Subgroups — #25254202

Order 11008.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147018	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136875	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136783	C₂	1	2	yes	maximal, complemented
4	#19136780	C₂	1	2	yes	maximal, complemented
5	#19136742	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136729	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136696	C₂	1	2	yes	maximal, complemented
8	C₄×#13063266	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136427	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136381	C₂	1	2	yes	maximal, complemented
11	C₄₃×#3334	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583614	—	2	1	no	—
13	C₂×#13063266	C₂²	1	1	yes	characteristic
14	C₂×#13063265	C₂²	1	2	yes	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583224	—	2	2	no	—
17	#14583190	—	2	1	no	—
18	#14583135	C₂²	1	2	yes	—
19	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
20	#14582937	C₂²	1	2	yes	—
21	#14582726	C₂²	1	2	yes	—
22	#14582726	C₂²	1	2	yes	—
23	#14582726	C₂²	1	2	yes	—
24	#14582724	C₂²	1	2	yes	—
25	#14582724	C₂²	1	1	yes	characteristic
26	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
27	#14582700	C₂²	1	1	yes	characteristic
28	#14582700	C₂²	1	2	yes	—
29	#14582699	C₂²	1	1	yes	characteristic
30	#14582699	C₂²	1	2	yes	—
31	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄₃×#547	—	2	1	no	—
34	C₄₃×#500	C₂²	1	1	yes	characteristic
35	C₄₃×#480	—	2	2	no	—
36	C₄₃×#474	—	2	1	no	—
37	C₄₃×#466	C₂²	1	2	yes	—
38	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
40	C₂×#107×C₄₃	C₂²	1	2	yes	—
41	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
42	#13063325	—	2	1	no	—
43	#13063311	—	4	2	no	—
44	#13063299	—	2	2	no	—
45	#13063296	—	4	1	no	—
46	C₄×D₁₇₂	—	2	1	no	—
47	#13063284	—	2	2	no	—
48	#13063283	—	4	1	no	—
49	#13063266	—	2	2	no	—
50	#13063265	—	2	4	no	—
51	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
54	#13063259	—	2	1	no	—
55	#13063259	C₂³	1	2	yes	—
56	#13063259	C₂³	1	1	yes	characteristic
57	#13063257	—	2	2	no	—
58	#13063247	—	2	2	no	—
59	#13063247	—	2	1	no	—
60	C₂×#41×C₄₃	—	4	1	no	—
61	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₄₃	—	2	1	no	—
64	#123×C₄₃	—	4	2	no	—
65	C₂²⋊Q₈×C₄₃	—	4	1	no	—
66	#121×C₄₃	—	2	2	no	—
67	C₄×D₄×C₄₃	—	2	1	no	—
68	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
70	#107×C₄₃	—	2	1	no	—
71	#107×C₄₃	—	2	2	no	—
72	#107×C₄₃	C₂³	1	2	yes	—
73	#107×C₄₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
75	#102×C₄₃	—	2	1	no	—
76	#102×C₄₃	—	2	2	no	—
77	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	#41×C₄₃	—	8	4	no	—
80	C₂×Q₈×C₄₃	—	4	1	no	—
81	C₂×D₄×C₄₃	—	2	1	no	—
82	C₂×D₄×C₄₃	—	4	2	no	—
83	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	—	4	1	no	—
85	C₂²×C₄×C₄₃	—	4	2	no	—
86	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
87	C₂×C₂²⋊D₄₃	—	4	2	no	—
88	C₂×D₁₇₂	—	2	1	no	—
89	C₂×C₄×D₄₃	—	4	1	no	—
90	C₂×Dic₈₆	—	4	1	no	—
91	D₈×C₄₃	—	2	1	no	—
92	D₈×C₄₃	—	2	1	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₄₃	—	2	1	no	—
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	4	2	no	—
101	C₄⋊C₄×C₄₃	—	4	1	no	—
102	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
103	C₂²⋊C₄×C₄₃	—	4	1	no	—
104	C₂²⋊C₄×C₄₃	—	4	2	no	—
105	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
106	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂²⋊Dic₄₃	—	4	2	no	—
108	(C₂×D₄₃)⋊C₄	—	4	2	no	—
109	(C₂×D₄₃)⋊C₄	—	4	1	no	—
110	C₄⋊Dic₄₃	—	2	1	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	4	1	no	—
113	C₄×Dic₄₃	—	4	2	no	—
114	D₃₄₄	—	2	1	no	—
115	D₃₄₄	—	2	1	no	—
116	C₄₃⋊C₁₆	—	2	2	no	—
117	C₄₃⋊C₁₆	—	2	1	no	—
118	C₄₃⋊C₁₆	#41	1	2	yes	complemented
119	C₂³×C₄₃	—	4	1	no	—
120	C₂³×C₄₃	C₂×D₈	1	2	yes	—
121	C₂²×D₄₃	—	4	1	no	—
122	Q₈×C₄₃	—	8	2	no	—
123	D₄×C₄₃	—	2	2	no	—
124	D₄×C₄₃	—	4	1	no	—
125	D₄×C₄₃	—	4	1	no	—
126	D₄×C₄₃	—	8	4	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	4	1	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	8	4	no	—
132	C₂×C₄×C₄₃	—	8	2	no	—
133	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₄₃	—	16	2	no	—
138	C₂×Dic₄₃	—	4	1	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	D₁₇₂	—	4	1	no	—
141	D₁₇₂	—	4	1	no	—
142	C₄×D₄₃	—	8	2	no	—
143	Dic₈₆	—	16	1	no	—
144	C₈×C₄₃	—	2	2	no	—
145	C₈×C₄₃	—	2	1	no	—
146	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
148	#33921	—	43	1	no	maximal, Sylow
149	C₂²×C₄₃	—	2	2	no	—
150	C₂²×C₄₃	—	2	2	no	—
151	C₂²×C₄₃	—	2	2	no	—
152	C₂²×C₄₃	—	8	1	no	—
153	C₂²×C₄₃	—	8	1	no	—
154	C₂²×C₄₃	—	8	1	no	—
155	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₄₃	—	8	1	no	—
157	C₂×D₄₃	—	8	1	no	—
158	C₂×D₄₃	—	8	1	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	2	1	no	—
161	C₄×C₄₃	—	8	1	no	—
162	C₄×C₄₃	—	8	2	no	—
163	C₄×C₄₃	#509	1	2	yes	—
164	C₄×C₄₃	#547	1	1	yes	characteristic
165	C₄×C₄₃	#578	1	1	yes	characteristic
166	Dic₄₃	—	8	1	no	—
167	Dic₄₃	—	8	2	no	—
168	#3341	—	43	1	no	—
169	#3334	D₄₃	1	1	yes	characteristic, complemented
170	#2254	—	43	1	no	—
171	#2234	—	43	2	no	—
172	#2233	—	43	2	no	—
173	#2226	—	43	1	no	—
174	#2224	—	43	1	no	—
175	#2218	—	43	2	no	—
176	C₄×D₁₆	—	43	1	no	—
177	#2169	—	43	1	no	—
178	#2159	—	43	2	no	—
179	C₂×C₄₃	—	4	2	no	—
180	C₂×C₄₃	—	16	1	no	—
181	C₂×C₄₃	#3334	1	1	yes	characteristic
182	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
183	C₂×C₄₃	#3416	1	1	yes	characteristic
184	D₄₃	—	16	1	no	—
185	#555	—	86	1	no	—
186	#547	—	2	1	no	—
187	C₂×D₁₆	—	43	1	no	—
188	C₂×M₃₂	—	43	2	no	—
189	#500	—	43	1	no	—
190	#500	C₂×D₄₃	1	1	yes	characteristic
191	#482	—	86	2	no	—
192	#480	—	2	2	no	—
193	#476	—	86	1	no	—
194	#474	—	2	1	no	—
195	#466	—	43	2	no	—
196	#466	C₂×D₄₃	1	2	yes	—
197	C₄×D₈	—	43	1	no	—
198	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
199	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
200	#427	—	43	2	no	—
201	C₂×#107	C₂×D₄₃	1	2	yes	—
202	#367	—	43	2	no	—
203	#367	—	43	2	no	—
204	#367	—	43	2	no	—
205	#366	—	43	1	no	—
206	#366	—	43	2	no	—
207	C₄⋊C₁₆	—	43	1	no	—
208	#357	—	43	1	no	—
209	#357	—	43	2	no	—
210	#357	—	43	1	no	—
211	#357	—	43	2	no	—
212	C₂²⋊C₁₆	—	43	2	no	—
213	C₄×C₁₆	—	43	1	no	—
214	C₄₃	#33921	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	—	4	1	no	—
216	C₂×D₈	—	43	1	no	—
217	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	86	1	no	—
221	#124	—	172	2	no	—
222	#123	—	4	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	#121	—	2	2	no	—
225	#121	—	86	2	no	—
226	#121	—	172	1	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	86	1	no	—
229	C₄×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
230	#117	—	86	2	no	—
231	#117	—	172	1	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	D₁₆	—	86	2	no	—
234	M₃₂	—	86	4	no	—
235	C₂×C₁₆	—	43	1	no	—
236	C₂×C₁₆	—	43	1	no	—
237	C₂×C₁₆	—	43	2	no	—
238	#107	—	2	2	no	—
239	#107	—	2	1	no	—
240	#107	—	43	1	no	—
241	#107	—	43	2	no	—
242	#107	—	86	1	no	—
243	#107	C₂²×D₄₃	1	1	yes	characteristic
244	#107	C₂²×D₄₃	1	2	yes	—
245	#106	—	86	2	no	—
246	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
247	#102	—	2	1	no	—
248	#102	—	2	2	no	—
249	#102	—	86	1	no	—
250	#102	—	86	2	no	—
251	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
252	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
253	#41	—	8	4	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	172	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	86	1	no	—
259	C₂×D₄	—	172	2	no	—
260	C₂×D₄	#10621882	1	1	yes	characteristic
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	172	1	no	—
264	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	86	1	no	—
268	D₈	—	86	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
273	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	86	1	no	—
279	C₄⋊C₄	—	86	2	no	—
280	C₄⋊C₄	—	172	1	no	—
281	C₄⋊C₄	#10621882	1	1	yes	characteristic
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	172	2	no	—
285	C₂²⋊C₄	—	172	2	no	—
286	C₂²⋊C₄	—	172	1	no	—
287	C₂²⋊C₄	#10621882	1	2	yes	—
288	C₄²	—	172	2	no	—
289	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
290	C₁₆	—	43	2	no	—
291	C₁₆	—	86	2	no	—
292	C₁₆	—	86	1	no	—
293	C₂³	—	4	1	no	—
294	C₂³	—	172	1	no	—
295	C₂³	C₂×#10621882	1	2	yes	—
296	Q₈	—	8	2	no	—
297	Q₈	—	688	1	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	8	4	no	—
302	D₄	—	172	1	no	—
303	D₄	—	172	1	no	—
304	D₄	—	688	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	344	2	no	—
312	C₂×C₄	—	172	1	no	—
313	C₂×C₄	—	172	2	no	—
314	C₂×C₄	C₂×#10621882	1	2	yes	—
315	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
316	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	#41×D₄₃	1	1	yes	characteristic
321	C₈	#13063372	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	344	1	no	—
329	C₂²	—	344	1	no	—
330	C₂²	—	344	1	no	—
331	C₂²	C₂²×#10621882	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	344	1	no	—
337	C₄	—	344	2	no	—
338	C₄	#14583335	1	2	yes	—
339	C₄	#14583548	1	1	yes	characteristic
340	C₄	#14583714	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	16	1	no	—
343	C₂	—	688	1	no	—
344	C₂	#19146938	1	1	yes	characteristic
345	C₂	C₂×#14583335	1	1	yes	characteristic
346	C₂	#19147739	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3334	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3334	D₄₃
hypocenter	C₄₃	#33921
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3334

Complements

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

N	H ≅ G/N	type
1. #19147018	C₂	semidirect
2. #19136875	C₂	semidirect
3. #19136783	C₂	semidirect
4. #19136780	C₂	semidirect
5. #19136742	C₂	semidirect
6. #19136729	C₂	semidirect
7. #19136696	C₂	semidirect
8. C₄×#13063266	C₂	semidirect
9. #19136427	C₂	semidirect
10. #19136381	C₂	semidirect
11. C₄₃×#3334	C₂	semidirect
118. C₄₃⋊C₁₆	#41	semidirect
169. #3334	D₄₃	semidirect
214. C₄₃	#33921	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19147018	2
#19136875	2
#19136783	2
#19136780	2
#19136742	2
#19136729	2
#19136696	2
C₄×#13063266	2
#19136427	2
#19136381	2
C₄₃×#3334	2
#33921	43