Subgroups — #25005029

Order 11008.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582948	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582669	C₂	1	2	yes	maximal
3	C₂×#14582664	C₂	1	2	yes	maximal
4	C₂×C₄₃⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂²×#10621877	C₂²	1	2	yes	—
8	#14582948	—	2	4	no	—
9	C₂×#13063261	—	2	2	no	—
10	#14582669	—	2	4	no	—
11	#14582664	—	2	4	no	—
12	C₄₃⋊#338	C₄	1	4	yes	complemented
13	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#336	—	2	2	no	—
17	C₂×#10621877	—	2	2	no	—
18	C₂×#10621877	—	2	2	no	—
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	—	2	2	no	—
25	C₂×#10621877	—	2	2	no	—
26	C₂×#10621877	—	2	2	no	—
27	C₂×#10621877	C₂×C₄	1	4	yes	—
28	C₂×#10621877	D₄	1	4	yes	—
29	C₂²×C₄₃⋊C₈	—	2	2	no	—
30	C₂²×C₄₃⋊C₈	—	2	2	no	—
31	#13063261	—	2	8	no	—
32	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₄₃	—	2	2	no	—
34	C₂×M₁₆×C₄₃	—	2	2	no	—
35	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₄₃	Q₈	1	2	yes	—
37	C₂²×C₈×C₄₃	—	2	1	no	—
38	C₂²×C₈×C₄₃	—	2	1	no	—
39	C₂²×C₈×C₄₃	—	2	2	no	—
40	C₂²×C₈×C₄₃	—	2	2	no	—
41	C₂²×C₈×C₄₃	D₄	1	2	yes	—
42	C₂²×C₈×C₄₃	D₄	1	2	yes	—
43	#117×C₄₃	—	2	2	no	—
44	#117×C₄₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₄₃	Q₈	1	2	yes	—
47	C₂×C₄²×C₄₃	—	2	1	no	—
48	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	—	2	2	no	—
50	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	#41	1	2	yes	—
58	C₂²×C₄×C₄₃	#41	1	2	yes	—
59	C₂²×C₄×C₄₃	#41	1	2	yes	—
60	M₁₆×C₄₃	—	2	4	no	—
61	M₁₆×C₄₃	—	2	4	no	—
62	C₂×C₈×C₄₃	—	2	2	no	—
63	C₂×C₈×C₄₃	—	2	2	no	—
64	C₂×C₈×C₄₃	—	2	2	no	—
65	C₂×C₈×C₄₃	—	2	4	no	—
66	C₂×C₈×C₄₃	—	4	1	no	—
67	C₂×C₈×C₄₃	—	4	1	no	—
68	C₂×C₈×C₄₃	—	4	2	no	—
69	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
70	C₄⋊C₄×C₄₃	—	4	2	no	—
71	C₂²⋊C₄×C₄₃	—	2	4	no	—
72	C₄²×C₄₃	—	4	1	no	—
73	C₄²×C₄₃	—	4	1	no	—
74	#10621877	—	2	4	no	—
75	#10621877	—	2	4	no	—
76	#10621877	—	2	4	no	—
77	#10621877	—	2	4	no	—
78	#10621877	—	4	2	no	—
79	#10621877	—	4	2	no	—
80	#10621877	—	4	2	no	—
81	#10621877	—	4	2	no	—
82	C₂×C₄₃⋊C₈	—	2	4	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	4	2	no	—
85	C₂×C₄₃⋊C₈	—	4	2	no	—
86	C₂×C₄₃⋊C₈	—	4	2	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂³×C₄₃	—	4	2	no	—
89	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
90	C₂³×C₄₃	C₂²⋊Q₈	1	2	yes	—
91	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂³×C₄₃	#125	1	2	yes	—
93	C₂³×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₄₃	—	4	4	no	—
95	C₂×C₄×C₄₃	—	4	2	no	—
96	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	#117	1	2	yes	—
98	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₄₃	#121	1	2	yes	—
103	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
105	C₂×C₄×C₄₃	#123	1	2	yes	—
106	C₂×C₄×C₄₃	#123	1	2	yes	—
107	C₂×C₄×C₄₃	#124	1	2	yes	—
108	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#125	1	2	yes	—
111	C₂×C₄×C₄₃	#125	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#126	1	2	yes	—
113	C₂×C₄×C₄₃	#126	1	2	yes	—
114	C₈×C₄₃	—	2	4	no	—
115	C₈×C₄₃	—	4	2	no	—
116	C₄₃⋊C₈	—	4	4	no	—
117	C₄₃⋊C₈	—	4	4	no	—
118	#11295	—	43	1	no	maximal, Sylow
119	C₂²×C₄₃	—	2	2	no	—
120	C₂²×C₄₃	—	2	4	no	—
121	C₂²×C₄₃	—	4	1	no	—
122	C₂²×C₄₃	—	4	2	no	—
123	C₂²×C₄₃	—	4	1	no	—
124	C₂²×C₄₃	#387	1	2	yes	—
125	C₂²×C₄₃	#389	1	1	yes	characteristic
126	C₂²×C₄₃	#393	1	1	yes	characteristic
127	C₂²×C₄₃	#398	1	2	yes	—
128	C₂²×C₄₃	#400	1	1	yes	characteristic
129	C₄×C₄₃	—	8	2	no	—
130	C₄×C₄₃	#382	1	1	yes	characteristic
131	C₄×C₄₃	#385	1	1	yes	characteristic
132	C₄×C₄₃	#388	1	2	yes	—
133	C₄×C₄₃	#394	1	1	yes	characteristic
134	C₄×C₄₃	#397	1	1	yes	characteristic
135	C₄×C₄₃	#400	1	2	yes	—
136	C₂×#430	—	43	1	no	—
137	C₂×#344	—	43	2	no	—
138	C₂×#341	—	43	2	no	—
139	C₂×#338	—	43	1	no	—
140	#1729	D₄₃	1	1	yes	characteristic
141	C₂×C₄₃	—	2	1	no	—
142	C₂×C₄₃	—	2	2	no	—
143	C₂×C₄₃	—	2	1	no	—
144	C₂×C₄₃	—	2	2	no	—
145	C₂×C₄₃	#1442	1	1	yes	characteristic
146	C₂×C₄₃	#2103	1	2	yes	—
147	C₂²×M₁₆	—	43	2	no	—
148	C₂²×M₁₆	—	43	2	no	—
149	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
150	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
151	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
152	#430	—	86	4	no	—
153	C₂×#108	—	86	2	no	—
154	#344	—	86	4	no	—
155	#341	—	86	4	no	—
156	#338	—	43	4	no	—
157	#336	—	2	2	no	—
158	C₄₃	#11295	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	43	4	no	—
163	C₂×M₁₆	—	43	4	no	—
164	C₂×M₁₆	—	86	2	no	—
165	C₂×M₁₆	—	86	2	no	—
166	C₂×M₁₆	—	86	2	no	—
167	C₂×M₁₆	—	86	2	no	—
168	C₂×M₁₆	—	86	2	no	—
169	C₂×M₁₆	—	86	2	no	—
170	C₂×M₁₆	—	86	2	no	—
171	C₂×M₁₆	—	86	2	no	—
172	C₂×M₁₆	—	86	2	no	—
173	C₂×M₁₆	—	86	2	no	—
174	C₂×M₁₆	Dic₈₆	1	2	yes	—
175	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	86	2	no	—
181	C₂²×C₈	—	86	2	no	—
182	C₂²×C₈	D₁₇₂	1	2	yes	—
183	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₂²⋊C₄	Dic₈₆	1	2	yes	—
188	C₂×C₄²	—	2	1	no	—
189	#108	—	86	8	no	—
190	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
195	C₂²×C₄	#10621898	1	1	yes	characteristic
196	C₂²×C₄	#10621898	1	1	yes	characteristic
197	C₂²×C₄	D₄×D₄₃	1	2	yes	—
198	C₂²×C₄	#10621900	1	2	yes	—
199	C₂²×C₄	#10621900	1	2	yes	—
200	C₂²×C₄	#10621902	1	2	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	86	4	no	—
205	M₁₆	—	86	4	no	—
206	M₁₆	—	86	4	no	—
207	M₁₆	—	86	4	no	—
208	M₁₆	—	172	2	no	—
209	M₁₆	—	172	2	no	—
210	M₁₆	—	172	2	no	—
211	M₁₆	—	172	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	86	4	no	—
220	C₂×C₈	—	86	4	no	—
221	C₂×C₈	—	172	2	no	—
222	C₂×C₈	—	172	2	no	—
223	C₂×C₈	—	172	2	no	—
224	C₂×C₈	—	172	2	no	—
225	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
226	C₄⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	C₄×Dic₈₆	1	1	yes	characteristic
232	C₂³	#13063302	1	2	yes	—
233	C₂³	#13063304	1	1	yes	characteristic
234	C₂³	#13063316	1	2	yes	—
235	C₂³	#13063325	1	1	yes	characteristic
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
239	C₂×C₄	#13063283	1	2	yes	—
240	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
241	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
242	C₂×C₄	#13063293	1	1	yes	characteristic
243	C₂×C₄	#13063299	1	1	yes	characteristic
244	C₂×C₄	#13063300	1	2	yes	—
245	C₂×C₄	#13063302	1	2	yes	—
246	C₂×C₄	#13063306	1	1	yes	characteristic
247	C₂×C₄	#13063309	1	2	yes	—
248	C₂×C₄	#13063310	1	2	yes	—
249	C₂×C₄	#13063311	1	2	yes	—
250	C₂×C₄	#13063312	1	1	yes	characteristic
251	C₂×C₄	#13063314	1	1	yes	characteristic
252	C₂×C₄	#13063316	1	2	yes	—
253	C₂×C₄	#13063317	1	1	yes	characteristic
254	C₂×C₄	#13063318	1	2	yes	—
255	C₂×C₄	#13063320	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	4	2	no	—
258	C₈	—	172	4	no	—
259	C₈	—	172	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	#14582786	1	2	yes	—
266	C₂²	#14582797	1	1	yes	characteristic
267	C₂²	#14582811	1	1	yes	characteristic
268	C₂²	#14582831	1	2	yes	—
269	C₂²	#14582838	1	1	yes	characteristic
270	C₄	—	8	2	no	—
271	C₄	#14582764	1	1	yes	characteristic
272	C₄	#14582771	1	1	yes	characteristic
273	C₄	#14582791	1	2	yes	—
274	C₄	#14582816	1	1	yes	characteristic
275	C₄	#14582824	1	1	yes	characteristic
276	C₄	#14582841	1	2	yes	—
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	#19132851	1	1	yes	characteristic
282	C₂	#19136175	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063314
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1729	C₂
socle	C₂²×C₄₃	#400
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1729	D₄₃
hypocenter	C₄₃	#11295
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
12. C₄₃⋊#338	C₄	semidirect
158. C₄₃	#11295	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582948	2
C₂×#14582669	2
C₂×#14582664	2
C₂×C₄₃⋊#338	2
C₄₃×#1729	2
#11295	43