Subgroups — #25254664

Order 11008.

510 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	#19147605	C₂	1	1	yes	maximal, characteristic
2	#19147599	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136863	C₂	1	2	yes	maximal, complemented
4	#19136852	C₂	1	1	yes	maximal, characteristic
5	#19136659	C₂	1	2	yes	maximal, complemented
6	#19136639	C₂	1	2	yes	maximal, complemented
7	#19136636	C₂	1	1	yes	maximal, characteristic
8	#19136500	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136462	C₂	1	2	yes	maximal, complemented
10	C₄×#13063271	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583634	—	2	1	no	—
13	#14583624	—	2	1	no	—
14	C₂×#13063271	C₂²	1	1	yes	characteristic
15	C₂×#13063270	C₂²	1	2	yes	—
16	C₂×#13063268	C₂²	1	2	yes	—
17	C₂×#13063268	C₂²	1	2	yes	—
18	#14583284	C₂²	1	2	yes	—
19	#14583277	C₂²	1	2	yes	—
20	#14583276	C₂²	1	1	yes	characteristic
21	#14583117	—	2	2	no	—
22	#14583107	—	2	2	no	—
23	#14583106	—	2	1	no	—
24	#14583099	—	2	1	no	—
25	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
27	Q₁₆×Dic₄₃	C₂²	1	2	yes	—
28	#14582981	C₂²	1	2	yes	—
29	#14582724	C₂²	1	1	yes	characteristic
30	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
31	#14582703	C₂²	1	1	yes	characteristic
32	#14582702	C₂²	1	1	yes	characteristic
33	#14582701	C₂²	1	2	yes	—
34	#14582674	C₂²	1	2	yes	—
35	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₂×Q₃₂×C₄₃	C₂²	1	2	yes	—
37	C₄₃×#494	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄₃×#358	C₂²	1	2	yes	—
41	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
42	C₂×#10621885	—	2	2	no	—
43	C₂×#10621884	—	2	2	no	—
44	C₂×D₃₄₄	—	2	1	no	—
45	C₂×D₃₄₄	C₂³	1	2	yes	—
46	#13063323	—	2	2	no	—
47	#13063322	—	2	1	no	—
48	#13063316	—	4	1	no	—
49	#13063315	—	4	2	no	—
50	#13063314	—	2	2	no	—
51	#13063313	—	4	1	no	—
52	#13063310	—	4	2	no	—
53	Q₈×Dic₄₃	—	2	2	no	—
54	C₄×Dic₈₆	—	2	1	no	—
55	C₄×Dic₈₆	D₄	1	1	yes	characteristic
56	#13063290	—	4	1	no	—
57	C₄×D₁₇₂	—	4	1	no	—
58	#13063287	—	2	2	no	—
59	#13063271	—	2	2	no	—
60	#13063270	—	4	2	no	—
61	#13063268	—	2	4	no	—
62	#13063268	—	4	2	no	—
63	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
65	#13063259	C₂³	1	1	yes	characteristic
66	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
67	Dic₄₃⋊C₈	D₄	1	2	yes	—
68	#13063250	—	2	1	no	—
69	#13063249	—	2	2	no	—
70	#13063245	—	2	2	no	—
71	C₈×Dic₄₃	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₄₃	—	2	1	no	—
73	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₄₃	—	2	1	no	—
76	C₄×Q₈×C₄₃	—	2	1	no	—
77	Q₃₂×C₄₃	—	4	2	no	—
78	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
79	#107×C₄₃	C₂³	1	1	yes	characteristic
80	#103×C₄₃	—	2	1	no	—
81	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₃	—	2	1	no	—
83	C₂×Q₈×C₄₃	—	2	2	no	—
84	C₂×D₁₇₂	—	2	2	no	—
85	C₂×D₁₇₂	—	4	1	no	—
86	C₂×D₁₇₂	—	4	1	no	—
87	C₂×C₄×D₄₃	—	4	2	no	—
88	C₂×Dic₈₆	—	2	1	no	—
89	C₂×Dic₈₆	D₈	1	1	yes	characteristic
90	Q₁₆×C₄₃	—	2	2	no	—
91	Q₁₆×C₄₃	—	2	1	no	—
92	Q₁₆×C₄₃	—	2	1	no	—
93	Q₁₆×C₄₃	—	2	2	no	—
94	Q₁₆×C₄₃	—	8	1	no	—
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	—	4	1	no	—
99	C₄⋊C₄×C₄₃	—	4	1	no	—
100	C₄⋊C₄×C₄₃	—	4	1	no	—
101	C₄²×C₄₃	—	4	1	no	—
102	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
103	#10621885	—	8	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	1	no	—
109	C₄⋊Dic₄₃	—	4	2	no	—
110	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
111	Dic₄₃⋊C₄	—	4	1	no	—
112	Dic₄₃⋊C₄	—	4	2	no	—
113	Dic₄₃⋊C₄	—	4	2	no	—
114	Dic₄₃⋊C₄	D₈	1	2	yes	—
115	C₄×Dic₄₃	—	4	1	no	—
116	C₄×Dic₄₃	—	4	2	no	—
117	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
118	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
119	Dic₁₇₂	—	2	1	no	—
120	Dic₁₇₂	—	2	1	no	—
121	D₃₄₄	—	2	2	no	—
122	D₃₄₄	—	2	2	no	—
123	D₃₄₄	—	8	1	no	—
124	C₁₆×C₄₃	—	2	2	no	—
125	C₄₃⋊C₁₆	—	2	1	no	—
126	C₄₃⋊C₁₆	#41	1	2	yes	—
127	C₂²×D₄₃	—	4	2	no	—
128	Q₈×C₄₃	—	4	2	no	—
129	Q₈×C₄₃	—	4	1	no	—
130	Q₈×C₄₃	—	4	1	no	—
131	Q₈×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₄₃	—	4	1	no	—
139	C₂×Dic₄₃	—	4	1	no	—
140	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
141	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
142	D₁₇₂	—	4	2	no	—
143	D₁₇₂	—	4	2	no	—
144	D₁₇₂	—	8	2	no	—
145	D₁₇₂	—	16	1	no	—
146	C₄×D₄₃	—	16	2	no	—
147	Dic₈₆	—	2	2	no	—
148	Dic₈₆	—	4	1	no	—
149	Dic₈₆	—	4	1	no	—
150	C₈×C₄₃	—	2	1	no	—
151	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
152	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
153	C₄₃⋊C₈	—	2	2	no	—
154	#33967	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₄₃	—	8	2	no	—
157	C₂×D₄₃	—	8	2	no	—
158	C₂×D₄₃	—	8	2	no	—
159	C₄×C₄₃	—	2	1	no	—
160	C₄×C₄₃	—	8	2	no	—
161	C₄×C₄₃	—	8	1	no	—
162	C₄×C₄₃	—	8	1	no	—
163	C₄×C₄₃	#510	1	2	yes	—
164	C₄×C₄₃	#550	1	1	yes	characteristic
165	C₄×C₄₃	#576	1	1	yes	characteristic
166	Dic₄₃	—	2	2	no	—
167	Dic₄₃	—	2	2	no	—
168	Dic₄₃	—	8	1	no	—
169	Dic₄₃	—	8	1	no	—
170	#3396	—	43	1	no	—
171	#3395	—	43	1	no	—
172	#2251	—	43	2	no	—
173	#2248	—	43	1	no	—
174	#2211	—	43	2	no	—
175	#2209	D₄₃	1	1	yes	characteristic, complemented
176	#2208	—	43	2	no	—
177	#2207	—	43	1	no	—
178	#2183	—	43	1	no	—
179	#2175	—	43	2	no	—
180	C₄×Q₃₂	—	43	1	no	—
181	C₂×C₄₃	#3391	1	1	yes	characteristic
182	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
183	C₂×C₄₃	#3416	1	1	yes	characteristic
184	D₄₃	—	16	2	no	—
185	#559	—	86	1	no	—
186	#557	—	86	1	no	—
187	C₂×Q₃₂	—	43	1	no	—
188	C₂×Q₃₂	C₂×D₄₃	1	2	yes	—
189	C₂×SD₃₂	—	43	2	no	—
190	C₂×SD₃₂	—	43	2	no	—
191	C₂×SD₃₂	—	43	2	no	—
192	#495	—	43	2	no	—
193	#494	—	43	2	no	—
194	#494	C₂×D₄₃	1	1	yes	characteristic
195	#493	—	43	1	no	—
196	#463	—	86	2	no	—
197	#462	—	86	1	no	—
198	#462	—	86	2	no	—
199	#460	—	86	1	no	—
200	C₈×Q₈	—	43	1	no	—
201	C₄×Q₁₆	—	43	2	no	—
202	C₄×Q₁₆	—	43	1	no	—
203	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
204	C₄×D₈	—	43	2	no	—
205	#366	—	43	1	no	—
206	C₄⋊C₁₆	—	43	2	no	—
207	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
208	#358	—	43	1	no	—
209	#358	—	43	1	no	—
210	#358	C₂×D₄₃	1	2	yes	—
211	#357	—	43	2	no	—
212	#346	—	43	2	no	—
213	C₄×C₁₆	—	43	1	no	—
214	C₄₃	#33967	1	1	yes	characteristic, complemented, Sylow
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	—	43	1	no	—
217	C₂×Q₁₆	—	86	2	no	—
218	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
219	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
220	C₂×SD₁₆	—	86	2	no	—
221	C₂×D₈	—	43	2	no	—
222	C₂×D₈	—	86	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	86	2	no	—
225	#127	—	86	1	no	—
226	#125	—	172	2	no	—
227	#125	—	172	1	no	—
228	#124	—	86	2	no	—
229	#124	—	172	1	no	—
230	#123	—	172	2	no	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	—	43	1	no	—
233	C₄×Q₈	—	86	2	no	—
234	C₄×Q₈	—	86	1	no	—
235	C₄×Q₈	—	172	1	no	—
236	C₄×D₄	—	86	2	no	—
237	C₄×D₄	—	172	1	no	—
238	Q₃₂	—	4	2	no	—
239	Q₃₂	—	86	2	no	—
240	SD₃₂	—	86	4	no	—
241	SD₃₂	—	172	2	no	—
242	SD₃₂	—	172	2	no	—
243	C₂×C₁₆	—	43	1	no	—
244	C₂×C₁₆	—	43	1	no	—
245	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
246	#107	—	43	1	no	—
247	#107	C₂²×D₄₃	1	1	yes	characteristic
248	C₄⋊C₈	—	43	2	no	—
249	C₄⋊C₈	—	43	1	no	—
250	#103	—	2	1	no	—
251	#103	—	86	2	no	—
252	#103	—	86	1	no	—
253	#102	—	86	2	no	—
254	C₄×C₈	—	43	2	no	—
255	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	43	1	no	—
259	C₂×Q₈	—	86	1	no	—
260	C₂×D₄	—	86	2	no	—
261	C₂×D₄	—	172	1	no	—
262	C₂×D₄	—	172	1	no	—
263	C₂²×C₄	—	172	2	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	8	1	no	—
269	Q₁₆	—	86	1	no	—
270	Q₁₆	—	86	1	no	—
271	Q₁₆	—	344	2	no	—
272	SD₁₆	—	344	2	no	—
273	D₈	—	86	2	no	—
274	D₈	—	86	2	no	—
275	D₈	—	344	1	no	—
276	C₂×C₈	—	43	2	no	—
277	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
278	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	43	1	no	—
284	C₄⋊C₄	—	43	2	no	—
285	C₄⋊C₄	—	86	1	no	—
286	C₄⋊C₄	—	172	2	no	—
287	C₄⋊C₄	—	172	2	no	—
288	C₄⋊C₄	—	172	2	no	—
289	C₄⋊C₄	—	172	1	no	—
290	C₂²⋊C₄	—	172	2	no	—
291	C₂²⋊C₄	—	172	2	no	—
292	C₂²⋊C₄	—	172	2	no	—
293	C₄²	—	4	1	no	—
294	C₄²	—	43	2	no	—
295	C₄²	—	172	2	no	—
296	C₄²	—	172	1	no	—
297	C₄²	D₄×D₄₃	1	1	yes	characteristic
298	C₁₆	—	2	2	no	—
299	C₁₆	—	43	2	no	—
300	C₁₆	—	86	1	no	—
301	C₂³	—	172	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	86	2	no	—
307	Q₈	—	172	1	no	—
308	Q₈	—	172	1	no	—
309	D₄	—	344	2	no	—
310	D₄	—	172	2	no	—
311	D₄	—	172	2	no	—
312	D₄	—	688	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	43	2	no	—
317	C₂×C₄	—	43	2	no	—
318	C₂×C₄	—	172	1	no	—
319	C₂×C₄	—	172	1	no	—
320	C₂×C₄	—	688	2	no	—
321	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
322	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	86	2	no	—
326	C₈	#41×D₄₃	1	1	yes	characteristic
327	C₈	#13063370	1	1	yes	characteristic
328	C₂²	—	344	2	no	—
329	C₂²	—	344	2	no	—
330	C₂²	—	344	2	no	—
331	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	86	2	no	—
337	C₄	—	86	2	no	—
338	C₄	—	344	1	no	—
339	C₄	—	344	1	no	—
340	C₄	#14583341	1	2	yes	—
341	C₄	#14583567	1	1	yes	characteristic
342	C₄	#14583703	1	1	yes	characteristic
343	C₂	—	688	2	no	—
344	C₂	#19147566	1	1	yes	characteristic
345	C₂	C₂×#14583341	1	1	yes	characteristic
346	C₂	#19147740	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2209	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#14583276	C₂²
hypercenter	#2209	D₄₃
hypocenter	C₄₃	#33967
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
2. #19147599	C₂	semidirect
3. #19136863	C₂	semidirect
5. #19136659	C₂	semidirect
6. #19136639	C₂	semidirect
8. #19136500	C₂	semidirect
9. #19136462	C₂	semidirect
10. C₄×#13063271	C₂	semidirect
11. C₄₃×#2209	C₂	semidirect
175. #2209	D₄₃	semidirect
214. C₄₃	#33967	semidirect

Maximal subgroups

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

subgroup	index
#19147605	2
#19147599	2
#19136863	2
#19136852	2
#19136659	2
#19136639	2
#19136636	2
#19136500	2
#19136462	2
C₄×#13063271	2
C₄₃×#2209	2
#33967	43