Subgroups — C₂×#14583433 #19147861

Order 5504.

1232 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583433	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13063319	C₂	1	2	yes	maximal, complemented
3	C₂×#13063317	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13063313	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063308	C₂	1	2	yes	maximal, complemented
6	C₂×#13063304	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13063299	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₂²⋊D₄₃	C₂	1	2	yes	maximal, complemented
11	C₂×#13063282	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄²×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
14	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
15	C₂²×C₂²⋊D₄₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
17	#13063319	C₂²	1	16	yes	complemented
18	#13063317	C₂²	1	8	yes	complemented
19	#13063313	C₂²	1	8	yes	complemented
20	#13063308	C₂²	1	16	yes	complemented
21	#13063304	C₂²	1	8	yes	complemented
22	#13063299	C₂²	1	8	yes	—
23	C₄×Dic₈₆	C₂²	1	8	yes	complemented
24	C₄×D₁₇₂	C₂²	1	8	yes	complemented
25	C₄×C₂²⋊D₄₃	C₂²	1	16	yes	complemented
26	#13063282	C₂²	1	16	yes	complemented
27	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
29	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
30	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
31	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
32	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
33	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
34	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
35	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
36	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
37	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
38	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
39	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
40	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
41	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	C₂²	1	8	yes	complemented
47	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
48	C₂³×D₄₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₄₃	—	2	4	no	—
50	C₂²×C₄×C₄₃	—	2	4	no	—
51	C₂²×C₄×C₄₃	—	2	4	no	—
52	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
58	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₄₃	—	2	8	no	—
62	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
63	C₂²×Dic₄₃	C₂³	1	2	yes	—
64	C₂²×Dic₄₃	C₂³	1	2	yes	—
65	C₂²×Dic₄₃	C₂³	1	2	yes	—
66	C₂×D₁₇₂	—	2	4	no	—
67	C₂×D₁₇₂	C₂³	1	4	yes	—
68	C₂×C₄×D₄₃	—	2	8	no	—
69	C₂×C₄×D₄₃	C₂³	1	8	yes	—
70	C₂×Dic₈₆	—	2	4	no	—
71	C₂×Dic₈₆	C₂³	1	4	yes	—
72	C₄²×C₄₃	C₂³	1	4	yes	—
73	C₄²×C₄₃	C₂³	1	4	yes	—
74	C₄²×C₄₃	C₂³	1	4	yes	—
75	C₄²×C₄₃	C₂³	1	4	yes	—
76	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
77	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
80	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
81	C₄⋊Dic₄₃	C₂³	1	4	yes	—
82	C₄⋊Dic₄₃	C₂³	1	4	yes	—
83	Dic₄₃⋊C₄	C₂³	1	8	yes	—
84	Dic₄₃⋊C₄	C₂³	1	8	yes	—
85	Dic₄₃⋊C₄	C₂³	1	8	yes	—
86	Dic₄₃⋊C₄	C₂³	1	8	yes	—
87	C₄×Dic₄₃	C₂³	1	8	yes	—
88	C₂³×C₄₃	—	2	2	no	—
89	C₂³×C₄₃	—	2	2	no	—
90	C₂³×C₄₃	#41	1	4	yes	—
91	C₂³×C₄₃	C₂⁴	1	2	yes	—
92	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₂²×D₄₃	—	2	4	no	—
94	C₂²×D₄₃	—	2	4	no	—
95	C₂²×D₄₃	—	2	4	no	—
96	C₂²×D₄₃	C₂⁴	1	4	yes	—
97	C₂×C₄×C₄₃	—	2	8	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	8	no	—
103	C₂×C₄×C₄₃	—	2	8	no	—
104	C₂×C₄×C₄₃	#41	1	4	yes	—
105	C₂×C₄×C₄₃	#41	1	4	yes	—
106	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
107	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
108	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
109	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
110	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
111	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
112	C₂²⋊D₄₃	—	2	16	no	—
113	C₂×Dic₄₃	—	2	4	no	—
114	C₂×Dic₄₃	—	2	4	no	—
115	C₂×Dic₄₃	—	2	4	no	—
116	C₂×Dic₄₃	C₂⁴	1	4	yes	—
117	C₂×Dic₄₃	C₂⁴	1	4	yes	—
118	C₂×Dic₄₃	C₂⁴	1	4	yes	—
119	D₁₇₂	—	2	8	no	—
120	C₄×D₄₃	—	2	16	no	—
121	Dic₈₆	—	2	8	no	—
122	C₂²×C₄₃	—	2	8	no	—
123	C₂²×C₄₃	—	2	2	no	—
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	C₂×#41	1	4	yes	—
126	C₂²×C₄₃	C₂×#41	1	2	yes	—
127	C₂²×C₄₃	C₂×#41	1	2	yes	—
128	C₂²×C₄₃	C₂×#41	1	2	yes	—
129	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
130	C₂×D₄₃	—	2	4	no	—
131	C₂×D₄₃	—	2	4	no	—
132	C₂×D₄₃	—	2	4	no	—
133	C₂×D₄₃	—	4	8	no	—
134	C₄×C₄₃	—	2	2	no	—
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	—	2	2	no	—
138	C₄×C₄₃	C₂×#41	1	4	yes	—
139	C₄×C₄₃	C₂×#41	1	4	yes	—
140	Dic₄₃	—	2	4	no	—
141	Dic₄₃	—	2	4	no	—
142	Dic₄₃	—	2	4	no	—
143	C₂×#529	—	43	1	no	maximal, Sylow
144	C₂×C₄₃	—	2	4	no	—
145	C₂×C₄₃	#529	1	4	yes	complemented
146	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
147	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
148	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
149	D₄₃	—	4	4	no	—
150	#529	—	43	16	no	—
151	C₂×#126	—	43	2	no	—
152	C₂×#125	—	43	1	no	—
153	C₂×#124	—	43	1	no	—
154	C₂×#123	—	43	2	no	—
155	C₂×C₂²⋊Q₈	—	43	1	no	—
156	C₂×#121	—	43	1	no	—
157	C₂×C₄×Q₈	—	43	1	no	—
158	C₂×C₄×D₄	—	43	2	no	—
159	C₂×C₄×D₄	—	43	1	no	—
160	C₂×#117	—	43	2	no	—
161	C₂²×C₄²	D₄₃	1	1	yes	characteristic, complemented
162	C₄₃	C₂×#529	1	1	yes	characteristic, complemented, Sylow
163	C₂²×Q₈	—	43	1	no	—
164	C₂²×D₄	—	43	2	no	—
165	C₂²×D₄	—	43	1	no	—
166	C₂³×C₄	—	43	2	no	—
167	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
168	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
169	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
170	#126	—	43	16	no	—
171	#125	—	43	8	no	—
172	#124	—	43	8	no	—
173	#123	—	43	16	no	—
174	C₂²⋊Q₈	—	43	8	no	—
175	#121	—	43	8	no	—
176	C₄×Q₈	—	43	8	no	—
177	C₄×D₄	—	43	16	no	—
178	C₄×D₄	—	43	8	no	—
179	#117	—	43	16	no	—
180	C₂×C₄⋊C₄	—	43	2	no	—
181	C₂×C₄⋊C₄	—	43	2	no	—
182	C₂×C₄⋊C₄	—	43	1	no	—
183	C₂×C₄⋊C₄	—	43	2	no	—
184	C₂×C₄⋊C₄	—	43	1	no	—
185	C₂×C₄⋊C₄	—	43	2	no	—
186	C₂×C₂²⋊C₄	—	43	2	no	—
187	C₂×C₂²⋊C₄	—	43	2	no	—
188	C₂×C₂²⋊C₄	—	43	2	no	—
189	C₂×C₂²⋊C₄	—	43	2	no	—
190	C₂×C₂²⋊C₄	—	43	2	no	—
191	C₂×C₄²	—	43	2	no	—
192	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
194	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
195	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
196	C₂×C₄²	C₂×D₄₃	1	8	yes	complemented
197	C₂⁴	—	43	2	no	—
198	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
199	C₂×Q₈	—	43	4	no	—
200	C₂×Q₈	—	86	4	no	—
201	C₂×D₄	—	43	8	no	—
202	C₂×D₄	—	43	4	no	—
203	C₂×D₄	—	86	8	no	—
204	C₂×D₄	—	86	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	43	2	no	—
209	C₂²×C₄	—	43	2	no	—
210	C₂²×C₄	—	43	2	no	—
211	C₂²×C₄	—	43	8	no	—
212	C₂²×C₄	—	86	8	no	—
213	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
218	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
219	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
220	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
222	C₄⋊C₄	—	43	8	no	—
223	C₄⋊C₄	—	43	8	no	—
224	C₄⋊C₄	—	43	4	no	—
225	C₄⋊C₄	—	43	8	no	—
226	C₄⋊C₄	—	43	4	no	—
227	C₄⋊C₄	—	43	8	no	—
228	C₂²⋊C₄	—	43	8	no	—
229	C₂²⋊C₄	—	43	8	no	—
230	C₂²⋊C₄	—	43	8	no	—
231	C₂²⋊C₄	—	43	8	no	—
232	C₂²⋊C₄	—	43	8	no	—
233	C₄²	—	43	8	no	—
234	C₄²	C₂²×D₄₃	1	4	yes	—
235	C₄²	C₂²×D₄₃	1	4	yes	—
236	C₄²	C₂²×D₄₃	1	4	yes	—
237	C₄²	C₂²×D₄₃	1	4	yes	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	43	4	no	—
241	C₂³	—	86	4	no	—
242	C₂³	—	86	4	no	—
243	C₂³	—	86	4	no	—
244	C₂³	#10621898	1	4	yes	—
245	C₂³	C₂³×D₄₃	1	2	yes	—
246	C₂³	C₂³×D₄₃	1	1	yes	characteristic
247	Q₈	—	86	8	no	—
248	D₄	—	86	16	no	—
249	D₄	—	86	8	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	43	4	no	—
258	C₂×C₄	—	43	4	no	—
259	C₂×C₄	—	43	4	no	—
260	C₂×C₄	—	86	4	no	—
261	C₂×C₄	—	86	4	no	—
262	C₂×C₄	—	86	4	no	—
263	C₂×C₄	—	86	16	no	—
264	C₂×C₄	#10621898	1	4	yes	—
265	C₂×C₄	#10621898	1	4	yes	—
266	C₂×C₄	C₂³×D₄₃	1	2	yes	—
267	C₂×C₄	C₂³×D₄₃	1	2	yes	—
268	C₂×C₄	C₂³×D₄₃	1	2	yes	—
269	C₂×C₄	C₂³×D₄₃	1	2	yes	—
270	C₂×C₄	C₂³×D₄₃	1	2	yes	—
271	C₂×C₄	C₂³×D₄₃	1	2	yes	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	86	4	no	—
276	C₂²	—	86	4	no	—
277	C₂²	—	86	4	no	—
278	C₂²	—	172	8	no	—
279	C₂²	C₂×#10621898	1	4	yes	—
280	C₂²	C₂×#10621898	1	2	yes	—
281	C₂²	C₂×#10621898	1	2	yes	—
282	C₂²	C₂×#10621898	1	2	yes	—
283	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	86	4	no	—
289	C₄	—	86	4	no	—
290	C₄	—	86	4	no	—
291	C₄	C₂×#10621898	1	4	yes	—
292	C₄	C₂×#10621898	1	4	yes	—
293	C₂	—	2	4	no	—
294	C₂	—	172	4	no	—
295	C₂	#14583433	1	4	yes	direct product factor
296	C₂	C₂²×#10621898	1	1	yes	characteristic
297	C₂	C₂²×#10621898	1	1	yes	characteristic
298	C₂	C₂²×#10621898	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₂²×C₄²×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂²×C₄²	D₄₃
hypocenter	C₄₃	C₂×#529
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #14583433	C₂	direct (N × H)
2. C₂×#13063319	C₂	semidirect
3. C₂×#13063317	C₂	semidirect
4. C₂×#13063313	C₂	semidirect
5. C₂×#13063308	C₂	semidirect
6. C₂×#13063304	C₂	semidirect
8. C₂×C₄×Dic₈₆	C₂	semidirect
9. C₂×C₄×D₁₇₂	C₂	semidirect
10. C₂×C₄×C₂²⋊D₄₃	C₂	semidirect
11. C₂×#13063282	C₂	semidirect
12. C₂²×C₄²×C₄₃	C₂	semidirect
17. #13063319	C₂²	semidirect
18. #13063317	C₂²	semidirect
19. #13063313	C₂²	semidirect
20. #13063308	C₂²	semidirect
21. #13063304	C₂²	semidirect
23. C₄×Dic₈₆	C₂²	semidirect
24. C₄×D₁₇₂	C₂²	semidirect
25. C₄×C₂²⋊D₄₃	C₂²	semidirect
26. #13063282	C₂²	semidirect
46. C₂×C₄²×C₄₃	C₂²	semidirect
145. C₂×C₄₃	#529	semidirect
161. C₂²×C₄²	D₄₃	semidirect
162. C₄₃	C₂×#529	semidirect
196. C₂×C₄²	C₂×D₄₃	semidirect
295. C₂	#14583433	direct (N × H)

Subgroups — C2×#14583433 #19147861

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583433 #19147861