Subgroups — #25115535

Order 11008.

822 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	C₂×C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×Dic₈₆	C₂	1	2	yes	maximal
3	#19135358	C₂	1	2	yes	maximal
4	#19135355	C₂	1	4	yes	maximal
5	#19134723	C₂	1	1	yes	maximal, characteristic
6	#19134721	C₂	1	2	yes	maximal
7	#19134483	C₂	1	2	yes	maximal
8	C₄×#106×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₈₆	—	2	2	no	—
10	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
11	C₈×Dic₈₆	C₄	1	8	yes	complemented
12	C₂×#13063256	C₂²	1	4	yes	—
13	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
16	#14582797	—	2	2	no	—
17	#14582796	—	2	4	no	—
18	#14582769	C₂²	1	2	yes	—
19	#14582768	C₂²	1	4	yes	—
20	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
21	#14582662	C₂²	1	2	yes	—
22	#14582662	C₂²	1	2	yes	—
23	#14582661	C₂²	1	4	yes	—
24	C₂×#106×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
27	C₄₃×#340	C₂²	1	1	yes	characteristic
28	C₄₃×#340	C₂²	1	1	yes	characteristic
29	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
30	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
31	C₄×Dic₈₆	—	4	8	no	—
32	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
33	C₄×Dic₈₆	Q₈	1	4	yes	—
34	C₂×C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄⋊Dic₄₃	—	2	2	no	—
36	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
38	C₂×Dic₄₃⋊C₄	—	2	4	no	—
39	C₂×Dic₄₃⋊C₄	—	2	4	no	—
40	C₂×Dic₄₃⋊C₄	D₄	1	2	yes	—
41	C₂×C₄×Dic₄₃	—	2	4	no	—
42	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
43	#13063256	—	2	8	no	—
44	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
45	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
46	C₈×Dic₄₃	C₂×C₄	1	8	yes	—
47	C₄₃⋊#95	—	2	4	no	—
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₄₃	—	2	2	no	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	#106×C₄₃	—	2	4	no	—
55	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
56	#95×C₄₃	—	2	2	no	—
57	C₂²×C₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₄₃	—	2	4	no	—
63	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
64	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
65	C₂×Dic₈₆	C₄⋊C₄	1	4	yes	—
66	C₂×Dic₈₆	SD₁₆	1	8	yes	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
71	C₄⋊C₄×C₄₃	—	2	2	no	—
72	C₄⋊C₄×C₄₃	—	2	2	no	—
73	C₄⋊C₄×C₄₃	—	2	2	no	—
74	C₄⋊C₄×C₄₃	#41	1	4	yes	—
75	C₄²×C₄₃	—	4	4	no	—
76	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	C₄⋊Dic₄₃	—	2	4	no	—
80	C₄⋊Dic₄₃	—	4	4	no	—
81	C₄⋊Dic₄₃	—	8	2	no	—
82	C₄⋊Dic₄₃	C₄⋊C₄	1	4	yes	—
83	Dic₄₃⋊C₄	—	4	8	no	—
84	Dic₄₃⋊C₄	—	8	4	no	—
85	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
86	C₄×Dic₄₃	—	4	8	no	—
87	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
88	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
89	C₂×C₄₃⋊C₈	—	2	4	no	—
90	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
91	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₃	—	2	1	no	—
93	C₂×C₄×C₄₃	—	2	1	no	—
94	C₂×C₄×C₄₃	—	2	4	no	—
95	C₂×C₄×C₄₃	—	4	2	no	—
96	C₂×C₄×C₄₃	—	4	2	no	—
97	C₂×C₄×C₄₃	—	4	2	no	—
98	C₂×C₄×C₄₃	—	4	2	no	—
99	C₂×C₄×C₄₃	—	4	2	no	—
100	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
103	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
106	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
107	C₂×Dic₄₃	—	2	2	no	—
108	C₂×Dic₄₃	—	2	2	no	—
109	C₂×Dic₄₃	—	4	4	no	—
110	C₂×Dic₄₃	—	4	4	no	—
111	C₂×Dic₄₃	—	4	4	no	—
112	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×Dic₄₃	C₂×SD₁₆	1	8	yes	—
115	Dic₈₆	#106	1	16	yes	complemented
116	C₈×C₄₃	—	2	2	no	—
117	C₈×C₄₃	—	2	2	no	—
118	C₄₃⋊C₈	—	2	4	no	—
119	#22738	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₃	#548	1	2	yes	—
122	C₂²×C₄₃	C₂²×SD₁₆	1	2	yes	—
123	C₂²×C₄₃	#576	1	1	yes	characteristic
124	C₂²×C₄₃	#578	1	1	yes	characteristic
125	C₄×C₄₃	—	2	2	no	—
126	C₄×C₄₃	—	4	4	no	—
127	C₄×C₄₃	—	8	2	no	—
128	C₄×C₄₃	C₂×#106	1	4	yes	—
129	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
130	C₄×C₄₃	#519	1	1	yes	characteristic
131	C₄×C₄₃	#557	1	2	yes	—
132	Dic₄₃	—	2	4	no	—
133	Dic₄₃	—	8	4	no	—
134	Dic₄₃	C₂×#106	1	8	yes	—
135	C₂×C₈×Q₈	—	43	1	no	—
136	Q₈×C₄⋊C₄	—	43	2	no	—
137	#1948	—	43	4	no	—
138	#1948	—	43	2	no	—
139	#1834	—	43	1	no	—
140	#1834	—	43	2	no	—
141	C₄×#106	—	43	2	no	—
142	C₄×#106	D₄₃	1	1	yes	characteristic
143	C₂×C₄₃	#2351	1	1	yes	characteristic
144	C₂×C₄₃	C₂²×#106	1	1	yes	characteristic
145	C₂×C₄₃	#2913	1	1	yes	characteristic
146	C₂×C₄₃	#3359	1	2	yes	—
147	C₂×C₄₃	#3362	1	2	yes	—
148	C₂×C₄×Q₈	—	43	1	no	—
149	C₂×C₄×Q₈	—	86	2	no	—
150	C₈×Q₈	—	43	8	no	—
151	C₂×#106	—	43	4	no	—
152	C₂×#106	C₂×D₄₃	1	2	yes	—
153	C₂×C₄⋊C₈	—	43	2	no	—
154	C₂×C₄⋊C₈	—	43	1	no	—
155	C₂×C₄×C₈	—	43	2	no	—
156	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
157	#389	—	86	4	no	—
158	#389	—	86	2	no	—
159	#384	—	43	4	no	—
160	#384	—	43	2	no	—
161	C₄×C₄⋊C₄	—	43	4	no	—
162	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
163	#340	—	43	2	no	—
164	#340	—	43	2	no	—
165	#340	—	43	4	no	—
166	#340	C₂×D₄₃	1	1	yes	characteristic
167	#340	C₂×D₄₃	1	1	yes	characteristic
168	C₄₃	#22738	1	1	yes	characteristic, complemented, Sylow
169	C₂²×Q₈	—	43	1	no	—
170	C₂²×C₈	—	43	2	no	—
171	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
173	C₄×Q₈	—	43	4	no	—
174	C₄×Q₈	—	43	4	no	—
175	C₄×Q₈	—	172	8	no	—
176	C₂×C₄⋊C₄	—	43	2	no	—
177	C₂×C₄⋊C₄	—	43	1	no	—
178	C₂×C₄⋊C₄	—	43	4	no	—
179	C₂×C₄⋊C₄	—	86	4	no	—
180	C₂×C₄⋊C₄	—	86	2	no	—
181	C₂×C₄⋊C₄	—	86	4	no	—
182	C₂×C₄⋊C₄	—	86	2	no	—
183	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	43	2	no	—
187	C₂×C₄²	—	86	4	no	—
188	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
189	#106	—	2	4	no	—
190	#106	—	86	8	no	—
191	C₄⋊C₈	—	43	8	no	—
192	C₄⋊C₈	—	43	4	no	—
193	C₄×C₈	—	43	8	no	—
194	C₄×C₈	C₄×D₄₃	1	4	yes	—
195	#95	—	2	2	no	—
196	#95	—	86	4	no	—
197	C₂×Q₈	—	43	4	no	—
198	C₂×Q₈	—	43	8	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	43	2	no	—
202	C₂²×C₄	—	43	2	no	—
203	C₂²×C₄	—	86	4	no	—
204	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
206	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	43	4	no	—
210	C₂×C₈	—	86	4	no	—
211	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	43	8	no	—
217	C₄⋊C₄	—	43	4	no	—
218	C₄⋊C₄	—	86	4	no	—
219	C₄⋊C₄	—	86	4	no	—
220	C₄⋊C₄	—	344	2	no	—
221	C₄⋊C₄	—	344	4	no	—
222	C₄⋊C₄	—	172	8	no	—
223	C₄⋊C₄	—	172	4	no	—
224	C₄⋊C₄	#10621898	1	4	yes	—
225	C₄²	—	4	4	no	—
226	C₄²	—	43	4	no	—
227	C₄²	—	43	4	no	—
228	C₄²	—	172	8	no	—
229	C₄²	C₂×C₄×D₄₃	1	2	yes	—
230	C₄²	Q₈×D₄₃	1	2	yes	—
231	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
232	Q₈	—	43	16	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	43	4	no	—
242	C₂×C₄	—	43	8	no	—
243	C₂×C₄	—	43	4	no	—
244	C₂×C₄	—	86	2	no	—
245	C₂×C₄	—	86	2	no	—
246	C₂×C₄	—	172	4	no	—
247	C₂×C₄	—	172	4	no	—
248	C₂×C₄	—	172	4	no	—
249	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
250	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
251	C₂×C₄	SD₁₆×D₄₃	1	4	yes	—
252	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
253	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#10621898	1	2	yes	—
255	C₂×C₄	#13063372	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	86	4	no	—
259	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
260	C₂²	#14583558	1	2	yes	—
261	C₂²	C₂×SD₁₆×D₄₃	1	2	yes	—
262	C₂²	#14583701	1	1	yes	characteristic
263	C₂²	#14583713	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	4	4	no	—
266	C₄	—	8	2	no	—
267	C₄	—	43	8	no	—
268	C₄	—	86	4	no	—
269	C₄	—	344	4	no	—
270	C₄	#106×D₄₃	1	4	yes	—
271	C₄	C₄×#10621898	1	1	yes	characteristic
272	C₄	#14583376	1	1	yes	characteristic
273	C₄	#14583627	1	2	yes	—
274	C₂	#19137515	1	1	yes	characteristic
275	C₂	C₂×#106×D₄₃	1	1	yes	characteristic
276	C₂	#19143192	1	1	yes	characteristic
277	C₂	#19147235	1	2	yes	—
278	C₂	#19147264	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#106×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#106	D₄₃
hypocenter	C₄₃	#22738
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₈₆	C₄	semidirect
115. Dic₈₆	#106	semidirect
168. C₄₃	#22738	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₈₆	2
C₄⋊C₄×Dic₈₆	2
#19135358	2
#19135355	2
#19134723	2
#19134721	2
#19134483	2
C₄×#106×C₄₃	2
#22738	43