Subgroups — C₂×#14583019 #19143489

Order 5504.

464 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×#13063345	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063331	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583019	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13063260	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063252	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063251	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13063235	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10621898	C₂²	1	1	yes	characteristic
10	#13063345	C₄	1	4	yes	complemented
11	#13063345	C₂²	1	4	yes	—
12	C₂×#10621885	C₄	1	1	yes	characteristic, complemented
13	C₂×#10621883	C₄	1	1	yes	characteristic, complemented
14	C₂×#10621884	C₄	1	1	yes	characteristic, complemented
15	C₂×#10621882	C₄	1	1	yes	characteristic, complemented
16	#13063331	—	2	2	no	—
17	#13063331	C₂²	1	4	yes	complemented
18	C₂×#10621877	C₂²	1	1	yes	characteristic
19	C₂×#10621872	—	2	1	no	—
20	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₄₃	—	2	1	no	—
22	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
23	#13063260	C₂²	1	4	yes	complemented
24	#13063252	C₂²	1	4	yes	complemented
25	#13063251	C₂²	1	4	yes	complemented
26	#13063235	C₂²	1	4	yes	complemented
27	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
28	#131×C₄₃	—	2	2	no	—
29	#131×C₄₃	C₂²	1	4	yes	complemented
30	C₂×M₁₆×C₄₃	—	2	1	no	—
31	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₄₃	—	2	1	no	—
33	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
34	#41×C₄₃	—	2	1	no	—
35	#41×C₄₃	—	2	1	no	—
36	#41×C₄₃	C₂×C₄	1	2	yes	—
37	#41×C₄₃	C₂³	1	2	yes	—
38	C₂×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₄₃	—	2	1	no	—
40	C₂×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	—	2	1	no	—
42	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊D₄₃	—	2	1	no	—
44	C₂²×Dic₄₃	—	2	1	no	—
45	#10621898	—	2	1	no	—
46	#10621898	—	2	1	no	—
47	#10621898	C₂×C₄	1	2	yes	—
48	#10621898	C₂³	1	2	yes	—
49	C₂×D₁₇₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₄₃	—	2	1	no	—
51	C₂×Dic₈₆	C₂×C₄	1	1	yes	characteristic
52	M₁₆×C₄₃	—	2	1	no	—
53	M₁₆×C₄₃	—	2	4	no	—
54	M₁₆×C₄₃	C₂³	1	2	yes	—
55	C₂×C₈×C₄₃	—	2	1	no	—
56	C₂×C₈×C₄₃	—	2	4	no	—
57	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
58	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
59	C₂×C₈×C₄₃	D₄	1	2	yes	—
60	C₂×C₈×C₄₃	C₂³	1	2	yes	—
61	#10621885	C₂×C₄	1	4	yes	complemented
62	#10621884	C₂×C₄	1	4	yes	complemented
63	#10621883	C₂×C₄	1	4	yes	complemented
64	#10621882	C₂×C₄	1	4	yes	complemented
65	C₄×Dic₄₃	—	2	1	no	—
66	C₄×Dic₄₃	C₂³	1	2	yes	—
67	#10621877	—	2	1	no	—
68	#10621877	C₂³	1	2	yes	—
69	C₂×C₄₃⋊C₈	—	2	1	no	—
70	C₂×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
73	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
74	#10621872	—	2	4	no	—
75	C₈×D₄₃	—	2	4	no	—
76	C₂³×C₄₃	—	2	1	no	—
77	C₂³×C₄₃	#41	1	1	yes	characteristic
78	C₂²×D₄₃	—	2	1	no	—
79	Q₈×C₄₃	—	2	1	no	—
80	Q₈×C₄₃	C₂²×C₄	1	2	yes	—
81	D₄×C₄₃	—	2	2	no	—
82	D₄×C₄₃	—	2	2	no	—
83	D₄×C₄₃	—	2	1	no	—
84	D₄×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₄₃	—	2	1	no	—
86	C₂×C₄×C₄₃	—	2	2	no	—
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₄₃	—	2	2	no	—
95	C₂²⋊D₄₃	—	2	2	no	—
96	C₂×Dic₄₃	—	2	1	no	—
97	C₂×Dic₄₃	—	2	2	no	—
98	C₂×Dic₄₃	—	4	2	no	—
99	D₁₇₂	—	2	1	no	—
100	D₁₇₂	C₂²×C₄	1	2	yes	—
101	C₄×D₄₃	—	2	2	no	—
102	C₄×D₄₃	—	2	2	no	—
103	Dic₈₆	—	2	1	no	—
104	Dic₈₆	C₂²×C₄	1	2	yes	—
105	C₈×C₄₃	—	2	2	no	—
106	C₈×C₄₃	C₂×D₄	1	2	yes	—
107	C₈×C₄₃	C₂×D₄	1	2	yes	—
108	C₄₃⋊C₈	—	2	2	no	—
109	C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
110	C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂²×C₄₃	—	2	2	no	—
112	C₂²×C₄₃	—	2	2	no	—
113	C₂²×C₄₃	—	4	2	no	—
114	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
116	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
117	C₂×D₄₃	—	2	2	no	—
118	C₂×D₄₃	—	4	2	no	—
119	C₄×C₄₃	—	2	2	no	—
120	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
121	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
122	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	Dic₄₃	—	2	2	no	—
125	Dic₄₃	—	4	2	no	—
126	C₂×#444	—	43	1	no	maximal, Sylow
127	C₂×C₄₃	—	2	1	no	—
128	C₂×C₄₃	—	2	1	no	—
129	C₂×C₄₃	—	4	2	no	—
130	C₂×C₄₃	#444	1	2	yes	complemented
131	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
132	D₄₃	—	4	2	no	—
133	C₂×#135	—	43	1	no	—
134	C₂×#131	—	43	1	no	—
135	C₂×#131	D₄₃	1	1	yes	characteristic, complemented
136	#444	—	43	8	no	—
137	C₂×#108	—	43	1	no	—
138	C₂×#104	—	43	1	no	—
139	C₂×#104	—	43	1	no	—
140	C₂×#97	—	43	1	no	—
141	C₄₃	C₂×#444	1	1	yes	characteristic, complemented, Sylow
142	C₂×#41	—	43	1	no	—
143	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
144	#135	—	43	4	no	—
145	#135	—	43	4	no	—
146	C₂×Q₁₆	—	43	1	no	—
147	C₂×SD₁₆	—	43	1	no	—
148	C₂×SD₁₆	—	43	1	no	—
149	C₂×D₈	—	43	1	no	—
150	#131	—	2	2	no	—
151	#131	—	43	4	no	—
152	#131	—	86	2	no	—
153	#131	C₂×D₄₃	1	4	yes	complemented
154	C₂×M₁₆	—	2	1	no	—
155	C₂×M₁₆	—	43	1	no	—
156	C₂×M₁₆	—	86	1	no	—
157	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	43	1	no	—
160	C₂²×C₈	—	86	1	no	—
161	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
162	C₂×C₄²	—	43	1	no	—
163	#108	—	43	4	no	—
164	#104	—	43	4	no	—
165	#104	—	43	4	no	—
166	#97	—	43	4	no	—
167	#41	—	2	1	no	—
168	#41	—	2	1	no	—
169	#41	—	43	2	no	—
170	#41	—	43	2	no	—
171	#41	—	86	1	no	—
172	#41	—	86	1	no	—
173	#41	C₄×D₄₃	1	2	yes	—
174	#41	C₂²×D₄₃	1	2	yes	—
175	C₂×Q₈	—	43	1	no	—
176	C₂×Q₈	C₄×D₄₃	1	1	yes	characteristic
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	43	1	no	—
179	C₂×D₄	—	86	1	no	—
180	C₂×D₄	C₄×D₄₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	86	1	no	—
183	C₂²×C₄	—	86	1	no	—
184	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
185	Q₁₆	—	43	4	no	—
186	SD₁₆	—	43	4	no	—
187	SD₁₆	—	43	4	no	—
188	D₈	—	43	4	no	—
189	M₁₆	—	2	1	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	43	2	no	—
192	M₁₆	—	86	1	no	—
193	M₁₆	—	86	4	no	—
194	M₁₆	C₂²×D₄₃	1	2	yes	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	43	1	no	—
198	C₂×C₈	—	43	1	no	—
199	C₂×C₈	—	43	2	no	—
200	C₂×C₈	—	43	2	no	—
201	C₂×C₈	—	86	1	no	—
202	C₂×C₈	—	86	4	no	—
203	C₂×C₈	C₂²⋊D₄₃	1	2	yes	—
204	C₂×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
206	C₂×C₈	C₂²×D₄₃	1	2	yes	—
207	C₄²	—	43	2	no	—
208	C₄²	—	86	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	86	1	no	—
211	C₂³	#10621898	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	43	2	no	—
214	Q₈	—	86	1	no	—
215	Q₈	C₂×C₄×D₄₃	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	43	2	no	—
220	D₄	—	86	2	no	—
221	D₄	—	86	2	no	—
222	D₄	—	86	1	no	—
223	D₄	C₂×C₄×D₄₃	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	86	1	no	—
228	C₂×C₄	—	86	2	no	—
229	C₂×C₄	—	86	2	no	—
230	C₂×C₄	—	86	2	no	—
231	C₂×C₄	—	172	2	no	—
232	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
234	C₂×C₄	#10621898	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
237	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	43	2	no	—
240	C₈	—	43	2	no	—
241	C₈	—	86	2	no	—
242	C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
243	C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	86	2	no	—
248	C₂²	—	172	2	no	—
249	C₂²	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂²	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
251	C₂²	C₂×#10621898	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	86	2	no	—
254	C₄	—	172	2	no	—
255	C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	—	172	2	no	—
263	C₂	#14583019	1	2	yes	direct product factor
264	C₂	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×#131×C₄₃	C₂
socle	C₂²×C₄₃	C₄×D₄
hypercenter	C₂×#131	D₄₃
hypocenter	C₄₃	C₂×#444
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#13063331	C₂	semidirect
3. #14583019	C₂	direct (N × H)
4. C₂×#13063260	C₂	semidirect
5. C₂×#13063252	C₂	semidirect
6. C₂×#13063251	C₂	semidirect
7. C₂×#13063235	C₂	semidirect
8. C₂×#131×C₄₃	C₂	semidirect
10. #13063345	C₄	semidirect
12. C₂×#10621885	C₄	semidirect
13. C₂×#10621883	C₄	semidirect
14. C₂×#10621884	C₄	semidirect
15. C₂×#10621882	C₄	semidirect
17. #13063331	C₂²	semidirect
23. #13063260	C₂²	semidirect
24. #13063252	C₂²	semidirect
25. #13063251	C₂²	semidirect
26. #13063235	C₂²	semidirect
29. #131×C₄₃	C₂²	semidirect
61. #10621885	C₂×C₄	semidirect
62. #10621884	C₂×C₄	semidirect
63. #10621883	C₂×C₄	semidirect
64. #10621882	C₂×C₄	semidirect
130. C₂×C₄₃	#444	semidirect
135. C₂×#131	D₄₃	semidirect
141. C₄₃	C₂×#444	semidirect
153. #131	C₂×D₄₃	semidirect
263. C₂	#14583019	direct (N × H)

Subgroups — C2×#14583019 #19143489

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583019 #19143489