Subgroups — #24995611

Order 11008.

526 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134448	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13063236	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582641	C₂	1	2	yes	maximal, complemented
4	C₂×#14582639	C₂	1	2	yes	maximal
5	C₄₃×#2861	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
7	C₂×#13063236	—	2	1	no	—
8	C₂×#13063236	C₂²	1	2	yes	—
9	C₂×#13063236	C₂²	1	2	yes	—
10	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
11	#14582654	—	2	1	no	—
12	#14582641	—	2	4	no	—
13	#14582639	—	2	4	no	—
14	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×#124×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#391	C₄	1	1	yes	characteristic
17	C₄₃×#391	C₄	1	1	yes	characteristic
18	C₄₃×#390	C₄	1	1	yes	characteristic
19	C₄₃×#390	C₄	1	1	yes	characteristic
20	C₄₃×#388	—	2	1	no	—
21	C₄₃×#388	—	2	1	no	—
22	C₄₃×#388	—	2	1	no	—
23	C₄₃×#388	—	2	1	no	—
24	C₄³×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
28	#13063236	—	2	4	no	—
29	#13063236	—	2	2	no	—
30	#13063236	—	2	2	no	—
31	#13063236	—	2	2	no	—
32	#13063236	D₄	1	4	yes	—
33	C₄×C₄₃⋊C₈	—	2	4	no	—
34	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
36	#125×C₄₃	—	2	2	no	—
37	#125×C₄₃	—	2	2	no	—
38	#124×C₄₃	—	2	2	no	—
39	#124×C₄₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
50	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	#95×C₄₃	—	2	1	no	—
58	#95×C₄₃	—	2	1	no	—
59	#95×C₄₃	—	2	1	no	—
60	#95×C₄₃	—	2	1	no	—
61	C₂⁴×C₄₃	—	2	1	no	—
62	C₂×Q₈×C₄₃	—	2	2	no	—
63	C₂×Q₈×C₄₃	—	4	1	no	—
64	C₂×Q₈×C₄₃	—	4	1	no	—
65	C₂×D₄×C₄₃	—	2	2	no	—
66	C₂×D₄×C₄₃	—	4	1	no	—
67	C₂×D₄×C₄₃	—	4	1	no	—
68	C₂²×C₄×C₄₃	—	2	1	no	—
69	C₂²×C₄×C₄₃	—	2	1	no	—
70	C₂²×C₄×C₄₃	—	2	1	no	—
71	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂²⋊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×D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	—	2	2	no	—
79	C₄⋊C₄×C₄₃	—	2	2	no	—
80	C₄⋊C₄×C₄₃	—	4	2	no	—
81	C₄⋊C₄×C₄₃	—	4	1	no	—
82	C₄⋊C₄×C₄₃	—	4	1	no	—
83	C₂²⋊C₄×C₄₃	—	4	2	no	—
84	C₂²⋊C₄×C₄₃	—	4	2	no	—
85	C₂²⋊C₄×C₄₃	—	4	2	no	—
86	C₂²⋊C₄×C₄₃	—	4	2	no	—
87	C₂²⋊C₄×C₄₃	—	4	2	no	—
88	C₂²⋊C₄×C₄₃	—	4	1	no	—
89	C₂²⋊C₄×C₄₃	—	4	1	no	—
90	C₄²×C₄₃	—	2	2	no	—
91	C₄²×C₄₃	—	2	2	no	—
92	C₄²×C₄₃	—	2	2	no	—
93	C₄²×C₄₃	—	4	1	no	—
94	C₄²×C₄₃	—	4	1	no	—
95	C₄²×C₄₃	—	4	1	no	—
96	C₄²×C₄₃	C₂×D₄	1	2	yes	—
97	C₄²×C₄₃	#41	1	2	yes	—
98	C₂×C₄₃⋊C₈	—	2	4	no	—
99	C₂×C₄₃⋊C₈	—	2	4	no	—
100	C₂×C₄₃⋊C₈	—	2	4	no	—
101	C₂×C₄₃⋊C₈	—	2	4	no	—
102	C₂×C₄₃⋊C₈	—	4	2	no	—
103	C₂×C₄₃⋊C₈	—	4	2	no	—
104	C₂³×C₄₃	—	4	1	no	—
105	C₂³×C₄₃	—	4	1	no	—
106	C₂³×C₄₃	—	4	1	no	—
107	C₂³×C₄₃	—	4	1	no	—
108	C₂³×C₄₃	—	4	1	no	—
109	C₂³×C₄₃	—	4	1	no	—
110	C₂³×C₄₃	—	4	1	no	—
111	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	Q₈×C₄₃	—	8	2	no	—
113	D₄×C₄₃	—	8	2	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	4	1	no	—
130	C₂×C₄×C₄₃	—	4	1	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	4	1	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	#104	1	2	yes	—
139	C₂×C₄×C₄₃	#104	1	2	yes	—
140	C₂×C₄×C₄₃	#104	1	2	yes	—
141	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₄₃	#121	1	2	yes	—
143	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
145	C₄₃⋊C₈	—	2	8	no	—
146	C₄₃⋊C₈	—	4	4	no	—
147	#9687	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	8	1	no	—
149	C₂²×C₄₃	—	8	1	no	—
150	C₂²×C₄₃	—	8	1	no	—
151	C₂²×C₄₃	—	8	1	no	—
152	C₂²×C₄₃	—	8	1	no	—
153	C₂²×C₄₃	—	8	1	no	—
154	C₂²×C₄₃	—	8	1	no	—
155	C₂²×C₄₃	#390	1	1	yes	characteristic
156	C₂²×C₄₃	#413	1	1	yes	characteristic
157	C₂²×C₄₃	#413	1	1	yes	characteristic
158	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
160	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
161	C₂²×C₄₃	#421	1	1	yes	characteristic
162	C₄×C₄₃	—	2	2	no	—
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	2	2	no	—
167	C₄×C₄₃	—	2	2	no	—
168	C₄×C₄₃	—	2	2	no	—
169	C₄×C₄₃	—	4	1	no	—
170	C₄×C₄₃	—	8	1	no	—
171	C₄×C₄₃	—	8	1	no	—
172	C₄×C₄₃	—	8	1	no	—
173	#2861	D₄₃	1	1	yes	characteristic
174	#1767	—	43	1	no	—
175	C₄×#97	—	43	1	no	—
176	C₂×#330	—	43	2	no	—
177	C₂×#329	—	43	2	no	—
178	C₂×C₄₃	—	16	1	no	—
179	C₂×C₄₃	#1524	1	1	yes	characteristic
180	C₂×C₄₃	#1524	1	1	yes	characteristic
181	C₂×C₄₃	#1529	1	1	yes	characteristic
182	C₂×C₄₃	#1532	1	1	yes	characteristic
183	C₂×C₄₃	#1962	1	1	yes	characteristic
184	C₂×C₄₃	#1967	1	1	yes	characteristic
185	C₂×C₄₃	#1980	1	1	yes	characteristic
186	C₂×#125	C₂×D₄₃	1	1	yes	characteristic
187	C₂×#124	C₂×D₄₃	1	1	yes	characteristic
188	C₂×C₄⋊C₈	—	86	1	no	—
189	C₂×#97	—	43	2	no	—
190	C₂×#97	—	43	2	no	—
191	C₂×#97	—	86	1	no	—
192	C₂×C₄×C₈	—	86	1	no	—
193	#391	Dic₄₃	1	1	yes	characteristic
194	#391	Dic₄₃	1	1	yes	characteristic
195	#390	Dic₄₃	1	1	yes	characteristic
196	#390	Dic₄₃	1	1	yes	characteristic
197	#388	—	2	1	no	—
198	#388	—	2	1	no	—
199	#388	—	2	1	no	—
200	#388	—	2	1	no	—
201	C₄³	C₂×D₄₃	1	1	yes	characteristic
202	#336	—	86	1	no	—
203	#330	—	86	4	no	—
204	#329	—	86	4	no	—
205	C₄₃	#9687	1	1	yes	characteristic, complemented, Sylow
206	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
207	C₂²×D₄	C₂×Dic₄₃	1	1	yes	characteristic
208	C₂²×C₈	—	86	2	no	—
209	C₂²×C₈	—	86	2	no	—
210	#125	—	2	2	no	—
211	#125	—	2	2	no	—
212	#124	—	2	2	no	—
213	#124	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	—	2	1	no	—
219	C₂×C₂²⋊C₄	—	2	1	no	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	2	1	no	—
223	C₂×C₂²⋊C₄	—	2	1	no	—
224	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
227	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
231	C₄⋊C₈	—	86	4	no	—
232	#97	—	43	4	no	—
233	#97	—	86	4	no	—
234	#97	—	86	2	no	—
235	#97	—	86	2	no	—
236	#97	—	86	2	no	—
237	C₄×C₈	—	86	4	no	—
238	#95	—	2	1	no	—
239	#95	—	2	1	no	—
240	#95	—	2	1	no	—
241	#95	—	2	1	no	—
242	C₂⁴	—	2	1	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
256	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
259	C₂×C₈	—	86	4	no	—
260	C₂×C₈	—	86	4	no	—
261	C₂×C₈	—	86	4	no	—
262	C₂×C₈	—	86	4	no	—
263	C₂×C₈	—	172	2	no	—
264	C₂×C₈	—	172	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	D₄×D₄₃	1	2	yes	—
284	C₄²	#10621900	1	2	yes	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
293	Q₈	—	8	2	no	—
294	D₄	—	8	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	#13063252	1	2	yes	—
320	C₂×C₄	#13063252	1	2	yes	—
321	C₂×C₄	#13063252	1	2	yes	—
322	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
323	C₂×C₄	#13063296	1	2	yes	—
324	C₂×C₄	#13063312	1	1	yes	characteristic
325	C₂×C₄	#13063321	1	1	yes	characteristic
326	C₈	—	86	8	no	—
327	C₈	—	172	4	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	1	no	—
334	C₂²	—	8	1	no	—
335	C₂²	#14582802	1	1	yes	characteristic
336	C₂²	#14582883	1	1	yes	characteristic
337	C₂²	#14582883	1	1	yes	characteristic
338	C₂²	C₂×#13063252	1	1	yes	characteristic
339	C₂²	C₂×#13063252	1	1	yes	characteristic
340	C₂²	C₂×#13063252	1	1	yes	characteristic
341	C₂²	#14582916	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	4	1	no	—
350	C₄	—	8	1	no	—
351	C₄	—	8	1	no	—
352	C₄	—	8	1	no	—
353	C₂	—	16	1	no	—
354	C₂	#19133183	1	1	yes	characteristic
355	C₂	#19133183	1	1	yes	characteristic
356	C₂	#19133202	1	1	yes	characteristic
357	C₂	#19133215	1	1	yes	characteristic
358	C₂	#19135424	1	1	yes	characteristic
359	C₂	#19135445	1	1	yes	characteristic
360	C₂	#19135515	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2861	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₃	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2861	D₄₃
hypocenter	C₄₃	#9687
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2861

Complements

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

N	H ≅ G/N	type
1. #19134448	C₂	semidirect
2. C₄×#13063236	C₂	semidirect
3. C₂×#14582641	C₂	semidirect
205. C₄₃	#9687	semidirect

Maximal subgroups

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

subgroup	index
#19134448	2
C₄×#13063236	2
C₂×#14582641	2
C₂×#14582639	2
C₄₃×#2861	2
#9687	43