Subgroups — #25065618

Order 11008.

710 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143223	C₂	1	2	yes	maximal
2	C₂×C₄×#10621872	C₂	1	1	yes	maximal, characteristic
3	#19133361	C₂	1	4	yes	maximal, complemented
4	#19133359	C₂	1	4	yes	maximal, complemented
5	#19133343	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133341	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621872	C₄	1	1	yes	characteristic
9	C₂²×#10621872	C₄	1	1	yes	characteristic
10	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
11	#14582955	C₂²	1	4	yes	—
12	#14582953	—	2	2	no	—
13	#14582952	C₂²	1	2	yes	—
14	#14582950	—	2	2	no	—
15	#14582921	C₂²	1	4	yes	—
16	#14582861	—	2	2	no	—
17	C₄×#10621872	—	2	2	no	—
18	C₄×#10621872	C₄	1	2	yes	—
19	C₄×#10621872	C₄	1	2	yes	—
20	C₄×#10621872	C₂²	1	2	yes	—
21	C₄×#10621872	C₂²	1	2	yes	—
22	C₂×#13063235	C₄	1	1	yes	characteristic
23	C₂×#13063235	C₄	1	1	yes	characteristic
24	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
25	#14582652	C₂²	1	2	yes	complemented
26	#14582651	C₂²	1	4	yes	complemented
27	#14582650	C₂²	1	2	yes	complemented
28	#14582649	C₂²	1	4	yes	complemented
29	C₄₃×#431	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#335	C₂²	1	2	yes	complemented
32	C₄₃×#334	C₂²	1	2	yes	complemented
33	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×D₄₃	Q₈	1	1	yes	characteristic
36	C₂×#10621872	—	2	4	no	—
37	C₂×#10621872	—	2	4	no	—
38	C₂×#10621872	—	2	4	no	—
39	C₂×#10621872	C₂×C₄	1	2	yes	—
40	C₂×#10621872	C₂×C₄	1	2	yes	—
41	C₂×#10621872	C₂×C₄	1	2	yes	—
42	C₂×#10621872	C₂×C₄	1	2	yes	—
43	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₄₃	Q₈	1	1	yes	characteristic
48	C₄²×D₄₃	C₂×C₄	1	2	yes	—
49	C₄²×D₄₃	D₄	1	2	yes	—
50	C₄²×D₄₃	Q₈	1	2	yes	—
51	C₄²×D₄₃	C₂³	1	2	yes	—
52	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
53	Dic₄₃⋊C₈	—	2	4	no	—
54	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
55	(C₂×D₄₃)⋊C₈	—	2	4	no	—
56	#13063235	—	2	4	no	—
57	#13063235	C₂×C₄	1	2	yes	—
58	#13063235	C₂×C₄	1	2	yes	—
59	#13063235	C₂×C₄	1	2	yes	—
60	#13063235	C₂×C₄	1	2	yes	—
61	C₄×C₄₃⋊C₈	—	2	2	no	—
62	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
63	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
68	C₂²⋊C₈×C₄₃	—	2	2	no	—
69	C₄×C₈×C₄₃	—	2	2	no	—
70	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₂³×D₄₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₄₃	—	2	2	no	—
82	C₂×C₄×D₄₃	—	2	2	no	—
83	C₂×C₄×D₄₃	—	4	2	no	—
84	C₂×C₄×D₄₃	—	4	2	no	—
85	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
86	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
87	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
88	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₄₃	—	2	2	no	—
92	C₂×C₈×C₄₃	—	2	4	no	—
93	C₂×C₈×C₄₃	M₁₆	1	4	yes	complemented
94	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
101	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
102	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
103	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
104	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
105	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
106	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
107	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
108	C₂×C₄₃⋊C₈	—	2	2	no	—
109	C₂×C₄₃⋊C₈	—	2	4	no	—
110	C₂×C₄₃⋊C₈	M₁₆	1	4	yes	complemented
111	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
113	#10621872	—	2	4	no	—
114	#10621872	—	2	4	no	—
115	#10621872	—	2	4	no	—
116	#10621872	—	2	4	no	—
117	#10621872	—	4	8	no	—
118	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×D₄₃	—	2	2	no	—
120	C₂²×D₄₃	—	4	1	no	—
121	C₂²×D₄₃	—	4	1	no	—
122	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×Dic₄₃	—	2	2	no	—
134	C₂×Dic₄₃	—	4	1	no	—
135	C₂×Dic₄₃	—	4	1	no	—
136	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₄×D₄₃	—	2	2	no	—
140	C₄×D₄₃	—	2	2	no	—
141	C₄×D₄₃	—	2	4	no	—
142	C₄×D₄₃	—	2	4	no	—
143	C₄×D₄₃	—	2	4	no	—
144	C₄×D₄₃	—	4	2	no	—
145	C₄×D₄₃	—	4	2	no	—
146	C₈×C₄₃	—	2	1	no	—
147	C₈×C₄₃	—	2	1	no	—
148	C₈×C₄₃	—	4	4	no	—
149	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
150	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
151	C₄₃⋊C₈	—	2	2	no	—
152	C₄₃⋊C₈	—	4	4	no	—
153	C₄₃⋊C₈	C₂×M₁₆	1	4	yes	—
154	#17795	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	—	2	1	no	—
156	C₂²×C₄₃	—	2	1	no	—
157	C₂²×C₄₃	#423	1	2	yes	—
158	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂×D₄₃	—	2	2	no	—
160	C₂×D₄₃	—	2	2	no	—
161	C₂×D₄₃	—	2	2	no	—
162	C₂×D₄₃	—	4	4	no	—
163	C₄×C₄₃	—	2	1	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	#423	1	2	yes	—
166	C₄×C₄₃	#430	1	2	yes	—
167	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
168	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
169	C₄×C₄₃	#575	1	2	yes	—
170	Dic₄₃	—	2	2	no	—
171	Dic₄₃	—	2	2	no	—
172	Dic₄₃	—	2	2	no	—
173	#2920	—	43	2	no	—
174	C₂×C₄×M₁₆	—	43	1	no	—
175	#1569	—	43	4	no	—
176	#1569	—	43	4	no	—
177	#1564	—	43	2	no	—
178	#1564	—	43	1	no	—
179	#1564	D₄₃	1	1	yes	characteristic, complemented
180	C₂×C₄₃	—	2	2	no	—
181	C₂×C₄₃	C₂×#423	1	1	yes	characteristic
182	C₂×C₄₃	#2911	1	1	yes	characteristic
183	C₂×C₄₃	C₂×#430	1	1	yes	characteristic
184	D₄₃	—	4	2	no	—
185	C₂²×M₁₆	—	43	1	no	—
186	C₂²×M₁₆	—	43	1	no	—
187	C₂²×C₄²	—	43	1	no	—
188	#432	—	43	4	no	—
189	#432	—	86	2	no	—
190	#431	—	43	2	no	—
191	#431	—	86	2	no	—
192	#431	C₂×D₄₃	1	2	yes	—
193	#423	—	43	4	no	—
194	#407	—	86	2	no	—
195	C₄×M₁₆	—	43	2	no	—
196	C₄×M₁₆	—	43	2	no	—
197	C₄×M₁₆	—	43	2	no	—
198	C₄×M₁₆	—	43	2	no	—
199	C₄×M₁₆	—	86	2	no	—
200	C₂×#97	—	43	1	no	—
201	C₂×#97	—	43	1	no	—
202	C₂×C₄×C₈	—	43	1	no	—
203	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
204	#335	—	43	4	no	—
205	#335	—	43	2	no	—
206	#335	C₂×D₄₃	1	2	yes	complemented
207	#334	—	43	4	no	—
208	#334	—	43	2	no	—
209	#334	C₂×D₄₃	1	2	yes	complemented
210	C₄₃	#17795	1	1	yes	characteristic, complemented, Sylow
211	C₂³×C₄	—	43	1	no	—
212	C₂³×C₄	—	43	1	no	—
213	C₂³×C₄	—	43	1	no	—
214	C₂×M₁₆	—	43	2	no	—
215	C₂×M₁₆	—	43	2	no	—
216	C₂×M₁₆	—	43	2	no	—
217	C₂×M₁₆	—	43	2	no	—
218	C₂×M₁₆	—	86	4	no	—
219	C₂×M₁₆	—	86	4	no	—
220	C₂×M₁₆	—	86	4	no	—
221	C₂²×C₈	—	43	1	no	—
222	C₂²×C₈	—	43	1	no	—
223	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
224	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
225	C₂×C₄²	—	43	1	no	—
226	C₂×C₄²	—	43	1	no	—
227	C₂×C₄²	—	43	1	no	—
228	C₂×C₄²	—	43	2	no	—
229	C₂×C₄²	—	43	2	no	—
230	C₂×C₄²	—	43	2	no	—
231	C₂×C₄²	—	43	2	no	—
232	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
233	C₄⋊C₈	—	43	4	no	—
234	C₄⋊C₈	—	86	4	no	—
235	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₂²⋊C₈	—	86	4	no	—
238	C₂²⋊C₈	—	86	2	no	—
239	#97	—	43	2	no	—
240	#97	—	43	2	no	—
241	#97	—	43	2	no	—
242	#97	—	43	2	no	—
243	#97	—	86	4	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	43	2	no	—
246	C₄×C₈	—	43	2	no	—
247	C₄×C₈	—	86	2	no	—
248	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
249	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
252	C₂⁴	—	43	1	no	—
253	C₂²×C₄	—	43	1	no	—
254	C₂²×C₄	—	43	1	no	—
255	C₂²×C₄	—	43	1	no	—
256	C₂²×C₄	—	43	2	no	—
257	C₂²×C₄	—	43	2	no	—
258	C₂²×C₄	—	43	2	no	—
259	C₂²×C₄	—	43	2	no	—
260	C₂²×C₄	—	43	2	no	—
261	C₂²×C₄	—	43	2	no	—
262	C₂²×C₄	—	86	2	no	—
263	C₂²×C₄	—	86	2	no	—
264	C₂²×C₄	—	172	2	no	—
265	C₂²×C₄	—	172	2	no	—
266	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
268	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
269	M₁₆	—	86	4	no	—
270	M₁₆	—	86	4	no	—
271	M₁₆	—	86	4	no	—
272	M₁₆	—	86	4	no	—
273	M₁₆	—	172	8	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	43	4	no	—
277	C₂×C₈	—	43	2	no	—
278	C₂×C₈	—	43	2	no	—
279	C₂×C₈	—	86	2	no	—
280	C₂×C₈	—	86	4	no	—
281	C₂×C₈	#10621872	1	4	yes	complemented
282	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
283	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
284	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
285	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
286	C₄²	—	43	2	no	—
287	C₄²	—	43	2	no	—
288	C₄²	—	43	2	no	—
289	C₄²	—	43	2	no	—
290	C₄²	—	43	2	no	—
291	C₄²	—	43	2	no	—
292	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
293	C₄²	D₄×D₄₃	1	1	yes	characteristic
294	C₄²	Q₈×D₄₃	1	1	yes	characteristic
295	C₄²	C₂³×D₄₃	1	1	yes	characteristic
296	C₂³	—	43	2	no	—
297	C₂³	—	86	2	no	—
298	C₂³	—	172	1	no	—
299	C₂³	—	172	1	no	—
300	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	43	2	no	—
304	C₂×C₄	—	43	2	no	—
305	C₂×C₄	—	43	2	no	—
306	C₂×C₄	—	86	2	no	—
307	C₂×C₄	—	86	2	no	—
308	C₂×C₄	—	86	2	no	—
309	C₂×C₄	—	86	4	no	—
310	C₂×C₄	—	86	4	no	—
311	C₂×C₄	—	86	4	no	—
312	C₂×C₄	—	172	2	no	—
313	C₂×C₄	—	172	2	no	—
314	C₂×C₄	—	172	1	no	—
315	C₂×C₄	—	172	1	no	—
316	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
317	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
318	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
319	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
320	C₂×C₄	C₂×#10621872	1	2	yes	—
321	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
323	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	4	4	no	—
327	C₈	—	43	4	no	—
328	C₈	—	86	2	no	—
329	C₈	—	172	4	no	—
330	C₈	C₂×#10621872	1	2	yes	—
331	C₈	C₂×#10621872	1	2	yes	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	86	2	no	—
335	C₂²	—	86	2	no	—
336	C₂²	—	86	2	no	—
337	C₂²	—	172	4	no	—
338	C₂²	#14582921	1	2	yes	—
339	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	86	2	no	—
343	C₄	—	86	2	no	—
344	C₄	—	86	2	no	—
345	C₄	#14582921	1	2	yes	—
346	C₄	#14582946	1	2	yes	—
347	C₄	C₂×M₁₆×D₄₃	1	1	yes	characteristic
348	C₄	C₂²×#10621872	1	1	yes	characteristic
349	C₄	#14583696	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	172	2	no	—
352	C₂	C₂×#14582921	1	1	yes	characteristic
353	C₂	#19143183	1	1	yes	characteristic
354	C₂	C₂×#14582946	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1564	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₃	C₄⋊C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1564	D₄₃
hypocenter	C₄₃	#17795
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #19133361	C₂	semidirect
4. #19133359	C₂	semidirect
5. #19133343	C₂	semidirect
6. #19133341	C₂	semidirect
7. C₄₃×#1564	C₂	semidirect
25. #14582652	C₂²	semidirect
26. #14582651	C₂²	semidirect
27. #14582650	C₂²	semidirect
28. #14582649	C₂²	semidirect
31. C₄₃×#335	C₂²	semidirect
32. C₄₃×#334	C₂²	semidirect
93. C₂×C₈×C₄₃	M₁₆	semidirect
110. C₂×C₄₃⋊C₈	M₁₆	semidirect
179. #1564	D₄₃	semidirect
206. #335	C₂×D₄₃	semidirect
209. #334	C₂×D₄₃	semidirect
210. C₄₃	#17795	semidirect
281. C₂×C₈	#10621872	semidirect

Maximal subgroups

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

subgroup	index
#19143223	2
C₂×C₄×#10621872	2
#19133361	2
#19133359	2
#19133343	2
#19133341	2
C₄₃×#1564	2
#17795	43