Subgroups — #25253025

Order 11008.

464 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145254	C₂	1	1	yes	maximal, characteristic
2	#19136759	C₂	1	2	yes	maximal, complemented
3	#19136756	C₂	1	2	yes	maximal, complemented
4	#19136701	C₂	1	2	yes	maximal
5	#19136684	C₂	1	2	yes	maximal
6	#19136645	C₂	1	2	yes	maximal, complemented
7	#19136642	C₂	1	2	yes	maximal, complemented
8	#19136354	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#3177	C₂	1	1	yes	maximal, characteristic
10	#14583511	—	2	1	no	—
11	C₂×#13063271	C₂²	1	2	yes	—
12	C₂×#13063271	C₂²	1	2	yes	—
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	#14583301	C₂²	1	1	yes	characteristic
16	#14583279	C₂²	1	1	yes	characteristic
17	#14583268	—	2	1	no	—
18	#14583157	—	2	1	no	—
19	#14583140	C₂²	1	2	yes	—
20	#14583007	—	2	1	no	—
21	C₄×Dic₁₇₂	C₂²	1	2	yes	—
22	#14582726	C₂²	1	2	yes	—
23	#14582724	C₂²	1	2	yes	—
24	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
25	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
26	#14582703	C₂²	1	2	yes	—
27	#14582703	C₂²	1	2	yes	—
28	#14582702	C₂²	1	2	yes	—
29	#14582702	C₂²	1	2	yes	—
30	C₄₃×#541	—	2	1	no	—
31	C₄₃×#499	C₂²	1	1	yes	characteristic
32	C₄₃×#494	C₂²	1	1	yes	characteristic
33	C₄₃×#491	—	2	1	no	—
34	C₄₃×#470	—	2	1	no	—
35	C₄₃×#467	C₂²	1	2	yes	—
36	C₄₃×#441	—	2	1	no	—
37	C₄×Q₁₆×C₄₃	C₂²	1	2	yes	—
38	C₄₃×#405	C₂²	1	1	yes	characteristic
39	C₂×Dic₁₇₂	—	2	1	no	—
40	C₂×Dic₁₇₂	C₂³	1	2	yes	—
41	#13063325	—	2	1	no	—
42	#13063318	—	4	2	no	—
43	#13063317	—	2	1	no	—
44	#13063315	—	4	2	no	—
45	#13063304	—	2	2	no	—
46	C₄×Dic₈₆	—	2	2	no	—
47	#13063271	—	4	2	no	—
48	#13063271	—	4	2	no	—
49	#13063265	—	4	1	no	—
50	C₂×C₄₃⋊C₁₆	—	4	1	no	—
51	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
53	#13063259	C₂³	1	2	yes	—
54	#13063257	—	2	1	no	—
55	#13063250	—	2	2	no	—
56	#13063250	—	2	2	no	—
57	C₂×Q₁₆×C₄₃	—	2	1	no	—
58	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
59	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₄₃	—	2	1	no	—
62	#126×C₄₃	—	4	2	no	—
63	#125×C₄₃	—	2	1	no	—
64	#125×C₄₃	—	4	2	no	—
65	C₂²⋊Q₈×C₄₃	—	2	2	no	—
66	C₄×Q₈×C₄₃	—	2	2	no	—
67	#117×C₄₃	D₄	1	1	yes	characteristic
68	#107×C₄₃	C₂³	1	2	yes	—
69	#106×C₄₃	—	2	1	no	—
70	#103×C₄₃	—	2	2	no	—
71	#103×C₄₃	—	2	2	no	—
72	#97×C₄₃	D₄	1	1	yes	characteristic
73	#97×C₄₃	D₄	1	1	yes	characteristic
74	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×Q₈×C₄₃	—	2	2	no	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂×Dic₈₆	—	2	2	no	—
79	Q₁₆×C₄₃	—	2	2	no	—
80	Q₁₆×C₄₃	—	2	2	no	—
81	Q₁₆×C₄₃	—	4	2	no	—
82	M₁₆×C₄₃	—	4	1	no	—
83	C₂×C₈×C₄₃	—	2	1	no	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₄₃	—	2	2	no	—
90	C₄⋊C₄×C₄₃	—	4	2	no	—
91	C₄⋊C₄×C₄₃	—	4	2	no	—
92	C₄⋊C₄×C₄₃	—	4	2	no	—
93	C₄⋊C₄×C₄₃	—	4	2	no	—
94	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
95	C₂²⋊C₄×C₄₃	—	4	2	no	—
96	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
97	C₄²×C₄₃	—	4	2	no	—
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂²⋊Dic₄₃	—	4	2	no	—
101	C₄⋊Dic₄₃	—	2	2	no	—
102	C₄⋊Dic₄₃	—	4	2	no	—
103	Dic₄₃⋊C₄	—	4	2	no	—
104	Dic₄₃⋊C₄	—	4	2	no	—
105	Dic₄₃⋊C₄	—	4	2	no	—
106	C₄×Dic₄₃	—	4	2	no	—
107	Dic₁₇₂	—	2	2	no	—
108	Dic₁₇₂	—	2	2	no	—
109	Dic₁₇₂	—	4	2	no	—
110	C₄₃⋊C₁₆	—	2	2	no	—
111	C₄₃⋊C₁₆	—	2	2	no	—
112	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₄₃	—	4	2	no	—
114	Q₈×C₄₃	—	4	2	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₄₃	—	4	2	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	Dic₈₆	—	4	2	no	—
126	Dic₈₆	—	4	2	no	—
127	C₈×C₄₃	—	2	1	no	—
128	C₈×C₄₃	—	2	1	no	—
129	C₈×C₄₃	—	2	1	no	—
130	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
131	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
132	#33827	—	43	1	no	maximal, Sylow
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	—	2	1	no	—
135	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	—	2	2	no	—
138	C₄×C₄₃	—	2	1	no	—
139	C₄×C₄₃	—	8	2	no	—
140	C₄×C₄₃	—	8	2	no	—
141	C₄×C₄₃	#537	1	1	yes	characteristic
142	C₄×C₄₃	#575	1	1	yes	characteristic
143	Dic₄₃	—	8	2	no	—
144	Dic₄₃	—	8	2	no	—
145	#3177	—	43	1	no	—
146	#3177	D₄₃	1	1	yes	characteristic
147	#2229	—	43	2	no	—
148	#2229	—	43	2	no	—
149	#2219	—	43	2	no	—
150	#2216	—	43	2	no	—
151	#2209	—	43	2	no	—
152	#2209	—	43	2	no	—
153	#2151	—	43	1	no	—
154	C₂×C₄₃	—	4	1	no	—
155	C₂×C₄₃	#3245	1	1	yes	characteristic
156	C₂×C₄₃	#3417	1	2	yes	—
157	#541	—	2	1	no	—
158	#541	—	86	1	no	—
159	C₂×Q₃₂	—	43	2	no	—
160	C₂×Q₃₂	—	43	2	no	—
161	C₂×M₃₂	—	43	1	no	—
162	C₂²×C₁₆	—	43	1	no	—
163	#499	—	43	1	no	—
164	#499	C₂×D₄₃	1	1	yes	characteristic
165	#494	—	43	1	no	—
166	#494	C₂×D₄₃	1	1	yes	characteristic
167	#491	—	2	1	no	—
168	#491	—	86	1	no	—
169	#470	—	2	1	no	—
170	#470	—	86	1	no	—
171	#467	—	43	2	no	—
172	#467	C₂×D₄₃	1	2	yes	—
173	#441	—	2	1	no	—
174	#441	—	86	1	no	—
175	C₄×Q₁₆	—	43	2	no	—
176	C₄×Q₁₆	C₂×D₄₃	1	2	yes	—
177	#405	C₂×D₄₃	1	1	yes	characteristic
178	#367	—	43	2	no	—
179	#366	—	43	2	no	—
180	C₄⋊C₁₆	—	43	2	no	—
181	C₄⋊C₁₆	—	43	2	no	—
182	#358	—	43	2	no	—
183	#358	—	43	2	no	—
184	#358	—	43	2	no	—
185	#358	—	43	2	no	—
186	C₄₃	#33827	1	1	yes	characteristic, complemented, Sylow
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	43	2	no	—
189	C₂×Q₁₆	—	86	1	no	—
190	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
191	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	86	1	no	—
195	#126	—	4	2	no	—
196	#126	—	172	2	no	—
197	#125	—	2	1	no	—
198	#125	—	4	2	no	—
199	#125	—	86	1	no	—
200	#125	—	172	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₂²⋊Q₈	—	86	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	86	2	no	—
205	#117	C₂²⋊D₄₃	1	1	yes	characteristic
206	Q₃₂	—	172	2	no	—
207	Q₃₂	—	172	2	no	—
208	M₃₂	—	172	1	no	—
209	C₂×C₁₆	—	43	2	no	—
210	C₂×C₁₆	—	43	2	no	—
211	C₂×C₁₆	—	172	1	no	—
212	#107	—	43	2	no	—
213	#107	C₂²×D₄₃	1	2	yes	—
214	#106	—	2	1	no	—
215	#106	—	86	1	no	—
216	#103	—	2	2	no	—
217	#103	—	2	2	no	—
218	#103	—	86	2	no	—
219	#103	—	86	2	no	—
220	#97	C₂²⋊D₄₃	1	1	yes	characteristic
221	#97	C₂²⋊D₄₃	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	86	2	no	—
226	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
227	Q₁₆	—	2	2	no	—
228	Q₁₆	—	2	2	no	—
229	Q₁₆	—	4	2	no	—
230	Q₁₆	—	86	2	no	—
231	Q₁₆	—	86	2	no	—
232	Q₁₆	—	172	2	no	—
233	M₁₆	—	4	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
239	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	86	2	no	—
246	C₄⋊C₄	—	172	2	no	—
247	C₄⋊C₄	—	172	2	no	—
248	C₄⋊C₄	—	172	2	no	—
249	C₄⋊C₄	—	172	2	no	—
250	C₄⋊C₄	#10621882	1	2	yes	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	172	2	no	—
253	C₂²⋊C₄	#10621882	1	2	yes	—
254	C₄²	—	4	2	no	—
255	C₄²	—	172	2	no	—
256	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
257	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
258	C₁₆	—	86	2	no	—
259	C₁₆	—	86	2	no	—
260	C₂³	C₂×#10621882	1	1	yes	characteristic
261	Q₈	—	4	2	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	172	2	no	—
264	Q₈	—	172	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	172	2	no	—
270	C₂×C₄	—	172	2	no	—
271	C₂×C₄	C₂×#10621882	1	2	yes	—
272	C₂×C₄	C₂×#10621882	1	2	yes	—
273	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	#13063371	1	1	yes	characteristic
279	C₈	#13063372	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	2	no	—
282	C₂²	C₂²×#10621882	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	8	2	no	—
287	C₄	—	8	2	no	—
288	C₄	—	344	2	no	—
289	C₄	—	344	2	no	—
290	C₄	#14583477	1	1	yes	characteristic
291	C₄	#14583697	1	1	yes	characteristic
292	C₂	—	4	1	no	—
293	C₂	#19145902	1	1	yes	characteristic
294	C₂	#19147746	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3177	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂³×C₄₃	C₂×D₈
hypercenter	#3177	D₄₃
hypocenter	C₄₃	#33827
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3177

Complements

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

N	H ≅ G/N	type
2. #19136759	C₂	semidirect
3. #19136756	C₂	semidirect
6. #19136645	C₂	semidirect
7. #19136642	C₂	semidirect
186. C₄₃	#33827	semidirect

Maximal subgroups

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

subgroup	index
#19145254	2
#19136759	2
#19136756	2
#19136701	2
#19136684	2
#19136645	2
#19136642	2
#19136354	2
C₄₃×#3177	2
#33827	43