Subgroups — #24985980

Order 11008.

798 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10621877	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄₃⋊#1273	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10621877	C₄	1	2	yes	—
6	C₂³×C₄₃⋊C₈	C₄	1	2	yes	—
7	C₄×#10621877	—	2	2	no	—
8	C₄×#10621877	—	2	2	no	—
9	C₂×#13063236	C₄	1	2	yes	—
10	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
11	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
12	C₂×C₄×C₄₃⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#431	—	2	2	no	—
17	C₄₃×#431	—	2	2	no	—
18	C₂×C₄⋊C₈×C₄₃	C₄	1	2	yes	—
19	C₂×C₂²⋊C₈×C₄₃	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
21	C₂×#10621877	—	2	4	no	—
22	C₂×#10621877	—	2	4	no	—
23	C₂×#10621877	C₈	1	8	yes	complemented
24	C₂²×C₄₃⋊C₈	—	2	4	no	—
25	C₂²×C₄₃⋊C₈	—	2	8	no	—
26	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
28	#13063236	—	2	4	no	—
29	C₄×C₄₃⋊C₈	—	2	4	no	—
30	C₄×C₄₃⋊C₈	—	2	2	no	—
31	C₄×C₄₃⋊C₈	—	2	2	no	—
32	C₄×C₄₃⋊C₈	—	2	2	no	—
33	C₄×C₄₃⋊C₈	—	2	2	no	—
34	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₃	D₄	1	2	yes	—
42	C₂×C₄²×C₄₃	D₄	1	2	yes	—
43	C₂×C₄²×C₄₃	D₄	1	2	yes	—
44	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
45	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₄₃	—	2	4	no	—
47	C₂²⋊C₈×C₄₃	—	2	4	no	—
48	C₄×C₈×C₄₃	—	2	2	no	—
49	C₄×C₈×C₄₃	—	2	2	no	—
50	C₂⁴×C₄₃	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	—	2	4	no	—
52	C₂²×C₄×C₄₃	—	2	4	no	—
53	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₄₃	M₁₆	1	4	yes	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₄₃	—	2	4	no	—
68	C₂×C₈×C₄₃	—	2	8	no	—
69	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
74	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
75	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
79	#10621877	—	2	16	no	—
80	C₂×C₄₃⋊C₈	—	2	16	no	—
81	C₂×C₄₃⋊C₈	—	2	4	no	—
82	C₂×C₄₃⋊C₈	—	2	4	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	C₂×C₈	1	8	yes	complemented
86	C₂×C₄₃⋊C₈	M₁₆	1	8	yes	complemented
87	C₂³×C₄₃	—	2	2	no	—
88	C₂³×C₄₃	—	2	2	no	—
89	C₂³×C₄₃	#95	1	2	yes	—
90	C₂³×C₄₃	C₄×C₈	1	2	yes	—
91	C₂³×C₄₃	#97	1	2	yes	—
92	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	2	no	—
95	C₂×C₄×C₄₃	—	2	4	no	—
96	C₂×C₄×C₄₃	—	2	4	no	—
97	C₂×C₄×C₄₃	—	2	4	no	—
98	C₂×C₄×C₄₃	—	2	4	no	—
99	C₂×C₄×C₄₃	#95	1	2	yes	—
100	C₂×C₄×C₄₃	#95	1	2	yes	—
101	C₂×C₄×C₄₃	#95	1	2	yes	—
102	C₂×C₄×C₄₃	C₄×C₈	1	2	yes	—
103	C₂×C₄×C₄₃	#97	1	2	yes	—
104	C₂×C₄×C₄₃	C₂²⋊C₈	1	8	yes	—
105	C₂×C₄×C₄₃	C₄⋊C₈	1	8	yes	—
106	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₄₃	C₂×M₁₆	1	4	yes	—
114	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₄₃	#131	1	4	yes	—
116	C₈×C₄₃	—	4	4	no	—
117	C₄₃⋊C₈	—	2	8	no	—
118	C₄₃⋊C₈	—	2	8	no	—
119	#7848	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	—	2	4	no	—
121	C₂²×C₄₃	—	2	4	no	—
122	C₂²×C₄₃	—	2	2	no	—
123	C₂²×C₄₃	—	2	2	no	—
124	C₂²×C₄₃	#336	1	4	yes	—
125	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
126	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
127	C₂²×C₄₃	C₂×#97	1	1	yes	characteristic
128	C₂²×C₄₃	C₄×M₁₆	1	2	yes	—
129	C₂²×C₄₃	#405	1	2	yes	—
130	C₄×C₄₃	—	2	2	no	—
131	C₄×C₄₃	—	2	2	no	—
132	C₄×C₄₃	#336	1	4	yes	—
133	C₄×C₄₃	C₂×C₂²⋊C₈	1	2	yes	—
134	C₄×C₄₃	#407	1	2	yes	—
135	C₄×C₄₃	#408	1	2	yes	—
136	C₄×C₄₃	C₂×C₄⋊C₈	1	2	yes	—
137	C₄×C₄₃	#423	1	2	yes	—
138	C₄×C₄₃	#424	1	2	yes	—
139	C₂×#431	D₄₃	1	1	yes	characteristic
140	C₂×C₄×M₁₆	—	43	1	no	—
141	C₂²×C₄×C₈	—	43	1	no	—
142	#1273	—	43	4	no	—
143	C₂×C₄₃	—	2	4	no	—
144	C₂×C₄₃	C₈×M₁₆	1	2	yes	—
145	C₂×C₄₃	#1451	1	2	yes	—
146	C₂×C₄₃	C₂×#336	1	1	yes	characteristic
147	C₂×C₄₃	#1728	1	1	yes	characteristic
148	C₂×C₄₃	#1729	1	1	yes	characteristic
149	C₂²×M₁₆	—	43	2	no	—
150	C₂³×C₈	—	43	2	no	—
151	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
152	#431	—	2	2	no	—
153	#431	—	2	2	no	—
154	C₂×C₄⋊C₈	Dic₄₃	1	2	yes	—
155	C₂×C₂²⋊C₈	Dic₄₃	1	2	yes	—
156	C₄×M₁₆	—	86	2	no	—
157	C₄×M₁₆	—	86	2	no	—
158	C₂×#97	—	43	2	no	—
159	C₂×C₄×C₈	—	43	2	no	—
160	C₂×C₄×C₈	—	43	2	no	—
161	C₂×C₄×C₈	—	43	2	no	—
162	C₂×C₄×C₈	—	86	2	no	—
163	C₂×C₄×C₈	—	86	2	no	—
164	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
165	C₄₃	#7848	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
167	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
169	C₂×M₁₆	—	43	8	no	—
170	C₂×M₁₆	—	86	4	no	—
171	C₂×M₁₆	—	86	4	no	—
172	C₂²×C₈	—	43	4	no	—
173	C₂²×C₈	—	43	4	no	—
174	C₂²×C₈	—	86	8	no	—
175	C₂²×C₈	—	86	4	no	—
176	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
177	C₂×C₄²	Dic₈₆	1	2	yes	—
178	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
179	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
180	C₂×C₄²	D₁₇₂	1	2	yes	—
181	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
184	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
185	C₄⋊C₈	—	2	4	no	—
186	C₂²⋊C₈	—	2	4	no	—
187	#97	—	86	4	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	86	4	no	—
191	C₄×C₈	—	86	2	no	—
192	C₄×C₈	—	86	2	no	—
193	C₄×C₈	—	86	2	no	—
194	C₄×C₈	—	86	2	no	—
195	C₂⁴	C₄×Dic₄₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₈×D₄₃	1	4	yes	—
199	C₂²×C₄	#10621872	1	4	yes	—
200	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
201	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
202	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
203	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
204	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
206	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
207	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
208	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
209	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
212	M₁₆	—	86	16	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	8	no	—
215	C₂×C₈	—	43	8	no	—
216	C₂×C₈	—	43	8	no	—
217	C₂×C₈	—	86	16	no	—
218	C₂×C₈	—	86	4	no	—
219	C₂×C₈	—	86	4	no	—
220	C₂×C₈	—	86	4	no	—
221	C₂×C₈	—	86	4	no	—
222	C₄²	Dic₄₃⋊C₄	1	2	yes	—
223	C₄²	Dic₄₃⋊C₄	1	2	yes	—
224	C₄²	C₄⋊Dic₄₃	1	2	yes	—
225	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
226	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
227	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
228	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
229	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
230	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₄₃⋊#95	1	2	yes	—
235	C₂³	C₈×Dic₄₃	1	2	yes	—
236	C₂³	#13063235	1	2	yes	—
237	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	C₄₃⋊#95	1	2	yes	—
245	C₂×C₄	C₄₃⋊#95	1	2	yes	—
246	C₂×C₄	C₄₃⋊#95	1	2	yes	—
247	C₂×C₄	C₈×Dic₄₃	1	2	yes	—
248	C₂×C₄	#13063235	1	2	yes	—
249	C₂×C₄	(C₂×D₄₃)⋊C₈	1	8	yes	—
250	C₂×C₄	Dic₄₃⋊C₈	1	8	yes	—
251	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
254	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
255	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
258	C₂×C₄	M₁₆×D₄₃	1	4	yes	—
259	C₂×C₄	C₂×#10621872	1	2	yes	—
260	C₂×C₄	#13063332	1	4	yes	—
261	C₈	—	4	4	no	—
262	C₈	—	86	8	no	—
263	C₈	—	86	8	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	#14582653	1	4	yes	—
269	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
270	C₂²	C₂×C₈×Dic₄₃	1	1	yes	characteristic
271	C₂²	C₂×#13063235	1	1	yes	characteristic
272	C₂²	M₁₆×Dic₄₃	1	2	yes	—
273	C₂²	#14582855	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	#14582653	1	4	yes	—
277	C₄	C₂×(C₂×D₄₃)⋊C₈	1	2	yes	—
278	C₄	#14582862	1	2	yes	—
279	C₄	#14582866	1	2	yes	—
280	C₄	C₂×Dic₄₃⋊C₈	1	2	yes	—
281	C₄	#14582922	1	2	yes	—
282	C₄	#14582926	1	2	yes	—
283	C₂	—	2	4	no	—
284	C₂	#19132876	1	2	yes	—
285	C₂	#19132885	1	2	yes	—
286	C₂	C₂×#14582653	1	1	yes	characteristic
287	C₂	#19134288	1	1	yes	characteristic
288	C₂	#19134292	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂×C₄×C₈
Fitting	C₂×C₄₃×#431	C₂
socle	C₂³×C₄₃	C₂×C₄²
prime radical	C₂⁴×C₄₃	C₄²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#431	D₄₃
hypocenter	C₄₃	#7848
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. C₄₃⋊#1273	C₂	semidirect
23. C₂×#10621877	C₈	semidirect
85. C₂×C₄₃⋊C₈	C₂×C₈	semidirect
86. C₂×C₄₃⋊C₈	M₁₆	semidirect
165. C₄₃	#7848	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10621877	2
C₂²×C₄×C₄₃⋊C₈	2
C₄₃⋊#1273	2
C₂×C₄₃×#431	2
#7848	43