Subgroups — #25094103

Order 11008.

908 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	#19143014	C₂	1	2	yes	maximal
2	#19143013	C₂	1	1	yes	maximal, characteristic
3	#19134349	C₂	1	8	yes	maximal, complemented
4	C₂×C₄₃⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582655	C₂	1	2	yes	maximal
6	C₂×C₄₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10621877	C₂²	1	2	yes	—
8	C₂²×#10621877	C₂²	1	1	yes	characteristic
9	C₂×#13063284	C₂²	1	1	yes	characteristic
10	#14582857	C₂²	1	4	yes	—
11	#14582855	C₄	1	8	yes	complemented
12	#14582855	C₂²	1	8	yes	—
13	#14582854	C₄	1	4	yes	complemented
14	C₄×#10621877	C₄	1	8	yes	complemented
15	M₁₆×Dic₄₃	C₄	1	4	yes	complemented
16	C₄₃⋊#338	C₂²	1	4	yes	—
17	#14582655	C₂²	1	8	yes	—
18	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×#117×C₄₃	C₂²	1	2	yes	—
20	C₄₃×H(3,ℤ/4)	C₂²	1	4	yes	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	C₂×C₄	1	4	yes	—
26	C₂×#10621877	C₂×C₄	1	2	yes	—
27	C₂×#10621877	C₂×C₄	1	4	yes	—
28	C₂×#10621877	C₂×C₄	1	2	yes	—
29	C₂×#10621877	C₂×C₄	1	4	yes	—
30	C₂×#10621877	C₂³	1	4	yes	—
31	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
33	#13063284	—	2	1	no	—
34	#13063284	—	2	1	no	—
35	#13063284	C₂×C₄	1	2	yes	—
36	#13063284	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₄₃	—	2	1	no	—
38	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
39	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
40	#13063236	C₂×C₄	1	16	yes	complemented
41	#13063235	C₂×C₄	1	8	yes	complemented
42	C₄×C₄₃⋊C₈	C₂×C₄	1	16	yes	complemented
43	C₈×Dic₄₃	C₂×C₄	1	8	yes	complemented
44	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₄₃	—	2	2	no	—
46	C₂×M₁₆×C₄₃	—	2	2	no	—
47	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
50	#117×C₄₃	—	2	2	no	—
51	#117×C₄₃	—	2	2	no	—
52	#117×C₄₃	C₂×C₄	1	4	yes	—
53	#117×C₄₃	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
56	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
57	C₂⁴×C₄₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	—	2	4	no	—
59	C₂²×C₄×C₄₃	C₂²×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	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	2	yes	—
64	C₂²×C₄×C₄₃	#41	1	2	yes	—
65	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	2	2	no	—
70	M₁₆×C₄₃	—	2	4	no	—
71	M₁₆×C₄₃	—	2	4	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
75	C₄⋊C₄×C₄₃	—	2	4	no	—
76	C₄⋊C₄×C₄₃	—	2	4	no	—
77	C₂²⋊C₄×C₄₃	—	2	4	no	—
78	C₂²⋊C₄×C₄₃	—	2	4	no	—
79	C₄²×C₄₃	—	2	4	no	—
80	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
81	C₂²⋊Dic₄₃	—	2	2	no	—
82	C₂²⋊Dic₄₃	—	2	2	no	—
83	C₄⋊Dic₄₃	—	2	2	no	—
84	C₄⋊Dic₄₃	—	2	2	no	—
85	C₄×Dic₄₃	—	2	2	no	—
86	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
87	#10621877	—	2	4	no	—
88	#10621877	—	2	4	no	—
89	#10621877	—	2	4	no	—
90	#10621877	—	2	4	no	—
91	#10621877	C₄²	1	8	yes	—
92	#10621877	C₄²	1	4	yes	—
93	#10621877	C₄²	1	4	yes	—
94	C₂×C₄₃⋊C₈	—	2	4	no	—
95	C₂×C₄₃⋊C₈	—	2	4	no	—
96	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
97	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
98	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
99	C₂³×C₄₃	—	8	1	no	—
100	C₂³×C₄₃	#117	1	2	yes	—
101	C₂³×C₄₃	#117	1	2	yes	—
102	C₂³×C₄₃	#117	1	1	yes	characteristic
103	C₂³×C₄₃	#117	1	1	yes	characteristic
104	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₄₃	—	2	8	no	—
106	C₂×C₄×C₄₃	—	8	4	no	—
107	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#117	1	2	yes	—
113	C₂×C₄×C₄₃	#117	1	2	yes	—
114	C₂×C₄×C₄₃	#117	1	2	yes	—
115	C₂×C₄×C₄₃	#117	1	2	yes	—
116	C₂×C₄×C₄₃	#117	1	2	yes	—
117	C₂×C₄×C₄₃	#117	1	2	yes	—
118	C₂×C₄×C₄₃	#117	1	2	yes	—
119	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
124	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₄₃	—	2	4	no	—
126	C₂×Dic₄₃	—	8	2	no	—
127	C₈×C₄₃	—	2	4	no	—
128	C₄₃⋊C₈	—	2	8	no	—
129	C₄₃⋊C₈	C₂×C₄²	1	8	yes	—
130	#20370	—	43	1	no	maximal, Sylow
131	C₂²×C₄₃	—	4	2	no	—
132	C₂²×C₄₃	—	4	1	no	—
133	C₂²×C₄₃	—	4	1	no	—
134	C₂²×C₄₃	—	4	2	no	—
135	C₂²×C₄₃	—	4	1	no	—
136	C₂²×C₄₃	#376	1	1	yes	characteristic
137	C₂²×C₄₃	#376	1	2	yes	—
138	C₂²×C₄₃	#376	1	1	yes	characteristic
139	C₂²×C₄₃	C₂×#117	1	2	yes	—
140	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
141	C₄×C₄₃	—	4	8	no	—
142	C₄×C₄₃	#376	1	1	yes	characteristic
143	C₄×C₄₃	#376	1	1	yes	characteristic
144	C₄×C₄₃	#376	1	2	yes	—
145	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
146	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
147	C₄×C₄₃	C₂×#117	1	2	yes	—
148	Dic₄₃	—	4	4	no	—
149	#2873	—	43	1	no	—
150	#2873	—	43	2	no	—
151	#1747	—	43	8	no	—
152	C₂×#338	—	43	1	no	—
153	C₂×H(3,ℤ/4)	—	43	2	no	—
154	C₂×H(3,ℤ/4)	D₄₃	1	1	yes	characteristic
155	C₂×C₄₃	—	2	1	no	—
156	C₂×C₄₃	—	2	2	no	—
157	C₂×C₄₃	—	2	1	no	—
158	C₂×C₄₃	—	2	2	no	—
159	C₂×C₄₃	—	2	1	no	—
160	C₂×C₄₃	C₂×#376	1	1	yes	characteristic
161	C₂²×M₁₆	—	43	1	no	—
162	C₂²×M₁₆	—	43	2	no	—
163	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
164	C₂×#117	—	43	1	no	—
165	C₂×#117	C₂×D₄₃	1	2	yes	—
166	#405	—	43	4	no	—
167	#405	—	43	4	no	—
168	#405	—	43	8	no	—
169	#405	—	43	8	no	—
170	C₄×M₁₆	—	43	4	no	—
171	C₄×M₁₆	—	43	8	no	—
172	#338	—	43	4	no	—
173	H(3,ℤ/4)	—	43	8	no	—
174	H(3,ℤ/4)	C₂×D₄₃	1	4	yes	—
175	C₄₃	#20370	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	43	2	no	—
180	C₂×M₁₆	—	43	4	no	—
181	C₂×M₁₆	—	43	2	no	—
182	C₂×M₁₆	—	43	4	no	—
183	C₂×M₁₆	—	43	4	no	—
184	C₂×M₁₆	—	43	4	no	—
185	C₂×M₁₆	—	86	2	no	—
186	C₂×M₁₆	—	86	2	no	—
187	C₂×M₁₆	—	86	2	no	—
188	C₂×M₁₆	—	86	2	no	—
189	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
190	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
191	C₂²×C₈	—	43	2	no	—
192	C₂²×C₈	—	43	4	no	—
193	C₂²×C₈	C₄×D₄₃	1	2	yes	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	43	2	no	—
197	#117	—	43	2	no	—
198	#117	—	86	1	no	—
199	#117	—	86	1	no	—
200	#117	C₄×D₄₃	1	4	yes	—
201	#117	C₂²×D₄₃	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	86	1	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	86	1	no	—
206	C₂×C₄²	—	43	2	no	—
207	C₂×C₄²	C₄×D₄₃	1	4	yes	—
208	#97	—	43	8	no	—
209	#97	—	43	16	no	—
210	C₄×C₈	—	43	8	no	—
211	C₄×C₈	—	43	16	no	—
212	C₂⁴	#10621900	1	1	yes	characteristic
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	86	2	no	—
215	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
219	C₂²×C₄	#10621900	1	1	yes	characteristic
220	C₂²×C₄	#10621900	1	2	yes	—
221	C₂²×C₄	#10621900	1	2	yes	—
222	C₂²×C₄	#10621902	1	1	yes	characteristic
223	C₂²×C₄	#10621902	1	1	yes	characteristic
224	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	43	8	no	—
228	M₁₆	—	43	4	no	—
229	M₁₆	—	43	4	no	—
230	M₁₆	—	86	4	no	—
231	M₁₆	—	86	4	no	—
232	M₁₆	—	86	4	no	—
233	M₁₆	—	86	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	43	4	no	—
237	C₂×C₈	—	43	8	no	—
238	C₂×C₈	—	43	8	no	—
239	C₂×C₈	—	86	4	no	—
240	C₂×C₈	—	86	4	no	—
241	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	86	2	no	—
245	C₄⋊C₄	—	86	2	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	86	2	no	—
249	C₂²⋊C₄	—	86	2	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	43	4	no	—
252	C₄²	—	86	2	no	—
253	C₄²	C₂×C₄×D₄₃	1	8	yes	—
254	C₂³	—	8	1	no	—
255	C₂³	#13063281	1	2	yes	—
256	C₂³	#13063281	1	2	yes	—
257	C₂³	#13063283	1	1	yes	characteristic
258	C₂³	#13063283	1	1	yes	characteristic
259	C₂³	C₂×#10621900	1	1	yes	characteristic
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	—	86	4	no	—
263	C₂×C₄	—	344	2	no	—
264	C₂×C₄	C₄²×D₄₃	1	2	yes	—
265	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
266	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
267	C₂×C₄	#13063281	1	1	yes	characteristic
268	C₂×C₄	#13063281	1	1	yes	characteristic
269	C₂×C₄	#13063281	1	2	yes	—
270	C₂×C₄	#13063281	1	2	yes	—
271	C₂×C₄	#13063281	1	2	yes	—
272	C₂×C₄	#13063283	1	2	yes	—
273	C₂×C₄	#13063283	1	2	yes	—
274	C₂×C₄	#13063283	1	2	yes	—
275	C₂×C₄	#13063283	1	2	yes	—
276	C₂×C₄	#13063283	1	1	yes	characteristic
277	C₂×C₄	#13063283	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
280	C₂×C₄	C₂×#10621900	1	2	yes	—
281	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	43	8	no	—
284	C₈	—	86	8	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	#14582744	1	1	yes	characteristic
291	C₂²	#14582744	1	1	yes	characteristic
292	C₂²	#14582744	1	2	yes	—
293	C₂²	C₂×#13063281	1	2	yes	—
294	C₂²	C₂×#13063283	1	1	yes	characteristic
295	C₄	—	4	8	no	—
296	C₄	—	172	4	no	—
297	C₄	#14582744	1	1	yes	characteristic
298	C₄	#14582744	1	1	yes	characteristic
299	C₄	#14582744	1	2	yes	—
300	C₄	C₂×C₄²×D₄₃	1	1	yes	characteristic
301	C₄	C₂×#13063281	1	1	yes	characteristic
302	C₄	C₂×#13063283	1	2	yes	—
303	C₂	—	2	2	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	2	1	no	—
308	C₂	C₂×#14582744	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄²×D₄₃
commutator	C₄×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×H(3,ℤ/4)	C₂
socle	C₂×C₄₃	C₂×#376
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×H(3,ℤ/4)	D₄₃
hypocenter	C₄₃	#20370
LCS	G ▷ C₄×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
3. #19134349	C₂	semidirect
11. #14582855	C₄	semidirect
13. #14582854	C₄	semidirect
14. C₄×#10621877	C₄	semidirect
15. M₁₆×Dic₄₃	C₄	semidirect
40. #13063236	C₂×C₄	semidirect
41. #13063235	C₂×C₄	semidirect
42. C₄×C₄₃⋊C₈	C₂×C₄	semidirect
43. C₈×Dic₄₃	C₂×C₄	semidirect
175. C₄₃	#20370	semidirect

Maximal subgroups

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

subgroup	index
#19143014	2
#19143013	2
#19134349	2
C₂×C₄₃⋊#338	2
C₂×#14582655	2
C₂×C₄₃×H(3,ℤ/4)	2
#20370	43