Subgroups — #24987885

Order 11008.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134464	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582652	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582621	C₂	1	1	yes	maximal, characteristic
5	#19132436	C₂	1	1	yes	maximal, characteristic
6	#19132436	C₂	1	1	yes	maximal, characteristic
7	C₄×#107×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#13063236	—	2	1	no	—
11	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
14	#14582654	—	2	1	no	—
15	#14582654	—	2	1	no	—
16	#14582652	C₄	1	4	yes	complemented
17	#14582621	C₄	1	4	yes	complemented
18	C₂×#107×C₄₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#340	C₄	1	2	yes	—
23	C₂²×C₄₃⋊C₈	—	2	1	no	—
24	C₂²×C₄₃⋊C₈	—	2	1	no	—
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
27	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
29	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
30	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
31	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
32	#13063236	—	4	2	no	—
33	C₄×C₄₃⋊C₈	—	2	1	no	—
34	C₄×C₄₃⋊C₈	—	4	2	no	—
35	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₄×C₄₃⋊C₈	Q₈	1	2	yes	—
38	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₃	—	2	1	no	—
43	C₂×C₄²×C₄₃	—	2	1	no	—
44	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
45	#107×C₄₃	—	2	4	no	—
46	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
48	#95×C₄₃	—	2	1	no	—
49	#95×C₄₃	—	2	1	no	—
50	C₂²×C₄×C₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	—	2	2	no	—
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₄₃	—	2	2	no	—
56	C₂×C₈×C₄₃	—	2	2	no	—
57	C₂×C₈×C₄₃	C₄²	1	2	yes	—
58	C₂×C₈×C₄₃	C₄²	1	2	yes	—
59	C₄⋊C₄×C₄₃	—	2	2	no	—
60	C₄⋊C₄×C₄₃	—	2	2	no	—
61	C₄⋊C₄×C₄₃	—	2	2	no	—
62	C₄⋊C₄×C₄₃	M₁₆	1	4	yes	—
63	C₄²×C₄₃	—	4	1	no	—
64	C₄²×C₄₃	—	4	1	no	—
65	C₄²×C₄₃	—	4	1	no	—
66	C₄²×C₄₃	—	4	1	no	—
67	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₄₃	#41	1	1	yes	characteristic
71	C₂×C₄₃⋊C₈	—	2	1	no	—
72	C₂×C₄₃⋊C₈	—	2	1	no	—
73	C₂×C₄₃⋊C₈	—	2	2	no	—
74	C₂×C₄₃⋊C₈	—	2	2	no	—
75	C₂×C₄₃⋊C₈	—	2	2	no	—
76	C₂×C₄₃⋊C₈	—	4	2	no	—
77	C₂×C₄₃⋊C₈	—	4	1	no	—
78	C₂×C₄₃⋊C₈	—	4	1	no	—
79	C₂×C₄₃⋊C₈	—	4	1	no	—
80	C₂×C₄₃⋊C₈	—	4	1	no	—
81	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
82	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
83	C₂×C₄₃⋊C₈	D₈	1	2	yes	—
84	C₂×C₄₃⋊C₈	Q₁₆	1	2	yes	—
85	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₃	—	2	2	no	—
87	C₂×C₄×C₄₃	—	2	4	no	—
88	C₂×C₄×C₄₃	—	4	2	no	—
89	C₂×C₄×C₄₃	—	4	2	no	—
90	C₂×C₄×C₄₃	—	4	2	no	—
91	C₂×C₄×C₄₃	—	4	2	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₄₃	#131	1	2	yes	—
101	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
104	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
105	C₈×C₄₃	—	2	2	no	—
106	C₈×C₄₃	—	2	2	no	—
107	C₄₃⋊C₈	—	2	2	no	—
108	C₄₃⋊C₈	—	4	2	no	—
109	C₄₃⋊C₈	—	4	2	no	—
110	C₄₃⋊C₈	—	4	2	no	—
111	C₄₃⋊C₈	#107	1	4	yes	complemented
112	#8291	—	43	1	no	maximal, Sylow
113	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₃	#435	1	2	yes	—
115	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
116	C₂²×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
117	C₂²×C₄₃	#444	1	2	yes	—
118	C₄×C₄₃	—	2	1	no	—
119	C₄×C₄₃	—	2	1	no	—
120	C₄×C₄₃	—	4	4	no	—
121	C₄×C₄₃	—	8	2	no	—
122	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
123	C₄×C₄₃	#405	1	1	yes	characteristic
124	C₄×C₄₃	C₂×#107	1	1	yes	characteristic
125	C₄×C₄₃	#427	1	1	yes	characteristic
126	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
127	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
128	C₄×C₄₃	#446	1	2	yes	—
129	C₄×#107	D₄₃	1	1	yes	characteristic
130	#1770	—	43	1	no	—
131	C₈×C₄⋊C₄	—	43	1	no	—
132	C₂×#335	—	43	1	no	—
133	C₂×#322	—	43	1	no	—
134	#1279	—	43	1	no	—
135	#1279	—	43	1	no	—
136	C₂×C₄₃	#1581	1	2	yes	—
137	C₂×C₄₃	#1584	1	2	yes	—
138	C₂×C₄₃	#1771	1	1	yes	characteristic
139	C₂×C₄₃	#1773	1	1	yes	characteristic
140	C₂×C₄₃	C₄×#107	1	1	yes	characteristic
141	C₂×#107	Dic₄₃	1	2	yes	—
142	C₂×C₄⋊C₈	—	43	1	no	—
143	C₂×C₄⋊C₈	—	43	1	no	—
144	C₂×#97	—	86	1	no	—
145	C₂×C₄×C₈	—	43	1	no	—
146	C₂×C₄×C₈	—	43	1	no	—
147	C₂×C₄×C₈	—	86	1	no	—
148	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
149	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
150	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
151	#340	Dic₄₃	1	2	yes	—
152	#336	—	86	1	no	—
153	#336	—	86	1	no	—
154	#335	—	43	4	no	—
155	#322	—	43	4	no	—
156	C₄₃	#8291	1	1	yes	characteristic, complemented, Sylow
157	C₂²×C₈	—	43	1	no	—
158	C₂²×C₈	—	43	1	no	—
159	C₂²×C₈	—	86	1	no	—
160	C₂²×C₈	—	86	1	no	—
161	C₂²×C₈	—	86	1	no	—
162	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
163	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
165	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
169	#107	—	2	4	no	—
170	C₄⋊C₈	—	43	2	no	—
171	C₄⋊C₈	—	43	2	no	—
172	C₄⋊C₈	—	86	1	no	—
173	C₄⋊C₈	—	86	1	no	—
174	#97	—	172	2	no	—
175	C₄×C₈	—	43	2	no	—
176	C₄×C₈	—	43	2	no	—
177	C₄×C₈	—	43	2	no	—
178	C₄×C₈	—	86	1	no	—
179	C₄×C₈	—	172	2	no	—
180	C₄×C₈	C₄×D₄₃	1	2	yes	—
181	C₄×C₈	C₄×D₄₃	1	2	yes	—
182	#95	—	2	1	no	—
183	#95	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
187	C₂²×C₄	#10621900	1	1	yes	characteristic
188	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	43	2	no	—
192	C₂×C₈	—	43	2	no	—
193	C₂×C₈	—	43	2	no	—
194	C₂×C₈	—	43	2	no	—
195	C₂×C₈	—	86	1	no	—
196	C₂×C₈	—	86	1	no	—
197	C₂×C₈	—	86	2	no	—
198	C₂×C₈	—	86	2	no	—
199	C₂×C₈	—	86	2	no	—
200	C₂×C₈	—	172	2	no	—
201	C₂×C₈	—	172	1	no	—
202	C₂×C₈	—	172	1	no	—
203	C₂×C₈	—	172	1	no	—
204	C₂×C₈	—	172	1	no	—
205	C₂×C₈	C₄×Dic₄₃	1	2	yes	—
206	C₂×C₈	C₄×Dic₄₃	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	#10621877	1	4	yes	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
216	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
217	C₄²	Q₈×D₄₃	1	1	yes	characteristic
218	C₄²	#10621902	1	1	yes	characteristic
219	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
228	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
229	C₂×C₄	#13063283	1	1	yes	characteristic
230	C₂×C₄	#13063287	1	1	yes	characteristic
231	C₂×C₄	#13063290	1	1	yes	characteristic
232	C₂×C₄	Q₈×Dic₄₃	1	1	yes	characteristic
233	C₂×C₄	C₂×#10621877	1	2	yes	—
234	C₂×C₄	#13063332	1	2	yes	—
235	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
236	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
237	C₂×C₄	#13063344	1	1	yes	characteristic
238	C₂×C₄	#13063348	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	—	43	4	no	—
242	C₈	—	86	2	no	—
243	C₈	—	172	2	no	—
244	C₈	—	172	2	no	—
245	C₈	—	172	2	no	—
246	C₂²	C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
247	C₂²	#14582971	1	2	yes	—
248	C₂²	D₈×Dic₄₃	1	1	yes	characteristic
249	C₂²	Q₁₆×Dic₄₃	1	1	yes	characteristic
250	C₂²	#14583021	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₄×#10621877	1	1	yes	characteristic
256	C₄	#14582855	1	1	yes	characteristic
257	C₄	#107×D₄₃	1	1	yes	characteristic
258	C₄	#14582936	1	1	yes	characteristic
259	C₄	#14582981	1	1	yes	characteristic
260	C₄	#14582993	1	1	yes	characteristic
261	C₄	#14583030	1	2	yes	—
262	C₂	#19133425	1	2	yes	—
263	C₂	#19133446	1	2	yes	—
264	C₂	#19134470	1	1	yes	characteristic
265	C₂	#19134481	1	1	yes	characteristic
266	C₂	#107×Dic₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#107×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#107	D₄₃
hypocenter	C₄₃	#8291
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
16. #14582652	C₄	semidirect
17. #14582621	C₄	semidirect
111. C₄₃⋊C₈	#107	semidirect
156. C₄₃	#8291	semidirect

Maximal subgroups

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

subgroup	index
#19134464	2
C₄⋊C₄×C₄₃⋊C₈	2
C₂×#14582652	2
C₂×#14582621	2
#19132436	2
C₄×#107×C₄₃	2
#8291	43