Subgroups — C₂×#20276556 #27537325

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×#16811041	C₂	1	2	yes	maximal, complemented
2	C₂×#16810741	C₂	1	1	yes	maximal, characteristic
3	#20276556	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810408	C₂	1	2	yes	maximal, complemented
5	C₂×#16810399	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133239	C₄	1	1	yes	characteristic
8	C₂×#13133223	C₂²	1	2	yes	—
9	#16811041	—	2	4	no	—
10	#16811041	C₂²	1	8	yes	complemented
11	C₂×#13133156	—	2	2	no	—
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₃	—	2	2	no	—
15	#16810741	C₄	1	4	yes	—
16	#16810741	C₂²	1	4	yes	—
17	C₂×#13133153	C₄	1	1	yes	characteristic
18	C₂×#13133145	C₄	1	2	yes	—
19	#16810408	C₂²	1	8	yes	complemented
20	#16810399	C₂²	1	4	yes	complemented
21	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#364	C₂²	1	4	yes	complemented
24	C₂×#11719945	C₂×C₄	1	2	yes	—
25	#13133239	C₈	1	4	yes	—
26	#13133239	C₂×C₄	1	4	yes	—
27	C₂×Dic₂₁₂	C₈	1	1	yes	characteristic
28	C₂×#11719913	C₈	1	2	yes	—
29	C₂×D₄₂₄	C₈	1	1	yes	characteristic
30	#13133223	—	2	2	no	—
31	#13133223	—	2	2	no	—
32	#13133223	C₂×C₄	1	4	yes	—
33	#13133223	C₂³	1	4	yes	—
34	C₂×#11719917	C₂×C₄	1	2	yes	—
35	C₂×#11719912	—	2	2	no	—
36	C₂×C₈×D₅₃	—	2	2	no	—
37	#13133156	—	2	8	no	—
38	C₂×#11719908	—	2	1	no	—
39	C₂×#11719908	—	2	1	no	—
40	C₂×#11719908	D₄	1	2	yes	—
41	C₂×#11719908	D₄	1	2	yes	—
42	C₂×#11719908	C₂³	1	2	yes	—
43	C₂×#11719908	C₂³	1	2	yes	—
44	C₁₆×D₅₃	—	2	8	no	—
45	#13133153	C₂×C₄	1	4	yes	—
46	#13133145	C₂×C₄	1	8	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	M₃₂×C₅₃	—	2	2	no	—
51	M₃₂×C₅₃	C₂³	1	4	yes	—
52	C₂×C₁₆×C₅₃	—	2	2	no	—
53	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₅₃	—	2	2	no	—
58	#11719945	—	2	2	no	—
59	#11719945	—	2	2	no	—
60	#11719945	C₂×C₈	1	4	yes	—
61	#11719945	C₂²×C₄	1	4	yes	—
62	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₅₃	—	2	2	no	—
64	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
77	#11719917	—	2	2	no	—
78	#11719917	C₂²×C₄	1	4	yes	—
79	C₂×#88322	—	2	2	no	—
80	Dic₂₁₂	C₂×C₈	1	4	yes	—
81	D₄₂₄	C₂×C₈	1	4	yes	—
82	#11719913	C₂×C₈	1	8	yes	—
83	#11719912	—	2	4	no	—
84	#11719912	—	2	4	no	—
85	C₈×D₅₃	—	2	4	no	—
86	C₈×D₅₃	—	2	4	no	—
87	C₁₆×C₅₃	—	2	4	no	—
88	#11719908	—	2	2	no	—
89	#11719908	C₂×D₄	1	4	yes	—
90	C₂³×C₅₃	#131	1	1	yes	characteristic
91	C₂²×D₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂²⋊D₅₃	—	2	4	no	—
101	C₂²⋊D₅₃	—	2	4	no	—
102	C₂×Dic₅₃	—	2	2	no	—
103	D₂₁₂	—	2	2	no	—
104	D₂₁₂	C₂²×C₈	1	4	yes	—
105	C₄×D₅₃	—	2	4	no	—
106	C₄×D₅₃	—	2	4	no	—
107	Dic₁₀₆	—	2	2	no	—
108	Dic₁₀₆	C₂²×C₈	1	4	yes	—
109	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₈×C₅₃	C₂²×C₈	1	2	yes	—
112	C₈×C₅₃	C₂²×C₈	1	2	yes	—
113	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
115	#88322	—	2	4	no	—
116	C₂×#2171	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
119	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
121	C₂×D₅₃	—	2	4	no	—
122	C₂×D₅₃	—	4	4	no	—
123	C₄×C₅₃	—	4	2	no	—
124	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
125	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
126	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
128	Dic₅₃	—	2	4	no	—
129	C₂×#504	—	53	2	no	—
130	C₂×#443	—	53	1	no	—
131	#2171	—	53	8	no	—
132	C₂×#364	D₅₃	1	1	yes	characteristic, complemented
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	#2171	1	2	yes	complemented
138	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
139	D₅₃	—	4	4	no	—
140	C₂×#135	—	53	1	no	—
141	C₂×#131	—	53	2	no	—
142	#504	—	53	8	no	—
143	#504	—	106	4	no	—
144	C₂×M₃₂	—	106	2	no	—
145	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
146	C₂²×C₁₆	—	53	1	no	—
147	C₂²×C₁₆	—	53	1	no	—
148	C₂²×C₁₆	—	106	2	no	—
149	#443	—	53	4	no	—
150	#443	—	53	4	no	—
151	C₂×#108	—	53	1	no	—
152	C₂×#104	—	53	2	no	—
153	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	#364	C₂×D₅₃	1	4	yes	complemented
155	#350	—	53	8	no	—
156	#346	—	53	4	no	—
157	C₅₃	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	53	2	no	—
159	#135	—	53	4	no	—
160	#135	—	53	4	no	—
161	C₂×Q₁₆	—	53	1	no	—
162	C₂×SD₁₆	—	53	2	no	—
163	C₂×D₈	—	53	1	no	—
164	#131	—	53	4	no	—
165	#131	—	53	4	no	—
166	#131	—	106	2	no	—
167	#131	—	106	2	no	—
168	C₂×M₁₆	—	53	2	no	—
169	C₂×M₁₆	—	106	2	no	—
170	C₂²×C₈	—	106	2	no	—
171	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
174	M₃₂	—	2	2	no	—
175	M₃₂	—	106	8	no	—
176	M₃₂	C₂²×D₅₃	1	4	yes	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	53	2	no	—
179	C₂×C₁₆	—	53	2	no	—
180	C₂×C₁₆	—	53	2	no	—
181	C₂×C₁₆	—	53	2	no	—
182	C₂×C₁₆	—	106	8	no	—
183	C₂×C₁₆	—	106	1	no	—
184	C₂×C₁₆	—	106	1	no	—
185	#108	—	53	4	no	—
186	#104	—	53	8	no	—
187	C₄×C₈	C₄×D₅₃	1	2	yes	—
188	C₄×C₈	C₂²×D₅₃	1	2	yes	—
189	#41	—	53	4	no	—
190	#41	—	53	4	no	—
191	#41	—	106	2	no	—
192	#41	—	106	2	no	—
193	C₂×Q₈	—	53	2	no	—
194	C₂×D₄	—	53	2	no	—
195	C₂×D₄	—	106	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	106	2	no	—
198	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
199	Q₁₆	—	53	4	no	—
200	SD₁₆	—	53	8	no	—
201	D₈	—	53	4	no	—
202	M₁₆	—	53	4	no	—
203	M₁₆	—	106	2	no	—
204	M₁₆	—	106	4	no	—
205	M₁₆	—	106	4	no	—
206	C₂×C₈	—	106	2	no	—
207	C₂×C₈	—	106	4	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
210	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
211	C₂×C₈	C₈×D₅₃	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
214	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
215	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
216	C₂×C₈	#11719949	1	1	yes	characteristic
217	C₂×C₈	#11719949	1	1	yes	characteristic
218	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	C₂×C₄×D₅₃	1	2	yes	—
221	C₁₆	—	2	4	no	—
222	C₁₆	—	53	4	no	—
223	C₁₆	—	106	2	no	—
224	C₂³	—	106	2	no	—
225	C₂³	#13133225	1	1	yes	characteristic
226	Q₈	—	53	4	no	—
227	Q₈	—	106	2	no	—
228	D₄	—	53	4	no	—
229	D₄	—	106	4	no	—
230	D₄	—	106	4	no	—
231	D₄	—	106	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	106	2	no	—
235	C₂×C₄	—	106	4	no	—
236	C₂×C₄	—	106	4	no	—
237	C₂×C₄	#13133179	1	1	yes	characteristic
238	C₂×C₄	#13133179	1	1	yes	characteristic
239	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
241	C₂×C₄	#13133225	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
243	C₈	—	106	4	no	—
244	C₈	#13133179	1	1	yes	characteristic
245	C₈	#13133179	1	1	yes	characteristic
246	C₈	C₂×C₈×D₅₃	1	2	yes	—
247	C₈	C₂×C₈×D₅₃	1	2	yes	—
248	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
249	C₈	C₂×#11719949	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	106	4	no	—
252	C₂²	—	212	4	no	—
253	C₂²	#16810688	1	1	yes	characteristic
254	C₂²	#16810688	1	1	yes	characteristic
255	C₂²	C₂×#13133225	1	1	yes	characteristic
256	C₄	—	4	2	no	—
257	C₄	—	106	4	no	—
258	C₄	#16810688	1	1	yes	characteristic
259	C₄	#16810688	1	1	yes	characteristic
260	C₄	C₂×#13133179	1	1	yes	characteristic
261	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	212	4	no	—
265	C₂	#20276556	1	2	yes	direct product factor
266	C₂	C₂×#16810688	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×#364	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	C₂×#2171
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#16811041	C₂	semidirect
3. #20276556	C₂	direct (N × H)
4. C₂×#16810408	C₂	semidirect
5. C₂×#16810399	C₂	semidirect
6. C₂×C₅₃×#364	C₂	semidirect
10. #16811041	C₂²	semidirect
19. #16810408	C₂²	semidirect
20. #16810399	C₂²	semidirect
23. C₅₃×#364	C₂²	semidirect
132. C₂×#364	D₅₃	semidirect
137. C₂×C₅₃	#2171	semidirect
154. #364	C₂×D₅₃	semidirect
157. C₅₃	C₂×#2171	semidirect
265. C₂	#20276556	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×#16811041	2
C₂×#16810741	2
#20276556	2
C₂×#16810408	2
C₂×#16810399	2
C₂×C₅₃×#364	2
C₂×#2171	53

Subgroups — C2×#20276556 #27537325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276556 #27537325