Subgroups — #19147277

Order 5504.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583626	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583618	C₂	1	1	yes	maximal, characteristic
3	#14583252	C₂	1	1	yes	maximal, characteristic
4	#14583248	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583208	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583203	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583177	C₂	1	1	yes	maximal, characteristic
8	#14583120	C₂	1	1	yes	maximal, characteristic
9	#14583118	C₂	1	1	yes	maximal, characteristic
10	#14583108	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582991	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582990	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#488	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
17	C₂×#10621885	C₂²	1	1	yes	characteristic
18	C₂×#10621883	C₂²	1	1	yes	characteristic
19	#13063324	—	2	1	no	—
20	#13063324	C₂²	1	1	yes	characteristic
21	#13063323	C₂²	1	1	yes	characteristic
22	#13063318	—	2	1	no	—
23	#13063318	—	2	1	no	—
24	#13063316	C₂²	1	1	yes	characteristic
25	#13063315	—	2	1	no	—
26	#13063315	—	2	1	no	—
27	#13063312	C₂²	1	1	yes	characteristic
28	#13063302	—	2	1	no	—
29	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
30	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
31	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
32	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
33	#13063290	—	2	1	no	—
34	#13063290	C₂²	1	1	yes	characteristic
35	#13063290	C₂²	1	1	yes	characteristic
36	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
37	#13063281	—	2	1	no	—
38	#13063259	C₂²	1	1	yes	characteristic
39	#13063258	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	C₄⋊(C₄₃⋊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	#13063249	C₂²	1	1	yes	characteristic
47	#13063248	C₂²	1	1	yes	characteristic
48	#13063248	C₂²	1	1	yes	characteristic
49	#13063246	C₂²	1	1	yes	characteristic
50	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	#125×C₄₃	C₂²	1	1	yes	characteristic
53	#124×C₄₃	C₂²	1	1	yes	characteristic
54	#103×C₄₃	C₂²	1	1	yes	characteristic
55	#103×C₄₃	C₂²	1	1	yes	characteristic
56	#102×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₄₃	—	2	1	no	—
62	C₂×Dic₈₆	—	2	1	no	—
63	C₂×Dic₈₆	D₄	1	1	yes	characteristic
64	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
65	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₄₃	—	2	1	no	—
69	C₄⋊C₄×C₄₃	—	2	1	no	—
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⋊C₄×C₄₃	—	2	1	no	—
74	C₄²×C₄₃	C₂³	1	1	yes	characteristic
75	C₂²⋊Dic₄₃	—	2	1	no	—
76	C₂²⋊Dic₄₃	—	2	1	no	—
77	C₂²⋊Dic₄₃	—	2	1	no	—
78	#10621885	—	2	2	no	—
79	#10621883	—	2	2	no	—
80	C₄⋊Dic₄₃	—	2	1	no	—
81	C₄⋊Dic₄₃	—	2	1	no	—
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	Dic₄₃⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
94	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₄₃	—	2	1	no	—
96	C₄×Dic₄₃	—	2	1	no	—
97	C₄×Dic₄₃	—	2	1	no	—
98	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
101	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
104	Dic₁₇₂	—	2	2	no	—
105	C₂³×C₄₃	—	2	1	no	—
106	Q₈×C₄₃	—	2	1	no	—
107	Q₈×C₄₃	—	2	1	no	—
108	D₄×C₄₃	—	2	1	no	—
109	D₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	4	2	no	—
121	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
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	Dic₈₆	—	2	1	no	—
126	Dic₈₆	—	2	2	no	—
127	Dic₈₆	—	2	1	no	—
128	Dic₈₆	—	2	1	no	—
129	Dic₈₆	—	4	2	no	—
130	Dic₈₆	#41	1	2	yes	—
131	C₈×C₄₃	—	2	1	no	—
132	C₈×C₄₃	—	2	1	no	—
133	C₄₃⋊C₈	—	2	1	no	—
134	C₄₃⋊C₈	—	2	1	no	—
135	C₂²×C₄₃	—	4	1	no	—
136	C₂²×C₄₃	—	4	1	no	—
137	C₂²×C₄₃	—	4	1	no	—
138	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₄₃	—	2	1	no	—
140	C₄×C₄₃	—	2	1	no	—
141	C₄×C₄₃	—	4	1	no	—
142	C₄×C₄₃	—	4	1	no	—
143	C₄×C₄₃	—	4	1	no	—
144	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
145	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
146	Dic₄₃	—	2	2	no	—
147	Dic₄₃	—	2	1	no	—
148	Dic₄₃	—	2	1	no	—
149	Dic₄₃	—	2	1	no	—
150	Dic₄₃	—	4	1	no	—
151	Dic₄₃	—	4	1	no	—
152	Dic₄₃	—	4	1	no	—
153	Dic₄₃	#135	1	2	yes	—
154	#3364	—	43	1	no	maximal, Sylow
155	C₂×C₄₃	—	8	1	no	—
156	C₂×C₄₃	#548	1	1	yes	characteristic
157	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
158	C₂×C₄₃	#578	1	1	yes	characteristic
159	#557	—	43	1	no	—
160	#556	—	43	1	no	—
161	#488	—	43	1	no	—
162	#488	D₄₃	1	1	yes	characteristic
163	#487	—	43	1	no	—
164	#479	—	43	1	no	—
165	#479	—	43	1	no	—
166	#474	—	43	1	no	—
167	#464	—	43	1	no	—
168	#464	—	43	1	no	—
169	#462	—	43	1	no	—
170	C₈×Q₈	—	43	1	no	—
171	C₄×Q₁₆	—	43	1	no	—
172	C₄×Q₁₆	—	43	1	no	—
173	C₄×SD₁₆	—	43	1	no	—
174	C₄₃	#3364	1	1	yes	characteristic, complemented, Sylow
175	C₂×Q₁₆	—	43	1	no	—
176	C₂×Q₁₆	—	43	1	no	—
177	C₂×SD₁₆	—	43	1	no	—
178	C₄⋊Q₈	—	43	1	no	—
179	C₄⋊Q₈	—	43	1	no	—
180	C₄⋊Q₈	—	86	1	no	—
181	#126	—	86	1	no	—
182	#126	—	86	1	no	—
183	#125	—	43	1	no	—
184	#125	—	86	1	no	—
185	#125	—	86	1	no	—
186	#125	C₂×D₄₃	1	1	yes	characteristic
187	#124	—	43	1	no	—
188	#124	C₂×D₄₃	1	1	yes	characteristic
189	C₂²⋊Q₈	—	86	1	no	—
190	C₄×Q₈	—	43	1	no	—
191	C₄×Q₈	—	43	1	no	—
192	C₄×Q₈	—	43	1	no	—
193	C₄×Q₈	—	43	1	no	—
194	C₄×Q₈	—	43	1	no	—
195	C₄×Q₈	—	43	1	no	—
196	C₄×Q₈	—	86	1	no	—
197	C₄×D₄	—	43	1	no	—
198	#117	—	86	1	no	—
199	#107	—	43	1	no	—
200	#107	—	43	1	no	—
201	#106	—	43	1	no	—
202	C₄⋊C₈	—	43	1	no	—
203	C₄⋊C₈	—	43	1	no	—
204	C₄⋊C₈	—	43	1	no	—
205	#103	—	43	1	no	—
206	#103	—	43	1	no	—
207	#103	—	43	1	no	—
208	#103	—	43	1	no	—
209	#103	—	43	1	no	—
210	#103	C₂×D₄₃	1	1	yes	characteristic
211	#103	C₂×D₄₃	1	1	yes	characteristic
212	#102	—	43	1	no	—
213	#102	C₂×D₄₃	1	1	yes	characteristic
214	#102	C₂×D₄₃	1	1	yes	characteristic
215	C₄×C₈	—	43	1	no	—
216	C₄×C₈	—	43	1	no	—
217	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
218	C₂×Q₈	—	43	1	no	—
219	C₂×Q₈	—	43	1	no	—
220	C₂×Q₈	—	43	1	no	—
221	C₂×Q₈	—	86	1	no	—
222	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
223	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
224	C₂²×C₄	—	86	1	no	—
225	Q₁₆	—	86	2	no	—
226	Q₁₆	—	86	2	no	—
227	SD₁₆	—	86	2	no	—
228	C₂×C₈	—	43	1	no	—
229	C₂×C₈	—	43	1	no	—
230	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	43	1	no	—
235	C₄⋊C₄	—	43	1	no	—
236	C₄⋊C₄	—	43	1	no	—
237	C₄⋊C₄	—	43	1	no	—
238	C₄⋊C₄	—	43	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	—	86	1	no	—
244	C₄⋊C₄	—	86	1	no	—
245	C₄⋊C₄	—	86	1	no	—
246	C₄⋊C₄	—	86	1	no	—
247	C₄⋊C₄	—	86	1	no	—
248	C₄⋊C₄	—	86	1	no	—
249	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	86	1	no	—
254	C₂²⋊C₄	—	86	1	no	—
255	C₂²⋊C₄	—	86	1	no	—
256	C₄²	—	43	1	no	—
257	C₄²	—	43	1	no	—
258	C₄²	—	86	1	no	—
259	C₄²	—	86	1	no	—
260	C₄²	—	86	1	no	—
261	C₄²	—	86	1	no	—
262	C₄²	—	86	1	no	—
263	C₄²	C₂²×D₄₃	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	43	2	no	—
268	Q₈	—	86	2	no	—
269	Q₈	—	86	1	no	—
270	Q₈	—	86	1	no	—
271	Q₈	—	86	1	no	—
272	Q₈	—	172	2	no	—
273	D₄	—	2	1	no	—
274	D₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	43	1	no	—
279	C₂×C₄	—	43	1	no	—
280	C₂×C₄	—	43	1	no	—
281	C₂×C₄	—	43	1	no	—
282	C₂×C₄	—	86	1	no	—
283	C₂×C₄	—	86	1	no	—
284	C₂×C₄	—	86	1	no	—
285	C₂×C₄	—	86	1	no	—
286	C₂×C₄	—	172	2	no	—
287	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
288	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	86	1	no	—
293	C₈	—	86	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	43	2	no	—
304	C₄	—	86	2	no	—
305	C₄	—	86	1	no	—
306	C₄	—	86	1	no	—
307	C₄	—	86	1	no	—
308	C₄	—	172	1	no	—
309	C₄	—	172	1	no	—
310	C₄	—	172	1	no	—
311	C₄	#41×D₄₃	1	1	yes	characteristic
312	C₄	#13063372	1	1	yes	characteristic
313	C₂	—	8	1	no	—
314	C₂	#14583557	1	1	yes	characteristic
315	C₂	#135×D₄₃	1	1	yes	characteristic
316	C₂	#14583713	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#488	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#488	D₄₃
hypocenter	C₄₃	#3364
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #14583626	C₂	semidirect
4. #14583248	C₂	semidirect
5. #14583208	C₂	semidirect
6. #14583203	C₂	semidirect
10. #14583108	C₂	semidirect
11. C₈×Dic₈₆	C₂	semidirect
12. C₄×Dic₁₇₂	C₂	semidirect
13. #14582991	C₂	semidirect
174. C₄₃	#3364	semidirect