Subgroups — #24995505

Order 11008.

586 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	#19135390	C₂	1	2	yes	maximal, complemented
2	C₂×#14582650	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂×#14582621	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	#14582650	—	2	2	no	—
11	D₄⋊(C₄₃⋊C₈)	—	2	4	no	—
12	#14582621	—	2	2	no	—
13	C₂²×D₈×C₄₃	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₃	C₂²	1	2	yes	—
15	C₄×D₈×C₄₃	—	2	2	no	—
16	C₄×D₈×C₄₃	—	2	2	no	—
17	C₂×#107×C₄₃	C₄	1	1	yes	characteristic
18	C₂×#102×C₄₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₈	—	2	1	no	—
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
23	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₄₃⋊C₈)	—	4	8	no	—
26	C₄×C₄₃⋊C₈	—	2	1	no	—
27	C₄×C₄₃⋊C₈	—	2	1	no	—
28	C₄×C₄₃⋊C₈	—	2	1	no	—
29	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
30	C₂²×D₄×C₄₃	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₃	—	2	2	no	—
32	C₂×D₈×C₄₃	—	2	4	no	—
33	C₂×D₈×C₄₃	—	2	2	no	—
34	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₄₃	—	2	2	no	—
37	C₄×D₄×C₄₃	—	2	2	no	—
38	C₄×D₄×C₄₃	—	2	2	no	—
39	C₄×D₄×C₄₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	#107×C₄₃	—	2	2	no	—
44	#102×C₄₃	—	2	4	no	—
45	C₄×C₈×C₄₃	D₄	1	2	yes	—
46	C₄×C₈×C₄₃	D₄	1	2	yes	—
47	C₂⁴×C₄₃	—	2	2	no	—
48	C₂×D₄×C₄₃	—	2	2	no	—
49	C₂×D₄×C₄₃	—	2	2	no	—
50	C₂×D₄×C₄₃	—	2	2	no	—
51	C₂×D₄×C₄₃	—	2	2	no	—
52	C₂×D₄×C₄₃	—	2	2	no	—
53	C₂×D₄×C₄₃	M₁₆	1	4	yes	—
54	C₂²×C₄×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	—	4	2	no	—
56	C₂²×C₄×C₄₃	—	4	2	no	—
57	C₂²×C₄×C₄₃	—	4	4	no	—
58	C₂²×C₄×C₄₃	—	4	4	no	—
59	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
62	D₈×C₄₃	—	2	8	no	—
63	C₂×C₈×C₄₃	—	2	2	no	—
64	C₂×C₈×C₄₃	—	2	2	no	—
65	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₄₃	—	2	2	no	—
68	C₄⋊C₄×C₄₃	—	2	2	no	—
69	C₂²⋊C₄×C₄₃	—	4	2	no	—
70	C₂²⋊C₄×C₄₃	—	4	2	no	—
71	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₃	#41	1	1	yes	characteristic
75	C₂×C₄₃⋊C₈	—	2	4	no	—
76	C₂×C₄₃⋊C₈	—	2	2	no	—
77	C₂×C₄₃⋊C₈	—	2	2	no	—
78	C₂×C₄₃⋊C₈	—	2	4	no	—
79	C₂×C₄₃⋊C₈	—	4	2	no	—
80	C₂×C₄₃⋊C₈	—	4	4	no	—
81	C₂³×C₄₃	—	2	4	no	—
82	C₂³×C₄₃	—	4	2	no	—
83	C₂³×C₄₃	—	4	2	no	—
84	C₂³×C₄₃	—	4	4	no	—
85	C₂³×C₄₃	—	4	4	no	—
86	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
87	D₄×C₄₃	—	2	4	no	—
88	D₄×C₄₃	—	2	4	no	—
89	D₄×C₄₃	—	4	2	no	—
90	D₄×C₄₃	—	4	2	no	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	—	4	4	no	—
94	C₂×C₄×C₄₃	—	8	4	no	—
95	C₂×C₄×C₄₃	—	8	2	no	—
96	C₂×C₄×C₄₃	—	8	2	no	—
97	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
102	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₄₃	#131	1	2	yes	—
105	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
106	C₈×C₄₃	—	2	2	no	—
107	C₈×C₄₃	—	2	2	no	—
108	C₄₃⋊C₈	—	2	4	no	—
109	C₄₃⋊C₈	—	4	2	no	—
110	C₄₃⋊C₈	—	8	2	no	—
111	#9665	—	43	1	no	maximal, Sylow
112	C₂²×C₄₃	—	4	8	no	—
113	C₂²×C₄₃	—	4	4	no	—
114	C₂²×C₄₃	—	8	4	no	—
115	C₂²×C₄₃	—	8	2	no	—
116	C₂²×C₄₃	—	8	2	no	—
117	C₂²×C₄₃	#390	1	1	yes	characteristic
118	C₂²×C₄₃	#435	1	2	yes	—
119	C₂²×C₄₃	#442	1	2	yes	—
120	C₂²×C₄₃	#443	1	2	yes	—
121	C₄×C₄₃	—	2	1	no	—
122	C₄×C₄₃	—	2	1	no	—
123	C₄×C₄₃	—	8	2	no	—
124	C₄×C₄₃	#407	1	1	yes	characteristic
125	C₄×C₄₃	#408	1	1	yes	characteristic
126	C₄×C₄₃	#435	1	2	yes	—
127	C₄×C₄₃	#469	1	2	yes	—
128	C₄×C₄₃	#496	1	2	yes	—
129	C₂×C₄×D₈	D₄₃	1	1	yes	characteristic
130	#1954	—	43	2	no	—
131	C₂×#334	—	43	1	no	—
132	C₂×D₄⋊C₈	—	43	2	no	—
133	C₂×#322	—	43	1	no	—
134	C₂×C₄₃	—	8	4	no	—
135	C₂×C₄₃	#1586	1	4	yes	—
136	C₂×C₄₃	#1955	1	1	yes	characteristic
137	C₂×C₄₃	#1972	1	1	yes	characteristic
138	C₂×C₄₃	#1974	1	1	yes	characteristic
139	C₂²×D₈	Dic₄₃	1	1	yes	characteristic
140	C₂×C₄×D₄	C₂×D₄₃	1	2	yes	—
141	C₄×D₈	—	2	2	no	—
142	C₄×D₈	—	2	2	no	—
143	C₂×#107	Dic₄₃	1	1	yes	characteristic
144	C₂×C₄⋊C₈	—	43	2	no	—
145	C₂×#102	Dic₄₃	1	2	yes	—
146	C₂×C₂²⋊C₈	—	86	4	no	—
147	C₂×C₄×C₈	—	43	1	no	—
148	C₂×C₄×C₈	—	43	1	no	—
149	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
150	#334	—	86	2	no	—
151	D₄⋊C₈	—	86	4	no	—
152	#322	—	86	2	no	—
153	C₄₃	#9665	1	1	yes	characteristic, complemented, Sylow
154	C₂²×D₄	C₂×Dic₄₃	1	2	yes	—
155	C₂³×C₄	—	2	2	no	—
156	C₂×D₈	—	2	4	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂²×C₈	—	43	2	no	—
159	C₂²×C₈	—	86	1	no	—
160	C₂²×C₈	—	86	2	no	—
161	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
168	C₂×C₂²⋊C₄	—	2	2	no	—
169	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
170	#107	—	2	2	no	—
171	C₄⋊C₈	—	86	2	no	—
172	C₄⋊C₈	—	86	2	no	—
173	#102	—	2	4	no	—
174	C₂²⋊C₈	—	172	8	no	—
175	C₄×C₈	—	43	2	no	—
176	C₄×C₈	—	86	1	no	—
177	C₄×C₈	—	86	1	no	—
178	C₄×C₈	—	86	1	no	—
179	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
180	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
181	C₂⁴	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	#10621877	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
194	C₂²×C₄	#10621900	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
196	D₈	—	2	8	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	86	4	no	—
200	C₂×C₈	—	86	2	no	—
201	C₂×C₈	—	86	2	no	—
202	C₂×C₈	—	86	4	no	—
203	C₂×C₈	—	172	2	no	—
204	C₂×C₈	—	172	4	no	—
205	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
206	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	D₄×D₄₃	1	1	yes	characteristic
212	C₄²	#10621900	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
215	C₂³	—	2	4	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	4	no	—
220	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
221	D₄	—	2	4	no	—
222	D₄	—	2	4	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
232	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
233	C₂×C₄	#13063296	1	1	yes	characteristic
234	C₂×C₄	#13063296	1	1	yes	characteristic
235	C₂×C₄	#13063312	1	1	yes	characteristic
236	C₂×C₄	#13063321	1	1	yes	characteristic
237	C₂×C₄	C₂×#10621877	1	2	yes	—
238	C₂×C₄	#13063332	1	2	yes	—
239	C₂×C₄	#13063349	1	4	yes	—
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	86	4	no	—
243	C₈	—	344	2	no	—
244	C₈	—	172	2	no	—
245	C₂²	—	4	8	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	#14582802	1	1	yes	characteristic
251	C₂²	#14582971	1	2	yes	—
252	C₂²	#14583011	1	2	yes	—
253	C₂²	#14583016	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	8	2	no	—
257	C₄	#14582863	1	1	yes	characteristic
258	C₄	#14582866	1	1	yes	characteristic
259	C₄	#14582971	1	2	yes	—
260	C₄	#14583149	1	2	yes	—
261	C₄	#14583287	1	2	yes	—
262	C₂	—	8	4	no	—
263	C₂	#19133460	1	4	yes	—
264	C₂	#19135394	1	1	yes	characteristic
265	C₂	#19135472	1	1	yes	characteristic
266	C₂	#19135484	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄×D₈×C₄₃	C₂
socle	C₂³×C₄₃	C₄×D₄
prime radical	C₂²×D₈×C₄₃	C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×C₄×D₈	D₄₃
hypocenter	C₄₃	#9665
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #19135390	C₂	semidirect
2. C₂×#14582650	C₂	semidirect
3. C₂×D₄⋊(C₄₃⋊C₈)	C₂	semidirect
4. C₂×#14582621	C₂	semidirect
153. C₄₃	#9665	semidirect

Maximal subgroups

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

subgroup	index
#19135390	2
C₂×#14582650	2
C₂×D₄⋊(C₄₃⋊C₈)	2
C₂×#14582621	2
C₂×C₄×D₈×C₄₃	2
#9665	43