Subgroups — #25110203

Order 11008.

790 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583302	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063325	C₂	1	1	yes	maximal, characteristic
3	#19135557	C₂	1	1	yes	maximal, characteristic
4	#19135555	C₂	1	1	yes	maximal, characteristic
5	#19135166	C₂	1	4	yes	maximal
6	#19135163	C₂	1	4	yes	maximal
7	#19134722	C₂	1	1	yes	maximal, characteristic
8	#19134341	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₆	—	2	1	no	—
12	#14583302	C₄	1	8	yes	complemented
13	C₂×#13063258	C₂²	1	4	yes	—
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
16	#14582805	—	2	1	no	—
17	#14582797	—	2	1	no	—
18	#14582797	C₂²	1	2	yes	—
19	#14582769	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊Dic₄₃	—	2	1	no	—
21	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
22	#14582661	C₂²	1	4	yes	—
23	#14582661	C₂²	1	4	yes	—
24	#14582661	C₂²	1	4	yes	—
25	#14582654	C₂²	1	2	yes	—
26	#14582654	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₄₃	—	2	1	no	—
29	C₄₃×#391	—	2	1	no	—
30	C₄₃×#389	—	2	1	no	—
31	C₄₃×#389	C₂²	1	2	yes	—
32	C₄₃×#384	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
34	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
35	C₄³×C₄₃	C₂²	1	1	yes	characteristic
36	C₂²×Dic₈₆	—	2	1	no	—
37	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
39	#13063325	—	2	2	no	—
40	#13063325	C₂×C₄	1	4	yes	—
41	C₄×Dic₈₆	—	4	4	no	—
42	C₂×C₄⋊Dic₄₃	—	2	1	no	—
43	C₂×C₄⋊Dic₄₃	—	2	1	no	—
44	C₂×C₄⋊Dic₄₃	—	2	1	no	—
45	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
47	C₂×Dic₄₃⋊C₄	—	2	2	no	—
48	C₂×C₄×Dic₄₃	—	2	2	no	—
49	#13063258	C₂×C₄	1	16	yes	—
50	C₄×C₄₃⋊C₈	—	2	2	no	—
51	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
52	C₄₃⋊#95	—	2	2	no	—
53	C₄₃⋊#95	—	2	2	no	—
54	C₂²×Q₈×C₄₃	—	2	1	no	—
55	C₄⋊Q₈×C₄₃	—	2	2	no	—
56	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
57	C₄×Q₈×C₄₃	—	4	4	no	—
58	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₃	—	2	2	no	—
65	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
72	#95×C₄₃	—	2	2	no	—
73	#95×C₄₃	—	2	2	no	—
74	C₂×Q₈×C₄₃	—	2	4	no	—
75	C₂×Q₈×C₄₃	—	4	2	no	—
76	C₂×Q₈×C₄₃	—	4	2	no	—
77	C₂²×C₄×C₄₃	—	2	2	no	—
78	C₂²×C₄×C₄₃	—	2	2	no	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₄₃	—	2	2	no	—
87	C₂²×Dic₄₃	—	2	2	no	—
88	C₂×Dic₈₆	—	2	4	no	—
89	C₂×Dic₈₆	—	4	2	no	—
90	C₂×Dic₈₆	—	4	2	no	—
91	C₄⋊C₄×C₄₃	—	2	4	no	—
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	2	4	no	—
94	C₄⋊C₄×C₄₃	—	4	4	no	—
95	C₄⋊C₄×C₄₃	—	4	2	no	—
96	C₄⋊C₄×C₄₃	—	4	2	no	—
97	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
98	C₄²×C₄₃	—	2	2	no	—
99	C₄²×C₄₃	—	2	2	no	—
100	C₄²×C₄₃	—	2	2	no	—
101	C₄²×C₄₃	—	2	2	no	—
102	C₄²×C₄₃	—	2	2	no	—
103	C₄²×C₄₃	—	4	4	no	—
104	C₄²×C₄₃	D₈	1	4	yes	—
105	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₄₃	#41	1	2	yes	—
107	C₄⋊Dic₄₃	—	2	4	no	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	2	4	no	—
110	C₄⋊Dic₄₃	—	4	2	no	—
111	C₄⋊Dic₄₃	—	4	2	no	—
112	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
113	Dic₄₃⋊C₄	—	4	4	no	—
114	C₄×Dic₄₃	—	4	4	no	—
115	C₂×C₄₃⋊C₈	—	2	2	no	—
116	C₂×C₄₃⋊C₈	—	2	2	no	—
117	C₂×C₄₃⋊C₈	—	2	2	no	—
118	C₂×C₄₃⋊C₈	—	2	2	no	—
119	C₂×C₄₃⋊C₈	—	2	2	no	—
120	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
121	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	Q₈×C₄₃	—	4	8	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	4	no	—
128	C₂×C₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	1	no	—
132	C₂×C₄×C₄₃	—	2	1	no	—
133	C₂×C₄×C₄₃	—	2	1	no	—
134	C₂×C₄×C₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₄₃	#135	1	2	yes	—
145	C₂×C₄×C₄₃	#137	1	2	yes	—
146	C₂×C₄×C₄₃	#137	1	2	yes	—
147	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
150	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
151	C₂×Dic₄₃	—	2	4	no	—
152	C₂×Dic₄₃	—	4	2	no	—
153	C₂×Dic₄₃	—	4	2	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	C₂×Dic₄₃	—	4	2	no	—
157	Dic₈₆	—	4	8	no	—
158	C₄₃⋊C₈	—	2	4	no	—
159	C₄₃⋊C₈	—	4	2	no	—
160	#22247	—	43	1	no	maximal, Sylow
161	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂²×C₄₃	#536	1	1	yes	characteristic
163	C₂²×C₄₃	#536	1	1	yes	characteristic
164	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
165	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
166	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
167	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
168	C₄×C₄₃	—	2	2	no	—
169	C₄×C₄₃	—	2	2	no	—
170	C₄×C₄₃	—	4	4	no	—
171	C₄×C₄₃	—	4	1	no	—
172	C₄×C₄₃	—	4	1	no	—
173	C₄×C₄₃	—	8	2	no	—
174	C₄×C₄₃	C₄×D₈	1	4	yes	—
175	C₄×C₄₃	#441	1	4	yes	—
176	C₄×C₄₃	#519	1	2	yes	—
177	C₄×C₄₃	#541	1	2	yes	—
178	Dic₄₃	—	4	4	no	—
179	Dic₄₃	—	8	2	no	—
180	C₂×#500	—	43	1	no	—
181	C₄×C₄⋊Q₈	—	43	1	no	—
182	C₄×C₄⋊Q₈	D₄₃	1	1	yes	characteristic
183	#1990	—	43	1	no	—
184	#1990	—	43	1	no	—
185	#1918	—	43	4	no	—
186	#1918	—	43	4	no	—
187	#1834	—	43	1	no	—
188	#1744	—	43	1	no	—
189	C₂×C₄₃	#2327	1	1	yes	characteristic
190	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
191	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
192	C₂×C₄₃	#3226	1	1	yes	characteristic
193	C₂×C₄₃	#3226	1	1	yes	characteristic
194	C₂×C₄₃	#3231	1	1	yes	characteristic
195	C₂×C₄₃	#3231	1	1	yes	characteristic
196	C₂×C₄⋊Q₈	—	43	1	no	—
197	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
198	C₂×C₄×Q₈	—	2	1	no	—
199	C₂×C₄×Q₈	—	86	1	no	—
200	#500	—	43	8	no	—
201	C₂×#107	—	43	4	no	—
202	C₂×C₄×C₈	—	43	1	no	—
203	C₂×C₄×C₈	—	43	1	no	—
204	#391	—	2	1	no	—
205	#391	—	86	1	no	—
206	#389	—	2	1	no	—
207	#389	—	43	2	no	—
208	#389	—	86	1	no	—
209	#389	C₂×D₄₃	1	2	yes	—
210	#384	—	43	1	no	—
211	#384	C₂×D₄₃	1	1	yes	characteristic
212	C₄×C₄⋊C₄	—	2	1	no	—
213	C₄×C₄⋊C₄	—	43	2	no	—
214	C₄×C₄⋊C₄	—	86	1	no	—
215	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
216	C₄³	C₂×D₄₃	1	1	yes	characteristic
217	#340	—	43	4	no	—
218	#340	—	43	4	no	—
219	#340	—	43	4	no	—
220	#336	—	43	2	no	—
221	#336	—	43	2	no	—
222	C₄₃	#22247	1	1	yes	characteristic, complemented, Sylow
223	C₂²×Q₈	—	2	1	no	—
224	C₂²×Q₈	—	86	1	no	—
225	C₂²×C₈	—	43	2	no	—
226	C₂²×C₈	—	43	2	no	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	—	43	4	no	—
229	C₄⋊Q₈	—	86	2	no	—
230	C₄⋊Q₈	C₄×D₄₃	1	4	yes	—
231	C₄×Q₈	—	4	4	no	—
232	C₄×Q₈	—	172	4	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	2	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	43	2	no	—
238	C₂×C₄⋊C₄	—	43	2	no	—
239	C₂×C₄⋊C₄	—	86	1	no	—
240	C₂×C₄⋊C₄	—	86	2	no	—
241	C₂×C₄⋊C₄	—	86	1	no	—
242	C₂×C₄⋊C₄	—	86	1	no	—
243	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
244	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
245	C₂×C₄²	—	2	2	no	—
246	C₂×C₄²	—	86	2	no	—
247	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
248	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
249	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
251	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
253	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
254	#107	—	43	16	no	—
255	C₄×C₈	—	43	4	no	—
256	C₄×C₈	—	86	2	no	—
257	#95	—	2	2	no	—
258	#95	—	2	2	no	—
259	#95	—	86	2	no	—
260	#95	—	86	2	no	—
261	C₂×Q₈	—	2	4	no	—
262	C₂×Q₈	—	4	2	no	—
263	C₂×Q₈	—	4	2	no	—
264	C₂×Q₈	—	86	4	no	—
265	C₂×Q₈	—	172	2	no	—
266	C₂×Q₈	—	172	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	86	2	no	—
270	C₂²×C₄	—	86	2	no	—
271	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
277	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
278	C₂×C₈	—	43	4	no	—
279	C₂×C₈	—	86	2	no	—
280	C₂×C₈	—	86	2	no	—
281	C₂×C₈	—	86	2	no	—
282	C₂×C₈	—	86	2	no	—
283	C₂×C₈	—	86	2	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	4	4	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	43	4	no	—
291	C₄⋊C₄	—	86	4	no	—
292	C₄⋊C₄	—	86	2	no	—
293	C₄⋊C₄	—	86	4	no	—
294	C₄⋊C₄	—	172	4	no	—
295	C₄⋊C₄	—	172	2	no	—
296	C₄⋊C₄	—	172	2	no	—
297	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	4	4	no	—
304	C₄²	—	172	4	no	—
305	C₄²	#10621882	1	4	yes	—
306	C₄²	C₂×C₄×D₄₃	1	2	yes	—
307	C₄²	#10621898	1	2	yes	—
308	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
309	Q₈	—	4	8	no	—
310	Q₈	—	172	8	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	86	4	no	—
328	C₂×C₄	—	172	2	no	—
329	C₂×C₄	—	172	2	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	—	172	2	no	—
333	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
334	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
335	C₂×C₄	C₂×#10621882	1	2	yes	—
336	C₂×C₄	C₂×#10621882	1	2	yes	—
337	C₂×C₄	C₂×#10621882	1	2	yes	—
338	C₂×C₄	#13063345	1	2	yes	—
339	C₂×C₄	#13063355	1	2	yes	—
340	C₂×C₄	#13063356	1	2	yes	—
341	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
342	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
343	C₂×C₄	#13063372	1	2	yes	—
344	C₂×C₄	#13063373	1	2	yes	—
345	C₈	—	86	4	no	—
346	C₈	—	172	2	no	—
347	C₂²	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
348	C₂²	#14583470	1	1	yes	characteristic
349	C₂²	#14583473	1	1	yes	characteristic
350	C₂²	C₂²×#10621882	1	1	yes	characteristic
351	C₂²	C₂×#13063345	1	1	yes	characteristic
352	C₂²	C₂×#13063355	1	1	yes	characteristic
353	C₂²	C₂×#13063356	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	4	no	—
359	C₄	—	8	2	no	—
360	C₄	—	344	2	no	—
361	C₄	—	172	4	no	—
362	C₄	C₄×#10621882	1	4	yes	—
363	C₄	#14583004	1	4	yes	—
364	C₄	#14583375	1	2	yes	—
365	C₄	#14583507	1	2	yes	—
366	C₂	#19137311	1	1	yes	characteristic
367	C₂	C₂×C₄×#10621882	1	1	yes	characteristic
368	C₂	C₂×#14583004	1	1	yes	characteristic
369	C₂	#19145713	1	1	yes	characteristic
370	C₂	#19145716	1	1	yes	characteristic
371	C₂	#19145750	1	1	yes	characteristic
372	C₂	#19145754	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×C₄⋊Q₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#22247
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
12. #14583302	C₄	semidirect
222. C₄₃	#22247	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583302	2
C₄×#13063325	2
#19135557	2
#19135555	2
#19135166	2
#19135163	2
#19134722	2
#19134341	2
C₄×C₄⋊Q₈×C₄₃	2
#22247	43