Subgroups — #25115548

Order 11008.

822 conjugacy classes of subgroups.

The subgroup lattice has 282 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	#19135353	C₂	1	2	yes	maximal
4	#19135350	C₂	1	4	yes	maximal
5	#19134723	C₂	1	1	yes	maximal, characteristic
6	#19134721	C₂	1	2	yes	maximal
7	#19134487	C₂	1	2	yes	maximal
8	C₄×#107×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×#13063258	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×#107×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	#13063258	—	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	#107×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	D₈	1	4	yes	—
67	C₂×Dic₈₆	Q₁₆	1	4	yes	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
72	C₄⋊C₄×C₄₃	—	2	2	no	—
73	C₄⋊C₄×C₄₃	—	2	2	no	—
74	C₄⋊C₄×C₄₃	—	2	2	no	—
75	C₄⋊C₄×C₄₃	#41	1	4	yes	—
76	C₄²×C₄₃	—	4	4	no	—
77	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
79	C₄⋊Dic₄₃	—	2	4	no	—
80	C₄⋊Dic₄₃	—	2	4	no	—
81	C₄⋊Dic₄₃	—	4	4	no	—
82	C₄⋊Dic₄₃	—	8	2	no	—
83	C₄⋊Dic₄₃	C₄⋊C₄	1	4	yes	—
84	Dic₄₃⋊C₄	—	4	8	no	—
85	Dic₄₃⋊C₄	—	8	4	no	—
86	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
87	C₄×Dic₄₃	—	4	8	no	—
88	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
89	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
90	C₂×C₄₃⋊C₈	—	2	4	no	—
91	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
92	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₃	—	2	2	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×D₈	1	2	yes	—
103	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
107	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
108	C₂×Dic₄₃	—	2	4	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×D₈	1	4	yes	—
115	C₂×Dic₄₃	C₂×Q₁₆	1	4	yes	—
116	Dic₈₆	#107	1	16	yes	complemented
117	C₈×C₄₃	—	2	2	no	—
118	C₈×C₄₃	—	2	2	no	—
119	C₄₃⋊C₈	—	2	4	no	—
120	#22740	—	43	1	no	maximal, Sylow
121	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₄₃	#548	1	2	yes	—
123	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
124	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
125	C₂²×C₄₃	#577	1	2	yes	—
126	C₄×C₄₃	—	2	2	no	—
127	C₄×C₄₃	—	4	4	no	—
128	C₄×C₄₃	—	8	2	no	—
129	C₄×C₄₃	C₂×#107	1	4	yes	—
130	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
131	C₄×C₄₃	#519	1	1	yes	characteristic
132	C₄×C₄₃	#557	1	2	yes	—
133	Dic₄₃	—	2	4	no	—
134	Dic₄₃	—	8	4	no	—
135	Dic₄₃	C₂×#107	1	8	yes	—
136	C₂×C₈×Q₈	—	43	1	no	—
137	Q₈×C₄⋊C₄	—	43	2	no	—
138	#1947	—	43	4	no	—
139	#1947	—	43	2	no	—
140	#1834	—	43	1	no	—
141	#1834	—	43	2	no	—
142	C₄×#107	—	43	2	no	—
143	C₄×#107	D₄₃	1	1	yes	characteristic
144	C₂×C₄₃	#2351	1	1	yes	characteristic
145	C₂×C₄₃	C₂²×#107	1	1	yes	characteristic
146	C₂×C₄₃	#2912	1	1	yes	characteristic
147	C₂×C₄₃	#3358	1	2	yes	—
148	C₂×C₄₃	#3363	1	2	yes	—
149	C₂×C₄×Q₈	—	43	1	no	—
150	C₂×C₄×Q₈	—	86	2	no	—
151	C₈×Q₈	—	43	8	no	—
152	C₂×#107	—	43	4	no	—
153	C₂×#107	C₂×D₄₃	1	2	yes	—
154	C₂×C₄⋊C₈	—	43	2	no	—
155	C₂×C₄⋊C₈	—	43	1	no	—
156	C₂×C₄×C₈	—	43	2	no	—
157	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
158	#389	—	86	4	no	—
159	#389	—	86	2	no	—
160	#384	—	43	4	no	—
161	#384	—	43	2	no	—
162	C₄×C₄⋊C₄	—	43	4	no	—
163	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
164	#340	—	43	2	no	—
165	#340	—	43	2	no	—
166	#340	—	43	4	no	—
167	#340	C₂×D₄₃	1	1	yes	characteristic
168	#340	C₂×D₄₃	1	1	yes	characteristic
169	C₄₃	#22740	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₈	—	43	1	no	—
171	C₂²×C₈	—	43	2	no	—
172	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
174	C₄×Q₈	—	43	4	no	—
175	C₄×Q₈	—	43	4	no	—
176	C₄×Q₈	—	172	8	no	—
177	C₂×C₄⋊C₄	—	43	2	no	—
178	C₂×C₄⋊C₄	—	43	1	no	—
179	C₂×C₄⋊C₄	—	43	4	no	—
180	C₂×C₄⋊C₄	—	86	4	no	—
181	C₂×C₄⋊C₄	—	86	2	no	—
182	C₂×C₄⋊C₄	—	86	4	no	—
183	C₂×C₄⋊C₄	—	86	2	no	—
184	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
185	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	43	2	no	—
188	C₂×C₄²	—	86	4	no	—
189	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
190	#107	—	2	4	no	—
191	#107	—	86	8	no	—
192	C₄⋊C₈	—	43	8	no	—
193	C₄⋊C₈	—	43	4	no	—
194	C₄×C₈	—	43	8	no	—
195	C₄×C₈	C₄×D₄₃	1	4	yes	—
196	#95	—	2	2	no	—
197	#95	—	86	4	no	—
198	C₂×Q₈	—	43	4	no	—
199	C₂×Q₈	—	43	4	no	—
200	C₂×Q₈	—	43	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	43	2	no	—
204	C₂²×C₄	—	43	2	no	—
205	C₂²×C₄	—	86	4	no	—
206	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
208	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	43	4	no	—
212	C₂×C₈	—	86	4	no	—
213	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	43	8	no	—
219	C₄⋊C₄	—	43	4	no	—
220	C₄⋊C₄	—	86	4	no	—
221	C₄⋊C₄	—	86	4	no	—
222	C₄⋊C₄	—	344	2	no	—
223	C₄⋊C₄	—	344	4	no	—
224	C₄⋊C₄	—	172	8	no	—
225	C₄⋊C₄	—	172	4	no	—
226	C₄⋊C₄	#10621898	1	4	yes	—
227	C₄²	—	4	4	no	—
228	C₄²	—	43	4	no	—
229	C₄²	—	43	4	no	—
230	C₄²	—	172	8	no	—
231	C₄²	C₂×C₄×D₄₃	1	2	yes	—
232	C₄²	Q₈×D₄₃	1	2	yes	—
233	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
234	Q₈	—	43	16	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	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	—	4	2	no	—
242	C₂×C₄	—	43	4	no	—
243	C₂×C₄	—	43	4	no	—
244	C₂×C₄	—	43	4	no	—
245	C₂×C₄	—	43	4	no	—
246	C₂×C₄	—	86	4	no	—
247	C₂×C₄	—	172	4	no	—
248	C₂×C₄	—	172	4	no	—
249	C₂×C₄	—	172	4	no	—
250	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
251	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
252	C₂×C₄	D₈×D₄₃	1	2	yes	—
253	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
254	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
255	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
256	C₂×C₄	C₂×#10621898	1	2	yes	—
257	C₂×C₄	#13063372	1	2	yes	—
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	86	4	no	—
261	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
262	C₂²	#14583558	1	2	yes	—
263	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
264	C₂²	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
265	C₂²	#14583706	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	2	no	—
269	C₄	—	43	8	no	—
270	C₄	—	86	4	no	—
271	C₄	—	344	4	no	—
272	C₄	#107×D₄₃	1	4	yes	—
273	C₄	C₄×#10621898	1	1	yes	characteristic
274	C₄	#14583376	1	1	yes	characteristic
275	C₄	#14583627	1	2	yes	—
276	C₂	#19137515	1	1	yes	characteristic
277	C₂	C₂×#107×D₄₃	1	1	yes	characteristic
278	C₂	#19143187	1	1	yes	characteristic
279	C₂	#19147228	1	2	yes	—
280	C₂	#19147271	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#107×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#107	D₄₃
hypocenter	C₄₃	#22740
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₈₆	C₄	semidirect
116. Dic₈₆	#107	semidirect
169. C₄₃	#22740	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
#19135353	2
#19135350	2
#19134723	2
#19134721	2
#19134487	2
C₄×#107×C₄₃	2
#22740	43