Subgroups — #25081780

Order 11008.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144388	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583176	C₂	1	1	yes	maximal, characteristic
3	#19143618	C₂	1	1	yes	maximal, characteristic
4	#19134043	C₂	1	4	yes	maximal, complemented
5	#19133957	C₂	1	4	yes	maximal, complemented
6	#19133370	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133280	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133252	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
11	#14583196	—	2	2	no	—
12	#14583186	C₂²	1	2	yes	—
13	#14583176	—	2	2	no	—
14	#14583176	C₂²	1	2	yes	—
15	#14583176	C₂²	1	4	yes	—
16	C₈×D₁₇₂	C₂²	1	4	yes	—
17	#14582964	—	2	2	no	—
18	#14582952	C₂²	1	1	yes	characteristic
19	#14582942	—	2	1	no	—
20	C₂×#13063258	—	2	1	no	—
21	#14582923	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
23	#14582903	—	2	1	no	—
24	C₂×#13063245	—	2	1	no	—
25	C₂²⋊C₈×D₄₃	—	2	2	no	—
26	#14582652	C₂²	1	4	yes	complemented
27	#14582648	C₂²	1	2	yes	complemented
28	#14582645	C₂²	1	2	yes	complemented
29	#14582645	C₂²	1	4	yes	complemented
30	C₄₃×#533	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₄₃×#494	C₂²	1	2	yes	complemented
33	C₄₃×#487	C₂²	1	2	yes	complemented
34	C₄₃×#462	—	2	2	no	—
35	C₄₃×#451	—	2	2	no	—
36	C₄₃×#431	C₂²	1	1	yes	characteristic
37	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₄₃	—	2	1	no	—
39	C₂×#10621877	—	2	1	no	—
40	C₂²×C₄₃⋊C₈	—	2	1	no	—
41	C₂×C₈×D₄₃	—	2	4	no	—
42	C₄×D₁₇₂	—	2	1	no	—
43	C₄×D₁₇₂	—	2	1	no	—
44	C₄×D₁₇₂	D₄	1	2	yes	—
45	C₄×D₁₇₂	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
47	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
48	#13063258	—	2	4	no	—
49	#13063258	—	2	2	no	—
50	#13063256	—	2	2	no	—
51	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
52	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
53	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
54	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
55	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
56	#13063245	—	2	4	no	—
57	#13063245	—	2	2	no	—
58	#13063245	—	2	2	no	—
59	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
60	(C₂×D₄₃)⋊C₈	—	2	4	no	—
61	C₄×C₄₃⋊C₈	—	2	1	no	—
62	C₂²×Q₈×C₄₃	—	2	1	no	—
63	C₂×Q₁₆×C₄₃	—	2	4	no	—
64	C₄⋊Q₈×C₄₃	—	2	1	no	—
65	C₄⋊Q₈×C₄₃	—	2	1	no	—
66	C₄⋊Q₈×C₄₃	—	2	1	no	—
67	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
68	C₄⋊Q₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₄₃	C₂³	1	1	yes	characteristic
70	#125×C₄₃	—	2	1	no	—
71	C₂²⋊Q₈×C₄₃	—	2	1	no	—
72	C₂²⋊Q₈×C₄₃	—	2	1	no	—
73	C₄×Q₈×C₄₃	—	2	1	no	—
74	C₄×Q₈×C₄₃	—	2	1	no	—
75	#117×C₄₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
78	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
80	#103×C₄₃	—	2	4	no	—
81	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
82	C₄×C₈×C₄₃	C₂³	1	2	yes	—
83	C₂³×D₄₃	—	2	1	no	—
84	C₂×Q₈×C₄₃	—	2	2	no	—
85	C₂×Q₈×C₄₃	—	2	1	no	—
86	C₂×Q₈×C₄₃	—	2	1	no	—
87	C₂×Q₈×C₄₃	—	4	1	no	—
88	C₂×Q₈×C₄₃	—	8	2	no	—
89	C₂²×C₄×C₄₃	—	4	1	no	—
90	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
93	C₂²×Dic₄₃	—	2	1	no	—
94	C₂×D₁₇₂	—	2	1	no	—
95	C₂×D₁₇₂	—	2	2	no	—
96	C₂×D₁₇₂	Q₁₆	1	4	yes	complemented
97	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
98	C₂×C₄×D₄₃	—	2	2	no	—
99	C₂×C₄×D₄₃	—	2	2	no	—
100	C₂×C₄×D₄₃	—	8	2	no	—
101	Q₁₆×C₄₃	—	8	4	no	—
102	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
103	C₄⋊C₄×C₄₃	—	2	1	no	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	—	2	1	no	—
106	C₄⋊C₄×C₄₃	—	2	1	no	—
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	—	2	1	no	—
109	C₄⋊C₄×C₄₃	—	4	1	no	—
110	C₄⋊C₄×C₄₃	—	4	1	no	—
111	C₄⋊C₄×C₄₃	—	4	1	no	—
112	C₄⋊C₄×C₄₃	—	4	1	no	—
113	C₂²⋊C₄×C₄₃	—	4	1	no	—
114	C₄²×C₄₃	—	4	1	no	—
115	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₃	#41	1	1	yes	characteristic
118	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
119	(C₂×D₄₃)⋊C₄	—	2	2	no	—
120	(C₂×D₄₃)⋊C₄	—	2	2	no	—
121	C₄⋊Dic₄₃	—	2	1	no	—
122	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
123	#10621877	—	4	2	no	—
124	C₂×C₄₃⋊C₈	—	2	2	no	—
125	C₂×C₄₃⋊C₈	—	2	2	no	—
126	C₂×C₄₃⋊C₈	—	2	2	no	—
127	C₂×C₄₃⋊C₈	—	4	2	no	—
128	C₈×D₄₃	—	8	4	no	—
129	C₂³×C₄₃	#121	1	1	yes	characteristic
130	C₂²×D₄₃	—	2	2	no	—
131	C₂²×D₄₃	—	2	4	no	—
132	C₂²×D₄₃	—	8	1	no	—
133	C₂²×D₄₃	—	8	1	no	—
134	Q₈×C₄₃	—	4	2	no	—
135	Q₈×C₄₃	—	4	4	no	—
136	Q₈×C₄₃	—	8	1	no	—
137	Q₈×C₄₃	—	8	2	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	4	1	no	—
140	C₂×C₄×C₄₃	—	4	1	no	—
141	C₂×C₄×C₄₃	—	4	1	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	—	8	2	no	—
144	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
148	C₂×C₄×C₄₃	#135	1	2	yes	—
149	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₄₃	—	2	2	no	—
153	C₂×Dic₄₃	—	8	1	no	—
154	D₁₇₂	—	2	2	no	—
155	D₁₇₂	—	2	4	no	—
156	D₁₇₂	C₂×Q₁₆	1	4	yes	—
157	C₄×D₄₃	—	4	4	no	—
158	C₄×D₄₃	—	8	2	no	—
159	C₈×C₄₃	—	2	4	no	—
160	C₄₃⋊C₈	—	4	2	no	—
161	C₄₃⋊C₈	—	4	2	no	—
162	C₄₃⋊C₈	—	4	2	no	—
163	#19387	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	#467	1	2	yes	—
167	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
168	C₂×D₄₃	—	2	4	no	—
169	C₂×D₄₃	—	4	2	no	—
170	C₂×D₄₃	—	4	2	no	—
171	C₂×D₄₃	—	8	4	no	—
172	C₄×C₄₃	—	2	1	no	—
173	C₄×C₄₃	—	2	1	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	8	2	no	—
176	C₄×C₄₃	—	8	1	no	—
177	C₄×C₄₃	—	8	1	no	—
178	C₄×C₄₃	#467	1	2	yes	—
179	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
180	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
181	C₄×C₄₃	#576	1	2	yes	—
182	Dic₄₃	—	4	2	no	—
183	#3209	D₄₃	1	1	yes	characteristic, complemented
184	#3077	—	43	1	no	—
185	C₂×#474	—	43	1	no	—
186	#2972	—	43	1	no	—
187	#1683	—	43	4	no	—
188	#1669	—	43	4	no	—
189	#1571	—	43	1	no	—
190	#1549	—	43	1	no	—
191	#1542	—	43	1	no	—
192	C₂×C₄₃	—	2	2	no	—
193	C₂×C₄₃	#3041	1	1	yes	characteristic
194	C₂×C₄₃	C₂×#467	1	1	yes	characteristic
195	C₂×C₄₃	#3069	1	1	yes	characteristic
196	D₄₃	—	4	4	no	—
197	#533	C₂×D₄₃	1	1	yes	characteristic
198	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
199	C₂×C₄×D₄	—	43	1	no	—
200	#494	C₂×D₄₃	1	2	yes	complemented
201	#487	C₂×D₄₃	1	2	yes	complemented
202	#478	—	86	2	no	—
203	#476	—	43	2	no	—
204	#474	—	43	2	no	—
205	#474	—	43	4	no	—
206	#474	—	86	2	no	—
207	#462	—	2	2	no	—
208	#451	—	2	2	no	—
209	C₈×D₄	—	43	4	no	—
210	C₈×D₄	—	86	2	no	—
211	#431	—	43	1	no	—
212	#431	C₂×D₄₃	1	1	yes	characteristic
213	#428	—	86	1	no	—
214	C₂×#107	—	86	1	no	—
215	#423	—	43	1	no	—
216	C₂×C₄⋊C₈	—	43	1	no	—
217	#418	—	86	1	no	—
218	C₂×#102	—	86	1	no	—
219	C₂×C₂²⋊C₈	—	86	2	no	—
220	#335	—	43	4	no	—
221	#333	—	43	2	no	—
222	#332	—	43	2	no	—
223	#332	—	43	4	no	—
224	C₄₃	#19387	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	—	2	1	no	—
226	C₂²×D₄	—	43	1	no	—
227	C₂³×C₄	—	86	1	no	—
228	C₂×Q₁₆	—	2	4	no	—
229	C₂×M₁₆	—	86	1	no	—
230	C₂²×C₈	—	86	1	no	—
231	C₂²×C₈	—	86	4	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
236	C₄⋊Q₈	C₂²×D₄₃	1	1	yes	characteristic
237	C₄⋊Q₈	C₂²×D₄₃	1	1	yes	characteristic
238	#125	—	2	1	no	—
239	C₂²⋊Q₈	—	2	1	no	—
240	C₂²⋊Q₈	—	2	1	no	—
241	C₄×Q₈	—	2	1	no	—
242	C₄×Q₈	—	2	1	no	—
243	C₄×D₄	—	43	2	no	—
244	C₄×D₄	—	43	2	no	—
245	C₄×D₄	—	86	1	no	—
246	C₄×D₄	—	86	1	no	—
247	#117	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	43	1	no	—
251	C₂×C₂²⋊C₄	—	86	1	no	—
252	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
253	#107	—	86	4	no	—
254	#107	—	86	2	no	—
255	#106	—	86	2	no	—
256	C₄⋊C₈	—	43	2	no	—
257	C₄⋊C₈	—	43	2	no	—
258	C₄⋊C₈	—	43	2	no	—
259	C₄⋊C₈	—	86	1	no	—
260	C₄⋊C₈	—	86	1	no	—
261	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
262	#103	—	2	4	no	—
263	#102	—	86	4	no	—
264	#102	—	86	2	no	—
265	#102	—	86	2	no	—
266	C₂²⋊C₈	—	86	1	no	—
267	C₂²⋊C₈	—	86	4	no	—
268	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
269	C₄×C₈	—	86	1	no	—
270	C₄×C₈	C₂²×D₄₃	1	2	yes	—
271	C₂⁴	—	86	1	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	4	1	no	—
276	C₂×Q₈	—	8	2	no	—
277	C₂×D₄	—	43	4	no	—
278	C₂×D₄	—	43	2	no	—
279	C₂×D₄	—	86	1	no	—
280	C₂×D₄	—	86	2	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	86	1	no	—
283	C₂²×C₄	—	86	2	no	—
284	C₂²×C₄	—	86	2	no	—
285	C₂²×C₄	—	344	2	no	—
286	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
287	C₂²×C₄	#10621900	1	1	yes	characteristic
288	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
289	Q₁₆	—	8	4	no	—
290	M₁₆	—	172	2	no	—
291	C₂×C₈	—	86	2	no	—
292	C₂×C₈	—	86	2	no	—
293	C₂×C₈	—	86	2	no	—
294	C₂×C₈	—	344	4	no	—
295	C₂×C₈	—	172	2	no	—
296	C₂×C₈	C₂×C₂²⋊D₄₃	1	4	yes	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	43	2	no	—
308	C₄⋊C₄	—	86	1	no	—
309	C₂²⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	86	2	no	—
311	C₂²⋊C₄	—	86	2	no	—
312	C₄²	—	4	1	no	—
313	C₄²	D₄×D₄₃	1	1	yes	characteristic
314	C₄²	#10621900	1	1	yes	characteristic
315	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
316	C₄²	C₂³×D₄₃	1	1	yes	characteristic
317	C₂³	—	86	4	no	—
318	C₂³	—	86	2	no	—
319	C₂³	—	344	1	no	—
320	C₂³	—	344	1	no	—
321	C₂³	#13063296	1	1	yes	characteristic
322	Q₈	—	4	2	no	—
323	Q₈	—	4	4	no	—
324	Q₈	—	8	1	no	—
325	Q₈	—	8	2	no	—
326	D₄	—	43	4	no	—
327	D₄	—	86	4	no	—
328	D₄	—	86	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	86	2	no	—
336	C₂×C₄	—	344	2	no	—
337	C₂×C₄	—	344	1	no	—
338	C₂×C₄	—	172	4	no	—
339	C₂×C₄	#13063296	1	1	yes	characteristic
340	C₂×C₄	#13063296	1	1	yes	characteristic
341	C₂×C₄	#13063296	1	1	yes	characteristic
342	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
343	C₂×C₄	#13063348	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
346	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
347	C₈	—	2	4	no	—
348	C₈	—	172	2	no	—
349	C₈	—	172	2	no	—
350	C₈	—	172	2	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	86	4	no	—
354	C₂²	—	344	4	no	—
355	C₂²	—	172	2	no	—
356	C₂²	—	172	2	no	—
357	C₂²	#14583137	1	2	yes	—
358	C₂²	C₂×#13063296	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	172	2	no	—
366	C₄	#14583137	1	2	yes	—
367	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
368	C₄	C₂×#13063348	1	1	yes	characteristic
369	C₄	#14583702	1	2	yes	—
370	C₂	—	2	2	no	—
371	C₂	—	172	4	no	—
372	C₂	#19144094	1	1	yes	characteristic
373	C₂	C₂×#14583137	1	1	yes	characteristic
374	C₂	#19144328	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063296
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#3209	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂²×D₁₇₂	D₄
hypercenter	#3209	D₄₃
hypocenter	C₄₃	#19387
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3209

Complements

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

N	H ≅ G/N	type
4. #19134043	C₂	semidirect
5. #19133957	C₂	semidirect
6. #19133370	C₂	semidirect
7. #19133280	C₂	semidirect
8. #19133252	C₂	semidirect
9. C₄₃×#3209	C₂	semidirect
26. #14582652	C₂²	semidirect
27. #14582648	C₂²	semidirect
28. #14582645	C₂²	semidirect
29. #14582645	C₂²	semidirect
32. C₄₃×#494	C₂²	semidirect
33. C₄₃×#487	C₂²	semidirect
96. C₂×D₁₇₂	Q₁₆	semidirect
183. #3209	D₄₃	semidirect
200. #494	C₂×D₄₃	semidirect
201. #487	C₂×D₄₃	semidirect
224. C₄₃	#19387	semidirect

Maximal subgroups

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

subgroup	index
#19144388	2
C₂×#14583176	2
#19143618	2
#19134043	2
#19133957	2
#19133370	2
#19133280	2
#19133252	2
C₄₃×#3209	2
#19387	43