Subgroups — #24995392

Order 11008.

698 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134712	C₂	1	1	yes	maximal, characteristic, complemented
2	#19134341	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₃⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₄×#127×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
7	#14582654	—	2	2	no	—
8	#14582654	—	2	2	no	—
9	D₄⋊(C₄₃⋊C₈)	—	2	8	no	—
10	C₂×#127×C₄₃	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₄₃	—	2	2	no	—
12	C₂×C₄×D₄×C₄₃	C₂²	1	2	yes	—
13	C₄₃×#390	—	2	2	no	—
14	C₄₃×#390	C₄	1	2	yes	—
15	C₄₃×#384	C₄	1	1	yes	characteristic
16	C₄³×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄₃⋊C₈	—	2	2	no	—
18	C₂²×C₄₃⋊C₈	—	2	2	no	—
19	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
20	C₄×C₄₃⋊C₈	—	2	2	no	—
21	C₄×C₄₃⋊C₈	D₄	1	4	yes	complemented
22	C₂²×D₄×C₄₃	—	2	2	no	—
23	C₂²×D₄×C₄₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₃	—	2	2	no	—
25	#127×C₄₃	—	2	2	no	—
26	#127×C₄₃	C₈	1	4	yes	complemented
27	C₄×D₄×C₄₃	—	2	4	no	—
28	C₄×D₄×C₄₃	—	2	4	no	—
29	C₄×D₄×C₄₃	—	4	8	no	—
30	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
33	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
34	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₃	D₄	1	2	yes	—
37	C₂×C₄²×C₄₃	D₄	1	2	yes	—
38	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	C₂⁴×C₄₃	—	2	2	no	—
40	C₂×D₄×C₄₃	—	2	8	no	—
41	C₂×D₄×C₄₃	—	2	2	no	—
42	C₂×D₄×C₄₃	—	2	4	no	—
43	C₂×D₄×C₄₃	—	2	4	no	—
44	C₂×D₄×C₄₃	—	4	4	no	—
45	C₂×D₄×C₄₃	—	4	4	no	—
46	C₂×D₄×C₄₃	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₄₃	—	2	2	no	—
48	C₂²×C₄×C₄₃	—	4	2	no	—
49	C₂²×C₄×C₄₃	—	4	2	no	—
50	C₂²×C₄×C₄₃	—	4	8	no	—
51	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₄₃	—	2	4	no	—
57	C₄⋊C₄×C₄₃	—	4	4	no	—
58	C₂²⋊C₄×C₄₃	—	4	8	no	—
59	C₂²⋊C₄×C₄₃	—	4	4	no	—
60	C₄²×C₄₃	—	2	2	no	—
61	C₄²×C₄₃	—	2	4	no	—
62	C₄²×C₄₃	—	4	2	no	—
63	C₄²×C₄₃	C₂×C₈	1	2	yes	—
64	C₄²×C₄₃	M₁₆	1	2	yes	—
65	C₄²×C₄₃	C₂×D₄	1	2	yes	—
66	C₄²×C₄₃	#41	1	2	yes	—
67	C₂×C₄₃⋊C₈	—	2	4	no	—
68	C₂×C₄₃⋊C₈	—	4	4	no	—
69	C₂×C₄₃⋊C₈	—	4	2	no	—
70	C₂×C₄₃⋊C₈	—	4	4	no	—
71	C₂³×C₄₃	—	2	4	no	—
72	C₂³×C₄₃	—	4	2	no	—
73	C₂³×C₄₃	—	4	2	no	—
74	C₂³×C₄₃	—	4	4	no	—
75	C₂³×C₄₃	—	4	4	no	—
76	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	D₄×C₄₃	—	2	8	no	—
78	D₄×C₄₃	—	4	16	no	—
79	D₄×C₄₃	—	4	4	no	—
80	C₂×C₄×C₄₃	—	2	2	no	—
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	4	2	no	—
87	C₂×C₄×C₄₃	—	4	4	no	—
88	C₂×C₄×C₄₃	—	8	4	no	—
89	C₂×C₄×C₄₃	—	8	4	no	—
90	C₂×C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
91	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₄₃	#121	1	2	yes	—
93	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
94	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₄₃	#131	1	2	yes	—
98	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
99	C₄₃⋊C₈	—	4	4	no	—
100	C₄₃⋊C₈	—	8	2	no	—
101	#9642	—	43	1	no	maximal, Sylow
102	C₂²×C₄₃	—	4	4	no	—
103	C₂²×C₄₃	—	4	8	no	—
104	C₂²×C₄₃	—	8	2	no	—
105	C₂²×C₄₃	—	8	2	no	—
106	C₂²×C₄₃	—	8	4	no	—
107	C₂²×C₄₃	#390	1	1	yes	characteristic
108	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
109	C₂²×C₄₃	#408	1	1	yes	characteristic
110	C₂²×C₄₃	#418	1	2	yes	—
111	C₂²×C₄₃	#421	1	2	yes	—
112	C₄×C₄₃	—	2	2	no	—
113	C₄×C₄₃	—	2	4	no	—
114	C₄×C₄₃	—	4	2	no	—
115	C₄×C₄₃	—	8	2	no	—
116	C₄×C₄₃	C₈×D₄	1	2	yes	—
117	C₄×C₄₃	#436	1	2	yes	—
118	C₄×C₄₃	#490	1	2	yes	—
119	C₄×C₄₃	#496	1	2	yes	—
120	C₄×#127	D₄₃	1	1	yes	characteristic
121	#1831	—	43	1	no	—
122	#1744	—	43	1	no	—
123	C₂×D₄⋊C₈	—	43	4	no	—
124	C₂×C₄₃	—	8	4	no	—
125	C₂×C₄₃	#1478	1	2	yes	—
126	C₂×C₄₃	#1496	1	2	yes	—
127	C₂×C₄₃	#1954	1	1	yes	characteristic
128	C₂×C₄₃	#1961	1	1	yes	characteristic
129	C₂×C₄₃	#1963	1	1	yes	characteristic
130	C₂×#127	Dic₄₃	1	1	yes	characteristic
131	C₂×C₄×D₄	—	2	2	no	—
132	C₂×C₄×D₄	C₂×D₄₃	1	2	yes	—
133	C₂×C₄⋊C₈	—	43	2	no	—
134	C₂×C₄×C₈	—	43	2	no	—
135	#390	—	2	2	no	—
136	#390	Dic₄₃	1	2	yes	—
137	#384	Dic₄₃	1	1	yes	characteristic
138	C₄³	C₂×D₄₃	1	1	yes	characteristic
139	#336	—	86	2	no	—
140	#336	—	86	2	no	—
141	D₄⋊C₈	—	86	8	no	—
142	C₄₃	#9642	1	1	yes	characteristic, complemented, Sylow
143	C₂²×D₄	—	2	2	no	—
144	C₂²×D₄	C₂×Dic₄₃	1	2	yes	—
145	C₂³×C₄	—	2	2	no	—
146	C₂²×C₈	—	86	2	no	—
147	C₂²×C₈	—	86	2	no	—
148	#127	—	2	2	no	—
149	#127	C₄₃⋊C₈	1	4	yes	complemented
150	C₄×D₄	—	2	4	no	—
151	C₄×D₄	—	2	4	no	—
152	C₄×D₄	—	4	8	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
155	C₂×C₂²⋊C₄	—	2	4	no	—
156	C₂×C₂²⋊C₄	—	2	2	no	—
157	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
158	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
160	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
161	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
162	C₄⋊C₈	—	86	4	no	—
163	C₄×C₈	—	43	4	no	—
164	C₄×C₈	—	86	2	no	—
165	C₂⁴	—	2	2	no	—
166	C₂×D₄	—	2	8	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	4	4	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	C₂×C₄₃⋊C₈	1	4	yes	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
182	C₂×C₈	—	86	4	no	—
183	C₂×C₈	—	172	2	no	—
184	C₂×C₈	—	172	4	no	—
185	C₂×C₈	—	172	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₂²⋊C₄	—	4	8	no	—
189	C₂²⋊C₄	—	4	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	4	2	no	—
193	C₄²	C₂×C₄₃⋊C₈	1	2	yes	—
194	C₄²	#10621877	1	2	yes	—
195	C₄²	D₄×D₄₃	1	2	yes	—
196	C₄²	#10621900	1	2	yes	—
197	C₂³	—	2	4	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	4	no	—
202	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
203	D₄	—	2	8	no	—
204	D₄	—	4	16	no	—
205	D₄	—	4	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	4	yes	—
217	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
218	C₂×C₄	#13063296	1	2	yes	—
219	C₂×C₄	#13063312	1	1	yes	characteristic
220	C₂×C₄	#13063321	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
222	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
223	C₂×C₄	#13063332	1	2	yes	—
224	C₂×C₄	#13063351	1	4	yes	—
225	C₈	—	344	2	no	—
226	C₈	—	172	4	no	—
227	C₂²	—	4	8	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	8	4	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	#14582802	1	1	yes	characteristic
233	C₂²	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
234	C₂²	#14582866	1	1	yes	characteristic
235	C₂²	#14582904	1	2	yes	—
236	C₂²	#14582916	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	2	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	8	2	no	—
241	C₄	D₄×C₄₃⋊C₈	1	2	yes	—
242	C₄	#14582977	1	2	yes	—
243	C₄	#14583263	1	2	yes	—
244	C₄	#14583286	1	2	yes	—
245	C₂	—	8	4	no	—
246	C₂	#19132986	1	2	yes	—
247	C₂	#19133064	1	2	yes	—
248	C₂	#19135390	1	1	yes	characteristic
249	C₂	#19135420	1	1	yes	characteristic
250	C₂	#19135428	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#127×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₃	C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#127	D₄₃
hypocenter	C₄₃	#9642
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #19134712	C₂	semidirect
2. #19134341	C₂	semidirect
3. C₂×D₄⋊(C₄₃⋊C₈)	C₂	semidirect
21. C₄×C₄₃⋊C₈	D₄	semidirect
26. #127×C₄₃	C₈	semidirect
142. C₄₃	#9642	semidirect
149. #127	C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#19134712	2
#19134341	2
C₂×D₄⋊(C₄₃⋊C₈)	2
C₄×#127×C₄₃	2
#9642	43