—
247	C₈	—	4	2	no	—
248	C₈	—	212	4	no	—
249	C₈	C₂×#11719912	1	4	yes	complemented
250	C₂²	—	424	8	no	—
251	C₂²	—	212	4	no	—
252	C₂²	—	212	4	no	—
253	C₂²	—	212	4	no	—
254	C₂²	#16810702	1	2	yes	—
255	C₂²	#16810736	1	2	yes	—
256	C₂²	#16810746	1	2	yes	—
257	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	212	4	no	—
261	C₄	#16810702	1	2	yes	—
262	C₄	#16810736	1	2	yes	—
263	C₄	C₂²×#11719912	1	1	yes	characteristic
264	C₄	C₂×#13133223	1	1	yes	characteristic
265	C₄	C₂×#13133245	1	2	yes	—
266	C₂	—	424	4	no	—
267	C₂	#20273630	1	4	yes	direct product factor
268	C₂	C₂×#16810702	1	1	yes	characteristic
269	C₂	C₂×#16810736	1	1	yes	characteristic
270	C₂	C₂×#16810746	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×#322	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₄₂₄	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	C₂×#1594
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. C₂×#16810702	C₂	semidirect
3. #20273630	C₂	direct (N × H)
4. C₂×#16810374	C₂	semidirect
5. C₂×#16810352	C₂	semidirect
6. C₂×C₅₃×#322	C₂	semidirect
13. #16810702	C₂²	semidirect
18. #16810374	C₂²	semidirect
19. #16810352	C₂²	semidirect
23. C₅₃×#322	C₂²	semidirect
70. C₂×C₈×C₅₃	M₁₆	semidirect
112. C₈×C₅₃	C₂×M₁₆	semidirect
136. C₂×#322	D₅₃	semidirect
137. C₂×C₅₃	#1594	semidirect
158. #322	C₂×D₅₃	semidirect
159. C₅₃	C₂×#1594	semidirect
210. C₂×C₈	#11719912	semidirect
249. C₈	C₂×#11719912	semidirect
267. C₂	#20273630	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×C₄×D₄₂₄	2
C₂×#16810702	2
#20273630	2
C₂×#16810374	2
C₂×#16810352	2
C₂×C₅₃×#322	2
C₂×#1594	53

Subgroups — C2×#20273630 #27353863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273630 #27353863