Subgroups — #25110304

Order 11008.

594 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145117	C₂	1	1	yes	maximal, characteristic
2	#19143536	C₂	1	1	yes	maximal, characteristic
3	#19143532	C₂	1	1	yes	maximal, characteristic
4	#19135572	C₂	1	1	yes	maximal, characteristic
5	#19135197	C₂	1	4	yes	maximal
6	#19135192	C₂	1	4	yes	maximal
7	#19134735	C₂	1	1	yes	maximal, characteristic
8	#19134344	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1994	C₂	1	1	yes	maximal, characteristic
10	#14583456	C₂²	1	1	yes	characteristic
11	#14583309	C₄	1	4	yes	complemented
12	#14583305	C₄	1	4	yes	complemented
13	#14583030	—	2	4	no	—
14	#14583028	—	2	4	no	—
15	#14582935	C₂²	1	4	yes	—
16	#14582927	C₂²	1	2	yes	—
17	#14582926	C₂²	1	2	yes	—
18	#14582923	C₂²	1	1	yes	characteristic
19	#14582857	C₂²	1	2	yes	—
20	#14582855	C₂²	1	2	yes	—
21	C₂×#13063236	C₂²	1	1	yes	characteristic
22	C₂×#13063236	C₂²	1	1	yes	characteristic
23	#14582663	C₂²	1	4	yes	—
24	#14582659	C₂²	1	4	yes	—
25	#14582654	C₂²	1	2	yes	—
26	#14582654	C₂²	1	2	yes	—
27	C₄₃×#533	C₂²	1	1	yes	characteristic
28	C₄₃×#423	C₂²	1	1	yes	characteristic
29	C₄³×C₄₃	C₂²	1	1	yes	characteristic
30	C₄₃×#339	C₂²	1	4	yes	—
31	C₂×#10621877	C₂³	1	2	yes	—
32	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
35	#13063325	C₂×C₄	1	2	yes	—
36	#13063317	C₂×C₄	1	2	yes	—
37	#13063304	—	2	2	no	—
38	C₄×Dic₈₆	—	2	2	no	—
39	#13063284	C₂³	1	2	yes	—
40	#13063258	C₂×C₄	1	8	yes	—
41	#13063256	C₂×C₄	1	8	yes	—
42	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
43	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
44	Dic₄₃⋊C₈	—	2	4	no	—
45	#13063236	—	2	2	no	—
46	#13063236	C₂×C₄	1	2	yes	—
47	#13063236	C₂×C₄	1	2	yes	—
48	#13063236	D₄	1	4	yes	—
49	#13063235	—	2	2	no	—
50	C₄×C₄₃⋊C₈	—	2	2	no	—
51	C₈×Dic₄₃	—	2	2	no	—
52	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
54	#125×C₄₃	C₂×C₄	1	2	yes	—
55	C₂²⋊Q₈×C₄₃	—	2	2	no	—
56	C₄×Q₈×C₄₃	—	2	2	no	—
57	#117×C₄₃	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₃	—	2	2	no	—
59	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₃	—	2	2	no	—
62	C₂×Q₈×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	—	2	2	no	—
64	C₂²×C₄×C₄₃	#41	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₈₆	—	2	2	no	—
67	M₁₆×C₄₃	—	2	4	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₄⋊C₄×C₄₃	—	2	4	no	—
70	C₄⋊C₄×C₄₃	—	2	2	no	—
71	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
72	C₂²⋊C₄×C₄₃	—	2	2	no	—
73	C₄²×C₄₃	—	2	2	no	—
74	C₄²×C₄₃	—	2	2	no	—
75	C₄²×C₄₃	—	2	2	no	—
76	C₄²×C₄₃	—	4	4	no	—
77	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₄₃	C₂×D₄	1	2	yes	—
80	C₂²⋊Dic₄₃	—	2	2	no	—
81	C₄⋊Dic₄₃	—	2	2	no	—
82	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
83	Dic₄₃⋊C₄	—	2	4	no	—
84	C₄×Dic₄₃	—	2	2	no	—
85	#10621877	—	2	4	no	—
86	C₂×C₄₃⋊C₈	—	2	2	no	—
87	C₂×C₄₃⋊C₈	—	2	2	no	—
88	C₂×C₄₃⋊C₈	—	2	2	no	—
89	C₂×C₄₃⋊C₈	—	2	2	no	—
90	C₂×C₄₃⋊C₈	—	2	2	no	—
91	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
92	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
93	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
94	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
95	Q₈×C₄₃	—	8	2	no	—
96	C₂×C₄×C₄₃	—	2	4	no	—
97	C₂×C₄×C₄₃	—	2	4	no	—
98	C₂×C₄×C₄₃	—	4	2	no	—
99	C₂×C₄×C₄₃	—	4	2	no	—
100	C₂×C₄×C₄₃	—	4	2	no	—
101	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
106	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
107	C₂×Dic₄₃	—	2	4	no	—
108	Dic₈₆	—	8	2	no	—
109	C₈×C₄₃	—	4	2	no	—
110	C₄₃⋊C₈	—	2	4	no	—
111	C₄₃⋊C₈	—	2	2	no	—
112	C₄₃⋊C₈	—	2	2	no	—
113	C₄₃⋊C₈	—	4	2	no	—
114	#22259	—	43	1	no	maximal, Sylow
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
117	C₂²×C₄₃	#541	1	2	yes	—
118	C₄×C₄₃	—	2	2	no	—
119	C₄×C₄₃	—	2	2	no	—
120	C₄×C₄₃	—	4	4	no	—
121	C₄×C₄₃	—	4	4	no	—
122	C₄×C₄₃	#519	1	2	yes	—
123	C₄×C₄₃	#536	1	1	yes	characteristic
124	C₄×C₄₃	#536	1	1	yes	characteristic
125	Dic₄₃	—	4	4	no	—
126	#3162	—	43	1	no	—
127	#2961	—	43	1	no	—
128	#2961	—	43	1	no	—
129	#1994	—	43	1	no	—
130	#1994	D₄₃	1	1	yes	characteristic
131	#1922	—	43	4	no	—
132	#1922	—	43	4	no	—
133	#1838	—	43	1	no	—
134	#1745	—	43	1	no	—
135	C₂×C₄₃	—	2	2	no	—
136	C₂×C₄₃	#2327	1	1	yes	characteristic
137	C₂×C₄₃	#2956	1	2	yes	—
138	#533	—	43	1	no	—
139	#533	C₂×D₄₃	1	1	yes	characteristic
140	#501	—	43	4	no	—
141	#501	—	43	4	no	—
142	#446	—	86	4	no	—
143	#446	—	86	4	no	—
144	#427	—	43	4	no	—
145	#424	—	43	2	no	—
146	#424	—	43	2	no	—
147	#423	—	43	1	no	—
148	#423	C₂×D₄₃	1	1	yes	characteristic
149	#405	—	43	2	no	—
150	#405	—	43	2	no	—
151	C₂×#97	—	43	1	no	—
152	C₂×#97	—	43	1	no	—
153	C₄³	C₂×D₄₃	1	1	yes	characteristic
154	#341	—	43	4	no	—
155	#339	—	43	4	no	—
156	#339	C₂×D₄₃	1	4	yes	—
157	#336	—	43	2	no	—
158	#336	—	43	2	no	—
159	C₄₃	#22259	1	1	yes	characteristic, complemented, Sylow
160	C₂×M₁₆	—	43	2	no	—
161	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
162	C₂²×C₈	—	43	2	no	—
163	C₂²×C₈	—	43	1	no	—
164	C₂²×C₈	—	43	1	no	—
165	C₄⋊Q₈	—	43	2	no	—
166	C₄⋊Q₈	C₄×D₄₃	1	2	yes	—
167	#125	—	43	2	no	—
168	#125	C₄×D₄₃	1	2	yes	—
169	C₂²⋊Q₈	—	2	2	no	—
170	C₂²⋊Q₈	—	86	2	no	—
171	C₄×Q₈	—	2	2	no	—
172	C₄×Q₈	—	86	2	no	—
173	#117	—	43	2	no	—
174	#117	C₂²×D₄₃	1	2	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
177	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
178	#107	—	43	8	no	—
179	#106	—	43	8	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	86	4	no	—
182	C₄⋊C₈	—	86	4	no	—
183	C₄⋊C₈	—	86	2	no	—
184	#97	—	43	2	no	—
185	#97	—	43	2	no	—
186	#97	—	43	4	no	—
187	#97	—	86	2	no	—
188	#97	—	86	2	no	—
189	C₄×C₈	—	86	2	no	—
190	C₄×C₈	—	86	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	86	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	#10621900	1	2	yes	—
195	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
196	M₁₆	—	2	4	no	—
197	M₁₆	—	86	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	43	4	no	—
200	C₂×C₈	—	43	2	no	—
201	C₂×C₈	—	43	2	no	—
202	C₂×C₈	—	86	2	no	—
203	C₂×C₈	—	86	2	no	—
204	C₂×C₈	—	86	2	no	—
205	C₂×C₈	—	86	2	no	—
206	C₂×C₈	—	86	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	43	4	no	—
210	C₄⋊C₄	—	86	4	no	—
211	C₄⋊C₄	—	86	2	no	—
212	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	86	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	4	no	—
219	C₄²	—	86	2	no	—
220	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
221	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
222	C₄²	D₄×D₄₃	1	2	yes	—
223	C₂³	C₂×#10621900	1	1	yes	characteristic
224	Q₈	—	8	2	no	—
225	Q₈	—	344	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	86	4	no	—
232	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
233	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
234	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
236	C₂×C₄	#13063372	1	2	yes	—
237	C₂×C₄	#13063373	1	2	yes	—
238	C₈	—	4	2	no	—
239	C₈	—	86	4	no	—
240	C₈	—	86	2	no	—
241	C₈	—	86	2	no	—
242	C₈	—	172	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
245	C₂²	#14583509	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	4	4	no	—
249	C₄	—	4	4	no	—
250	C₄	—	172	4	no	—
251	C₄	#14583375	1	2	yes	—
252	C₄	#14583469	1	1	yes	characteristic
253	C₄	#14583474	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	#19137313	1	1	yes	characteristic
256	C₂	#19143501	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1994	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1994	D₄₃
hypocenter	C₄₃	#22259
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1994

Complements

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

N	H ≅ G/N	type
11. #14583309	C₄	semidirect
12. #14583305	C₄	semidirect
159. C₄₃	#22259	semidirect

Maximal subgroups

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

subgroup	index
#19145117	2
#19143536	2
#19143532	2
#19135572	2
#19135197	2
#19135192	2
#19134735	2
#19134344	2
C₄₃×#1994	2
#22259	43