Subgroups — #25094567

Order 11008.

1166 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582857	C₂	1	2	yes	maximal
2	C₂²×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#19137002	C₂	1	1	yes	maximal, characteristic
4	#19134465	C₂	1	4	yes	maximal, complemented
5	#19134460	C₂	1	4	yes	maximal, complemented
6	#19134293	C₂	1	2	yes	maximal
7	C₂×C₄₃×#423	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×#13063284	C₂²	1	2	yes	—
11	#14582857	—	2	8	no	—
12	C₂×#13063235	C₂²	1	4	yes	—
13	C₂×C₄×C₄₃⋊C₈	C₄	1	8	yes	complemented
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₃	C₂²	1	4	yes	—
17	#14582769	C₄	1	2	yes	—
18	#14582765	C₄	1	2	yes	—
19	#14582759	C₄	1	4	yes	—
20	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
21	#14582653	C₂²	1	8	yes	—
22	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#423	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₄₃	C₂²	1	4	yes	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	C₂²×C₄₃⋊C₈	—	2	2	no	—
28	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
31	C₂²×C₄₃⋊C₈	Q₈	1	4	yes	—
32	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
35	#13063284	—	2	4	no	—
36	#13063284	—	2	4	no	—
37	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊Dic₄₃	C₂×C₄	1	8	yes	—
39	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₄₃	C₂³	1	4	yes	—
41	#13063235	—	2	8	no	—
42	C₄×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
43	C₄×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
44	C₈×Dic₄₃	—	2	8	no	—
45	C₄₃⋊#95	C₂×C₄	1	8	yes	—
46	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
47	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₄₃	—	2	4	no	—
49	C₂×M₁₆×C₄₃	—	2	4	no	—
50	C₂×M₁₆×C₄₃	C₂×C₄	1	8	yes	—
51	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
52	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
57	C₄⋊C₈×C₄₃	C₂×C₄	1	16	yes	—
58	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	—	2	8	no	—
60	C₂²×C₄×C₄₃	C₄²	1	8	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₃	#41	1	2	yes	—
69	C₂²×C₄×C₄₃	#41	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₃	C₂²×C₄	1	8	yes	—
72	M₁₆×C₄₃	—	2	16	no	—
73	C₂×C₈×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	—	2	4	no	—
75	C₂×C₈×C₄₃	C₂²×C₄	1	8	yes	—
76	C₄²×C₄₃	C₄²	1	4	yes	—
77	C₄²×C₄₃	C₄²	1	4	yes	—
78	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
80	C₂²⋊Dic₄₃	—	2	8	no	—
81	C₄⋊Dic₄₃	—	2	8	no	—
82	C₄⋊Dic₄₃	—	2	16	no	—
83	C₄×Dic₄₃	—	2	4	no	—
84	C₄×Dic₄₃	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	2	8	no	—
86	C₂×C₄₃⋊C₈	—	2	2	no	—
87	C₂×C₄₃⋊C₈	—	2	2	no	—
88	C₂×C₄₃⋊C₈	C₄⋊C₄	1	16	yes	complemented
89	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
90	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
93	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
94	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
95	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
96	C₂³×C₄₃	—	2	4	no	—
97	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
98	C₂³×C₄₃	C₄×D₄	1	2	yes	—
99	C₂³×C₄₃	C₄×Q₈	1	2	yes	—
100	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	2	16	no	—
103	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₄₃	#117	1	4	yes	—
108	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
110	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₄₃	#131	1	8	yes	—
113	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₄₃	—	2	8	no	—
120	C₂×Dic₄₃	—	2	16	no	—
121	C₈×C₄₃	—	2	8	no	—
122	C₄₃⋊C₈	—	2	4	no	—
123	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
124	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
125	#20437	—	43	1	no	maximal, Sylow
126	C₂²×C₄₃	—	2	4	no	—
127	C₂²×C₄₃	—	2	4	no	—
128	C₂²×C₄₃	—	2	4	no	—
129	C₂²×C₄₃	C₄×C₄⋊C₄	1	4	yes	—
130	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
131	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₂²×C₄₃	C₂×#131	1	4	yes	—
134	C₄×C₄₃	—	2	4	no	—
135	C₄×C₄₃	C₄×C₄⋊C₄	1	4	yes	—
136	C₄×C₄₃	#405	1	8	yes	complemented
137	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
139	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
141	Dic₄₃	—	4	8	no	—
142	C₂×#423	D₄₃	1	1	yes	characteristic
143	C₂×#405	—	43	2	no	—
144	C₂²×C₄×C₈	—	43	1	no	—
145	#2285	—	43	1	no	—
146	#1770	—	43	4	no	—
147	C₈×C₄⋊C₄	—	43	4	no	—
148	#1729	—	43	2	no	—
149	C₂×C₄₃	—	2	4	no	—
150	C₂×C₄₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄₃	C₂×#405	1	2	yes	—
152	C₂×C₄₃	#2931	1	2	yes	—
153	C₂×C₄₃	#2960	1	2	yes	—
154	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
155	C₂³×C₈	—	43	1	no	—
156	C₂³×C₈	—	43	1	no	—
157	C₂×#117	—	43	2	no	—
158	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
159	#423	Dic₄₃	1	8	yes	complemented
160	C₂×C₄⋊C₈	C₂×D₄₃	1	4	yes	—
161	#405	—	86	8	no	—
162	C₂×#97	—	43	4	no	—
163	C₂×C₄×C₈	—	43	2	no	—
164	C₂×C₄×C₈	—	43	2	no	—
165	C₂×C₄×C₈	—	43	8	no	—
166	C₂×C₄×C₈	—	43	4	no	—
167	#384	—	43	2	no	—
168	#382	—	43	2	no	—
169	#380	—	43	4	no	—
170	C₄×C₄⋊C₄	—	43	4	no	—
171	#336	—	43	8	no	—
172	C₄₃	#20437	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	C₄×D₄₃	1	2	yes	—
174	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	C₂×Dic₄₃	1	8	yes	—
178	C₂²×C₈	—	43	4	no	—
179	C₂²×C₈	—	43	4	no	—
180	C₂²×C₈	—	43	4	no	—
181	C₂²×C₈	—	43	1	no	—
182	C₂²×C₈	—	43	1	no	—
183	C₂²×C₈	—	43	4	no	—
184	C₂²×C₈	—	43	2	no	—
185	C₂²×C₈	—	86	2	no	—
186	C₂²×C₈	—	86	2	no	—
187	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
188	#117	—	86	4	no	—
189	#117	—	86	4	no	—
190	C₂×C₄⋊C₄	—	43	4	no	—
191	C₂×C₄⋊C₄	—	43	8	no	—
192	C₂×C₂²⋊C₄	—	43	4	no	—
193	C₂×C₄²	—	43	4	no	—
194	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₄₃	1	4	yes	—
197	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
198	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
199	C₄⋊C₈	C₂×Dic₄₃	1	16	yes	—
200	#97	—	86	8	no	—
201	C₄×C₈	—	43	8	no	—
202	C₄×C₈	—	43	8	no	—
203	C₄×C₈	—	86	8	no	—
204	#95	—	43	8	no	—
205	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	43	8	no	—
208	C₂²×C₄	C₄×Dic₄₃	1	8	yes	—
209	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
210	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
213	C₂²×C₄	D₄×D₄₃	1	2	yes	—
214	C₂²×C₄	#10621900	1	2	yes	—
215	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
216	C₂²×C₄	#10621902	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
218	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
219	M₁₆	—	2	16	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	43	16	no	—
223	C₂×C₈	—	43	2	no	—
224	C₂×C₈	—	43	2	no	—
225	C₂×C₈	—	43	2	no	—
226	C₂×C₈	—	43	2	no	—
227	C₂×C₈	—	43	2	no	—
228	C₂×C₈	—	43	2	no	—
229	C₂×C₈	—	43	4	no	—
230	C₂×C₈	—	86	8	no	—
231	C₂×C₈	—	86	2	no	—
232	C₂×C₈	—	86	2	no	—
233	C₂×C₈	C₂²×Dic₄₃	1	8	yes	—
234	C₄⋊C₄	—	86	8	no	—
235	C₄⋊C₄	—	86	16	no	—
236	C₂²⋊C₄	—	86	8	no	—
237	C₄²	—	86	4	no	—
238	C₄²	—	86	4	no	—
239	C₄²	C₄×Dic₄₃	1	4	yes	—
240	C₄²	C₄×Dic₄₃	1	4	yes	—
241	C₄²	C₂×C₄×D₄₃	1	4	yes	—
242	C₄²	C₂²×Dic₄₃	1	4	yes	—
243	C₂³	—	2	4	no	—
244	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
245	C₂³	#13063287	1	2	yes	—
246	C₂³	#13063290	1	2	yes	—
247	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	86	8	no	—
251	C₂×C₄	—	86	16	no	—
252	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
253	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
254	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
255	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
256	C₂×C₄	#13063283	1	4	yes	—
257	C₂×C₄	#13063287	1	2	yes	—
258	C₂×C₄	D₄×Dic₄₃	1	4	yes	—
259	C₂×C₄	#13063290	1	2	yes	—
260	C₂×C₄	Q₈×Dic₄₃	1	4	yes	—
261	C₂×C₄	#13063333	1	8	yes	—
262	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
263	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
265	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
267	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	43	4	no	—
270	C₈	—	43	4	no	—
271	C₈	—	86	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₄⋊C₄×Dic₄₃	1	4	yes	—
276	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
277	C₂²	C₂×#13063287	1	1	yes	characteristic
278	C₂²	C₂×#13063290	1	1	yes	characteristic
279	C₂²	C₂×#13063333	1	4	yes	—
280	C₄	—	2	4	no	—
281	C₄	—	172	8	no	—
282	C₄	C₄⋊C₄×Dic₄₃	1	4	yes	—
283	C₄	#14582857	1	8	yes	complemented
284	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
285	C₄	C₂×#13063283	1	1	yes	characteristic
286	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
287	C₄	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
288	C₂	—	2	4	no	—
289	C₂	C₂×C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
290	C₂	C₂×#14582857	1	2	yes	—
291	C₂	#19143306	1	2	yes	—
292	C₂	#19143525	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₂×C₄₃×#423	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#423	D₄₃
hypocenter	C₄₃	#20437
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #19134465	C₂	semidirect
5. #19134460	C₂	semidirect
13. C₂×C₄×C₄₃⋊C₈	C₄	semidirect
24. C₄₃×#423	C₄	semidirect
88. C₂×C₄₃⋊C₈	C₄⋊C₄	semidirect
136. C₄×C₄₃	#405	semidirect
159. #423	Dic₄₃	semidirect
172. C₄₃	#20437	semidirect
283. C₄	#14582857	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582857	2
C₂²×C₄×C₄₃⋊C₈	2
#19137002	2
#19134465	2
#19134460	2
#19134293	2
C₂×C₄₃×#423	2
#20437	43