Subgroups — #25105453

Order 11008.

654 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145109	C₂	1	1	yes	maximal, characteristic
2	#19143535	C₂	1	2	yes	maximal
3	#19135576	C₂	1	2	yes	maximal
4	#19135348	C₂	1	2	yes	maximal
5	#19135343	C₂	1	2	yes	maximal
6	#19134767	C₂	1	1	yes	maximal, characteristic
7	#19134479	C₂	1	2	yes	maximal
8	#19134474	C₂	1	2	yes	maximal
9	C₄₃×#1726	C₂	1	1	yes	maximal, characteristic
10	#14583456	C₂²	1	2	yes	—
11	#14583304	C₄	1	2	yes	complemented
12	#14583301	C₄	1	2	yes	complemented
13	#14583301	C₄	1	2	yes	complemented
14	#14583297	C₄	1	2	yes	complemented
15	#14583027	—	2	8	no	—
16	C₂×#13063261	C₂²	1	2	yes	—
17	C₂×#13063261	C₂²	1	2	yes	—
18	#14582937	C₂²	1	2	yes	—
19	#14582937	C₂²	1	2	yes	—
20	#14582925	C₂²	1	4	yes	—
21	#14582923	C₂²	1	2	yes	—
22	#14582854	C₂²	1	4	yes	—
23	#14582659	C₂²	1	4	yes	—
24	#14582659	C₂²	1	4	yes	—
25	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#376	C₂²	1	1	yes	characteristic
32	C₂×#10621877	C₂³	1	4	yes	—
33	#13063325	C₂×C₄	1	4	yes	—
34	#13063317	C₂×C₄	1	4	yes	—
35	#13063304	—	2	4	no	—
36	C₄×Dic₈₆	—	2	4	no	—
37	#13063284	C₂³	1	4	yes	—
38	#13063261	—	2	4	no	—
39	#13063261	—	2	4	no	—
40	#13063259	C₂×C₄	1	4	yes	—
41	#13063259	C₂×C₄	1	4	yes	—
42	#13063257	C₂×C₄	1	4	yes	—
43	#13063257	C₂×C₄	1	4	yes	—
44	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
45	Dic₄₃⋊C₈	—	2	8	no	—
46	#13063235	—	2	4	no	—
47	C₈×Dic₄₃	—	2	4	no	—
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	#97×C₄₃	—	2	2	no	—
56	#97×C₄₃	D₄	1	4	yes	—
57	#97×C₄₃	Q₈	1	4	yes	complemented
58	C₄×C₈×C₄₃	—	2	2	no	—
59	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₄₃	Q₈	1	4	yes	complemented
64	#95×C₄₃	—	2	2	no	—
65	C₂²×C₄×C₄₃	—	2	1	no	—
66	C₂²×C₄×C₄₃	—	2	1	no	—
67	C₂²×C₄×C₄₃	#41	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
69	C₂×Dic₈₆	—	2	4	no	—
70	C₂×C₈×C₄₃	—	2	1	no	—
71	C₂×C₈×C₄₃	—	2	1	no	—
72	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₄₃	#41	1	2	yes	—
83	C₂×C₈×C₄₃	#41	1	2	yes	—
84	C₄²×C₄₃	—	2	1	no	—
85	C₄²×C₄₃	—	2	1	no	—
86	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₄₃	C₂×D₄	1	2	yes	—
89	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
91	C₂²⋊Dic₄₃	—	2	4	no	—
92	C₄⋊Dic₄₃	—	2	4	no	—
93	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
94	Dic₄₃⋊C₄	—	2	8	no	—
95	C₄×Dic₄₃	—	2	4	no	—
96	#10621877	—	2	4	no	—
97	#10621877	—	2	4	no	—
98	C₂×C₄₃⋊C₈	—	2	4	no	—
99	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	4	2	no	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
113	C₂×Dic₄₃	—	2	8	no	—
114	Dic₈₆	—	8	4	no	—
115	C₈×C₄₃	—	2	2	no	—
116	C₈×C₄₃	C₄×Q₈	1	2	yes	—
117	C₈×C₄₃	C₄×Q₈	1	2	yes	—
118	C₈×C₄₃	C₄×Q₈	1	2	yes	—
119	C₈×C₄₃	C₄×Q₈	1	2	yes	—
120	C₈×C₄₃	C₄⋊Q₈	1	4	yes	—
121	C₄₃⋊C₈	—	4	4	no	—
122	#21733	—	43	1	no	maximal, Sylow
123	C₂²×C₄₃	—	2	2	no	—
124	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₄₃	#533	1	1	yes	characteristic
126	C₂²×C₄₃	#533	1	1	yes	characteristic
127	C₄×C₄₃	—	2	2	no	—
128	C₄×C₄₃	—	2	2	no	—
129	C₄×C₄₃	—	4	2	no	—
130	C₄×C₄₃	—	4	2	no	—
131	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₄×C₄₃	#532	1	1	yes	characteristic
135	Dic₄₃	—	4	8	no	—
136	#3160	—	43	1	no	—
137	#2961	—	43	2	no	—
138	#1995	—	43	2	no	—
139	#1946	—	43	2	no	—
140	#1945	—	43	2	no	—
141	#1848	—	43	1	no	—
142	#1773	—	43	2	no	—
143	#1772	—	43	2	no	—
144	#1726	D₄₃	1	1	yes	characteristic
145	C₂×C₄₃	—	2	1	no	—
146	C₂×C₄₃	—	2	1	no	—
147	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₄₃	#2957	1	2	yes	—
149	#533	—	43	2	no	—
150	#500	—	43	2	no	—
151	#499	—	43	2	no	—
152	#499	—	43	2	no	—
153	#498	—	43	2	no	—
154	#446	—	86	8	no	—
155	C₂×#108	—	43	2	no	—
156	C₂×#108	—	43	2	no	—
157	#427	—	43	2	no	—
158	#427	—	43	2	no	—
159	#424	—	43	4	no	—
160	#423	—	43	2	no	—
161	#405	—	43	4	no	—
162	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
163	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
164	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
168	#376	C₂×D₄₃	1	1	yes	characteristic
169	#339	—	43	4	no	—
170	#339	—	43	4	no	—
171	C₄₃	#21733	1	1	yes	characteristic, complemented, Sylow
172	C₂×M₁₆	—	43	4	no	—
173	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
177	C₄⋊Q₈	—	43	4	no	—
178	#125	—	43	4	no	—
179	C₂²⋊Q₈	—	86	4	no	—
180	C₄×Q₈	—	86	4	no	—
181	#117	—	43	4	no	—
182	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
185	#108	—	86	4	no	—
186	#108	—	86	4	no	—
187	#107	—	43	4	no	—
188	#107	—	43	4	no	—
189	#106	—	43	4	no	—
190	#106	—	43	4	no	—
191	C₄⋊C₈	—	86	8	no	—
192	C₄⋊C₈	—	86	4	no	—
193	#97	—	2	2	no	—
194	#97	—	86	4	no	—
195	#97	Dic₈₆	1	4	yes	complemented
196	#97	D₁₇₂	1	4	yes	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	86	4	no	—
199	C₄×C₈	Dic₈₆	1	4	yes	complemented
200	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
201	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
202	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
203	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
204	#95	—	2	2	no	—
205	C₂×Q₈	—	86	4	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	#10621898	1	2	yes	—
209	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
210	M₁₆	—	86	4	no	—
211	M₁₆	—	86	4	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	86	4	no	—
215	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
216	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
217	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
218	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
221	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
223	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
224	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
225	C₂×C₈	#10621898	1	2	yes	—
226	C₂×C₈	#10621898	1	2	yes	—
227	C₄⋊C₄	—	43	8	no	—
228	C₄⋊C₄	—	86	8	no	—
229	C₄⋊C₄	—	86	4	no	—
230	C₂²⋊C₄	—	86	4	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	86	4	no	—
234	C₄²	C₂×Dic₈₆	1	2	yes	—
235	C₄²	C₂×Dic₈₆	1	2	yes	—
236	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
238	C₄²	C₂×D₁₇₂	1	2	yes	—
239	C₂³	C₂×#10621898	1	1	yes	characteristic
240	Q₈	—	344	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	86	8	no	—
247	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
248	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
249	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
250	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
251	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
252	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
253	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
254	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	172	4	no	—
257	C₈	C₄×Dic₈₆	1	2	yes	—
258	C₈	C₄×Dic₈₆	1	2	yes	—
259	C₈	C₄×Dic₈₆	1	2	yes	—
260	C₈	C₄×Dic₈₆	1	2	yes	—
261	C₈	#13063325	1	4	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
264	C₂²	#14583456	1	1	yes	characteristic
265	C₂²	#14583456	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	172	8	no	—
271	C₄	C₂×C₄×Dic₈₆	1	1	yes	characteristic
272	C₄	C₂×C₄×Dic₈₆	1	1	yes	characteristic
273	C₄	C₂×#13063325	1	1	yes	characteristic
274	C₄	#14583449	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	C₄×#13063325	1	1	yes	characteristic
278	C₂	#19143513	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1726	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1726	D₄₃
hypocenter	C₄₃	#21733
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
11. #14583304	C₄	semidirect
12. #14583301	C₄	semidirect
13. #14583301	C₄	semidirect
14. #14583297	C₄	semidirect
57. #97×C₄₃	Q₈	semidirect
63. C₄×C₈×C₄₃	Q₈	semidirect
171. C₄₃	#21733	semidirect
195. #97	Dic₈₆	semidirect
199. C₄×C₈	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
#19143535	2
#19135576	2
#19135348	2
#19135343	2
#19134767	2
#19134479	2
#19134474	2
C₄₃×#1726	2
#21733	43