Subgroups — #19144646

Order 5504.

454 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583436	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583233	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583219	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583198	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583197	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583117	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583116	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583048	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583034	C₂	1	1	yes	maximal, characteristic
11	C₄×#10621884	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#10621882	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582950	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₄₃	—	2	1	no	—
18	C₂×#10621884	C₂²	1	1	yes	characteristic
19	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
20	C₂×#10621882	C₂²	1	1	yes	characteristic
21	#13063320	—	2	1	no	—
22	#13063316	C₂²	1	1	yes	characteristic, complemented
23	#13063314	C₂²	1	1	yes	characteristic, complemented
24	#13063310	—	2	1	no	—
25	#13063301	C₂²	1	1	yes	characteristic
26	#13063300	C₂²	1	1	yes	characteristic
27	C₄×D₁₇₂	—	2	1	no	—
28	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
29	C₄×D₁₇₂	C₂²	1	1	yes	characteristic, complemented
30	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
31	#13063287	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₄₃	—	2	1	no	—
33	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
34	#13063282	—	2	1	no	—
35	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
36	C₂²⋊C₄×D₄₃	—	2	1	no	—
37	C₄²×D₄₃	C₂²	1	1	yes	characteristic
38	#13063259	C₂²	1	1	yes	characteristic, complemented
39	#13063258	C₂²	1	1	yes	characteristic, complemented
40	#13063256	C₂²	1	1	yes	characteristic, complemented
41	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#13063249	C₂²	1	1	yes	characteristic, complemented
44	#13063247	C₂²	1	1	yes	characteristic
45	#13063247	C₂²	1	1	yes	characteristic, complemented
46	#13063246	C₂²	1	1	yes	characteristic
47	#13063245	C₂²	1	1	yes	characteristic
48	#13063245	C₂²	1	1	yes	characteristic, complemented
49	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
54	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic, complemented
56	#103×C₄₃	C₂²	1	1	yes	characteristic, complemented
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	—	2	1	no	—
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂²×Dic₄₃	—	2	1	no	—
65	D₄×D₄₃	—	2	2	no	—
66	D₄×D₄₃	—	2	2	no	—
67	C₂×D₁₇₂	—	2	1	no	—
68	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
69	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	—	2	1	no	—
74	C₂×C₄×D₄₃	—	4	4	no	—
75	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
76	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
77	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
78	SD₁₆×C₄₃	—	2	2	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⋊C₄×C₄₃	—	2	1	no	—
85	C₄²×C₄₃	—	2	1	no	—
86	C₄²×C₄₃	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₄₃	—	2	1	no	—
88	#10621884	—	2	2	no	—
89	#10621882	—	2	2	no	—
90	(C₂×D₄₃)⋊C₄	—	2	1	no	—
91	(C₂×D₄₃)⋊C₄	—	2	1	no	—
92	(C₂×D₄₃)⋊C₄	—	2	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	C₄⋊Dic₄₃	—	2	1	no	—
98	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
100	Dic₄₃⋊C₄	—	2	1	no	—
101	Dic₄₃⋊C₄	—	2	1	no	—
102	Dic₄₃⋊C₄	—	2	1	no	—
103	C₄×Dic₄₃	D₄	1	1	yes	characteristic
104	C₄×Dic₄₃	D₄	1	1	yes	characteristic
105	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
108	D₃₄₄	—	2	2	no	—
109	C₂³×C₄₃	—	2	1	no	—
110	C₂²×D₄₃	—	2	4	no	—
111	C₂²×D₄₃	—	2	1	no	—
112	C₂²×D₄₃	—	2	1	no	—
113	C₂²×D₄₃	—	4	2	no	—
114	C₂²×D₄₃	—	4	2	no	—
115	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₄₃	—	2	1	no	—
117	Q₈×C₄₃	—	2	1	no	—
118	D₄×C₄₃	—	2	1	no	—
119	D₄×C₄₃	#41	1	2	yes	—
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	—	4	2	no	—
124	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₄₃	—	2	4	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	2	1	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
134	D₁₇₂	—	2	1	no	—
135	D₁₇₂	—	2	1	no	—
136	D₁₇₂	—	2	1	no	—
137	D₁₇₂	—	4	2	no	—
138	D₁₇₂	#41	1	2	yes	—
139	C₄×D₄₃	—	2	2	no	—
140	C₄×D₄₃	—	2	2	no	—
141	C₄×D₄₃	—	2	1	no	—
142	C₄×D₄₃	—	2	1	no	—
143	C₄×D₄₃	—	4	2	no	—
144	C₄×D₄₃	—	4	2	no	—
145	C₄×D₄₃	—	4	2	no	—
146	C₄×D₄₃	—	4	2	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	—	4	1	no	—
153	C₂²×C₄₃	—	4	1	no	—
154	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₄₃	—	2	1	no	—
156	C₂×D₄₃	—	2	1	no	—
157	C₂×D₄₃	—	2	2	no	—
158	C₂×D₄₃	—	4	8	no	—
159	C₂×D₄₃	—	4	1	no	—
160	C₂×D₄₃	—	4	1	no	—
161	C₂×D₄₃	—	4	1	no	—
162	C₂×D₄₃	—	4	1	no	—
163	C₂×D₄₃	—	4	1	no	—
164	C₂×D₄₃	#135	1	2	yes	—
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	4	1	no	—
167	C₄×C₄₃	—	4	1	no	—
168	C₄×C₄₃	—	4	1	no	—
169	C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
170	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
171	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
172	Dic₄₃	—	2	1	no	—
173	Dic₄₃	—	2	1	no	—
174	Dic₄₃	—	2	1	no	—
175	Dic₄₃	—	4	1	no	—
176	Dic₄₃	—	4	1	no	—
177	#3105	—	43	1	no	maximal, Sylow
178	C₂×C₄₃	—	4	2	no	—
179	C₂×C₄₃	#525	1	1	yes	characteristic
180	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
181	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	D₄₃	—	2	2	no	—
183	D₄₃	—	4	2	no	—
184	D₄₃	—	8	1	no	—
185	#529	—	43	1	no	—
186	C₂×C₄×D₄	—	43	1	no	—
187	#484	—	43	1	no	—
188	#481	—	43	1	no	—
189	#478	—	43	1	no	—
190	#478	—	43	1	no	—
191	#463	—	43	1	no	—
192	#463	—	43	1	no	—
193	#450	—	43	1	no	—
194	#447	—	43	1	no	—
195	C₄×SD₁₆	—	43	1	no	—
196	C₄×SD₁₆	D₄₃	1	1	yes	characteristic, complemented
197	C₄×D₈	—	43	1	no	—
198	C₄×D₈	—	43	1	no	—
199	#431	—	43	1	no	—
200	C₄₃	#3105	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₄	—	43	1	no	—
202	C₂³×C₄	—	86	1	no	—
203	C₂×SD₁₆	—	43	1	no	—
204	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
205	C₂×D₈	—	43	1	no	—
206	C₂×D₈	—	43	1	no	—
207	#126	—	86	1	no	—
208	#125	—	43	1	no	—
209	#124	—	43	1	no	—
210	#123	—	86	1	no	—
211	C₂²⋊Q₈	—	43	1	no	—
212	#121	—	43	1	no	—
213	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
214	C₄×D₄	—	43	1	no	—
215	C₄×D₄	—	43	1	no	—
216	C₄×D₄	—	43	1	no	—
217	C₄×D₄	—	43	1	no	—
218	C₄×D₄	—	86	1	no	—
219	C₄×D₄	—	86	1	no	—
220	C₄×D₄	—	86	1	no	—
221	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
222	#117	—	86	1	no	—
223	C₂×C₄⋊C₄	—	43	1	no	—
224	C₂×C₂²⋊C₄	—	86	1	no	—
225	C₂×C₄²	—	43	1	no	—
226	#107	—	43	1	no	—
227	#107	—	43	1	no	—
228	#106	—	43	1	no	—
229	#106	C₂×D₄₃	1	1	yes	characteristic, complemented
230	C₄⋊C₈	—	43	1	no	—
231	C₄⋊C₈	—	43	1	no	—
232	#103	—	43	1	no	—
233	#103	C₂×D₄₃	1	1	yes	characteristic, complemented
234	#102	—	43	1	no	—
235	#102	—	43	1	no	—
236	#102	—	43	1	no	—
237	#102	—	43	1	no	—
238	#102	—	43	1	no	—
239	#102	C₂×D₄₃	1	1	yes	characteristic
240	C₂²⋊C₈	—	43	1	no	—
241	C₂²⋊C₈	—	43	1	no	—
242	C₄×C₈	—	43	1	no	—
243	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
244	C₂⁴	—	86	1	no	—
245	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
246	C₂×D₄	—	43	1	no	—
247	C₂×D₄	—	43	1	no	—
248	C₂×D₄	—	86	1	no	—
249	C₂×D₄	—	86	1	no	—
250	C₂×D₄	—	86	2	no	—
251	C₂×D₄	—	86	2	no	—
252	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	43	1	no	—
255	C₂²×C₄	—	43	1	no	—
256	C₂²×C₄	—	43	1	no	—
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	—	172	4	no	—
263	SD₁₆	—	2	2	no	—
264	SD₁₆	—	86	2	no	—
265	D₈	—	86	2	no	—
266	D₈	—	86	2	no	—
267	C₂×C₈	—	43	1	no	—
268	C₂×C₈	—	43	1	no	—
269	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	43	1	no	—
273	C₄⋊C₄	—	43	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	C₂²×D₄₃	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	86	1	no	—
282	C₂²⋊C₄	—	86	1	no	—
283	C₂²⋊C₄	—	86	1	no	—
284	C₂²⋊C₄	—	86	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	C₄²	—	2	1	no	—
290	C₄²	—	43	1	no	—
291	C₄²	—	43	1	no	—
292	C₄²	—	43	1	no	—
293	C₄²	C₂²×D₄₃	1	1	yes	characteristic
294	C₂³	—	2	1	no	—
295	C₂³	—	43	1	no	—
296	C₂³	—	86	4	no	—
297	C₂³	—	86	1	no	—
298	C₂³	—	86	1	no	—
299	C₂³	—	172	2	no	—
300	C₂³	—	172	2	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	43	2	no	—
305	D₄	—	86	4	no	—
306	D₄	—	86	1	no	—
307	D₄	—	86	1	no	—
308	D₄	—	86	1	no	—
309	D₄	—	172	2	no	—
310	D₄	#10621898	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	43	1	no	—
316	C₂×C₄	—	43	1	no	—
317	C₂×C₄	—	43	1	no	—
318	C₂×C₄	—	86	2	no	—
319	C₂×C₄	—	86	2	no	—
320	C₂×C₄	—	86	1	no	—
321	C₂×C₄	—	86	1	no	—
322	C₂×C₄	—	86	1	no	—
323	C₂×C₄	—	86	1	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	—	172	2	no	—
328	C₂×C₄	—	172	2	no	—
329	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
330	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
331	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	86	1	no	—
335	C₈	—	86	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	43	2	no	—
340	C₂²	—	86	1	no	—
341	C₂²	—	86	1	no	—
342	C₂²	—	86	2	no	—
343	C₂²	—	172	8	no	—
344	C₂²	—	172	1	no	—
345	C₂²	—	172	1	no	—
346	C₂²	—	172	1	no	—
347	C₂²	—	172	1	no	—
348	C₂²	—	172	1	no	—
349	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	86	1	no	—
355	C₄	—	86	1	no	—
356	C₄	—	86	1	no	—
357	C₄	—	172	1	no	—
358	C₄	—	172	1	no	—
359	C₄	#13063352	1	2	yes	—
360	C₄	#41×D₄₃	1	1	yes	characteristic
361	C₄	C₂×#10621898	1	1	yes	characteristic
362	C₂	—	4	2	no	—
363	C₂	—	86	2	no	—
364	C₂	—	344	1	no	—
365	C₂	—	172	2	no	—
366	C₂	#14583408	1	1	yes	characteristic
367	C₂	#135×D₄₃	1	1	yes	characteristic
368	C₂	C₂×#13063352	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄×SD₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₄×SD₁₆	D₄₃
hypocenter	C₄₃	#3105
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #14583436	C₂	semidirect
2. C₄×D₄×D₄₃	C₂	semidirect
3. #14583233	C₂	semidirect
4. #14583219	C₂	semidirect
5. #14583198	C₂	semidirect
6. #14583197	C₂	semidirect
7. #14583117	C₂	semidirect
8. #14583116	C₂	semidirect
9. #14583048	C₂	semidirect
11. C₄×#10621884	C₂	semidirect
12. C₄×D₃₄₄	C₂	semidirect
13. C₄×#10621882	C₂	semidirect
14. #14582950	C₂	semidirect
15. C₄×SD₁₆×C₄₃	C₂	semidirect
22. #13063316	C₂²	semidirect
23. #13063314	C₂²	semidirect
29. C₄×D₁₇₂	C₂²	semidirect
38. #13063259	C₂²	semidirect
39. #13063258	C₂²	semidirect
40. #13063256	C₂²	semidirect
41. Dic₄₃⋊C₈	C₂²	semidirect
42. Dic₄₃⋊C₈	C₂²	semidirect
43. #13063249	C₂²	semidirect
45. #13063247	C₂²	semidirect
48. #13063245	C₂²	semidirect
51. C₄×C₄₃⋊C₈	C₂²	semidirect
53. C₄×Q₈×C₄₃	C₂²	semidirect
55. #106×C₄₃	C₂²	semidirect
56. #103×C₄₃	C₂²	semidirect
58. C₄×C₈×C₄₃	C₂²	semidirect
196. C₄×SD₁₆	D₄₃	semidirect
200. C₄₃	#3105	semidirect
213. C₄×Q₈	C₂×D₄₃	semidirect
229. #106	C₂×D₄₃	semidirect
233. #103	C₂×D₄₃	semidirect
243. C₄×C₈	C₂×D₄₃	semidirect