Subgroups — #24986145

Order 11008.

734 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#19136993	C₂	1	1	yes	maximal, characteristic
3	#19132435	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₃⋊C₈	—	2	1	no	—
6	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	#14582767	—	2	1	no	—
11	#14582765	C₄	1	2	yes	—
12	#14582762	—	2	1	no	—
13	#14582759	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
15	C₄×C₂²⋊Dic₄₃	—	2	1	no	—
16	#14582745	—	2	1	no	—
17	C₂²×M₁₆×C₄₃	—	2	1	no	—
18	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₄₃	—	2	2	no	—
20	C₄×M₁₆×C₄₃	C₄	1	4	yes	complemented
21	C₂×#97×C₄₃	—	2	1	no	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
23	C₂²×C₄₃⋊C₈	—	2	4	no	—
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₂²×C₄₃⋊C₈	—	4	4	no	—
26	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄×Dic₄₃	—	2	2	no	—
29	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
32	C₄₃⋊#95	—	2	2	no	—
33	C₄₃⋊#95	—	2	2	no	—
34	C₄₃⋊#95	—	2	2	no	—
35	C₄₃⋊#95	—	2	2	no	—
36	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
39	C₂×M₁₆×C₄₃	—	2	8	no	—
40	C₂×M₁₆×C₄₃	—	2	4	no	—
41	C₂²×C₈×C₄₃	—	2	2	no	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	D₄	1	2	yes	—
48	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
49	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
50	#97×C₄₃	—	2	4	no	—
51	C₄×C₈×C₄₃	—	2	2	no	—
52	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
53	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	—	2	2	no	—
57	C₂²×C₄×C₄₃	—	2	4	no	—
58	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₄₃	—	2	4	no	—
71	M₁₆×C₄₃	—	2	16	no	—
72	C₂×C₈×C₄₃	—	2	8	no	—
73	C₂×C₈×C₄₃	—	2	4	no	—
74	C₄²×C₄₃	C₄²	1	2	yes	—
75	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
79	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
80	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
84	C₂²⋊Dic₄₃	—	4	4	no	—
85	C₄⋊Dic₄₃	—	2	8	no	—
86	C₄×Dic₄₃	—	4	2	no	—
87	C₄×Dic₄₃	—	4	2	no	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	4	8	no	—
90	C₂×C₄₃⋊C₈	—	4	4	no	—
91	C₂³×C₄₃	—	2	2	no	—
92	C₂³×C₄₃	—	2	2	no	—
93	C₂³×C₄₃	#95	1	2	yes	—
94	C₂³×C₄₃	#104	1	4	yes	—
95	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	—	2	4	no	—
97	C₂×C₄×C₄₃	—	2	4	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	2	4	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	#95	1	2	yes	—
108	C₂×C₄×C₄₃	#95	1	2	yes	—
109	C₂×C₄×C₄₃	#95	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#131	1	4	yes	—
117	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₄₃	#137	1	2	yes	—
119	C₂×Dic₄₃	—	4	4	no	—
120	C₂×Dic₄₃	—	4	4	no	—
121	C₂×Dic₄₃	—	4	4	no	—
122	C₈×C₄₃	—	2	8	no	—
123	C₄₃⋊C₈	—	4	4	no	—
124	#7895	—	43	1	no	maximal, Sylow
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	—	2	2	no	—
127	C₂²×C₄₃	—	2	8	no	—
128	C₂²×C₄₃	#339	1	4	yes	—
129	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
130	C₂²×C₄₃	#408	1	2	yes	—
131	C₂²×C₄₃	#417	1	2	yes	—
132	C₂²×C₄₃	C₂×#104	1	2	yes	—
133	C₄×C₄₃	—	2	2	no	—
134	C₄×C₄₃	—	2	1	no	—
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	2	2	no	—
138	C₄×C₄₃	#405	1	2	yes	—
139	C₄×C₄₃	#418	1	1	yes	characteristic
140	C₄×C₄₃	#419	1	1	yes	characteristic
141	C₄×C₄₃	#424	1	2	yes	—
142	C₄×C₄₃	#428	1	2	yes	—
143	Dic₄₃	—	8	4	no	—
144	C₂²×C₄⋊C₈	—	43	1	no	—
145	C₂×C₄×M₁₆	D₄₃	1	1	yes	characteristic
146	#2283	—	43	1	no	—
147	#1279	—	43	4	no	—
148	C₂×C₄₃	—	2	4	no	—
149	C₂×C₄₃	#1483	1	4	yes	—
150	C₂×C₄₃	#1729	1	1	yes	characteristic
151	C₂×C₄₃	C₂×#339	1	1	yes	characteristic
152	C₂×C₄₃	#1736	1	1	yes	characteristic
153	C₂²×M₁₆	—	2	1	no	—
154	C₂³×C₈	—	86	1	no	—
155	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
156	C₂×C₄⋊C₈	—	43	2	no	—
157	C₂×C₄⋊C₈	—	43	4	no	—
158	C₂×C₄⋊C₈	—	86	1	no	—
159	C₂×C₄⋊C₈	—	86	2	no	—
160	C₄×M₁₆	—	2	2	no	—
161	C₄×M₁₆	Dic₄₃	1	4	yes	complemented
162	C₂×#97	—	2	1	no	—
163	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
164	#383	—	86	1	no	—
165	#382	—	43	2	no	—
166	#381	—	86	1	no	—
167	#380	—	43	2	no	—
168	C₄×C₄⋊C₄	—	43	2	no	—
169	C₄×C₂²⋊C₄	—	86	1	no	—
170	#376	—	86	1	no	—
171	C₄₃	#7895	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
173	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
174	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
175	C₂×M₁₆	—	2	8	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	86	4	no	—
179	C₂²×C₈	—	86	2	no	—
180	C₂²×C₈	—	172	4	no	—
181	C₂×C₄⋊C₄	—	43	4	no	—
182	C₂×C₂²⋊C₄	—	86	2	no	—
183	C₂×C₄²	—	86	2	no	—
184	C₂×C₄²	Dic₈₆	1	2	yes	—
185	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₄₃	1	2	yes	—
187	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
189	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
191	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
192	C₄⋊C₈	—	43	4	no	—
193	C₄⋊C₈	—	86	4	no	—
194	C₄⋊C₈	—	86	2	no	—
195	#97	—	2	4	no	—
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
198	#95	—	86	2	no	—
199	#95	—	86	2	no	—
200	#95	—	86	2	no	—
201	#95	—	86	2	no	—
202	C₂⁴	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	86	4	no	—
208	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
209	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
210	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
211	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
212	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
216	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
220	M₁₆	—	2	16	no	—
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	86	4	no	—
224	C₂×C₈	—	172	8	no	—
225	C₂×C₈	—	172	4	no	—
226	C₄⋊C₄	—	86	8	no	—
227	C₂²⋊C₄	—	172	4	no	—
228	C₄²	—	172	2	no	—
229	C₄²	—	172	2	no	—
230	C₄²	C₄×Dic₄₃	1	2	yes	—
231	C₄²	Dic₄₃⋊C₄	1	2	yes	—
232	C₄²	Dic₄₃⋊C₄	1	2	yes	—
233	C₄²	C₄⋊Dic₄₃	1	2	yes	—
234	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
235	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
236	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
238	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
239	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	C₄₃⋊#95	1	2	yes	—
243	C₂³	#13063253	1	4	yes	—
244	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	172	4	no	—
257	C₂×C₄	—	172	4	no	—
258	C₂×C₄	—	172	4	no	—
259	C₂×C₄	C₄₃⋊#95	1	2	yes	—
260	C₂×C₄	C₄₃⋊#95	1	2	yes	—
261	C₂×C₄	C₄₃⋊#95	1	2	yes	—
262	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
263	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
265	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
268	C₂×C₄	#13063333	1	4	yes	—
269	C₂×C₄	#13063353	1	2	yes	—
270	C₂×C₄	#13063358	1	2	yes	—
271	C₈	—	2	8	no	—
272	C₈	—	172	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#14582659	1	4	yes	—
277	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
278	C₂²	#14582867	1	2	yes	—
279	C₂²	#14582901	1	2	yes	—
280	C₂²	C₂×#13063253	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	344	4	no	—
287	C₄	#14582857	1	2	yes	—
288	C₄	#14582905	1	1	yes	characteristic
289	C₄	#14582909	1	1	yes	characteristic
290	C₄	#14582927	1	2	yes	—
291	C₄	#14582941	1	2	yes	—
292	C₂	—	2	4	no	—
293	C₂	#19133009	1	4	yes	—
294	C₂	#19134293	1	1	yes	characteristic
295	C₂	C₂×#14582659	1	1	yes	characteristic
296	C₂	#19134315	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄×M₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₃
hypocenter	C₄₃	#7895
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #19132435	C₂	semidirect
8. C₂×C₄⋊(C₄₃⋊C₈)	C₄	semidirect
20. C₄×M₁₆×C₄₃	C₄	semidirect
161. C₄×M₁₆	Dic₄₃	semidirect
171. C₄₃	#7895	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₄₃⋊C₈)	2
#19136993	2
#19132435	2
C₂×C₄×M₁₆×C₄₃	2
#7895	43