Subgroups — #25002330

Order 11008.

718 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582664	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582663	C₂	1	2	yes	maximal
4	#19134288	C₂	1	2	yes	maximal
5	C₂×C₄₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₄₃	—	2	2	no	—
10	C₂×C₈×Dic₄₃	—	2	2	no	—
11	C₂×C₈×Dic₄₃	C₄	1	4	yes	—
12	#14582664	—	2	2	no	—
13	#14582663	C₄	1	8	yes	complemented
14	#14582653	—	2	4	no	—
15	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
16	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#341	—	2	2	no	—
18	C₂³×Dic₄₃	Q₈	1	2	yes	—
19	C₂×#10621877	—	2	2	no	—
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	C₂×C₄	1	4	yes	—
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	—	2	1	no	—
27	C₂²×C₄₃⋊C₈	—	2	2	no	—
28	C₂²×C₄₃⋊C₈	—	2	2	no	—
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₄₃	—	2	1	no	—
32	C₂×C₄×Dic₄₃	—	2	1	no	—
33	C₂×C₄×Dic₄₃	—	2	1	no	—
34	C₂×C₄×Dic₄₃	—	2	1	no	—
35	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
37	C₈×Dic₄₃	—	2	8	no	—
38	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₄₃	—	2	2	no	—
40	C₂×M₁₆×C₄₃	—	2	2	no	—
41	C₂×M₁₆×C₄₃	—	2	2	no	—
42	C₂×M₁₆×C₄₃	—	2	2	no	—
43	C₂×M₁₆×C₄₃	—	2	2	no	—
44	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₄₃	—	2	2	no	—
46	C₂²×C₈×C₄₃	—	2	2	no	—
47	C₂²×C₈×C₄₃	—	2	2	no	—
48	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₄₃	D₄	1	2	yes	—
50	C₂²×C₈×C₄₃	D₄	1	2	yes	—
51	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	#41	1	2	yes	—
58	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	#41	1	2	yes	—
61	C₂²×C₄×C₄₃	#41	1	2	yes	—
62	C₂²×Dic₄₃	—	2	8	no	—
63	C₂²×Dic₄₃	—	2	2	no	—
64	C₂²×Dic₄₃	—	2	2	no	—
65	C₂²×Dic₄₃	C₄⋊C₄	1	4	yes	—
66	M₁₆×C₄₃	—	2	4	no	—
67	M₁₆×C₄₃	—	2	4	no	—
68	M₁₆×C₄₃	—	4	2	no	—
69	M₁₆×C₄₃	—	4	2	no	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	—	2	4	no	—
75	C₂×C₈×C₄₃	—	4	2	no	—
76	C₂×C₈×C₄₃	—	4	2	no	—
77	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
78	C₄×Dic₄₃	—	2	2	no	—
79	C₄×Dic₄₃	—	2	2	no	—
80	C₄×Dic₄₃	—	2	2	no	—
81	C₄×Dic₄₃	—	2	2	no	—
82	#10621877	—	2	4	no	—
83	#10621877	—	2	4	no	—
84	#10621877	—	4	2	no	—
85	#10621877	—	4	2	no	—
86	C₂×C₄₃⋊C₈	—	2	2	no	—
87	C₂×C₄₃⋊C₈	—	2	2	no	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	2	4	no	—
90	C₂×C₄₃⋊C₈	—	2	4	no	—
91	C₂×C₄₃⋊C₈	—	4	2	no	—
92	C₂×C₄₃⋊C₈	—	4	2	no	—
93	C₂×C₄₃⋊C₈	C₄²	1	4	yes	—
94	C₂³×C₄₃	—	2	4	no	—
95	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₃	C₄×Q₈	1	2	yes	—
97	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₄₃	#125	1	2	yes	—
100	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#117	1	2	yes	—
104	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#117	1	2	yes	—
108	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#123	1	2	yes	—
114	C₂×C₄×C₄₃	#123	1	2	yes	—
115	C₂×C₄×C₄₃	#124	1	2	yes	—
116	C₂×C₄×C₄₃	#126	1	2	yes	—
117	C₂×C₄×C₄₃	#126	1	2	yes	—
118	C₂×Dic₄₃	—	2	8	no	—
119	C₂×Dic₄₃	—	2	8	no	—
120	C₂×Dic₄₃	—	2	4	no	—
121	C₂×Dic₄₃	—	2	4	no	—
122	C₂×Dic₄₃	#108	1	8	yes	—
123	C₈×C₄₃	—	2	4	no	—
124	C₈×C₄₃	—	4	4	no	—
125	C₄₃⋊C₈	—	2	4	no	—
126	C₄₃⋊C₈	—	4	4	no	—
127	#10822	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	2	no	—
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	—	2	4	no	—
131	C₂²×C₄₃	—	2	4	no	—
132	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₄₃	#380	1	1	yes	characteristic
134	C₂²×C₄₃	#382	1	1	yes	characteristic
135	C₂²×C₄₃	#387	1	2	yes	—
136	C₂²×C₄₃	#387	1	2	yes	—
137	C₂²×C₄₃	C₂×#108	1	4	yes	—
138	C₄×C₄₃	#376	1	1	yes	characteristic
139	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₄₃	#381	1	1	yes	characteristic
141	C₄×C₄₃	#383	1	1	yes	characteristic
142	C₄×C₄₃	#388	1	2	yes	—
143	C₄×C₄₃	#388	1	2	yes	—
144	Dic₄₃	—	2	8	no	—
145	C₂²×C₄×C₈	—	43	1	no	—
146	C₂×#341	—	43	2	no	—
147	C₂×#341	—	43	1	no	—
148	C₂×#341	D₄₃	1	1	yes	characteristic
149	#1728	—	43	2	no	—
150	C₂×C₄₃	—	2	2	no	—
151	C₂×C₄₃	—	2	2	no	—
152	C₂×C₄₃	#1434	1	1	yes	characteristic
153	C₂×C₄₃	C₄×#108	1	2	yes	—
154	C₂×C₄₃	#1810	1	2	yes	—
155	C₂×C₄₃	#1837	1	2	yes	—
156	C₂²×M₁₆	—	43	2	no	—
157	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
158	C₂³×C₈	—	43	1	no	—
159	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
160	C₂²×C₄²	—	43	1	no	—
161	C₂×C₄×C₈	—	43	4	no	—
162	C₂×C₄×C₈	—	86	2	no	—
163	C₂×C₄×C₈	—	86	2	no	—
164	#341	—	2	2	no	—
165	#341	—	43	8	no	—
166	#341	—	86	2	no	—
167	#336	—	86	4	no	—
168	C₄₃	#10822	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	43	2	no	—
170	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	43	4	no	—
177	C₂×M₁₆	—	86	2	no	—
178	C₂×M₁₆	—	86	2	no	—
179	C₂×M₁₆	—	86	2	no	—
180	C₂×M₁₆	—	86	2	no	—
181	C₂×M₁₆	—	86	2	no	—
182	C₂×M₁₆	C₄×D₄₃	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	43	4	no	—
187	C₂²×C₈	—	43	2	no	—
188	C₂²×C₈	—	86	1	no	—
189	C₂²×C₈	—	86	2	no	—
190	C₂²×C₈	—	86	1	no	—
191	C₂²×C₈	—	86	2	no	—
192	C₂²×C₈	C₄×D₄₃	1	2	yes	—
193	C₂²×C₈	D₁₇₂	1	2	yes	—
194	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
195	C₂×C₄²	—	43	2	no	—
196	C₂×C₄²	—	43	2	no	—
197	C₂×C₄²	—	86	1	no	—
198	C₂×C₄²	—	86	1	no	—
199	C₂×C₄²	—	86	1	no	—
200	C₂×C₄²	—	86	1	no	—
201	C₄×C₈	—	86	8	no	—
202	C₂⁴	Q₈×D₄₃	1	1	yes	characteristic
203	C₂²×C₄	—	43	4	no	—
204	C₂²×C₄	—	86	8	no	—
205	C₂²×C₄	—	86	2	no	—
206	C₂²×C₄	—	86	2	no	—
207	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
209	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
210	C₂²×C₄	#10621898	1	2	yes	—
211	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
212	C₂²×C₄	#10621900	1	1	yes	characteristic
213	C₂²×C₄	#10621900	1	2	yes	—
214	C₂²×C₄	#10621900	1	2	yes	—
215	C₂²×C₄	#10621902	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	86	4	no	—
222	M₁₆	—	86	4	no	—
223	M₁₆	—	172	2	no	—
224	M₁₆	—	172	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	43	4	no	—
233	C₂×C₈	—	86	2	no	—
234	C₂×C₈	—	86	2	no	—
235	C₂×C₈	—	86	4	no	—
236	C₂×C₈	—	86	4	no	—
237	C₂×C₈	—	86	4	no	—
238	C₂×C₈	—	172	2	no	—
239	C₂×C₈	—	172	2	no	—
240	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
241	C₄²	—	86	2	no	—
242	C₄²	—	86	2	no	—
243	C₄²	—	86	2	no	—
244	C₄²	—	86	2	no	—
245	C₂³	—	2	4	no	—
246	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
247	C₂³	#13063290	1	2	yes	—
248	C₂³	#13063301	1	1	yes	characteristic
249	C₂³	#13063305	1	1	yes	characteristic
250	C₂³	#13063315	1	2	yes	—
251	C₂×C₄	—	43	8	no	—
252	C₂×C₄	—	86	8	no	—
253	C₂×C₄	—	86	8	no	—
254	C₂×C₄	—	86	4	no	—
255	C₂×C₄	—	86	4	no	—
256	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
257	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
258	C₂×C₄	#13063281	1	2	yes	—
259	C₂×C₄	#13063281	1	1	yes	characteristic
260	C₂×C₄	#13063281	1	1	yes	characteristic
261	C₂×C₄	#13063282	1	1	yes	characteristic
262	C₂×C₄	#13063283	1	2	yes	—
263	C₂×C₄	#13063283	1	1	yes	characteristic
264	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
265	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
266	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
267	C₂×C₄	#13063296	1	1	yes	characteristic
268	C₂×C₄	#13063300	1	1	yes	characteristic
269	C₂×C₄	#13063306	1	2	yes	—
270	C₂×C₄	#13063309	1	2	yes	—
271	C₂×C₄	#13063311	1	2	yes	—
272	C₂×C₄	#13063318	1	2	yes	—
273	C₂×C₄	#13063320	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	4	4	no	—
276	C₈	—	86	4	no	—
277	C₈	—	172	4	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	#14582750	1	1	yes	characteristic
283	C₂²	#14582756	1	1	yes	characteristic
284	C₂²	#14582763	1	1	yes	characteristic
285	C₂²	#14582780	1	2	yes	—
286	C₂²	#14582784	1	2	yes	—
287	C₂²	#108×D₄₃	1	4	yes	—
288	C₄	—	86	8	no	—
289	C₄	#14582744	1	1	yes	characteristic
290	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
291	C₄	#14582760	1	1	yes	characteristic
292	C₄	#14582766	1	1	yes	characteristic
293	C₄	#14582787	1	2	yes	—
294	C₄	#14582791	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	#19132819	1	1	yes	characteristic
298	C₂	#19134496	1	2	yes	—
299	C₂	#19134630	1	2	yes	—
300	C₂	#19134731	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×#341	C₂
socle	C₂³×C₄₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#341	D₄₃
hypocenter	C₄₃	#10822
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
13. #14582663	C₄	semidirect
168. C₄₃	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₄₃	2
C₂×#14582664	2
C₂×#14582663	2
#19134288	2
C₂×C₄₃×#341	2
#10822	43