Subgroups — #25011451

Order 11008.

586 conjugacy classes of subgroups.

The subgroup lattice has 268 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	#19132677	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132676	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10621872	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
8	C₂×#13063282	C₄	1	1	yes	characteristic
9	C₂×#13063265	C₂²	1	2	yes	—
10	C₂×#13063264	—	2	2	no	—
11	M₃₂×D₄₃	—	2	4	no	—
12	M₃₂×D₄₃	C₄	1	8	yes	complemented
13	C₂×C₁₆×D₄₃	—	2	2	no	—
14	#14582853	—	2	4	no	—
15	C₂×#13063236	C₄	1	1	yes	characteristic
16	C₂×#13063235	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#10621872	—	2	2	no	—
22	C₂×#10621872	—	2	2	no	—
23	C₂×#10621872	C₈	1	4	yes	—
24	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
27	C₂×C₈×D₄₃	D₄	1	2	yes	—
28	C₂×C₈×D₄₃	D₄	1	4	yes	—
29	C₂×C₈×D₄₃	Q₈	1	2	yes	—
30	#13063282	—	2	2	no	—
31	#13063282	C₈	1	4	yes	—
32	C₂×Dic₄₃⋊C₄	—	2	1	no	—
33	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
34	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
35	#13063265	—	2	2	no	—
36	#13063265	C₂×C₄	1	4	yes	—
37	#13063264	—	2	8	no	—
38	C₂×C₄₃⋊C₁₆	—	2	2	no	—
39	C₁₆×D₄₃	—	2	8	no	—
40	#13063236	—	2	2	no	—
41	#13063235	—	2	2	no	—
42	C₈×Dic₄₃	—	2	2	no	—
43	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
46	M₃₂×C₄₃	—	2	2	no	—
47	M₃₂×C₄₃	C₂×C₄	1	4	yes	—
48	C₂×C₁₆×C₄₃	—	2	2	no	—
49	C₄×C₈×C₄₃	—	2	2	no	—
50	C₂³×D₄₃	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	—	2	1	no	—
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₄₃	—	2	1	no	—
54	C₂²×Dic₄₃	M₁₆	1	1	yes	characteristic
55	C₂×C₄×D₄₃	C₄²	1	2	yes	—
56	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
60	C₂×C₄×D₄₃	M₁₆	1	2	yes	—
61	C₂×C₈×C₄₃	—	2	1	no	—
62	C₂×C₈×C₄₃	—	2	1	no	—
63	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₄₃	—	2	1	no	—
69	C₄²×C₄₃	C₂×C₈	1	2	yes	—
70	(C₂×D₄₃)⋊C₄	—	2	4	no	—
71	Dic₄₃⋊C₄	—	2	4	no	—
72	C₄×Dic₄₃	—	2	1	no	—
73	C₄×Dic₄₃	C₂×C₈	1	2	yes	—
74	C₂×C₄₃⋊C₈	—	2	1	no	—
75	C₂×C₄₃⋊C₈	—	2	1	no	—
76	C₂×C₄₃⋊C₈	C₄²	1	2	yes	—
77	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
79	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
80	#10621872	—	2	8	no	—
81	C₈×D₄₃	C₂²⋊C₄	1	4	yes	—
82	C₈×D₄₃	C₂²⋊C₄	1	4	yes	—
83	C₈×D₄₃	C₄⋊C₄	1	8	yes	—
84	C₁₆×C₄₃	—	2	4	no	—
85	C₄₃⋊C₁₆	—	2	4	no	—
86	C₂³×C₄₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂²×D₄₃	—	4	2	no	—
88	C₂²×D₄₃	#97	1	2	yes	—
89	C₂²×D₄₃	C₂²⋊C₈	1	2	yes	—
90	C₂²×D₄₃	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×Dic₄₃	—	2	2	no	—
100	C₂×Dic₄₃	—	4	2	no	—
101	C₂×Dic₄₃	#97	1	2	yes	—
102	C₂×Dic₄₃	C₂²⋊C₈	1	2	yes	—
103	C₂×Dic₄₃	C₄⋊C₈	1	2	yes	—
104	C₄×D₄₃	#95	1	4	yes	—
105	C₄×D₄₃	C₄×C₈	1	4	yes	—
106	C₄×D₄₃	C₂²⋊C₈	1	4	yes	—
107	C₄×D₄₃	C₄⋊C₈	1	4	yes	—
108	C₈×C₄₃	—	2	2	no	—
109	C₈×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
110	C₈×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₄₃⋊C₈	—	2	2	no	—
112	C₄₃⋊C₈	#95	1	4	yes	—
113	#12549	—	43	1	no	maximal, Sylow
114	C₂²×C₄₃	—	2	2	no	—
115	C₂²×C₄₃	C₂×#97	1	1	yes	characteristic
116	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂×D₄₃	—	2	4	no	—
119	C₂×D₄₃	—	4	2	no	—
120	C₂×D₄₃	—	4	2	no	—
121	C₂×D₄₃	#336	1	4	yes	—
122	C₄×C₄₃	—	4	2	no	—
123	C₄×C₄₃	C₂×#95	1	1	yes	characteristic
124	C₄×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
125	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
127	Dic₄₃	—	4	2	no	—
128	Dic₄₃	#336	1	4	yes	—
129	C₂²×M₃₂	—	43	2	no	—
130	C₂×#405	—	43	1	no	—
131	#1376	—	43	2	no	—
132	#1376	—	43	1	no	—
133	#1376	D₄₃	1	1	yes	characteristic, complemented
134	C₂×C₄₃	—	2	1	no	—
135	C₂×C₄₃	—	2	1	no	—
136	C₂×C₄₃	C₂×#336	1	1	yes	characteristic
137	C₂×C₄₃	#2153	1	2	yes	—
138	D₄₃	—	2	4	no	—
139	C₂²×M₁₆	—	43	1	no	—
140	C₂³×C₈	—	43	1	no	—
141	C₂×#117	—	43	1	no	—
142	C₂×M₃₂	—	43	2	no	—
143	C₂×M₃₂	—	43	8	no	—
144	C₂×M₃₂	—	86	2	no	—
145	C₂×M₃₂	—	86	4	no	—
146	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
147	C₂²×C₁₆	—	86	2	no	—
148	#405	—	86	4	no	—
149	C₂×#97	—	43	1	no	—
150	C₂×#97	—	43	1	no	—
151	C₂×C₄×C₈	—	43	1	no	—
152	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
153	C₄₃	#12549	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	—	43	1	no	—
155	C₂×M₁₆	—	43	4	no	—
156	C₂×M₁₆	—	86	2	no	—
157	C₂×M₁₆	—	86	2	no	—
158	C₂²×C₈	—	43	1	no	—
159	C₂²×C₈	—	43	4	no	—
160	C₂²×C₈	—	43	4	no	—
161	C₂²×C₈	—	43	2	no	—
162	C₂²×C₈	—	43	2	no	—
163	C₂²×C₈	—	43	1	no	—
164	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
166	#117	—	43	4	no	—
167	#117	—	86	2	no	—
168	C₂×C₄⋊C₄	—	86	1	no	—
169	C₂×C₂²⋊C₄	—	86	1	no	—
170	C₂×C₄²	—	43	1	no	—
171	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
172	M₃₂	—	2	2	no	—
173	M₃₂	—	43	4	no	—
174	M₃₂	—	86	8	no	—
175	M₃₂	—	86	2	no	—
176	M₃₂	C₄×D₄₃	1	4	yes	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	86	8	no	—
179	C₂×C₁₆	—	86	2	no	—
180	#97	—	86	2	no	—
181	#97	—	86	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	86	2	no	—
184	C₂⁴	—	43	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	43	1	no	—
187	C₂²×C₄	—	43	2	no	—
188	C₂²×C₄	—	43	2	no	—
189	C₂²×C₄	—	43	2	no	—
190	C₂²×C₄	—	43	2	no	—
191	C₂²×C₄	—	43	2	no	—
192	C₂²×C₄	—	43	2	no	—
193	C₂²×C₄	—	86	1	no	—
194	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
195	M₁₆	—	86	8	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	43	2	no	—
199	C₂×C₈	—	43	2	no	—
200	C₂×C₈	—	43	2	no	—
201	C₂×C₈	—	43	4	no	—
202	C₂×C₈	—	43	4	no	—
203	C₂×C₈	—	43	8	no	—
204	C₂×C₈	—	43	2	no	—
205	C₂×C₈	—	86	1	no	—
206	C₂×C₈	—	86	1	no	—
207	C₂×C₈	C₈×D₄₃	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
209	C₂×C₈	D₄×D₄₃	1	2	yes	—
210	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
211	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
212	C₄⋊C₄	—	86	4	no	—
213	C₂²⋊C₄	—	86	4	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	43	2	no	—
216	C₄²	—	86	1	no	—
217	C₄²	C₈×D₄₃	1	2	yes	—
218	C₁₆	—	2	4	no	—
219	C₁₆	—	86	4	no	—
220	C₂³	—	43	2	no	—
221	C₂³	—	43	2	no	—
222	C₂³	—	43	2	no	—
223	C₂³	—	172	2	no	—
224	C₂³	M₁₆×D₄₃	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	43	2	no	—
228	C₂×C₄	—	43	2	no	—
229	C₂×C₄	—	43	2	no	—
230	C₂×C₄	—	43	4	no	—
231	C₂×C₄	—	43	4	no	—
232	C₂×C₄	—	43	4	no	—
233	C₂×C₄	—	43	4	no	—
234	C₂×C₄	—	86	2	no	—
235	C₂×C₄	—	172	2	no	—
236	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
237	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
238	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
241	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	43	4	no	—
244	C₈	—	86	2	no	—
245	C₈	C₂²⋊C₄×D₄₃	1	2	yes	—
246	C₈	C₄⋊C₄×D₄₃	1	2	yes	—
247	C₂²	—	2	2	no	—
248	C₂²	—	43	4	no	—
249	C₂²	—	86	4	no	—
250	C₂²	—	172	2	no	—
251	C₂²	—	172	2	no	—
252	C₂²	#97×D₄₃	1	1	yes	characteristic
253	C₂²	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
254	C₂²	C₄⋊C₈×D₄₃	1	1	yes	characteristic
255	C₄	—	4	2	no	—
256	C₄	—	43	4	no	—
257	C₄	—	172	2	no	—
258	C₄	#95×D₄₃	1	1	yes	characteristic
259	C₄	C₄×C₈×D₄₃	1	1	yes	characteristic
260	C₄	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
261	C₄	C₄⋊C₈×D₄₃	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	86	4	no	—
265	C₂	#336×D₄₃	1	1	yes	characteristic
266	C₂	#19136359	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₄₃
commutator	C₄×C₄₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1376	C₂
socle	C₂²×C₄₃	C₂×C₄⋊C₈
prime radical	C₂³×D₄₃	M₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1376	D₄₃
hypocenter	C₄₃	#12549
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
3. #19132677	C₂	semidirect
4. #19132676	C₂	semidirect
5. C₄₃×#1376	C₂	semidirect
12. M₃₂×D₄₃	C₄	semidirect
133. #1376	D₄₃	semidirect
153. C₄₃	#12549	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
#19132677	2
#19132676	2
C₄₃×#1376	2
#12549	43