Subgroups — #24987285

Order 11008.

428 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143240	C₂	1	1	yes	maximal, characteristic
2	#19132978	C₂	1	1	yes	maximal, characteristic
3	#19132975	C₂	1	1	yes	maximal, characteristic
4	#19132888	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132433	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10621898	C₂²	1	1	yes	characteristic
9	#14582961	—	2	2	no	—
10	#14582952	C₂²	1	1	yes	characteristic
11	#14582950	—	2	1	no	—
12	#14582950	—	2	1	no	—
13	#14582927	C₄	1	2	yes	—
14	#14582867	C₄	1	2	yes	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	#14582631	C₄	1	2	yes	—
18	#14582629	C₄	1	2	yes	—
19	#14582628	C₄	1	2	yes	—
20	#14582626	C₄	1	2	yes	—
21	#14582621	—	2	1	no	—
22	#14582620	—	2	1	no	—
23	C₈×C₄₃⋊C₈	—	2	2	no	—
24	C₄₃×#431	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₃⋊C₈	—	2	1	no	—
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
32	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
33	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₄₃	—	2	1	no	—
35	C₄×C₂²⋊D₄₃	—	2	1	no	—
36	#13063284	C₂×C₄	1	1	yes	characteristic
37	#13063282	—	2	1	no	—
38	C₄²×D₄₃	—	2	1	no	—
39	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
40	Dic₄₃⋊C₈	—	2	2	no	—
41	Dic₄₃⋊C₈	—	2	2	no	—
42	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
43	(C₂×D₄₃)⋊C₈	—	2	2	no	—
44	(C₂×D₄₃)⋊C₈	—	2	2	no	—
45	C₄×C₄₃⋊C₈	—	2	1	no	—
46	C₄×C₄₃⋊C₈	—	2	1	no	—
47	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
48	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
49	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
50	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
51	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₃	—	2	1	no	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₄₃	—	2	1	no	—
56	#97×C₄₃	—	2	1	no	—
57	C₄×C₈×C₄₃	—	2	1	no	—
58	C₄×C₈×C₄₃	—	2	1	no	—
59	C₄×C₈×C₄₃	—	2	1	no	—
60	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₃	—	2	1	no	—
65	#10621898	—	2	2	no	—
66	#10621898	—	2	2	no	—
67	C₂×D₁₇₂	C₄²	1	1	yes	characteristic
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×Dic₈₆	C₄²	1	1	yes	characteristic
72	M₁₆×C₄₃	—	2	2	no	—
73	M₁₆×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	—	2	4	no	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	2	no	—
78	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²⋊Dic₄₃	—	2	1	no	—
83	(C₂×D₄₃)⋊C₄	—	2	1	no	—
84	(C₂×D₄₃)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₄₃	C₄²	1	1	yes	characteristic
86	C₄⋊Dic₄₃	C₄²	1	1	yes	characteristic
87	Dic₄₃⋊C₄	—	2	1	no	—
88	Dic₄₃⋊C₄	—	2	1	no	—
89	C₄×Dic₄₃	—	2	1	no	—
90	C₄×Dic₄₃	—	2	1	no	—
91	C₄×Dic₄₃	—	2	1	no	—
92	C₂×C₄₃⋊C₈	—	2	4	no	—
93	C₂×C₄₃⋊C₈	—	2	2	no	—
94	C₂×C₄₃⋊C₈	—	2	2	no	—
95	C₂×C₄₃⋊C₈	—	2	2	no	—
96	C₂×C₄₃⋊C₈	—	2	2	no	—
97	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
98	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
99	C₂³×C₄₃	#117	1	1	yes	characteristic
100	C₂²×D₄₃	—	2	1	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#131	1	2	yes	—
110	C₂×C₄×C₄₃	#131	1	2	yes	—
111	C₂²⋊D₄₃	—	4	2	no	—
112	C₂×Dic₄₃	—	2	1	no	—
113	C₂×Dic₄₃	—	2	1	no	—
114	C₂×Dic₄₃	—	2	1	no	—
115	D₁₇₂	—	2	1	no	—
116	D₁₇₂	—	2	1	no	—
117	C₄×D₄₃	—	4	2	no	—
118	C₄×D₄₃	—	4	2	no	—
119	C₄×D₄₃	—	4	2	no	—
120	Dic₈₆	—	2	1	no	—
121	Dic₈₆	—	2	1	no	—
122	C₈×C₄₃	—	2	4	no	—
123	C₈×C₄₃	—	4	2	no	—
124	C₈×C₄₃	—	4	2	no	—
125	C₄₃⋊C₈	—	2	4	no	—
126	C₄₃⋊C₈	—	2	2	no	—
127	C₄₃⋊C₈	—	2	2	no	—
128	C₄₃⋊C₈	—	4	2	no	—
129	#8187	—	43	1	no	maximal, Sylow
130	C₂²×C₄₃	—	2	1	no	—
131	C₂²×C₄₃	—	2	1	no	—
132	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄₃	#433	1	2	yes	—
134	C₂×D₄₃	—	4	1	no	—
135	C₂×D₄₃	—	4	1	no	—
136	C₂×D₄₃	—	4	1	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	—	2	1	no	—
139	C₄×C₄₃	#405	1	1	yes	characteristic
140	C₄×C₄₃	#405	1	1	yes	characteristic
141	C₄×C₄₃	#408	1	2	yes	—
142	C₄×C₄₃	#443	1	2	yes	—
143	C₄×C₄₃	#443	1	2	yes	—
144	Dic₄₃	—	4	1	no	—
145	Dic₄₃	—	4	1	no	—
146	Dic₄₃	—	4	1	no	—
147	#2922	—	43	1	no	—
148	#1476	—	43	1	no	—
149	#1476	—	43	1	no	—
150	#1452	—	43	1	no	—
151	C₂×C₈²	—	43	1	no	—
152	#1278	—	43	1	no	—
153	#1278	D₄₃	1	1	yes	characteristic, complemented
154	C₂×C₄₃	—	2	2	no	—
155	C₂×C₄₃	#1750	1	1	yes	characteristic
156	C₂×C₄₃	#1764	1	1	yes	characteristic
157	C₂×C₄₃	#1764	1	1	yes	characteristic
158	D₄₃	—	8	1	no	—
159	C₄×#41	—	43	1	no	—
160	#433	—	86	2	no	—
161	#431	—	43	1	no	—
162	#431	—	86	1	no	—
163	#431	—	86	1	no	—
164	#431	C₂×D₄₃	1	1	yes	characteristic
165	#424	—	43	2	no	—
166	#408	—	43	2	no	—
167	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
168	C₂×C₄×C₈	—	43	1	no	—
169	C₂×C₄×C₈	—	43	1	no	—
170	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
171	Q₈⋊C₈	—	43	2	no	—
172	Q₈⋊C₈	—	43	2	no	—
173	D₄⋊C₈	—	43	2	no	—
174	D₄⋊C₈	—	43	2	no	—
175	#322	—	86	1	no	—
176	#322	—	86	1	no	—
177	C₈²	—	86	2	no	—
178	C₄₃	#8187	1	1	yes	characteristic, complemented, Sylow
179	C₂×#41	—	43	1	no	—
180	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	43	2	no	—
183	C₂²×C₈	—	86	1	no	—
184	C₄×Q₈	—	43	1	no	—
185	C₄×Q₈	—	43	1	no	—
186	C₄×D₄	—	43	1	no	—
187	C₄×D₄	—	43	1	no	—
188	C₄×D₄	—	86	1	no	—
189	C₄×D₄	—	86	1	no	—
190	#117	—	43	1	no	—
191	#117	—	86	1	no	—
192	C₂×C₄²	—	86	1	no	—
193	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
194	C₄⋊C₈	—	43	2	no	—
195	C₄⋊C₈	—	86	2	no	—
196	C₄⋊C₈	—	86	2	no	—
197	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	86	2	no	—
200	C₂²⋊C₈	—	86	2	no	—
201	C₂²⋊C₈	—	86	1	no	—
202	#97	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	43	2	no	—
207	C₄×C₈	—	43	2	no	—
208	C₄×C₈	—	43	1	no	—
209	C₄×C₈	—	43	1	no	—
210	C₄×C₈	—	86	1	no	—
211	C₄×C₈	—	86	1	no	—
212	C₄×C₈	C₄×D₄₃	1	2	yes	—
213	#41	—	86	2	no	—
214	#41	—	86	2	no	—
215	C₂×Q₈	—	43	1	no	—
216	C₂×D₄	—	43	1	no	—
217	C₂×D₄	—	86	1	no	—
218	C₂²×C₄	—	86	1	no	—
219	C₂²×C₄	—	86	1	no	—
220	C₂²×C₄	—	86	1	no	—
221	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
222	C₂²×C₄	#10621900	1	1	yes	characteristic
223	C₂²×C₄	#10621902	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	43	2	no	—
231	C₂×C₈	—	43	2	no	—
232	C₂×C₈	—	86	4	no	—
233	C₂×C₈	—	86	2	no	—
234	C₂×C₈	—	86	2	no	—
235	C₂×C₈	—	86	2	no	—
236	C₂×C₈	—	86	2	no	—
237	C₄⋊C₄	—	43	1	no	—
238	C₄⋊C₄	—	43	1	no	—
239	C₄⋊C₄	—	86	1	no	—
240	C₄⋊C₄	—	86	1	no	—
241	C₂²⋊C₄	—	86	1	no	—
242	C₂²⋊C₄	—	86	1	no	—
243	C₂²⋊C₄	—	86	1	no	—
244	C₄²	—	86	1	no	—
245	C₄²	—	86	1	no	—
246	C₄²	—	86	1	no	—
247	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
249	C₄²	D₄×D₄₃	1	1	yes	characteristic
250	C₄²	D₄×D₄₃	1	1	yes	characteristic
251	C₂³	—	86	1	no	—
252	C₂³	#13063283	1	1	yes	characteristic
253	Q₈	—	86	1	no	—
254	Q₈	—	86	1	no	—
255	D₄	—	86	1	no	—
256	D₄	—	86	1	no	—
257	D₄	—	172	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	86	1	no	—
261	C₂×C₄	—	86	1	no	—
262	C₂×C₄	—	86	1	no	—
263	C₂×C₄	—	172	2	no	—
264	C₂×C₄	—	172	2	no	—
265	C₂×C₄	—	172	2	no	—
266	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
267	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
268	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
270	C₂×C₄	#13063287	1	1	yes	characteristic
271	C₂×C₄	#13063287	1	1	yes	characteristic
272	C₂×C₄	#13063331	1	2	yes	—
273	C₂×C₄	#13063333	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	2	no	—
277	C₈	—	86	4	no	—
278	C₈	—	86	2	no	—
279	C₈	—	86	2	no	—
280	C₈	—	172	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	172	1	no	—
284	C₂²	—	172	1	no	—
285	C₂²	—	172	1	no	—
286	C₂²	#14582746	1	1	yes	characteristic
287	C₂²	#14582959	1	2	yes	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	172	1	no	—
291	C₄	—	172	1	no	—
292	C₄	—	172	1	no	—
293	C₄	#14582853	1	1	yes	characteristic
294	C₄	#14582856	1	1	yes	characteristic
295	C₄	#14582864	1	2	yes	—
296	C₄	#14583013	1	2	yes	—
297	C₄	#14583015	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	344	1	no	—
300	C₂	#19134360	1	1	yes	characteristic
301	C₂	#19134429	1	1	yes	characteristic
302	C₂	#19134431	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1278	C₂
socle	C₂²×C₄₃	C₄×C₂²⋊C₄
prime radical	C₂×#10621898	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1278	D₄₃
hypocenter	C₄₃	#8187
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
4. #19132888	C₂	semidirect
5. C₂×C₈×C₄₃⋊C₈	C₂	semidirect
6. #19132433	C₂	semidirect
7. C₄₃×#1278	C₂	semidirect
153. #1278	D₄₃	semidirect
178. C₄₃	#8187	semidirect

Maximal subgroups

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

subgroup	index
#19143240	2
#19132978	2
#19132975	2
#19132888	2
C₂×C₈×C₄₃⋊C₈	2
#19132433	2
C₄₃×#1278	2
#8187	43