Subgroups — #25254397

Order 11008.

520 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147605	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136833	C₂	1	2	yes	maximal, complemented
4	#19136819	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136741	C₂	1	2	yes	maximal, complemented
6	#19136737	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136652	C₂	1	2	yes	maximal, complemented
8	#19136630	C₂	1	1	yes	maximal, characteristic
9	#19136503	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136468	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583634	—	2	1	no	—
13	D₄×Dic₈₆	—	2	1	no	—
14	C₂×#13063266	C₂²	1	2	yes	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583303	C₂²	1	2	yes	—
17	#14583284	C₂²	1	2	yes	—
18	#14583276	C₂²	1	1	yes	characteristic
19	#14583179	—	2	1	no	—
20	#14583177	—	2	2	no	—
21	#14583117	—	2	2	no	—
22	#14583099	—	2	1	no	—
23	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
24	D₈×Dic₄₃	C₂²	1	2	yes	—
25	#14582981	C₂²	1	2	yes	—
26	#14582726	C₂²	1	2	yes	—
27	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
28	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
29	#14582703	C₂²	1	2	yes	—
30	#14582701	C₂²	1	2	yes	—
31	#14582701	C₂²	1	2	yes	—
32	#14582699	C₂²	1	2	yes	—
33	#14582674	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₄₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₄₃×#367	C₂²	1	1	yes	characteristic
38	C₄₃×#358	C₂²	1	1	yes	characteristic
39	C₄₃×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₂²×Dic₈₆	—	4	1	no	—
42	C₂×#10621884	—	2	2	no	—
43	C₂×D₃₄₄	—	2	1	no	—
44	C₂×D₃₄₄	C₂³	1	2	yes	—
45	#13063325	—	2	1	no	—
46	#13063324	—	2	2	no	—
47	#13063322	—	2	1	no	—
48	#13063314	—	2	2	no	—
49	#13063313	—	4	1	no	—
50	#13063310	—	4	2	no	—
51	#13063304	—	4	1	no	—
52	#13063302	—	4	2	no	—
53	C₄×Dic₈₆	D₄	1	1	yes	characteristic
54	C₄×D₁₇₂	—	4	1	no	—
55	D₄×Dic₄₃	—	2	2	no	—
56	#13063287	—	2	2	no	—
57	#13063266	—	2	4	no	—
58	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
59	#13063259	—	2	1	no	—
60	#13063259	C₂³	1	2	yes	—
61	#13063258	—	2	2	no	—
62	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
63	Dic₄₃⋊C₈	D₄	1	2	yes	—
64	#13063246	—	2	2	no	—
65	#13063245	—	2	2	no	—
66	C₈×Dic₄₃	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₄₃	—	2	1	no	—
70	C₄×D₄×C₄₃	—	2	1	no	—
71	SD₃₂×C₄₃	—	2	2	no	—
72	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
74	#107×C₄₃	C₂³	1	1	yes	characteristic
75	#107×C₄₃	C₂³	1	1	yes	characteristic
76	#103×C₄₃	—	2	1	no	—
77	#102×C₄₃	—	2	1	no	—
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₃	—	2	1	no	—
80	C₂×D₄×C₄₃	—	2	1	no	—
81	C₂²×C₄×C₄₃	—	4	1	no	—
82	C₂²×Dic₄₃	—	4	2	no	—
83	C₂×D₁₇₂	—	2	2	no	—
84	C₂×D₁₇₂	—	4	1	no	—
85	C₂×D₁₇₂	—	4	1	no	—
86	C₂×C₄×D₄₃	—	4	2	no	—
87	C₂×Dic₈₆	—	4	1	no	—
88	C₂×Dic₈₆	—	4	2	no	—
89	C₂×Dic₈₆	—	8	4	no	—
90	C₂×Dic₈₆	D₈	1	1	yes	characteristic
91	Q₁₆×C₄₃	—	2	1	no	—
92	Q₁₆×C₄₃	—	2	1	no	—
93	D₈×C₄₃	—	2	1	no	—
94	D₈×C₄₃	#41	1	2	yes	—
95	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₄₃	—	2	1	no	—
98	C₄⋊C₄×C₄₃	—	2	1	no	—
99	C₄⋊C₄×C₄₃	—	4	1	no	—
100	C₂²⋊C₄×C₄₃	—	4	1	no	—
101	C₄²×C₄₃	—	4	1	no	—
102	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂²⋊Dic₄₃	—	4	2	no	—
104	#10621884	—	8	2	no	—
105	(C₂×D₄₃)⋊C₄	—	4	2	no	—
106	(C₂×D₄₃)⋊C₄	—	4	2	no	—
107	(C₂×D₄₃)⋊C₄	—	4	2	no	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	4	1	no	—
110	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
111	Dic₄₃⋊C₄	—	4	2	no	—
112	Dic₄₃⋊C₄	D₈	1	2	yes	—
113	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
114	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
115	D₃₄₄	—	2	2	no	—
116	D₃₄₄	—	2	2	no	—
117	D₃₄₄	—	8	1	no	—
118	C₁₆×C₄₃	—	2	1	no	—
119	C₁₆×C₄₃	—	2	1	no	—
120	C₄₃⋊C₁₆	—	2	2	no	—
121	C₂³×C₄₃	—	4	1	no	—
122	C₂²×D₄₃	—	4	2	no	—
123	Q₈×C₄₃	—	4	1	no	—
124	Q₈×C₄₃	—	4	1	no	—
125	D₄×C₄₃	—	2	2	no	—
126	D₄×C₄₃	—	4	1	no	—
127	C₂×C₄×C₄₃	—	4	1	no	—
128	C₂×C₄×C₄₃	—	4	1	no	—
129	C₂×C₄×C₄₃	—	4	1	no	—
130	C₂×C₄×C₄₃	—	8	2	no	—
131	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	—	8	4	no	—
136	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
137	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
138	D₁₇₂	—	4	2	no	—
139	D₁₇₂	—	4	2	no	—
140	D₁₇₂	—	8	2	no	—
141	D₁₇₂	—	16	1	no	—
142	C₄×D₄₃	—	16	2	no	—
143	Dic₈₆	—	2	2	no	—
144	Dic₈₆	—	8	4	no	—
145	Dic₈₆	—	8	2	no	—
146	C₈×C₄₃	—	2	1	no	—
147	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
149	C₄₃⋊C₈	—	2	2	no	—
150	#33940	—	43	1	no	maximal, Sylow
151	C₂²×C₄₃	—	4	2	no	—
152	C₂²×C₄₃	—	8	1	no	—
153	C₂²×C₄₃	—	8	1	no	—
154	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₄₃	—	8	2	no	—
156	C₂×D₄₃	—	8	2	no	—
157	C₂×D₄₃	—	8	2	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	8	1	no	—
160	C₄×C₄₃	—	8	1	no	—
161	C₄×C₄₃	—	8	1	no	—
162	C₄×C₄₃	#509	1	2	yes	—
163	C₄×C₄₃	#548	1	1	yes	characteristic
164	C₄×C₄₃	#577	1	1	yes	characteristic
165	Dic₄₃	—	2	2	no	—
166	Dic₄₃	—	2	2	no	—
167	Dic₄₃	—	8	2	no	—
168	#3396	—	43	1	no	—
169	Q₈×D₈	—	43	1	no	—
170	#2243	—	43	2	no	—
171	#2240	—	43	1	no	—
172	#2226	—	43	2	no	—
173	#2225	—	43	1	no	—
174	#2210	—	43	2	no	—
175	#2206	—	43	1	no	—
176	#2183	—	43	1	no	—
177	#2177	—	43	2	no	—
178	C₄×SD₃₂	D₄₃	1	1	yes	characteristic, complemented
179	C₂×C₄₃	—	8	2	no	—
180	C₂×C₄₃	#3358	1	1	yes	characteristic
181	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
182	C₂×C₄₃	#3416	1	1	yes	characteristic
183	D₄₃	—	16	2	no	—
184	#559	—	86	1	no	—
185	D₄×Q₈	—	86	1	no	—
186	C₂×SD₃₂	C₂×D₄₃	1	1	yes	characteristic
187	C₂×D₁₆	—	43	2	no	—
188	#500	—	43	2	no	—
189	#500	—	43	1	no	—
190	#495	—	43	2	no	—
191	#493	—	43	1	no	—
192	#474	—	86	2	no	—
193	#474	—	86	1	no	—
194	#463	—	86	2	no	—
195	#460	—	86	1	no	—
196	C₈×Q₈	—	43	1	no	—
197	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
198	C₄×D₈	—	43	2	no	—
199	C₄×D₈	—	43	2	no	—
200	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
201	#367	—	43	2	no	—
202	#367	C₂×D₄₃	1	1	yes	characteristic
203	C₄⋊C₁₆	—	43	2	no	—
204	C₄⋊C₁₆	—	43	1	no	—
205	#358	—	43	2	no	—
206	#358	C₂×D₄₃	1	1	yes	characteristic
207	#357	—	43	2	no	—
208	#357	—	43	2	no	—
209	#357	—	43	2	no	—
210	#357	C₂×D₄₃	1	1	yes	characteristic
211	#346	—	43	2	no	—
212	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
213	C₄₃	#33940	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	—	172	1	no	—
215	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
216	C₂×SD₁₆	—	86	2	no	—
217	C₂×D₈	—	43	2	no	—
218	C₂×D₈	—	86	1	no	—
219	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
220	C₄⋊Q₈	—	86	2	no	—
221	C₄⋊Q₈	—	86	1	no	—
222	#127	—	86	1	no	—
223	#124	—	86	2	no	—
224	#124	—	172	1	no	—
225	#123	—	172	2	no	—
226	C₂²⋊Q₈	—	172	2	no	—
227	C₂²⋊Q₈	—	172	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	43	1	no	—
230	C₄×D₄	—	2	1	no	—
231	C₄×D₄	—	86	2	no	—
232	C₄×D₄	—	86	2	no	—
233	C₄×D₄	—	172	1	no	—
234	SD₃₂	—	2	2	no	—
235	D₁₆	—	86	4	no	—
236	C₂×C₁₆	—	43	2	no	—
237	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
238	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
239	#107	—	43	2	no	—
240	#107	—	86	2	no	—
241	#107	—	86	1	no	—
242	#107	C₂²×D₄₃	1	1	yes	characteristic
243	#107	C₂²×D₄₃	1	1	yes	characteristic
244	C₄⋊C₈	—	43	2	no	—
245	C₄⋊C₈	—	43	1	no	—
246	#103	—	2	1	no	—
247	#102	—	2	1	no	—
248	#102	—	86	2	no	—
249	#102	—	86	2	no	—
250	C₄×C₈	—	43	2	no	—
251	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	43	1	no	—
254	C₂×Q₈	—	344	4	no	—
255	C₂×Q₈	—	172	2	no	—
256	C₂×Q₈	—	172	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	86	2	no	—
259	C₂×D₄	—	172	1	no	—
260	C₂×D₄	—	172	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	172	2	no	—
263	C₂²×C₄	—	172	2	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	—	2	1	no	—
266	SD₁₆	—	344	2	no	—
267	D₈	—	2	1	no	—
268	D₈	—	86	2	no	—
269	D₈	—	86	2	no	—
270	D₈	—	344	1	no	—
271	D₈	#10621898	1	2	yes	—
272	C₂×C₈	—	43	2	no	—
273	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	43	1	no	—
279	C₄⋊C₄	—	43	2	no	—
280	C₄⋊C₄	—	86	2	no	—
281	C₄⋊C₄	—	172	2	no	—
282	C₄⋊C₄	—	172	1	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	172	2	no	—
285	C₂²⋊C₄	—	172	2	no	—
286	C₂²⋊C₄	—	172	2	no	—
287	C₂²⋊C₄	—	172	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	43	2	no	—
290	C₄²	D₄×D₄₃	1	1	yes	characteristic
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	86	2	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	172	2	no	—
296	Q₈	—	4	1	no	—
297	Q₈	—	4	1	no	—
298	Q₈	—	86	2	no	—
299	Q₈	—	344	4	no	—
300	Q₈	—	344	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	4	1	no	—
303	D₄	—	344	2	no	—
304	D₄	—	172	2	no	—
305	D₄	—	172	2	no	—
306	D₄	—	688	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	43	2	no	—
312	C₂×C₄	—	43	2	no	—
313	C₂×C₄	—	344	4	no	—
314	C₂×C₄	—	172	2	no	—
315	C₂×C₄	—	688	2	no	—
316	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
317	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	86	2	no	—
321	C₈	C₂×#10621898	1	1	yes	characteristic
322	C₈	#13063372	1	1	yes	characteristic
323	C₂²	—	4	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	344	2	no	—
327	C₂²	—	344	2	no	—
328	C₂²	—	344	2	no	—
329	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	86	2	no	—
335	C₄	—	86	2	no	—
336	C₄	—	344	2	no	—
337	C₄	#14583336	1	2	yes	—
338	C₄	#14583558	1	1	yes	characteristic
339	C₄	#14583706	1	1	yes	characteristic
340	C₂	—	8	2	no	—
341	C₂	—	688	2	no	—
342	C₂	#19147228	1	1	yes	characteristic
343	C₂	C₂×#14583336	1	1	yes	characteristic
344	C₂	#19147738	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×SD₃₂×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19136630	C₂
hypercenter	C₄×SD₃₂	D₄₃
hypocenter	C₄₃	#33940
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #19147605	C₂	semidirect
2. D₈×Dic₈₆	C₂	semidirect
3. #19136833	C₂	semidirect
4. #19136819	C₂	semidirect
5. #19136741	C₂	semidirect
6. #19136737	C₂	semidirect
7. #19136652	C₂	semidirect
9. #19136503	C₂	semidirect
10. #19136468	C₂	semidirect
11. C₄×SD₃₂×C₄₃	C₂	semidirect
178. C₄×SD₃₂	D₄₃	semidirect
213. C₄₃	#33940	semidirect

Maximal subgroups

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

subgroup	index
#19147605	2
D₈×Dic₈₆	2
#19136833	2
#19136819	2
#19136741	2
#19136737	2
#19136652	2
#19136630	2
#19136503	2
#19136468	2
C₄×SD₃₂×C₄₃	2
#33940	43