Subgroups — #19145867

Order 5504.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583530	C₂	1	1	yes	maximal, characteristic
2	#14583509	C₂	1	1	yes	maximal, characteristic
3	#14583307	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583300	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583266	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583248	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583241	C₂	1	1	yes	maximal, characteristic
8	#14583237	C₂	1	1	yes	maximal, characteristic
9	#14583229	C₂	1	1	yes	maximal, characteristic
10	#14583208	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583205	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583193	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583182	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582957	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#484	C₂	1	1	yes	maximal, characteristic
16	#13063324	—	2	1	no	—
17	#13063324	C₂²	1	1	yes	characteristic
18	#13063324	C₂²	1	1	yes	characteristic
19	#13063323	C₂²	1	1	yes	characteristic
20	#13063318	—	2	1	no	—
21	#13063318	—	2	1	no	—
22	#13063316	—	2	1	no	—
23	#13063316	—	2	1	no	—
24	#13063316	C₂²	1	1	yes	characteristic
25	#13063316	C₂²	1	1	yes	characteristic
26	#13063316	C₂²	1	1	yes	characteristic
27	#13063315	—	2	1	no	—
28	#13063304	—	2	1	no	—
29	#13063304	—	2	1	no	—
30	#13063304	C₂²	1	1	yes	characteristic
31	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
32	#13063290	C₂²	1	1	yes	characteristic
33	#13063284	C₂²	1	1	yes	characteristic
34	#13063284	C₂²	1	1	yes	characteristic
35	#13063259	C₂²	1	1	yes	characteristic
36	#13063258	C₂²	1	1	yes	characteristic
37	#13063257	C₂²	1	1	yes	characteristic
38	#13063256	C₂²	1	1	yes	characteristic
39	#13063256	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
43	#13063250	C₂²	1	1	yes	characteristic
44	#13063250	C₂²	1	1	yes	characteristic
45	#13063249	C₂²	1	1	yes	characteristic
46	#13063249	C₂²	1	1	yes	characteristic
47	#13063248	C₂²	1	1	yes	characteristic
48	#13063248	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
54	#107×C₄₃	C₂²	1	1	yes	characteristic
55	#107×C₄₃	C₂²	1	1	yes	characteristic
56	#103×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×Dic₈₆	—	2	1	no	—
63	C₂×Dic₈₆	—	2	1	no	—
64	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₄₃	—	2	1	no	—
68	C₄⋊C₄×C₄₃	—	2	1	no	—
69	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₄₃	—	2	1	no	—
73	C₂²⋊Dic₄₃	—	2	1	no	—
74	C₂²⋊Dic₄₃	—	2	1	no	—
75	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
76	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
77	C₄⋊Dic₄₃	—	2	1	no	—
78	C₄⋊Dic₄₃	—	2	1	no	—
79	C₄⋊Dic₄₃	—	2	1	no	—
80	C₄⋊Dic₄₃	—	2	1	no	—
81	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
82	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
83	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
86	Dic₄₃⋊C₄	—	2	1	no	—
87	Dic₄₃⋊C₄	—	2	1	no	—
88	Dic₄₃⋊C₄	—	2	1	no	—
89	Dic₄₃⋊C₄	—	2	1	no	—
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	C₄×Dic₄₃	—	2	1	no	—
96	C₄×Dic₄₃	—	2	1	no	—
97	C₄×Dic₄₃	—	2	1	no	—
98	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
100	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
102	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₄₃	—	2	1	no	—
104	Q₈×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	1	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	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	—	8	1	no	—
112	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
114	C₂×Dic₄₃	—	2	1	no	—
115	C₂×Dic₄₃	—	2	1	no	—
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
122	Dic₈₆	—	2	1	no	—
123	Dic₈₆	—	2	1	no	—
124	Dic₈₆	—	8	1	no	—
125	Dic₈₆	—	8	1	no	—
126	C₈×C₄₃	—	2	1	no	—
127	C₈×C₄₃	—	2	1	no	—
128	C₄₃⋊C₈	—	2	1	no	—
129	C₄₃⋊C₈	—	2	1	no	—
130	C₂²×C₄₃	—	2	1	no	—
131	C₂²×C₄₃	—	2	1	no	—
132	C₂²×C₄₃	—	2	1	no	—
133	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₄×C₄₃	—	2	1	no	—
135	C₄×C₄₃	—	4	1	no	—
136	C₄×C₄₃	—	4	1	no	—
137	C₄×C₄₃	—	4	1	no	—
138	C₄×C₄₃	—	4	1	no	—
139	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
140	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
141	Dic₄₃	—	2	1	no	—
142	Dic₄₃	—	2	1	no	—
143	Dic₄₃	—	2	1	no	—
144	Dic₄₃	—	2	1	no	—
145	Dic₄₃	—	4	1	no	—
146	Dic₄₃	—	4	1	no	—
147	Dic₄₃	—	4	1	no	—
148	Dic₄₃	—	4	1	no	—
149	#3242	—	43	1	no	maximal, Sylow
150	C₂×C₄₃	—	4	1	no	—
151	C₂×C₄₃	#536	1	1	yes	characteristic
152	C₂×C₄₃	#577	1	1	yes	characteristic
153	C₂×C₄₃	#578	1	1	yes	characteristic
154	#544	—	43	1	no	—
155	#541	—	43	1	no	—
156	#501	—	43	1	no	—
157	#499	—	43	1	no	—
158	#491	—	43	1	no	—
159	#487	—	43	1	no	—
160	#485	—	43	1	no	—
161	#485	—	43	1	no	—
162	#484	D₄₃	1	1	yes	characteristic
163	#483	—	43	1	no	—
164	#479	—	43	1	no	—
165	#479	—	43	1	no	—
166	#477	—	43	1	no	—
167	#475	—	43	1	no	—
168	#433	—	43	1	no	—
169	C₄₃	#3242	1	1	yes	characteristic, complemented, Sylow
170	C₄⋊Q₈	—	43	1	no	—
171	C₄⋊Q₈	—	43	1	no	—
172	C₄⋊Q₈	—	43	1	no	—
173	C₄⋊Q₈	—	86	1	no	—
174	#126	—	86	1	no	—
175	#126	—	86	1	no	—
176	#125	—	43	1	no	—
177	#125	—	43	1	no	—
178	#125	—	43	1	no	—
179	#125	—	86	1	no	—
180	#125	—	86	1	no	—
181	#125	—	86	1	no	—
182	C₂²⋊Q₈	—	43	1	no	—
183	C₂²⋊Q₈	—	86	1	no	—
184	C₂²⋊Q₈	—	86	1	no	—
185	C₂²⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
186	C₄×Q₈	—	43	1	no	—
187	C₄×Q₈	—	43	1	no	—
188	#117	—	43	1	no	—
189	#117	—	43	1	no	—
190	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
191	#107	—	43	1	no	—
192	#107	—	43	1	no	—
193	#107	C₂×D₄₃	1	1	yes	characteristic
194	#107	C₂×D₄₃	1	1	yes	characteristic
195	#106	—	43	1	no	—
196	#106	—	43	1	no	—
197	#106	—	43	1	no	—
198	#106	—	43	1	no	—
199	C₄⋊C₈	—	43	1	no	—
200	C₄⋊C₈	—	43	1	no	—
201	#103	—	43	1	no	—
202	#103	—	43	1	no	—
203	#103	—	43	1	no	—
204	#103	—	43	1	no	—
205	#103	—	43	1	no	—
206	#103	—	43	1	no	—
207	#103	C₂×D₄₃	1	1	yes	characteristic
208	#103	C₂×D₄₃	1	1	yes	characteristic
209	C₂²⋊C₈	—	43	1	no	—
210	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
211	#97	—	43	1	no	—
212	C₄×C₈	—	43	1	no	—
213	C₂×Q₈	—	43	1	no	—
214	C₂×Q₈	—	86	1	no	—
215	C₂×Q₈	—	86	1	no	—
216	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
219	C₂×C₈	—	43	1	no	—
220	C₂×C₈	—	43	1	no	—
221	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	43	1	no	—
226	C₄⋊C₄	—	43	1	no	—
227	C₄⋊C₄	—	43	1	no	—
228	C₄⋊C₄	—	43	1	no	—
229	C₄⋊C₄	—	43	1	no	—
230	C₄⋊C₄	—	86	1	no	—
231	C₄⋊C₄	—	86	1	no	—
232	C₄⋊C₄	—	86	1	no	—
233	C₄⋊C₄	—	86	1	no	—
234	C₄⋊C₄	—	86	1	no	—
235	C₄⋊C₄	—	86	1	no	—
236	C₄⋊C₄	—	86	1	no	—
237	C₄⋊C₄	—	86	1	no	—
238	C₄⋊C₄	—	86	1	no	—
239	C₄⋊C₄	—	86	1	no	—
240	C₄⋊C₄	—	86	1	no	—
241	C₄⋊C₄	—	86	1	no	—
242	C₄⋊C₄	—	86	1	no	—
243	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
244	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	43	1	no	—
248	C₂²⋊C₄	—	43	1	no	—
249	C₂²⋊C₄	—	86	1	no	—
250	C₂²⋊C₄	—	86	1	no	—
251	C₄²	—	43	1	no	—
252	C₄²	—	43	1	no	—
253	C₄²	—	86	1	no	—
254	C₄²	—	86	1	no	—
255	C₄²	—	86	1	no	—
256	C₂³	D₄×D₄₃	1	1	yes	characteristic
257	Q₈	—	2	1	no	—
258	Q₈	—	2	1	no	—
259	Q₈	—	86	1	no	—
260	Q₈	—	86	1	no	—
261	Q₈	—	344	1	no	—
262	Q₈	—	344	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	43	1	no	—
271	C₂×C₄	—	43	1	no	—
272	C₂×C₄	—	43	1	no	—
273	C₂×C₄	—	43	1	no	—
274	C₂×C₄	—	86	1	no	—
275	C₂×C₄	—	86	1	no	—
276	C₂×C₄	—	86	1	no	—
277	C₂×C₄	—	86	1	no	—
278	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	86	1	no	—
283	C₈	—	86	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	86	1	no	—
294	C₄	—	86	1	no	—
295	C₄	—	86	1	no	—
296	C₄	—	86	1	no	—
297	C₄	—	172	1	no	—
298	C₄	—	172	1	no	—
299	C₄	—	172	1	no	—
300	C₄	—	172	1	no	—
301	C₄	#13063372	1	1	yes	characteristic
302	C₄	#13063372	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	#14583471	1	1	yes	characteristic
305	C₂	#14583709	1	1	yes	characteristic
306	C₂	#14583712	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#484	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#484	D₄₃
hypocenter	C₄₃	#3242
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
3. #14583307	C₂	semidirect
4. #14583300	C₂	semidirect
5. #14583266	C₂	semidirect
6. #14583248	C₂	semidirect
10. #14583208	C₂	semidirect
11. #14583205	C₂	semidirect
12. #14583193	C₂	semidirect
13. #14583182	C₂	semidirect
169. C₄₃	#3242	semidirect