Subgroups — #25253540

Order 11008.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19146355	C₂	1	1	yes	maximal, characteristic
2	#19136820	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136786	C₂	1	2	yes	maximal, complemented
4	#19136734	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136693	C₂	1	2	yes	maximal, complemented
6	#19136635	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136533	C₂	1	2	yes	maximal, complemented
8	C₄×#13063271	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136300	C₂	1	2	yes	maximal, complemented
11	C₄₃×#3342	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	2	yes	—
13	#14583570	—	2	1	no	—
14	C₂×#13063271	C₂²	1	1	yes	characteristic
15	C₂×#13063268	C₂²	1	2	yes	—
16	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
17	#14583276	C₂²	1	1	yes	characteristic
18	#14583135	C₂²	1	2	yes	—
19	#14583100	—	2	1	no	—
20	#14583046	—	2	2	no	—
21	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
22	#14582726	C₂²	1	2	yes	—
23	#14582724	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#14582703	C₂²	1	1	yes	characteristic
26	#14582702	C₂²	1	2	yes	—
27	#14582702	C₂²	1	1	yes	characteristic
28	#14582699	C₂²	1	2	yes	—
29	#14582699	C₂²	1	2	yes	—
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×#556	—	2	1	no	—
34	C₄₃×#500	C₂²	1	1	yes	characteristic
35	C₄₃×#485	—	2	2	no	—
36	C₄₃×#475	—	2	1	no	—
37	C₄₃×#467	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
40	C₄₃×#427	C₂²	1	2	yes	—
41	C₂×#10621898	—	2	2	no	—
42	#13063347	—	2	4	no	—
43	#13063347	—	4	2	no	—
44	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
45	C₂×#10621873	—	2	2	no	—
46	C₂×D₃₄₄	—	2	1	no	—
47	C₂×D₃₄₄	C₂³	1	2	yes	—
48	#13063322	—	2	1	no	—
49	#13063321	—	4	1	no	—
50	#13063299	—	2	2	no	—
51	#13063298	—	4	2	no	—
52	C₄×Dic₈₆	—	2	1	no	—
53	#13063287	—	4	1	no	—
54	#13063271	—	2	2	no	—
55	#13063268	—	4	2	no	—
56	C₂×C₄₃⋊C₁₆	—	4	2	no	—
57	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
60	#13063259	C₂³	1	1	yes	characteristic
61	#13063250	—	2	1	no	—
62	#13063247	—	2	2	no	—
63	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₃	—	2	1	no	—
66	#126×C₄₃	—	4	2	no	—
67	#124×C₄₃	—	4	1	no	—
68	C₂²⋊Q₈×C₄₃	—	2	2	no	—
69	C₄×Q₈×C₄₃	—	2	1	no	—
70	C₄×Q₈×C₄₃	—	4	1	no	—
71	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
72	#117×C₄₃	—	2	2	no	—
73	#107×C₄₃	—	2	1	no	—
74	#107×C₄₃	C₂³	1	2	yes	—
75	#107×C₄₃	C₂³	1	1	yes	characteristic
76	#106×C₄₃	—	2	2	no	—
77	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
78	#103×C₄₃	—	2	1	no	—
79	#103×C₄₃	—	2	2	no	—
80	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
81	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₃	—	2	1	no	—
83	C₂×Q₈×C₄₃	—	4	1	no	—
84	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₂²⋊D₄₃	—	4	2	no	—
87	C₂×C₂²⋊D₄₃	—	4	2	no	—
88	#10621898	—	4	4	no	—
89	#10621898	—	8	2	no	—
90	C₂×D₁₇₂	—	2	2	no	—
91	C₂×D₁₇₂	—	4	1	no	—
92	C₂×D₁₇₂	—	4	1	no	—
93	C₂×C₄×D₄₃	—	4	2	no	—
94	C₂×Dic₈₆	—	2	1	no	—
95	Q₁₆×C₄₃	—	2	1	no	—
96	Q₁₆×C₄₃	—	2	1	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₄₃	—	2	1	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	4	1	no	—
105	C₄⋊C₄×C₄₃	—	4	2	no	—
106	C₄⋊C₄×C₄₃	—	4	1	no	—
107	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₄₃	—	4	2	no	—
109	C₂²⋊C₄×C₄₃	—	4	2	no	—
110	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
111	C₄²×C₄₃	—	4	2	no	—
112	C₄²×C₄₃	—	4	1	no	—
113	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
114	(C₂×D₄₃)⋊C₄	—	4	2	no	—
115	C₄⋊Dic₄₃	—	2	1	no	—
116	Dic₄₃⋊C₄	—	4	1	no	—
117	C₄×Dic₄₃	—	4	1	no	—
118	Dic₁₇₂	—	2	1	no	—
119	Dic₁₇₂	#41	1	2	yes	—
120	D₃₄₄	—	2	2	no	—
121	D₃₄₄	—	2	2	no	—
122	D₃₄₄	—	8	1	no	—
123	#10621873	—	4	4	no	—
124	C₄₃⋊C₁₆	—	2	1	no	—
125	C₄₃⋊C₁₆	—	2	1	no	—
126	C₄₃⋊C₁₆	—	2	2	no	—
127	C₂³×C₄₃	C₂×D₈	1	2	yes	—
128	C₂²×D₄₃	—	4	2	no	—
129	Q₈×C₄₃	—	4	1	no	—
130	Q₈×C₄₃	—	4	1	no	—
131	Q₈×C₄₃	—	8	2	no	—
132	D₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₄₃	—	8	4	no	—
143	C₂²⋊D₄₃	—	16	2	no	—
144	C₂×Dic₄₃	—	4	1	no	—
145	C₂×Dic₄₃	—	4	1	no	—
146	D₁₇₂	—	4	2	no	—
147	D₁₇₂	—	4	2	no	—
148	D₁₇₂	—	8	2	no	—
149	D₁₇₂	—	16	1	no	—
150	C₄×D₄₃	—	8	4	no	—
151	Dic₈₆	—	2	2	no	—
152	Dic₈₆	—	4	1	no	—
153	C₈×C₄₃	—	2	2	no	—
154	C₈×C₄₃	—	2	1	no	—
155	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
157	#33871	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	2	no	—
159	C₂²×C₄₃	—	2	2	no	—
160	C₂²×C₄₃	—	2	2	no	—
161	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₄₃	—	8	2	no	—
163	C₂×D₄₃	—	8	2	no	—
164	C₂×D₄₃	—	8	2	no	—
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	2	2	no	—
167	C₄×C₄₃	—	8	2	no	—
168	C₄×C₄₃	—	8	1	no	—
169	C₄×C₄₃	—	8	1	no	—
170	C₄×C₄₃	#508	1	2	yes	—
171	C₄×C₄₃	#546	1	1	yes	characteristic
172	C₄×C₄₃	#577	1	1	yes	characteristic
173	Dic₄₃	—	4	2	no	—
174	Dic₄₃	—	8	1	no	—
175	#3342	D₄₃	1	1	yes	characteristic, complemented
176	#3287	—	43	1	no	—
177	#2240	—	43	1	no	—
178	#2234	—	43	2	no	—
179	#2225	—	43	1	no	—
180	#2217	—	43	2	no	—
181	#2207	—	43	1	no	—
182	#2189	—	43	2	no	—
183	C₄×Q₃₂	—	43	1	no	—
184	D₄×C₁₆	—	43	1	no	—
185	#2138	—	43	2	no	—
186	C₂×C₄₃	—	4	2	no	—
187	C₂×C₄₃	#3294	1	1	yes	characteristic
188	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
189	C₂×C₄₃	#3416	1	1	yes	characteristic
190	D₄₃	—	16	2	no	—
191	C₂×#135	—	43	2	no	—
192	#556	—	2	1	no	—
193	#550	—	86	1	no	—
194	C₂×Q₃₂	—	43	1	no	—
195	C₂×SD₃₂	—	43	2	no	—
196	C₂²×C₁₆	—	43	2	no	—
197	#500	C₂×D₄₃	1	1	yes	characteristic
198	#493	—	43	1	no	—
199	#485	—	2	2	no	—
200	#475	—	2	1	no	—
201	#467	C₂×D₄₃	1	2	yes	—
202	#466	—	43	2	no	—
203	#460	—	86	1	no	—
204	#449	—	86	2	no	—
205	C₄×Q₁₆	—	43	1	no	—
206	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
207	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
208	#427	C₂×D₄₃	1	2	yes	—
209	#367	—	43	2	no	—
210	#366	—	43	1	no	—
211	C₄⋊C₁₆	—	43	1	no	—
212	#358	—	43	1	no	—
213	#358	—	43	1	no	—
214	#358	—	43	2	no	—
215	#357	—	43	2	no	—
216	#357	—	43	2	no	—
217	#357	—	43	2	no	—
218	C₂²⋊C₁₆	—	43	2	no	—
219	C₄×C₁₆	—	43	1	no	—
220	C₄₃	#33871	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	86	2	no	—
222	#135	—	86	4	no	—
223	#135	—	172	2	no	—
224	C₂×Q₁₆	—	43	1	no	—
225	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
226	C₂×SD₁₆	—	86	2	no	—
227	C₂×D₈	—	43	2	no	—
228	C₂×D₈	—	86	1	no	—
229	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
230	C₄⋊Q₈	—	2	1	no	—
231	#127	—	86	1	no	—
232	#127	—	172	1	no	—
233	#126	—	4	2	no	—
234	#124	—	4	1	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	#121	—	86	2	no	—
237	#121	—	172	2	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	4	1	no	—
240	C₄×Q₈	—	86	1	no	—
241	C₄×D₄	—	172	1	no	—
242	C₄×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
243	#117	—	2	2	no	—
244	Q₃₂	—	86	2	no	—
245	SD₃₂	—	172	2	no	—
246	C₂×C₁₆	—	43	1	no	—
247	C₂×C₁₆	—	43	1	no	—
248	C₂×C₁₆	—	43	2	no	—
249	C₂×C₁₆	—	172	2	no	—
250	#107	—	2	1	no	—
251	#107	—	43	1	no	—
252	#107	C₂²×D₄₃	1	1	yes	characteristic
253	#107	C₂²×D₄₃	1	2	yes	—
254	#106	—	2	2	no	—
255	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
256	#103	—	2	1	no	—
257	#103	—	2	2	no	—
258	#103	—	86	1	no	—
259	#102	—	86	2	no	—
260	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
261	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
262	#41	—	344	2	no	—
263	#41	—	172	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	86	1	no	—
267	C₂×D₄	—	86	2	no	—
268	C₂×D₄	—	172	2	no	—
269	C₂×D₄	—	172	2	no	—
270	C₂×D₄	—	172	1	no	—
271	C₂×D₄	—	172	1	no	—
272	C₂×D₄	#10621882	1	1	yes	characteristic
273	C₂²×C₄	—	172	2	no	—
274	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	43	2	no	—
278	Q₁₆	—	86	1	no	—
279	SD₁₆	—	172	4	no	—
280	D₈	—	86	2	no	—
281	D₈	—	86	2	no	—
282	D₈	—	344	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
287	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	86	1	no	—
294	C₄⋊C₄	—	172	1	no	—
295	C₄⋊C₄	#10621882	1	1	yes	characteristic
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	172	2	no	—
299	C₂²⋊C₄	#10621882	1	2	yes	—
300	C₄²	—	4	2	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	172	1	no	—
303	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
304	C₁₆	—	86	1	no	—
305	C₁₆	—	86	1	no	—
306	C₁₆	—	86	2	no	—
307	C₂³	—	172	2	no	—
308	C₂³	C₂×#10621882	1	2	yes	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	8	2	no	—
312	Q₈	—	86	2	no	—
313	Q₈	—	172	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	344	4	no	—
316	D₄	—	344	2	no	—
317	D₄	—	172	2	no	—
318	D₄	—	172	2	no	—
319	D₄	—	688	1	no	—
320	D₄	—	688	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	344	4	no	—
327	C₂×C₄	—	172	1	no	—
328	C₂×C₄	—	172	1	no	—
329	C₂×C₄	C₂×#10621882	1	2	yes	—
330	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
331	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	2	no	—
335	C₈	#41×D₄₃	1	1	yes	characteristic
336	C₈	#13063371	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	344	2	no	—
341	C₂²	—	344	2	no	—
342	C₂²	—	344	2	no	—
343	C₂²	C₂²×#10621882	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	344	1	no	—
350	C₄	—	172	2	no	—
351	C₄	#14583329	1	2	yes	—
352	C₄	#14583537	1	1	yes	characteristic
353	C₄	#14583707	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	688	2	no	—
356	C₂	#19146402	1	1	yes	characteristic
357	C₂	C₂×#14583329	1	1	yes	characteristic
358	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×#3342	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19146355	C₂
hypercenter	#3342	D₄₃
hypocenter	C₄₃	#33871
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3342

Complements

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

N	H ≅ G/N	type
2. #19136820	C₂	semidirect
3. #19136786	C₂	semidirect
4. #19136734	C₂	semidirect
5. #19136693	C₂	semidirect
6. #19136635	C₂	semidirect
7. #19136533	C₂	semidirect
8. C₄×#13063271	C₂	semidirect
9. D₄×C₄₃⋊C₁₆	C₂	semidirect
10. #19136300	C₂	semidirect
11. C₄₃×#3342	C₂	semidirect
175. #3342	D₄₃	semidirect
220. C₄₃	#33871	semidirect

Maximal subgroups

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

subgroup	index
#19146355	2
#19136820	2
#19136786	2
#19136734	2
#19136693	2
#19136635	2
#19136533	2
C₄×#13063271	2
D₄×C₄₃⋊C₁₆	2
#19136300	2
C₄₃×#3342	2
#33871	43