Subgroups — #19146554

Order 5504.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583550	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583540	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583260	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583219	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583188	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583153	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583130	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583117	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583047	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583035	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583010	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582967	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582899	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#13063245	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621902	C₂²	1	1	yes	characteristic
17	C₂×#10621898	—	2	1	no	—
18	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
19	C₂×#10621884	C₂²	1	1	yes	characteristic, complemented
20	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
21	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
22	C₂×#10621872	C₂²	1	1	yes	characteristic, complemented
23	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
24	#13063324	C₂²	1	1	yes	characteristic, complemented
25	#13063314	C₂²	1	1	yes	characteristic
26	#13063310	—	2	1	no	—
27	#13063310	—	2	1	no	—
28	#13063308	—	2	1	no	—
29	#13063303	—	2	1	no	—
30	#13063300	—	2	1	no	—
31	#13063300	C₂²	1	1	yes	characteristic
32	#13063300	C₂²	1	1	yes	characteristic, complemented
33	#13063299	C₂²	1	1	yes	characteristic
34	#13063295	—	2	1	no	—
35	#13063287	C₂²	1	1	yes	characteristic
36	#13063287	C₂²	1	1	yes	characteristic, complemented
37	C₄×C₂²⋊D₄₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊C₄×D₄₃	—	2	1	no	—
40	#13063256	C₂²	1	1	yes	characteristic, complemented
41	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	#13063249	C₂²	1	1	yes	characteristic, complemented
43	#13063247	C₂²	1	1	yes	characteristic
44	#13063247	C₂²	1	1	yes	characteristic
45	#13063245	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	#13063245	C₂²	1	1	yes	characteristic, complemented
50	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	#13063235	C₂²	1	1	yes	characteristic, complemented
52	C₂²⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
54	#106×C₄₃	C₂²	1	1	yes	characteristic, complemented
55	#106×C₄₃	C₂²	1	1	yes	characteristic, complemented
56	#103×C₄₃	C₂²	1	1	yes	characteristic, complemented
57	#103×C₄₃	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
59	C₂³×D₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂×C₂²⋊D₄₃	D₄	1	1	yes	characteristic
65	#10621902	—	2	2	no	—
66	#10621902	—	4	1	no	—
67	#10621898	—	4	4	no	—
68	C₂×D₁₇₂	—	2	1	no	—
69	C₂×D₁₇₂	—	2	1	no	—
70	C₂×D₁₇₂	—	2	1	no	—
71	C₂×D₁₇₂	—	2	1	no	—
72	C₂×D₁₇₂	—	2	2	no	—
73	C₂×D₁₇₂	—	4	1	no	—
74	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
75	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
76	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₄₃	—	2	1	no	—
78	C₂×C₄×D₄₃	—	2	1	no	—
79	C₂×C₄×D₄₃	—	2	1	no	—
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
82	C₂×Dic₈₆	—	2	1	no	—
83	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
84	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₄₃	—	2	1	no	—
86	C₄⋊C₄×C₄₃	—	2	1	no	—
87	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
90	C₂²⋊C₄×C₄₃	—	2	1	no	—
91	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
92	#10621884	—	4	1	no	—
93	(C₂×D₄₃)⋊C₄	—	2	1	no	—
94	(C₂×D₄₃)⋊C₄	—	2	1	no	—
95	(C₂×D₄₃)⋊C₄	—	2	1	no	—
96	(C₂×D₄₃)⋊C₄	—	2	1	no	—
97	(C₂×D₄₃)⋊C₄	—	2	1	no	—
98	(C₂×D₄₃)⋊C₄	—	2	1	no	—
99	(C₂×D₄₃)⋊C₄	—	2	1	no	—
100	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
101	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
102	Dic₄₃⋊C₄	—	2	1	no	—
103	Dic₄₃⋊C₄	—	2	1	no	—
104	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
105	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	—	4	1	no	—
107	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
109	D₃₄₄	—	2	2	no	—
110	D₃₄₄	—	4	1	no	—
111	#10621872	—	4	1	no	—
112	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₄₃	—	2	1	no	—
114	C₂²×D₄₃	—	2	1	no	—
115	C₂²×D₄₃	—	2	1	no	—
116	C₂²×D₄₃	—	4	2	no	—
117	C₂²×D₄₃	—	4	2	no	—
118	C₂²×D₄₃	—	8	1	no	—
119	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
120	Q₈×C₄₃	—	2	1	no	—
121	Q₈×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	4	2	no	—
129	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₄₃	—	2	2	no	—
132	C₂²⋊D₄₃	—	4	2	no	—
133	C₂×Dic₄₃	—	2	1	no	—
134	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
137	D₁₇₂	—	2	1	no	—
138	D₁₇₂	—	2	1	no	—
139	D₁₇₂	—	2	1	no	—
140	D₁₇₂	—	2	1	no	—
141	D₁₇₂	—	2	1	no	—
142	D₁₇₂	—	4	2	no	—
143	D₁₇₂	—	4	2	no	—
144	D₁₇₂	—	4	2	no	—
145	D₁₇₂	—	8	1	no	—
146	D₁₇₂	#41	1	2	yes	complemented
147	C₄×D₄₃	—	2	1	no	—
148	C₄×D₄₃	—	2	1	no	—
149	C₄×D₄₃	—	4	2	no	—
150	C₄×D₄₃	—	4	2	no	—
151	C₄×D₄₃	—	4	2	no	—
152	C₄×D₄₃	—	8	1	no	—
153	Dic₈₆	—	4	2	no	—
154	C₈×C₄₃	—	2	1	no	—
155	C₈×C₄₃	—	2	1	no	—
156	C₄₃⋊C₈	—	2	1	no	—
157	C₄₃⋊C₈	—	2	1	no	—
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
161	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×D₄₃	—	2	2	no	—
163	C₂×D₄₃	—	2	1	no	—
164	C₂×D₄₃	—	2	1	no	—
165	C₂×D₄₃	—	2	1	no	—
166	C₂×D₄₃	—	4	1	no	—
167	C₂×D₄₃	—	4	1	no	—
168	C₂×D₄₃	—	4	1	no	—
169	C₂×D₄₃	—	4	1	no	—
170	C₂×D₄₃	—	4	1	no	—
171	C₂×D₄₃	—	4	1	no	—
172	C₂×D₄₃	—	4	1	no	—
173	C₂×D₄₃	—	4	1	no	—
174	C₂×D₄₃	—	8	4	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	4	1	no	—
177	C₄×C₄₃	—	4	1	no	—
178	C₄×C₄₃	—	4	1	no	—
179	C₄×C₄₃	—	4	1	no	—
180	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
181	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
182	Dic₄₃	—	2	1	no	—
183	Dic₄₃	—	2	1	no	—
184	Dic₄₃	—	2	1	no	—
185	Dic₄₃	—	4	1	no	—
186	#3306	—	43	1	no	maximal, Sylow
187	C₂×C₄₃	—	2	2	no	—
188	C₂×C₄₃	#546	1	1	yes	characteristic
189	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×C₄₃	#577	1	1	yes	characteristic
191	D₄₃	—	4	1	no	—
192	D₄₃	—	4	2	no	—
193	D₄₃	—	8	1	no	—
194	D₄₃	—	8	1	no	—
195	#547	—	43	1	no	—
196	#546	—	43	1	no	—
197	#489	—	43	1	no	—
198	#483	D₄₃	1	1	yes	characteristic, complemented
199	#481	—	43	1	no	—
200	#476	—	43	1	no	—
201	#469	—	43	1	no	—
202	#465	—	43	1	no	—
203	#463	—	43	1	no	—
204	#449	—	43	1	no	—
205	#447	—	43	1	no	—
206	#442	—	43	1	no	—
207	#435	—	43	1	no	—
208	#417	—	43	1	no	—
209	C₂×#102	—	43	1	no	—
210	C₄₃	#3306	1	1	yes	characteristic, complemented, Sylow
211	C₂×#41	—	43	1	no	—
212	C₂×#41	—	86	1	no	—
213	C₂²×D₄	—	43	1	no	—
214	C₂×SD₁₆	—	43	1	no	—
215	C₂×D₈	—	43	1	no	—
216	C₂×D₈	—	43	1	no	—
217	C₂×M₁₆	—	43	1	no	—
218	C₂²×C₈	—	43	1	no	—
219	C₄⋊Q₈	—	43	1	no	—
220	#124	—	43	1	no	—
221	#123	—	86	1	no	—
222	#123	—	86	1	no	—
223	#123	—	86	1	no	—
224	C₂²⋊Q₈	—	86	1	no	—
225	C₂²⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
226	#121	—	43	1	no	—
227	#121	—	43	1	no	—
228	#121	—	43	1	no	—
229	#121	—	86	1	no	—
230	#120	—	86	1	no	—
231	C₄×D₄	—	43	1	no	—
232	C₄×D₄	—	43	1	no	—
233	C₄×D₄	—	43	1	no	—
234	C₂×C₄⋊C₄	—	43	1	no	—
235	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
236	C₂×C₂²⋊C₄	—	86	1	no	—
237	#106	—	43	1	no	—
238	#106	C₂×D₄₃	1	1	yes	characteristic, complemented
239	#106	C₂×D₄₃	1	1	yes	characteristic, complemented
240	C₄⋊C₈	—	43	1	no	—
241	#103	—	43	1	no	—
242	#103	C₂×D₄₃	1	1	yes	characteristic, complemented
243	#103	C₂×D₄₃	1	1	yes	characteristic, complemented
244	#102	—	43	1	no	—
245	#102	—	43	1	no	—
246	#102	—	43	1	no	—
247	#102	—	43	1	no	—
248	#102	—	43	1	no	—
249	#102	—	43	1	no	—
250	#102	—	43	1	no	—
251	C₂²⋊C₈	—	43	1	no	—
252	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
253	#97	—	43	1	no	—
254	C₂⁴	—	86	1	no	—
255	#41	—	86	2	no	—
256	#41	—	172	4	no	—
257	#41	—	172	1	no	—
258	C₂×Q₈	—	86	1	no	—
259	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
260	C₂×D₄	—	43	1	no	—
261	C₂×D₄	—	43	1	no	—
262	C₂×D₄	—	43	1	no	—
263	C₂×D₄	—	43	1	no	—
264	C₂×D₄	—	86	2	no	—
265	C₂×D₄	—	86	1	no	—
266	C₂×D₄	—	86	1	no	—
267	C₂×D₄	—	86	1	no	—
268	C₂×D₄	—	86	1	no	—
269	C₂×D₄	—	86	1	no	—
270	C₂×D₄	—	172	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	43	1	no	—
273	C₂²×C₄	—	86	1	no	—
274	C₂²×C₄	—	86	1	no	—
275	C₂²×C₄	—	86	1	no	—
276	C₂²×C₄	—	86	1	no	—
277	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
278	SD₁₆	—	172	1	no	—
279	D₈	—	86	2	no	—
280	D₈	—	172	1	no	—
281	M₁₆	—	172	1	no	—
282	C₂×C₈	—	43	1	no	—
283	C₂×C₈	—	43	1	no	—
284	C₂×C₈	—	172	1	no	—
285	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
286	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	43	1	no	—
290	C₄⋊C₄	—	43	1	no	—
291	C₄⋊C₄	—	86	1	no	—
292	C₄⋊C₄	—	86	1	no	—
293	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
294	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
295	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
296	C₂²⋊C₄	—	2	1	no	—
297	C₂²⋊C₄	—	43	1	no	—
298	C₂²⋊C₄	—	43	1	no	—
299	C₂²⋊C₄	—	86	1	no	—
300	C₂²⋊C₄	—	86	1	no	—
301	C₂²⋊C₄	—	86	1	no	—
302	C₂²⋊C₄	—	86	1	no	—
303	C₂²⋊C₄	—	86	1	no	—
304	C₂²⋊C₄	—	86	1	no	—
305	C₂²⋊C₄	—	86	1	no	—
306	C₄²	—	43	1	no	—
307	C₂³	—	43	1	no	—
308	C₂³	—	86	1	no	—
309	C₂³	—	86	1	no	—
310	C₂³	—	86	1	no	—
311	C₂³	—	344	1	no	—
312	C₂³	—	172	2	no	—
313	C₂³	—	172	2	no	—
314	C₂³	D₄×D₄₃	1	1	yes	characteristic
315	Q₈	—	2	1	no	—
316	Q₈	—	2	1	no	—
317	Q₈	—	172	2	no	—
318	D₄	—	43	2	no	—
319	D₄	—	86	2	no	—
320	D₄	—	86	1	no	—
321	D₄	—	86	1	no	—
322	D₄	—	86	1	no	—
323	D₄	—	86	1	no	—
324	D₄	—	86	1	no	—
325	D₄	—	344	1	no	—
326	D₄	—	172	2	no	—
327	D₄	—	172	2	no	—
328	D₄	—	172	2	no	—
329	D₄	—	172	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	43	1	no	—
338	C₂×C₄	—	43	1	no	—
339	C₂×C₄	—	43	1	no	—
340	C₂×C₄	—	86	1	no	—
341	C₂×C₄	—	86	1	no	—
342	C₂×C₄	—	86	1	no	—
343	C₂×C₄	—	344	1	no	—
344	C₂×C₄	—	172	2	no	—
345	C₂×C₄	—	172	2	no	—
346	C₂×C₄	—	172	2	no	—
347	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
348	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	86	1	no	—
352	C₈	—	86	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	86	1	no	—
356	C₂²	—	86	1	no	—
357	C₂²	—	86	1	no	—
358	C₂²	—	86	2	no	—
359	C₂²	—	344	4	no	—
360	C₂²	—	172	1	no	—
361	C₂²	—	172	1	no	—
362	C₂²	—	172	1	no	—
363	C₂²	—	172	1	no	—
364	C₂²	—	172	1	no	—
365	C₂²	—	172	1	no	—
366	C₂²	—	172	1	no	—
367	C₂²	—	172	1	no	—
368	C₂²	#13063352	1	2	yes	—
369	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	1	no	—
373	C₄	—	4	1	no	—
374	C₄	—	4	1	no	—
375	C₄	—	86	1	no	—
376	C₄	—	86	1	no	—
377	C₄	—	86	1	no	—
378	C₄	—	172	1	no	—
379	C₄	#41×D₄₃	1	1	yes	characteristic
380	C₄	#13063371	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	344	1	no	—
383	C₂	—	344	1	no	—
384	C₂	—	172	1	no	—
385	C₂	—	172	2	no	—
386	C₂	#14583541	1	1	yes	characteristic
387	C₂	C₂×#13063352	1	1	yes	characteristic
388	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×#483	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#483	D₄₃
hypocenter	C₄₃	#3306
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #483

Complements

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

N	H ≅ G/N	type
1. #14583550	C₂	semidirect
2. #14583540	C₂	semidirect
3. #14583260	C₂	semidirect
4. #14583219	C₂	semidirect
5. #14583188	C₂	semidirect
6. #14583153	C₂	semidirect
7. #14583130	C₂	semidirect
8. #14583117	C₂	semidirect
9. #14583047	C₂	semidirect
10. #14583035	C₂	semidirect
11. #14583010	C₂	semidirect
12. #14582967	C₂	semidirect
13. #14582899	C₂	semidirect
14. C₂×#13063245	C₂	semidirect
15. C₄₃×#483	C₂	semidirect
19. C₂×#10621884	C₂²	semidirect
22. C₂×#10621872	C₂²	semidirect
24. #13063324	C₂²	semidirect
32. #13063300	C₂²	semidirect
36. #13063287	C₂²	semidirect
38. C₄⋊C₄×D₄₃	C₂²	semidirect
40. #13063256	C₂²	semidirect
41. Dic₄₃⋊C₈	C₂²	semidirect
42. #13063249	C₂²	semidirect
49. #13063245	C₂²	semidirect
50. (C₂×D₄₃)⋊C₈	C₂²	semidirect
51. #13063235	C₂²	semidirect
54. #106×C₄₃	C₂²	semidirect
55. #106×C₄₃	C₂²	semidirect
56. #103×C₄₃	C₂²	semidirect
57. #103×C₄₃	C₂²	semidirect
146. D₁₇₂	#41	semidirect
198. #483	D₄₃	semidirect
210. C₄₃	#3306	semidirect
238. #106	C₂×D₄₃	semidirect
239. #106	C₂×D₄₃	semidirect
242. #103	C₂×D₄₃	semidirect
243. #103	C₂×D₄₃	semidirect