Subgroups — #25105252

Order 11008.

886 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583023	C₂	1	2	yes	maximal
2	C₄×#13063325	C₂	1	1	yes	maximal, characteristic
3	#19135530	C₂	1	2	yes	maximal
4	#19135311	C₂	1	4	yes	maximal
5	#19134460	C₂	1	4	yes	maximal
6	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1842	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063325	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
10	#14583023	—	2	8	no	—
11	C₂×Dic₄₃⋊C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₃	C₂²	1	4	yes	—
17	#14582805	C₄	1	2	yes	—
18	#14582797	C₄	1	4	yes	—
19	#14582769	C₄	1	1	yes	characteristic
20	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
21	#14582653	C₂²	1	8	yes	—
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	4	yes	—
24	C₄³×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₈₆	C₂×C₄	1	2	yes	—
26	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
28	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
29	#13063325	C₈	1	8	yes	complemented
30	C₄×Dic₈₆	—	2	4	no	—
31	C₄×Dic₈₆	—	2	4	no	—
32	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
35	C₂×Dic₄₃⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₃	C₂³	1	4	yes	—
37	Dic₄₃⋊C₈	—	2	8	no	—
38	C₄×C₄₃⋊C₈	—	2	4	no	—
39	C₄×C₄₃⋊C₈	—	2	2	no	—
40	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
41	C₄×C₄₃⋊C₈	Q₈	1	8	yes	complemented
42	C₈×Dic₄₃	—	2	8	no	—
43	C₄₃⋊#95	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
50	C₄⋊C₈×C₄₃	—	2	4	no	—
51	C₄⋊C₈×C₄₃	—	2	8	no	—
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
58	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
59	C₂×Dic₈₆	—	2	4	no	—
60	C₂×Dic₈₆	—	2	4	no	—
61	C₂×Dic₈₆	C₂×C₈	1	8	yes	—
62	C₂×C₈×C₄₃	—	2	4	no	—
63	C₂×C₈×C₄₃	—	2	8	no	—
64	C₄²×C₄₃	—	2	2	no	—
65	C₄²×C₄₃	—	2	4	no	—
66	C₄²×C₄₃	C₂×C₈	1	4	yes	—
67	C₄²×C₄₃	C₂×D₄	1	2	yes	—
68	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
69	C₄²×C₄₃	#41	1	4	yes	—
70	C₄²×C₄₃	#41	1	2	yes	—
71	C₄⋊Dic₄₃	—	2	4	no	—
72	C₄⋊Dic₄₃	—	2	8	no	—
73	C₄⋊Dic₄₃	C₂×C₈	1	16	yes	—
74	Dic₄₃⋊C₄	—	2	8	no	—
75	C₄×Dic₄₃	—	2	4	no	—
76	C₄×Dic₄₃	—	2	4	no	—
77	C₂×C₄₃⋊C₈	—	2	2	no	—
78	C₂×C₄₃⋊C₈	—	2	2	no	—
79	C₂×C₄₃⋊C₈	—	2	2	no	—
80	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
81	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
82	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
83	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
84	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	2	2	no	—
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
91	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
92	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₄₃	#131	1	2	yes	—
95	C₂×C₄×C₄₃	#131	1	4	yes	—
96	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
98	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
99	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
100	C₂×Dic₄₃	—	2	4	no	—
101	C₂×Dic₄₃	—	2	4	no	—
102	C₂×Dic₄₃	—	2	4	no	—
103	C₂×Dic₄₃	—	2	8	no	—
104	C₂×Dic₄₃	C₂²×C₈	1	8	yes	—
105	Dic₈₆	—	2	16	no	—
106	C₈×C₄₃	—	4	4	no	—
107	C₄₃⋊C₈	—	2	2	no	—
108	C₄₃⋊C₈	—	2	4	no	—
109	C₄₃⋊C₈	C₄⋊Q₈	1	4	yes	complemented
110	C₈×C₄⋊Q₈	—	43	1	no	maximal, Sylow
111	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂²×C₄₃	C₂×C₄×Q₈	1	2	yes	—
113	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
114	C₂²×C₄₃	C₂×#131	1	1	yes	characteristic
115	C₂²×C₄₃	C₂×#131	1	2	yes	—
116	C₄×C₄₃	—	2	2	no	—
117	C₄×C₄₃	—	2	4	no	—
118	C₄×C₄₃	C₈×D₄	1	4	yes	—
119	C₄×C₄₃	C₈×Q₈	1	8	yes	complemented
120	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₄×C₄₃	#532	1	1	yes	characteristic
122	C₄×C₄₃	#533	1	2	yes	—
123	Dic₄₃	—	2	8	no	—
124	Dic₄₃	—	4	4	no	—
125	C₂×C₈×Q₈	—	43	2	no	—
126	C₄×C₄⋊Q₈	—	43	1	no	—
127	#1984	—	43	2	no	—
128	#1939	—	43	4	no	—
129	#1842	D₄₃	1	1	yes	characteristic
130	C₈×C₄⋊C₄	—	43	4	no	—
131	C₄²×C₈	—	43	1	no	—
132	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
133	C₂×C₄₃	C₂×C₈×D₄	1	1	yes	characteristic
134	C₂×C₄₃	#2931	1	1	yes	characteristic
135	C₂×C₄₃	C₂×C₈×Q₈	1	2	yes	—
136	C₂×C₄₃	#2960	1	2	yes	—
137	C₂×C₄⋊Q₈	—	43	1	no	—
138	C₂×C₄×Q₈	—	43	2	no	—
139	C₈×Q₈	—	86	8	no	—
140	C₂×C₄⋊C₈	—	43	4	no	—
141	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
142	C₂×C₄⋊C₈	C₂×D₄₃	1	4	yes	—
143	C₂×C₄×C₈	—	43	1	no	—
144	C₂×C₄×C₈	—	43	2	no	—
145	C₂×C₄×C₈	—	43	4	no	—
146	C₂×C₄×C₈	—	43	2	no	—
147	C₂×C₄×C₈	—	43	1	no	—
148	#391	—	43	2	no	—
149	#389	—	43	4	no	—
150	#384	—	43	1	no	—
151	C₄×C₄⋊C₄	—	43	4	no	—
152	C₄³	C₂×D₄₃	1	1	yes	characteristic
153	#336	—	43	8	no	—
154	C₄₃	C₈×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	43	2	no	—
156	C₂²×C₈	—	43	1	no	—
157	C₂²×C₈	—	43	2	no	—
158	C₂²×C₈	—	43	1	no	—
159	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
160	C₄⋊Q₈	—	43	8	no	—
161	C₄×Q₈	—	86	4	no	—
162	C₄×Q₈	—	86	4	no	—
163	C₂×C₄⋊C₄	—	43	4	no	—
164	C₂×C₄⋊C₄	—	43	4	no	—
165	C₂×C₄⋊C₄	—	43	2	no	—
166	C₂×C₄⋊C₄	—	43	4	no	—
167	C₂×C₄²	—	43	4	no	—
168	C₂×C₄²	C₄×D₄₃	1	2	yes	—
169	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
173	C₄⋊C₈	—	2	4	no	—
174	C₄⋊C₈	—	2	8	no	—
175	C₄⋊C₈	—	86	8	no	—
176	C₄×C₈	—	43	4	no	—
177	C₄×C₈	—	43	8	no	—
178	C₄×C₈	—	86	8	no	—
179	C₄×C₈	—	86	4	no	—
180	C₄×C₈	—	86	2	no	—
181	#95	—	43	8	no	—
182	C₂×Q₈	—	43	8	no	—
183	C₂×Q₈	—	86	4	no	—
184	C₂×Q₈	—	86	4	no	—
185	C₂²×C₄	—	43	4	no	—
186	C₂²×C₄	—	43	4	no	—
187	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
189	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
191	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	—	43	2	no	—
195	C₂×C₈	—	43	4	no	—
196	C₂×C₈	—	43	2	no	—
197	C₂×C₈	—	43	4	no	—
198	C₂×C₈	—	86	2	no	—
199	C₂×C₈	—	86	2	no	—
200	C₂×C₈	—	86	2	no	—
201	C₄⋊C₄	—	43	16	no	—
202	C₄⋊C₄	—	86	8	no	—
203	C₄⋊C₄	—	86	4	no	—
204	C₄⋊C₄	—	86	8	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	86	4	no	—
208	C₄²	—	86	4	no	—
209	C₄²	C₈×D₄₃	1	4	yes	—
210	C₄²	D₄×D₄₃	1	2	yes	—
211	C₄²	#10621900	1	4	yes	—
212	C₄²	Q₈×D₄₃	1	4	yes	—
213	C₄²	#10621902	1	2	yes	—
214	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
215	Q₈	—	86	16	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	43	8	no	—
220	C₂×C₄	—	86	4	no	—
221	C₂×C₄	—	86	4	no	—
222	C₂×C₄	—	86	4	no	—
223	C₂×C₄	—	86	8	no	—
224	C₂×C₄	#13063287	1	2	yes	—
225	C₂×C₄	#13063287	1	2	yes	—
226	C₂×C₄	#13063290	1	4	yes	—
227	C₂×C₄	#13063290	1	4	yes	—
228	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
229	C₂×C₄	C₂×C₈×D₄₃	1	4	yes	—
230	C₂×C₄	#13063333	1	2	yes	—
231	C₂×C₄	#13063333	1	4	yes	—
232	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
233	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
234	C₂×C₄	C₂×#10621900	1	2	yes	—
235	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
236	C₈	—	4	4	no	—
237	C₈	—	43	4	no	—
238	C₈	—	86	2	no	—
239	C₈	—	86	4	no	—
240	C₂²	C₂×#13063287	1	1	yes	characteristic
241	C₂²	C₂×#13063290	1	2	yes	—
242	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
243	C₂²	C₂×#13063333	1	2	yes	—
244	C₂²	C₂×#13063333	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	—	86	8	no	—
248	C₄	—	172	4	no	—
249	C₄	#14582964	1	4	yes	—
250	C₄	#14583023	1	8	yes	complemented
251	C₄	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
252	C₄	#14583447	1	1	yes	characteristic
253	C₄	#14583454	1	2	yes	—
254	C₂	#19137162	1	1	yes	characteristic
255	C₂	C₂×#14582964	1	1	yes	characteristic
256	C₂	#19143306	1	1	yes	characteristic
257	C₂	C₂×#14583023	1	2	yes	—
258	C₂	#19143525	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₈
Fitting	C₄₃×#1842	C₂
socle, prime radical	C₂³×C₄₃	C₂³×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1842	D₄₃
hypocenter	C₄₃	C₈×C₄⋊Q₈
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1842

Complements

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

N	H ≅ G/N	type
29. #13063325	C₈	semidirect
41. C₄×C₄₃⋊C₈	Q₈	semidirect
109. C₄₃⋊C₈	C₄⋊Q₈	semidirect
119. C₄×C₄₃	C₈×Q₈	semidirect
154. C₄₃	C₈×C₄⋊Q₈	semidirect
250. C₄	#14583023	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583023	2
C₄×#13063325	2
#19135530	2
#19135311	2
#19134460	2
C₄²×C₄₃⋊C₈	2
C₄₃×#1842	2
C₈×C₄⋊Q₈	43