Subgroups — C₂×#20272760 #27295327

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811000	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272760	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₂₄	—	2	1	no	—
6	C₂²×D₄₂₄	C₄	1	2	yes	—
7	C₂×#13133214	—	2	1	no	—
8	C₂²×#11719908	—	2	1	no	—
9	#16811000	—	2	1	no	—
10	#16811000	—	2	1	no	—
11	#16811000	C₄	1	2	yes	—
12	#16811000	C₂²	1	2	yes	—
13	C₄⋊#11719908	—	2	1	no	—
14	C₄⋊#11719908	C₂²	1	2	yes	complemented
15	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	—	2	1	no	—
18	C₅₃×#335	C₂²	1	2	yes	complemented
19	C₂²×D₂₁₂	—	2	2	no	—
20	C₂²×D₂₁₂	—	4	1	no	—
21	C₂²×D₂₁₂	—	4	1	no	—
22	C₂×D₄₂₄	—	2	2	no	—
23	C₂×D₄₂₄	—	2	2	no	—
24	C₂×D₄₂₄	—	2	2	no	—
25	C₂×D₄₂₄	—	4	2	no	—
26	C₂×D₄₂₄	—	4	2	no	—
27	C₂×D₄₂₄	—	4	4	no	—
28	C₂×D₄₂₄	C₂×C₄	1	4	yes	—
29	#13133214	—	2	2	no	—
30	#13133214	—	2	2	no	—
31	#13133214	—	4	1	no	—
32	#13133214	—	4	1	no	—
33	C₂×#11719908	—	2	2	no	—
34	C₂×#11719908	—	4	2	no	—
35	C₂²×C₈×C₅₃	—	4	1	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	—	4	1	no	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂³×D₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂×D₂₁₂	—	2	4	no	—
50	C₂×D₂₁₂	—	4	4	no	—
51	C₂×D₂₁₂	—	4	4	no	—
52	C₂×D₂₁₂	—	4	2	no	—
53	C₂×D₂₁₂	—	4	2	no	—
54	C₂×D₂₁₂	—	4	2	no	—
55	C₂×D₂₁₂	—	4	2	no	—
56	C₂×D₂₁₂	—	4	2	no	—
57	C₂×D₂₁₂	—	8	4	no	—
58	C₂×D₂₁₂	—	8	2	no	—
59	C₂×D₂₁₂	—	8	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	8	2	no	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	D₈	1	2	yes	—
66	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₃	D₈	1	1	yes	characteristic
71	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	D₄₂₄	—	4	4	no	—
74	D₄₂₄	—	4	4	no	—
75	D₄₂₄	—	4	4	no	—
76	D₄₂₄	—	4	2	no	—
77	D₄₂₄	—	4	2	no	—
78	#11719908	—	4	2	no	—
79	C₂³×C₅₃	#102	1	1	yes	characteristic
80	C₂²×D₅₃	—	4	4	no	—
81	C₂²×D₅₃	—	8	4	no	—
82	C₂²×D₅₃	—	8	4	no	—
83	C₂²×D₅₃	—	8	4	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	#100	1	2	yes	—
86	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#102	1	2	yes	—
90	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
95	D₂₁₂	—	4	4	no	—
96	D₂₁₂	—	4	4	no	—
97	D₂₁₂	—	8	4	no	—
98	D₂₁₂	—	8	4	no	—
99	D₂₁₂	—	8	4	no	—
100	D₂₁₂	—	8	4	no	—
101	D₂₁₂	—	8	2	no	—
102	D₂₁₂	—	8	2	no	—
103	C₈×C₅₃	—	2	1	no	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	8	2	no	—
106	C₈×C₅₃	#102	1	2	yes	—
107	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
108	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×#1361	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	#331	1	2	yes	—
111	C₂²×C₅₃	#357	1	2	yes	—
112	C₂²×C₅₃	#361	1	2	yes	—
113	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
114	C₂×D₅₃	—	8	4	no	—
115	C₂×D₅₃	—	8	4	no	—
116	C₂×D₅₃	—	8	4	no	—
117	C₂×D₅₃	—	16	8	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	#331	1	2	yes	—
121	C₄×C₅₃	#357	1	2	yes	—
122	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
123	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
124	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
125	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
126	C₂×#493	—	53	1	no	—
127	C₂×C₄⋊C₁₆	—	53	1	no	—
128	C₂×#335	D₅₃	1	1	yes	characteristic, complemented
129	#1361	—	53	4	no	—
130	C₂×C₅₃	#1361	1	4	yes	complemented
131	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
132	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
133	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
134	D₅₃	—	16	4	no	—
135	C₂²×D₈	—	53	2	no	—
136	C₂²×D₈	—	106	1	no	—
137	C₂×#127	—	106	1	no	—
138	C₂²×C₁₆	—	106	1	no	—
139	#493	—	53	2	no	—
140	#493	—	53	2	no	—
141	#493	—	106	1	no	—
142	#493	—	106	1	no	—
143	C₂×C₄⋊C₈	—	2	1	no	—
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	C₄⋊C₁₆	—	53	2	no	—
146	C₄⋊C₁₆	—	106	1	no	—
147	#335	—	2	1	no	—
148	#335	C₂×D₅₃	1	2	yes	complemented
149	C₅₃	C₂×#1361	1	1	yes	characteristic, complemented, Sylow
150	C₂²×D₄	—	106	2	no	—
151	C₂²×D₄	—	212	1	no	—
152	C₂²×D₄	—	212	1	no	—
153	C₂×D₈	—	53	4	no	—
154	C₂×D₈	—	106	2	no	—
155	C₂×D₈	—	106	2	no	—
156	C₂×D₈	—	106	2	no	—
157	C₂×D₈	—	212	4	no	—
158	C₂×D₈	—	212	2	no	—
159	C₂×D₈	—	212	2	no	—
160	C₂²×C₈	—	4	1	no	—
161	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
162	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
163	#127	—	106	2	no	—
164	#127	—	106	2	no	—
165	#127	—	212	1	no	—
166	#127	—	212	1	no	—
167	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
168	C₂×C₁₆	—	106	2	no	—
169	C₂×C₁₆	—	212	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	4	1	no	—
172	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
173	C₄×C₈	D₂₁₂	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
176	C₂⁴	—	212	2	no	—
177	C₂×D₄	—	106	4	no	—
178	C₂×D₄	—	424	4	no	—
179	C₂×D₄	—	424	2	no	—
180	C₂×D₄	—	424	2	no	—
181	C₂×D₄	—	212	4	no	—
182	C₂×D₄	—	212	2	no	—
183	C₂×D₄	—	212	2	no	—
184	C₂×D₄	—	212	2	no	—
185	C₂×D₄	—	212	4	no	—
186	C₂×D₄	—	212	2	no	—
187	C₂×D₄	—	212	2	no	—
188	C₂²×C₄	#11719921	1	1	yes	characteristic
189	C₂²×C₄	#11719922	1	1	yes	characteristic
190	C₂²×C₄	#11719924	1	1	yes	characteristic
191	D₈	—	212	4	no	—
192	D₈	—	212	4	no	—
193	D₈	—	212	4	no	—
194	D₈	—	212	2	no	—
195	D₈	—	212	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	8	2	no	—
199	C₂×C₈	#11719913	1	2	yes	—
200	C₂×C₈	D₄₂₄	1	2	yes	—
201	C₂×C₈	#11719921	1	1	yes	characteristic
202	C₂×C₈	#11719921	1	1	yes	characteristic
203	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
205	C₄²	#11719921	1	1	yes	characteristic
206	C₄²	#11719922	1	1	yes	characteristic
207	C₄²	#11719924	1	1	yes	characteristic
208	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
209	C₁₆	—	212	2	no	—
210	C₂³	—	424	4	no	—
211	C₂³	—	424	4	no	—
212	C₂³	—	424	4	no	—
213	C₂³	—	212	4	no	—
214	C₂³	#13133137	1	1	yes	characteristic
215	D₄	—	424	4	no	—
216	D₄	—	424	4	no	—
217	D₄	—	424	2	no	—
218	D₄	—	424	2	no	—
219	D₄	—	424	4	no	—
220	D₄	—	424	4	no	—
221	D₄	—	212	4	no	—
222	D₄	—	212	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	#13133133	1	2	yes	—
225	C₂×C₄	#13133137	1	1	yes	characteristic
226	C₂×C₄	#13133137	1	1	yes	characteristic
227	C₂×C₄	#13133137	1	1	yes	characteristic
228	C₂×C₄	#13133139	1	2	yes	—
229	C₂×C₄	#13133158	1	2	yes	—
230	C₂×C₄	#13133160	1	2	yes	—
231	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
233	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	8	2	no	—
237	C₈	#13133139	1	2	yes	—
238	C₈	C₂×D₄₂₄	1	1	yes	characteristic
239	C₈	C₂×#11719913	1	1	yes	characteristic
240	C₂²	—	848	8	no	—
241	C₂²	—	424	4	no	—
242	C₂²	—	424	4	no	—
243	C₂²	—	424	4	no	—
244	C₂²	#16810366	1	2	yes	—
245	C₂²	#16810423	1	2	yes	—
246	C₂²	#16810435	1	2	yes	—
247	C₂²	C₂×#13133137	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	#16810366	1	2	yes	—
251	C₄	#16810423	1	2	yes	—
252	C₄	C₂×#13133133	1	1	yes	characteristic
253	C₄	C₂×#13133139	1	1	yes	characteristic
254	C₄	C₂×#13133158	1	1	yes	characteristic
255	C₄	C₂×#13133160	1	1	yes	characteristic
256	C₂	—	848	4	no	—
257	C₂	#20272760	1	4	yes	direct product factor
258	C₂	C₂×#16810366	1	1	yes	characteristic
259	C₂	C₂×#16810423	1	1	yes	characteristic
260	C₂	C₂×#16810435	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133137
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle	C₂³×C₅₃	#102
prime radical	C₂×#16811000	C₂
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	C₂×#1361
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#11719908	C₂	semidirect
3. #20272760	C₂	direct (N × H)
4. C₂×C₅₃×#335	C₂	semidirect
14. C₄⋊#11719908	C₂²	semidirect
18. C₅₃×#335	C₂²	semidirect
128. C₂×#335	D₅₃	semidirect
130. C₂×C₅₃	#1361	semidirect
148. #335	C₂×D₅₃	semidirect
149. C₅₃	C₂×#1361	semidirect
257. C₂	#20272760	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×#16811000	2
C₂×C₄⋊#11719908	2
#20272760	2
C₂×C₅₃×#335	2
C₂×#1361	53

Subgroups — C2×#20272760 #27295327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272760 #27295327