Subgroups — #25115540

Order 11008.

734 conjugacy classes of subgroups.

The subgroup lattice has 274 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	#19137550	C₂	1	2	yes	maximal
3	#19135364	C₂	1	4	yes	maximal
4	#19135353	C₂	1	2	yes	maximal
5	#19134724	C₂	1	2	yes	maximal
6	#19134723	C₂	1	1	yes	maximal, characteristic
7	#19134487	C₂	1	2	yes	maximal
8	C₄×#106×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
10	C₈×Dic₈₆	C₄	1	8	yes	complemented
11	C₂×#13063258	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
15	#14582805	—	2	2	no	—
16	#14582786	—	2	4	no	—
17	#14582769	C₂²	1	2	yes	—
18	#14582764	C₂²	1	4	yes	—
19	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
20	C₄×C₄⋊Dic₄₃	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
32	C₄×Dic₈₆	Q₈	1	4	yes	—
33	C₂×C₄⋊Dic₄₃	—	2	2	no	—
34	C₂×C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
37	C₂×Dic₄₃⋊C₄	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₃	—	2	4	no	—
39	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
40	#13063258	—	2	8	no	—
41	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
42	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
43	C₈×Dic₄₃	C₂×C₄	1	8	yes	—
44	C₄₃⋊#95	—	2	4	no	—
45	C₄₃⋊#95	—	2	4	no	—
46	C₄₃⋊#95	—	2	4	no	—
47	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₄₃	—	2	2	no	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	#106×C₄₃	—	2	4	no	—
54	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
55	#95×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	—	2	2	no	—
57	C₂²×C₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₃	—	2	4	no	—
62	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
63	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
64	C₂×Dic₈₆	—	2	4	no	—
65	C₂×Dic₈₆	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₄₃	—	2	2	no	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
70	C₄⋊C₄×C₄₃	—	2	2	no	—
71	C₄⋊C₄×C₄₃	—	2	2	no	—
72	C₄⋊C₄×C₄₃	—	2	2	no	—
73	C₄⋊C₄×C₄₃	#41	1	4	yes	—
74	C₄²×C₄₃	—	4	4	no	—
75	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
77	C₄⋊Dic₄₃	—	2	4	no	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	C₄⋊Dic₄₃	—	4	4	no	—
80	C₄⋊Dic₄₃	—	8	2	no	—
81	C₄⋊Dic₄₃	C₄⋊C₄	1	4	yes	—
82	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
83	C₄×Dic₄₃	—	8	4	no	—
84	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
85	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
86	C₂×C₄₃⋊C₈	—	2	4	no	—
87	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
88	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₃	—	2	1	no	—
90	C₂×C₄×C₄₃	—	2	1	no	—
91	C₂×C₄×C₄₃	—	2	4	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	—	4	2	no	—
94	C₂×C₄×C₄₃	—	4	2	no	—
95	C₂×C₄×C₄₃	—	4	2	no	—
96	C₂×C₄×C₄₃	—	4	2	no	—
97	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₃	#135	1	4	yes	—
100	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
103	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
104	C₂×Dic₄₃	—	2	4	no	—
105	C₂×Dic₄₃	—	2	2	no	—
106	C₂×Dic₄₃	—	2	2	no	—
107	C₂×Dic₄₃	—	4	4	no	—
108	C₂×Dic₄₃	—	4	4	no	—
109	C₂×Dic₄₃	—	4	4	no	—
110	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
112	Dic₈₆	—	2	8	no	—
113	C₈×C₄₃	—	2	2	no	—
114	C₈×C₄₃	—	2	2	no	—
115	C₄₃⋊C₈	—	2	4	no	—
116	#22739	—	43	1	no	maximal, Sylow
117	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₄₃	#548	1	2	yes	—
119	C₂²×C₄₃	C₂×#135	1	2	yes	—
120	C₂²×C₄₃	#576	1	1	yes	characteristic
121	C₂²×C₄₃	#578	1	1	yes	characteristic
122	C₄×C₄₃	—	2	2	no	—
123	C₄×C₄₃	—	4	4	no	—
124	C₄×C₄₃	—	8	2	no	—
125	C₄×C₄₃	#427	1	4	yes	—
126	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
127	C₄×C₄₃	#519	1	1	yes	characteristic
128	C₄×C₄₃	#557	1	2	yes	—
129	Dic₄₃	—	2	4	no	—
130	Dic₄₃	—	2	4	no	—
131	Dic₄₃	—	8	4	no	—
132	C₂×C₈×Q₈	—	43	1	no	—
133	#2355	—	43	2	no	—
134	#1949	—	43	4	no	—
135	#1947	—	43	2	no	—
136	#1835	—	43	2	no	—
137	#1834	—	43	1	no	—
138	C₄×#107	—	43	2	no	—
139	C₄×#106	D₄₃	1	1	yes	characteristic
140	C₂×C₄₃	#2351	1	1	yes	characteristic
141	C₂×C₄₃	C₂×#427	1	1	yes	characteristic
142	C₂×C₄₃	#2913	1	1	yes	characteristic
143	C₂×C₄₃	#3361	1	2	yes	—
144	C₂×C₄₃	#3364	1	2	yes	—
145	C₂×C₄×Q₈	—	43	1	no	—
146	C₈×Q₈	—	43	8	no	—
147	C₂×#107	—	43	4	no	—
148	C₂×#106	C₂×D₄₃	1	2	yes	—
149	C₂×C₄⋊C₈	—	43	2	no	—
150	C₂×C₄⋊C₈	—	43	1	no	—
151	C₂×C₄×C₈	—	43	2	no	—
152	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
153	#391	—	86	2	no	—
154	#387	—	86	4	no	—
155	#384	—	43	2	no	—
156	#382	—	43	4	no	—
157	C₄×C₄⋊C₄	—	43	4	no	—
158	C₄×C₄⋊C₄	—	86	2	no	—
159	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
160	#340	—	43	2	no	—
161	#340	—	43	2	no	—
162	#340	—	43	4	no	—
163	#340	C₂×D₄₃	1	1	yes	characteristic
164	#340	C₂×D₄₃	1	1	yes	characteristic
165	C₄₃	#22739	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	43	1	no	—
167	C₂²×C₈	—	43	2	no	—
168	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
170	C₄×Q₈	—	43	4	no	—
171	C₄×Q₈	—	43	4	no	—
172	C₂×C₄⋊C₄	—	43	2	no	—
173	C₂×C₄⋊C₄	—	43	1	no	—
174	C₂×C₄⋊C₄	—	43	4	no	—
175	C₂×C₄⋊C₄	—	86	2	no	—
176	C₂×C₄⋊C₄	—	86	2	no	—
177	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
178	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	—	43	2	no	—
181	C₂×C₄²	—	86	4	no	—
182	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
183	#107	—	86	8	no	—
184	#106	—	2	4	no	—
185	C₄⋊C₈	—	43	8	no	—
186	C₄⋊C₈	—	43	4	no	—
187	C₄×C₈	—	43	8	no	—
188	C₄×C₈	C₄×D₄₃	1	4	yes	—
189	#95	—	2	2	no	—
190	#95	—	86	4	no	—
191	#95	—	86	4	no	—
192	#95	—	86	4	no	—
193	C₂×Q₈	—	43	4	no	—
194	C₂×Q₈	—	86	4	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	43	2	no	—
198	C₂²×C₄	—	43	2	no	—
199	C₂²×C₄	—	86	4	no	—
200	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
201	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	43	4	no	—
206	C₂×C₈	—	86	4	no	—
207	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	43	8	no	—
213	C₄⋊C₄	—	43	4	no	—
214	C₄⋊C₄	—	86	4	no	—
215	C₄⋊C₄	—	86	4	no	—
216	C₄⋊C₄	—	344	2	no	—
217	C₄⋊C₄	—	172	4	no	—
218	C₄⋊C₄	#10621898	1	4	yes	—
219	C₄²	—	4	4	no	—
220	C₄²	—	43	4	no	—
221	C₄²	—	43	4	no	—
222	C₄²	—	344	4	no	—
223	C₄²	C₂×C₄×D₄₃	1	2	yes	—
224	C₄²	Q₈×D₄₃	1	2	yes	—
225	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
226	Q₈	—	86	8	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	43	4	no	—
236	C₂×C₄	—	43	4	no	—
237	C₂×C₄	—	86	4	no	—
238	C₂×C₄	—	86	2	no	—
239	C₂×C₄	—	86	2	no	—
240	C₂×C₄	—	172	4	no	—
241	C₂×C₄	—	172	4	no	—
242	C₂×C₄	—	172	4	no	—
243	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
244	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
245	C₂×C₄	#13063346	1	4	yes	—
246	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
247	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
248	C₂×C₄	C₂×#10621898	1	2	yes	—
249	C₂×C₄	#13063372	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	86	4	no	—
253	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
254	C₂²	#14583558	1	2	yes	—
255	C₂²	C₂×#13063346	1	2	yes	—
256	C₂²	#14583701	1	1	yes	characteristic
257	C₂²	#14583713	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	4	4	no	—
260	C₄	—	8	2	no	—
261	C₄	—	86	4	no	—
262	C₄	—	86	4	no	—
263	C₄	—	344	4	no	—
264	C₄	#14582934	1	4	yes	—
265	C₄	C₄×#10621898	1	1	yes	characteristic
266	C₄	#14583376	1	1	yes	characteristic
267	C₄	#14583627	1	2	yes	—
268	C₂	#19137515	1	1	yes	characteristic
269	C₂	C₂×#14582934	1	1	yes	characteristic
270	C₂	#19143192	1	1	yes	characteristic
271	C₂	#19147253	1	2	yes	—
272	C₂	#19147278	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	#22739
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
10. C₈×Dic₈₆	C₄	semidirect
165. C₄₃	#22739	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
#19137550	2
#19135364	2
#19135353	2
#19134724	2
#19134723	2
#19134487	2
C₄×#106×C₄₃	2
#22739	43