Subgroups — #19144390

Order 5504.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#14583200	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583199	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583196	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583188	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583187	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583178	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583177	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582940	C₂	1	1	yes	maximal, characteristic, complemented
11	#14582938	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582921	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582902	C₂	1	1	yes	maximal, characteristic
14	#14582902	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#478	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×#10621872	C₂²	1	1	yes	characteristic
19	C₂×#10621872	C₂²	1	1	yes	characteristic
20	#13063324	C₂²	1	1	yes	characteristic, complemented
21	#13063324	C₂²	1	1	yes	characteristic, complemented
22	#13063317	C₂²	1	1	yes	characteristic, complemented
23	#13063303	—	2	1	no	—
24	#13063303	—	2	1	no	—
25	#13063290	—	2	1	no	—
26	#13063290	—	2	1	no	—
27	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
28	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
29	#13063287	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₄₃	—	2	1	no	—
31	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
32	#13063283	C₂²	1	1	yes	characteristic
33	#13063283	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
35	C₂²⋊C₄×D₄₃	—	2	1	no	—
36	C₄²×D₄₃	C₂²	1	1	yes	characteristic
37	#13063259	C₂²	1	1	yes	characteristic, complemented
38	#13063258	C₂²	1	1	yes	characteristic, complemented
39	#13063258	C₂²	1	1	yes	characteristic, complemented
40	#13063257	C₂²	1	1	yes	characteristic, complemented
41	#13063256	C₂²	1	1	yes	characteristic, complemented
42	#13063256	C₂²	1	1	yes	characteristic, complemented
43	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
44	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	#13063247	C₂²	1	1	yes	characteristic
47	#13063247	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	#13063246	C₂²	1	1	yes	characteristic
50	#13063245	C₂²	1	1	yes	characteristic
51	#13063245	C₂²	1	1	yes	characteristic
52	#125×C₄₃	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
54	#107×C₄₃	C₂²	1	1	yes	characteristic, complemented
55	#106×C₄₃	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	#102×C₄₃	C₂²	1	1	yes	characteristic
59	C₂³×D₄₃	—	2	1	no	—
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂×C₂²⋊D₄₃	—	2	1	no	—
63	C₂²×Dic₄₃	—	2	1	no	—
64	D₄×D₄₃	—	2	2	no	—
65	D₄×D₄₃	Q₈	1	4	yes	complemented
66	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	—	4	4	no	—
71	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
72	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
73	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
74	C₂×Dic₈₆	—	2	1	no	—
75	C₂×Dic₈₆	—	2	1	no	—
76	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	—	2	1	no	—
79	C₄⋊C₄×C₄₃	—	2	1	no	—
80	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₄₃	—	2	1	no	—
84	C₄²×C₄₃	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₄₃	—	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	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	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	D₄	1	1	yes	characteristic
101	C₄×Dic₄₃	D₄	1	1	yes	characteristic
102	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
105	#10621872	—	2	2	no	—
106	#10621872	—	2	2	no	—
107	C₂³×C₄₃	—	2	1	no	—
108	C₂²×D₄₃	—	2	1	no	—
109	C₂²×D₄₃	—	2	4	no	—
110	C₂²×D₄₃	—	4	2	no	—
111	C₂²×D₄₃	—	4	2	no	—
112	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
113	D₄×C₄₃	—	2	1	no	—
114	D₄×C₄₃	C₂×Q₈	1	2	yes	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₄₃	—	2	4	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	D₁₇₂	—	2	1	no	—
131	D₁₇₂	C₂×Q₈	1	2	yes	—
132	C₄×D₄₃	—	2	2	no	—
133	C₄×D₄₃	—	4	1	no	—
134	C₄×D₄₃	—	4	2	no	—
135	C₄×D₄₃	—	4	2	no	—
136	C₄×D₄₃	—	4	2	no	—
137	C₄×D₄₃	C₂×Q₈	1	2	yes	—
138	C₄×D₄₃	#41	1	2	yes	—
139	Dic₈₆	—	4	2	no	—
140	Dic₈₆	—	4	2	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	—	2	1	no	—
143	C₄₃⋊C₈	—	2	1	no	—
144	C₄₃⋊C₈	—	2	1	no	—
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	—	4	1	no	—
148	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₄₃	—	2	1	no	—
150	C₂×D₄₃	—	2	1	no	—
151	C₂×D₄₃	—	2	2	no	—
152	C₂×D₄₃	—	4	8	no	—
153	C₂×D₄₃	—	4	1	no	—
154	C₂×D₄₃	—	4	1	no	—
155	C₂×D₄₃	C₂²⋊Q₈	1	2	yes	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	4	1	no	—
159	C₄×C₄₃	—	4	1	no	—
160	C₄×C₄₃	—	4	1	no	—
161	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
162	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
163	Dic₄₃	—	2	1	no	—
164	Dic₄₃	—	4	1	no	—
165	Dic₄₃	—	4	1	no	—
166	Dic₄₃	—	4	1	no	—
167	Dic₄₃	C₂²⋊Q₈	1	2	yes	—
168	Dic₄₃	AGL(1,ℤ/8)	1	2	yes	—
169	#3077	—	43	1	no	maximal, Sylow
170	C₂×C₄₃	—	4	2	no	—
171	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂×C₄₃	#575	1	1	yes	characteristic
174	D₄₃	—	2	2	no	—
175	D₄₃	—	4	2	no	—
176	#533	—	43	1	no	—
177	C₂×C₄×D₄	—	43	1	no	—
178	#478	—	43	1	no	—
179	#478	—	43	1	no	—
180	#478	—	43	1	no	—
181	#478	D₄₃	1	1	yes	characteristic, complemented
182	#476	—	43	1	no	—
183	#476	—	43	1	no	—
184	#474	—	43	1	no	—
185	#474	—	43	1	no	—
186	#428	—	43	1	no	—
187	#428	—	43	1	no	—
188	#423	—	43	1	no	—
189	#418	—	43	1	no	—
190	#418	—	43	1	no	—
191	C₄₃	#3077	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	—	43	1	no	—
193	C₂³×C₄	—	86	1	no	—
194	C₂×M₁₆	—	43	1	no	—
195	C₂×M₁₆	—	43	1	no	—
196	C₄⋊Q₈	—	43	1	no	—
197	C₄⋊Q₈	—	43	1	no	—
198	#125	—	43	1	no	—
199	#125	C₂×D₄₃	1	1	yes	characteristic, complemented
200	C₂²⋊Q₈	—	86	1	no	—
201	C₂²⋊Q₈	—	86	1	no	—
202	C₄×Q₈	—	86	1	no	—
203	C₄×Q₈	—	86	1	no	—
204	C₄×D₄	—	43	1	no	—
205	C₄×D₄	—	43	1	no	—
206	C₄×D₄	—	43	1	no	—
207	C₄×D₄	—	86	1	no	—
208	C₄×D₄	—	86	1	no	—
209	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
210	#117	—	43	1	no	—
211	#117	—	43	1	no	—
212	C₂×C₄⋊C₄	—	43	1	no	—
213	C₂×C₂²⋊C₄	—	86	1	no	—
214	C₂×C₄²	—	43	1	no	—
215	#107	—	43	1	no	—
216	#107	—	43	1	no	—
217	#107	—	43	1	no	—
218	#107	C₂×D₄₃	1	1	yes	characteristic, complemented
219	#106	—	43	1	no	—
220	#106	—	43	1	no	—
221	#106	—	43	1	no	—
222	#106	C₂×D₄₃	1	1	yes	characteristic, complemented
223	C₄⋊C₈	—	43	1	no	—
224	C₄⋊C₈	—	43	1	no	—
225	C₄⋊C₈	—	43	1	no	—
226	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
227	#102	—	43	1	no	—
228	#102	—	43	1	no	—
229	#102	—	43	1	no	—
230	#102	—	43	1	no	—
231	#102	—	43	1	no	—
232	#102	—	43	1	no	—
233	#102	C₂×D₄₃	1	1	yes	characteristic
234	#102	C₂×D₄₃	1	1	yes	characteristic
235	C₂⁴	—	86	1	no	—
236	C₂×Q₈	—	86	1	no	—
237	C₂×Q₈	—	86	1	no	—
238	C₂×D₄	—	43	4	no	—
239	C₂×D₄	—	43	1	no	—
240	C₂×D₄	—	86	2	no	—
241	C₂×D₄	—	86	1	no	—
242	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	43	1	no	—
245	C₂²×C₄	—	43	1	no	—
246	C₂²×C₄	—	43	1	no	—
247	C₂²×C₄	—	86	1	no	—
248	C₂²×C₄	—	86	1	no	—
249	C₂²×C₄	—	86	1	no	—
250	C₂²×C₄	—	86	1	no	—
251	C₂²×C₄	—	172	4	no	—
252	M₁₆	—	86	2	no	—
253	M₁₆	—	86	2	no	—
254	C₂×C₈	—	43	1	no	—
255	C₂×C₈	—	43	1	no	—
256	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	43	1	no	—
261	C₄⋊C₄	—	43	1	no	—
262	C₄⋊C₄	—	43	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	C₄⋊C₄	—	86	1	no	—
268	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	86	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	C₄²	—	43	1	no	—
278	C₄²	—	43	1	no	—
279	C₄²	—	43	1	no	—
280	C₄²	—	86	1	no	—
281	C₄²	—	86	1	no	—
282	C₄²	C₂²×D₄₃	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	43	1	no	—
285	C₂³	—	86	4	no	—
286	C₂³	—	86	1	no	—
287	C₂³	—	172	2	no	—
288	C₂³	—	172	2	no	—
289	Q₈	—	172	2	no	—
290	Q₈	—	172	2	no	—
291	D₄	—	2	1	no	—
292	D₄	—	43	2	no	—
293	D₄	—	86	4	no	—
294	D₄	—	86	1	no	—
295	D₄	Q₈×D₄₃	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	43	1	no	—
301	C₂×C₄	—	43	1	no	—
302	C₂×C₄	—	43	2	no	—
303	C₂×C₄	—	43	2	no	—
304	C₂×C₄	—	43	1	no	—
305	C₂×C₄	—	86	2	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	86	1	no	—
308	C₂×C₄	—	86	1	no	—
309	C₂×C₄	—	172	1	no	—
310	C₂×C₄	—	172	2	no	—
311	C₂×C₄	—	172	2	no	—
312	C₂×C₄	—	172	2	no	—
313	C₂×C₄	—	172	2	no	—
314	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
315	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	86	1	no	—
320	C₈	—	86	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	43	2	no	—
325	C₂²	—	86	1	no	—
326	C₂²	—	86	1	no	—
327	C₂²	—	86	2	no	—
328	C₂²	—	172	8	no	—
329	C₂²	—	172	1	no	—
330	C₂²	—	172	1	no	—
331	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	43	2	no	—
338	C₄	—	43	2	no	—
339	C₄	—	86	1	no	—
340	C₄	—	172	1	no	—
341	C₄	—	172	1	no	—
342	C₄	—	172	1	no	—
343	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
344	C₄	#41×D₄₃	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	86	2	no	—
347	C₂	—	172	2	no	—
348	C₂	C₂²⋊Q₈×D₄₃	1	1	yes	characteristic
349	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
350	C₂	#14583696	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#478	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#478	D₄₃
hypocenter	C₄₃	#3077
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #14583455	C₂	semidirect
3. #14583200	C₂	semidirect
4. #14583199	C₂	semidirect
5. #14583196	C₂	semidirect
6. #14583188	C₂	semidirect
7. #14583187	C₂	semidirect
8. #14583178	C₂	semidirect
9. #14583177	C₂	semidirect
10. #14582940	C₂	semidirect
11. #14582938	C₂	semidirect
12. #14582921	C₂	semidirect
15. C₄₃×#478	C₂	semidirect
20. #13063324	C₂²	semidirect
21. #13063324	C₂²	semidirect
22. #13063317	C₂²	semidirect
37. #13063259	C₂²	semidirect
38. #13063258	C₂²	semidirect
39. #13063258	C₂²	semidirect
40. #13063257	C₂²	semidirect
41. #13063256	C₂²	semidirect
42. #13063256	C₂²	semidirect
43. C₄⋊(C₄₃⋊C₈)	C₂²	semidirect
44. Dic₄₃⋊C₈	C₂²	semidirect
45. Dic₄₃⋊C₈	C₂²	semidirect
52. #125×C₄₃	C₂²	semidirect
54. #107×C₄₃	C₂²	semidirect
55. #106×C₄₃	C₂²	semidirect
56. C₄⋊C₈×C₄₃	C₂²	semidirect
65. D₄×D₄₃	Q₈	semidirect
181. #478	D₄₃	semidirect
191. C₄₃	#3077	semidirect
199. #125	C₂×D₄₃	semidirect
218. #107	C₂×D₄₃	semidirect
222. #106	C₂×D₄₃	semidirect
226. C₄⋊C₈	C₂×D₄₃	semidirect