Subgroups — #25249971

Order 11008.

646 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063271	C₂	1	1	yes	maximal, characteristic
2	#19143463	C₂	1	1	yes	maximal, characteristic
3	#19143462	C₂	1	1	yes	maximal, characteristic
4	#19136463	C₂	1	4	yes	maximal, complemented
5	#19136451	C₂	1	4	yes	maximal, complemented
6	#19136377	C₂	1	1	yes	maximal, characteristic
7	#19136316	C₂	1	1	yes	maximal, characteristic
8	#19136194	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#2141	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
11	#14583377	—	2	1	no	—
12	#14583376	—	2	1	no	—
13	C₂×#13063271	C₄	1	8	yes	complemented
14	C₂×#13063271	C₂²	1	2	yes	—
15	C₂×#13063271	C₂²	1	2	yes	—
16	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	#14583006	—	2	2	no	—
18	#14583005	—	2	2	no	—
19	Q₁₆×Dic₄₃	C₂²	1	4	yes	—
20	#14582993	C₂²	1	4	yes	—
21	#14582937	C₂²	1	1	yes	characteristic
22	#14582908	—	2	1	no	—
23	#14582907	—	2	1	no	—
24	#14582854	C₂²	1	1	yes	characteristic
25	#14582726	C₂²	1	2	yes	—
26	#14582724	C₂²	1	2	yes	—
27	#14582704	C₂²	1	4	yes	—
28	#14582674	C₂²	1	2	yes	—
29	C₁₆×Dic₄₃	C₂²	1	2	yes	—
30	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
32	C₄₃×#427	C₂²	1	1	yes	characteristic
33	C₄₃×#358	C₂²	1	4	yes	—
34	C₂²×Dic₈₆	—	2	1	no	—
35	C₂×Dic₁₇₂	—	2	1	no	—
36	C₂×Dic₁₇₂	—	2	2	no	—
37	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
38	C₂×Dic₁₇₂	C₂³	1	2	yes	—
39	C₂×#10621877	D₄	1	1	yes	characteristic
40	Q₈×Dic₄₃	—	2	4	no	—
41	Q₈×Dic₄₃	—	4	2	no	—
42	C₄×Dic₈₆	—	4	2	no	—
43	#13063290	—	2	4	no	—
44	#13063284	—	2	1	no	—
45	#13063284	—	4	2	no	—
46	#13063284	D₄	1	1	yes	characteristic
47	#13063281	—	4	2	no	—
48	#13063271	C₂×C₄	1	8	yes	—
49	#13063271	C₂×C₄	1	8	yes	—
50	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
51	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
53	#13063259	C₂³	1	2	yes	—
54	#13063257	—	2	1	no	—
55	#13063249	—	2	4	no	—
56	#13063248	—	2	4	no	—
57	#13063235	D₄	1	2	yes	—
58	C₈×Dic₄₃	C₂³	1	2	yes	—
59	C₂²×Q₈×C₄₃	—	2	1	no	—
60	C₂×Q₁₆×C₄₃	—	2	1	no	—
61	C₂×Q₁₆×C₄₃	—	2	2	no	—
62	C₂×Q₁₆×C₄₃	C₂×C₄	1	4	yes	—
63	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	#117×C₄₃	—	2	1	no	—
66	M₃₂×C₄₃	—	2	2	no	—
67	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
68	#107×C₄₃	C₂³	1	2	yes	—
69	#106×C₄₃	—	2	1	no	—
70	C₂×Q₈×C₄₃	—	2	4	no	—
71	C₂×Q₈×C₄₃	—	2	2	no	—
72	C₂×Q₈×C₄₃	—	4	1	no	—
73	C₂×Q₈×C₄₃	—	4	2	no	—
74	C₂²×C₄×C₄₃	—	4	1	no	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₄₃	—	4	1	no	—
77	C₂×Dic₈₆	—	2	4	no	—
78	C₂×Dic₈₆	—	2	2	no	—
79	C₂×Dic₈₆	—	4	1	no	—
80	C₂×Dic₈₆	—	4	2	no	—
81	Q₁₆×C₄₃	—	2	4	no	—
82	Q₁₆×C₄₃	—	2	2	no	—
83	Q₁₆×C₄₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	#41	1	2	yes	—
87	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₄₃	—	2	2	no	—
89	C₂²⋊C₄×C₄₃	—	4	1	no	—
90	C₄²×C₄₃	—	4	1	no	—
91	C₂²⋊Dic₄₃	—	4	1	no	—
92	C₂²⋊Dic₄₃	D₈	1	2	yes	—
93	C₄⋊Dic₄₃	—	2	2	no	—
94	C₄⋊Dic₄₃	—	4	4	no	—
95	C₄⋊Dic₄₃	D₈	1	2	yes	—
96	Dic₄₃⋊C₄	—	4	4	no	—
97	C₄×Dic₄₃	—	4	1	no	—
98	C₄×Dic₄₃	—	4	4	no	—
99	C₄×Dic₄₃	—	4	4	no	—
100	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
101	#10621877	—	2	2	no	—
102	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
103	Dic₁₇₂	—	2	4	no	—
104	Dic₁₇₂	—	2	2	no	—
105	Dic₁₇₂	C₂²×C₄	1	4	yes	—
106	C₁₆×C₄₃	—	2	2	no	—
107	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
109	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
110	Q₈×C₄₃	—	2	4	no	—
111	Q₈×C₄₃	—	4	2	no	—
112	Q₈×C₄₃	—	4	4	no	—
113	C₂×C₄×C₄₃	—	4	2	no	—
114	C₂×C₄×C₄₃	—	4	4	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#135	1	2	yes	—
119	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×Dic₄₃	—	4	2	no	—
121	C₂×Dic₄₃	—	4	4	no	—
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	C₂×D₈	1	4	yes	—
124	Dic₈₆	—	2	4	no	—
125	Dic₈₆	—	4	2	no	—
126	Dic₈₆	—	4	4	no	—
127	C₈×C₄₃	C₄×D₄	1	2	yes	—
128	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
129	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
130	C₄₃⋊C₈	—	2	2	no	—
131	#33458	—	43	1	no	maximal, Sylow
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	C₄×D₈	1	2	yes	—
134	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₄₃	—	4	4	no	—
136	C₄×C₄₃	—	8	2	no	—
137	C₄×C₄₃	C₄×D₈	1	2	yes	—
138	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
140	Dic₄₃	—	2	4	no	—
141	Dic₄₃	—	4	4	no	—
142	Dic₄₃	—	8	2	no	—
143	C₂²×Q₃₂	—	43	1	no	—
144	#2950	—	43	1	no	—
145	#2950	—	43	1	no	—
146	#2176	—	43	4	no	—
147	C₄×Q₃₂	—	43	4	no	—
148	#2158	—	43	1	no	—
149	#2141	—	43	1	no	—
150	#2141	D₄₃	1	1	yes	characteristic
151	#2108	—	43	1	no	—
152	C₂×C₄₃	—	2	2	no	—
153	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
154	C₂×C₄₃	#3417	1	2	yes	—
155	C₂²×Q₁₆	—	43	1	no	—
156	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
157	#519	—	86	1	no	—
158	#519	—	86	1	no	—
159	C₂×Q₃₂	—	43	2	no	—
160	C₂×Q₃₂	—	43	2	no	—
161	C₂×Q₃₂	—	43	8	no	—
162	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
163	C₂²×C₁₆	—	43	1	no	—
164	#441	—	86	2	no	—
165	#441	—	86	2	no	—
166	C₄×Q₁₆	—	43	4	no	—
167	C₄×Q₁₆	—	43	4	no	—
168	#427	—	43	1	no	—
169	#427	C₂×D₄₃	1	1	yes	characteristic
170	#419	—	86	1	no	—
171	#419	—	86	1	no	—
172	#405	—	43	1	no	—
173	#367	—	43	2	no	—
174	#366	—	43	2	no	—
175	#358	—	43	4	no	—
176	#358	C₂×D₄₃	1	4	yes	—
177	#346	—	43	2	no	—
178	C₄×C₁₆	—	43	2	no	—
179	C₄₃	#33458	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×Q₈	—	86	1	no	—
182	C₂×Q₁₆	—	2	1	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	43	4	no	—
185	C₂×Q₁₆	—	43	2	no	—
186	C₂×Q₁₆	—	86	1	no	—
187	C₂×Q₁₆	—	86	2	no	—
188	C₂×Q₁₆	C₄×D₄₃	1	4	yes	—
189	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
190	C₂×M₁₆	—	43	1	no	—
191	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
192	C₄×Q₈	—	86	4	no	—
193	C₄×Q₈	—	86	4	no	—
194	C₄×Q₈	—	172	2	no	—
195	C₄×Q₈	—	172	2	no	—
196	#117	—	2	1	no	—
197	#117	—	43	1	no	—
198	#117	—	86	1	no	—
199	#117	—	172	2	no	—
200	#117	—	172	2	no	—
201	Q₃₂	—	43	8	no	—
202	Q₃₂	—	43	8	no	—
203	M₃₂	—	2	2	no	—
204	C₂×C₁₆	—	43	4	no	—
205	C₂×C₁₆	—	43	1	no	—
206	C₂×C₁₆	—	43	1	no	—
207	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
208	#107	—	43	2	no	—
209	#107	C₂²×D₄₃	1	2	yes	—
210	#106	—	2	1	no	—
211	#106	—	86	1	no	—
212	#103	—	86	4	no	—
213	#103	—	86	4	no	—
214	#97	—	43	2	no	—
215	C₄×C₈	—	43	2	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	86	4	no	—
221	C₂×Q₈	—	86	2	no	—
222	C₂×Q₈	—	172	1	no	—
223	C₂×Q₈	—	172	2	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	172	1	no	—
226	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
227	Q₁₆	—	2	4	no	—
228	Q₁₆	—	2	2	no	—
229	Q₁₆	—	43	4	no	—
230	Q₁₆	—	86	4	no	—
231	Q₁₆	—	86	2	no	—
232	Q₁₆	C₂×C₄×D₄₃	1	4	yes	—
233	M₁₆	—	86	2	no	—
234	C₂×C₈	—	43	2	no	—
235	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
236	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
237	C₂×C₈	#10621900	1	2	yes	—
238	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	43	2	no	—
241	C₄⋊C₄	—	86	2	no	—
242	C₄⋊C₄	—	172	4	no	—
243	C₄⋊C₄	—	172	4	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	43	2	no	—
246	C₂²⋊C₄	—	172	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	43	2	no	—
249	C₄²	—	172	4	no	—
250	C₄²	—	172	4	no	—
251	C₄²	—	172	1	no	—
252	C₁₆	—	2	2	no	—
253	C₁₆	—	43	2	no	—
254	C₁₆	—	43	2	no	—
255	C₂³	D₈×D₄₃	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	4	2	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	86	4	no	—
260	Q₈	—	172	2	no	—
261	Q₈	—	172	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	43	4	no	—
266	C₂×C₄	—	172	2	no	—
267	C₂×C₄	—	172	2	no	—
268	C₂×C₄	—	172	4	no	—
269	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
270	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
271	C₂×C₄	#13063346	1	2	yes	—
272	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
273	C₈	—	86	2	no	—
274	C₈	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
275	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
276	C₈	C₂×#10621900	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	#14582979	1	2	yes	—
279	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
280	C₄	—	4	4	no	—
281	C₄	—	8	2	no	—
282	C₄	—	86	4	no	—
283	C₄	—	344	2	no	—
284	C₄	—	172	4	no	—
285	C₄	#14582979	1	2	yes	—
286	C₄	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
287	C₄	C₂×#13063346	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	C₂×#14582979	1	1	yes	characteristic
290	C₂	#19147747	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2141	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2141	D₄₃
hypocenter	C₄₃	#33458
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2141

Complements

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

N	H ≅ G/N	type
4. #19136463	C₂	semidirect
5. #19136451	C₂	semidirect
13. C₂×#13063271	C₄	semidirect
179. C₄₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063271	2
#19143463	2
#19143462	2
#19136463	2
#19136451	2
#19136377	2
#19136316	2
#19136194	2
C₄₃×#2141	2
#33458	43