Subgroups — #25011885

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	#19143397	C₂	1	1	yes	maximal, characteristic
2	#19136330	C₂	1	1	yes	maximal, characteristic
3	#19136322	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582677	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132709	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132696	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	1	yes	characteristic
9	C₄×#10621898	—	2	1	no	—
10	C₂×#13063265	C₂²	1	1	yes	characteristic
11	C₂×#13063265	C₂²	1	1	yes	characteristic
12	#14583012	C₄	1	2	yes	complemented
13	#14583007	C₄	1	2	yes	complemented
14	#14583001	—	2	2	no	—
15	#14582937	C₂²	1	1	yes	characteristic
16	#14582896	—	2	1	no	—
17	#14582708	C₄	1	4	yes	complemented
18	#14582705	C₄	1	4	yes	complemented
19	#14582677	—	2	2	no	—
20	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#427	C₂²	1	1	yes	characteristic
22	C₄₃×#405	—	2	1	no	—
23	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#339	—	2	1	no	—
25	C₂×#10621898	—	2	1	no	—
26	#13063347	—	2	1	no	—
27	#13063347	—	2	1	no	—
28	#13063347	C₂×C₄	1	2	yes	—
29	#13063347	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
31	C₂×#10621873	—	2	1	no	—
32	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₈₆	—	2	2	no	—
34	C₄×D₁₇₂	—	2	2	no	—
35	C₄×C₂²⋊D₄₃	—	4	2	no	—
36	#13063284	—	2	1	no	—
37	#13063282	—	4	1	no	—
38	C₄²×D₄₃	—	4	1	no	—
39	#13063265	—	2	1	no	—
40	#13063265	—	2	1	no	—
41	#13063265	—	2	1	no	—
42	#13063265	C₂×C₄	1	2	yes	complemented
43	C₂×C₄₃⋊C₁₆	—	2	1	no	—
44	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
45	#13063259	—	2	1	no	—
46	#13063257	C₂×C₄	1	2	yes	—
47	#13063250	—	2	2	no	—
48	#13063247	—	2	2	no	—
49	C₂×M₁₆×C₄₃	—	2	1	no	—
50	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	#117×C₄₃	—	2	1	no	—
53	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
54	#108×C₄₃	—	2	1	no	—
55	#108×C₄₃	C₂×C₄	1	2	yes	complemented
56	#107×C₄₃	—	2	1	no	—
57	#106×C₄₃	C₂×C₄	1	2	yes	complemented
58	#97×C₄₃	—	4	1	no	—
59	#97×C₄₃	C₂×C₄	1	2	yes	—
60	#97×C₄₃	D₄	1	2	yes	—
61	C₄×C₈×C₄₃	—	4	1	no	—
62	C₂²×C₄×C₄₃	—	2	1	no	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₃	—	4	1	no	—
65	#10621898	—	2	2	no	—
66	#10621898	—	2	2	no	—
67	#10621898	—	4	1	no	—
68	#10621898	—	4	1	no	—
69	C₂×D₁₇₂	—	2	1	no	—
70	C₂×C₄×D₄₃	—	4	1	no	—
71	C₂×C₄×D₄₃	—	4	2	no	—
72	C₂×Dic₈₆	—	2	1	no	—
73	M₁₆×C₄₃	—	2	2	no	—
74	M₁₆×C₄₃	—	4	1	no	—
75	C₂×C₈×C₄₃	—	2	1	no	—
76	C₂×C₈×C₄₃	—	2	1	no	—
77	C₂×C₈×C₄₃	—	4	1	no	—
78	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₄₃	—	2	2	no	—
86	C₂²⋊C₄×C₄₃	—	4	1	no	—
87	C₄²×C₄₃	—	2	1	no	—
88	C₄²×C₄₃	—	4	1	no	—
89	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
90	C₂²⋊Dic₄₃	—	4	1	no	—
91	(C₂×D₄₃)⋊C₄	—	4	2	no	—
92	C₄⋊Dic₄₃	—	2	2	no	—
93	Dic₄₃⋊C₄	—	4	2	no	—
94	C₄×Dic₄₃	—	4	2	no	—
95	C₄×Dic₄₃	—	4	1	no	—
96	Dic₁₇₂	—	2	1	no	—
97	Dic₁₇₂	C₄²	1	2	yes	—
98	D₃₄₄	—	2	1	no	—
99	D₃₄₄	C₄²	1	2	yes	—
100	#10621873	—	2	2	no	—
101	#10621873	—	2	2	no	—
102	C₄₃⋊C₁₆	—	2	2	no	—
103	C₄₃⋊C₁₆	—	2	2	no	—
104	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂²×D₄₃	—	4	1	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	4	2	no	—
108	C₂×C₄×C₄₃	—	8	1	no	—
109	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
115	C₂²⋊D₄₃	—	4	2	no	—
116	C₂²⋊D₄₃	—	4	2	no	—
117	C₂×Dic₄₃	—	4	1	no	—
118	C₂×Dic₄₃	—	4	2	no	—
119	D₁₇₂	—	2	2	no	—
120	D₁₇₂	—	4	1	no	—
121	C₄×D₄₃	—	4	2	no	—
122	C₄×D₄₃	—	4	2	no	—
123	C₄×D₄₃	—	8	4	no	—
124	Dic₈₆	—	2	2	no	—
125	Dic₈₆	—	4	1	no	—
126	C₈×C₄₃	—	2	2	no	—
127	C₈×C₄₃	—	4	2	no	—
128	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₄₃	#117	1	1	yes	characteristic
130	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
132	#12649	—	43	1	no	maximal, Sylow
133	C₂²×C₄₃	—	4	1	no	—
134	C₂²×C₄₃	#418	1	1	yes	characteristic
135	C₂²×C₄₃	#440	1	1	yes	characteristic
136	C₂²×C₄₃	#442	1	1	yes	characteristic
137	C₂×D₄₃	—	4	2	no	—
138	C₂×D₄₃	—	8	2	no	—
139	C₄×C₄₃	—	4	2	no	—
140	C₄×C₄₃	—	8	2	no	—
141	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₄₃	#417	1	1	yes	characteristic
143	C₄×C₄₃	#440	1	1	yes	characteristic
144	C₄×C₄₃	#442	1	1	yes	characteristic
145	Dic₄₃	—	4	2	no	—
146	Dic₄₃	—	8	2	no	—
147	#2943	—	43	1	no	—
148	#2145	—	43	1	no	—
149	#2143	—	43	1	no	—
150	C₂×#347	—	43	1	no	—
151	#1773	D₄₃	1	1	yes	characteristic, complemented
152	#1389	—	43	1	no	—
153	#1384	—	43	1	no	—
154	C₂×C₄₃	—	2	1	no	—
155	C₂×C₄₃	—	2	1	no	—
156	C₂×C₄₃	—	2	1	no	—
157	C₂×C₄₃	#1762	1	1	yes	characteristic
158	D₄₃	—	8	2	no	—
159	C₂×#135	—	43	1	no	—
160	C₄×#41	—	86	1	no	—
161	C₂×M₃₂	—	43	1	no	—
162	C₂×M₃₂	—	43	1	no	—
163	#442	—	43	2	no	—
164	#441	—	43	2	no	—
165	#440	—	86	2	no	—
166	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
167	#427	—	43	1	no	—
168	#427	C₂×D₄₃	1	1	yes	characteristic
169	#416	—	86	1	no	—
170	#405	—	2	1	no	—
171	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
172	#360	—	43	4	no	—
173	#359	—	43	4	no	—
174	#347	—	86	2	no	—
175	#339	—	2	1	no	—
176	C₄₃	#12649	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	86	1	no	—
178	#135	—	43	2	no	—
179	#135	—	43	2	no	—
180	#135	—	86	1	no	—
181	#135	—	86	1	no	—
182	C₂×Q₁₆	—	43	1	no	—
183	C₂×SD₁₆	—	86	1	no	—
184	C₂×D₈	—	43	1	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
188	C₄×Q₈	—	86	2	no	—
189	C₄×D₄	—	86	2	no	—
190	C₄×D₄	—	172	2	no	—
191	#117	—	2	1	no	—
192	#117	—	86	1	no	—
193	#117	—	172	1	no	—
194	C₂×C₄²	—	172	1	no	—
195	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
196	M₃₂	—	43	2	no	—
197	M₃₂	—	86	1	no	—
198	M₃₂	—	86	1	no	—
199	M₃₂	—	86	1	no	—
200	C₂×C₁₆	—	43	2	no	—
201	C₂×C₁₆	—	86	1	no	—
202	#108	—	2	1	no	—
203	#108	C₄×D₄₃	1	2	yes	complemented
204	#107	—	2	1	no	—
205	#107	—	86	1	no	—
206	#106	—	43	2	no	—
207	#106	C₄×D₄₃	1	2	yes	complemented
208	#103	—	86	2	no	—
209	#102	—	86	2	no	—
210	#97	—	4	1	no	—
211	#97	C₄×D₄₃	1	2	yes	—
212	#97	D₁₇₂	1	2	yes	—
213	C₄×C₈	—	4	1	no	—
214	#41	—	86	2	no	—
215	#41	—	86	2	no	—
216	#41	—	172	1	no	—
217	#41	—	172	1	no	—
218	C₂×Q₈	—	86	1	no	—
219	C₂×D₄	—	86	1	no	—
220	C₂×D₄	—	172	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	172	2	no	—
223	C₂²×C₄	—	172	1	no	—
224	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
225	Q₁₆	—	43	2	no	—
226	Q₁₆	—	86	1	no	—
227	SD₁₆	—	86	2	no	—
228	SD₁₆	—	86	2	no	—
229	D₈	—	43	2	no	—
230	D₈	—	86	1	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	4	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
240	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
241	C₂×C₈	#10621898	1	1	yes	characteristic
242	C₂×C₈	#10621898	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	86	2	no	—
245	C₄⋊C₄	—	172	2	no	—
246	C₂²⋊C₄	—	4	1	no	—
247	C₂²⋊C₄	—	172	2	no	—
248	C₂²⋊C₄	—	172	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	172	2	no	—
252	C₄²	—	172	1	no	—
253	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
254	C₁₆	—	86	2	no	—
255	C₁₆	—	86	2	no	—
256	C₂³	—	172	1	no	—
257	C₂³	#13063350	1	1	yes	characteristic
258	Q₈	—	86	2	no	—
259	Q₈	—	172	1	no	—
260	D₄	—	86	2	no	—
261	D₄	—	172	2	no	—
262	D₄	—	172	2	no	—
263	D₄	—	172	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	1	no	—
267	C₂×C₄	—	344	4	no	—
268	C₂×C₄	—	172	2	no	—
269	C₂×C₄	—	172	1	no	—
270	C₂×C₄	—	172	2	no	—
271	C₂×C₄	—	172	2	no	—
272	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
274	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
275	C₂×C₄	#13063350	1	1	yes	characteristic
276	C₂×C₄	#13063351	1	1	yes	characteristic
277	C₂×C₄	#13063356	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	C₄²×D₄₃	1	1	yes	characteristic
281	C₈	#13063282	1	1	yes	characteristic
282	C₈	C₄×D₁₇₂	1	1	yes	characteristic
283	C₈	C₄×D₁₇₂	1	1	yes	characteristic
284	C₂²	—	4	1	no	—
285	C₂²	—	344	2	no	—
286	C₂²	—	172	2	no	—
287	C₂²	#14582903	1	1	yes	characteristic
288	C₂²	#14582997	1	1	yes	characteristic
289	C₂²	#14583009	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	8	2	no	—
292	C₄	—	344	2	no	—
293	C₄	—	172	2	no	—
294	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
295	C₄	#14582898	1	1	yes	characteristic
296	C₄	#14582997	1	1	yes	characteristic
297	C₄	#14583009	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	344	2	no	—
302	C₂	#19134423	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14583009
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1773	C₂
socle	C₂×C₄₃	#1762
prime radical	C₂×#13063347	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1773	D₄₃
hypocenter	C₄₃	#12649
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1773

Complements

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

N	H ≅ G/N	type
4. C₂×#14582677	C₂	semidirect
5. #19132709	C₂	semidirect
6. #19132696	C₂	semidirect
7. C₄₃×#1773	C₂	semidirect
12. #14583012	C₄	semidirect
13. #14583007	C₄	semidirect
17. #14582708	C₄	semidirect
18. #14582705	C₄	semidirect
42. #13063265	C₂×C₄	semidirect
44. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
55. #108×C₄₃	C₂×C₄	semidirect
57. #106×C₄₃	C₂×C₄	semidirect
151. #1773	D₄₃	semidirect
176. C₄₃	#12649	semidirect
203. #108	C₄×D₄₃	semidirect
207. #106	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19143397	2
#19136330	2
#19136322	2
C₂×#14582677	2
#19132709	2
#19132696	2
C₄₃×#1773	2
#12649	43