Subgroups — #25001847

Order 11008.

1646 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582759	C₂	1	2	yes	maximal
2	C₂×#14582662	C₂	1	4	yes	maximal, complemented
3	C₂⁴×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₄₃	C₂²	1	4	yes	—
5	C₂²×C₂²⋊Dic₄₃	—	2	2	no	—
6	#14582759	—	2	4	no	—
7	#14582759	C₄	1	8	yes	complemented
8	C₂×C₄₃⋊#95	—	2	2	no	—
9	#14582662	—	2	8	no	—
10	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₄₃	—	2	4	no	—
12	C₂³×C₈×C₄₃	—	2	8	no	—
13	C₂³×C₈×C₄₃	C₄	1	4	yes	complemented
14	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
15	C₂³×Dic₄₃	—	2	4	no	—
16	C₂×C₄⋊Dic₄₃	—	2	8	no	—
17	C₂×C₄⋊Dic₄₃	—	2	8	no	—
18	C₂×C₄⋊Dic₄₃	—	2	4	no	—
19	C₂×C₄⋊Dic₄₃	C₂×C₄	1	8	yes	—
20	C₂×C₂²⋊Dic₄₃	—	2	8	no	—
21	C₂×C₂²⋊Dic₄₃	—	4	4	no	—
22	C₂×C₂²⋊Dic₄₃	—	4	4	no	—
23	C₄₃⋊#95	—	2	8	no	—
24	C₂⁵×C₄₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₃	—	2	4	no	—
26	C₂³×C₄×C₄₃	—	2	4	no	—
27	C₂³×C₄×C₄₃	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₃	D₄	1	4	yes	—
31	C₂³×C₄×C₄₃	Q₈	1	2	yes	—
32	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₄₃	—	2	32	no	—
34	C₂²×C₈×C₄₃	—	2	8	no	—
35	C₂²×C₈×C₄₃	—	2	4	no	—
36	C₂²×C₈×C₄₃	—	2	4	no	—
37	C₂²×C₈×C₄₃	—	2	16	no	—
38	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₄₃	D₄	1	8	yes	—
40	C₂⁴×C₄₃	—	2	4	no	—
41	C₂⁴×C₄₃	—	2	4	no	—
42	C₂⁴×C₄₃	C₂²⋊C₄	1	4	yes	—
43	C₂⁴×C₄₃	C₄⋊C₄	1	2	yes	—
44	C₂⁴×C₄₃	D₈	1	2	yes	—
45	C₂⁴×C₄₃	SD₁₆	1	4	yes	—
46	C₂⁴×C₄₃	Q₁₆	1	2	yes	—
47	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	—	2	4	no	—
49	C₂²×C₄×C₄₃	—	2	4	no	—
50	C₂²×C₄×C₄₃	—	2	16	no	—
51	C₂²×C₄×C₄₃	—	2	8	no	—
52	C₂²×C₄×C₄₃	—	2	8	no	—
53	C₂²×C₄×C₄₃	—	2	8	no	—
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
56	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
59	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	#41	1	4	yes	—
66	C₂²×Dic₄₃	—	2	8	no	—
67	C₂²×Dic₄₃	—	4	8	no	—
68	C₂²×Dic₄₃	—	4	4	no	—
69	C₂²×Dic₄₃	—	4	4	no	—
70	C₂²×Dic₄₃	—	4	8	no	—
71	C₂×C₈×C₄₃	—	2	32	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	—	2	16	no	—
75	C₂×C₈×C₄₃	C₂²⋊C₄	1	8	yes	complemented
76	C₂²⋊Dic₄₃	—	4	16	no	—
77	C₄⋊Dic₄₃	—	2	16	no	—
78	C₄⋊Dic₄₃	—	4	4	no	—
79	C₄⋊Dic₄₃	—	4	4	no	—
80	C₂³×C₄₃	—	2	4	no	—
81	C₂³×C₄₃	—	2	8	no	—
82	C₂³×C₄₃	—	2	8	no	—
83	C₂³×C₄₃	—	2	4	no	—
84	C₂³×C₄₃	—	2	16	no	—
85	C₂³×C₄₃	—	2	8	no	—
86	C₂³×C₄₃	—	2	8	no	—
87	C₂³×C₄₃	#95	1	4	yes	—
88	C₂³×C₄₃	#102	1	8	yes	—
89	C₂³×C₄₃	#103	1	8	yes	—
90	C₂³×C₄₃	#106	1	4	yes	—
91	C₂³×C₄₃	#107	1	4	yes	—
92	C₂³×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₄₃	#121	1	4	yes	—
95	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
96	C₂³×C₄₃	C₂×SD₁₆	1	2	yes	—
97	C₂³×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
98	C₂³×C₄₃	#135	1	4	yes	—
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	2	8	no	—
101	C₂×C₄×C₄₃	—	2	8	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	2	16	no	—
105	C₂×C₄×C₄₃	#95	1	4	yes	—
106	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₃	#117	1	2	yes	—
111	C₂×C₄×C₄₃	C₄×D₄	1	8	yes	—
112	C₂×C₄×C₄₃	#120	1	4	yes	—
113	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
114	C₂×C₄×C₄₃	#123	1	4	yes	—
115	C₂×Dic₄₃	—	4	8	no	—
116	C₂×Dic₄₃	—	4	8	no	—
117	C₂×Dic₄₃	—	4	8	no	—
118	C₂×Dic₄₃	—	8	8	no	—
119	C₂×Dic₄₃	—	8	8	no	—
120	C₈×C₄₃	—	2	8	no	—
121	#10721	—	43	1	no	maximal, Sylow
122	C₂²×C₄₃	—	2	8	no	—
123	C₂²×C₄₃	—	2	8	no	—
124	C₂²×C₄₃	—	2	16	no	—
125	C₂²×C₄₃	—	2	16	no	—
126	C₂²×C₄₃	—	2	4	no	—
127	C₂²×C₄₃	—	2	4	no	—
128	C₂²×C₄₃	#340	1	8	yes	complemented
129	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
130	C₂²×C₄₃	#380	1	2	yes	—
131	C₂²×C₄₃	#386	1	4	yes	—
132	C₂²×C₄₃	C₂×#102	1	2	yes	—
133	C₂²×C₄₃	C₂×#103	1	2	yes	—
134	C₂²×C₄₃	#416	1	4	yes	—
135	C₂²×C₄₃	C₂×#106	1	1	yes	characteristic
136	C₂²×C₄₃	C₂×#107	1	1	yes	characteristic
137	C₂²×C₄₃	#427	1	2	yes	—
138	C₂²×C₄₃	#465	1	8	yes	—
139	C₂²×C₄₃	#466	1	4	yes	—
140	C₂²×C₄₃	#467	1	4	yes	—
141	C₄×C₄₃	—	2	4	no	—
142	C₄×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
143	C₄×C₄₃	#379	1	1	yes	characteristic
144	C₄×C₄₃	#381	1	1	yes	characteristic
145	C₄×C₄₃	#385	1	4	yes	—
146	Dic₄₃	—	8	8	no	—
147	C₂⁴×C₈	D₄₃	1	1	yes	characteristic
148	C₂×#380	—	43	2	no	—
149	C₂×#340	—	43	4	no	—
150	C₂×C₄₃	—	2	8	no	—
151	C₂×C₄₃	C₂²⋊#95	1	1	yes	characteristic
152	C₂×C₄₃	C₂×#340	1	1	yes	characteristic
153	C₂×C₄₃	#1735	1	1	yes	characteristic
154	C₂×C₄₃	#1807	1	2	yes	—
155	C₂×C₄₃	#1808	1	2	yes	—
156	C₂×C₄₃	#1892	1	4	yes	—
157	C₂×C₄₃	#1893	1	4	yes	—
158	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
159	C₂³×C₈	—	2	4	no	—
160	C₂³×C₈	—	2	8	no	—
161	C₂³×C₈	Dic₄₃	1	4	yes	complemented
162	C₂³×C₈	C₂×D₄₃	1	2	yes	—
163	C₂²×C₄⋊C₄	—	43	4	no	—
164	C₂²×C₂²⋊C₄	—	86	2	no	—
165	#380	—	43	8	no	—
166	#380	—	86	4	no	—
167	C₂×#95	—	86	2	no	—
168	#340	—	86	8	no	—
169	C₄₃	#10721	1	1	yes	characteristic, complemented, Sylow
170	C₂⁵	D₁₇₂	1	1	yes	characteristic
171	C₂³×C₄	—	2	4	no	—
172	C₂³×C₄	—	2	4	no	—
173	C₂³×C₄	—	86	4	no	—
174	C₂³×C₄	Dic₈₆	1	2	yes	—
175	C₂³×C₄	C₄×D₄₃	1	4	yes	—
176	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
177	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
178	C₂³×C₄	C₂²⋊D₄₃	1	4	yes	—
179	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
180	C₂²×C₈	—	2	32	no	—
181	C₂²×C₈	—	2	8	no	—
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	—	2	16	no	—
185	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
186	C₂²×C₈	C₂²⋊D₄₃	1	8	yes	—
187	C₂×C₄⋊C₄	—	43	8	no	—
188	C₂×C₄⋊C₄	—	86	8	no	—
189	C₂×C₄⋊C₄	—	86	8	no	—
190	C₂×C₄⋊C₄	—	86	4	no	—
191	C₂×C₂²⋊C₄	—	86	8	no	—
192	C₂×C₂²⋊C₄	—	172	4	no	—
193	C₂×C₂²⋊C₄	—	172	4	no	—
194	#95	—	86	8	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	—	2	4	no	—
197	C₂⁴	#10621873	1	4	yes	—
198	C₂⁴	D₃₄₄	1	2	yes	—
199	C₂⁴	Dic₁₇₂	1	2	yes	—
200	C₂⁴	C₄⋊Dic₄₃	1	2	yes	—
201	C₂⁴	(C₂×D₄₃)⋊C₄	1	4	yes	—
202	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	16	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	86	8	no	—
211	C₂²×C₄	—	172	8	no	—
212	C₂²×C₄	—	172	4	no	—
213	C₂²×C₄	—	172	4	no	—
214	C₂²×C₄	—	172	8	no	—
215	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
216	C₂²×C₄	Dic₄₃⋊C₄	1	8	yes	—
217	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
218	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
219	C₂²×C₄	C₂²⋊Dic₄₃	1	4	yes	—
220	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
222	C₂²×C₄	#10621898	1	4	yes	—
223	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
226	C₂×C₈	—	2	32	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	16	no	—
230	C₂×C₈	C₂²⋊Dic₄₃	1	8	yes	complemented
231	C₄⋊C₄	—	86	16	no	—
232	C₄⋊C₄	—	172	4	no	—
233	C₄⋊C₄	—	172	4	no	—
234	C₂²⋊C₄	—	172	16	no	—
235	C₂³	—	2	16	no	—
236	C₂³	—	2	8	no	—
237	C₂³	—	2	8	no	—
238	C₂³	—	2	8	no	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	2	8	no	—
242	C₂³	C₄₃⋊#95	1	4	yes	—
243	C₂³	#13063247	1	8	yes	—
244	C₂³	#13063250	1	8	yes	—
245	C₂³	#13063257	1	4	yes	—
246	C₂³	#13063259	1	4	yes	—
247	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	2	yes	—
248	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
249	C₂³	#13063299	1	4	yes	—
250	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
251	C₂³	C₂×#10621873	1	2	yes	—
252	C₂³	C₂×Dic₁₇₂	1	1	yes	characteristic
253	C₂³	#13063347	1	4	yes	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	344	8	no	—
261	C₂×C₄	—	344	8	no	—
262	C₂×C₄	—	172	8	no	—
263	C₂×C₄	—	172	8	no	—
264	C₂×C₄	—	172	8	no	—
265	C₂×C₄	C₄₃⋊#95	1	4	yes	—
266	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
269	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
270	C₂×C₄	#13063284	1	2	yes	—
271	C₂×C₄	C₄×C₂²⋊D₄₃	1	8	yes	—
272	C₂×C₄	#13063294	1	4	yes	—
273	C₂×C₄	#13063304	1	4	yes	—
274	C₂×C₄	#13063308	1	4	yes	—
275	C₈	—	2	8	no	—
276	C₂²	—	2	16	no	—
277	C₂²	—	2	16	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	8	no	—
281	C₂²	—	2	8	no	—
282	C₂²	#14582662	1	8	yes	complemented
283	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
284	C₂²	#14582759	1	2	yes	—
285	C₂²	#14582778	1	4	yes	—
286	C₂²	C₂×#13063247	1	2	yes	—
287	C₂²	C₂×#13063250	1	2	yes	—
288	C₂²	#14582896	1	4	yes	—
289	C₂²	C₂×#13063257	1	1	yes	characteristic
290	C₂²	C₂×#13063259	1	1	yes	characteristic
291	C₂²	#14582937	1	2	yes	—
292	C₂²	#14583128	1	8	yes	—
293	C₂²	#14583135	1	4	yes	—
294	C₂²	#14583140	1	4	yes	—
295	C₄	—	2	4	no	—
296	C₄	—	344	8	no	—
297	C₄	C₄×C₂²⋊Dic₄₃	1	2	yes	—
298	C₄	#14582755	1	1	yes	characteristic
299	C₄	#14582762	1	1	yes	characteristic
300	C₄	#14582772	1	4	yes	—
301	C₂	—	2	8	no	—
302	C₂	#19132818	1	1	yes	characteristic
303	C₂	C₂×#14582662	1	1	yes	characteristic
304	C₂	#19134312	1	1	yes	characteristic
305	C₂	#19134620	1	2	yes	—
306	C₂	#19134624	1	2	yes	—
307	C₂	#19135018	1	4	yes	—
308	C₂	#19135023	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂⁴×C₈×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂×D₄
prime radical	C₂⁵×C₄₃	D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂⁴×C₈	D₄₃
hypocenter	C₄₃	#10721
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#14582662	C₂	semidirect
7. #14582759	C₄	semidirect
13. C₂³×C₈×C₄₃	C₄	semidirect
75. C₂×C₈×C₄₃	C₂²⋊C₄	semidirect
128. C₂²×C₄₃	#340	semidirect
161. C₂³×C₈	Dic₄₃	semidirect
169. C₄₃	#10721	semidirect
230. C₂×C₈	C₂²⋊Dic₄₃	semidirect
282. C₂²	#14582662	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582759	2
C₂×#14582662	2
C₂⁴×C₈×C₄₃	2
#10721	43