Subgroups — #24987356

Order 11008.

468 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143561	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063251	C₂	1	1	yes	maximal, characteristic
3	#19134340	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132975	C₂	1	1	yes	maximal, characteristic
5	#19132433	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10621898	C₂²	1	1	yes	characteristic
9	#14583029	C₄	1	2	yes	complemented
10	#14582978	C₄	1	2	yes	complemented
11	#14582970	—	2	2	no	—
12	#14582952	C₂²	1	1	yes	characteristic
13	#14582950	—	2	1	no	—
14	C₂×#13063251	C₄	1	2	yes	—
15	C₄×#10621872	—	2	1	no	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
18	#14582660	C₄	1	2	yes	—
19	#14582653	—	2	2	no	—
20	#14582629	C₄	1	2	yes	complemented
21	#14582626	C₄	1	2	yes	complemented
22	C₄₃×#431	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#322	—	2	1	no	—
26	C₈²×C₄₃	—	2	1	no	—
27	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
28	C₂×#10621872	—	2	2	no	—
29	C₂²×C₄₃⋊C₈	—	2	1	no	—
30	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
32	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
33	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₄₃	—	2	1	no	—
35	C₄×C₂²⋊D₄₃	—	2	1	no	—
36	#13063284	C₂×C₄	1	1	yes	characteristic
37	#13063282	—	2	1	no	—
38	C₂×C₄×Dic₄₃	—	2	1	no	—
39	C₂×C₄×Dic₄₃	—	2	1	no	—
40	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
41	C₄²×D₄₃	—	2	1	no	—
42	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
43	Dic₄₃⋊C₈	—	2	2	no	—
44	#13063251	—	2	4	no	—
45	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
46	(C₂×D₄₃)⋊C₈	—	2	2	no	—
47	#13063235	—	2	2	no	—
48	C₄×C₄₃⋊C₈	—	2	1	no	—
49	C₄×C₄₃⋊C₈	—	2	1	no	—
50	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
51	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₃	D₄	1	2	yes	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₄₃	—	2	1	no	—
56	#97×C₄₃	—	2	1	no	—
57	C₄×C₈×C₄₃	—	2	1	no	—
58	C₄×C₈×C₄₃	—	2	1	no	—
59	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
65	C₂×C₂²⋊D₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	—	2	2	no	—
67	#10621898	—	2	2	no	—
68	#10621898	—	2	2	no	—
69	C₂×D₁₇₂	C₄²	1	1	yes	characteristic
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×Dic₈₆	C₄²	1	1	yes	characteristic
74	M₁₆×C₄₃	—	2	2	no	—
75	M₁₆×C₄₃	M₁₆	1	4	yes	complemented
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	2	no	—
78	C₂×C₈×C₄₃	—	4	2	no	—
79	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
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	C₂²⋊Dic₄₃	—	2	1	no	—
86	(C₂×D₄₃)⋊C₄	—	2	1	no	—
87	(C₂×D₄₃)⋊C₄	—	2	1	no	—
88	C₄⋊Dic₄₃	C₄²	1	1	yes	characteristic
89	C₄⋊Dic₄₃	C₄²	1	1	yes	characteristic
90	Dic₄₃⋊C₄	—	2	1	no	—
91	Dic₄₃⋊C₄	—	2	1	no	—
92	C₄×Dic₄₃	—	2	4	no	—
93	C₄×Dic₄₃	—	2	1	no	—
94	C₄×Dic₄₃	—	2	1	no	—
95	C₄×Dic₄₃	—	2	1	no	—
96	C₄×Dic₄₃	—	2	2	no	—
97	C₄×Dic₄₃	—	2	2	no	—
98	C₄×Dic₄₃	—	4	2	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	—	4	2	no	—
102	#10621872	—	4	4	no	—
103	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
104	C₂²×D₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₄₃	#131	1	2	yes	—
115	C₂²⋊D₄₃	—	4	2	no	—
116	C₂×Dic₄₃	—	2	4	no	—
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	4	2	no	—
121	C₂×Dic₄₃	—	4	2	no	—
122	D₁₇₂	—	2	1	no	—
123	D₁₇₂	—	2	1	no	—
124	C₄×D₄₃	—	4	2	no	—
125	C₄×D₄₃	—	4	2	no	—
126	C₄×D₄₃	—	4	2	no	—
127	Dic₈₆	—	2	1	no	—
128	Dic₈₆	—	2	1	no	—
129	C₈×C₄₃	—	2	2	no	—
130	C₈×C₄₃	—	2	2	no	—
131	C₈×C₄₃	—	2	4	no	—
132	C₈×C₄₃	—	4	2	no	—
133	C₄₃⋊C₈	—	4	2	no	—
134	C₄₃⋊C₈	—	4	2	no	—
135	#8197	—	43	1	no	maximal, Sylow
136	C₂²×C₄₃	—	2	1	no	—
137	C₂²×C₄₃	—	2	1	no	—
138	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₃	#436	1	2	yes	—
140	C₂×D₄₃	—	4	1	no	—
141	C₂×D₄₃	—	4	1	no	—
142	C₂×D₄₃	—	4	1	no	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
146	C₄×C₄₃	#405	1	1	yes	characteristic
147	C₄×C₄₃	#421	1	2	yes	—
148	C₄×C₄₃	#435	1	2	yes	—
149	C₄×C₄₃	#443	1	2	yes	—
150	Dic₄₃	—	4	4	no	—
151	Dic₄₃	—	4	1	no	—
152	Dic₄₃	—	4	1	no	—
153	Dic₄₃	—	4	1	no	—
154	#2965	—	43	1	no	—
155	C₄×#104	—	43	1	no	—
156	#1744	—	43	1	no	—
157	#1476	—	43	1	no	—
158	C₈×M₁₆	D₄₃	1	1	yes	characteristic, complemented
159	#1278	—	43	1	no	—
160	#1276	—	43	1	no	—
161	C₂×C₄₃	—	2	2	no	—
162	C₂×C₄₃	#1753	1	1	yes	characteristic
163	C₂×C₄₃	#1759	1	1	yes	characteristic
164	C₂×C₄₃	#1764	1	1	yes	characteristic
165	D₄₃	—	8	1	no	—
166	C₄×#41	—	43	1	no	—
167	#446	—	43	2	no	—
168	#436	—	43	2	no	—
169	#435	—	86	2	no	—
170	#431	—	43	1	no	—
171	#431	—	86	1	no	—
172	#431	C₂×D₄₃	1	1	yes	characteristic
173	C₂×#104	—	43	2	no	—
174	C₄×M₁₆	—	86	1	no	—
175	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
176	C₂×C₄×C₈	—	43	1	no	—
177	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
178	C₄³	—	43	1	no	—
179	#339	—	43	2	no	—
180	#336	—	86	2	no	—
181	Q₈⋊C₈	—	43	2	no	—
182	D₄⋊C₈	—	43	2	no	—
183	#322	—	2	1	no	—
184	C₈²	—	2	1	no	—
185	C₄₃	#8197	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	—	43	1	no	—
187	C₂×M₁₆	—	86	2	no	—
188	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
189	C₂²×C₈	—	86	1	no	—
190	C₂²×C₈	D₁₇₂	1	2	yes	—
191	C₄×Q₈	—	43	1	no	—
192	C₄×Q₈	—	43	1	no	—
193	C₄×D₄	—	43	1	no	—
194	C₄×D₄	—	43	1	no	—
195	C₄×D₄	—	86	1	no	—
196	C₄×D₄	—	86	1	no	—
197	#117	—	43	1	no	—
198	#117	—	86	1	no	—
199	C₂×C₄²	—	43	2	no	—
200	C₂×C₄²	—	86	1	no	—
201	C₂×C₄²	—	86	1	no	—
202	C₂×C₄²	—	86	1	no	—
203	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
204	C₄⋊C₈	—	43	2	no	—
205	C₄⋊C₈	—	86	2	no	—
206	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
207	#104	—	86	4	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	86	1	no	—
210	C₂²⋊C₈	—	86	2	no	—
211	#97	—	2	1	no	—
212	#97	—	86	2	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	43	2	no	—
216	C₄×C₈	—	86	1	no	—
217	C₄×C₈	—	86	1	no	—
218	C₄×C₈	C₄×D₄₃	1	2	yes	—
219	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
220	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
221	#41	—	86	2	no	—
222	#41	—	86	2	no	—
223	C₂×Q₈	—	43	1	no	—
224	C₂×D₄	—	43	1	no	—
225	C₂×D₄	—	86	1	no	—
226	C₂²×C₄	—	86	2	no	—
227	C₂²×C₄	—	86	1	no	—
228	C₂²×C₄	—	86	1	no	—
229	C₂²×C₄	—	86	1	no	—
230	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
232	C₂²×C₄	#10621898	1	1	yes	characteristic
233	M₁₆	—	2	2	no	—
234	M₁₆	—	172	4	no	—
235	M₁₆	#10621872	1	4	yes	complemented
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	86	2	no	—
240	C₂×C₈	—	86	2	no	—
241	C₂×C₈	—	172	2	no	—
242	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
243	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
244	C₄⋊C₄	—	43	1	no	—
245	C₄⋊C₄	—	43	1	no	—
246	C₄⋊C₄	—	86	1	no	—
247	C₄⋊C₄	—	86	1	no	—
248	C₂²⋊C₄	—	86	1	no	—
249	C₂²⋊C₄	—	86	1	no	—
250	C₂²⋊C₄	—	86	1	no	—
251	C₄²	—	86	4	no	—
252	C₄²	—	86	1	no	—
253	C₄²	—	86	1	no	—
254	C₄²	—	86	1	no	—
255	C₄²	—	86	2	no	—
256	C₄²	—	86	2	no	—
257	C₄²	—	172	2	no	—
258	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
260	C₄²	#10621898	1	1	yes	characteristic
261	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
262	C₂³	—	86	1	no	—
263	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
264	Q₈	—	86	1	no	—
265	Q₈	—	86	1	no	—
266	D₄	—	86	1	no	—
267	D₄	—	86	1	no	—
268	D₄	—	172	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	86	4	no	—
272	C₂×C₄	—	86	1	no	—
273	C₂×C₄	—	86	1	no	—
274	C₂×C₄	—	86	1	no	—
275	C₂×C₄	—	172	2	no	—
276	C₂×C₄	—	172	2	no	—
277	C₂×C₄	—	172	2	no	—
278	C₂×C₄	—	172	2	no	—
279	C₂×C₄	—	172	2	no	—
280	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
281	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
282	C₂×C₄	#13063282	1	1	yes	characteristic
283	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
284	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
285	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
286	C₂×C₄	C₂×#10621872	1	2	yes	—
287	C₂×C₄	#13063331	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	—	2	2	no	—
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	172	2	no	—
293	C₈	—	172	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	172	1	no	—
297	C₂²	—	172	1	no	—
298	C₂²	—	172	1	no	—
299	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
300	C₂²	#14582978	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	172	4	no	—
304	C₄	—	172	1	no	—
305	C₄	—	172	1	no	—
306	C₄	—	172	1	no	—
307	C₄	C₄×#10621872	1	1	yes	characteristic
308	C₄	#14582853	1	1	yes	characteristic
309	C₄	#14582915	1	2	yes	—
310	C₄	#14582970	1	2	yes	—
311	C₄	#14583014	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	344	1	no	—
314	C₂	#19134378	1	1	yes	characteristic
315	C₂	#19134408	1	1	yes	characteristic
316	C₂	#19134434	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₈×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₄×C₂²⋊C₄
prime radical	C₂×#10621898	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₈×M₁₆	D₄₃
hypocenter	C₄₃	#8197
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #19134340	C₂	semidirect
5. #19132433	C₂	semidirect
6. #19132427	C₂	semidirect
7. C₈×M₁₆×C₄₃	C₂	semidirect
9. #14583029	C₄	semidirect
10. #14582978	C₄	semidirect
20. #14582629	C₄	semidirect
21. #14582626	C₄	semidirect
75. M₁₆×C₄₃	M₁₆	semidirect
158. C₈×M₁₆	D₄₃	semidirect
185. C₄₃	#8197	semidirect
235. M₁₆	#10621872	semidirect

Maximal subgroups

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

subgroup	index
#19143561	2
C₄×#13063251	2
#19134340	2
#19132975	2
#19132433	2
#19132427	2
C₈×M₁₆×C₄₃	2
#8197	43