Subgroups — #25110287

Order 11008.

614 conjugacy classes of subgroups.

The subgroup lattice has 250 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	#19143529	C₂	1	2	yes	maximal
3	#19135572	C₂	1	2	yes	maximal
4	#19135195	C₂	1	8	yes	maximal
5	#19134733	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#1744	C₂	1	1	yes	maximal, characteristic
7	#14583456	C₂²	1	2	yes	—
8	#14583304	C₄	1	1	yes	characteristic, complemented
9	#14583304	C₄	1	1	yes	characteristic, complemented
10	#14583301	C₄	1	4	yes	complemented
11	#14583297	C₄	1	2	yes	complemented
12	#14583029	—	2	2	no	—
13	#14583029	—	2	2	no	—
14	C₈×Dic₈₆	—	2	2	no	—
15	C₈×Dic₈₆	—	2	2	no	—
16	#14582937	C₂²	1	4	yes	—
17	#14582925	C₂²	1	4	yes	—
18	#14582923	C₂²	1	2	yes	—
19	#14582854	C₂²	1	4	yes	—
20	#14582664	C₂²	1	4	yes	—
21	#14582659	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄³×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#336	C₂²	1	4	yes	—
26	C₂×#10621877	C₂³	1	4	yes	—
27	#13063325	C₂×C₄	1	2	yes	—
28	#13063325	C₂×C₄	1	2	yes	—
29	#13063317	C₂×C₄	1	4	yes	—
30	#13063304	—	2	2	no	—
31	#13063304	—	2	2	no	—
32	C₄×Dic₈₆	—	2	2	no	—
33	C₄×Dic₈₆	—	2	2	no	—
34	#13063284	C₂³	1	4	yes	—
35	#13063259	C₂×C₄	1	4	yes	—
36	#13063259	C₂×C₄	1	4	yes	—
37	#13063257	C₂×C₄	1	8	yes	—
38	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
39	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
40	Dic₄₃⋊C₈	—	2	4	no	—
41	Dic₄₃⋊C₈	—	2	4	no	—
42	#13063235	—	2	4	no	—
43	C₈×Dic₄₃	—	2	4	no	—
44	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₃	—	2	1	no	—
47	C₂×C₄²×C₄₃	—	2	1	no	—
48	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₄₃	D₄	1	2	yes	—
55	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	—	2	2	no	—
57	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₈₆	—	2	2	no	—
61	C₂×Dic₈₆	—	2	2	no	—
62	C₂×C₈×C₄₃	—	2	4	no	—
63	C₂×C₈×C₄₃	—	2	4	no	—
64	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
68	C₄²×C₄₃	—	2	4	no	—
69	C₄²×C₄₃	—	2	4	no	—
70	C₄²×C₄₃	—	4	2	no	—
71	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²⋊Dic₄₃	—	2	4	no	—
76	C₄⋊Dic₄₃	—	2	2	no	—
77	C₄⋊Dic₄₃	—	2	2	no	—
78	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
79	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
80	Dic₄₃⋊C₄	—	2	4	no	—
81	Dic₄₃⋊C₄	—	2	4	no	—
82	C₄×Dic₄₃	—	2	4	no	—
83	#10621877	—	2	8	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
86	C₂×C₄×C₄₃	—	2	2	no	—
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	—	2	4	no	—
89	C₂×C₄×C₄₃	—	4	2	no	—
90	C₂×C₄×C₄₃	—	4	2	no	—
91	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	2₋¹⁺⁴	1	4	yes	—
98	C₂×Dic₄₃	—	2	4	no	—
99	C₂×Dic₄₃	—	2	4	no	—
100	Dic₈₆	—	4	4	no	—
101	Dic₈₆	—	4	4	no	—
102	C₈×C₄₃	—	2	2	no	—
103	C₈×C₄₃	—	2	2	no	—
104	C₈×C₄₃	—	2	2	no	—
105	C₈×C₄₃	—	2	2	no	—
106	C₄₃⋊C₈	—	4	4	no	—
107	#22256	—	43	1	no	maximal, Sylow
108	C₂²×C₄₃	—	2	1	no	—
109	C₂²×C₄₃	—	2	1	no	—
110	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂²×C₄₃	#541	1	2	yes	—
112	C₄×C₄₃	—	2	1	no	—
113	C₄×C₄₃	—	2	1	no	—
114	C₄×C₄₃	—	4	4	no	—
115	C₄×C₄₃	#443	1	2	yes	—
116	C₄×C₄₃	#444	1	2	yes	—
117	C₄×C₄₃	#519	1	2	yes	—
118	C₄×C₄₃	#536	1	2	yes	—
119	Dic₄₃	—	4	4	no	—
120	Dic₄₃	—	4	4	no	—
121	#3160	—	43	1	no	—
122	#2960	—	43	2	no	—
123	#1994	—	43	2	no	—
124	#1922	—	43	8	no	—
125	#1837	—	43	1	no	—
126	#1744	D₄₃	1	1	yes	characteristic
127	C₂×C₄₃	—	2	2	no	—
128	C₂×C₄₃	#2327	1	1	yes	characteristic
129	C₂×C₄₃	C₂×#443	1	1	yes	characteristic
130	C₂×C₄₃	C₂×#444	1	1	yes	characteristic
131	#533	—	43	2	no	—
132	#500	—	43	1	no	—
133	#500	—	43	1	no	—
134	#499	—	43	4	no	—
135	#498	—	43	2	no	—
136	#446	—	86	2	no	—
137	#446	—	86	2	no	—
138	C₈×Q₈	—	86	2	no	—
139	C₈×Q₈	—	86	2	no	—
140	#427	—	43	4	no	—
141	#424	—	43	4	no	—
142	#423	—	43	2	no	—
143	#405	—	43	4	no	—
144	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
146	C₄³	C₂×D₄₃	1	1	yes	characteristic
147	#341	—	43	4	no	—
148	#339	—	43	8	no	—
149	#336	C₂×D₄₃	1	4	yes	—
150	C₄₃	#22256	1	1	yes	characteristic, complemented, Sylow
151	C₂×M₁₆	—	43	4	no	—
152	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
153	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
154	C₄⋊Q₈	—	43	2	no	—
155	C₄⋊Q₈	—	43	2	no	—
156	#125	—	43	4	no	—
157	C₂²⋊Q₈	—	86	2	no	—
158	C₂²⋊Q₈	—	86	2	no	—
159	C₄×Q₈	—	86	2	no	—
160	C₄×Q₈	—	86	2	no	—
161	#117	—	43	4	no	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
165	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
166	#107	—	43	4	no	—
167	#107	—	43	4	no	—
168	#106	—	43	8	no	—
169	C₄⋊C₈	—	86	4	no	—
170	C₄⋊C₈	—	86	4	no	—
171	C₄⋊C₈	—	86	2	no	—
172	C₄⋊C₈	—	86	2	no	—
173	#97	—	86	4	no	—
174	C₄×C₈	—	86	4	no	—
175	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
176	C₄×C₈	C₄×D₄₃	1	2	yes	—
177	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
178	C₄×C₈	D₁₇₂	1	1	yes	characteristic
179	C₄×C₈	D₁₇₂	1	2	yes	—
180	C₄×C₈	D₁₇₂	1	1	yes	characteristic
181	C₂×Q₈	—	86	2	no	—
182	C₂×Q₈	—	86	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	#10621898	1	1	yes	characteristic
185	C₂²×C₄	#10621898	1	1	yes	characteristic
186	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
187	M₁₆	—	86	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	86	4	no	—
191	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
192	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
193	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
194	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
195	C₄⋊C₄	—	43	4	no	—
196	C₄⋊C₄	—	43	4	no	—
197	C₄⋊C₄	—	86	4	no	—
198	C₄⋊C₄	—	86	4	no	—
199	C₄⋊C₄	—	86	2	no	—
200	C₄⋊C₄	—	86	2	no	—
201	C₂²⋊C₄	—	86	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	4	2	no	—
205	C₄²	—	86	4	no	—
206	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
207	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
208	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
209	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
210	C₂³	C₂×#10621898	1	1	yes	characteristic
211	Q₈	—	172	4	no	—
212	Q₈	—	172	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	86	4	no	—
219	C₂×C₄	—	86	4	no	—
220	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
221	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
222	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
223	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
226	C₂×C₄	#13063372	1	4	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	172	4	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
235	C₂²	#14583511	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	4	no	—
239	C₄	—	172	4	no	—
240	C₄	—	172	4	no	—
241	C₄	#14583017	1	2	yes	—
242	C₄	#14583022	1	2	yes	—
243	C₄	#14583376	1	2	yes	—
244	C₄	#14583475	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	#19137314	1	1	yes	characteristic
247	C₂	C₂×#14583017	1	1	yes	characteristic
248	C₂	C₂×#14583022	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1744	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1744	D₄₃
hypocenter	C₄₃	#22256
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1744

Complements

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

N	H ≅ G/N	type
8. #14583304	C₄	semidirect
9. #14583304	C₄	semidirect
10. #14583301	C₄	semidirect
11. #14583297	C₄	semidirect
150. C₄₃	#22256	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
#19143529	2
#19135572	2
#19135195	2
#19134733	2
C₄₃×#1744	2
#22256	43