Subgroups — #25253471

Order 11008.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19146942	C₂	1	1	yes	maximal, characteristic
2	#19146345	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136791	C₂	1	2	yes	maximal, complemented
4	#19136734	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136694	C₂	1	2	yes	maximal, complemented
6	#19136637	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136513	C₂	1	2	yes	maximal, complemented
8	C₄×#13063270	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₆×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#358×D₄₃	C₂	1	2	yes	maximal, complemented
11	C₄₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₃	C₂²	1	2	yes	—
13	#14583564	—	2	1	no	—
14	#14583552	—	2	1	no	—
15	C₂×#13063270	C₂²	1	1	yes	characteristic
16	C₂×#13063268	C₂²	1	2	yes	—
17	C₂×C₁₆×D₄₃	C₂²	1	2	yes	—
18	#14583214	—	2	2	no	—
19	#14583176	—	2	1	no	—
20	#14583137	C₂²	1	2	yes	—
21	#14583136	C₂²	1	2	yes	—
22	#14583106	—	2	1	no	—
23	#14583053	—	2	2	no	—
24	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
25	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
26	C₈×D₁₇₂	C₂²	1	1	yes	characteristic
27	#107×D₄₃	C₂²	1	2	yes	—
28	#14582727	C₂²	1	1	yes	characteristic
29	#14582726	C₂²	1	2	yes	—
30	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
31	#14582704	C₂²	1	1	yes	characteristic
32	#14582703	C₂²	1	2	yes	—
33	#14582702	C₂²	1	2	yes	—
34	#14582701	C₂²	1	1	yes	characteristic
35	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
36	C₄₃×#500	C₂²	1	1	yes	characteristic
37	C₄₃×#494	C₂²	1	1	yes	characteristic
38	C₄₃×#358	C₂²	1	2	yes	—
39	C₄₃×#358	C₂²	1	2	yes	—
40	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×#10621902	—	4	1	no	—
42	C₂×Q₈×D₄₃	—	2	2	no	—
43	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
44	C₂×#10621885	—	2	2	no	—
45	Q₁₆×D₄₃	—	2	4	no	—
46	Q₁₆×D₄₃	—	4	2	no	—
47	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₄₃	C₂³	1	2	yes	—
49	#13063324	—	4	1	no	—
50	#13063310	—	4	2	no	—
51	#13063303	—	4	2	no	—
52	#13063301	—	2	2	no	—
53	#13063301	—	4	1	no	—
54	#13063300	—	2	2	no	—
55	C₄×Dic₈₆	—	2	1	no	—
56	C₄×D₁₇₂	—	2	1	no	—
57	C₄×D₁₇₂	D₄	1	1	yes	characteristic
58	#13063287	—	4	1	no	—
59	C₄⋊C₄×D₄₃	—	2	2	no	—
60	#13063270	—	2	2	no	—
61	#13063268	—	4	2	no	—
62	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
63	C₁₆×D₄₃	—	4	2	no	—
64	#13063259	C₂³	1	1	yes	characteristic
65	#13063259	C₂³	1	1	yes	characteristic
66	#13063258	—	2	2	no	—
67	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
68	#13063250	—	2	1	no	—
69	#13063249	—	2	2	no	—
70	#13063247	—	2	1	no	—
71	#13063245	—	2	2	no	—
72	(C₂×D₄₃)⋊C₈	D₄	1	2	yes	—
73	C₂×Q₁₆×C₄₃	—	2	1	no	—
74	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₄₃	—	2	1	no	—
76	C₄⋊Q₈×C₄₃	—	2	1	no	—
77	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
79	#107×C₄₃	—	2	1	no	—
80	#107×C₄₃	C₂³	1	2	yes	—
81	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₄₃	—	2	2	no	—
83	C₂×Q₈×C₄₃	—	4	1	no	—
84	#10621902	—	8	4	no	—
85	Q₈×D₄₃	—	4	4	no	—
86	Q₈×D₄₃	—	8	2	no	—
87	C₂×D₁₇₂	—	2	1	no	—
88	C₂×D₁₇₂	—	4	2	no	—
89	C₂×D₁₇₂	D₈	1	1	yes	characteristic
90	C₂×C₄×D₄₃	—	4	1	no	—
91	C₂×C₄×D₄₃	—	4	2	no	—
92	C₂×C₄×D₄₃	—	4	2	no	—
93	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
94	C₂×Dic₈₆	—	2	1	no	—
95	C₂×Dic₈₆	—	4	2	no	—
96	Q₁₆×C₄₃	—	2	2	no	—
97	Q₁₆×C₄₃	—	2	2	no	—
98	Q₁₆×C₄₃	—	8	1	no	—
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₄₃	—	2	2	no	—
102	C₄⋊C₄×C₄₃	—	4	1	no	—
103	C₄⋊C₄×C₄₃	—	4	1	no	—
104	C₄⋊C₄×C₄₃	—	4	1	no	—
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	#10621885	—	4	4	no	—
107	(C₂×D₄₃)⋊C₄	—	4	1	no	—
108	(C₂×D₄₃)⋊C₄	—	4	2	no	—
109	(C₂×D₄₃)⋊C₄	—	4	2	no	—
110	(C₂×D₄₃)⋊C₄	D₈	1	2	yes	—
111	C₄⋊Dic₄₃	—	2	1	no	—
112	C₄⋊Dic₄₃	—	2	1	no	—
113	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
114	Dic₄₃⋊C₄	—	4	1	no	—
115	Dic₄₃⋊C₄	—	4	2	no	—
116	Dic₄₃⋊C₄	—	4	2	no	—
117	C₄×Dic₄₃	—	4	1	no	—
118	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
119	Dic₁₇₂	—	2	1	no	—
120	Dic₁₇₂	#41	1	2	yes	complemented
121	D₃₄₄	—	2	1	no	—
122	D₃₄₄	—	2	1	no	—
123	C₈×D₄₃	—	2	2	no	—
124	C₈×D₄₃	—	2	2	no	—
125	C₁₆×C₄₃	—	2	1	no	—
126	C₁₆×C₄₃	—	2	1	no	—
127	C₄₃⋊C₁₆	—	2	2	no	—
128	C₂²×D₄₃	—	4	1	no	—
129	C₂²×D₄₃	C₂×D₈	1	2	yes	—
130	Q₈×C₄₃	—	4	2	no	—
131	Q₈×C₄₃	—	4	2	no	—
132	Q₈×C₄₃	—	8	2	no	—
133	C₂×C₄×C₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	—	4	2	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	—	4	1	no	—
141	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
142	D₁₇₂	—	4	1	no	—
143	D₁₇₂	—	4	1	no	—
144	D₁₇₂	—	8	4	no	—
145	D₁₇₂	SD₃₂	1	4	yes	complemented
146	C₄×D₄₃	—	2	2	no	—
147	C₄×D₄₃	—	2	2	no	—
148	C₄×D₄₃	—	8	4	no	—
149	C₄×D₄₃	—	8	4	no	—
150	C₄×D₄₃	—	8	2	no	—
151	Dic₈₆	—	2	2	no	—
152	Dic₈₆	—	4	1	no	—
153	Dic₈₆	—	8	4	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	C₄₃⋊C₈	—	2	2	no	—
158	#33865	—	43	1	no	maximal, Sylow
159	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
160	C₂×D₄₃	—	2	2	no	—
161	C₂×D₄₃	—	2	2	no	—
162	C₂×D₄₃	—	8	1	no	—
163	C₂×D₄₃	—	8	1	no	—
164	C₂×D₄₃	—	8	1	no	—
165	C₂×D₄₃	C₂×SD₃₂	1	4	yes	—
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	—	8	2	no	—
168	C₄×C₄₃	—	8	2	no	—
169	C₄×C₄₃	C₂×SD₃₂	1	2	yes	—
170	C₄×C₄₃	#546	1	1	yes	characteristic
171	C₄×C₄₃	#577	1	1	yes	characteristic
172	Dic₄₃	—	2	2	no	—
173	Dic₄₃	—	4	2	no	—
174	Dic₄₃	—	8	1	no	—
175	Dic₄₃	—	8	1	no	—
176	#3334	—	43	1	no	—
177	D₄×Q₁₆	—	43	1	no	—
178	#2241	D₄₃	1	1	yes	characteristic, complemented
179	#2235	—	43	2	no	—
180	#2225	—	43	1	no	—
181	#2217	—	43	2	no	—
182	#2208	—	43	1	no	—
183	#2185	—	43	2	no	—
184	C₄×SD₃₂	—	43	1	no	—
185	D₄×C₁₆	—	43	1	no	—
186	C₂×#358	—	43	2	no	—
187	C₂×C₄₃	#3294	1	1	yes	characteristic
188	C₂×C₄₃	C₂²×SD₃₂	1	1	yes	characteristic
189	C₂×C₄₃	#3417	1	1	yes	characteristic
190	D₄₃	—	2	4	no	—
191	D₄₃	—	16	1	no	—
192	C₂²×Q₁₆	—	43	2	no	—
193	D₄×Q₈	—	86	1	no	—
194	#547	—	86	1	no	—
195	C₂×SD₃₂	—	43	2	no	—
196	C₂×SD₃₂	—	43	1	no	—
197	C₂²×C₁₆	—	43	2	no	—
198	#500	C₂×D₄₃	1	1	yes	characteristic
199	#494	C₂×D₄₃	1	1	yes	characteristic
200	#480	—	86	2	no	—
201	#474	—	86	1	no	—
202	#467	—	43	2	no	—
203	#466	—	43	2	no	—
204	#462	—	86	1	no	—
205	#451	—	86	2	no	—
206	C₄×Q₁₆	—	43	1	no	—
207	C₄×D₈	—	43	1	no	—
208	C₈×D₄	—	43	1	no	—
209	C₂×#107	—	43	2	no	—
210	#367	—	43	1	no	—
211	#367	—	43	2	no	—
212	C₄⋊C₁₆	—	43	1	no	—
213	#358	—	43	2	no	—
214	#358	—	43	1	no	—
215	#358	—	43	2	no	—
216	#358	C₂×D₄₃	1	2	yes	—
217	#358	C₂×D₄₃	1	2	yes	—
218	#357	—	43	1	no	—
219	C₂²⋊C₁₆	—	43	2	no	—
220	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
221	C₄₃	#33865	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	—	172	1	no	—
223	C₂²×Q₈	—	86	2	no	—
224	C₂×Q₁₆	—	2	1	no	—
225	C₂×Q₁₆	—	43	1	no	—
226	C₂×Q₁₆	—	86	2	no	—
227	C₂×Q₁₆	—	86	4	no	—
228	C₂×Q₁₆	—	172	2	no	—
229	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
230	C₂×D₈	—	43	1	no	—
231	C₂²×C₈	—	43	2	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	172	1	no	—
235	#123	—	172	2	no	—
236	C₂²⋊Q₈	—	86	2	no	—
237	C₂²⋊Q₈	—	172	2	no	—
238	C₂²⋊Q₈	—	172	1	no	—
239	#121	—	86	2	no	—
240	C₄×Q₈	—	86	1	no	—
241	C₄×D₄	—	43	1	no	—
242	C₄×D₄	—	86	1	no	—
243	C₄×D₄	—	172	1	no	—
244	C₂×C₄⋊C₄	—	86	2	no	—
245	SD₃₂	—	86	2	no	—
246	SD₃₂	—	172	2	no	—
247	C₂×C₁₆	—	43	2	no	—
248	C₂×C₁₆	—	172	2	no	—
249	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
251	#107	—	2	1	no	—
252	#107	—	43	1	no	—
253	#107	—	43	1	no	—
254	#107	—	86	2	no	—
255	#107	C₂²×D₄₃	1	2	yes	—
256	C₄⋊C₈	—	43	1	no	—
257	#103	—	86	2	no	—
258	#103	—	86	1	no	—
259	#102	—	86	2	no	—
260	#102	—	86	1	no	—
261	C₂²⋊C₈	—	43	2	no	—
262	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
263	#41	—	344	4	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	86	1	no	—
267	C₂×Q₈	—	344	2	no	—
268	C₂×Q₈	—	172	2	no	—
269	C₂×Q₈	—	172	4	no	—
270	C₂×D₄	—	43	1	no	—
271	C₂×D₄	—	86	1	no	—
272	C₂×D₄	—	172	2	no	—
273	C₂²×C₄	—	43	2	no	—
274	C₂²×C₄	—	172	2	no	—
275	C₂²×C₄	—	172	2	no	—
276	C₂²×C₄	—	172	1	no	—
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	8	1	no	—
280	Q₁₆	—	43	2	no	—
281	Q₁₆	—	86	1	no	—
282	Q₁₆	—	172	4	no	—
283	D₈	—	86	1	no	—
284	D₈	—	86	1	no	—
285	C₂×C₈	—	43	2	no	—
286	C₂×C₈	—	86	2	no	—
287	C₂×C₈	—	86	2	no	—
288	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	43	1	no	—
295	C₄⋊C₄	—	86	1	no	—
296	C₄⋊C₄	—	86	1	no	—
297	C₄⋊C₄	—	172	2	no	—
298	C₄⋊C₄	—	172	2	no	—
299	C₄⋊C₄	—	172	1	no	—
300	C₂²⋊C₄	—	43	2	no	—
301	C₂²⋊C₄	—	172	2	no	—
302	C₂²⋊C₄	—	172	2	no	—
303	C₂²⋊C₄	—	172	1	no	—
304	C₄²	—	172	1	no	—
305	C₄²	D₄×D₄₃	1	1	yes	characteristic
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	86	2	no	—
309	C₂³	—	43	2	no	—
310	C₂³	—	172	1	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	2	no	—
314	Q₈	—	86	2	no	—
315	Q₈	—	344	4	no	—
316	Q₈	—	172	1	no	—
317	D₄	—	43	4	no	—
318	D₄	—	344	4	no	—
319	D₄	—	172	1	no	—
320	D₄	—	172	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	43	2	no	—
324	C₂×C₄	—	86	2	no	—
325	C₂×C₄	—	86	2	no	—
326	C₂×C₄	—	344	4	no	—
327	C₂×C₄	—	344	4	no	—
328	C₂×C₄	—	344	2	no	—
329	C₂×C₄	—	172	1	no	—
330	C₂×C₄	—	172	1	no	—
331	C₂×C₄	—	172	1	no	—
332	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
333	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	86	2	no	—
337	C₈	#41×D₄₃	1	1	yes	characteristic
338	C₈	#13063371	1	1	yes	characteristic
339	C₂²	—	43	4	no	—
340	C₂²	—	86	2	no	—
341	C₂²	—	86	2	no	—
342	C₂²	—	344	1	no	—
343	C₂²	—	344	1	no	—
344	C₂²	—	344	1	no	—
345	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	86	2	no	—
350	C₄	—	344	1	no	—
351	C₄	—	344	1	no	—
352	C₄	—	172	2	no	—
353	C₄	SD₃₂×D₄₃	1	2	yes	—
354	C₄	#14583542	1	1	yes	characteristic
355	C₄	#14583705	1	1	yes	characteristic
356	C₂	—	86	4	no	—
357	C₂	—	688	1	no	—
358	C₂	#19146407	1	1	yes	characteristic
359	C₂	C₂×SD₃₂×D₄₃	1	1	yes	characteristic
360	C₂	#19147744	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×#2241	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19146942	C₂
hypercenter	#2241	D₄₃
hypocenter	C₄₃	#33865
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
2. #19146345	C₂	semidirect
3. #19136791	C₂	semidirect
4. #19136734	C₂	semidirect
5. #19136694	C₂	semidirect
6. #19136637	C₂	semidirect
7. #19136513	C₂	semidirect
8. C₄×#13063270	C₂	semidirect
9. C₁₆×D₁₇₂	C₂	semidirect
10. #358×D₄₃	C₂	semidirect
11. C₄₃×#2241	C₂	semidirect
120. Dic₁₇₂	#41	semidirect
145. D₁₇₂	SD₃₂	semidirect
178. #2241	D₄₃	semidirect
221. C₄₃	#33865	semidirect

Maximal subgroups

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

subgroup	index
#19146942	2
#19146345	2
#19136791	2
#19136734	2
#19136694	2
#19136637	2
#19136513	2
C₄×#13063270	2
C₁₆×D₁₇₂	2
#358×D₄₃	2
C₄₃×#2241	2
#33865	43