Subgroups — #25110192

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583030	C₂	1	2	yes	maximal
2	#19135532	C₂	1	2	yes	maximal
3	#19135161	C₂	1	2	yes	maximal
4	#19135161	C₂	1	2	yes	maximal
5	#19135154	C₂	1	2	yes	maximal
6	#19135154	C₂	1	2	yes	maximal
7	#19134715	C₂	1	1	yes	maximal, characteristic
8	#19134347	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#2399	C₂	1	1	yes	maximal, characteristic
10	#14583030	—	2	8	no	—
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×#13063236	C₂²	1	2	yes	—
15	C₂×#13063236	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
18	#14582654	C₂²	1	2	yes	—
19	#14582654	C₂²	1	2	yes	—
20	#14582654	C₂²	1	2	yes	—
21	#14582654	C₂²	1	2	yes	—
22	#14582654	C₂²	1	2	yes	—
23	#14582654	C₂²	1	2	yes	—
24	#14582654	C₂²	1	1	yes	characteristic
25	#14582654	C₂²	1	2	yes	—
26	#14582654	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
28	C₄₃×#398	C₄	1	2	yes	—
29	C₄₃×#398	C₄	1	2	yes	—
30	C₄₃×#395	C₄	1	2	yes	—
31	C₄₃×#395	C₄	1	2	yes	—
32	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
35	C₄₃×#376	C₂²	1	1	yes	characteristic
36	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
40	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
41	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
42	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
43	#13063236	—	2	4	no	—
44	#13063236	D₄	1	4	yes	—
45	C₄×C₄₃⋊C₈	—	2	2	no	—
46	C₄×C₄₃⋊C₈	—	2	4	no	—
47	C₂²×Q₈×C₄₃	C₂×C₄	1	2	yes	—
48	C₄×Q₈×C₄₃	—	2	4	no	—
49	C₄×Q₈×C₄₃	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	#95×C₄₃	C₂×C₄	1	2	yes	—
63	#95×C₄₃	C₂×C₄	1	2	yes	—
64	#95×C₄₃	C₂×C₄	1	1	yes	characteristic
65	#95×C₄₃	C₂×C₄	1	2	yes	—
66	#95×C₄₃	C₂×C₄	1	1	yes	characteristic
67	#95×C₄₃	C₂×C₄	1	1	yes	characteristic
68	#95×C₄₃	C₂×C₄	1	2	yes	—
69	#95×C₄₃	C₂×C₄	1	1	yes	characteristic
70	C₂×Q₈×C₄₃	—	2	4	no	—
71	C₂×Q₈×C₄₃	—	2	4	no	—
72	C₂×Q₈×C₄₃	—	4	2	no	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₄₃	—	2	4	no	—
83	C₄⋊C₄×C₄₃	—	2	4	no	—
84	C₄⋊C₄×C₄₃	—	2	4	no	—
85	C₄⋊C₄×C₄₃	—	2	4	no	—
86	C₄⋊C₄×C₄₃	—	2	4	no	—
87	C₄⋊C₄×C₄₃	—	4	2	no	—
88	C₄⋊C₄×C₄₃	—	4	2	no	—
89	C₄²×C₄₃	—	2	2	no	—
90	C₄²×C₄₃	—	2	2	no	—
91	C₄²×C₄₃	—	2	2	no	—
92	C₄²×C₄₃	—	2	2	no	—
93	C₄²×C₄₃	—	2	2	no	—
94	C₄²×C₄₃	—	2	2	no	—
95	C₄²×C₄₃	C₂×D₄	1	2	yes	—
96	C₄²×C₄₃	#41	1	2	yes	—
97	C₂×C₄₃⋊C₈	—	2	2	no	—
98	C₂×C₄₃⋊C₈	—	2	2	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	—	2	2	no	—
102	C₂×C₄₃⋊C₈	—	2	2	no	—
103	C₂×C₄₃⋊C₈	—	2	2	no	—
104	C₂×C₄₃⋊C₈	—	2	2	no	—
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	C₂×C₄₃⋊C₈	—	2	2	no	—
107	C₂×C₄₃⋊C₈	—	2	2	no	—
108	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
109	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
110	Q₈×C₄₃	—	4	8	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
132	C₂×C₄×C₄₃	#131	1	4	yes	—
133	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
136	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
137	C₄₃⋊C₈	—	2	4	no	—
138	C₄₃⋊C₈	—	4	2	no	—
139	C₄₃⋊C₈	—	4	2	no	—
140	C₄₃⋊C₈	—	4	2	no	—
141	#22245	—	43	1	no	maximal, Sylow
142	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄₃	#519	1	1	yes	characteristic
144	C₂²×C₄₃	#519	1	1	yes	characteristic
145	C₂²×C₄₃	C₂×#131	1	2	yes	—
146	C₂²×C₄₃	#568	1	2	yes	—
147	C₄×C₄₃	—	2	4	no	—
148	C₄×C₄₃	—	2	4	no	—
149	C₄×C₄₃	—	4	2	no	—
150	C₄×C₄₃	—	4	2	no	—
151	C₄×C₄₃	—	4	2	no	—
152	C₄×C₄₃	—	4	2	no	—
153	C₄×C₄₃	—	4	1	no	—
154	C₄×C₄₃	—	4	1	no	—
155	C₄×C₄₃	#536	1	2	yes	—
156	C₄×C₄₃	#541	1	2	yes	—
157	C₂×#446	—	43	2	no	—
158	#2399	D₄₃	1	1	yes	characteristic
159	#1984	—	43	2	no	—
160	#1917	—	43	2	no	—
161	#1917	—	43	2	no	—
162	#1916	—	43	2	no	—
163	#1916	—	43	2	no	—
164	#1832	—	43	1	no	—
165	#1746	—	43	1	no	—
166	C₂×C₄₃	#2327	1	1	yes	characteristic
167	C₂×C₄₃	#2933	1	2	yes	—
168	C₂×C₄₃	#2985	1	2	yes	—
169	C₂×C₄₃	#2986	1	1	yes	characteristic
170	C₂×C₄₃	#2987	1	1	yes	characteristic
171	C₂×C₄×Q₈	C₂×D₄₃	1	2	yes	—
172	#446	—	86	8	no	—
173	C₂×C₄⋊C₈	—	43	2	no	—
174	C₂×C₄⋊C₈	—	43	2	no	—
175	C₂×C₄⋊C₈	—	43	2	no	—
176	C₂×#97	—	43	1	no	—
177	C₂×#97	—	43	2	no	—
178	C₂×C₄×C₈	—	43	2	no	—
179	C₂×C₄×C₈	—	43	1	no	—
180	#398	Dic₄₃	1	2	yes	—
181	#398	Dic₄₃	1	2	yes	—
182	#395	Dic₄₃	1	2	yes	—
183	#395	Dic₄₃	1	2	yes	—
184	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
185	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
186	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
187	#376	C₂×D₄₃	1	1	yes	characteristic
188	#336	—	43	1	no	—
189	#336	—	43	2	no	—
190	#336	—	43	1	no	—
191	#336	—	43	2	no	—
192	#336	—	43	2	no	—
193	#336	—	43	2	no	—
194	#336	—	43	2	no	—
195	#336	—	43	2	no	—
196	#336	—	43	2	no	—
197	C₄₃	#22245	1	1	yes	characteristic, complemented, Sylow
198	C₂²×Q₈	C₂×Dic₄₃	1	2	yes	—
199	C₂²×C₈	—	43	2	no	—
200	C₂²×C₈	—	43	2	no	—
201	C₂²×C₈	—	43	2	no	—
202	C₂²×C₈	—	43	2	no	—
203	C₄×Q₈	—	2	4	no	—
204	C₄×Q₈	—	2	4	no	—
205	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
211	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
212	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
213	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
214	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
217	C₄⋊C₈	—	86	4	no	—
218	C₄⋊C₈	—	86	4	no	—
219	C₄⋊C₈	—	86	4	no	—
220	#97	—	43	4	no	—
221	#97	—	86	4	no	—
222	C₄×C₈	—	86	4	no	—
223	C₄×C₈	—	86	2	no	—
224	#95	C₂×Dic₄₃	1	2	yes	—
225	#95	C₂×Dic₄₃	1	2	yes	—
226	#95	C₂×Dic₄₃	1	1	yes	characteristic
227	#95	C₂×Dic₄₃	1	2	yes	—
228	#95	C₂×Dic₄₃	1	1	yes	characteristic
229	#95	C₂×Dic₄₃	1	1	yes	characteristic
230	#95	C₂×Dic₄₃	1	2	yes	—
231	#95	C₂×Dic₄₃	1	1	yes	characteristic
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
236	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
239	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
240	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
243	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
244	C₂×C₈	—	43	4	no	—
245	C₂×C₈	—	86	2	no	—
246	C₂×C₈	—	86	2	no	—
247	C₂×C₈	—	86	2	no	—
248	C₂×C₈	—	86	2	no	—
249	C₂×C₈	—	86	2	no	—
250	C₂×C₈	—	86	2	no	—
251	C₂×C₈	—	86	2	no	—
252	C₂×C₈	—	86	2	no	—
253	C₂×C₈	—	86	2	no	—
254	C₂×C₈	—	86	2	no	—
255	C₂×C₈	—	86	2	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	D₄×D₄₃	1	2	yes	—
270	C₄²	#10621900	1	2	yes	—
271	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
272	Q₈	—	4	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	D₄×Dic₄₃	1	4	yes	—
294	C₂×C₄	#13063332	1	4	yes	—
295	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
296	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
297	C₂×C₄	#13063373	1	2	yes	—
298	C₂×C₄	#13063373	1	2	yes	—
299	C₈	—	86	4	no	—
300	C₈	—	172	2	no	—
301	C₈	—	172	2	no	—
302	C₈	—	172	2	no	—
303	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
304	C₂²	#14583377	1	1	yes	characteristic
305	C₂²	#14583377	1	1	yes	characteristic
306	C₂²	C₂×#13063332	1	2	yes	—
307	C₂²	#14583666	1	2	yes	—
308	C₄	—	2	4	no	—
309	C₄	—	2	4	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	#14583474	1	2	yes	—
317	C₄	#14583510	1	2	yes	—
318	C₂	#19137315	1	1	yes	characteristic
319	C₂	#19143322	1	2	yes	—
320	C₂	#19143690	1	2	yes	—
321	C₂	#19143697	1	1	yes	characteristic
322	C₂	#19143701	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle, prime radical	C₂³×C₄₃	C₂³×C₄
Fitting	C₄₃×#2399	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2399	D₄₃
hypocenter	C₄₃	#22245
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2399

Complements

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

N	H ≅ G/N	type
197. C₄₃	#22245	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583030	2
#19135532	2
#19135161	2
#19135154	2
#19134715	2
#19134347	2
C₄₃×#2399	2
#22245	43