Subgroups — C₂×#20276557 #27537326

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811040	C₂	1	2	yes	maximal, complemented
2	#20276557	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810407	C₂	1	2	yes	maximal, complemented
5	C₂×#16810399	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
7	#16811040	—	2	4	no	—
8	#16811040	C₂²	1	8	yes	complemented
9	C₂×#13133157	—	2	2	no	—
10	C₂×#13133157	C₂²	1	2	yes	—
11	C₂²×#11719908	—	2	2	no	—
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	#16810444	C₂²	1	4	yes	complemented
15	#16810407	C₂²	1	8	yes	complemented
16	#16810399	C₂²	1	4	yes	complemented
17	C₂×#135×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#131×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#443	C₄	1	4	yes	—
20	C₅₃×#443	C₂²	1	4	yes	—
21	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
22	C₂×#104×C₅₃	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#13133157	—	2	2	no	—
25	#13133157	—	2	8	no	—
26	#13133157	C₂³	1	4	yes	—
27	C₂×#11719908	—	2	2	no	—
28	C₂×#11719908	—	2	1	no	—
29	C₂×#11719908	—	2	1	no	—
30	C₂×#11719908	—	2	8	no	—
31	C₂×#11719908	D₄	1	2	yes	—
32	C₂×#11719908	D₄	1	2	yes	—
33	C₂×#11719908	C₂³	1	2	yes	—
34	C₂×#11719908	C₂³	1	2	yes	—
35	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
36	#135×C₅₃	C₈	1	4	yes	—
37	#135×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×Q₁₆×C₅₃	C₈	1	1	yes	characteristic
39	C₂×SD₁₆×C₅₃	C₈	1	2	yes	—
40	C₂×D₈×C₅₃	C₈	1	1	yes	characteristic
41	#131×C₅₃	—	2	2	no	—
42	#131×C₅₃	—	2	2	no	—
43	#131×C₅₃	C₂×C₄	1	4	yes	—
44	#131×C₅₃	C₂³	1	4	yes	—
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	#108×C₅₃	C₂×C₄	1	4	yes	—
52	#104×C₅₃	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₅₃	C₂³	1	2	yes	—
55	#41×C₅₃	—	2	2	no	—
56	#41×C₅₃	—	2	2	no	—
57	#41×C₅₃	C₂×C₈	1	4	yes	—
58	#41×C₅₃	C₂²×C₄	1	4	yes	—
59	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	Q₁₆×C₅₃	C₂×C₈	1	4	yes	—
66	SD₁₆×C₅₃	C₂×C₈	1	8	yes	—
67	D₈×C₅₃	C₂×C₈	1	4	yes	—
68	M₁₆×C₅₃	—	2	2	no	—
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	4	no	—
71	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₄²×C₅₃	—	2	1	no	—
86	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
87	#11719908	—	2	4	no	—
88	#11719908	—	2	2	no	—
89	#11719908	C₂×D₄	1	4	yes	—
90	C₂³×C₅₃	—	2	2	no	—
91	C₂³×C₅₃	#131	1	1	yes	characteristic
92	Q₈×C₅₃	—	2	2	no	—
93	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
94	D₄×C₅₃	—	2	4	no	—
95	D₄×C₅₃	—	2	4	no	—
96	D₄×C₅₃	—	2	2	no	—
97	D₄×C₅₃	C₂²×C₈	1	4	yes	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₅₃	—	2	4	no	—
110	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₈×C₅₃	C₂²×C₈	1	2	yes	—
113	C₈×C₅₃	C₂²×C₈	1	2	yes	—
114	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
115	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
116	C₂×#2171	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	—	2	4	no	—
119	C₂²×C₅₃	—	4	4	no	—
120	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
123	C₄×C₅₃	—	2	4	no	—
124	C₄×C₅₃	—	4	2	no	—
125	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
126	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
127	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
129	C₂×#504	—	53	2	no	—
130	C₂×#443	D₅₃	1	1	yes	characteristic
131	#2171	—	53	8	no	—
132	C₂×#364	—	53	1	no	—
133	C₂×#350	—	53	2	no	—
134	C₂×#346	—	53	1	no	—
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	—	2	1	no	—
137	C₂×C₅₃	—	4	4	no	—
138	C₂×C₅₃	#2171	1	2	yes	complemented
139	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
140	C₂×#135	Dic₅₃	1	1	yes	characteristic
141	C₂×#131	C₂×D₅₃	1	2	yes	—
142	#504	—	53	8	no	—
143	#504	—	106	4	no	—
144	C₂×M₃₂	—	53	2	no	—
145	C₂×M₃₂	—	106	2	no	—
146	C₂²×C₁₆	—	53	1	no	—
147	C₂²×C₁₆	—	53	1	no	—
148	C₂²×C₁₆	—	106	2	no	—
149	#443	Dic₅₃	1	4	yes	—
150	#443	C₂×D₅₃	1	4	yes	—
151	C₂×#108	Dic₅₃	1	1	yes	characteristic
152	C₂×#104	Dic₅₃	1	2	yes	—
153	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	#364	—	53	4	no	—
155	#350	—	53	8	no	—
156	#346	—	53	4	no	—
157	C₅₃	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	C₂×Dic₅₃	1	2	yes	—
159	#135	#88322	1	4	yes	—
160	#135	C₂×Dic₅₃	1	4	yes	—
161	C₂×Q₁₆	#88322	1	1	yes	characteristic
162	C₂×SD₁₆	#88322	1	2	yes	—
163	C₂×D₈	#88322	1	1	yes	characteristic
164	#131	—	2	2	no	—
165	#131	—	2	2	no	—
166	#131	C₂×Dic₅₃	1	4	yes	—
167	#131	C₂²×D₅₃	1	4	yes	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
174	M₃₂	—	53	4	no	—
175	M₃₂	—	106	8	no	—
176	M₃₂	—	106	2	no	—
177	C₂×C₁₆	—	53	2	no	—
178	C₂×C₁₆	—	53	2	no	—
179	C₂×C₁₆	—	53	2	no	—
180	C₂×C₁₆	—	53	2	no	—
181	C₂×C₁₆	—	106	8	no	—
182	C₂×C₁₆	—	106	2	no	—
183	C₂×C₁₆	—	106	1	no	—
184	C₂×C₁₆	—	106	1	no	—
185	#108	C₂×Dic₅₃	1	4	yes	—
186	#104	C₂×Dic₅₃	1	8	yes	—
187	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
188	C₄×C₈	C₂²×D₅₃	1	2	yes	—
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	C₂×#88322	1	4	yes	—
192	#41	C₂²×Dic₅₃	1	4	yes	—
193	C₂×Q₈	C₂×#88322	1	2	yes	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	C₂×#88322	1	2	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
199	Q₁₆	C₂×#88322	1	4	yes	—
200	SD₁₆	C₂×#88322	1	8	yes	—
201	D₈	C₂×#88322	1	4	yes	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	C₂²×Dic₅₃	1	4	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	C₂×#88322	1	1	yes	characteristic
210	C₂×C₈	C₂×#88322	1	1	yes	characteristic
211	C₂×C₈	C₂×#88322	1	2	yes	—
212	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
213	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
214	C₂×C₈	#11719947	1	1	yes	characteristic
215	C₂×C₈	#11719947	1	1	yes	characteristic
216	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
217	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
218	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	C₂²×Dic₅₃	1	2	yes	—
221	C₁₆	—	53	4	no	—
222	C₁₆	—	106	4	no	—
223	C₁₆	—	106	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	#13133224	1	1	yes	characteristic
226	Q₈	—	2	2	no	—
227	Q₈	C₂²×#88322	1	4	yes	—
228	D₄	—	2	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	D₄	C₂²×#88322	1	4	yes	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
238	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
240	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
241	C₂×C₄	#13133224	1	1	yes	characteristic
242	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	D₄×Dic₅₃	1	1	yes	characteristic
245	C₈	D₄×Dic₅₃	1	1	yes	characteristic
246	C₈	C₂²×#88322	1	2	yes	—
247	C₈	C₂²×#88322	1	2	yes	—
248	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
249	C₈	C₂×#11719947	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	D₄×#88322	1	1	yes	characteristic
254	C₂²	D₄×#88322	1	1	yes	characteristic
255	C₂²	C₂×#13133224	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	D₄×#88322	1	1	yes	characteristic
259	C₄	D₄×#88322	1	1	yes	characteristic
260	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
261	C₄	C₂³×#88322	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	4	4	no	—
265	C₂	#20276557	1	2	yes	direct product factor
266	C₂	C₂×D₄×#88322	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#135×C₅₃	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	C₂×#2171
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#16811040	C₂	semidirect
2. #20276557	C₂	direct (N × H)
3. C₂×#16810444	C₂	semidirect
4. C₂×#16810407	C₂	semidirect
5. C₂×#16810399	C₂	semidirect
8. #16811040	C₂²	semidirect
14. #16810444	C₂²	semidirect
15. #16810407	C₂²	semidirect
16. #16810399	C₂²	semidirect
138. C₂×C₅₃	#2171	semidirect
157. C₅₃	C₂×#2171	semidirect
265. C₂	#20276557	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×#16811040	2
#20276557	2
C₂×#16810444	2
C₂×#16810407	2
C₂×#16810399	2
C₂×C₅₃×#443	2
C₂×#2171	53

Subgroups — C2×#20276557 #27537326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276557 #27537326