Subgroups — #25249236

Order 11008.

826 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₄₃	C₂	1	2	yes	maximal
2	C₂×#14582853	C₂	1	1	yes	maximal, characteristic
3	#19136193	C₂	1	4	yes	maximal, complemented
4	#19136190	C₂	1	4	yes	maximal, complemented
5	C₄×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
6	M₃₂×Dic₄₃	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621872	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
10	C₂×#13063282	C₄	1	1	yes	characteristic
11	C₂×#13063265	C₂²	1	2	yes	—
12	C₂×#13063264	C₂²	1	4	yes	—
13	M₃₂×D₄₃	C₄	1	16	yes	complemented
14	C₂×C₁₆×D₄₃	C₂²	1	4	yes	—
15	#14582853	C₄	1	2	yes	—
16	#14582853	C₄	1	2	yes	—
17	#14582853	C₂²	1	2	yes	—
18	#14582853	C₂²	1	2	yes	—
19	C₂×#13063236	C₄	1	1	yes	characteristic
20	C₂×#13063235	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
22	#14582675	C₂²	1	2	yes	complemented
23	#14582674	C₂²	1	4	yes	complemented
24	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	complemented
25	C₁₆×Dic₄₃	C₂²	1	4	yes	complemented
26	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#346	C₂²	1	2	yes	complemented
29	C₄×C₁₆×C₄₃	C₂²	1	2	yes	complemented
30	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#10621872	C₈	1	8	yes	—
32	C₂×#10621872	C₂×C₄	1	2	yes	—
33	C₂×#10621872	C₂×C₄	1	2	yes	—
34	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₄₃	C₂×C₄	1	8	yes	—
38	C₂×C₈×D₄₃	C₂³	1	2	yes	—
39	#13063282	C₈	1	4	yes	—
40	#13063282	C₂×C₄	1	2	yes	—
41	#13063282	C₂×C₄	1	2	yes	—
42	C₂×Dic₄₃⋊C₄	C₈	1	2	yes	—
43	C₂×(C₂×D₄₃)⋊C₄	C₈	1	2	yes	—
44	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
45	#13063265	C₂×C₄	1	8	yes	—
46	#13063264	C₂×C₄	1	16	yes	complemented
47	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
48	C₁₆×D₄₃	C₂×C₄	1	16	yes	complemented
49	#13063236	C₂×C₄	1	2	yes	—
50	#13063236	C₂×C₄	1	2	yes	—
51	#13063235	C₂×C₄	1	2	yes	—
52	#13063235	C₂×C₄	1	2	yes	—
53	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₄₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
58	M₃₂×C₄₃	C₂×C₄	1	8	yes	—
59	C₂×C₁₆×C₄₃	C₂³	1	4	yes	—
60	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂³×D₄₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
68	C₂²×Dic₄₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₄₃	C₄²	1	4	yes	—
70	C₂×C₄×D₄₃	C₂×C₈	1	4	yes	—
71	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
79	C₄²×C₄₃	C₂×C₈	1	2	yes	—
80	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	(C₂×D₄₃)⋊C₄	C₂×C₈	1	4	yes	—
83	(C₂×D₄₃)⋊C₄	C₂×C₈	1	4	yes	—
84	Dic₄₃⋊C₄	C₂×C₈	1	4	yes	—
85	Dic₄₃⋊C₄	C₂×C₈	1	4	yes	—
86	C₄×Dic₄₃	C₂×C₈	1	2	yes	—
87	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄₃⋊C₈	C₄²	1	4	yes	—
89	C₂×C₄₃⋊C₈	C₂×C₈	1	4	yes	—
90	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
92	#10621872	C₂×C₈	1	8	yes	—
93	#10621872	C₂×C₈	1	8	yes	—
94	C₈×D₄₃	C₄²	1	8	yes	—
95	C₈×D₄₃	C₂²×C₄	1	4	yes	—
96	C₈×D₄₃	C₂²×C₄	1	4	yes	—
97	C₁₆×C₄₃	C₂²×C₄	1	8	yes	—
98	C₄₃⋊C₁₆	C₂²×C₄	1	8	yes	—
99	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
100	C₂²×D₄₃	—	4	2	no	—
101	C₂²×D₄₃	C₄×C₈	1	4	yes	—
102	C₂²×D₄₃	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₄₃	—	4	2	no	—
111	C₂×Dic₄₃	C₄×C₈	1	4	yes	—
112	C₂×Dic₄₃	C₂²×C₈	1	4	yes	—
113	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
114	C₄×D₄₃	C₄×C₈	1	8	yes	—
115	C₄×D₄₃	C₂×C₄²	1	4	yes	—
116	C₄×D₄₃	C₂²×C₈	1	4	yes	—
117	C₈×C₄₃	C₂×C₄²	1	2	yes	—
118	C₈×C₄₃	C₂²×C₈	1	2	yes	—
119	C₈×C₄₃	C₂²×C₈	1	2	yes	—
120	C₈×C₄₃	C₂³×C₄	1	2	yes	—
121	C₄₃⋊C₈	C₂×C₄²	1	4	yes	—
122	C₄₃⋊C₈	C₂²×C₈	1	4	yes	—
123	#33324	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	C₂×C₄×C₈	1	2	yes	—
126	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
127	C₂×D₄₃	—	2	4	no	—
128	C₂×D₄₃	—	4	4	no	—
129	C₂×D₄₃	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₄₃	—	2	2	no	—
131	C₄×C₄₃	—	2	2	no	—
132	C₄×C₄₃	C₂×C₄×C₈	1	2	yes	—
133	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
134	C₄×C₄₃	C₂³×C₈	1	1	yes	characteristic
135	Dic₄₃	—	2	4	no	—
136	Dic₄₃	C₂×C₄×C₈	1	4	yes	—
137	C₂²×M₃₂	—	43	2	no	—
138	C₂×#405	—	43	1	no	—
139	#2108	—	43	4	no	—
140	#2108	—	43	4	no	—
141	C₄×M₃₂	—	43	2	no	—
142	C₄×M₃₂	—	43	1	no	—
143	C₄×M₃₂	D₄₃	1	1	yes	characteristic, complemented
144	C₂×C₄₃	—	2	2	no	—
145	C₂×C₄₃	C₂²×C₄×C₈	1	1	yes	characteristic
146	C₂×C₄₃	#3407	1	2	yes	—
147	D₄₃	—	2	4	no	—
148	C₂²×M₁₆	—	43	1	no	—
149	C₂³×C₈	—	43	1	no	—
150	C₂×#117	—	43	1	no	—
151	C₂×M₃₂	—	43	2	no	—
152	C₂×M₃₂	—	43	16	no	—
153	C₂×M₃₂	—	43	4	no	—
154	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
155	C₂²×C₁₆	—	43	4	no	—
156	#405	—	43	2	no	—
157	#405	—	43	2	no	—
158	#405	—	43	2	no	—
159	#405	—	43	2	no	—
160	C₂×#97	—	43	1	no	—
161	C₂×#97	—	43	1	no	—
162	C₂×C₄×C₈	—	43	1	no	—
163	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
164	#346	—	43	4	no	—
165	#346	—	43	2	no	—
166	#346	C₂×D₄₃	1	2	yes	complemented
167	C₄×C₁₆	—	43	4	no	—
168	C₄×C₁₆	—	43	2	no	—
169	C₄×C₁₆	C₂×D₄₃	1	2	yes	complemented
170	C₄₃	#33324	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	—	43	1	no	—
172	C₂×M₁₆	—	43	8	no	—
173	C₂×M₁₆	—	43	2	no	—
174	C₂×M₁₆	—	43	2	no	—
175	C₂²×C₈	—	43	1	no	—
176	C₂²×C₈	—	43	1	no	—
177	C₂²×C₈	—	43	2	no	—
178	C₂²×C₈	—	43	8	no	—
179	C₂²×C₈	—	43	2	no	—
180	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
182	#117	—	43	4	no	—
183	#117	—	43	2	no	—
184	#117	—	43	2	no	—
185	C₂×C₄⋊C₄	—	43	2	no	—
186	C₂×C₂²⋊C₄	—	43	2	no	—
187	C₂×C₄²	—	43	1	no	—
188	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
189	M₃₂	—	43	8	no	—
190	M₃₂	—	43	16	no	—
191	M₃₂	C₄×D₄₃	1	8	yes	—
192	C₂×C₁₆	—	43	4	no	—
193	C₂×C₁₆	—	43	16	no	—
194	C₂×C₁₆	C₂²×D₄₃	1	4	yes	—
195	#97	—	43	2	no	—
196	#97	—	43	2	no	—
197	#97	—	43	2	no	—
198	#97	—	43	2	no	—
199	C₄×C₈	—	43	2	no	—
200	C₄×C₈	—	43	2	no	—
201	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
202	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
205	C₂⁴	—	43	1	no	—
206	C₂²×C₄	—	43	1	no	—
207	C₂²×C₄	—	43	4	no	—
208	C₂²×C₄	—	43	2	no	—
209	C₂²×C₄	—	43	4	no	—
210	C₂²×C₄	—	43	2	no	—
211	C₂²×C₄	—	43	2	no	—
212	C₂²×C₄	C₈×D₄₃	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
214	M₁₆	—	43	8	no	—
215	M₁₆	—	43	8	no	—
216	C₂×C₈	—	43	4	no	—
217	C₂×C₈	—	43	4	no	—
218	C₂×C₈	—	43	2	no	—
219	C₂×C₈	—	43	2	no	—
220	C₂×C₈	—	43	8	no	—
221	C₂×C₈	—	43	4	no	—
222	C₂×C₈	—	43	4	no	—
223	C₂×C₈	C₈×D₄₃	1	4	yes	—
224	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
226	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
228	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
229	C₄⋊C₄	—	43	4	no	—
230	C₄⋊C₄	—	43	4	no	—
231	C₂²⋊C₄	—	43	4	no	—
232	C₂²⋊C₄	—	43	4	no	—
233	C₄²	—	43	2	no	—
234	C₄²	—	43	2	no	—
235	C₄²	C₈×D₄₃	1	2	yes	—
236	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
238	C₁₆	—	43	8	no	—
239	C₁₆	C₂×C₄×D₄₃	1	8	yes	—
240	C₂³	—	43	2	no	—
241	C₂³	—	43	4	no	—
242	C₂³	—	172	2	no	—
243	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	43	4	no	—
246	C₂×C₄	—	43	2	no	—
247	C₂×C₄	—	43	4	no	—
248	C₂×C₄	—	43	8	no	—
249	C₂×C₄	—	43	4	no	—
250	C₂×C₄	—	43	4	no	—
251	C₂×C₄	—	172	2	no	—
252	C₂×C₄	C₄²×D₄₃	1	2	yes	—
253	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
254	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
255	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
257	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
258	C₈	—	43	4	no	—
259	C₈	—	43	4	no	—
260	C₈	C₄²×D₄₃	1	2	yes	—
261	C₈	C₂×C₈×D₄₃	1	2	yes	—
262	C₈	C₂×C₈×D₄₃	1	2	yes	—
263	C₈	C₂²×C₄×D₄₃	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	43	4	no	—
266	C₂²	—	86	4	no	—
267	C₂²	—	172	4	no	—
268	C₂²	C₄×C₈×D₄₃	1	2	yes	—
269	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	43	4	no	—
273	C₄	—	86	4	no	—
274	C₄	C₄×C₈×D₄₃	1	2	yes	—
275	C₄	C₂×C₄²×D₄₃	1	1	yes	characteristic
276	C₄	C₂²×C₈×D₄₃	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	86	4	no	—
279	C₂	C₂×C₄×C₈×D₄₃	1	1	yes	characteristic
280	C₂	#19147697	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator	C₂×C₄₃	C₂²×C₄×C₈
Fitting	C₄×M₃₂×C₄₃	C₂
socle	C₂²×C₄₃	C₂³×C₈
prime radical	C₂³×D₄₃	C₂×C₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₄×M₃₂	D₄₃
hypocenter	C₄₃	#33324
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #19136193	C₂	semidirect
4. #19136190	C₂	semidirect
5. C₄×#13063265	C₂	semidirect
6. M₃₂×Dic₄₃	C₂	semidirect
7. C₄×M₃₂×C₄₃	C₂	semidirect
13. M₃₂×D₄₃	C₄	semidirect
22. #14582675	C₂²	semidirect
23. #14582674	C₂²	semidirect
24. C₄×C₄₃⋊C₁₆	C₂²	semidirect
25. C₁₆×Dic₄₃	C₂²	semidirect
28. C₄₃×#346	C₂²	semidirect
29. C₄×C₁₆×C₄₃	C₂²	semidirect
46. #13063264	C₂×C₄	semidirect
48. C₁₆×D₄₃	C₂×C₄	semidirect
143. C₄×M₃₂	D₄₃	semidirect
166. #346	C₂×D₄₃	semidirect
169. C₄×C₁₆	C₂×D₄₃	semidirect
170. C₄₃	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₄₃	2
C₂×#14582853	2
#19136193	2
#19136190	2
C₄×#13063265	2
M₃₂×Dic₄₃	2
C₄×M₃₂×C₄₃	2
#33324	43