Subgroups — #24989441

Order 11008.

546 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10621872	C₂	1	1	yes	maximal, characteristic
2	#19134572	C₂	1	2	yes	maximal
3	C₂×#14582650	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132429	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10621872	C₄	1	1	yes	characteristic
7	C₂²×#10621872	C₄	1	1	yes	characteristic
8	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×(C₂×D₄₃)⋊C₈	—	2	2	no	—
11	C₄×#10621872	—	2	2	no	—
12	C₄×#10621872	—	2	2	no	—
13	C₂×#13063235	C₄	1	1	yes	characteristic
14	C₂×#13063235	C₄	1	1	yes	characteristic
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
16	#14582653	—	2	2	no	—
17	#14582650	—	2	2	no	—
18	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#335	—	2	2	no	—
21	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×D₄₃	Q₈	1	1	yes	characteristic
24	C₂×#10621872	—	2	2	no	—
25	C₂×#10621872	—	2	2	no	—
26	C₂×#10621872	—	2	2	no	—
27	C₂×#10621872	—	2	2	no	—
28	C₂×#10621872	—	2	2	no	—
29	C₂×#10621872	C₈	1	4	yes	complemented
30	C₂²×C₄₃⋊C₈	—	2	2	no	—
31	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₄₃	Q₈	1	1	yes	characteristic
36	C₄²×D₄₃	—	2	1	no	—
37	C₄²×D₄₃	—	2	1	no	—
38	C₄²×D₄₃	—	2	1	no	—
39	C₄²×D₄₃	—	2	1	no	—
40	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
42	(C₂×D₄₃)⋊C₈	—	4	4	no	—
43	#13063235	—	2	2	no	—
44	#13063235	—	2	2	no	—
45	C₄×C₄₃⋊C₈	—	2	1	no	—
46	C₄×C₄₃⋊C₈	—	2	1	no	—
47	C₂²×C₈×C₄₃	—	2	2	no	—
48	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₄₃	—	2	2	no	—
52	C₄⋊C₈×C₄₃	—	2	2	no	—
53	C₄×C₈×C₄₃	D₄	1	2	yes	—
54	C₄×C₈×C₄₃	D₄	1	2	yes	—
55	C₂³×D₄₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×D₄₃	—	2	1	no	—
63	C₂×C₄×D₄₃	—	2	1	no	—
64	C₂×C₄×D₄₃	—	2	1	no	—
65	C₂×C₄×D₄₃	—	2	1	no	—
66	C₂×C₄×D₄₃	—	2	1	no	—
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	—	2	2	no	—
71	C₂×C₄×D₄₃	—	2	2	no	—
72	C₂×C₄×D₄₃	—	4	2	no	—
73	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₄₃	M₁₆	1	2	yes	—
75	C₂×C₈×C₄₃	—	2	1	no	—
76	C₂×C₈×C₄₃	—	2	1	no	—
77	C₂×C₈×C₄₃	—	4	2	no	—
78	C₂×C₈×C₄₃	—	4	2	no	—
79	C₂×C₈×C₄₃	—	4	2	no	—
80	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
83	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
84	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₃	#41	1	1	yes	characteristic
87	C₄²×C₄₃	#41	1	1	yes	characteristic
88	C₄×Dic₄₃	—	2	1	no	—
89	C₄×Dic₄₃	—	2	1	no	—
90	C₄×Dic₄₃	—	2	1	no	—
91	C₄×Dic₄₃	—	2	1	no	—
92	C₄×Dic₄₃	—	2	1	no	—
93	C₄×Dic₄₃	—	2	1	no	—
94	C₂×C₄₃⋊C₈	—	2	1	no	—
95	C₂×C₄₃⋊C₈	—	2	1	no	—
96	C₂×C₄₃⋊C₈	—	2	1	no	—
97	C₂×C₄₃⋊C₈	—	2	1	no	—
98	C₂×C₄₃⋊C₈	—	4	2	no	—
99	C₂×C₄₃⋊C₈	—	4	2	no	—
100	C₂×C₄₃⋊C₈	—	4	2	no	—
101	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
102	C₂×C₄₃⋊C₈	M₁₆	1	2	yes	—
103	#10621872	—	2	8	no	—
104	#10621872	—	2	8	no	—
105	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×D₄₃	—	2	1	no	—
107	C₂²×D₄₃	—	2	1	no	—
108	C₂²×D₄₃	—	2	2	no	—
109	C₂²×D₄₃	—	2	2	no	—
110	C₂²×D₄₃	C₄⋊C₈	1	2	yes	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₄₃	#137	1	2	yes	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	2	no	—
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	C₄⋊C₈	1	2	yes	—
131	C₄×D₄₃	—	2	2	no	—
132	C₄×D₄₃	—	2	2	no	—
133	C₄×D₄₃	—	2	2	no	—
134	C₄×D₄₃	—	2	2	no	—
135	C₄×D₄₃	—	4	2	no	—
136	C₄×D₄₃	—	4	2	no	—
137	C₄×D₄₃	—	4	2	no	—
138	C₄×D₄₃	—	4	1	no	—
139	C₄×D₄₃	—	4	1	no	—
140	C₈×C₄₃	—	2	2	no	—
141	C₈×C₄₃	—	8	2	no	—
142	C₈×C₄₃	C₂²⋊C₈	1	4	yes	complemented
143	C₄₃⋊C₈	—	2	2	no	—
144	C₄₃⋊C₈	—	2	2	no	—
145	C₄₃⋊C₈	—	8	2	no	—
146	#8597	—	43	1	no	maximal, Sylow
147	C₂²×C₄₃	#380	1	1	yes	characteristic
148	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
149	C₂²×C₄₃	#423	1	1	yes	characteristic
150	C₂²×C₄₃	#428	1	2	yes	—
151	C₂²×C₄₃	#430	1	2	yes	—
152	C₂×D₄₃	—	2	2	no	—
153	C₂×D₄₃	—	2	4	no	—
154	C₂×D₄₃	—	4	1	no	—
155	C₂×D₄₃	—	4	1	no	—
156	C₂×D₄₃	—	4	2	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₄×C₄₃	#407	1	1	yes	characteristic
161	C₄×C₄₃	#431	1	1	yes	characteristic
162	C₄×C₄₃	#432	1	1	yes	characteristic
163	C₄×C₄₃	#468	1	2	yes	—
164	C₄×C₄₃	#469	1	1	yes	characteristic
165	C₄×C₄₃	#470	1	1	yes	characteristic
166	Dic₄₃	—	2	2	no	—
167	Dic₄₃	—	4	1	no	—
168	Dic₄₃	—	4	1	no	—
169	C₂×C₄×M₁₆	—	43	1	no	—
170	#1797	—	43	2	no	—
171	C₂×#335	D₄₃	1	1	yes	characteristic, complemented
172	C₂×#334	—	43	1	no	—
173	#1277	—	43	2	no	—
174	C₂×C₄₃	#1565	1	2	yes	—
175	C₂×C₄₃	#1569	1	2	yes	—
176	C₂×C₄₃	#1797	1	1	yes	characteristic
177	C₂×C₄₃	#1809	1	1	yes	characteristic
178	C₂×C₄₃	#1811	1	1	yes	characteristic
179	D₄₃	—	4	2	no	—
180	C₂²×M₁₆	—	43	1	no	—
181	C₂²×M₁₆	—	43	1	no	—
182	C₂²×C₄²	—	43	1	no	—
183	C₂×C₄⋊C₈	—	43	2	no	—
184	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
185	C₂×C₂²⋊C₈	—	86	2	no	—
186	C₄×M₁₆	—	86	2	no	—
187	C₄×M₁₆	—	86	2	no	—
188	C₂×#97	—	43	1	no	—
189	C₂×#97	—	43	1	no	—
190	C₂×C₄×C₈	—	43	1	no	—
191	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
192	#336	—	86	2	no	—
193	#335	—	2	2	no	—
194	#334	—	86	2	no	—
195	C₄₃	#8597	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	—	43	1	no	—
197	C₂³×C₄	—	43	1	no	—
198	C₂³×C₄	—	43	1	no	—
199	C₂×M₁₆	—	43	4	no	—
200	C₂×M₁₆	—	86	2	no	—
201	C₂×M₁₆	—	86	2	no	—
202	C₂×M₁₆	—	86	2	no	—
203	C₂×M₁₆	—	86	2	no	—
204	C₂×M₁₆	—	86	2	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	43	1	no	—
207	C₂²×C₈	—	43	1	no	—
208	C₂²×C₈	—	86	2	no	—
209	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
210	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
211	C₂×C₄²	—	43	1	no	—
212	C₂×C₄²	—	43	1	no	—
213	C₂×C₄²	—	43	1	no	—
214	C₂×C₄²	—	86	1	no	—
215	C₂×C₄²	—	86	1	no	—
216	C₂×C₄²	—	86	1	no	—
217	C₂×C₄²	—	86	1	no	—
218	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	86	2	no	—
222	C₄⋊C₈	—	86	2	no	—
223	C₂²⋊C₈	—	172	4	no	—
224	#97	—	86	2	no	—
225	#97	—	86	2	no	—
226	C₄×C₈	—	86	1	no	—
227	C₄×C₈	—	86	1	no	—
228	C₄×C₈	D₁₇₂	1	2	yes	—
229	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
230	C₂⁴	—	43	1	no	—
231	C₂²×C₄	—	43	1	no	—
232	C₂²×C₄	—	43	2	no	—
233	C₂²×C₄	—	43	2	no	—
234	C₂²×C₄	—	43	1	no	—
235	C₂²×C₄	—	43	1	no	—
236	C₂²×C₄	—	86	1	no	—
237	C₂²×C₄	—	86	1	no	—
238	C₂²×C₄	—	86	1	no	—
239	C₂²×C₄	—	86	1	no	—
240	C₂²×C₄	—	86	1	no	—
241	C₂²×C₄	—	86	1	no	—
242	C₂²×C₄	—	86	1	no	—
243	C₂²×C₄	—	86	1	no	—
244	C₂²×C₄	—	86	2	no	—
245	C₂²×C₄	—	86	2	no	—
246	C₂²×C₄	—	172	2	no	—
247	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
249	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
250	M₁₆	—	86	8	no	—
251	M₁₆	—	86	8	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	43	2	no	—
258	C₂×C₈	—	43	2	no	—
259	C₂×C₈	—	86	1	no	—
260	C₂×C₈	—	86	1	no	—
261	C₂×C₈	—	86	1	no	—
262	C₂×C₈	—	86	1	no	—
263	C₂×C₈	—	172	2	no	—
264	C₂×C₈	—	172	2	no	—
265	C₂×C₈	—	172	2	no	—
266	C₂×C₈	C₈×D₄₃	1	2	yes	—
267	C₂×C₈	#10621872	1	2	yes	—
268	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
269	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
270	C₄²	—	86	1	no	—
271	C₄²	—	86	1	no	—
272	C₄²	—	86	1	no	—
273	C₄²	—	86	1	no	—
274	C₄²	—	86	1	no	—
275	C₄²	—	86	1	no	—
276	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
277	C₄²	#10621900	1	1	yes	characteristic
278	C₄²	#10621902	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
280	C₂³	—	43	2	no	—
281	C₂³	—	86	1	no	—
282	C₂³	—	86	2	no	—
283	C₂³	—	86	2	no	—
284	C₂³	—	86	1	no	—
285	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	43	2	no	—
288	C₂×C₄	—	86	1	no	—
289	C₂×C₄	—	86	1	no	—
290	C₂×C₄	—	86	2	no	—
291	C₂×C₄	—	86	2	no	—
292	C₂×C₄	—	86	2	no	—
293	C₂×C₄	—	86	2	no	—
294	C₂×C₄	—	86	2	no	—
295	C₂×C₄	—	86	2	no	—
296	C₂×C₄	—	86	1	no	—
297	C₂×C₄	—	86	1	no	—
298	C₂×C₄	—	172	2	no	—
299	C₂×C₄	—	172	2	no	—
300	C₂×C₄	—	172	2	no	—
301	C₂×C₄	—	172	1	no	—
302	C₂×C₄	—	172	1	no	—
303	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
304	C₂×C₄	#13063283	1	1	yes	characteristic
305	C₂×C₄	#13063296	1	1	yes	characteristic
306	C₂×C₄	#13063300	1	1	yes	characteristic
307	C₂×C₄	#13063301	1	1	yes	characteristic
308	C₂×C₄	#13063305	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
310	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
311	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
312	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
313	C₂×C₄	#13063349	1	2	yes	—
314	C₂×C₄	#13063357	1	2	yes	—
315	C₈	—	2	2	no	—
316	C₈	—	8	2	no	—
317	C₈	—	86	2	no	—
318	C₈	—	86	2	no	—
319	C₈	—	344	2	no	—
320	C₈	(C₂×D₄₃)⋊C₈	1	4	yes	complemented
321	C₂²	—	86	2	no	—
322	C₂²	—	86	4	no	—
323	C₂²	—	172	1	no	—
324	C₂²	—	172	1	no	—
325	C₂²	—	172	2	no	—
326	C₂²	#14582756	1	1	yes	characteristic
327	C₂²	C₄⋊C₈×D₄₃	1	1	yes	characteristic
328	C₂²	#14582921	1	1	yes	characteristic
329	C₂²	#14582940	1	2	yes	—
330	C₂²	#14582946	1	2	yes	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	86	2	no	—
334	C₄	—	172	1	no	—
335	C₄	—	172	1	no	—
336	C₄	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
337	C₄	#14582862	1	1	yes	characteristic
338	C₄	#14582951	1	1	yes	characteristic
339	C₄	#14582955	1	1	yes	characteristic
340	C₄	#14583146	1	2	yes	—
341	C₄	#14583149	1	1	yes	characteristic
342	C₄	#14583154	1	1	yes	characteristic
343	C₂	—	172	2	no	—
344	C₂	#19133346	1	2	yes	—
345	C₂	#19133361	1	2	yes	—
346	C₂	#19134572	1	1	yes	characteristic
347	C₂	#19134629	1	1	yes	characteristic
348	C₂	#19134634	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#335	C₂
socle	C₂³×C₄₃	C₂×C₄⋊C₄
prime radical	C₂³×D₄₃	C₄⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#335	D₄₃
hypocenter	C₄₃	#8597
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. C₂×#14582650	C₂	semidirect
4. #19132429	C₂	semidirect
5. C₂×C₄₃×#335	C₂	semidirect
29. C₂×#10621872	C₈	semidirect
142. C₈×C₄₃	C₂²⋊C₈	semidirect
171. C₂×#335	D₄₃	semidirect
195. C₄₃	#8597	semidirect
320. C₈	(C₂×D₄₃)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10621872	2
#19134572	2
C₂×#14582650	2
#19132429	2
C₂×C₄₃×#335	2
#8597	43