Subgroups — #19147892

Order 5504.

1212 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583454	C₂	1	2	yes	maximal, complemented
2	#14583450	C₂	1	2	yes	maximal, complemented
3	#14583449	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583436	C₂	1	4	yes	maximal, complemented
5	#14583435	C₂	1	4	yes	maximal, complemented
6	#14583434	C₂	1	4	yes	maximal, complemented
7	#14583412	C₂	1	4	yes	maximal, complemented
8	#14583411	C₂	1	2	yes	maximal, complemented
9	C₄×#10621902	C₂	1	4	yes	maximal, complemented
10	#14583367	C₂	1	2	yes	maximal, complemented
11	C₂×C₄²×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#10621902	C₂²	1	4	yes	—
14	C₂×#10621898	C₂²	1	2	yes	—
15	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
17	#13063325	C₂²	1	1	yes	characteristic, complemented
18	#13063322	C₂²	1	1	yes	characteristic, complemented
19	#13063320	C₂²	1	8	yes	complemented
20	#13063316	C₂²	1	4	yes	complemented
21	#13063314	C₂²	1	4	yes	complemented
22	#13063313	C₂²	1	2	yes	complemented
23	#13063310	C₂²	1	8	yes	—
24	#13063309	C₂²	1	4	yes	—
25	#13063305	C₂²	1	4	yes	—
26	#13063301	C₂²	1	4	yes	—
27	#13063300	C₂²	1	4	yes	—
28	#13063300	C₂²	1	4	yes	—
29	#13063299	C₂²	1	4	yes	—
30	#13063295	C₂²	1	4	yes	—
31	Q₈×Dic₄₃	C₂²	1	4	yes	complemented
32	#13063290	C₂²	1	4	yes	complemented
33	C₄×D₁₇₂	C₂²	1	8	yes	—
34	C₄×D₁₇₂	C₂²	1	4	yes	complemented
35	#13063287	C₂²	1	8	yes	—
36	#13063287	C₂²	1	4	yes	complemented
37	#13063287	C₂²	1	4	yes	complemented
38	(C₂²⋊D₄₃)⋊C₄	C₂²	1	8	yes	—
39	#13063283	C₂²	1	4	yes	—
40	#13063283	C₂²	1	8	yes	complemented
41	#13063282	C₂²	1	4	yes	complemented
42	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
43	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
44	C₄²×D₄₃	C₂²	1	2	yes	—
45	C₄²×D₄₃	C₂²	1	2	yes	—
46	C₄²×D₄₃	C₂²	1	2	yes	—
47	C₄²×D₄₃	C₂²	1	2	yes	—
48	C₄²×D₄₃	C₂²	1	4	yes	complemented
49	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	complemented
50	#125×C₄₃	C₂²	1	2	yes	complemented
51	C₂²⋊Q₈×C₄₃	C₂²	1	4	yes	—
52	C₄×Q₈×C₄₃	C₂²	1	4	yes	complemented
53	#117×C₄₃	C₂²	1	2	yes	—
54	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
55	C₂³×D₄₃	C₂³	1	1	yes	characteristic
56	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
60	C₂²×Dic₄₃	C₂³	1	2	yes	—
61	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
62	#10621902	—	2	16	no	—
63	#10621898	—	2	8	no	—
64	C₂×D₁₇₂	C₂³	1	8	yes	—
65	C₂×D₁₇₂	C₂³	1	2	yes	—
66	C₂×D₁₇₂	C₂³	1	4	yes	—
67	C₂×C₄×D₄₃	—	2	4	no	—
68	C₂×C₄×D₄₃	—	2	8	no	—
69	C₂×C₄×D₄₃	C₂³	1	2	yes	—
70	C₂×C₄×D₄₃	C₂³	1	2	yes	—
71	C₂×C₄×D₄₃	C₂³	1	2	yes	—
72	C₂×C₄×D₄₃	C₂³	1	2	yes	—
73	C₂×C₄×D₄₃	C₂³	1	2	yes	—
74	C₂×C₄×D₄₃	C₂³	1	4	yes	—
75	C₂×C₄×D₄₃	C₂³	1	8	yes	—
76	C₂×C₄×D₄₃	C₂³	1	2	yes	—
77	C₂×Dic₈₆	C₂³	1	2	yes	—
78	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
79	C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
80	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
81	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
82	C₄²×C₄₃	C₂³	1	2	yes	—
83	C₄²×C₄₃	C₂³	1	1	yes	characteristic
84	C₄²×C₄₃	C₂³	1	1	yes	characteristic
85	C₄²×C₄₃	C₂³	1	4	yes	—
86	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
87	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
88	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
89	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
90	C₄⋊Dic₄₃	C₂³	1	4	yes	—
91	C₄⋊Dic₄₃	C₂³	1	8	yes	—
92	Dic₄₃⋊C₄	C₂³	1	8	yes	—
93	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
94	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
95	C₄×Dic₄₃	C₂³	1	2	yes	—
96	C₄×Dic₄₃	C₂³	1	2	yes	—
97	C₄×Dic₄₃	C₂³	1	2	yes	—
98	C₄×Dic₄₃	C₂³	1	2	yes	—
99	C₄×Dic₄₃	C₂³	1	2	yes	—
100	C₄×Dic₄₃	C₂³	1	8	yes	—
101	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
102	C₂²×D₄₃	—	2	2	no	—
103	C₂²×D₄₃	—	2	4	no	—
104	C₂²×D₄₃	C₂⁴	1	4	yes	—
105	C₂²×D₄₃	C₂⁴	1	2	yes	—
106	Q₈×C₄₃	—	2	8	no	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	#41	1	4	yes	complemented
109	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
110	C₂×C₄×C₄₃	C₂⁴	1	8	yes	—
111	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₄₃	—	2	8	no	—
115	C₂×Dic₄₃	—	2	2	no	—
116	C₂×Dic₄₃	—	2	4	no	—
117	C₂×Dic₄₃	C₂⁴	1	2	yes	—
118	C₂×Dic₄₃	C₂⁴	1	2	yes	—
119	C₂×Dic₄₃	C₂⁴	1	4	yes	—
120	C₂×Dic₄₃	C₂⁴	1	2	yes	—
121	D₁₇₂	—	2	16	no	—
122	D₁₇₂	—	2	4	no	—
123	D₁₇₂	—	2	8	no	—
124	C₄×D₄₃	—	2	4	no	—
125	C₄×D₄₃	—	2	4	no	—
126	C₄×D₄₃	—	2	4	no	—
127	C₄×D₄₃	—	2	4	no	—
128	C₄×D₄₃	—	2	8	no	—
129	C₄×D₄₃	—	2	4	no	—
130	C₄×D₄₃	—	2	16	no	—
131	C₄×D₄₃	#41	1	8	yes	complemented
132	Dic₈₆	—	2	4	no	—
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	C₂×#41	1	2	yes	—
135	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
136	C₂×D₄₃	—	2	8	no	—
137	C₂×D₄₃	—	2	2	no	—
138	C₂×D₄₃	—	2	2	no	—
139	C₂×D₄₃	—	2	8	no	—
140	C₂×D₄₃	—	2	4	no	—
141	C₂×D₄₃	C₂×#41	1	4	yes	—
142	C₄×C₄₃	—	2	2	no	—
143	C₄×C₄₃	—	2	8	no	—
144	C₄×C₄₃	C₂×#41	1	4	yes	—
145	C₄×C₄₃	C₂×#41	1	2	yes	—
146	C₄×C₄₃	C₂×#41	1	2	yes	—
147	Dic₄₃	—	2	4	no	—
148	Dic₄₃	—	2	2	no	—
149	Dic₄₃	—	2	2	no	—
150	Dic₄₃	C₂×#41	1	4	yes	—
151	#3444	—	43	1	no	maximal, Sylow
152	C₂×C₄₃	—	2	2	no	—
153	C₂×C₄₃	C₂²×#41	1	2	yes	—
154	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
155	D₄₃	—	2	4	no	—
156	D₄₃	—	4	4	no	—
157	#533	—	43	2	no	—
158	#533	D₄₃	1	1	yes	characteristic, complemented
159	#532	—	43	2	no	—
160	#532	—	43	1	no	—
161	#529	—	43	4	no	—
162	#529	—	43	4	no	—
163	#529	—	43	4	no	—
164	#525	—	43	4	no	—
165	#525	—	43	2	no	—
166	C₄×#41	—	43	4	no	—
167	C₄×#41	—	43	2	no	—
168	C₂²×C₄²	—	43	1	no	—
169	C₄₃	#3444	1	1	yes	characteristic, complemented, Sylow
170	C₂×#41	—	43	4	no	—
171	C₂×#41	—	43	2	no	—
172	C₂³×C₄	—	43	2	no	—
173	C₂³×C₄	—	43	1	no	—
174	C₄⋊Q₈	—	43	1	no	—
175	C₄⋊Q₈	C₂×D₄₃	1	2	yes	complemented
176	#127	—	43	1	no	—
177	#126	—	43	8	no	—
178	#125	—	43	4	no	—
179	#125	C₂×D₄₃	1	2	yes	complemented
180	#124	—	43	4	no	—
181	#124	—	43	2	no	—
182	#123	—	43	8	no	—
183	#123	—	43	4	no	—
184	C₂²⋊Q₈	—	43	4	no	—
185	C₂²⋊Q₈	—	43	4	no	—
186	C₂²⋊Q₈	C₂×D₄₃	1	4	yes	—
187	#121	—	43	4	no	—
188	#121	—	43	4	no	—
189	#121	—	43	4	no	—
190	#120	—	43	4	no	—
191	C₄×Q₈	—	43	4	no	—
192	C₄×Q₈	—	43	4	no	—
193	C₄×Q₈	C₂×D₄₃	1	4	yes	complemented
194	C₄×D₄	—	43	8	no	—
195	C₄×D₄	—	43	4	no	—
196	C₄×D₄	—	43	4	no	—
197	C₄×D₄	—	43	8	no	—
198	C₄×D₄	—	43	8	no	—
199	C₄×D₄	—	43	4	no	—
200	#117	—	43	4	no	—
201	#117	—	43	8	no	—
202	#117	—	43	4	no	—
203	#117	C₂×D₄₃	1	2	yes	—
204	C₂×C₄²	—	43	1	no	—
205	C₂×C₄²	—	43	2	no	—
206	C₂×C₄²	—	43	4	no	—
207	C₂×C₄²	—	43	2	no	—
208	C₂×C₄²	—	43	2	no	—
209	C₂×C₄²	—	43	2	no	—
210	C₂×C₄²	—	43	2	no	—
211	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
212	C₂⁴	—	43	1	no	—
213	#41	—	86	16	no	—
214	#41	—	86	8	no	—
215	C₂×Q₈	—	43	2	no	—
216	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
217	C₂×D₄	—	43	4	no	—
218	C₂×D₄	—	43	8	no	—
219	C₂×D₄	—	43	4	no	—
220	C₂×D₄	—	43	2	no	—
221	C₂²×C₄	—	43	2	no	—
222	C₂²×C₄	—	43	2	no	—
223	C₂²×C₄	—	43	1	no	—
224	C₂²×C₄	—	43	2	no	—
225	C₂²×C₄	—	43	2	no	—
226	C₂²×C₄	—	43	2	no	—
227	C₂²×C₄	—	43	2	no	—
228	C₂²×C₄	—	43	8	no	—
229	C₂²×C₄	—	43	2	no	—
230	C₂²×C₄	—	43	4	no	—
231	C₂²×C₄	—	86	8	no	—
232	C₂²×C₄	—	86	4	no	—
233	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
234	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
235	C₄⋊C₄	—	43	4	no	—
236	C₄⋊C₄	—	43	8	no	—
237	C₄⋊C₄	—	43	8	no	—
238	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
239	C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
240	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
241	C₂²⋊C₄	—	43	8	no	—
242	C₂²⋊C₄	—	43	8	no	—
243	C₂²⋊C₄	—	43	8	no	—
244	C₂²⋊C₄	—	43	8	no	—
245	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
246	C₄²	—	43	2	no	—
247	C₄²	—	43	2	no	—
248	C₄²	—	43	2	no	—
249	C₄²	—	43	2	no	—
250	C₄²	—	43	2	no	—
251	C₄²	—	43	8	no	—
252	C₄²	—	43	1	no	—
253	C₄²	—	43	1	no	—
254	C₄²	C₂²×D₄₃	1	1	yes	characteristic
255	C₄²	C₂²×D₄₃	1	1	yes	characteristic
256	C₄²	C₂²×D₄₃	1	2	yes	—
257	C₄²	C₂²×D₄₃	1	4	yes	—
258	C₂³	—	43	2	no	—
259	C₂³	—	43	4	no	—
260	C₂³	—	86	4	no	—
261	C₂³	—	86	2	no	—
262	C₂³	C₂³×D₄₃	1	1	yes	characteristic
263	Q₈	—	2	8	no	—
264	Q₈	—	86	4	no	—
265	D₄	—	86	16	no	—
266	D₄	—	86	8	no	—
267	D₄	—	86	8	no	—
268	D₄	—	86	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	43	2	no	—
271	C₂×C₄	—	43	8	no	—
272	C₂×C₄	—	43	2	no	—
273	C₂×C₄	—	43	2	no	—
274	C₂×C₄	—	43	4	no	—
275	C₂×C₄	—	86	4	no	—
276	C₂×C₄	—	86	16	no	—
277	C₂×C₄	—	86	4	no	—
278	C₂×C₄	—	86	4	no	—
279	C₂×C₄	—	86	2	no	—
280	C₂×C₄	—	86	4	no	—
281	C₂×C₄	—	86	4	no	—
282	C₂×C₄	—	86	4	no	—
283	C₂×C₄	—	86	8	no	—
284	C₂×C₄	#10621902	1	4	yes	complemented
285	C₂×C₄	C₂³×D₄₃	1	2	yes	—
286	C₂×C₄	C₂³×D₄₃	1	8	yes	—
287	C₂×C₄	C₂³×D₄₃	1	2	yes	—
288	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	43	4	no	—
292	C₂²	—	86	8	no	—
293	C₂²	—	86	2	no	—
294	C₂²	—	86	2	no	—
295	C₂²	—	86	8	no	—
296	C₂²	—	86	4	no	—
297	C₂²	C₂×#10621902	1	2	yes	—
298	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	8	no	—
301	C₄	—	43	4	no	—
302	C₄	—	86	2	no	—
303	C₄	—	86	2	no	—
304	C₄	—	86	4	no	—
305	C₄	#41×D₄₃	1	4	yes	—
306	C₄	C₂×#10621902	1	2	yes	—
307	C₄	C₂×#10621902	1	2	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	86	4	no	—
310	C₂	—	172	4	no	—
311	C₂	C₂×#41×D₄₃	1	2	yes	—
312	C₂	C₂²×#10621902	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₄₃×#533	C₂
hypercenter	#533	D₄₃
hypocenter	C₄₃	#3444
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #14583454	C₂	semidirect
2. #14583450	C₂	semidirect
3. #14583449	C₂	semidirect
4. #14583436	C₂	semidirect
5. #14583435	C₂	semidirect
6. #14583434	C₂	semidirect
7. #14583412	C₂	semidirect
8. #14583411	C₂	semidirect
9. C₄×#10621902	C₂	semidirect
10. #14583367	C₂	semidirect
11. C₂×C₄²×D₄₃	C₂	semidirect
12. C₄₃×#533	C₂	semidirect
17. #13063325	C₂²	semidirect
18. #13063322	C₂²	semidirect
19. #13063320	C₂²	semidirect
20. #13063316	C₂²	semidirect
21. #13063314	C₂²	semidirect
22. #13063313	C₂²	semidirect
31. Q₈×Dic₄₃	C₂²	semidirect
32. #13063290	C₂²	semidirect
34. C₄×D₁₇₂	C₂²	semidirect
36. #13063287	C₂²	semidirect
37. #13063287	C₂²	semidirect
40. #13063283	C₂²	semidirect
41. #13063282	C₂²	semidirect
48. C₄²×D₄₃	C₂²	semidirect
49. C₄⋊Q₈×C₄₃	C₂²	semidirect
50. #125×C₄₃	C₂²	semidirect
52. C₄×Q₈×C₄₃	C₂²	semidirect
108. C₂×C₄×C₄₃	#41	semidirect
131. C₄×D₄₃	#41	semidirect
158. #533	D₄₃	semidirect
169. C₄₃	#3444	semidirect
175. C₄⋊Q₈	C₂×D₄₃	semidirect
179. #125	C₂×D₄₃	semidirect
193. C₄×Q₈	C₂×D₄₃	semidirect
284. C₂×C₄	#10621902	semidirect