Subgroups — #25015149

Order 11008.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 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	#19134474	C₂	1	1	yes	maximal, characteristic
3	#19132710	C₂	1	2	yes	maximal
4	#19132697	C₂	1	2	yes	maximal
5	C₂×C₄₃×#364	C₂	1	1	yes	maximal, characteristic
6	#14583456	—	2	1	no	—
7	#14583304	—	2	1	no	—
8	#14583301	—	2	1	no	—
9	#14583301	C₄	1	2	yes	complemented
10	#14583297	C₄	1	2	yes	complemented
11	C₂×#13063261	C₂²	1	1	yes	characteristic
12	#14582937	—	2	1	no	—
13	#14582937	C₂²	1	2	yes	—
14	#14582937	C₂²	1	1	yes	characteristic
15	#14582854	—	2	1	no	—
16	#14582659	—	2	1	no	—
17	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#364	—	2	1	no	—
20	C₄₃×#364	—	2	1	no	—
21	C₂×#10621877	—	2	1	no	—
22	#13063325	—	2	2	no	—
23	#13063317	—	2	2	no	—
24	#13063304	—	4	2	no	—
25	C₄×Dic₈₆	—	4	2	no	—
26	#13063284	—	2	1	no	—
27	#13063284	—	2	2	no	—
28	#13063261	—	2	1	no	—
29	#13063261	Q₈	1	2	yes	complemented
30	#13063259	—	2	2	no	—
31	#13063259	—	2	2	no	—
32	#13063259	—	2	1	no	—
33	#13063257	—	2	2	no	—
34	#13063257	C₂×C₄	1	4	yes	—
35	#13063257	Q₈	1	2	yes	complemented
36	#13063235	—	4	1	no	—
37	C₈×Dic₄₃	—	4	1	no	—
38	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
41	M₃₂×C₄₃	—	2	2	no	—
42	M₃₂×C₄₃	—	2	2	no	—
43	C₂×C₁₆×C₄₃	—	2	2	no	—
44	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	—	2	1	no	—
49	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₈₆	—	4	2	no	—
51	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₄₃	Q₁₆	1	1	yes	characteristic
56	C₂×C₈×C₄₃	Q₁₆	1	1	yes	characteristic
57	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
62	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
63	C₄²×C₄₃	—	2	1	no	—
64	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²⋊Dic₄₃	—	4	1	no	—
67	C₂²⋊Dic₄₃	—	4	2	no	—
68	C₄⋊Dic₄₃	—	2	4	no	—
69	C₄⋊Dic₄₃	—	2	2	no	—
70	C₄⋊Dic₄₃	—	4	2	no	—
71	Dic₄₃⋊C₄	—	4	4	no	—
72	C₄×Dic₄₃	—	4	1	no	—
73	C₄×Dic₄₃	—	4	2	no	—
74	#10621877	—	2	2	no	—
75	#10621877	—	4	1	no	—
76	C₂×C₄₃⋊C₈	—	4	1	no	—
77	C₁₆×C₄₃	—	4	2	no	—
78	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₄₃	—	2	1	no	—
80	C₂×C₄×C₄₃	—	2	1	no	—
81	C₂×C₄×C₄₃	—	4	2	no	—
82	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×Dic₄₃	—	4	4	no	—
89	C₂×Dic₄₃	—	4	2	no	—
90	Dic₈₆	—	8	4	no	—
91	C₈×C₄₃	#103	1	1	yes	characteristic
92	C₈×C₄₃	#103	1	1	yes	characteristic
93	C₈×C₄₃	#103	1	1	yes	characteristic
94	C₈×C₄₃	#103	1	1	yes	characteristic
95	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
96	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₄₃	#124	1	1	yes	characteristic
98	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
99	C₄₃⋊C₈	—	4	2	no	—
100	#13271	—	43	1	no	maximal, Sylow
101	C₂²×C₄₃	—	4	1	no	—
102	C₂²×C₄₃	#418	1	1	yes	characteristic
103	C₂²×C₄₃	#476	1	1	yes	characteristic
104	C₂²×C₄₃	#477	1	1	yes	characteristic
105	C₄×C₄₃	—	2	2	no	—
106	C₄×C₄₃	—	4	1	no	—
107	C₄×C₄₃	#391	1	1	yes	characteristic
108	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
109	C₄×C₄₃	#476	1	1	yes	characteristic
110	C₄×C₄₃	#477	1	1	yes	characteristic
111	Dic₄₃	—	8	4	no	—
112	Dic₄₃	—	8	2	no	—
113	#3160	—	43	1	no	—
114	C₂×#364	D₄₃	1	1	yes	characteristic
115	#1772	—	43	1	no	—
116	#1389	—	43	2	no	—
117	#1384	—	43	2	no	—
118	C₂×C₄₃	—	2	1	no	—
119	C₂×C₄₃	—	2	1	no	—
120	C₂×C₄₃	—	2	1	no	—
121	C₂×C₄₃	#1987	1	1	yes	characteristic
122	#533	—	86	1	no	—
123	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
124	#500	—	86	1	no	—
125	#499	—	43	2	no	—
126	#499	—	86	1	no	—
127	#498	—	43	2	no	—
128	C₂×#108	—	43	1	no	—
129	#427	—	43	2	no	—
130	#427	—	43	1	no	—
131	#427	—	86	1	no	—
132	#405	—	86	1	no	—
133	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
134	#364	—	2	1	no	—
135	#364	—	2	1	no	—
136	#339	—	86	1	no	—
137	C₄₃	#13271	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	—	86	1	no	—
139	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
141	C₄⋊Q₈	—	86	2	no	—
142	#125	—	86	2	no	—
143	C₂²⋊Q₈	—	172	2	no	—
144	C₄×Q₈	—	172	2	no	—
145	#117	—	86	2	no	—
146	#117	—	86	1	no	—
147	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#108	—	43	2	no	—
152	#108	—	86	1	no	—
153	#107	—	86	2	no	—
154	#107	—	86	1	no	—
155	#107	—	86	2	no	—
156	#106	—	43	4	no	—
157	#106	—	43	2	no	—
158	#106	—	86	2	no	—
159	#97	—	172	1	no	—
160	C₄×C₈	—	172	1	no	—
161	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
162	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
165	C₂×Q₈	—	172	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
168	M₁₆	—	86	2	no	—
169	M₁₆	—	172	1	no	—
170	C₂×C₈	—	172	1	no	—
171	C₂×C₈	#10621873	1	1	yes	characteristic
172	C₂×C₈	#10621873	1	1	yes	characteristic
173	C₂×C₈	Dic₁₇₂	1	1	yes	characteristic
174	C₂×C₈	Dic₁₇₂	1	1	yes	characteristic
175	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
176	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
178	C₂×C₈	#10621900	1	1	yes	characteristic
179	C₂×C₈	#10621900	1	1	yes	characteristic
180	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
181	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
182	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
183	C₄⋊C₄	—	86	4	no	—
184	C₄⋊C₄	—	86	2	no	—
185	C₄⋊C₄	—	172	4	no	—
186	C₄⋊C₄	—	172	2	no	—
187	C₂²⋊C₄	—	172	2	no	—
188	C₂²⋊C₄	—	172	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	172	1	no	—
191	C₄²	—	172	2	no	—
192	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
193	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
194	C₁₆	—	4	2	no	—
195	C₂³	#13063353	1	1	yes	characteristic
196	Q₈	—	344	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	172	4	no	—
201	C₂×C₄	—	172	2	no	—
202	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#13063305	1	1	yes	characteristic
204	C₂×C₄	#13063305	1	1	yes	characteristic
205	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
206	C₂×C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
207	C₂×C₄	#13063353	1	1	yes	characteristic
208	C₈	—	172	2	no	—
209	C₈	#13063250	1	1	yes	characteristic
210	C₈	#13063250	1	1	yes	characteristic
211	C₈	#13063250	1	1	yes	characteristic
212	C₈	#13063250	1	1	yes	characteristic
213	C₈	#13063290	1	1	yes	characteristic
214	C₈	#13063290	1	1	yes	characteristic
215	C₈	#13063312	1	1	yes	characteristic
216	C₈	#13063323	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#14582905	1	1	yes	characteristic
219	C₂²	#14583190	1	1	yes	characteristic
220	C₂²	#14583195	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	344	4	no	—
224	C₄	—	344	2	no	—
225	C₄	#14582804	1	1	yes	characteristic
226	C₄	C₂×#13063250	1	1	yes	characteristic
227	C₄	#14583190	1	1	yes	characteristic
228	C₄	#14583195	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#19135542	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14583190
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#364	C₂
socle	C₂×C₄₃	#1987
prime radical	C₂³×C₄₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#364	D₄₃
hypocenter	C₄₃	#13271
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
9. #14583301	C₄	semidirect
10. #14583297	C₄	semidirect
29. #13063261	Q₈	semidirect
35. #13063257	Q₈	semidirect
137. C₄₃	#13271	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
#19134474	2
#19132710	2
#19132697	2
C₂×C₄₃×#364	2
#13271	43