Subgroups — #19147247

Order 5504.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583621	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583605	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583256	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583255	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583248	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583205	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583188	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583117	C₂	1	1	yes	maximal, characteristic
9	#14583115	C₂	1	1	yes	maximal, characteristic
10	#14583103	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583023	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#10621884	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582986	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582981	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621873	C₂²	1	1	yes	characteristic
17	C₂×#10621884	C₂²	1	1	yes	characteristic
18	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063323	C₂²	1	1	yes	characteristic
21	#13063320	—	2	1	no	—
22	#13063319	—	2	1	no	—
23	#13063316	C₂²	1	1	yes	characteristic
24	#13063314	C₂²	1	1	yes	characteristic
25	#13063313	C₂²	1	1	yes	characteristic
26	#13063305	—	2	1	no	—
27	#13063303	—	2	1	no	—
28	#13063303	—	2	1	no	—
29	#13063303	—	2	1	no	—
30	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
31	#13063290	C₂²	1	1	yes	characteristic
32	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
33	#13063287	C₂²	1	1	yes	characteristic
34	#13063287	C₂²	1	1	yes	characteristic
35	#13063287	C₂²	1	1	yes	characteristic
36	#13063282	—	2	1	no	—
37	C₄⋊C₄×D₄₃	—	2	1	no	—
38	#13063256	C₂²	1	1	yes	characteristic
39	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
41	#13063250	C₂²	1	1	yes	characteristic
42	#13063249	C₂²	1	1	yes	characteristic
43	#13063248	C₂²	1	1	yes	characteristic
44	#13063248	C₂²	1	1	yes	characteristic
45	#13063247	C₂²	1	1	yes	characteristic
46	#13063245	C₂²	1	1	yes	characteristic
47	#13063245	C₂²	1	1	yes	characteristic
48	#13063245	C₂²	1	1	yes	characteristic
49	C₄×C₄₃⋊C₈	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	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
54	#107×C₄₃	C₂²	1	1	yes	characteristic
55	#106×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×D₁₇₂	C₂³	1	1	yes	characteristic
61	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₄₃	—	2	1	no	—
63	C₂×C₄×D₄₃	—	2	1	no	—
64	C₂×C₄×D₄₃	—	2	1	no	—
65	C₂×C₄×D₄₃	—	2	1	no	—
66	C₂×Dic₈₆	—	2	1	no	—
67	C₂×Dic₈₆	D₄	1	1	yes	characteristic
68	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₄₃	—	2	1	no	—
72	C₄⋊C₄×C₄₃	—	2	1	no	—
73	C₄⋊C₄×C₄₃	—	2	1	no	—
74	C₄⋊C₄×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	C₄²×C₄₃	—	2	1	no	—
78	C₄²×C₄₃	C₂³	1	1	yes	characteristic
79	#10621884	—	2	2	no	—
80	(C₂×D₄₃)⋊C₄	—	2	1	no	—
81	(C₂×D₄₃)⋊C₄	—	2	1	no	—
82	(C₂×D₄₃)⋊C₄	—	2	1	no	—
83	(C₂×D₄₃)⋊C₄	—	2	1	no	—
84	(C₂×D₄₃)⋊C₄	—	2	1	no	—
85	(C₂×D₄₃)⋊C₄	—	2	1	no	—
86	(C₂×D₄₃)⋊C₄	—	2	1	no	—
87	(C₂×D₄₃)⋊C₄	—	2	1	no	—
88	C₄⋊Dic₄₃	—	2	1	no	—
89	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
90	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
97	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
100	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
101	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
103	D₃₄₄	—	2	2	no	—
104	#10621873	—	2	2	no	—
105	C₂²×D₄₃	—	2	1	no	—
106	C₂²×D₄₃	—	2	1	no	—
107	Q₈×C₄₃	—	2	1	no	—
108	Q₈×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	—	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	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	D₁₇₂	—	2	1	no	—
123	D₁₇₂	—	2	1	no	—
124	D₁₇₂	—	2	1	no	—
125	D₁₇₂	#41	1	2	yes	—
126	C₄×D₄₃	—	4	2	no	—
127	C₄×D₄₃	—	4	2	no	—
128	C₄×D₄₃	—	8	1	no	—
129	C₄×D₄₃	—	8	1	no	—
130	Dic₈₆	—	2	2	no	—
131	Dic₈₆	—	2	1	no	—
132	Dic₈₆	—	2	1	no	—
133	Dic₈₆	—	8	1	no	—
134	C₈×C₄₃	—	2	1	no	—
135	C₈×C₄₃	—	2	1	no	—
136	C₄₃⋊C₈	—	2	1	no	—
137	C₄₃⋊C₈	—	2	1	no	—
138	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₄₃	—	2	2	no	—
140	C₂×D₄₃	—	4	1	no	—
141	C₂×D₄₃	—	4	1	no	—
142	C₂×D₄₃	—	4	1	no	—
143	C₂×D₄₃	—	4	1	no	—
144	C₂×D₄₃	—	4	1	no	—
145	C₄×C₄₃	—	2	1	no	—
146	C₄×C₄₃	—	2	1	no	—
147	C₄×C₄₃	—	4	1	no	—
148	C₄×C₄₃	—	4	1	no	—
149	C₄×C₄₃	—	4	1	no	—
150	C₄×C₄₃	—	4	1	no	—
151	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
152	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₄₃	—	2	1	no	—
154	Dic₄₃	—	2	1	no	—
155	Dic₄₃	—	2	1	no	—
156	Dic₄₃	—	4	1	no	—
157	Dic₄₃	—	4	1	no	—
158	Dic₄₃	#135	1	2	yes	—
159	#3360	—	43	1	no	maximal, Sylow
160	C₂×C₄₃	#548	1	1	yes	characteristic
161	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
162	C₂×C₄₃	#577	1	1	yes	characteristic
163	D₄₃	—	4	2	no	—
164	D₄₃	—	8	1	no	—
165	#556	—	43	1	no	—
166	#554	—	43	1	no	—
167	#488	—	43	1	no	—
168	#488	—	43	1	no	—
169	#487	—	43	1	no	—
170	#479	—	43	1	no	—
171	#479	D₄₃	1	1	yes	characteristic, complemented
172	#476	—	43	1	no	—
173	#463	—	43	1	no	—
174	#463	—	43	1	no	—
175	#461	—	43	1	no	—
176	C₈×Q₈	—	43	1	no	—
177	C₄×SD₁₆	—	43	1	no	—
178	C₄×SD₁₆	—	43	1	no	—
179	C₄×D₈	—	43	1	no	—
180	C₄₃	#3360	1	1	yes	characteristic, complemented, Sylow
181	C₂×SD₁₆	—	43	1	no	—
182	C₂×SD₁₆	—	43	1	no	—
183	C₂×D₈	—	43	1	no	—
184	C₄⋊Q₈	—	43	1	no	—
185	C₄⋊Q₈	—	43	1	no	—
186	#126	—	86	1	no	—
187	#126	—	86	1	no	—
188	#125	—	43	1	no	—
189	#125	C₂×D₄₃	1	1	yes	characteristic
190	#124	—	43	1	no	—
191	#124	—	43	1	no	—
192	C₂²⋊Q₈	—	86	1	no	—
193	C₂²⋊Q₈	—	86	1	no	—
194	C₂²⋊Q₈	—	86	1	no	—
195	C₂²⋊Q₈	—	86	1	no	—
196	C₄×Q₈	—	43	1	no	—
197	C₄×Q₈	—	43	1	no	—
198	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
199	C₄×D₄	—	43	1	no	—
200	C₄×D₄	—	43	1	no	—
201	C₄×D₄	—	43	1	no	—
202	C₄×D₄	—	43	1	no	—
203	#117	—	86	1	no	—
204	C₂×C₄⋊C₄	—	86	1	no	—
205	#107	C₂×D₄₃	1	1	yes	characteristic
206	#106	—	43	1	no	—
207	#106	C₂×D₄₃	1	1	yes	characteristic
208	C₄⋊C₈	—	43	1	no	—
209	C₄⋊C₈	—	43	1	no	—
210	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
211	#103	—	43	1	no	—
212	#103	—	43	1	no	—
213	#103	—	43	1	no	—
214	#103	—	43	1	no	—
215	#103	C₂×D₄₃	1	1	yes	characteristic
216	#103	C₂×D₄₃	1	1	yes	characteristic
217	#102	—	43	1	no	—
218	#102	—	43	1	no	—
219	#102	—	43	1	no	—
220	#102	—	43	1	no	—
221	C₄×C₈	—	43	1	no	—
222	C₄×C₈	—	43	1	no	—
223	C₄×C₈	—	43	1	no	—
224	C₂×Q₈	—	43	1	no	—
225	C₂×Q₈	—	43	1	no	—
226	C₂×Q₈	—	86	1	no	—
227	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
228	C₂×D₄	—	43	1	no	—
229	C₂×D₄	—	43	1	no	—
230	C₂²×C₄	—	86	1	no	—
231	C₂²×C₄	—	86	1	no	—
232	C₂²×C₄	—	86	1	no	—
233	C₂²×C₄	—	86	1	no	—
234	SD₁₆	—	86	2	no	—
235	SD₁₆	—	86	2	no	—
236	D₈	—	86	2	no	—
237	C₂×C₈	—	43	1	no	—
238	C₂×C₈	—	43	1	no	—
239	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	43	1	no	—
245	C₄⋊C₄	—	43	1	no	—
246	C₄⋊C₄	—	43	1	no	—
247	C₄⋊C₄	—	43	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	C₂²⋊C₄	—	86	1	no	—
258	C₂²⋊C₄	—	86	1	no	—
259	C₂²⋊C₄	—	86	1	no	—
260	C₂²⋊C₄	—	86	1	no	—
261	C₂²⋊C₄	—	86	1	no	—
262	C₂²⋊C₄	—	86	1	no	—
263	C₂²⋊C₄	—	86	1	no	—
264	C₂²⋊C₄	—	86	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	43	1	no	—
267	C₄²	—	43	1	no	—
268	C₄²	—	86	1	no	—
269	C₄²	C₂²×D₄₃	1	1	yes	characteristic
270	C₂³	—	86	1	no	—
271	C₂³	—	86	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	86	2	no	—
275	Q₈	—	86	1	no	—
276	Q₈	—	86	1	no	—
277	Q₈	—	344	1	no	—
278	D₄	—	43	2	no	—
279	D₄	—	86	1	no	—
280	D₄	—	86	1	no	—
281	D₄	—	86	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	43	1	no	—
287	C₂×C₄	—	43	1	no	—
288	C₂×C₄	—	43	1	no	—
289	C₂×C₄	—	43	1	no	—
290	C₂×C₄	—	86	1	no	—
291	C₂×C₄	—	86	1	no	—
292	C₂×C₄	—	344	1	no	—
293	C₂×C₄	—	344	1	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	172	2	no	—
296	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
297	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	86	1	no	—
302	C₈	—	86	1	no	—
303	C₂²	—	86	2	no	—
304	C₂²	—	172	1	no	—
305	C₂²	—	172	1	no	—
306	C₂²	—	172	1	no	—
307	C₂²	—	172	1	no	—
308	C₂²	—	172	1	no	—
309	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
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	1	no	—
318	C₄	—	86	1	no	—
319	C₄	—	86	1	no	—
320	C₄	—	172	1	no	—
321	C₄	—	172	1	no	—
322	C₄	#41×D₄₃	1	1	yes	characteristic
323	C₄	#13063373	1	1	yes	characteristic
324	C₂	—	344	1	no	—
325	C₂	—	172	2	no	—
326	C₂	#14583559	1	1	yes	characteristic
327	C₂	#135×D₄₃	1	1	yes	characteristic
328	C₂	#14583708	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#479	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#479	D₄₃
hypocenter	C₄₃	#3360
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #14583621	C₂	semidirect
2. #14583605	C₂	semidirect
3. #14583256	C₂	semidirect
4. #14583255	C₂	semidirect
5. #14583248	C₂	semidirect
6. #14583205	C₂	semidirect
7. #14583188	C₂	semidirect
10. #14583103	C₂	semidirect
11. #14583023	C₂	semidirect
12. C₄×#10621884	C₂	semidirect
13. #14582986	C₂	semidirect
15. C₄₃×#479	C₂	semidirect
171. #479	D₄₃	semidirect
180. C₄₃	#3360	semidirect