Subgroups — #25011627

Order 11008.

590 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₄₃)⋊C₁₆	C₂	1	2	yes	maximal
3	C₂×#14582674	C₂	1	2	yes	maximal, complemented
4	#19132673	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₄₃	C₄	1	1	yes	characteristic
9	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₄₃⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₄₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₄₃	—	2	4	no	—
14	(C₂×D₄₃)⋊C₁₆	C₄	1	8	yes	complemented
15	#14582674	—	2	4	no	—
16	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₄₃	C₈	1	2	yes	—
19	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₄₃	—	2	2	no	—
23	C₂×C₈×D₄₃	—	2	1	no	—
24	C₂×C₈×D₄₃	—	2	1	no	—
25	C₂×C₈×D₄₃	—	2	2	no	—
26	C₂×C₈×D₄₃	—	2	2	no	—
27	C₂×C₈×D₄₃	C₈	1	4	yes	—
28	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₄₃	C₈	1	2	yes	—
30	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×D₄₃	—	2	2	no	—
32	C₄²×D₄₃	—	2	2	no	—
33	C₂×C₄₃⋊C₁₆	—	2	2	no	—
34	C₂×C₄₃⋊C₁₆	—	2	2	no	—
35	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
36	C₄×C₄₃⋊C₈	—	2	2	no	—
37	C₈×Dic₄₃	—	2	2	no	—
38	C₈×Dic₄₃	D₄	1	4	yes	—
39	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₄₃	—	2	2	no	—
43	C₂×C₁₆×C₄₃	—	2	2	no	—
44	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
45	C₄×C₈×C₄₃	—	2	2	no	—
46	C₂³×D₄₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₄₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₄₃	—	2	4	no	—
52	C₂×C₄×D₄₃	—	2	1	no	—
53	C₂×C₄×D₄₃	—	2	1	no	—
54	C₂×C₄×D₄₃	—	2	1	no	—
55	C₂×C₄×D₄₃	—	2	1	no	—
56	C₂×C₄×D₄₃	—	2	4	no	—
57	C₂×C₄×D₄₃	—	2	4	no	—
58	C₂×C₄×D₄₃	C₄²	1	2	yes	—
59	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₄₃	—	2	1	no	—
61	C₂×C₈×C₄₃	—	2	1	no	—
62	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
63	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
66	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
67	C₄²×C₄₃	—	2	1	no	—
68	C₄²×C₄₃	—	2	1	no	—
69	C₄×Dic₄₃	—	2	4	no	—
70	C₄×Dic₄₃	C₂²⋊C₄	1	2	yes	—
71	C₄×Dic₄₃	M₁₆	1	2	yes	—
72	C₂×C₄₃⋊C₈	—	2	1	no	—
73	C₂×C₄₃⋊C₈	—	2	1	no	—
74	C₂×C₄₃⋊C₈	C₄²	1	2	yes	—
75	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
77	C₂×C₄₃⋊C₈	M₁₆	1	2	yes	—
78	C₈×D₄₃	—	2	4	no	—
79	C₈×D₄₃	—	2	4	no	—
80	C₈×D₄₃	—	2	8	no	—
81	C₁₆×C₄₃	—	2	4	no	—
82	C₄₃⋊C₁₆	—	2	4	no	—
83	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
84	C₂²×D₄₃	—	2	1	no	—
85	C₂²×D₄₃	—	2	1	no	—
86	C₂²×D₄₃	—	2	2	no	—
87	C₂²×D₄₃	C₄×C₈	1	2	yes	—
88	C₂²×D₄₃	M₃₂	1	4	yes	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	M₃₂	1	4	yes	—
92	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×Dic₄₃	—	2	1	no	—
99	C₂×Dic₄₃	—	2	1	no	—
100	C₂×Dic₄₃	—	2	2	no	—
101	C₂×Dic₄₃	—	2	2	no	—
102	C₂×Dic₄₃	C₄×C₈	1	2	yes	—
103	C₂×Dic₄₃	C₂²⋊C₈	1	4	yes	—
104	C₄×D₄₃	—	2	16	no	—
105	C₄×D₄₃	—	2	2	no	—
106	C₄×D₄₃	—	2	2	no	—
107	C₄×D₄₃	—	2	2	no	—
108	C₄×D₄₃	—	2	2	no	—
109	C₈×C₄₃	—	2	2	no	—
110	C₈×C₄₃	C₄×D₄	1	2	yes	—
111	C₈×C₄₃	C₄×D₄	1	2	yes	—
112	C₄₃⋊C₈	—	2	2	no	—
113	C₄₃⋊C₈	C₂²⋊C₈	1	4	yes	—
114	#12599	—	43	1	no	maximal, Sylow
115	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
116	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₄₃	#431	1	1	yes	characteristic
118	C₂²×C₄₃	C₂×M₃₂	1	2	yes	—
119	C₂²×C₄₃	C₂×M₃₂	1	2	yes	—
120	C₂×D₄₃	—	2	2	no	—
121	C₂×D₄₃	—	2	4	no	—
122	C₂×D₄₃	—	2	4	no	—
123	C₂×D₄₃	—	2	2	no	—
124	C₂×D₄₃	#346	1	4	yes	complemented
125	C₄×C₄₃	—	2	4	no	—
126	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
128	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₄×C₄₃	#431	1	1	yes	characteristic
130	Dic₄₃	—	2	4	no	—
131	Dic₄₃	—	2	2	no	—
132	C₂²×C₄×C₈	—	43	1	no	—
133	C₂×C₂²⋊C₁₆	—	43	2	no	—
134	C₂×#346	—	43	2	no	—
135	#1374	—	43	1	no	—
136	#1374	D₄₃	1	1	yes	characteristic, complemented
137	C₂×C₄₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
138	C₂×C₄₃	C₂×#346	1	1	yes	characteristic
139	C₂×C₄₃	C₄×M₃₂	1	1	yes	characteristic
140	C₂×C₄₃	#2112	1	2	yes	—
141	C₂×C₄₃	#2163	1	2	yes	—
142	D₄₃	—	2	4	no	—
143	C₂³×C₈	—	43	1	no	—
144	C₂³×C₈	—	43	1	no	—
145	C₂²×C₄²	—	43	1	no	—
146	C₂²×C₁₆	—	43	2	no	—
147	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
148	C₂×C₄×C₈	—	43	1	no	—
149	C₂×C₄×C₈	—	43	1	no	—
150	C₂×C₄×C₈	—	43	1	no	—
151	C₂×C₄×C₈	—	86	4	no	—
152	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
153	C₂²⋊C₁₆	—	43	8	no	—
154	#346	—	86	4	no	—
155	C₄₃	#12599	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₄	—	43	1	no	—
157	C₂³×C₄	—	43	2	no	—
158	C₂²×C₈	—	43	1	no	—
159	C₂²×C₈	—	43	1	no	—
160	C₂²×C₈	—	43	4	no	—
161	C₂²×C₈	—	43	4	no	—
162	C₂²×C₈	—	86	2	no	—
163	C₂²×C₈	—	86	2	no	—
164	C₂²×C₈	—	86	2	no	—
165	C₂²×C₈	—	86	1	no	—
166	C₂²×C₈	—	86	1	no	—
167	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
169	C₂×C₄²	—	43	1	no	—
170	C₂×C₄²	—	43	2	no	—
171	C₂×C₄²	—	86	2	no	—
172	C₂×C₄²	—	86	2	no	—
173	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	43	4	no	—
177	C₂×C₁₆	—	86	2	no	—
178	C₂×C₁₆	—	86	2	no	—
179	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	43	4	no	—
182	C₄×C₈	—	86	2	no	—
183	C₄×C₈	—	86	2	no	—
184	C₂⁴	—	43	1	no	—
185	C₂²×C₄	—	43	1	no	—
186	C₂²×C₄	—	43	2	no	—
187	C₂²×C₄	—	43	2	no	—
188	C₂²×C₄	—	43	2	no	—
189	C₂²×C₄	—	86	4	no	—
190	C₂²×C₄	—	86	4	no	—
191	C₂²×C₄	—	86	4	no	—
192	C₂²×C₄	—	86	1	no	—
193	C₂²×C₄	—	86	1	no	—
194	C₂²×C₄	—	86	1	no	—
195	C₂²×C₄	—	86	1	no	—
196	C₂²×C₄	C₈×D₄₃	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	43	2	no	—
201	C₂×C₈	—	43	2	no	—
202	C₂×C₈	—	43	2	no	—
203	C₂×C₈	—	43	2	no	—
204	C₂×C₈	—	86	8	no	—
205	C₂×C₈	—	86	1	no	—
206	C₂×C₈	—	86	1	no	—
207	C₂×C₈	—	86	4	no	—
208	C₂×C₈	—	86	4	no	—
209	C₂×C₈	C₈×D₄₃	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
211	C₂×C₈	D₄×D₄₃	1	2	yes	—
212	C₂×C₈	#10621900	1	1	yes	characteristic
213	C₂×C₈	#10621902	1	1	yes	characteristic
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	43	2	no	—
217	C₄²	—	43	2	no	—
218	C₄²	—	86	4	no	—
219	C₁₆	—	2	4	no	—
220	C₁₆	—	86	4	no	—
221	C₂³	—	43	4	no	—
222	C₂³	—	43	2	no	—
223	C₂³	—	86	2	no	—
224	C₂³	—	86	1	no	—
225	C₂³	—	86	1	no	—
226	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	43	2	no	—
230	C₂×C₄	—	43	4	no	—
231	C₂×C₄	—	86	16	no	—
232	C₂×C₄	—	86	2	no	—
233	C₂×C₄	—	86	2	no	—
234	C₂×C₄	—	86	1	no	—
235	C₂×C₄	—	86	1	no	—
236	C₂×C₄	—	86	2	no	—
237	C₂×C₄	—	86	2	no	—
238	C₂×C₄	—	86	2	no	—
239	C₂×C₄	—	86	2	no	—
240	C₂×C₄	#13063264	1	4	yes	—
241	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
242	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
243	C₂×C₄	#13063283	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
245	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
246	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	43	4	no	—
249	C₈	—	86	2	no	—
250	C₈	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
251	C₈	#13063287	1	2	yes	—
252	C₂²	—	43	4	no	—
253	C₂²	—	86	4	no	—
254	C₂²	—	86	2	no	—
255	C₂²	—	86	2	no	—
256	C₂²	—	86	4	no	—
257	C₂²	C₄×C₈×D₄₃	1	1	yes	characteristic
258	C₂²	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
259	C₂²	#14582951	1	1	yes	characteristic
260	C₂²	M₃₂×D₄₃	1	2	yes	—
261	C₂²	C₂×#13063264	1	2	yes	—
262	C₄	—	2	4	no	—
263	C₄	—	86	2	no	—
264	C₄	—	86	4	no	—
265	C₄	#14582746	1	1	yes	characteristic
266	C₄	#14582851	1	1	yes	characteristic
267	C₄	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
268	C₄	#14582951	1	1	yes	characteristic
269	C₂	—	86	4	no	—
270	C₂	#19134352	1	1	yes	characteristic
271	C₂	#346×D₄₃	1	1	yes	characteristic
272	C₂	C₄×#13063264	1	1	yes	characteristic
273	C₂	#19136204	1	2	yes	—
274	C₂	#19136397	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂×C₄×C₈
Fitting	C₄₃×#1374	C₂
socle	C₂³×C₄₃	C₂²×C₈
prime radical	C₂³×D₄₃	C₂×C₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1374	D₄₃
hypocenter	C₄₃	#12599
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
3. C₂×#14582674	C₂	semidirect
4. #19132673	C₂	semidirect
5. C₄₃×#1374	C₂	semidirect
14. (C₂×D₄₃)⋊C₁₆	C₄	semidirect
35. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
44. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
124. C₂×D₄₃	#346	semidirect
136. #1374	D₄₃	semidirect
155. C₄₃	#12599	semidirect
179. C₂×C₁₆	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₄₃	2
C₂×(C₂×D₄₃)⋊C₁₆	2
C₂×#14582674	2
#19132673	2
C₄₃×#1374	2
#12599	43