Subgroups — #25105219

Order 11008.

1022 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583021	C₂	1	4	yes	maximal, complemented
2	#19143307	C₂	1	2	yes	maximal, complemented
3	#19135489	C₂	1	4	yes	maximal, complemented
4	#19135424	C₂	1	2	yes	maximal, complemented
5	#19134767	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13063235	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#3147	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063332	C₂²	1	4	yes	—
9	#14583021	—	2	16	no	—
10	#14582977	—	2	4	no	—
11	D₄×C₄₃⋊C₈	—	2	4	no	—
12	C₂×#13063261	C₂²	1	4	yes	—
13	#14582923	C₂²	1	2	yes	—
14	C₂×#13063252	C₂²	1	8	yes	—
15	#14582866	C₂²	1	4	yes	—
16	C₂×#13063235	C₂²	1	2	yes	—
17	C₂×#13063235	C₂²	1	2	yes	—
18	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
20	C₂×#135×C₄₃	C₂²	1	4	yes	—
21	C₄₃×#532	C₂²	1	2	yes	—
22	C₄₃×#495	C₄	1	4	yes	complemented
23	C₄₃×#494	C₄	1	1	yes	characteristic, complemented
24	C₄₃×#493	C₄	1	1	yes	characteristic, complemented
25	C₄₃×#492	C₄	1	2	yes	complemented
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	#13063332	—	2	16	no	—
28	C₂×#10621877	—	2	4	no	—
29	C₂×#10621877	C₂³	1	4	yes	—
30	C₂²×C₄₃⋊C₈	—	2	4	no	—
31	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
33	C₂×C₄×Dic₄₃	—	2	2	no	—
34	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
35	#13063261	—	2	8	no	—
36	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
37	#13063252	—	2	16	no	—
38	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
39	#13063235	—	2	4	no	—
40	#13063235	D₄	1	8	yes	complemented
41	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
42	C₂×#41×C₄₃	C₂³	1	4	yes	—
43	#135×C₄₃	—	2	8	no	—
44	#135×C₄₃	—	2	8	no	—
45	C₂×Q₁₆×C₄₃	C₂×C₄	1	4	yes	—
46	C₂×SD₁₆×C₄₃	C₂×C₄	1	8	yes	—
47	C₂×D₈×C₄₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
50	#127×C₄₃	C₂×C₄	1	2	yes	—
51	#124×C₄₃	C₂×C₄	1	4	yes	—
52	#121×C₄₃	—	2	4	no	—
53	C₄×D₄×C₄₃	—	2	4	no	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
58	#41×C₄₃	—	2	8	no	—
59	#41×C₄₃	—	2	8	no	—
60	C₂×Q₈×C₄₃	C₂²×C₄	1	4	yes	—
61	C₂×D₄×C₄₃	—	2	4	no	—
62	C₂×D₄×C₄₃	—	2	4	no	—
63	C₂×D₄×C₄₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	#41	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	2	no	—
68	Q₁₆×C₄₃	—	2	8	no	—
69	SD₁₆×C₄₃	—	2	16	no	—
70	D₈×C₄₃	—	2	8	no	—
71	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
74	C₂×C₈×C₄₃	#41	1	4	yes	—
75	C₄⋊C₄×C₄₃	—	2	4	no	—
76	C₂²⋊C₄×C₄₃	—	2	8	no	—
77	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₄₃	C₂×D₄	1	2	yes	—
80	C₄×Dic₄₃	—	2	2	no	—
81	C₄×Dic₄₃	—	4	4	no	—
82	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
83	#10621877	—	2	8	no	—
84	#10621877	—	4	8	no	—
85	C₂×C₄₃⋊C₈	—	2	2	no	—
86	C₂×C₄₃⋊C₈	—	2	4	no	—
87	C₂×C₄₃⋊C₈	—	4	8	no	—
88	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
89	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
90	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
91	C₂³×C₄₃	—	2	4	no	—
92	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
93	Q₈×C₄₃	—	2	8	no	—
94	D₄×C₄₃	—	2	8	no	—
95	D₄×C₄₃	—	4	8	no	—
96	D₄×C₄₃	—	4	8	no	—
97	C₂×C₄×C₄₃	—	2	4	no	—
98	C₂×C₄×C₄₃	—	2	4	no	—
99	C₂×C₄×C₄₃	—	4	4	no	—
100	C₂×C₄×C₄₃	—	4	4	no	—
101	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
107	C₂×Dic₄₃	—	2	4	no	—
108	C₂×Dic₄₃	—	4	4	no	—
109	C₈×C₄₃	C₄×D₄	1	4	yes	—
110	C₈×C₄₃	C₄×D₄	1	4	yes	—
111	C₄₃⋊C₈	—	2	2	no	—
112	C₄₃⋊C₈	—	4	4	no	—
113	C₄₃⋊C₈	#127	1	4	yes	—
114	#21706	—	43	1	no	maximal, Sylow
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	—	4	4	no	—
117	C₂²×C₄₃	—	4	8	no	—
118	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₄₃	#532	1	2	yes	—
120	C₄×C₄₃	—	2	2	no	—
121	C₄×C₄₃	—	4	4	no	—
122	C₄×C₄₃	#444	1	4	yes	complemented
123	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
124	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
125	C₄×C₄₃	#532	1	1	yes	characteristic
126	Dic₄₃	—	4	4	no	—
127	#3147	D₄₃	1	1	yes	characteristic
128	C₂×#444	—	43	4	no	—
129	#2931	—	43	2	no	—
130	#1975	—	43	4	no	—
131	#1962	—	43	2	no	—
132	#1848	—	43	1	no	—
133	C₄×#97	—	43	1	no	—
134	C₂×C₄₃	—	2	2	no	—
135	C₂×C₄₃	—	8	4	no	—
136	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
137	C₂×C₄₃	C₂×#444	1	2	yes	—
138	C₂×#135	C₂×D₄₃	1	4	yes	—
139	C₂×#131	—	43	4	no	—
140	#532	C₂×D₄₃	1	2	yes	—
141	#495	Dic₄₃	1	4	yes	complemented
142	#494	Dic₄₃	1	1	yes	characteristic, complemented
143	#493	Dic₄₃	1	1	yes	characteristic, complemented
144	#492	Dic₄₃	1	2	yes	complemented
145	#444	—	86	16	no	—
146	#436	—	86	4	no	—
147	C₈×D₄	—	86	4	no	—
148	C₂×#108	—	43	4	no	—
149	#423	—	43	2	no	—
150	C₂×#104	—	43	8	no	—
151	#408	—	43	4	no	—
152	C₂×#97	—	43	2	no	—
153	C₂×#97	—	43	2	no	—
154	C₂×C₄×C₈	—	43	1	no	—
155	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
156	C₄³	—	43	1	no	—
157	C₄₃	#21706	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	C₂²×D₄₃	1	4	yes	—
159	#135	—	2	8	no	—
160	#135	—	2	8	no	—
161	C₂×Q₁₆	C₂×Dic₄₃	1	4	yes	—
162	C₂×SD₁₆	C₂×Dic₄₃	1	8	yes	—
163	C₂×D₈	C₂×Dic₄₃	1	4	yes	—
164	#131	—	86	16	no	—
165	C₂×M₁₆	—	43	4	no	—
166	C₂×M₁₆	—	86	4	no	—
167	C₂²×C₈	—	43	1	no	—
168	C₂²×C₈	—	43	1	no	—
169	C₂²×C₈	—	86	4	no	—
170	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
171	C₄⋊Q₈	C₂×Dic₄₃	1	2	yes	—
172	#127	C₂×Dic₄₃	1	2	yes	—
173	#124	C₂×Dic₄₃	1	4	yes	—
174	#121	—	2	4	no	—
175	C₄×D₄	—	2	4	no	—
176	C₂×C₄²	—	43	2	no	—
177	C₂×C₄²	—	86	2	no	—
178	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
179	#108	—	86	8	no	—
180	C₄⋊C₈	—	86	4	no	—
181	#104	—	86	16	no	—
182	C₂²⋊C₈	—	86	8	no	—
183	#97	—	43	8	no	—
184	#97	—	86	4	no	—
185	C₄×C₈	—	43	4	no	—
186	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
187	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
188	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
189	#41	—	2	8	no	—
190	#41	—	2	8	no	—
191	C₂×Q₈	C₂²×Dic₄₃	1	4	yes	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	C₂²×Dic₄₃	1	4	yes	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	86	2	no	—
197	C₂²×C₄	#10621900	1	2	yes	—
198	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
199	Q₁₆	—	2	8	no	—
200	SD₁₆	—	2	16	no	—
201	D₈	—	2	8	no	—
202	M₁₆	—	86	8	no	—
203	M₁₆	—	172	8	no	—
204	C₂×C₈	—	43	4	no	—
205	C₂×C₈	—	43	2	no	—
206	C₂×C₈	—	43	2	no	—
207	C₂×C₈	—	86	2	no	—
208	C₂×C₈	—	86	4	no	—
209	C₂×C₈	—	172	8	no	—
210	C₂×C₈	D₄×D₄₃	1	4	yes	—
211	C₂×C₈	#10621900	1	4	yes	—
212	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
213	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₄²	—	43	4	no	—
217	C₄²	—	86	2	no	—
218	C₄²	—	172	4	no	—
219	C₄²	D₄×D₄₃	1	2	yes	—
220	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
221	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	C₂×#10621900	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	D₄	—	2	8	no	—
226	D₄	—	4	8	no	—
227	D₄	—	4	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	86	4	no	—
233	C₂×C₄	—	172	4	no	—
234	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
235	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
236	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
238	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
239	C₂×C₄	C₂×#10621900	1	2	yes	—
240	C₈	—	43	4	no	—
241	C₈	—	86	2	no	—
242	C₈	—	172	4	no	—
243	C₈	D₄×Dic₄₃	1	4	yes	—
244	C₈	D₄×Dic₄₃	1	4	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	4	4	no	—
248	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
249	C₂²	#14583447	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	4	4	no	—
252	C₄	—	172	4	no	—
253	C₄	#14583021	1	4	yes	complemented
254	C₄	C₂×D₄×Dic₄₃	1	2	yes	—
255	C₄	#127×D₄₃	1	1	yes	characteristic
256	C₄	#14583447	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	8	4	no	—
259	C₂	#127×Dic₄₃	1	1	yes	characteristic
260	C₂	C₂×#14583021	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting, prime radical	C₄₃×#3147	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#3147	D₄₃
hypocenter	C₄₃	#21706
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#14583021	C₂	semidirect
2. #19143307	C₂	semidirect
3. #19135489	C₂	semidirect
4. #19135424	C₂	semidirect
5. #19134767	C₂	semidirect
6. C₄×#13063235	C₂	semidirect
22. C₄₃×#495	C₄	semidirect
23. C₄₃×#494	C₄	semidirect
24. C₄₃×#493	C₄	semidirect
25. C₄₃×#492	C₄	semidirect
40. #13063235	D₄	semidirect
122. C₄×C₄₃	#444	semidirect
141. #495	Dic₄₃	semidirect
142. #494	Dic₄₃	semidirect
143. #493	Dic₄₃	semidirect
144. #492	Dic₄₃	semidirect
157. C₄₃	#21706	semidirect
253. C₄	#14583021	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583021	2
#19143307	2
#19135489	2
#19135424	2
#19134767	2
C₄×#13063235	2
C₄₃×#3147	2
#21706	43