Subgroups — #25012793

Order 11008.

358 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583317	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582728	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136377	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134650	C₂	1	1	yes	maximal, characteristic
5	#19134646	C₂	1	1	yes	maximal, characteristic
6	#19132710	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063331	C₂²	1	1	yes	characteristic
9	#14583317	—	2	1	no	—
10	#14583317	—	2	1	no	—
11	#14583317	C₄	1	2	yes	complemented
12	#14583317	C₄	1	2	yes	complemented
13	C₂×#13063264	—	2	1	no	—
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₄₃	—	2	1	no	—
16	C₂×#13063261	C₂²	1	1	yes	characteristic
17	#14582937	C₂²	1	1	yes	characteristic
18	#14582917	—	2	1	no	—
19	#14582915	—	2	1	no	—
20	#14582728	—	2	1	no	—
21	#14582728	—	2	1	no	—
22	#14582726	—	2	1	no	—
23	#14582724	—	2	1	no	—
24	#14582670	—	2	1	no	—
25	#14582669	—	2	1	no	—
26	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#427	C₂²	1	1	yes	characteristic
29	C₂×#10621898	D₄	1	1	yes	characteristic
30	#13063331	—	2	1	no	—
31	#13063331	—	2	1	no	—
32	#13063331	C₂×C₄	1	2	yes	—
33	#13063331	Q₈	1	2	yes	—
34	C₂×#10621877	—	2	1	no	—
35	C₂×#10621877	D₄	1	1	yes	characteristic
36	C₂×#10621872	—	2	1	no	—
37	C₂×C₈×D₄₃	—	2	1	no	—
38	#13063284	—	2	1	no	—
39	#13063264	—	2	2	no	—
40	#13063264	—	2	2	no	—
41	C₂×C₄₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
44	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
45	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
46	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
47	C₁₆×D₄₃	—	2	2	no	—
48	C₁₆×D₄₃	—	2	2	no	—
49	#13063261	—	2	1	no	—
50	#13063261	—	2	1	no	—
51	#13063259	—	2	1	no	—
52	#13063257	—	2	1	no	—
53	#13063253	—	4	2	no	—
54	#13063251	—	4	2	no	—
55	C₂×M₁₆×C₄₃	—	2	1	no	—
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	#117×C₄₃	—	2	1	no	—
58	M₃₂×C₄₃	—	2	1	no	—
59	M₃₂×C₄₃	C₂×C₄	1	2	yes	—
60	C₂×C₁₆×C₄₃	—	2	1	no	—
61	#108×C₄₃	—	2	1	no	—
62	#108×C₄₃	—	2	1	no	—
63	#107×C₄₃	—	2	1	no	—
64	#106×C₄₃	—	2	1	no	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₄₃	—	2	1	no	—
67	#10621898	—	2	1	no	—
68	#10621898	—	2	1	no	—
69	#10621898	C₄⋊C₄	1	2	yes	—
70	#10621898	D₈	1	2	yes	—
71	C₂×D₁₇₂	Q₁₆	1	1	yes	characteristic
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×Dic₈₆	Q₁₆	1	1	yes	characteristic
74	M₁₆×C₄₃	—	4	1	no	—
75	M₁₆×C₄₃	—	4	1	no	—
76	C₂×C₈×C₄₃	—	4	1	no	—
77	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₄₃	—	4	1	no	—
84	C₂²⋊C₄×C₄₃	—	4	1	no	—
85	C₄²×C₄₃	—	4	1	no	—
86	C₂²⋊Dic₄₃	—	4	1	no	—
87	C₄⋊Dic₄₃	—	4	1	no	—
88	C₄×Dic₄₃	—	4	1	no	—
89	#10621877	—	2	1	no	—
90	#10621877	—	4	1	no	—
91	#10621877	—	4	1	no	—
92	#10621877	C₄⋊C₄	1	2	yes	—
93	C₂×C₄₃⋊C₈	—	2	1	no	—
94	C₂×C₄₃⋊C₈	—	4	1	no	—
95	#10621872	—	2	2	no	—
96	#10621872	—	2	2	no	—
97	C₈×D₄₃	—	2	2	no	—
98	C₈×D₄₃	—	2	2	no	—
99	C₁₆×C₄₃	—	2	2	no	—
100	C₄₃⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
101	C₄₃⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
102	C₄₃⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
103	C₄₃⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
104	C₂³×C₄₃	#135	1	1	yes	characteristic
105	C₂²×D₄₃	—	2	1	no	—
106	C₂×C₄×C₄₃	—	8	1	no	—
107	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
113	C₂²⋊D₄₃	—	2	2	no	—
114	C₂²⋊D₄₃	—	2	2	no	—
115	C₂×Dic₄₃	—	2	1	no	—
116	C₂×Dic₄₃	—	8	1	no	—
117	D₁₇₂	—	2	1	no	—
118	D₁₇₂	#107	1	2	yes	—
119	C₄×D₄₃	—	2	2	no	—
120	C₄×D₄₃	—	2	2	no	—
121	Dic₈₆	—	2	1	no	—
122	Dic₈₆	#107	1	2	yes	—
123	C₈×C₄₃	—	8	1	no	—
124	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₈×C₄₃	#117	1	1	yes	characteristic
126	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₄₃⋊C₈	—	2	2	no	—
129	C₄₃⋊C₈	—	8	1	no	—
130	#12841	—	43	1	no	maximal, Sylow
131	C₂²×C₄₃	—	4	1	no	—
132	C₂²×C₄₃	#427	1	1	yes	characteristic
133	C₂²×C₄₃	#466	1	1	yes	characteristic
134	C₂²×C₄₃	#467	1	1	yes	characteristic
135	C₂×D₄₃	—	2	2	no	—
136	C₂×D₄₃	—	4	2	no	—
137	C₄×C₄₃	—	16	1	no	—
138	C₄×C₄₃	#380	1	1	yes	characteristic
139	C₄×C₄₃	C₂×#107	1	1	yes	characteristic
140	C₄×C₄₃	#466	1	1	yes	characteristic
141	C₄×C₄₃	#467	1	1	yes	characteristic
142	Dic₄₃	—	2	2	no	—
143	Dic₄₃	—	16	1	no	—
144	C₂×#504	—	43	1	no	—
145	C₂×#368	—	43	1	no	—
146	#2158	—	43	1	no	—
147	#1814	—	43	1	no	—
148	#1814	—	43	1	no	—
149	#1389	—	43	1	no	—
150	#1389	D₄₃	1	1	yes	characteristic, complemented
151	C₂×C₄₃	—	2	1	no	—
152	C₂×C₄₃	—	2	1	no	—
153	C₂×C₄₃	—	2	1	no	—
154	C₂×C₄₃	#1808	1	1	yes	characteristic
155	D₄₃	—	4	2	no	—
156	C₂×#131	—	43	1	no	—
157	#504	—	43	2	no	—
158	#504	—	43	2	no	—
159	#504	—	86	1	no	—
160	#504	—	86	1	no	—
161	C₂×M₃₂	—	86	1	no	—
162	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
163	C₂²×C₁₆	—	43	1	no	—
164	C₂²×C₁₆	—	86	1	no	—
165	C₂×#108	—	43	1	no	—
166	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
167	#427	—	43	1	no	—
168	#427	C₂×D₄₃	1	1	yes	characteristic
169	#421	—	86	1	no	—
170	#421	—	86	1	no	—
171	#368	—	86	1	no	—
172	#368	—	86	1	no	—
173	#367	—	86	1	no	—
174	#366	—	86	1	no	—
175	#344	—	86	1	no	—
176	#344	—	86	1	no	—
177	C₄₃	#12841	1	1	yes	characteristic, complemented, Sylow
178	C₂×#41	—	43	1	no	—
179	#131	—	43	2	no	—
180	#131	—	43	2	no	—
181	#131	—	86	1	no	—
182	#131	—	86	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	43	1	no	—
185	C₂×M₁₆	—	86	1	no	—
186	C₂×M₁₆	—	86	1	no	—
187	C₂²×C₈	—	86	1	no	—
188	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
189	#117	—	2	1	no	—
190	#117	—	86	1	no	—
191	M₃₂	—	2	1	no	—
192	M₃₂	—	86	2	no	—
193	M₃₂	—	86	2	no	—
194	M₃₂	C₄×D₄₃	1	2	yes	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	43	1	no	—
197	C₂×C₁₆	—	43	1	no	—
198	C₂×C₁₆	—	43	1	no	—
199	C₂×C₁₆	—	43	1	no	—
200	C₂×C₁₆	—	43	1	no	—
201	C₂×C₁₆	—	43	1	no	—
202	C₂×C₁₆	—	86	2	no	—
203	C₂×C₁₆	—	86	2	no	—
204	#108	—	2	1	no	—
205	#108	—	2	1	no	—
206	#108	—	86	1	no	—
207	#108	—	86	1	no	—
208	#107	—	2	1	no	—
209	#107	—	86	1	no	—
210	#106	—	2	1	no	—
211	#106	—	86	1	no	—
212	#104	—	172	2	no	—
213	#104	—	172	2	no	—
214	#41	—	43	2	no	—
215	#41	—	43	2	no	—
216	#41	—	86	1	no	—
217	#41	—	86	1	no	—
218	C₂×Q₈	—	43	1	no	—
219	C₂×D₄	—	43	1	no	—
220	C₂×D₄	—	86	1	no	—
221	C₂²×C₄	—	86	1	no	—
222	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	43	2	no	—
226	M₁₆	—	86	1	no	—
227	M₁₆	—	86	2	no	—
228	M₁₆	—	86	2	no	—
229	M₁₆	—	172	1	no	—
230	M₁₆	—	172	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	86	1	no	—
233	C₂×C₈	—	86	2	no	—
234	C₂×C₈	—	86	2	no	—
235	C₂×C₈	—	172	1	no	—
236	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
237	C₂×C₈	#10621898	1	1	yes	characteristic
238	C₂×C₈	#10621898	1	1	yes	characteristic
239	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
240	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
241	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	172	1	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	172	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	172	1	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	43	1	no	—
250	C₁₆	—	43	1	no	—
251	C₁₆	—	43	1	no	—
252	C₁₆	—	43	1	no	—
253	C₂³	—	86	1	no	—
254	C₂³	#13063346	1	1	yes	characteristic
255	Q₈	—	43	2	no	—
256	Q₈	—	86	1	no	—
257	D₄	—	43	2	no	—
258	D₄	—	86	2	no	—
259	D₄	—	86	2	no	—
260	D₄	—	86	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	86	1	no	—
263	C₂×C₄	—	86	2	no	—
264	C₂×C₄	—	86	2	no	—
265	C₂×C₄	—	344	1	no	—
266	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
267	C₂×C₄	#13063298	1	1	yes	characteristic
268	C₂×C₄	#13063298	1	1	yes	characteristic
269	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
270	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
271	C₂×C₄	#13063346	1	1	yes	characteristic
272	C₈	—	8	1	no	—
273	C₈	—	86	2	no	—
274	C₈	—	344	1	no	—
275	C₈	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
276	C₈	#13063282	1	1	yes	characteristic
277	C₈	#13063303	1	1	yes	characteristic
278	C₈	#13063303	1	1	yes	characteristic
279	C₂²	—	4	1	no	—
280	C₂²	—	86	2	no	—
281	C₂²	—	172	2	no	—
282	C₂²	#14582934	1	1	yes	characteristic
283	C₂²	#14583134	1	1	yes	characteristic
284	C₂²	#14583139	1	1	yes	characteristic
285	C₄	—	16	1	no	—
286	C₄	—	86	2	no	—
287	C₄	—	688	1	no	—
288	C₄	#14582758	1	1	yes	characteristic
289	C₄	#107×D₄₃	1	1	yes	characteristic
290	C₄	#14583134	1	1	yes	characteristic
291	C₄	#14583139	1	1	yes	characteristic
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	172	2	no	—
296	C₂	#19134623	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14583134
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1389	C₂
socle	C₂×C₄₃	#1808
prime radical	C₂×#10621898	D₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1389	D₄₃
hypocenter	C₄₃	#12841
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1389

Complements

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

N	H ≅ G/N	type
2. C₂×#14582728	C₂	semidirect
3. #19136377	C₂	semidirect
6. #19132710	C₂	semidirect
7. C₄₃×#1389	C₂	semidirect
11. #14583317	C₄	semidirect
12. #14583317	C₄	semidirect
150. #1389	D₄₃	semidirect
177. C₄₃	#12841	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583317	2
C₂×#14582728	2
#19136377	2
#19134650	2
#19134646	2
#19132710	2
C₄₃×#1389	2
#12841	43