Subgroups — #25114894

Order 11008.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583010	C₂	1	2	yes	maximal
2	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
3	#19137503	C₂	1	2	yes	maximal, complemented
4	#19135331	C₂	1	2	yes	maximal, complemented
5	#19134839	C₂	1	4	yes	maximal, complemented
6	#19134602	C₂	1	2	yes	maximal, complemented
7	#19134462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
11	C₂×#13063287	C₂²	1	4	yes	—
12	#14583010	—	2	8	no	—
13	C₄×D₃₄₄	C₄	1	8	yes	complemented
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×#13063247	C₂²	1	2	yes	—
17	C₂×#13063245	C₂²	1	4	yes	—
18	C₂×#13063235	C₂²	1	2	yes	—
19	#14582795	C₂²	1	4	yes	—
20	#14582773	—	2	4	no	—
21	#14582758	—	2	2	no	—
22	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
23	#95×D₄₃	—	2	2	no	—
24	#14582661	C₂²	1	4	yes	—
25	#14582653	C₂²	1	2	yes	—
26	C₂×#107×C₄₃	C₂²	1	2	yes	—
27	C₂×#106×C₄₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#382	C₂²	1	2	yes	—
30	C₂²×D₁₇₂	C₂³	1	2	yes	—
31	C₂²×C₄×D₄₃	—	2	2	no	—
32	C₂²×C₄×D₄₃	—	2	4	no	—
33	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
34	C₂×D₃₄₄	D₄	1	4	yes	—
35	C₂×D₃₄₄	Q₈	1	4	yes	—
36	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
37	C₄×D₁₇₂	—	2	4	no	—
38	C₄×D₁₇₂	C₂×C₄	1	8	yes	complemented
39	#13063287	—	2	8	no	—
40	#13063287	—	4	4	no	—
41	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
43	C₂×Dic₄₃⋊C₄	—	2	4	no	—
44	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
45	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
46	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
47	#13063259	C₂×C₄	1	4	yes	complemented
48	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
49	#13063247	C₂×C₄	1	8	yes	complemented
50	#13063245	—	2	8	no	—
51	#13063235	—	2	4	no	—
52	C₄₃⋊#95	—	2	4	no	—
53	C₄₃⋊#95	D₄	1	2	yes	—
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	#107×C₄₃	—	4	2	no	—
59	#106×C₄₃	—	2	4	no	—
60	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
61	#95×C₄₃	—	2	2	no	—
62	#95×C₄₃	—	2	2	no	—
63	C₂³×D₄₃	—	2	2	no	—
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₃	—	2	2	no	—
69	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
70	C₂×D₁₇₂	—	2	2	no	—
71	C₂×D₁₇₂	—	2	2	no	—
72	C₂×D₁₇₂	—	2	2	no	—
73	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
74	C₂×D₁₇₂	C₂×D₄	1	4	yes	—
75	C₂×D₁₇₂	C₂×Q₈	1	4	yes	—
76	C₂×C₄×D₄₃	—	2	8	no	—
77	C₂×C₄×D₄₃	—	4	8	no	—
78	C₂×C₄×D₄₃	—	4	8	no	—
79	C₂×C₄×D₄₃	—	8	4	no	—
80	C₂×C₄×D₄₃	—	8	4	no	—
81	C₂×C₈×C₄₃	—	2	1	no	—
82	C₂×C₈×C₄₃	—	2	1	no	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
87	C₄⋊C₄×C₄₃	—	2	4	no	—
88	C₄⋊C₄×C₄₃	—	2	4	no	—
89	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₃	#41	1	2	yes	—
91	(C₂×D₄₃)⋊C₄	—	2	8	no	—
92	(C₂×D₄₃)⋊C₄	—	4	8	no	—
93	C₄⋊Dic₄₃	—	2	2	no	—
94	C₄⋊Dic₄₃	—	4	2	no	—
95	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
96	Dic₄₃⋊C₄	—	8	4	no	—
97	C₄×Dic₄₃	—	2	2	no	—
98	C₄×Dic₄₃	—	2	2	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	—	2	2	no	—
102	D₃₄₄	C₄⋊C₄	1	16	yes	complemented
103	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
104	C₂²×D₄₃	—	2	4	no	—
105	C₂²×D₄₃	—	2	4	no	—
106	C₂²×D₄₃	—	2	4	no	—
107	C₂²×D₄₃	—	4	4	no	—
108	C₂²×D₄₃	—	4	4	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	4	4	no	—
112	C₂×C₄×C₄₃	—	4	4	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×Dic₄₃	—	2	4	no	—
123	C₂×Dic₄₃	—	2	4	no	—
124	C₂×Dic₄₃	—	2	4	no	—
125	C₂×Dic₄₃	—	2	4	no	—
126	C₂×Dic₄₃	—	4	4	no	—
127	D₁₇₂	—	2	4	no	—
128	D₁₇₂	—	2	4	no	—
129	D₁₇₂	—	2	4	no	—
130	D₁₇₂	C₂×C₄⋊C₄	1	8	yes	—
131	C₄×D₄₃	—	8	8	no	—
132	C₄×D₄₃	—	8	16	no	—
133	C₈×C₄₃	—	2	2	no	—
134	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
135	C₄₃⋊C₈	—	4	2	no	—
136	#22672	—	43	1	no	maximal, Sylow
137	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₄₃	D₄²	1	1	yes	characteristic
139	C₂²×C₄₃	D₄×Q₈	1	1	yes	characteristic
140	C₂²×C₄₃	#575	1	2	yes	—
141	C₂²×C₄₃	#576	1	2	yes	—
142	C₂×D₄₃	—	2	8	no	—
143	C₂×D₄₃	—	4	8	no	—
144	C₂×D₄₃	—	4	4	no	—
145	C₂×D₄₃	—	4	4	no	—
146	C₂×D₄₃	—	4	8	no	—
147	C₄×C₄₃	—	2	2	no	—
148	C₄×C₄₃	—	2	2	no	—
149	C₄×C₄₃	—	8	4	no	—
150	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₄₃	#520	1	1	yes	characteristic
152	C₄×C₄₃	#547	1	1	yes	characteristic
153	C₄×C₄₃	#554	1	1	yes	characteristic
154	Dic₄₃	—	4	4	no	—
155	Dic₄₃	—	4	4	no	—
156	C₂×#442	—	43	2	no	—
157	C₂×C₄×D₈	—	43	1	no	—
158	#2349	—	43	2	no	—
159	#1943	—	43	2	no	—
160	#1862	—	43	4	no	—
161	#1846	D₄₃	1	1	yes	characteristic, complemented
162	#1804	—	43	2	no	—
163	#1770	—	43	1	no	—
164	C₂×C₄₃	D₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄₃	#2943	1	1	yes	characteristic
166	C₂×C₄₃	#2948	1	1	yes	characteristic
167	C₂×C₄₃	#3273	1	2	yes	—
168	C₂×C₄₃	#3389	1	2	yes	—
169	D₄₃	—	4	8	no	—
170	C₂²×D₈	—	43	1	no	—
171	C₂×C₄×D₄	—	43	4	no	—
172	C₂×C₄×D₄	—	43	2	no	—
173	#442	—	86	8	no	—
174	C₄×D₈	—	43	8	no	—
175	C₂×#107	—	43	1	no	—
176	C₂×#107	C₂×D₄₃	1	2	yes	—
177	C₂×#106	C₂×D₄₃	1	2	yes	—
178	C₂×C₄⋊C₈	—	43	1	no	—
179	C₂×#102	—	43	2	no	—
180	C₂×#102	—	43	4	no	—
181	C₂×#97	—	43	2	no	—
182	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
183	#389	—	43	4	no	—
184	#385	—	86	4	no	—
185	#382	C₂×D₄₃	1	2	yes	—
186	#380	—	86	2	no	—
187	C₄×C₄⋊C₄	—	43	1	no	—
188	C₂×#95	—	86	2	no	—
189	#340	—	43	4	no	—
190	#336	—	43	2	no	—
191	C₄₃	#22672	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	—	43	2	no	—
193	C₂³×C₄	—	86	4	no	—
194	C₂³×C₄	—	86	2	no	—
195	C₂×D₈	—	43	4	no	—
196	C₂×D₈	—	43	4	no	—
197	C₂×D₈	—	43	4	no	—
198	C₂²×C₈	—	43	2	no	—
199	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
201	C₄×D₄	—	43	8	no	—
202	C₄×D₄	—	86	8	no	—
203	C₄×D₄	—	86	4	no	—
204	C₄×D₄	—	172	4	no	—
205	C₂×C₄⋊C₄	—	43	2	no	—
206	C₂×C₄⋊C₄	—	43	2	no	—
207	C₂×C₄⋊C₄	—	86	4	no	—
208	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
209	C₂×C₂²⋊C₄	—	86	2	no	—
210	C₂×C₂²⋊C₄	—	86	4	no	—
211	C₂×C₄²	—	43	2	no	—
212	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
213	#107	—	4	2	no	—
214	#107	—	43	4	no	—
215	#106	—	2	4	no	—
216	C₄⋊C₈	—	86	2	no	—
217	#102	—	43	8	no	—
218	#102	—	86	8	no	—
219	#97	—	86	4	no	—
220	C₄×C₈	C₄×D₄₃	1	4	yes	complemented
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	43	2	no	—
224	#95	—	86	4	no	—
225	C₂⁴	—	86	2	no	—
226	C₂×D₄	—	43	4	no	—
227	C₂×D₄	—	43	4	no	—
228	C₂×D₄	—	43	4	no	—
229	C₂×D₄	—	86	2	no	—
230	C₂×D₄	—	86	2	no	—
231	C₂×D₄	—	86	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	43	4	no	—
234	C₂²×C₄	—	86	2	no	—
235	C₂²×C₄	—	86	8	no	—
236	C₂²×C₄	—	344	4	no	—
237	C₂²×C₄	—	344	4	no	—
238	C₂²×C₄	—	172	8	no	—
239	C₂²×C₄	—	172	8	no	—
240	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
243	D₈	—	43	16	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	86	2	no	—
247	C₂×C₈	—	86	2	no	—
248	C₂×C₈	—	86	2	no	—
249	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
251	C₂×C₈	D₄×D₄₃	1	2	yes	—
252	C₂×C₈	Q₈×D₄₃	1	2	yes	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	43	4	no	—
256	C₄⋊C₄	—	86	2	no	—
257	C₄⋊C₄	—	344	4	no	—
258	C₄⋊C₄	—	172	2	no	—
259	C₂²⋊C₄	—	86	8	no	—
260	C₂²⋊C₄	—	172	8	no	—
261	C₄²	—	86	2	no	—
262	C₄²	—	86	2	no	—
263	C₄²	C₂×C₄×D₄₃	1	2	yes	—
264	C₄²	#10621902	1	2	yes	—
265	C₂³	—	86	4	no	—
266	C₂³	—	86	4	no	—
267	C₂³	—	86	4	no	—
268	C₂³	—	172	4	no	—
269	C₂³	—	172	4	no	—
270	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
271	D₄	—	43	8	no	—
272	D₄	—	86	4	no	—
273	D₄	—	86	4	no	—
274	D₄	—	86	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	86	4	no	—
281	C₂×C₄	—	86	4	no	—
282	C₂×C₄	—	86	4	no	—
283	C₂×C₄	—	86	4	no	—
284	C₂×C₄	—	344	8	no	—
285	C₂×C₄	—	344	16	no	—
286	C₂×C₄	—	172	4	no	—
287	C₂×C₄	#13063287	1	2	yes	—
288	C₂×C₄	#13063287	1	2	yes	—
289	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
291	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
292	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
293	C₂×C₄	#13063370	1	1	yes	characteristic
294	C₂×C₄	#13063373	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	172	2	no	—
297	C₈	C₄⋊C₄×D₄₃	1	4	yes	—
298	C₂²	—	86	8	no	—
299	C₂²	—	172	4	no	—
300	C₂²	—	172	8	no	—
301	C₂²	—	172	8	no	—
302	C₂²	—	172	4	no	—
303	C₂²	C₂×#13063287	1	1	yes	characteristic
304	C₂²	#14583534	1	1	yes	characteristic
305	C₂²	#14583564	1	1	yes	characteristic
306	C₂²	#14583696	1	2	yes	—
307	C₂²	#14583701	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	8	4	no	—
311	C₄	—	172	4	no	—
312	C₄	—	172	4	no	—
313	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
314	C₄	#14583380	1	1	yes	characteristic
315	C₄	#14583552	1	1	yes	characteristic
316	C₄	#14583610	1	1	yes	characteristic
317	C₂	—	172	8	no	—
318	C₂	#19137495	1	1	yes	characteristic
319	C₂	#19143395	1	1	yes	characteristic
320	C₂	#19143446	1	1	yes	characteristic
321	C₂	#19146204	1	2	yes	—
322	C₂	#19147556	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1846	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1846	D₄₃
hypocenter	C₄₃	#22672
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
3. #19137503	C₂	semidirect
4. #19135331	C₂	semidirect
5. #19134839	C₂	semidirect
6. #19134602	C₂	semidirect
7. #19134462	C₂	semidirect
8. C₄₃×#1846	C₂	semidirect
13. C₄×D₃₄₄	C₄	semidirect
38. C₄×D₁₇₂	C₂×C₄	semidirect
47. #13063259	C₂×C₄	semidirect
49. #13063247	C₂×C₄	semidirect
60. C₄×C₈×C₄₃	C₂×C₄	semidirect
102. D₃₄₄	C₄⋊C₄	semidirect
161. #1846	D₄₃	semidirect
191. C₄₃	#22672	semidirect
220. C₄×C₈	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×#14583010	2
C₂×C₄×D₃₄₄	2
#19137503	2
#19135331	2
#19134839	2
#19134602	2
#19134462	2
C₄₃×#1846	2
#22672	43