Subgroups — #19144806

Order 5504.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583510	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583378	C₂	1	1	yes	maximal, characteristic
3	#14583233	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583228	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583207	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583199	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583106	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583058	C₂	1	1	yes	maximal, characteristic
9	#14583058	C₂	1	1	yes	maximal, characteristic
10	#14583003	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583000	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582991	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582959	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621900	C₂²	1	1	yes	characteristic
17	C₂×#10621885	C₂²	1	1	yes	characteristic
18	C₂×#10621885	C₂²	1	1	yes	characteristic
19	#13063320	—	2	1	no	—
20	#13063320	—	2	1	no	—
21	#13063317	C₂²	1	1	yes	characteristic
22	#13063315	—	2	1	no	—
23	#13063315	—	2	1	no	—
24	#13063301	C₂²	1	1	yes	characteristic
25	#13063301	C₂²	1	1	yes	characteristic
26	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
27	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
28	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
29	#13063290	C₂²	1	1	yes	characteristic
30	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₄₃	—	2	1	no	—
32	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
33	#13063282	C₂²	1	1	yes	characteristic
34	#13063281	—	2	1	no	—
35	C₂×Dic₄₃⋊C₄	—	2	1	no	—
36	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
37	#13063259	C₂²	1	1	yes	characteristic
38	#13063258	C₂²	1	1	yes	characteristic
39	#13063257	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#13063250	C₂²	1	1	yes	characteristic
43	#13063250	C₂²	1	1	yes	characteristic
44	#13063249	C₂²	1	1	yes	characteristic
45	#13063249	C₂²	1	1	yes	characteristic
46	#13063246	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
54	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#103×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	—	2	1	no	—
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂²×Dic₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	2	1	no	—
66	#10621900	—	2	2	no	—
67	#10621900	—	2	2	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
70	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
71	SD₁₆×C₄₃	—	2	2	no	—
72	SD₁₆×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₄₃	—	2	1	no	—
76	C₄⋊C₄×C₄₃	—	2	1	no	—
77	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₄₃	—	2	1	no	—
79	C₄²×C₄₃	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₄₃	—	2	1	no	—
81	#10621885	—	2	2	no	—
82	#10621885	—	2	2	no	—
83	(C₂×D₄₃)⋊C₄	—	2	1	no	—
84	(C₂×D₄₃)⋊C₄	—	2	1	no	—
85	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
86	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
87	C₄⋊Dic₄₃	—	2	1	no	—
88	C₄⋊Dic₄₃	—	2	1	no	—
89	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
92	Dic₄₃⋊C₄	—	2	1	no	—
93	Dic₄₃⋊C₄	—	2	1	no	—
94	Dic₄₃⋊C₄	—	2	1	no	—
95	Dic₄₃⋊C₄	—	2	1	no	—
96	Dic₄₃⋊C₄	—	2	1	no	—
97	Dic₄₃⋊C₄	—	2	1	no	—
98	Dic₄₃⋊C₄	—	2	1	no	—
99	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
100	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	—	2	1	no	—
103	C₄×Dic₄₃	—	2	1	no	—
104	C₄×Dic₄₃	—	2	1	no	—
105	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₄₃	—	2	1	no	—
109	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₄₃	—	2	1	no	—
111	Q₈×C₄₃	—	2	1	no	—
112	Q₈×C₄₃	—	2	1	no	—
113	Q₈×C₄₃	—	2	1	no	—
114	D₄×C₄₃	—	2	1	no	—
115	D₄×C₄₃	#41	1	2	yes	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	8	1	no	—
120	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₄₃	—	2	4	no	—
124	C₂×Dic₄₃	—	2	4	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	8	1	no	—
130	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
133	C₄×D₄₃	—	2	1	no	—
134	C₄×D₄₃	—	2	1	no	—
135	C₄×D₄₃	—	8	1	no	—
136	Dic₈₆	—	2	1	no	—
137	Dic₈₆	#41	1	2	yes	—
138	C₈×C₄₃	—	2	1	no	—
139	C₈×C₄₃	—	2	1	no	—
140	C₄₃⋊C₈	—	2	1	no	—
141	C₄₃⋊C₈	—	2	1	no	—
142	C₂²×C₄₃	—	2	2	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	—	4	1	no	—
145	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₄₃	—	2	1	no	—
147	C₂×D₄₃	—	2	1	no	—
148	C₂×D₄₃	—	2	1	no	—
149	C₄×C₄₃	—	2	1	no	—
150	C₄×C₄₃	—	2	1	no	—
151	C₄×C₄₃	—	4	1	no	—
152	C₄×C₄₃	—	4	1	no	—
153	C₄×C₄₃	—	4	1	no	—
154	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
155	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
156	Dic₄₃	—	2	2	no	—
157	Dic₄₃	—	2	1	no	—
158	Dic₄₃	—	2	1	no	—
159	Dic₄₃	—	2	1	no	—
160	Dic₄₃	—	4	1	no	—
161	Dic₄₃	—	4	1	no	—
162	Dic₄₃	—	4	1	no	—
163	#3120	—	43	1	no	maximal, Sylow
164	C₂×C₄₃	—	4	2	no	—
165	C₂×C₄₃	#525	1	1	yes	characteristic
166	C₂×C₄₃	#577	1	1	yes	characteristic
167	C₂×C₄₃	#578	1	1	yes	characteristic
168	D₄₃	—	4	1	no	—
169	#541	—	43	1	no	—
170	#520	—	43	1	no	—
171	#484	—	43	1	no	—
172	#483	—	43	1	no	—
173	#479	—	43	1	no	—
174	#478	—	43	1	no	—
175	#462	—	43	1	no	—
176	#461	D₄₃	1	1	yes	characteristic, complemented
177	#452	—	43	1	no	—
178	#452	—	43	1	no	—
179	#441	—	43	1	no	—
180	#440	—	43	1	no	—
181	C₄×Q₁₆	—	43	1	no	—
182	C₄×SD₁₆	—	43	1	no	—
183	#433	—	43	1	no	—
184	C₄₃	#3120	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	43	1	no	—
186	C₂×Q₁₆	—	43	1	no	—
187	C₂×Q₁₆	—	43	1	no	—
188	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
189	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
190	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
191	#126	—	86	1	no	—
192	#126	—	86	1	no	—
193	#125	—	43	1	no	—
194	#125	—	86	1	no	—
195	#125	—	86	1	no	—
196	C₂²⋊Q₈	—	43	1	no	—
197	C₂²⋊Q₈	—	43	1	no	—
198	C₄×Q₈	—	43	1	no	—
199	C₄×Q₈	—	43	1	no	—
200	C₄×Q₈	—	43	1	no	—
201	C₄×Q₈	—	43	1	no	—
202	C₄×D₄	—	43	1	no	—
203	C₄×D₄	—	86	1	no	—
204	C₄×D₄	—	86	1	no	—
205	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
206	#117	—	43	1	no	—
207	#117	—	86	1	no	—
208	C₂×C₄⋊C₄	—	43	1	no	—
209	C₂×C₄⋊C₄	—	86	1	no	—
210	#107	—	43	1	no	—
211	#107	—	43	1	no	—
212	#106	—	43	1	no	—
213	#106	—	43	1	no	—
214	C₄⋊C₈	—	43	1	no	—
215	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
216	#103	—	43	1	no	—
217	#103	—	43	1	no	—
218	#103	—	43	1	no	—
219	#103	—	43	1	no	—
220	#103	C₂×D₄₃	1	1	yes	characteristic
221	#103	C₂×D₄₃	1	1	yes	characteristic
222	#102	—	43	1	no	—
223	#102	—	43	1	no	—
224	C₂²⋊C₈	—	43	1	no	—
225	C₂²⋊C₈	—	43	1	no	—
226	#97	—	43	1	no	—
227	C₄×C₈	—	43	1	no	—
228	#41	—	86	2	no	—
229	#41	—	86	2	no	—
230	C₂×Q₈	—	43	1	no	—
231	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
233	C₂×D₄	—	86	1	no	—
234	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	43	1	no	—
237	C₂²×C₄	—	86	1	no	—
238	C₂²×C₄	—	86	1	no	—
239	C₂²×C₄	—	86	1	no	—
240	Q₁₆	—	86	2	no	—
241	Q₁₆	—	86	2	no	—
242	SD₁₆	—	2	2	no	—
243	SD₁₆	—	2	2	no	—
244	C₂×C₈	—	43	1	no	—
245	C₂×C₈	—	43	1	no	—
246	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	43	1	no	—
251	C₄⋊C₄	—	43	1	no	—
252	C₄⋊C₄	—	43	1	no	—
253	C₄⋊C₄	—	43	1	no	—
254	C₄⋊C₄	—	43	1	no	—
255	C₄⋊C₄	—	86	1	no	—
256	C₄⋊C₄	—	86	1	no	—
257	C₄⋊C₄	—	86	1	no	—
258	C₄⋊C₄	—	86	1	no	—
259	C₄⋊C₄	—	86	1	no	—
260	C₄⋊C₄	—	86	1	no	—
261	C₄⋊C₄	—	86	1	no	—
262	C₄⋊C₄	—	86	1	no	—
263	C₄⋊C₄	—	86	1	no	—
264	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	43	1	no	—
267	C₂²⋊C₄	—	43	1	no	—
268	C₂²⋊C₄	—	86	1	no	—
269	C₂²⋊C₄	—	86	1	no	—
270	C₂²⋊C₄	—	86	1	no	—
271	C₄²	—	43	1	no	—
272	C₄²	—	86	1	no	—
273	C₄²	—	86	1	no	—
274	C₄²	—	86	1	no	—
275	C₄²	—	86	1	no	—
276	C₄²	C₂²×D₄₃	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	—	43	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	43	2	no	—
284	Q₈	—	86	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	86	4	no	—
287	D₄	#10621900	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	43	1	no	—
293	C₂×C₄	—	43	1	no	—
294	C₂×C₄	—	43	1	no	—
295	C₂×C₄	—	86	4	no	—
296	C₂×C₄	—	86	1	no	—
297	C₂×C₄	—	86	1	no	—
298	C₂×C₄	—	86	1	no	—
299	C₂×C₄	—	86	1	no	—
300	C₂×C₄	—	86	1	no	—
301	C₂×C₄	—	86	1	no	—
302	C₂×C₄	—	344	1	no	—
303	C₂×C₄	—	344	1	no	—
304	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
305	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	86	1	no	—
310	C₈	—	86	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	86	1	no	—
315	C₂²	—	86	1	no	—
316	C₂²	—	86	1	no	—
317	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	86	1	no	—
324	C₄	—	86	2	no	—
325	C₄	—	86	1	no	—
326	C₄	—	86	1	no	—
327	C₄	—	172	1	no	—
328	C₄	—	172	1	no	—
329	C₄	—	172	1	no	—
330	C₄	#41×D₄₃	1	1	yes	characteristic
331	C₄	C₂×#10621900	1	1	yes	characteristic
332	C₂	—	4	2	no	—
333	C₂	—	172	1	no	—
334	C₂	#14583409	1	1	yes	characteristic
335	C₂	#14583705	1	1	yes	characteristic
336	C₂	#14583715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#461	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#461	D₄₃
hypocenter	C₄₃	#3120
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

Complements

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

N	H ≅ G/N	type
1. #14583510	C₂	semidirect
3. #14583233	C₂	semidirect
4. #14583228	C₂	semidirect
5. #14583207	C₂	semidirect
6. #14583199	C₂	semidirect
7. #14583106	C₂	semidirect
10. #14583003	C₂	semidirect
11. #14583000	C₂	semidirect
12. #14582991	C₂	semidirect
13. SD₁₆×Dic₄₃	C₂	semidirect
14. #14582959	C₂	semidirect
15. C₄₃×#461	C₂	semidirect
176. #461	D₄₃	semidirect
184. C₄₃	#3120	semidirect