Subgroups — #19144381

Order 5504.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583452	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583367	C₂	1	1	yes	maximal, characteristic
3	#14583209	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583205	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583202	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583198	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583193	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583188	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583183	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583178	C₂	1	1	yes	maximal, characteristic, complemented
11	#14582935	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582896	C₂	1	1	yes	maximal, characteristic
14	#14582893	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	C₂²	1	1	yes	characteristic
17	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	#13063324	C₂²	1	1	yes	characteristic
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063316	C₂²	1	1	yes	characteristic
21	#13063316	C₂²	1	1	yes	characteristic
22	#13063302	—	2	1	no	—
23	#13063302	—	2	1	no	—
24	C₄×Dic₈₆	—	2	1	no	—
25	C₄×Dic₈₆	—	2	1	no	—
26	#13063290	C₂²	1	1	yes	characteristic
27	#13063290	C₂²	1	1	yes	characteristic
28	#13063287	C₂²	1	1	yes	characteristic
29	#13063287	C₂²	1	1	yes	characteristic
30	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
31	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
32	#13063284	C₂²	1	1	yes	characteristic
33	#13063282	—	2	1	no	—
34	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	C₄²×D₄₃	—	2	1	no	—
36	#13063258	C₂²	1	1	yes	characteristic
37	#13063258	C₂²	1	1	yes	characteristic
38	#13063256	C₂²	1	1	yes	characteristic
39	#13063256	C₂²	1	1	yes	characteristic
40	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
44	#13063250	C₂²	1	1	yes	characteristic
45	#13063250	C₂²	1	1	yes	characteristic
46	#13063248	C₂²	1	1	yes	characteristic
47	#13063248	C₂²	1	1	yes	characteristic
48	#13063247	C₂²	1	1	yes	characteristic
49	#13063247	C₂²	1	1	yes	characteristic
50	#13063245	C₂²	1	1	yes	characteristic
51	#13063245	C₂²	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
53	#117×C₄₃	C₂²	1	1	yes	characteristic
54	#117×C₄₃	C₂²	1	1	yes	characteristic
55	#107×C₄₃	C₂²	1	1	yes	characteristic
56	#107×C₄₃	C₂²	1	1	yes	characteristic
57	#106×C₄₃	C₂²	1	1	yes	characteristic
58	#106×C₄₃	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₄₃	—	2	1	no	—
61	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
62	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
63	#10621898	—	2	2	no	—
64	#10621898	Q₈	1	4	yes	complemented
65	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₄₃	—	2	1	no	—
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×Dic₈₆	—	2	1	no	—
70	C₂×Dic₈₆	—	2	1	no	—
71	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
72	C₂×C₈×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	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₄₃	—	2	1	no	—
80	C₂²⋊C₄×C₄₃	—	2	1	no	—
81	C₄²×C₄₃	—	2	1	no	—
82	C₄²×C₄₃	—	2	1	no	—
83	C₂²⋊Dic₄₃	—	2	1	no	—
84	(C₂×D₄₃)⋊C₄	—	2	1	no	—
85	(C₂×D₄₃)⋊C₄	—	2	1	no	—
86	C₄⋊Dic₄₃	—	2	1	no	—
87	C₄⋊Dic₄₃	—	2	1	no	—
88	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
90	Dic₄₃⋊C₄	—	2	1	no	—
91	Dic₄₃⋊C₄	—	2	1	no	—
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	C₄×Dic₄₃	—	2	1	no	—
97	C₄×Dic₄₃	—	2	1	no	—
98	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	D₄	1	1	yes	characteristic
100	C₄×Dic₄₃	D₄	1	1	yes	characteristic
101	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
102	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	—	2	2	no	—
104	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
114	C₂×C₄×C₄₃	#41	1	2	yes	—
115	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₄₃	—	2	4	no	—
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	—	2	2	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	—	4	1	no	—
122	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
126	D₁₇₂	—	2	1	no	—
127	D₁₇₂	C₂×Q₈	1	2	yes	—
128	C₄×D₄₃	—	2	4	no	—
129	C₄×D₄₃	—	4	2	no	—
130	C₄×D₄₃	—	4	2	no	—
131	Dic₈₆	—	2	1	no	—
132	Dic₈₆	—	4	2	no	—
133	Dic₈₆	—	4	2	no	—
134	Dic₈₆	C₂×Q₈	1	2	yes	—
135	C₈×C₄₃	—	2	1	no	—
136	C₈×C₄₃	—	2	1	no	—
137	C₄₃⋊C₈	—	2	1	no	—
138	C₄₃⋊C₈	—	2	1	no	—
139	C₂²×C₄₃	—	2	1	no	—
140	C₂²×C₄₃	—	2	1	no	—
141	C₂²×C₄₃	C₂²⋊Q₈	1	2	yes	—
142	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₄₃	—	2	2	no	—
144	C₂×D₄₃	—	4	1	no	—
145	C₂×D₄₃	—	4	1	no	—
146	C₄×C₄₃	—	4	1	no	—
147	C₄×C₄₃	—	4	1	no	—
148	C₄×C₄₃	—	4	1	no	—
149	C₄×C₄₃	—	4	1	no	—
150	C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
151	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
153	Dic₄₃	—	2	2	no	—
154	Dic₄₃	—	2	1	no	—
155	Dic₄₃	—	2	1	no	—
156	Dic₄₃	—	2	1	no	—
157	Dic₄₃	—	4	1	no	—
158	Dic₄₃	—	4	1	no	—
159	Dic₄₃	#135	1	2	yes	—
160	#3076	—	43	1	no	maximal, Sylow
161	C₂×C₄₃	—	2	2	no	—
162	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
164	C₂×C₄₃	#575	1	1	yes	characteristic
165	D₄₃	—	4	2	no	—
166	#533	—	43	1	no	—
167	C₄×#41	—	43	1	no	—
168	#479	—	43	1	no	—
169	#479	—	43	1	no	—
170	#478	—	43	1	no	—
171	#478	—	43	1	no	—
172	#477	—	43	1	no	—
173	#476	—	43	1	no	—
174	#475	—	43	1	no	—
175	#474	—	43	1	no	—
176	#427	—	43	1	no	—
177	#427	D₄₃	1	1	yes	characteristic, complemented
178	C₂×C₄⋊C₈	—	43	1	no	—
179	#416	—	43	1	no	—
180	#416	—	43	1	no	—
181	C₄₃	#3076	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	—	43	1	no	—
183	C₂²×C₈	—	43	1	no	—
184	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
185	C₄⋊Q₈	—	43	1	no	—
186	C₄⋊Q₈	—	43	1	no	—
187	#125	—	43	1	no	—
188	#125	—	43	1	no	—
189	C₂²⋊Q₈	—	86	1	no	—
190	C₂²⋊Q₈	—	86	1	no	—
191	C₄×Q₈	—	43	1	no	—
192	C₄×Q₈	—	43	1	no	—
193	C₄×Q₈	—	86	1	no	—
194	C₄×Q₈	—	86	1	no	—
195	C₄×D₄	—	43	1	no	—
196	C₄×D₄	—	43	1	no	—
197	C₄×D₄	—	86	1	no	—
198	C₄×D₄	—	86	1	no	—
199	#117	—	43	1	no	—
200	#117	—	86	1	no	—
201	#117	C₂×D₄₃	1	1	yes	characteristic
202	#117	C₂×D₄₃	1	1	yes	characteristic
203	C₂×C₄²	—	43	1	no	—
204	C₂×C₄²	—	86	1	no	—
205	#107	—	43	1	no	—
206	#107	—	43	1	no	—
207	#107	C₂×D₄₃	1	1	yes	characteristic
208	#107	C₂×D₄₃	1	1	yes	characteristic
209	#106	—	43	1	no	—
210	#106	—	43	1	no	—
211	#106	C₂×D₄₃	1	1	yes	characteristic
212	#106	C₂×D₄₃	1	1	yes	characteristic
213	C₄⋊C₈	—	43	1	no	—
214	C₄⋊C₈	—	43	1	no	—
215	C₄⋊C₈	—	43	1	no	—
216	C₄⋊C₈	—	43	1	no	—
217	#103	—	43	1	no	—
218	#103	—	43	1	no	—
219	#103	—	43	1	no	—
220	#103	—	43	1	no	—
221	#102	—	43	1	no	—
222	#102	—	43	1	no	—
223	#102	—	43	1	no	—
224	#102	—	43	1	no	—
225	#41	—	43	4	no	—
226	#41	—	86	2	no	—
227	C₂×Q₈	—	43	1	no	—
228	C₂×Q₈	—	86	1	no	—
229	C₂×Q₈	—	86	1	no	—
230	C₂×D₄	—	43	1	no	—
231	C₂×D₄	—	86	1	no	—
232	C₂²×C₄	—	43	1	no	—
233	C₂²×C₄	—	43	1	no	—
234	C₂²×C₄	—	86	1	no	—
235	C₂²×C₄	—	86	1	no	—
236	C₂²×C₄	—	86	1	no	—
237	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	43	1	no	—
240	C₂×C₈	—	43	1	no	—
241	C₂×C₈	—	86	2	no	—
242	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
244	C₄⋊C₄	—	43	1	no	—
245	C₄⋊C₄	—	43	1	no	—
246	C₄⋊C₄	—	86	1	no	—
247	C₄⋊C₄	—	86	1	no	—
248	C₄⋊C₄	—	86	1	no	—
249	C₄⋊C₄	—	86	1	no	—
250	C₄⋊C₄	—	86	1	no	—
251	C₄⋊C₄	—	86	1	no	—
252	C₄⋊C₄	—	86	1	no	—
253	C₄⋊C₄	—	86	1	no	—
254	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	86	1	no	—
261	C₂²⋊C₄	—	86	1	no	—
262	C₂²⋊C₄	—	86	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	43	1	no	—
266	C₄²	—	43	1	no	—
267	C₄²	—	43	1	no	—
268	C₄²	—	43	1	no	—
269	C₄²	—	86	1	no	—
270	C₄²	—	86	1	no	—
271	C₄²	—	86	1	no	—
272	C₂³	—	86	1	no	—
273	C₂³	D₄×D₄₃	1	1	yes	characteristic
274	Q₈	—	43	2	no	—
275	Q₈	—	86	1	no	—
276	Q₈	—	172	2	no	—
277	Q₈	—	172	2	no	—
278	D₄	—	43	2	no	—
279	D₄	—	86	4	no	—
280	D₄	—	86	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	43	1	no	—
286	C₂×C₄	—	43	1	no	—
287	C₂×C₄	—	43	1	no	—
288	C₂×C₄	—	43	1	no	—
289	C₂×C₄	—	86	2	no	—
290	C₂×C₄	—	86	1	no	—
291	C₂×C₄	—	86	4	no	—
292	C₂×C₄	—	86	1	no	—
293	C₂×C₄	—	86	1	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	172	2	no	—
296	C₂×C₄	—	172	1	no	—
297	C₂×C₄	#10621898	1	2	yes	—
298	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
299	C₂×C₄	Q₈×D₄₃	1	2	yes	—
300	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	86	1	no	—
304	C₈	—	86	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	86	2	no	—
308	C₂²	—	172	1	no	—
309	C₂²	—	172	1	no	—
310	C₂²	#13063303	1	2	yes	—
311	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	43	2	no	—
317	C₄	—	86	2	no	—
318	C₄	—	86	1	no	—
319	C₄	—	86	1	no	—
320	C₄	—	86	1	no	—
321	C₄	—	172	1	no	—
322	C₄	—	172	1	no	—
323	C₄	#13063303	1	2	yes	—
324	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
325	C₄	C₂×#10621898	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	172	2	no	—
328	C₂	C₂×#13063303	1	1	yes	characteristic
329	C₂	#135×D₄₃	1	1	yes	characteristic
330	C₂	#14583696	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#427	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#427	D₄₃
hypocenter	C₄₃	#3076
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
1. #14583452	C₂	semidirect
3. #14583209	C₂	semidirect
4. #14583205	C₂	semidirect
5. #14583202	C₂	semidirect
6. #14583198	C₂	semidirect
7. #14583193	C₂	semidirect
8. #14583188	C₂	semidirect
9. #14583183	C₂	semidirect
10. #14583178	C₂	semidirect
11. #14582935	C₂	semidirect
12. C₂×Dic₄₃⋊C₈	C₂	semidirect
15. C₄₃×#427	C₂	semidirect
64. #10621898	Q₈	semidirect
177. #427	D₄₃	semidirect
181. C₄₃	#3076	semidirect