Subgroups — #19144706

Order 5504.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583433	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#10621898	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583241	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583224	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583207	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583200	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583122	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583115	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583063	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583046	C₂	1	1	yes	maximal, characteristic
11	#14583012	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583007	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583001	C₂	1	1	yes	maximal, characteristic, complemented
14	#14583001	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	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	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
21	#13063319	—	2	1	no	—
22	#13063317	C₂²	1	1	yes	characteristic
23	#13063313	C₂²	1	1	yes	characteristic
24	#13063308	—	2	1	no	—
25	#13063304	C₂²	1	1	yes	characteristic
26	#13063299	C₂²	1	1	yes	characteristic
27	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
28	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
29	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
30	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
31	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
32	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₄₃	—	2	1	no	—
34	C₄×C₂²⋊D₄₃	—	2	1	no	—
35	C₄×C₂²⋊D₄₃	—	2	1	no	—
36	#13063284	C₂²	1	1	yes	characteristic
37	#13063282	—	2	1	no	—
38	#13063282	—	2	1	no	—
39	C₄²×D₄₃	—	2	1	no	—
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	#13063247	C₂²	1	1	yes	characteristic
49	#13063247	C₂²	1	1	yes	characteristic
50	#13063247	C₂²	1	1	yes	characteristic
51	#13063247	C₂²	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	#97×C₄₃	C₂²	1	1	yes	characteristic
58	#97×C₄₃	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₃	—	2	1	no	—
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	#10621898	—	2	2	no	—
65	#10621898	—	2	2	no	—
66	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
67	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
73	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄²×C₄₃	D₄	1	1	yes	characteristic
79	C₄²×C₄₃	D₄	1	1	yes	characteristic
80	C₄²×C₄₃	C₂³	1	1	yes	characteristic
81	C₄²×C₄₃	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₄₃	—	2	1	no	—
83	C₂²⋊Dic₄₃	—	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	(C₂×D₄₃)⋊C₄	—	2	1	no	—
89	(C₂×D₄₃)⋊C₄	—	2	1	no	—
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	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	C₄×Dic₄₃	—	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	D₃₄₄	—	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	C₂²×D₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	4	1	no	—
114	C₂×C₄×C₄₃	—	4	1	no	—
115	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₄₃	—	2	4	no	—
122	C₂²⋊D₄₃	—	4	2	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	D₁₇₂	—	2	1	no	—
130	D₁₇₂	—	2	1	no	—
131	D₁₇₂	—	2	1	no	—
132	D₁₇₂	#41	1	2	yes	—
133	C₄×D₄₃	—	2	4	no	—
134	C₄×D₄₃	—	4	2	no	—
135	C₄×D₄₃	—	4	2	no	—
136	C₄×D₄₃	—	4	2	no	—
137	Dic₈₆	—	2	1	no	—
138	Dic₈₆	—	2	1	no	—
139	Dic₈₆	—	2	1	no	—
140	Dic₈₆	#41	1	2	yes	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	—	2	1	no	—
143	C₈×C₄₃	—	2	1	no	—
144	C₈×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₄₃	—	2	2	no	—
150	C₂×D₄₃	—	4	1	no	—
151	C₂×D₄₃	—	4	1	no	—
152	C₂×D₄₃	—	4	1	no	—
153	C₂×D₄₃	—	4	1	no	—
154	C₂×D₄₃	—	4	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	2	1	no	—
160	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
161	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
162	Dic₄₃	—	2	2	no	—
163	Dic₄₃	—	4	1	no	—
164	Dic₄₃	—	4	1	no	—
165	Dic₄₃	—	4	1	no	—
166	Dic₄₃	—	4	1	no	—
167	Dic₄₃	—	4	1	no	—
168	#3111	—	43	1	no	maximal, Sylow
169	C₂×C₄₃	—	4	1	no	—
170	C₂×C₄₃	#525	1	1	yes	characteristic
171	C₂×C₄₃	#575	1	1	yes	characteristic
172	C₂×C₄₃	#575	1	1	yes	characteristic
173	D₄₃	—	4	2	no	—
174	D₄₃	—	8	1	no	—
175	#529	—	43	1	no	—
176	C₄×#41	—	43	1	no	—
177	#485	—	43	1	no	—
178	#482	—	43	1	no	—
179	#479	—	43	1	no	—
180	#478	—	43	1	no	—
181	#464	—	43	1	no	—
182	#463	—	43	1	no	—
183	#452	—	43	1	no	—
184	#449	—	43	1	no	—
185	#442	—	43	1	no	—
186	#441	—	43	1	no	—
187	#440	—	43	1	no	—
188	#440	—	43	1	no	—
189	#432	D₄₃	1	1	yes	characteristic, complemented
190	C₄₃	#3111	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	—	43	1	no	—
192	C₂×Q₁₆	—	43	1	no	—
193	C₂×SD₁₆	—	43	1	no	—
194	C₂×SD₁₆	—	43	1	no	—
195	C₂×D₈	—	43	1	no	—
196	#126	—	86	1	no	—
197	#125	—	43	1	no	—
198	#124	—	43	1	no	—
199	#123	—	86	1	no	—
200	C₂²⋊Q₈	—	43	1	no	—
201	#121	—	43	1	no	—
202	C₄×Q₈	—	43	1	no	—
203	C₄×Q₈	—	43	1	no	—
204	C₄×Q₈	—	43	1	no	—
205	C₄×D₄	—	43	1	no	—
206	C₄×D₄	—	43	1	no	—
207	C₄×D₄	—	43	1	no	—
208	C₄×D₄	—	86	1	no	—
209	C₄×D₄	—	86	1	no	—
210	C₄×D₄	—	86	1	no	—
211	#117	—	43	1	no	—
212	#117	—	86	1	no	—
213	#117	—	86	1	no	—
214	C₂×C₄²	—	86	1	no	—
215	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
216	#107	—	43	1	no	—
217	#107	—	43	1	no	—
218	#106	—	43	1	no	—
219	#106	—	43	1	no	—
220	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
221	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
222	#103	—	43	1	no	—
223	#103	—	43	1	no	—
224	#103	—	43	1	no	—
225	#103	—	43	1	no	—
226	#102	—	43	1	no	—
227	#102	—	43	1	no	—
228	#102	—	43	1	no	—
229	#102	—	43	1	no	—
230	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
231	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
232	#97	C₂×D₄₃	1	1	yes	characteristic
233	#97	C₂×D₄₃	1	1	yes	characteristic
234	#41	—	86	2	no	—
235	#41	—	86	2	no	—
236	C₂×Q₈	—	43	1	no	—
237	C₂×Q₈	—	43	1	no	—
238	C₂×D₄	—	43	1	no	—
239	C₂×D₄	—	43	1	no	—
240	C₂×D₄	—	86	1	no	—
241	C₂×D₄	—	86	1	no	—
242	C₂²×C₄	—	86	1	no	—
243	C₂²×C₄	—	86	1	no	—
244	C₂²×C₄	—	86	1	no	—
245	C₂²×C₄	—	86	1	no	—
246	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
247	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
249	Q₁₆	—	86	2	no	—
250	SD₁₆	—	86	2	no	—
251	SD₁₆	—	86	2	no	—
252	D₈	—	86	2	no	—
253	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
257	C₄⋊C₄	—	43	1	no	—
258	C₄⋊C₄	—	43	1	no	—
259	C₄⋊C₄	—	43	1	no	—
260	C₄⋊C₄	—	43	1	no	—
261	C₄⋊C₄	—	86	1	no	—
262	C₄⋊C₄	—	86	1	no	—
263	C₄⋊C₄	—	86	1	no	—
264	C₄⋊C₄	—	86	1	no	—
265	C₄⋊C₄	—	86	1	no	—
266	C₄⋊C₄	—	86	1	no	—
267	C₂²⋊C₄	—	86	1	no	—
268	C₂²⋊C₄	—	86	1	no	—
269	C₂²⋊C₄	—	86	1	no	—
270	C₂²⋊C₄	—	86	1	no	—
271	C₂²⋊C₄	—	86	1	no	—
272	C₂²⋊C₄	—	86	1	no	—
273	C₂²⋊C₄	—	86	1	no	—
274	C₂²⋊C₄	—	86	1	no	—
275	C₄²	—	86	1	no	—
276	C₄²	—	86	1	no	—
277	C₄²	—	86	1	no	—
278	C₄²	—	86	1	no	—
279	C₄²	D₁₇₂	1	1	yes	characteristic
280	C₄²	D₁₇₂	1	1	yes	characteristic
281	C₄²	C₂²×D₄₃	1	1	yes	characteristic
282	C₄²	C₂²×D₄₃	1	1	yes	characteristic
283	C₂³	—	86	1	no	—
284	C₂³	—	86	1	no	—
285	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
286	Q₈	—	43	2	no	—
287	Q₈	—	86	1	no	—
288	Q₈	—	86	1	no	—
289	Q₈	—	86	1	no	—
290	D₄	—	43	2	no	—
291	D₄	—	86	4	no	—
292	D₄	—	86	1	no	—
293	D₄	—	86	1	no	—
294	D₄	—	86	1	no	—
295	D₄	—	172	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	86	4	no	—
301	C₂×C₄	—	86	1	no	—
302	C₂×C₄	—	86	1	no	—
303	C₂×C₄	—	86	1	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	1	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	172	2	no	—
308	C₂×C₄	—	172	2	no	—
309	C₂×C₄	—	172	2	no	—
310	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
311	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
312	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
313	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
314	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	86	2	no	—
324	C₂²	—	172	1	no	—
325	C₂²	—	172	1	no	—
326	C₂²	—	172	1	no	—
327	C₂²	—	172	1	no	—
328	C₂²	—	172	1	no	—
329	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	86	2	no	—
336	C₄	—	172	1	no	—
337	C₄	—	172	1	no	—
338	C₄	—	172	1	no	—
339	C₄	—	172	1	no	—
340	C₄	—	172	1	no	—
341	C₄	#41×D₄₃	1	1	yes	characteristic
342	C₄	#41×D₄₃	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	344	1	no	—
345	C₂	—	172	2	no	—
346	C₂	#14583411	1	1	yes	characteristic
347	C₂	#14583699	1	1	yes	characteristic
348	C₂	#14583699	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#432	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#432	D₄₃
hypocenter	C₄₃	#3111
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

Complements

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

N	H ≅ G/N	type
1. #14583433	C₂	semidirect
2. C₄×#10621898	C₂	semidirect
3. #14583241	C₂	semidirect
4. #14583224	C₂	semidirect
5. #14583207	C₂	semidirect
6. #14583200	C₂	semidirect
7. #14583122	C₂	semidirect
8. #14583115	C₂	semidirect
9. #14583063	C₂	semidirect
11. #14583012	C₂	semidirect
12. #14583007	C₂	semidirect
13. #14583001	C₂	semidirect
14. #14583001	C₂	semidirect
15. C₄₃×#432	C₂	semidirect
189. #432	D₄₃	semidirect
190. C₄₃	#3111	semidirect