Subgroups — #19145263

Order 5504.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583520	C₂	1	1	yes	maximal, characteristic
2	#14583511	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583308	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583293	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583254	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583250	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583157	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583145	C₂	1	1	yes	maximal, characteristic
9	#14583140	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583128	C₂	1	1	yes	maximal, characteristic
11	#14583007	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583001	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#10621873	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
18	C₂×#10621873	C₂²	1	1	yes	characteristic
19	C₂×#10621873	C₂²	1	1	yes	characteristic
20	#13063325	C₂²	1	1	yes	characteristic
21	#13063325	C₂²	1	1	yes	characteristic
22	#13063320	—	2	1	no	—
23	#13063318	—	2	1	no	—
24	#13063318	—	2	1	no	—
25	#13063317	C₂²	1	1	yes	characteristic
26	#13063315	—	2	1	no	—
27	#13063315	—	2	1	no	—
28	#13063313	C₂²	1	1	yes	characteristic
29	#13063311	—	2	1	no	—
30	#13063307	—	2	1	no	—
31	#13063304	C₂²	1	1	yes	characteristic
32	#13063304	C₂²	1	1	yes	characteristic
33	#13063304	C₂²	1	1	yes	characteristic
34	#13063303	—	2	1	no	—
35	#13063299	C₂²	1	1	yes	characteristic
36	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
37	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
38	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
39	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
40	#13063259	C₂²	1	1	yes	characteristic
41	#13063259	C₂²	1	1	yes	characteristic
42	#13063257	C₂²	1	1	yes	characteristic
43	#13063257	C₂²	1	1	yes	characteristic
44	#13063250	C₂²	1	1	yes	characteristic
45	#13063250	C₂²	1	1	yes	characteristic
46	#13063250	C₂²	1	1	yes	characteristic
47	#13063250	C₂²	1	1	yes	characteristic
48	#13063250	C₂²	1	1	yes	characteristic
49	#13063250	C₂²	1	1	yes	characteristic
50	#13063247	C₂²	1	1	yes	characteristic
51	#13063247	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
54	#117×C₄₃	C₂²	1	1	yes	characteristic
55	#97×C₄₃	C₂²	1	1	yes	characteristic
56	#97×C₄₃	C₂²	1	1	yes	characteristic
57	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₄₃	—	2	1	no	—
61	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₄₃	—	2	1	no	—
63	C₂×Dic₈₆	—	2	1	no	—
64	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
65	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
66	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
67	M₁₆×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
76	C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
77	C₄²×C₄₃	C₂³	1	1	yes	characteristic
78	C₄²×C₄₃	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₄₃	—	2	1	no	—
80	C₂²⋊Dic₄₃	—	2	1	no	—
81	C₂²⋊Dic₄₃	—	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	C₄⋊Dic₄₃	—	2	1	no	—
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	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
100	Dic₄₃⋊C₄	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	—	2	1	no	—
103	C₄×Dic₄₃	—	2	1	no	—
104	Dic₁₇₂	—	2	2	no	—
105	Dic₁₇₂	—	2	2	no	—
106	#10621873	—	2	2	no	—
107	#10621873	—	2	2	no	—
108	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₄₃	—	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×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₄₃	—	8	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	2	1	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	D₁₇₂	—	2	1	no	—
127	D₁₇₂	—	2	1	no	—
128	C₄×D₄₃	—	8	1	no	—
129	Dic₈₆	—	2	1	no	—
130	Dic₈₆	—	2	1	no	—
131	Dic₈₆	—	2	1	no	—
132	Dic₈₆	—	2	1	no	—
133	Dic₈₆	—	2	1	no	—
134	Dic₈₆	—	2	1	no	—
135	Dic₈₆	—	8	1	no	—
136	C₈×C₄₃	—	2	1	no	—
137	C₈×C₄₃	—	2	1	no	—
138	C₈×C₄₃	#41	1	2	yes	—
139	C₈×C₄₃	#41	1	2	yes	—
140	C₂²×C₄₃	—	2	1	no	—
141	C₂²×C₄₃	—	2	1	no	—
142	C₂²×C₄₃	—	2	1	no	—
143	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₄₃	—	4	1	no	—
145	C₂×D₄₃	—	4	1	no	—
146	C₂×D₄₃	—	4	1	no	—
147	C₄×C₄₃	—	2	1	no	—
148	C₄×C₄₃	—	2	1	no	—
149	C₄×C₄₃	—	2	1	no	—
150	C₄×C₄₃	—	2	1	no	—
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
153	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
154	Dic₄₃	—	4	1	no	—
155	Dic₄₃	—	4	1	no	—
156	Dic₄₃	—	4	1	no	—
157	Dic₄₃	—	4	1	no	—
158	Dic₄₃	—	4	1	no	—
159	Dic₄₃	—	4	1	no	—
160	Dic₄₃	—	4	1	no	—
161	#3178	—	43	1	no	maximal, Sylow
162	C₂×C₄₃	—	4	1	no	—
163	C₂×C₄₃	#532	1	1	yes	characteristic
164	C₂×C₄₃	#577	1	1	yes	characteristic
165	C₂×C₄₃	#578	1	1	yes	characteristic
166	D₄₃	—	8	1	no	—
167	#542	—	43	1	no	—
168	#541	—	43	1	no	—
169	#501	—	43	1	no	—
170	#497	—	43	1	no	—
171	#488	—	43	1	no	—
172	#487	—	43	1	no	—
173	#470	—	43	1	no	—
174	#468	—	43	1	no	—
175	#467	—	43	1	no	—
176	#465	—	43	1	no	—
177	#441	—	43	1	no	—
178	#440	—	43	1	no	—
179	C₄×Q₁₆	—	43	1	no	—
180	C₄×SD₁₆	—	43	1	no	—
181	#405	D₄₃	1	1	yes	characteristic, complemented
182	C₄₃	#3178	1	1	yes	characteristic, complemented, Sylow
183	C₂×Q₁₆	—	43	1	no	—
184	C₂×Q₁₆	—	43	1	no	—
185	C₂×SD₁₆	—	43	1	no	—
186	C₂×SD₁₆	—	43	1	no	—
187	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
188	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
189	C₄⋊Q₈	—	43	1	no	—
190	C₄⋊Q₈	—	43	1	no	—
191	#126	—	86	1	no	—
192	#126	—	86	1	no	—
193	#126	—	86	1	no	—
194	#125	—	43	1	no	—
195	#125	—	86	1	no	—
196	#125	—	86	1	no	—
197	#124	—	43	1	no	—
198	#124	—	86	1	no	—
199	#123	—	86	1	no	—
200	C₂²⋊Q₈	—	43	1	no	—
201	C₂²⋊Q₈	—	43	1	no	—
202	C₂²⋊Q₈	—	43	1	no	—
203	C₂²⋊Q₈	—	86	1	no	—
204	#121	—	43	1	no	—
205	C₄×Q₈	—	43	1	no	—
206	C₄×Q₈	—	43	1	no	—
207	C₄×Q₈	—	43	1	no	—
208	C₄×D₄	—	43	1	no	—
209	#117	C₂×D₄₃	1	1	yes	characteristic
210	#107	—	43	1	no	—
211	#107	—	43	1	no	—
212	#106	—	43	1	no	—
213	#106	—	43	1	no	—
214	#103	—	43	1	no	—
215	#103	—	43	1	no	—
216	#103	—	43	1	no	—
217	#103	—	43	1	no	—
218	#103	—	43	1	no	—
219	#103	—	43	1	no	—
220	#102	—	43	1	no	—
221	#102	—	43	1	no	—
222	#97	C₂×D₄₃	1	1	yes	characteristic
223	#97	C₂×D₄₃	1	1	yes	characteristic
224	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
225	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
226	C₂×Q₈	—	43	1	no	—
227	C₂×Q₈	—	43	1	no	—
228	C₂×Q₈	—	43	1	no	—
229	C₂×Q₈	—	86	1	no	—
230	C₂×D₄	—	43	1	no	—
231	C₂×D₄	—	86	1	no	—
232	C₂²×C₄	—	86	1	no	—
233	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
234	Q₁₆	—	86	2	no	—
235	Q₁₆	—	86	2	no	—
236	SD₁₆	—	86	2	no	—
237	SD₁₆	—	86	2	no	—
238	M₁₆	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
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	—	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	—	86	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	D₁₇₂	1	1	yes	characteristic
260	C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
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	C₂²⋊C₄	—	86	1	no	—
266	C₂²⋊C₄	—	86	1	no	—
267	C₂²⋊C₄	—	86	1	no	—
268	C₂²⋊C₄	D₁₇₂	1	1	yes	characteristic
269	C₂²⋊C₄	D₁₇₂	1	1	yes	characteristic
270	C₄²	—	86	1	no	—
271	C₄²	—	86	1	no	—
272	C₄²	—	86	1	no	—
273	C₄²	C₂²×D₄₃	1	1	yes	characteristic
274	C₄²	C₂²×D₄₃	1	1	yes	characteristic
275	C₂³	—	86	1	no	—
276	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
277	Q₈	—	86	1	no	—
278	Q₈	—	86	1	no	—
279	Q₈	—	86	1	no	—
280	Q₈	—	86	1	no	—
281	Q₈	—	86	1	no	—
282	Q₈	—	86	1	no	—
283	Q₈	—	344	1	no	—
284	D₄	—	86	1	no	—
285	D₄	—	86	1	no	—
286	D₄	—	344	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	86	1	no	—
290	C₂×C₄	—	86	1	no	—
291	C₂×C₄	—	86	1	no	—
292	C₂×C₄	—	86	1	no	—
293	C₂×C₄	—	86	1	no	—
294	C₂×C₄	—	86	1	no	—
295	C₂×C₄	—	86	1	no	—
296	C₂×C₄	—	344	1	no	—
297	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
298	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
299	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
300	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
301	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	#10621898	1	2	yes	—
306	C₈	#10621898	1	2	yes	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	172	1	no	—
311	C₂²	—	172	1	no	—
312	C₂²	—	172	1	no	—
313	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	172	1	no	—
320	C₄	—	172	1	no	—
321	C₄	—	172	1	no	—
322	C₄	—	172	1	no	—
323	C₄	—	172	1	no	—
324	C₄	—	172	1	no	—
325	C₄	—	172	1	no	—
326	C₄	C₂×#10621898	1	1	yes	characteristic
327	C₄	C₂×#10621898	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	344	1	no	—
330	C₂	#14583449	1	1	yes	characteristic
331	C₂	#14583711	1	1	yes	characteristic
332	C₂	#14583716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#405	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#405	D₄₃
hypocenter	C₄₃	#3178
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #14583511	C₂	semidirect
3. #14583308	C₂	semidirect
4. #14583293	C₂	semidirect
5. #14583254	C₂	semidirect
6. #14583250	C₂	semidirect
7. #14583157	C₂	semidirect
9. #14583140	C₂	semidirect
11. #14583007	C₂	semidirect
12. #14583001	C₂	semidirect
13. C₄×Dic₁₇₂	C₂	semidirect
14. C₄×#10621873	C₂	semidirect
15. C₄₃×#405	C₂	semidirect
181. #405	D₄₃	semidirect
182. C₄₃	#3178	semidirect