Subgroups — #25249908

Order 11008.

682 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	#135×Dic₄₃	C₂	1	2	yes	maximal, complemented
2	#19136471	C₂	1	2	yes	maximal, complemented
3	#19136466	C₂	1	2	yes	maximal, complemented
4	#19136460	C₂	1	4	yes	maximal, complemented
5	#19136379	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136302	C₂	1	2	yes	maximal, complemented
7	C₂×#14582674	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#508	C₂	1	1	yes	maximal, characteristic
9	#41×Dic₄₃	—	2	2	no	—
10	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₄₃	C₂²	1	4	yes	—
12	SD₁₆×Dic₄₃	—	2	4	no	—
13	D₈×Dic₄₃	C₂²	1	4	yes	—
14	#14582937	C₂²	1	2	yes	—
15	#14582895	—	2	2	no	—
16	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
17	#14582727	C₂²	1	2	yes	—
18	#14582725	C₂²	1	2	yes	—
19	#14582703	C₂²	1	4	yes	—
20	#14582700	C₂²	1	4	yes	—
21	#14582674	C₂²	1	2	yes	—
22	#14582674	C₂²	1	2	yes	—
23	C₂×#135×C₄₃	C₂²	1	2	yes	—
24	C₄₃×#508	C₄	1	8	yes	complemented
25	C₂×Q₃₂×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×SD₃₂×C₄₃	C₂²	1	2	yes	—
27	C₂×D₁₆×C₄₃	C₂²	1	1	yes	characteristic
28	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
30	Q₈×Dic₄₃	—	2	4	no	—
31	D₄×Dic₄₃	—	2	4	no	—
32	D₄×Dic₄₃	—	4	4	no	—
33	#13063284	—	2	2	no	—
34	#13063284	—	4	2	no	—
35	C₂×C₄×Dic₄₃	—	4	2	no	—
36	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
37	C₂×C₄₃⋊C₁₆	—	2	2	no	—
38	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
39	#13063259	C₂³	1	4	yes	—
40	#13063257	—	2	2	no	—
41	#13063249	—	2	4	no	—
42	#13063246	—	2	4	no	—
43	C₈×Dic₄₃	D₄	1	2	yes	—
44	C₈×Dic₄₃	C₂³	1	2	yes	—
45	C₂×#41×C₄₃	—	2	2	no	—
46	#135×C₄₃	—	2	4	no	—
47	#135×C₄₃	C₂×C₄	1	8	yes	complemented
48	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
49	C₂×SD₁₆×C₄₃	—	2	2	no	—
50	C₂×D₈×C₄₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	Q₃₂×C₄₃	C₂×C₄	1	4	yes	complemented
53	SD₃₂×C₄₃	C₂×C₄	1	8	yes	complemented
54	D₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
58	#41×C₄₃	—	2	8	no	—
59	#41×C₄₃	—	4	4	no	—
60	C₂×Q₈×C₄₃	—	2	2	no	—
61	C₂×D₄×C₄₃	—	2	2	no	—
62	C₂×D₄×C₄₃	—	4	2	no	—
63	C₂²×C₄×C₄₃	—	4	2	no	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₄₃	—	4	4	no	—
66	C₂²×Dic₄₃	D₈	1	2	yes	—
67	Q₁₆×C₄₃	—	2	2	no	—
68	Q₁₆×C₄₃	C₂²×C₄	1	4	yes	—
69	SD₁₆×C₄₃	—	2	8	no	—
70	D₈×C₄₃	—	2	2	no	—
71	D₈×C₄₃	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	#41	1	2	yes	—
75	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
76	C₂²⋊Dic₄₃	—	4	4	no	—
77	C₂²⋊Dic₄₃	—	4	2	no	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	C₄⋊Dic₄₃	—	4	4	no	—
80	C₄×Dic₄₃	—	4	2	no	—
81	C₄×Dic₄₃	—	4	4	no	—
82	C₄×Dic₄₃	D₈	1	2	yes	—
83	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
84	C₂×C₄₃⋊C₈	—	2	2	no	—
85	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
86	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
87	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
88	C₄₃⋊C₁₆	—	2	2	no	—
89	C₂³×C₄₃	—	4	2	no	—
90	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₄₃	—	2	4	no	—
92	Q₈×C₄₃	—	4	2	no	—
93	D₄×C₄₃	—	2	4	no	—
94	D₄×C₄₃	—	4	8	no	—
95	D₄×C₄₃	—	4	2	no	—
96	C₂×C₄×C₄₃	—	4	2	no	—
97	C₂×C₄×C₄₃	—	4	8	no	—
98	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₄₃	#135	1	2	yes	—
101	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×Dic₄₃	—	4	2	no	—
103	C₂×Dic₄₃	—	4	4	no	—
104	C₂×Dic₄₃	—	8	8	no	—
105	C₂×Dic₄₃	C₂×D₈	1	4	yes	—
106	C₈×C₄₃	C₄×D₄	1	2	yes	—
107	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
109	C₄₃⋊C₈	—	2	2	no	—
110	#33451	—	43	1	no	maximal, Sylow
111	C₂²×C₄₃	—	2	2	no	—
112	C₂²×C₄₃	—	4	4	no	—
113	C₂²×C₄₃	—	8	4	no	—
114	C₂²×C₄₃	C₄×D₈	1	2	yes	—
115	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₄₃	—	4	4	no	—
117	C₄×C₄₃	C₄×D₈	1	2	yes	—
118	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
120	Dic₄₃	—	2	4	no	—
121	Dic₄₃	—	8	4	no	—
122	C₂×#508	D₄₃	1	1	yes	characteristic
123	C₄×#135	—	43	2	no	—
124	#2177	—	43	2	no	—
125	#2176	—	43	2	no	—
126	#2175	—	43	4	no	—
127	#2158	—	43	1	no	—
128	#2138	—	43	2	no	—
129	C₂×#346	—	43	1	no	—
130	C₂×C₄₃	—	2	2	no	—
131	C₂×C₄₃	—	8	4	no	—
132	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
133	C₂×C₄₃	#3414	1	2	yes	—
134	C₂×#135	C₂×D₄₃	1	2	yes	—
135	C₄×#41	—	86	2	no	—
136	#508	Dic₄₃	1	8	yes	complemented
137	C₂×Q₃₂	C₂×D₄₃	1	1	yes	characteristic
138	C₂×SD₃₂	C₂×D₄₃	1	2	yes	—
139	C₂×D₁₆	C₂×D₄₃	1	1	yes	characteristic
140	C₂²×C₁₆	—	43	1	no	—
141	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
142	C₄×Q₁₆	—	43	4	no	—
143	C₄×SD₁₆	—	86	4	no	—
144	C₄×D₈	—	43	4	no	—
145	#427	—	43	2	no	—
146	#416	—	86	2	no	—
147	C₂×C₄×C₈	—	43	1	no	—
148	#367	—	43	2	no	—
149	#366	—	43	2	no	—
150	#358	—	43	4	no	—
151	#357	—	43	4	no	—
152	#346	—	43	2	no	—
153	#346	—	43	2	no	—
154	C₄₃	#33451	1	1	yes	characteristic, complemented, Sylow
155	C₂×#41	—	2	2	no	—
156	#135	—	2	4	no	—
157	#135	C₂×Dic₄₃	1	8	yes	complemented
158	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	C₂²×D₄₃	1	2	yes	—
161	C₂²×C₈	—	43	1	no	—
162	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
163	C₄×Q₈	—	86	4	no	—
164	C₄×D₄	—	86	4	no	—
165	C₄×D₄	—	172	4	no	—
166	#117	—	86	2	no	—
167	#117	—	172	2	no	—
168	C₂×C₄²	—	43	1	no	—
169	C₂×C₄²	—	172	2	no	—
170	Q₃₂	C₂×Dic₄₃	1	4	yes	complemented
171	SD₃₂	C₂×Dic₄₃	1	8	yes	complemented
172	D₁₆	C₂×Dic₄₃	1	4	yes	complemented
173	C₂×C₁₆	—	43	2	no	—
174	C₂×C₁₆	—	86	2	no	—
175	C₂×C₁₆	C₂×Dic₄₃	1	4	yes	complemented
176	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
178	#107	—	43	4	no	—
179	#106	—	86	2	no	—
180	#103	—	86	4	no	—
181	#102	—	86	4	no	—
182	C₄×C₈	—	43	2	no	—
183	C₄×C₈	—	43	2	no	—
184	#41	—	2	8	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	43	2	no	—
191	C₂²×C₄	—	172	4	no	—
192	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
193	Q₁₆	—	2	2	no	—
194	Q₁₆	C₂²×Dic₄₃	1	4	yes	—
195	SD₁₆	—	2	8	no	—
196	D₈	—	2	2	no	—
197	D₈	C₂²×Dic₄₃	1	4	yes	—
198	C₂×C₈	—	43	2	no	—
199	C₂×C₈	—	86	2	no	—
200	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
201	C₂×C₈	#10621900	1	2	yes	—
202	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
203	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
204	C₄⋊C₄	—	86	4	no	—
205	C₄⋊C₄	—	172	4	no	—
206	C₂²⋊C₄	—	172	4	no	—
207	C₂²⋊C₄	—	172	2	no	—
208	C₄²	—	43	2	no	—
209	C₄²	—	43	2	no	—
210	C₄²	—	172	4	no	—
211	C₄²	—	172	2	no	—
212	C₁₆	—	86	2	no	—
213	C₁₆	C₂²×Dic₄₃	1	2	yes	—
214	C₁₆	C₂²×Dic₄₃	1	2	yes	—
215	C₂³	—	4	2	no	—
216	C₂³	D₈×D₄₃	1	1	yes	characteristic
217	Q₈	—	2	4	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	8	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	43	4	no	—
225	C₂×C₄	—	344	8	no	—
226	C₂×C₄	—	172	2	no	—
227	C₂×C₄	—	172	4	no	—
228	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
229	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
230	C₂×C₄	#13063344	1	2	yes	—
231	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
232	C₈	—	86	2	no	—
233	C₈	D₄×Dic₄₃	1	2	yes	—
234	C₈	C₂³×Dic₄₃	1	1	yes	characteristic
235	C₈	C₂×#10621900	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	D₈×Dic₄₃	1	2	yes	—
240	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
241	C₄	—	4	4	no	—
242	C₄	—	86	4	no	—
243	C₄	—	344	4	no	—
244	C₄	D₈×Dic₄₃	1	2	yes	—
245	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
246	C₄	C₂×#13063344	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	8	4	no	—
249	C₂	C₂×D₈×Dic₄₃	1	1	yes	characteristic
250	C₂	#19147728	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting, prime radical	C₂×C₄₃×#508	C₂
socle	C₂²×C₄₃	C₂²×D₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#508	D₄₃
hypocenter	C₄₃	#33451
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #135×Dic₄₃	C₂	semidirect
2. #19136471	C₂	semidirect
3. #19136466	C₂	semidirect
4. #19136460	C₂	semidirect
5. #19136379	C₂	semidirect
6. #19136302	C₂	semidirect
7. C₂×#14582674	C₂	semidirect
24. C₄₃×#508	C₄	semidirect
47. #135×C₄₃	C₂×C₄	semidirect
52. Q₃₂×C₄₃	C₂×C₄	semidirect
53. SD₃₂×C₄₃	C₂×C₄	semidirect
54. D₁₆×C₄₃	C₂×C₄	semidirect
55. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
136. #508	Dic₄₃	semidirect
154. C₄₃	#33451	semidirect
157. #135	C₂×Dic₄₃	semidirect
170. Q₃₂	C₂×Dic₄₃	semidirect
171. SD₃₂	C₂×Dic₄₃	semidirect
172. D₁₆	C₂×Dic₄₃	semidirect
175. C₂×C₁₆	C₂×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
#135×Dic₄₃	2
#19136471	2
#19136466	2
#19136460	2
#19136379	2
#19136302	2
C₂×#14582674	2
C₂×C₄₃×#508	2
#33451	43