Subgroups — C₂×#20276602 #27537499

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811055	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810741	C₂	1	2	yes	maximal, complemented
3	#20276602	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810453	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810431	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133239	C₂²	1	2	yes	—
8	C₂×#13133223	—	2	2	no	—
9	#16811055	C₄	1	4	yes	complemented
10	#16811055	C₂²	1	4	yes	—
11	C₂×Dic₄₂₄	C₄	1	1	yes	characteristic, complemented
12	C₂×#13133162	C₄	1	2	yes	complemented
13	C₂×D₈₄₈	C₄	1	1	yes	characteristic, complemented
14	#16810741	—	2	4	no	—
15	#16810741	C₂²	1	8	yes	complemented
16	C₂×#13133153	C₂²	1	2	yes	—
17	C₂×#13133145	—	2	2	no	—
18	#16810453	C₂²	1	4	yes	complemented
19	#16810431	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#346	C₂²	1	4	yes	complemented
24	C₂×#11719945	—	2	2	no	—
25	#13133239	—	2	2	no	—
26	#13133239	—	2	2	no	—
27	#13133239	C₂×C₄	1	4	yes	—
28	#13133239	C₂³	1	4	yes	—
29	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
30	C₂×#11719913	—	2	2	no	—
31	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
32	#13133223	—	2	8	no	—
33	#13133223	—	4	4	no	—
34	C₂×#11719917	—	2	2	no	—
35	C₂×#11719912	—	4	2	no	—
36	C₂×C₈×D₅₃	—	4	2	no	—
37	Dic₄₂₄	C₂×C₄	1	4	yes	complemented
38	#13133162	C₂×C₄	1	8	yes	complemented
39	D₈₄₈	C₂×C₄	1	4	yes	complemented
40	#13133153	—	2	2	no	—
41	#13133153	C₂³	1	4	yes	—
42	#13133145	—	2	8	no	—
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₁₆×C₅₃	—	2	1	no	—
47	C₂×C₁₆×C₅₃	—	2	1	no	—
48	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
52	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
53	C₄×C₈×C₅₃	D₄	1	2	yes	—
54	C₄×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₅₃	—	4	2	no	—
58	#11719945	—	2	4	no	—
59	#11719945	—	2	4	no	—
60	#11719945	—	4	2	no	—
61	#11719945	—	4	2	no	—
62	C₂×D₂₁₂	—	2	2	no	—
63	C₂×C₄×D₅₃	—	4	2	no	—
64	C₂×Dic₁₀₆	—	2	2	no	—
65	C₂×C₈×C₅₃	D₈	1	2	yes	—
66	C₂×C₈×C₅₃	D₈	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
72	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	C₂×D₄	1	2	yes	—
76	#11719917	—	2	4	no	—
77	#11719917	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	Dic₂₁₂	—	2	2	no	—
80	Dic₂₁₂	C₂²×C₄	1	4	yes	—
81	D₄₂₄	—	2	2	no	—
82	D₄₂₄	C₂²×C₄	1	4	yes	—
83	#11719913	—	2	4	no	—
84	#11719913	—	2	4	no	—
85	#11719912	—	4	8	no	—
86	C₈×D₅₃	—	4	8	no	—
87	C₁₆×C₅₃	—	2	2	no	—
88	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂³×C₅₃	#135	1	1	yes	characteristic
91	C₂²×D₅₃	—	4	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	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₂²⋊D₅₃	—	4	4	no	—
101	C₂²⋊D₅₃	—	4	4	no	—
102	C₂×Dic₅₃	—	4	2	no	—
103	D₂₁₂	—	2	4	no	—
104	D₂₁₂	—	4	2	no	—
105	C₄×D₅₃	—	4	4	no	—
106	C₄×D₅₃	—	4	4	no	—
107	Dic₁₀₆	—	2	4	no	—
108	Dic₁₀₆	—	4	2	no	—
109	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₈×C₅₃	C₂×D₈	1	4	yes	—
112	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
114	#88322	—	4	4	no	—
115	C₂×#2179	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
118	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
120	C₂×D₅₃	—	4	4	no	—
121	C₂×D₅₃	—	8	4	no	—
122	C₄×C₅₃	—	4	2	no	—
123	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
124	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
125	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
127	Dic₅₃	—	4	4	no	—
128	C₂×#508	—	53	1	no	—
129	C₂×#443	—	53	2	no	—
130	#2179	—	53	8	no	—
131	C₂×#368	—	53	1	no	—
132	C₂×#359	—	53	2	no	—
133	C₂×#346	D₅₃	1	1	yes	characteristic, complemented
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	#2179	1	2	yes	complemented
137	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
138	D₅₃	—	8	4	no	—
139	C₂×#135	—	53	2	no	—
140	C₂×#131	—	106	2	no	—
141	#508	—	53	4	no	—
142	#508	—	53	4	no	—
143	C₂×Q₃₂	—	53	1	no	—
144	C₂×SD₃₂	—	53	2	no	—
145	C₂×D₁₆	—	53	1	no	—
146	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
147	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
148	#443	—	53	8	no	—
149	#443	—	106	4	no	—
150	C₂×#108	—	53	2	no	—
151	C₂×#104	—	106	2	no	—
152	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
153	#368	—	53	4	no	—
154	#359	—	53	8	no	—
155	#346	C₂×D₅₃	1	4	yes	complemented
156	C₅₃	C₂×#2179	1	1	yes	characteristic, complemented, Sylow
157	C₂×#41	—	106	2	no	—
158	#135	—	53	4	no	—
159	#135	—	53	4	no	—
160	#135	—	106	2	no	—
161	#135	—	106	2	no	—
162	C₂×Q₁₆	—	53	2	no	—
163	C₂×SD₁₆	—	106	2	no	—
164	C₂×D₈	—	53	2	no	—
165	#131	—	106	8	no	—
166	#131	—	212	4	no	—
167	C₂×M₁₆	—	106	2	no	—
168	C₂×M₁₆	—	212	2	no	—
169	C₂²×C₈	—	212	2	no	—
170	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
173	Q₃₂	—	53	4	no	—
174	SD₃₂	—	53	8	no	—
175	D₁₆	—	53	4	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
181	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
182	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
183	#108	—	53	4	no	—
184	#108	—	106	2	no	—
185	#104	—	106	8	no	—
186	C₄×C₈	D₂₁₂	1	2	yes	—
187	C₄×C₈	C₂²×D₅₃	1	2	yes	—
188	#41	—	106	4	no	—
189	#41	—	106	4	no	—
190	#41	—	212	2	no	—
191	#41	—	212	2	no	—
192	C₂×Q₈	—	106	2	no	—
193	C₂×D₄	—	106	2	no	—
194	C₂×D₄	—	212	2	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	212	2	no	—
197	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
198	Q₁₆	—	53	4	no	—
199	Q₁₆	—	106	2	no	—
200	SD₁₆	—	106	4	no	—
201	SD₁₆	—	106	4	no	—
202	D₈	—	53	4	no	—
203	D₈	—	106	2	no	—
204	M₁₆	—	106	4	no	—
205	M₁₆	—	212	2	no	—
206	M₁₆	—	212	8	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	—	212	8	no	—
209	C₂×C₈	D₄₂₄	1	2	yes	—
210	C₂×C₈	D₄₂₄	1	2	yes	—
211	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
214	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
215	C₂×C₈	#11719945	1	1	yes	characteristic
216	C₂×C₈	#11719945	1	1	yes	characteristic
217	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
218	C₄²	—	2	1	no	—
219	C₄²	C₂×D₂₁₂	1	2	yes	—
220	C₁₆	—	2	2	no	—
221	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
222	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
223	C₂³	—	212	2	no	—
224	C₂³	#13133239	1	1	yes	characteristic
225	Q₈	—	106	4	no	—
226	Q₈	—	212	2	no	—
227	D₄	—	106	4	no	—
228	D₄	—	212	4	no	—
229	D₄	—	212	4	no	—
230	D₄	—	212	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	212	2	no	—
234	C₂×C₄	—	212	4	no	—
235	C₂×C₄	—	212	4	no	—
236	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
237	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
240	C₂×C₄	#13133239	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
242	C₈	—	212	4	no	—
243	C₈	C₄×D₂₁₂	1	1	yes	characteristic
244	C₈	C₄×D₂₁₂	1	1	yes	characteristic
245	C₈	C₂×D₄₂₄	1	4	yes	—
246	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
247	C₈	C₂×#11719945	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	424	4	no	—
250	C₂²	—	212	4	no	—
251	C₂²	C₄×D₄₂₄	1	1	yes	characteristic
252	C₂²	C₄×D₄₂₄	1	1	yes	characteristic
253	C₂²	C₂×#13133239	1	1	yes	characteristic
254	C₄	—	4	2	no	—
255	C₄	—	212	4	no	—
256	C₄	C₄×D₄₂₄	1	1	yes	characteristic
257	C₄	C₄×D₄₂₄	1	1	yes	characteristic
258	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
259	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	424	4	no	—
263	C₂	#20276602	1	2	yes	direct product factor
264	C₂	C₂×C₄×D₄₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂×#16811055	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	C₂×#2179
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×#16810741	C₂	semidirect
3. #20276602	C₂	direct (N × H)
4. C₂×#16810453	C₂	semidirect
5. C₂×#16810431	C₂	semidirect
6. C₂×C₅₃×#346	C₂	semidirect
9. #16811055	C₄	semidirect
11. C₂×Dic₄₂₄	C₄	semidirect
12. C₂×#13133162	C₄	semidirect
13. C₂×D₈₄₈	C₄	semidirect
15. #16810741	C₂²	semidirect
18. #16810453	C₂²	semidirect
19. #16810431	C₂²	semidirect
23. C₅₃×#346	C₂²	semidirect
37. Dic₄₂₄	C₂×C₄	semidirect
38. #13133162	C₂×C₄	semidirect
39. D₈₄₈	C₂×C₄	semidirect
133. C₂×#346	D₅₃	semidirect
136. C₂×C₅₃	#2179	semidirect
155. #346	C₂×D₅₃	semidirect
156. C₅₃	C₂×#2179	semidirect
263. C₂	#20276602	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×#16811055	2
C₂×#16810741	2
#20276602	2
C₂×#16810453	2
C₂×#16810431	2
C₂×C₅₃×#346	2
C₂×#2179	53

Subgroups — C2×#20276602 #27537499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276602 #27537499