Subgroups — #25068415

Order 11008.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145181	C₂	1	1	yes	maximal, characteristic
2	#19143542	C₂	1	2	yes	maximal
3	#19133505	C₂	1	2	yes	maximal, complemented
4	#19133468	C₂	1	4	yes	maximal, complemented
5	#19133415	C₂	1	2	yes	maximal, complemented
6	C₂×#14582650	C₂	1	1	yes	maximal, characteristic
7	#19132999	C₂	1	2	yes	maximal
8	C₄₃×#1451	C₂	1	1	yes	maximal, characteristic
9	#14583456	C₂²	1	2	yes	—
10	#14583304	C₄	1	1	yes	characteristic
11	#14583279	C₄	1	1	yes	characteristic
12	#14583250	C₄	1	2	yes	—
13	#14583140	C₄	1	2	yes	—
14	#14583140	C₄	1	2	yes	—
15	#14583029	C₂²	1	4	yes	—
16	#14583027	—	2	4	no	—
17	C₈×Dic₈₆	C₂²	1	4	yes	—
18	C₄×Dic₁₇₂	C₂²	1	2	yes	—
19	C₄×Dic₁₇₂	C₂²	1	2	yes	—
20	#14582927	—	2	2	no	—
21	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
22	#14582857	—	2	2	no	—
23	#14582650	C₂²	1	2	yes	—
24	#14582650	C₂²	1	1	yes	characteristic
25	#14582650	C₂²	1	1	yes	characteristic
26	#14582631	C₂²	1	4	yes	—
27	#14582631	C₂²	1	4	yes	—
28	C₄₃×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#322	C₂²	1	2	yes	—
32	C₄₃×#322	C₂²	1	2	yes	—
33	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
35	C₂²×C₄₃⋊C₈	—	2	2	no	—
36	#13063325	C₂×C₄	1	2	yes	—
37	#13063325	C₂×C₄	1	2	yes	—
38	#13063317	—	2	2	no	—
39	#13063304	—	2	2	no	—
40	#13063304	C₂×C₄	1	4	yes	—
41	C₄×Dic₈₆	—	2	2	no	—
42	C₄×Dic₈₆	C₂³	1	4	yes	—
43	#13063284	—	2	2	no	—
44	#13063259	C₂×C₄	1	2	yes	—
45	#13063259	C₂×C₄	1	2	yes	—
46	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
47	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
49	Dic₄₃⋊C₈	—	2	4	no	—
50	Dic₄₃⋊C₈	—	2	4	no	—
51	#13063250	C₂×C₄	1	4	yes	—
52	#13063250	C₂×C₄	1	4	yes	—
53	#13063235	—	2	4	no	—
54	C₈×Dic₄₃	—	2	4	no	—
55	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
56	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₃	—	2	1	no	—
60	C₂²⋊C₈×C₄₃	—	2	1	no	—
61	#97×C₄₃	D₄	1	2	yes	—
62	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₄₃	C₂³	1	2	yes	—
65	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₃	C₂³	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
71	C₂×Dic₈₆	—	2	2	no	—
72	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
73	M₁₆×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	—	4	1	no	—
77	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
82	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₃	#41	1	1	yes	characteristic
85	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₄₃	—	2	4	no	—
87	C₄⋊Dic₄₃	—	2	2	no	—
88	C₄⋊Dic₄₃	—	2	2	no	—
89	C₄⋊Dic₄₃	—	2	2	no	—
90	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
91	Dic₄₃⋊C₄	—	2	4	no	—
92	Dic₄₃⋊C₄	—	2	4	no	—
93	C₄×Dic₄₃	—	2	4	no	—
94	C₂×C₄₃⋊C₈	—	2	2	no	—
95	C₂×C₄₃⋊C₈	—	2	2	no	—
96	C₂×C₄₃⋊C₈	—	4	4	no	—
97	Dic₁₇₂	—	2	4	no	—
98	Dic₁₇₂	—	4	2	no	—
99	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	—	2	1	no	—
101	C₂×C₄×C₄₃	—	2	1	no	—
102	C₂×C₄×C₄₃	—	4	1	no	—
103	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	#137	1	4	yes	—
107	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₄₃	—	2	4	no	—
111	C₂×Dic₄₃	—	2	4	no	—
112	Dic₈₆	—	2	4	no	—
113	Dic₈₆	—	2	4	no	—
114	Dic₈₆	—	8	2	no	—
115	C₈×C₄₃	—	2	1	no	—
116	C₈×C₄₃	—	2	1	no	—
117	C₈×C₄₃	—	2	1	no	—
118	C₈×C₄₃	—	2	4	no	—
119	C₈×C₄₃	—	4	2	no	—
120	C₈×C₄₃	#131	1	2	yes	—
121	C₄₃⋊C₈	—	4	2	no	—
122	C₄₃⋊C₈	—	4	2	no	—
123	#18072	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	—	2	1	no	—
125	C₂²×C₄₃	—	2	1	no	—
126	C₂²×C₄₃	#441	1	2	yes	—
127	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₄₃	—	2	1	no	—
129	C₄×C₄₃	—	2	1	no	—
130	C₄×C₄₃	—	2	1	no	—
131	C₄×C₄₃	#441	1	2	yes	—
132	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
133	C₄×C₄₃	#568	1	1	yes	characteristic
134	C₄×C₄₃	C₂×#137	1	2	yes	—
135	Dic₄₃	—	4	4	no	—
136	Dic₄₃	—	4	4	no	—
137	#3170	—	43	1	no	—
138	#2962	—	43	2	no	—
139	#1593	—	43	2	no	—
140	#1587	—	43	4	no	—
141	#1579	—	43	2	no	—
142	C₂×#334	—	43	1	no	—
143	#1481	—	43	2	no	—
144	#1451	D₄₃	1	1	yes	characteristic
145	C₂×C₄₃	—	2	2	no	—
146	C₂×C₄₃	#2935	1	1	yes	characteristic
147	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
148	C₂×C₄₃	#2957	1	1	yes	characteristic
149	#533	—	43	2	no	—
150	#500	—	43	1	no	—
151	#494	—	43	1	no	—
152	#487	—	43	2	no	—
153	#467	—	43	2	no	—
154	#467	—	43	2	no	—
155	#446	—	43	4	no	—
156	#446	—	86	4	no	—
157	C₈×Q₈	—	43	4	no	—
158	C₄×Q₁₆	—	43	2	no	—
159	C₄×Q₁₆	—	43	2	no	—
160	#431	C₂×D₄₃	1	1	yes	characteristic
161	#424	—	86	2	no	—
162	C₂×C₄⋊C₈	—	43	2	no	—
163	#405	—	86	2	no	—
164	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
166	#334	—	43	1	no	—
167	#334	—	43	2	no	—
168	#334	—	43	1	no	—
169	Q₈⋊C₈	—	43	4	no	—
170	Q₈⋊C₈	—	43	4	no	—
171	#322	C₂×D₄₃	1	2	yes	—
172	#322	C₂×D₄₃	1	2	yes	—
173	C₄₃	#18072	1	1	yes	characteristic, complemented, Sylow
174	C₂×Q₁₆	—	43	2	no	—
175	C₂×Q₁₆	—	43	2	no	—
176	C₂×M₁₆	D₁₇₂	1	2	yes	—
177	C₂²×C₈	—	86	2	no	—
178	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
180	C₄⋊Q₈	—	43	2	no	—
181	C₄⋊Q₈	—	43	2	no	—
182	#125	—	86	2	no	—
183	C₂²⋊Q₈	—	43	4	no	—
184	C₂²⋊Q₈	—	86	2	no	—
185	C₄×Q₈	—	43	4	no	—
186	C₄×Q₈	—	86	2	no	—
187	#117	—	86	2	no	—
188	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
189	#107	—	43	2	no	—
190	#107	—	43	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	43	2	no	—
193	C₄⋊C₈	—	43	2	no	—
194	C₄⋊C₈	—	86	4	no	—
195	C₄⋊C₈	—	86	4	no	—
196	C₄⋊C₈	—	86	2	no	—
197	#103	—	43	4	no	—
198	#103	—	43	4	no	—
199	C₂²⋊C₈	—	2	1	no	—
200	#97	—	86	4	no	—
201	#97	D₁₇₂	1	2	yes	—
202	C₄×C₈	—	86	4	no	—
203	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
204	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₄₃	1	2	yes	—
206	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₄₃	1	2	yes	—
209	C₂×Q₈	—	43	4	no	—
210	C₂×Q₈	—	86	2	no	—
211	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
213	C₂²×C₄	#10621898	1	1	yes	characteristic
214	Q₁₆	—	86	4	no	—
215	Q₁₆	—	172	2	no	—
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	86	2	no	—
221	C₂×C₈	—	86	2	no	—
222	C₂×C₈	—	172	4	no	—
223	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
228	C₄⋊C₄	—	43	4	no	—
229	C₄⋊C₄	—	86	2	no	—
230	C₄⋊C₄	—	86	2	no	—
231	C₄⋊C₄	—	86	4	no	—
232	C₄⋊C₄	—	86	2	no	—
233	C₄⋊C₄	—	86	4	no	—
234	C₂²⋊C₄	—	86	4	no	—
235	C₄²	—	86	4	no	—
236	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
237	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
238	C₄²	#10621898	1	1	yes	characteristic
239	C₄²	C₂³×D₄₃	1	1	yes	characteristic
240	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
241	Q₈	—	86	4	no	—
242	Q₈	—	86	4	no	—
243	Q₈	—	344	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	86	4	no	—
248	C₂×C₄	—	86	4	no	—
249	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
250	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
251	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
252	C₂×C₄	#13063358	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
255	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	172	2	no	—
262	C₈	—	172	2	no	—
263	C₈	#13063333	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#14583007	1	2	yes	—
267	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	172	4	no	—
272	C₄	—	172	4	no	—
273	C₄	#14583007	1	2	yes	—
274	C₄	C₂×#13063333	1	1	yes	characteristic
275	C₄	#14583665	1	1	yes	characteristic
276	C₄	C₂×#13063358	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#19143341	1	1	yes	characteristic
279	C₂	C₂×#14583007	1	1	yes	characteristic
280	C₂	#19143513	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1451	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1451	D₄₃
hypocenter	C₄₃	#18072
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #19133505	C₂	semidirect
4. #19133468	C₂	semidirect
5. #19133415	C₂	semidirect
173. C₄₃	#18072	semidirect

Maximal subgroups

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

subgroup	index
#19145181	2
#19143542	2
#19133505	2
#19133468	2
#19133415	2
C₂×#14582650	2
#19132999	2
C₄₃×#1451	2
#18072	43