Subgroups — #19146907

Order 5504.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583604	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583548	C₂	1	1	yes	maximal, characteristic
3	#14583248	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583229	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583228	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583187	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583157	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583156	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583108	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583062	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583061	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582990	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582975	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582897	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621900	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
18	C₂×#10621885	C₂²	1	1	yes	characteristic
19	C₂×#10621883	C₂²	1	1	yes	characteristic
20	C₂×#10621872	C₂²	1	1	yes	characteristic
21	#13063324	C₂²	1	1	yes	characteristic
22	#13063323	C₂²	1	1	yes	characteristic
23	#13063315	—	2	1	no	—
24	#13063308	—	2	1	no	—
25	#13063306	—	2	1	no	—
26	#13063305	C₂²	1	1	yes	characteristic
27	#13063304	—	2	1	no	—
28	#13063304	C₂²	1	1	yes	characteristic
29	#13063304	C₂²	1	1	yes	characteristic
30	#13063303	—	2	1	no	—
31	#13063301	C₂²	1	1	yes	characteristic
32	#13063297	—	2	1	no	—
33	#13063290	C₂²	1	1	yes	characteristic
34	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊D₄₃	—	2	1	no	—
36	C₄×C₂²⋊D₄₃	C₂²	1	1	yes	characteristic
37	C₂×Dic₄₃⋊C₄	—	2	1	no	—
38	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
39	#13063257	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	#13063256	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	#13063248	C₂²	1	1	yes	characteristic
47	#13063248	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
55	#103×C₄₃	C₂²	1	1	yes	characteristic
56	#103×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	#102×C₄₃	C₂²	1	1	yes	characteristic
59	#41×C₄₃	—	2	2	no	—
60	#41×C₄₃	—	4	1	no	—
61	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₄₃	—	2	1	no	—
63	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	—	2	1	no	—
65	C₂²×C₄×C₄₃	—	2	1	no	—
66	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₄₃	—	2	1	no	—
68	C₂×C₂²⋊D₄₃	D₄	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	2	1	no	—
70	C₂²×Dic₄₃	—	2	1	no	—
71	#10621900	—	2	2	no	—
72	#10621900	—	4	1	no	—
73	C₂×C₄×D₄₃	—	2	1	no	—
74	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
75	C₂×Dic₈₆	—	2	1	no	—
76	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
77	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
78	M₁₆×C₄₃	—	4	1	no	—
79	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	—	2	1	no	—
82	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₄₃	—	2	1	no	—
85	C₂²⋊Dic₄₃	—	2	1	no	—
86	C₂²⋊Dic₄₃	—	2	1	no	—
87	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
88	#10621885	—	4	1	no	—
89	#10621883	—	2	2	no	—
90	(C₂×D₄₃)⋊C₄	—	2	1	no	—
91	(C₂×D₄₃)⋊C₄	—	2	1	no	—
92	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
93	C₄⋊Dic₄₃	—	2	1	no	—
94	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	Dic₄₃⋊C₄	—	2	1	no	—
102	Dic₄₃⋊C₄	—	2	1	no	—
103	Dic₄₃⋊C₄	—	2	1	no	—
104	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
105	C₄×Dic₄₃	—	2	1	no	—
106	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
107	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
109	Dic₁₇₂	—	4	1	no	—
110	#10621872	—	4	1	no	—
111	C₂³×C₄₃	—	2	1	no	—
112	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₄₃	—	2	1	no	—
115	Q₈×C₄₃	—	2	1	no	—
116	D₄×C₄₃	—	2	1	no	—
117	D₄×C₄₃	—	4	2	no	—
118	D₄×C₄₃	#41	1	2	yes	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	—	8	1	no	—
126	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₄₃	—	2	2	no	—
129	C₂²⋊D₄₃	—	4	2	no	—
130	C₂×Dic₄₃	—	2	1	no	—
131	C₂×Dic₄₃	—	2	1	no	—
132	C₂×Dic₄₃	—	2	1	no	—
133	C₂×Dic₄₃	—	2	1	no	—
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	—	4	2	no	—
136	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
139	C₄×D₄₃	—	2	1	no	—
140	C₄×D₄₃	—	2	1	no	—
141	C₄×D₄₃	—	8	1	no	—
142	Dic₈₆	—	2	1	no	—
143	Dic₈₆	—	2	1	no	—
144	Dic₈₆	—	2	1	no	—
145	Dic₈₆	—	2	1	no	—
146	Dic₈₆	—	8	1	no	—
147	C₈×C₄₃	—	2	1	no	—
148	C₈×C₄₃	—	2	1	no	—
149	C₄₃⋊C₈	—	2	1	no	—
150	C₄₃⋊C₈	—	2	1	no	—
151	C₂²×C₄₃	—	2	2	no	—
152	C₂²×C₄₃	—	2	1	no	—
153	C₂²×C₄₃	—	2	1	no	—
154	C₂²×C₄₃	—	2	1	no	—
155	C₂²×C₄₃	—	4	1	no	—
156	C₂²×C₄₃	—	4	1	no	—
157	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₄₃	—	2	1	no	—
159	C₂×D₄₃	—	2	1	no	—
160	C₂×D₄₃	—	2	1	no	—
161	C₄×C₄₃	—	2	1	no	—
162	C₄×C₄₃	—	4	1	no	—
163	C₄×C₄₃	—	4	1	no	—
164	C₄×C₄₃	—	4	1	no	—
165	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
166	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
167	Dic₄₃	—	2	1	no	—
168	Dic₄₃	—	2	1	no	—
169	Dic₄₃	—	4	1	no	—
170	Dic₄₃	—	4	1	no	—
171	Dic₄₃	—	4	1	no	—
172	Dic₄₃	—	4	1	no	—
173	Dic₄₃	#137	1	2	yes	—
174	#3331	—	43	1	no	maximal, Sylow
175	C₂×C₄₃	—	4	1	no	—
176	C₂×C₄₃	—	4	2	no	—
177	C₂×C₄₃	#546	1	1	yes	characteristic
178	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
179	C₂×C₄₃	#577	1	1	yes	characteristic
180	D₄₃	—	4	1	no	—
181	#554	—	43	1	no	—
182	#547	—	43	1	no	—
183	#487	—	43	1	no	—
184	#483	—	43	1	no	—
185	#483	—	43	1	no	—
186	#476	—	43	1	no	—
187	#470	—	43	1	no	—
188	#470	—	43	1	no	—
189	#462	—	43	1	no	—
190	#452	—	43	1	no	—
191	#452	—	43	1	no	—
192	C₄×SD₁₆	—	43	1	no	—
193	#436	—	43	1	no	—
194	#417	—	43	1	no	—
195	#417	D₄₃	1	1	yes	characteristic, complemented
196	C₄₃	#3331	1	1	yes	characteristic, complemented, Sylow
197	C₂×#41	—	43	1	no	—
198	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
199	C₂×Q₁₆	—	43	1	no	—
200	C₂×Q₁₆	—	43	1	no	—
201	C₂×SD₁₆	—	43	1	no	—
202	C₂×M₁₆	—	43	1	no	—
203	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
204	C₄⋊Q₈	—	43	1	no	—
205	C₄⋊Q₈	—	43	1	no	—
206	#125	—	86	1	no	—
207	#123	—	86	1	no	—
208	#123	—	86	1	no	—
209	C₂²⋊Q₈	—	43	1	no	—
210	C₂²⋊Q₈	—	43	1	no	—
211	C₂²⋊Q₈	—	43	1	no	—
212	C₂²⋊Q₈	—	43	1	no	—
213	C₂²⋊Q₈	—	86	1	no	—
214	C₂²⋊Q₈	—	86	1	no	—
215	#121	—	86	1	no	—
216	C₄×Q₈	—	43	1	no	—
217	C₄×D₄	—	43	1	no	—
218	C₄×D₄	—	43	1	no	—
219	C₄×D₄	—	86	1	no	—
220	C₂×C₄⋊C₄	—	43	1	no	—
221	C₂×C₄⋊C₄	—	86	1	no	—
222	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
223	#106	—	43	1	no	—
224	#106	—	43	1	no	—
225	#106	—	43	1	no	—
226	C₄⋊C₈	—	43	1	no	—
227	#103	—	43	1	no	—
228	#103	—	43	1	no	—
229	#103	—	43	1	no	—
230	#103	—	43	1	no	—
231	#103	—	43	1	no	—
232	#103	C₂×D₄₃	1	1	yes	characteristic
233	#103	C₂×D₄₃	1	1	yes	characteristic
234	#102	—	43	1	no	—
235	#102	C₂×D₄₃	1	1	yes	characteristic
236	#102	C₂×D₄₃	1	1	yes	characteristic
237	C₂²⋊C₈	—	43	1	no	—
238	C₂²⋊C₈	—	43	1	no	—
239	C₄×C₈	—	43	1	no	—
240	#41	—	2	2	no	—
241	#41	—	4	1	no	—
242	#41	—	86	2	no	—
243	#41	—	172	1	no	—
244	C₂×Q₈	—	43	1	no	—
245	C₂×Q₈	—	43	1	no	—
246	C₂×Q₈	—	86	1	no	—
247	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	43	1	no	—
250	C₂×D₄	—	86	1	no	—
251	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	43	1	no	—
255	C₂²×C₄	—	86	1	no	—
256	C₂²×C₄	—	86	1	no	—
257	C₂²×C₄	—	86	1	no	—
258	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
259	Q₁₆	—	172	1	no	—
260	Q₁₆	—	172	1	no	—
261	SD₁₆	—	86	2	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	172	1	no	—
264	C₂×C₈	—	43	1	no	—
265	C₂×C₈	—	43	1	no	—
266	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	43	1	no	—
270	C₄⋊C₄	—	43	1	no	—
271	C₄⋊C₄	—	43	1	no	—
272	C₄⋊C₄	—	86	1	no	—
273	C₄⋊C₄	—	86	1	no	—
274	C₄⋊C₄	—	86	1	no	—
275	C₄⋊C₄	—	86	1	no	—
276	C₄⋊C₄	—	86	1	no	—
277	C₄⋊C₄	—	86	1	no	—
278	C₄⋊C₄	—	86	1	no	—
279	C₄⋊C₄	—	86	1	no	—
280	C₄⋊C₄	—	86	1	no	—
281	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
283	C₂²⋊C₄	—	43	1	no	—
284	C₂²⋊C₄	—	43	1	no	—
285	C₂²⋊C₄	—	86	1	no	—
286	C₂²⋊C₄	—	86	1	no	—
287	C₂²⋊C₄	—	86	1	no	—
288	C₂²⋊C₄	—	86	1	no	—
289	C₂²⋊C₄	—	86	1	no	—
290	C₄²	—	43	1	no	—
291	C₄²	—	86	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	43	1	no	—
294	C₂³	D₄×D₄₃	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	86	1	no	—
298	Q₈	—	86	1	no	—
299	Q₈	—	86	1	no	—
300	Q₈	—	86	1	no	—
301	Q₈	—	344	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	4	2	no	—
304	D₄	—	86	2	no	—
305	D₄	—	172	2	no	—
306	D₄	#10621898	1	2	yes	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	—	43	1	no	—
315	C₂×C₄	—	43	1	no	—
316	C₂×C₄	—	43	1	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	86	1	no	—
319	C₂×C₄	—	86	1	no	—
320	C₂×C₄	—	86	1	no	—
321	C₂×C₄	—	86	1	no	—
322	C₂×C₄	—	86	1	no	—
323	C₂×C₄	—	344	1	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	86	1	no	—
331	C₈	—	86	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	86	1	no	—
339	C₂²	—	86	1	no	—
340	C₂²	—	86	1	no	—
341	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	43	2	no	—
347	C₄	—	86	1	no	—
348	C₄	—	86	1	no	—
349	C₄	—	172	1	no	—
350	C₄	—	172	1	no	—
351	C₄	—	172	1	no	—
352	C₄	—	172	1	no	—
353	C₄	C₂×#10621898	1	1	yes	characteristic
354	C₄	#13063370	1	1	yes	characteristic
355	C₂	—	4	1	no	—
356	C₂	—	4	2	no	—
357	C₂	—	172	1	no	—
358	C₂	#14583539	1	1	yes	characteristic
359	C₂	#137×D₄₃	1	1	yes	characteristic
360	C₂	#14583709	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#417	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#417	D₄₃
hypocenter	C₄₃	#3331
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
1. #14583604	C₂	semidirect
3. #14583248	C₂	semidirect
4. #14583229	C₂	semidirect
5. #14583228	C₂	semidirect
6. #14583187	C₂	semidirect
7. #14583157	C₂	semidirect
8. #14583156	C₂	semidirect
9. #14583108	C₂	semidirect
10. #14583062	C₂	semidirect
11. #14583061	C₂	semidirect
12. #14582990	C₂	semidirect
13. #14582975	C₂	semidirect
14. #14582897	C₂	semidirect
15. C₄₃×#417	C₂	semidirect
195. #417	D₄₃	semidirect
196. C₄₃	#3331	semidirect