Subgroups — #555×D₄₃ #19148131

Order 5504.

1240 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583615	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583614	C₂	1	2	yes	maximal, complemented
3	#14583613	C₂	1	4	yes	maximal, complemented
4	#14583612	C₂	1	4	yes	maximal, complemented
5	#14583611	C₂	1	4	yes	maximal, complemented
6	C₄⋊Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#124×D₄₃	C₂	1	4	yes	maximal, complemented
8	#121×D₄₃	C₂	1	4	yes	maximal, complemented
9	C₄×D₄×D₄₃	C₂	1	2	yes	maximal, complemented
10	#117×D₄₃	C₂	1	4	yes	maximal, complemented
11	C₄₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₄₃	C₂²	1	2	yes	—
13	C₂×D₄×D₄₃	C₂²	1	2	yes	—
14	C₂×D₄×D₄₃	C₂²	1	4	yes	—
15	C₂²×C₄×D₄₃	C₂²	1	4	yes	—
16	#13063325	C₂²	1	1	yes	characteristic, complemented
17	#13063324	C₂²	1	4	yes	complemented
18	#13063323	C₂²	1	2	yes	complemented
19	#13063314	C₂²	1	4	yes	complemented
20	#13063313	C₂²	1	4	yes	complemented
21	#13063312	C₂²	1	4	yes	complemented
22	#13063311	C₂²	1	8	yes	complemented
23	#13063311	C₂²	1	8	yes	complemented
24	#13063300	C₂²	1	4	yes	complemented
25	#13063299	C₂²	1	4	yes	complemented
26	#13063298	C₂²	1	8	yes	complemented
27	#13063297	C₂²	1	8	yes	complemented
28	#13063296	C₂²	1	4	yes	complemented
29	C₄×D₁₇₂	C₂²	1	2	yes	complemented
30	D₄×Dic₄₃	C₂²	1	2	yes	complemented
31	#13063287	C₂²	1	2	yes	complemented
32	C₄×C₂²⋊D₄₃	C₂²	1	4	yes	complemented
33	(C₂²⋊D₄₃)⋊C₄	C₂²	1	4	yes	complemented
34	#13063284	C₂²	1	4	yes	complemented
35	#13063283	C₂²	1	4	yes	complemented
36	#13063283	C₂²	1	4	yes	complemented
37	#13063282	C₂²	1	8	yes	complemented
38	#13063281	C₂²	1	8	yes	complemented
39	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
40	C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
41	C₂²⋊C₄×D₄₃	C₂²	1	8	yes	—
42	C₂²⋊C₄×D₄₃	C₂²	1	4	yes	—
43	C₄²×D₄₃	C₂²	1	4	yes	—
44	C₄²×D₄₃	C₂²	1	1	yes	characteristic
45	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
46	#124×C₄₃	C₂²	1	4	yes	complemented
47	#121×C₄₃	C₂²	1	4	yes	complemented
48	C₄×D₄×C₄₃	C₂²	1	2	yes	complemented
49	#117×C₄₃	C₂²	1	4	yes	complemented
50	C₂³×D₄₃	C₂³	1	4	yes	—
51	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
52	C₂×D₄×C₄₃	C₂³	1	2	yes	—
53	C₂×D₄×C₄₃	C₂³	1	4	yes	—
54	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
56	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
57	C₂²×Dic₄₃	C₂³	1	4	yes	—
58	Q₈×D₄₃	—	4	4	no	—
59	D₄×D₄₃	—	2	8	no	—
60	D₄×D₄₃	—	4	8	no	—
61	C₂×D₁₇₂	C₂³	1	2	yes	—
62	C₂×D₁₇₂	C₂³	1	4	yes	—
63	C₂×C₄×D₄₃	—	2	16	no	—
64	C₂×C₄×D₄₃	C₂³	1	4	yes	—
65	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₄₃	C₂³	1	4	yes	—
67	C₂×C₄×D₄₃	C₂³	1	4	yes	—
68	C₂×C₄×D₄₃	C₂³	1	4	yes	—
69	C₂×C₄×D₄₃	C₂³	1	2	yes	—
70	C₂×Dic₈₆	C₂³	1	2	yes	—
71	C₂×Dic₈₆	C₂³	1	4	yes	—
72	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
73	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
74	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
75	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
76	C₄²×C₄₃	C₂³	1	4	yes	—
77	C₄²×C₄₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
79	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
80	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
81	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
82	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
83	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
84	C₄⋊Dic₄₃	C₂³	1	2	yes	—
85	C₄⋊Dic₄₃	C₂³	1	4	yes	—
86	Dic₄₃⋊C₄	C₂³	1	4	yes	—
87	Dic₄₃⋊C₄	C₂³	1	8	yes	—
88	C₄×Dic₄₃	C₂³	1	4	yes	—
89	C₄×Dic₄₃	C₂³	1	8	yes	—
90	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
91	C₄×Dic₄₃	C₂³	1	2	yes	—
92	C₂³×C₄₃	C₂⁴	1	4	yes	—
93	C₂²×D₄₃	—	2	8	no	—
94	C₂²×D₄₃	—	2	8	no	—
95	C₂²×D₄₃	—	2	8	no	—
96	C₂²×D₄₃	C₂⁴	1	4	yes	—
97	C₂²×D₄₃	C₂⁴	1	1	yes	characteristic
98	Q₈×C₄₃	—	4	2	no	—
99	D₄×C₄₃	—	2	4	no	—
100	D₄×C₄₃	—	4	4	no	—
101	C₂×C₄×C₄₃	—	2	8	no	—
102	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
104	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
106	C₂²⋊D₄₃	—	2	8	no	—
107	C₂²⋊D₄₃	—	4	8	no	—
108	C₂×Dic₄₃	—	2	8	no	—
109	C₂×Dic₄₃	C₂⁴	1	4	yes	—
110	C₂×Dic₄₃	C₂⁴	1	4	yes	—
111	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
112	C₂×Dic₄₃	C₂⁴	1	2	yes	—
113	D₁₇₂	—	2	4	no	—
114	D₁₇₂	—	4	4	no	—
115	C₄×D₄₃	—	2	8	no	—
116	C₄×D₄₃	—	2	8	no	—
117	C₄×D₄₃	—	2	2	no	—
118	C₄×D₄₃	—	2	8	no	—
119	C₄×D₄₃	—	2	8	no	—
120	C₄×D₄₃	#41	1	8	yes	—
121	Dic₈₆	—	4	4	no	—
122	Dic₈₆	—	4	2	no	—
123	C₂²×C₄₃	—	2	4	no	—
124	C₂²×C₄₃	—	2	4	no	—
125	C₂²×C₄₃	—	2	4	no	—
126	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
127	C₂×D₄₃	—	2	4	no	—
128	C₂×D₄₃	—	2	4	no	—
129	C₂×D₄₃	—	2	4	no	—
130	C₂×D₄₃	—	4	16	no	—
131	C₂×D₄₃	C₂×#41	1	4	yes	—
132	C₂×D₄₃	H(5,𝔽₂)	1	2	yes	—
133	C₄×C₄₃	—	2	4	no	—
134	C₄×C₄₃	—	2	1	no	—
135	C₄×C₄₃	—	2	4	no	—
136	C₄×C₄₃	C₂×#41	1	4	yes	—
137	Dic₄₃	—	2	4	no	—
138	Dic₄₃	—	2	1	no	—
139	Dic₄₃	—	2	4	no	—
140	Dic₄₃	C₂×#41	1	4	yes	—
141	C₂×#555	—	43	1	no	maximal, Sylow
142	C₂×C₄₃	—	4	4	no	—
143	C₂×C₄₃	C₂²×#41	1	2	yes	—
144	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
145	D₄₃	—	4	4	no	—
146	D₄₃	#555	1	4	yes	direct product factor
147	#555	—	43	1	no	—
148	#555	—	43	4	no	—
149	#555	—	43	4	no	—
150	#555	—	43	2	no	—
151	#555	—	43	4	no	—
152	#555	D₄₃	1	1	yes	characteristic, direct product factor
153	C₂×C₄⋊Q₈	—	43	1	no	—
154	C₂×#124	—	43	4	no	—
155	C₂×#121	—	43	4	no	—
156	C₂×C₄×D₄	—	43	2	no	—
157	C₂×#117	—	43	4	no	—
158	C₄₃	C₂×#555	1	1	yes	characteristic, complemented, Sylow
159	C₂²×Q₈	—	43	2	no	—
160	C₂²×D₄	—	43	2	no	—
161	C₂²×D₄	—	43	4	no	—
162	C₂³×C₄	—	43	4	no	—
163	C₄⋊Q₈	—	43	2	no	—
164	C₄⋊Q₈	—	43	4	no	—
165	C₄⋊Q₈	—	43	1	no	—
166	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
167	#124	—	43	4	no	—
168	#124	—	43	4	no	—
169	#124	—	43	8	no	—
170	#124	—	43	8	no	—
171	#124	—	43	4	no	—
172	#124	C₂×D₄₃	1	4	yes	complemented
173	#121	—	43	4	no	—
174	#121	—	43	8	no	—
175	#121	—	43	4	no	—
176	#121	—	43	4	no	—
177	#121	—	43	8	no	—
178	#121	C₂×D₄₃	1	4	yes	complemented
179	C₄×D₄	—	43	2	no	—
180	C₄×D₄	—	43	4	no	—
181	C₄×D₄	—	43	2	no	—
182	C₄×D₄	—	43	4	no	—
183	C₄×D₄	—	43	2	no	—
184	C₄×D₄	C₂×D₄₃	1	2	yes	complemented
185	#117	—	43	4	no	—
186	#117	—	43	4	no	—
187	#117	—	43	4	no	—
188	#117	—	43	8	no	—
189	#117	—	43	8	no	—
190	#117	C₂×D₄₃	1	4	yes	complemented
191	C₂×C₄⋊C₄	—	43	2	no	—
192	C₂×C₄⋊C₄	—	43	4	no	—
193	C₂×C₂²⋊C₄	—	43	8	no	—
194	C₂×C₂²⋊C₄	—	43	4	no	—
195	C₂×C₄²	—	43	4	no	—
196	C₂×C₄²	—	43	1	no	—
197	C₂⁴	—	43	4	no	—
198	C₂×Q₈	—	43	4	no	—
199	C₂×Q₈	—	43	2	no	—
200	C₂×Q₈	—	172	4	no	—
201	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
202	C₂×D₄	—	43	4	no	—
203	C₂×D₄	—	43	8	no	—
204	C₂×D₄	—	43	2	no	—
205	C₂×D₄	—	43	4	no	—
206	C₂×D₄	—	86	8	no	—
207	C₂×D₄	—	172	8	no	—
208	C₂×D₄	C₂²×D₄₃	1	2	yes	—
209	C₂×D₄	C₂²×D₄₃	1	4	yes	—
210	C₂²×C₄	—	43	2	no	—
211	C₂²×C₄	—	43	4	no	—
212	C₂²×C₄	—	43	4	no	—
213	C₂²×C₄	—	43	1	no	—
214	C₂²×C₄	—	43	4	no	—
215	C₂²×C₄	—	43	4	no	—
216	C₂²×C₄	—	43	4	no	—
217	C₂²×C₄	—	86	16	no	—
218	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
219	C₄⋊C₄	—	43	4	no	—
220	C₄⋊C₄	—	43	2	no	—
221	C₄⋊C₄	—	43	8	no	—
222	C₄⋊C₄	—	43	4	no	—
223	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
224	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
225	C₂²⋊C₄	—	43	8	no	—
226	C₂²⋊C₄	—	43	4	no	—
227	C₂²⋊C₄	—	43	8	no	—
228	C₂²⋊C₄	—	43	8	no	—
229	C₂²⋊C₄	—	43	4	no	—
230	C₂²⋊C₄	—	43	4	no	—
231	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
232	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
233	C₄²	—	43	4	no	—
234	C₄²	—	43	8	no	—
235	C₄²	—	43	1	no	—
236	C₄²	—	43	2	no	—
237	C₄²	C₂²×D₄₃	1	4	yes	—
238	C₄²	C₂²×D₄₃	1	1	yes	characteristic
239	C₂³	—	43	1	no	—
240	C₂³	—	43	4	no	—
241	C₂³	—	86	8	no	—
242	C₂³	—	86	8	no	—
243	C₂³	—	86	8	no	—
244	C₂³	C₂³×D₄₃	1	4	yes	—
245	Q₈	—	4	2	no	—
246	Q₈	—	172	4	no	—
247	Q₈	—	172	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	86	8	no	—
251	D₄	—	86	4	no	—
252	D₄	—	172	4	no	—
253	D₄	—	172	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	43	2	no	—
256	C₂×C₄	—	43	8	no	—
257	C₂×C₄	—	43	4	no	—
258	C₂×C₄	—	43	1	no	—
259	C₂×C₄	—	43	4	no	—
260	C₂×C₄	—	86	8	no	—
261	C₂×C₄	—	86	8	no	—
262	C₂×C₄	—	86	8	no	—
263	C₂×C₄	—	86	2	no	—
264	C₂×C₄	—	86	8	no	—
265	C₂×C₄	—	86	8	no	—
266	C₂×C₄	C₂³×D₄₃	1	4	yes	—
267	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
268	C₂×C₄	C₂³×D₄₃	1	4	yes	—
269	C₂×C₄	C₂³×D₄₃	1	2	yes	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	43	4	no	—
274	C₂²	—	43	2	no	—
275	C₂²	—	86	4	no	—
276	C₂²	—	86	4	no	—
277	C₂²	—	86	4	no	—
278	C₂²	—	172	16	no	—
279	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	1	no	—
283	C₄	—	43	4	no	—
284	C₄	—	86	4	no	—
285	C₄	—	86	4	no	—
286	C₄	—	86	1	no	—
287	C₄	#41×D₄₃	1	4	yes	—
288	C₂	—	4	4	no	—
289	C₂	—	43	4	no	—
290	C₂	—	172	4	no	—
291	C₂	C₂×#41×D₄₃	1	2	yes	—
292	C₂	H(5,𝔽₂)×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#555	C₂
hypercenter	#555	D₄₃
hypocenter	C₄₃	C₂×#555
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #555

Complements

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

N	H ≅ G/N	type
1. #14583615	C₂	semidirect
2. #14583614	C₂	semidirect
3. #14583613	C₂	semidirect
4. #14583612	C₂	semidirect
5. #14583611	C₂	semidirect
6. C₄⋊Q₈×D₄₃	C₂	semidirect
7. #124×D₄₃	C₂	semidirect
8. #121×D₄₃	C₂	semidirect
9. C₄×D₄×D₄₃	C₂	semidirect
10. #117×D₄₃	C₂	semidirect
11. C₄₃×#555	C₂	semidirect
16. #13063325	C₂²	semidirect
17. #13063324	C₂²	semidirect
18. #13063323	C₂²	semidirect
19. #13063314	C₂²	semidirect
20. #13063313	C₂²	semidirect
21. #13063312	C₂²	semidirect
22. #13063311	C₂²	semidirect
23. #13063311	C₂²	semidirect
24. #13063300	C₂²	semidirect
25. #13063299	C₂²	semidirect
26. #13063298	C₂²	semidirect
27. #13063297	C₂²	semidirect
28. #13063296	C₂²	semidirect
29. C₄×D₁₇₂	C₂²	semidirect
30. D₄×Dic₄₃	C₂²	semidirect
31. #13063287	C₂²	semidirect
32. C₄×C₂²⋊D₄₃	C₂²	semidirect
33. (C₂²⋊D₄₃)⋊C₄	C₂²	semidirect
34. #13063284	C₂²	semidirect
35. #13063283	C₂²	semidirect
36. #13063283	C₂²	semidirect
37. #13063282	C₂²	semidirect
38. #13063281	C₂²	semidirect
45. C₄⋊Q₈×C₄₃	C₂²	semidirect
46. #124×C₄₃	C₂²	semidirect
47. #121×C₄₃	C₂²	semidirect
48. C₄×D₄×C₄₃	C₂²	semidirect
49. #117×C₄₃	C₂²	semidirect
146. D₄₃	#555	direct (N × H)
152. #555	D₄₃	direct (N × H)
158. C₄₃	C₂×#555	semidirect
166. C₄⋊Q₈	C₂×D₄₃	semidirect
172. #124	C₂×D₄₃	semidirect
178. #121	C₂×D₄₃	semidirect
184. C₄×D₄	C₂×D₄₃	semidirect
190. #117	C₂×D₄₃	semidirect

Subgroups — #555×D43 #19148131

Conjugacy classes of subgroups

Complements

Subgroups — #555×D₄₃ #19148131