Subgroups — C₂×#14583410 #19147837

Order 5504.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10621898	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583410	C₂	1	16	yes	maximal, direct product factor
3	C₂×#13063308	C₂	1	2	yes	maximal, complemented
4	C₂×#13063304	C₂	1	2	yes	maximal, complemented
5	C₂×#13063299	C₂	1	2	yes	maximal, complemented
6	C₂×#13063294	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₂²⋊D₄₃	C₂	1	4	yes	maximal, complemented
8	C₂×#13063284	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621898	—	2	8	no	—
11	C₂×#10621898	C₂²	1	8	yes	complemented
12	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₂²⋊D₄₃	C₂²	1	4	yes	—
15	C₂²×C₂²⋊D₄₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₄₃	C₂²	1	2	yes	complemented
17	#13063308	C₂²	1	16	yes	complemented
18	#13063304	C₂²	1	16	yes	complemented
19	#13063299	C₂²	1	16	yes	complemented
20	#13063294	C₂²	1	16	yes	complemented
21	C₄×C₂²⋊D₄₃	C₂²	1	32	yes	complemented
22	#13063284	C₂²	1	8	yes	complemented
23	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	complemented
24	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	complemented
25	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
27	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	complemented
28	C₂×C₄×Dic₄₃	C₂²	1	2	yes	complemented
29	C₂⁵×C₄₃	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₄₃	—	2	8	no	—
31	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
32	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
33	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
34	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₄₃	C₂²	1	8	yes	complemented
36	C₂⁴×C₄₃	—	2	4	no	—
37	C₂⁴×C₄₃	—	2	8	no	—
38	C₂⁴×C₄₃	C₂³	1	4	yes	—
39	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
40	C₂⁴×C₄₃	C₂³	1	2	yes	—
41	C₂³×D₄₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₄₃	—	2	8	no	—
43	C₂²×C₄×C₄₃	—	2	8	no	—
44	C₂²×C₄×C₄₃	—	2	32	no	—
45	C₂²×C₄×C₄₃	—	2	4	no	—
46	C₂²×C₄×C₄₃	D₄	1	8	yes	complemented
47	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₄₃	C₂³	1	8	yes	—
49	C₂²×C₄×C₄₃	C₂³	1	8	yes	—
50	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
53	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₄₃	—	2	8	no	—
55	C₂×C₂²⋊D₄₃	—	2	16	no	—
56	C₂×C₂²⋊D₄₃	C₂³	1	16	yes	—
57	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
58	C₂²×Dic₄₃	C₂³	1	4	yes	—
59	C₂²×Dic₄₃	C₂³	1	2	yes	—
60	#10621898	—	2	32	no	—
61	C₂×D₁₇₂	—	2	4	no	—
62	C₂×D₁₇₂	C₂³	1	4	yes	complemented
63	C₂×C₄×D₄₃	—	2	8	no	—
64	C₂×C₄×D₄₃	C₂³	1	8	yes	complemented
65	C₂×Dic₈₆	—	2	4	no	—
66	C₂×Dic₈₆	C₂³	1	4	yes	complemented
67	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
68	C₂²⋊Dic₄₃	C₂³	1	16	yes	—
69	(C₂×D₄₃)⋊C₄	C₂³	1	16	yes	complemented
70	C₄⋊Dic₄₃	C₂³	1	8	yes	complemented
71	Dic₄₃⋊C₄	C₂³	1	16	yes	complemented
72	C₄×Dic₄₃	C₂³	1	8	yes	complemented
73	C₂³×C₄₃	—	2	32	no	—
74	C₂³×C₄₃	—	2	8	no	—
75	C₂³×C₄₃	—	2	4	no	—
76	C₂³×C₄₃	—	2	4	no	—
77	C₂³×C₄₃	—	2	4	no	—
78	C₂³×C₄₃	—	2	16	no	—
79	C₂³×C₄₃	C₂×D₄	1	4	yes	—
80	C₂³×C₄₃	#41	1	8	yes	complemented
81	C₂³×C₄₃	C₂⁴	1	2	yes	—
82	C₂³×C₄₃	C₂⁴	1	4	yes	—
83	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
84	C₂²×D₄₃	—	2	4	no	—
85	C₂²×D₄₃	—	2	8	no	—
86	C₂²×D₄₃	C₂⁴	1	4	yes	—
87	C₂×C₄×C₄₃	—	2	8	no	—
88	C₂×C₄×C₄₃	—	2	4	no	—
89	C₂×C₄×C₄₃	—	2	16	no	—
90	C₂×C₄×C₄₃	—	2	16	no	—
91	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₄₃	C₂×D₄	1	16	yes	complemented
94	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
95	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
96	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
97	C₂²⋊D₄₃	—	2	16	no	—
98	C₂²⋊D₄₃	—	2	32	no	—
99	C₂×Dic₄₃	—	2	4	no	—
100	C₂×Dic₄₃	—	2	8	no	—
101	C₂×Dic₄₃	C₂⁴	1	8	yes	—
102	C₂×Dic₄₃	C₂⁴	1	4	yes	—
103	D₁₇₂	—	2	8	no	—
104	C₄×D₄₃	—	2	16	no	—
105	Dic₈₆	—	2	8	no	—
106	C₂²×C₄₃	—	2	32	no	—
107	C₂²×C₄₃	—	2	8	no	—
108	C₂²×C₄₃	—	2	16	no	—
109	C₂²×C₄₃	—	2	4	no	—
110	C₂²×C₄₃	—	2	4	no	—
111	C₂²×C₄₃	—	2	4	no	—
112	C₂²×C₄₃	C₂²×D₄	1	4	yes	—
113	C₂²×C₄₃	C₂²×D₄	1	2	yes	—
114	C₂²×C₄₃	C₂×#41	1	4	yes	—
115	C₂²×C₄₃	C₂×#41	1	8	yes	complemented
116	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
117	C₂×D₄₃	—	2	4	no	—
118	C₂×D₄₃	—	2	8	no	—
119	C₂×D₄₃	—	4	8	no	—
120	C₄×C₄₃	—	2	4	no	—
121	C₄×C₄₃	C₂²×D₄	1	4	yes	—
122	C₄×C₄₃	C₂²×D₄	1	4	yes	—
123	Dic₄₃	—	2	4	no	—
124	Dic₄₃	—	2	8	no	—
125	C₂×#525	—	43	1	no	maximal, Sylow
126	C₂×C₄₃	—	2	4	no	—
127	C₂×C₄₃	—	2	8	no	—
128	C₂×C₄₃	#525	1	4	yes	complemented
129	C₂×C₄₃	C₂³×D₄	1	1	yes	characteristic
130	C₂×C₄₃	C₂²×#41	1	2	yes	—
131	D₄₃	—	4	4	no	—
132	C₂²×#41	—	43	1	no	—
133	C₂⁴×C₄	D₄₃	1	1	yes	characteristic, complemented
134	#525	—	43	16	no	—
135	C₂×#123	—	43	2	no	—
136	C₂×C₂²⋊Q₈	—	43	2	no	—
137	C₂×#121	—	43	2	no	—
138	C₂×#120	—	43	2	no	—
139	C₂×C₄×D₄	—	43	4	no	—
140	C₂×#117	—	43	1	no	—
141	C₄₃	C₂×#525	1	1	yes	characteristic, complemented, Sylow
142	C₂⁵	C₂×D₄₃	1	1	yes	characteristic
143	C₂×#41	—	43	8	no	—
144	C₂×#41	—	86	8	no	—
145	C₂²×Q₈	—	43	1	no	—
146	C₂²×D₄	—	43	4	no	—
147	C₂²×D₄	—	43	2	no	—
148	C₂²×D₄	—	43	1	no	—
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	—	43	2	no	—
151	C₂³×C₄	C₂×D₄₃	1	2	yes	—
152	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
153	C₂³×C₄	C₂×D₄₃	1	2	yes	—
154	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
155	C₂³×C₄	C₂×D₄₃	1	8	yes	complemented
156	#123	—	43	16	no	—
157	C₂²⋊Q₈	—	43	16	no	—
158	#121	—	43	16	no	—
159	#120	—	43	16	no	—
160	C₄×D₄	—	43	32	no	—
161	#117	—	43	8	no	—
162	C₂×C₄⋊C₄	—	43	2	no	—
163	C₂×C₄⋊C₄	—	43	4	no	—
164	C₂×C₂²⋊C₄	—	43	2	no	—
165	C₂×C₂²⋊C₄	—	43	4	no	—
166	C₂×C₂²⋊C₄	—	43	4	no	—
167	C₂×C₄²	—	43	2	no	—
168	C₂⁴	—	2	8	no	—
169	C₂⁴	—	2	4	no	—
170	C₂⁴	—	43	2	no	—
171	C₂⁴	C₂²×D₄₃	1	4	yes	—
172	C₂⁴	C₂²×D₄₃	1	2	yes	—
173	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
174	#41	—	86	32	no	—
175	C₂×Q₈	—	43	4	no	—
176	C₂×Q₈	—	86	4	no	—
177	C₂×D₄	—	43	16	no	—
178	C₂×D₄	—	43	8	no	—
179	C₂×D₄	—	43	4	no	—
180	C₂×D₄	—	86	16	no	—
181	C₂×D₄	—	86	8	no	—
182	C₂×D₄	—	86	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	32	no	—
187	C₂²×C₄	—	43	4	no	—
188	C₂²×C₄	—	43	2	no	—
189	C₂²×C₄	—	43	8	no	—
190	C₂²×C₄	—	86	8	no	—
191	C₂²×C₄	C₂²⋊D₄₃	1	8	yes	complemented
192	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
193	C₂²×C₄	C₂²×D₄₃	1	8	yes	—
194	C₂²×C₄	C₂²×D₄₃	1	8	yes	—
195	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
196	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
198	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
199	C₄⋊C₄	—	43	8	no	—
200	C₄⋊C₄	—	43	16	no	—
201	C₂²⋊C₄	—	43	8	no	—
202	C₂²⋊C₄	—	43	16	no	—
203	C₂²⋊C₄	—	43	16	no	—
204	C₄²	—	43	8	no	—
205	C₂³	—	2	32	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	16	no	—
211	C₂³	—	43	4	no	—
212	C₂³	—	86	4	no	—
213	C₂³	—	86	8	no	—
214	C₂³	#10621898	1	8	yes	complemented
215	C₂³	C₂×C₂²⋊D₄₃	1	4	yes	—
216	C₂³	C₂³×D₄₃	1	2	yes	—
217	C₂³	C₂³×D₄₃	1	4	yes	—
218	C₂³	C₂³×D₄₃	1	1	yes	characteristic
219	Q₈	—	86	8	no	—
220	D₄	—	86	16	no	—
221	D₄	—	86	32	no	—
222	D₄	—	86	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	2	16	no	—
227	C₂×C₄	—	43	8	no	—
228	C₂×C₄	—	43	4	no	—
229	C₂×C₄	—	86	4	no	—
230	C₂×C₄	—	86	8	no	—
231	C₂×C₄	—	86	16	no	—
232	C₂×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊D₄₃	1	16	yes	complemented
235	C₂×C₄	C₂³×D₄₃	1	4	yes	—
236	C₂×C₄	C₂³×D₄₃	1	2	yes	—
237	C₂×C₄	C₂³×D₄₃	1	2	yes	—
238	C₂²	—	2	32	no	—
239	C₂²	—	2	16	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	86	4	no	—
245	C₂²	—	86	8	no	—
246	C₂²	—	172	8	no	—
247	C₂²	C₂²×C₂²⋊D₄₃	1	4	yes	—
248	C₂²	C₂²×C₂²⋊D₄₃	1	2	yes	—
249	C₂²	C₂×#10621898	1	4	yes	—
250	C₂²	C₂×#10621898	1	8	yes	complemented
251	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	86	4	no	—
254	C₄	—	86	8	no	—
255	C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
256	C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
257	C₂	—	2	4	no	—
258	C₂	—	2	8	no	—
259	C₂	—	172	4	no	—
260	C₂	#14583410	1	4	yes	direct product factor
261	C₂	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
262	C₂	C₂²×#10621898	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₂⁴×C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂⁴×C₄	D₄₃
hypocenter	C₄₃	C₂×#525
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#10621898	C₂	semidirect
2. #14583410	C₂	direct (N × H)
3. C₂×#13063308	C₂	semidirect
4. C₂×#13063304	C₂	semidirect
5. C₂×#13063299	C₂	semidirect
6. C₂×#13063294	C₂	semidirect
7. C₂×C₄×C₂²⋊D₄₃	C₂	semidirect
8. C₂×#13063284	C₂	semidirect
9. C₂⁴×C₄×C₄₃	C₂	semidirect
11. C₂×#10621898	C₂²	semidirect
12. C₂²×Dic₈₆	C₂²	semidirect
13. C₂²×D₁₇₂	C₂²	semidirect
16. C₂²×C₄×D₄₃	C₂²	semidirect
17. #13063308	C₂²	semidirect
18. #13063304	C₂²	semidirect
19. #13063299	C₂²	semidirect
20. #13063294	C₂²	semidirect
21. C₄×C₂²⋊D₄₃	C₂²	semidirect
22. #13063284	C₂²	semidirect
23. C₂×C₄⋊Dic₄₃	C₂²	semidirect
24. C₂×Dic₄₃⋊C₄	C₂²	semidirect
27. C₂×(C₂×D₄₃)⋊C₄	C₂²	semidirect
28. C₂×C₄×Dic₄₃	C₂²	semidirect
35. C₂³×C₄×C₄₃	C₂²	semidirect
46. C₂²×C₄×C₄₃	D₄	semidirect
62. C₂×D₁₇₂	C₂³	semidirect
64. C₂×C₄×D₄₃	C₂³	semidirect
66. C₂×Dic₈₆	C₂³	semidirect
69. (C₂×D₄₃)⋊C₄	C₂³	semidirect
70. C₄⋊Dic₄₃	C₂³	semidirect
71. Dic₄₃⋊C₄	C₂³	semidirect
72. C₄×Dic₄₃	C₂³	semidirect
80. C₂³×C₄₃	#41	semidirect
93. C₂×C₄×C₄₃	C₂×D₄	semidirect
115. C₂²×C₄₃	C₂×#41	semidirect
128. C₂×C₄₃	#525	semidirect
133. C₂⁴×C₄	D₄₃	semidirect
141. C₄₃	C₂×#525	semidirect
155. C₂³×C₄	C₂×D₄₃	semidirect
191. C₂²×C₄	C₂²⋊D₄₃	semidirect
214. C₂³	#10621898	semidirect
234. C₂×C₄	C₂×C₂²⋊D₄₃	semidirect
250. C₂²	C₂×#10621898	semidirect
260. C₂	#14583410	direct (N × H)

Subgroups — C2×#14583410 #19147837

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583410 #19147837