Subgroups — #25004532

Order 11008.

714 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143159	C₂	1	1	yes	maximal, characteristic
2	#19134328	C₂	1	1	yes	maximal, characteristic
3	#19134304	C₂	1	2	yes	maximal
4	C₂×#14582659	C₂	1	2	yes	maximal
5	C₄₃×#2285	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂×#13063284	C₂²	1	2	yes	—
8	#14582926	—	2	4	no	—
9	#14582923	—	2	2	no	—
10	#14582666	—	2	2	no	—
11	#14582659	—	2	4	no	—
12	#14582658	—	2	4	no	—
13	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#384	C₄	1	2	yes	complemented
17	C₄₃×#382	C₄	1	2	yes	complemented
18	C₄₃×#380	—	2	2	no	—
19	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	D₄	1	4	yes	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	#13063284	—	2	2	no	—
28	#13063284	—	2	2	no	—
29	#13063284	—	2	2	no	—
30	#13063284	Q₈	1	4	yes	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄×Dic₄₃	—	2	2	no	—
34	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
35	C₂³×C₄×C₄₃	—	2	1	no	—
36	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	#117×C₄₃	—	2	2	no	—
38	#117×C₄₃	—	2	2	no	—
39	#117×C₄₃	—	2	2	no	—
40	#117×C₄₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
46	C₂×C₄²×C₄₃	—	2	1	no	—
47	C₂×C₄²×C₄₃	—	2	1	no	—
48	C₂×C₄²×C₄₃	—	2	1	no	—
49	C₂×C₄²×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	2	yes	—
53	C₂×C₄²×C₄₃	D₄	1	2	yes	—
54	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
55	#95×C₄₃	—	2	4	no	—
56	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	—	2	1	no	—
58	C₂²×C₄×C₄₃	—	2	1	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	—	4	4	no	—
63	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	#41	1	2	yes	—
72	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	#41	1	2	yes	—
74	C₂²×Dic₄₃	—	4	2	no	—
75	C₄⋊C₄×C₄₃	—	2	4	no	—
76	C₄⋊C₄×C₄₃	—	2	4	no	—
77	C₄⋊C₄×C₄₃	—	4	8	no	—
78	C₂²⋊C₄×C₄₃	—	2	4	no	—
79	C₂²⋊C₄×C₄₃	—	4	2	no	—
80	C₄²×C₄₃	—	2	1	no	—
81	C₄²×C₄₃	—	2	1	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₄²×C₄₃	—	2	2	no	—
84	C₄²×C₄₃	—	2	2	no	—
85	C₄²×C₄₃	—	2	2	no	—
86	C₄²×C₄₃	—	4	1	no	—
87	C₄²×C₄₃	—	4	1	no	—
88	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
90	C₂²⋊Dic₄₃	—	2	4	no	—
91	C₂²⋊Dic₄₃	—	4	2	no	—
92	C₄⋊Dic₄₃	—	2	4	no	—
93	C₄⋊Dic₄₃	—	4	2	no	—
94	C₄×Dic₄₃	—	2	4	no	—
95	C₄×Dic₄₃	—	4	2	no	—
96	#10621877	—	2	4	no	—
97	#10621877	—	2	4	no	—
98	#10621877	—	4	2	no	—
99	#10621877	—	4	2	no	—
100	C₂×C₄₃⋊C₈	—	2	4	no	—
101	C₂×C₄₃⋊C₈	—	4	2	no	—
102	C₂×C₄₃⋊C₈	—	4	2	no	—
103	C₂³×C₄₃	—	4	2	no	—
104	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₄₃	#121	1	2	yes	—
106	C₂³×C₄₃	#121	1	1	yes	characteristic
107	C₂³×C₄₃	C₂²⋊Q₈	1	2	yes	—
108	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₃	—	2	4	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	—	4	4	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	4	4	no	—
117	C₂×C₄×C₄₃	—	4	4	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#121	1	2	yes	—
126	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#123	1	2	yes	—
131	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#125	1	2	yes	—
133	C₂×C₄×C₄₃	#125	1	2	yes	—
134	C₂×C₄×C₄₃	#126	1	2	yes	—
135	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
137	C₂×Dic₄₃	—	4	4	no	—
138	C₂×Dic₄₃	—	8	4	no	—
139	C₄₃⋊C₈	—	4	4	no	—
140	#11212	—	43	1	no	maximal, Sylow
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	2	4	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	—	4	2	no	—
145	C₂²×C₄₃	—	4	1	no	—
146	C₂²×C₄₃	#380	1	1	yes	characteristic
147	C₂²×C₄₃	#386	1	1	yes	characteristic
148	C₂²×C₄₃	#391	1	1	yes	characteristic
149	C₂²×C₄₃	#396	1	2	yes	—
150	C₂²×C₄₃	#398	1	2	yes	—
151	C₄×C₄₃	—	2	4	no	—
152	C₄×C₄₃	—	4	2	no	—
153	C₄×C₄₃	—	4	4	no	—
154	C₄×C₄₃	—	8	2	no	—
155	C₄×C₄₃	#387	1	2	yes	—
156	C₄×C₄₃	#389	1	2	yes	—
157	C₄×C₄₃	#394	1	1	yes	characteristic
158	C₄×C₄₃	#395	1	1	yes	characteristic
159	C₄×C₄₃	#398	1	1	yes	characteristic
160	C₄×C₄₃	#400	1	1	yes	characteristic
161	Dic₄₃	—	8	4	no	—
162	#2905	—	43	1	no	—
163	#2285	D₄₃	1	1	yes	characteristic
164	#1740	—	43	1	no	—
165	#1733	—	43	2	no	—
166	C₂×#339	—	43	2	no	—
167	C₂×C₄₃	—	2	2	no	—
168	C₂×C₄₃	—	2	1	no	—
169	C₂×C₄₃	—	2	1	no	—
170	C₂×C₄₃	—	2	2	no	—
171	C₂×C₄₃	#1441	1	1	yes	characteristic
172	C₂×C₄₃	#1884	1	2	yes	—
173	C₂²×M₁₆	—	43	2	no	—
174	C₂×#117	—	43	2	no	—
175	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
176	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
177	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
178	#424	—	86	4	no	—
179	#423	—	86	2	no	—
180	#384	Dic₄₃	1	2	yes	complemented
181	#382	Dic₄₃	1	2	yes	complemented
182	#380	—	2	2	no	—
183	C₄×C₄⋊C₄	—	2	2	no	—
184	#342	—	86	2	no	—
185	#339	—	86	4	no	—
186	#339	—	86	4	no	—
187	C₄₃	#11212	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	2	1	no	—
189	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
190	C₂×M₁₆	—	43	4	no	—
191	C₂×M₁₆	—	86	2	no	—
192	C₂×M₁₆	—	86	2	no	—
193	C₂×M₁₆	—	86	2	no	—
194	C₂×M₁₆	—	86	2	no	—
195	C₂×M₁₆	—	86	2	no	—
196	C₂²×C₈	—	86	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	43	4	no	—
202	#117	—	86	2	no	—
203	#117	—	86	2	no	—
204	#117	—	86	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	86	2	no	—
207	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	86	2	no	—
211	C₂×C₂²⋊C₄	C₂²⋊D₄₃	1	2	yes	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	—	86	2	no	—
217	C₂×C₄²	Dic₈₆	1	2	yes	—
218	C₂×C₄²	D₁₇₂	1	2	yes	—
219	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
222	C₄⋊C₈	—	86	8	no	—
223	#95	—	2	4	no	—
224	C₂⁴	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	172	2	no	—
232	C₂²×C₄	C₂²⋊Dic₄₃	1	4	yes	—
233	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
235	C₂²×C₄	#10621898	1	1	yes	characteristic
236	C₂²×C₄	#10621898	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₄₃	1	2	yes	—
238	C₂²×C₄	#10621900	1	2	yes	—
239	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
240	C₂²×C₄	#10621902	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
243	M₁₆	—	86	4	no	—
244	M₁₆	—	86	4	no	—
245	M₁₆	—	172	2	no	—
246	M₁₆	—	172	2	no	—
247	C₂×C₈	—	86	4	no	—
248	C₂×C₈	—	172	2	no	—
249	C₂×C₈	—	172	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	8	no	—
253	C₄⋊C₄	—	86	4	no	—
254	C₄⋊C₄	—	172	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	86	4	no	—
258	C₂²⋊C₄	—	172	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	4	1	no	—
266	C₄²	—	4	1	no	—
267	C₄²	—	86	4	no	—
268	C₄²	—	172	2	no	—
269	C₄²	C₄⋊Dic₄₃	1	2	yes	—
270	C₄²	C₄⋊Dic₄₃	1	2	yes	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
273	C₂³	#13063297	1	2	yes	—
274	C₂³	#13063299	1	1	yes	characteristic
275	C₂³	#13063301	1	2	yes	—
276	C₂³	#13063304	1	1	yes	characteristic
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	344	4	no	—
288	C₂×C₄	—	172	4	no	—
289	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
290	C₂×C₄	#13063284	1	1	yes	characteristic
291	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
292	C₂×C₄	Q₈×Dic₄₃	1	2	yes	—
293	C₂×C₄	#13063293	1	1	yes	characteristic
294	C₂×C₄	#13063299	1	1	yes	characteristic
295	C₂×C₄	#13063300	1	2	yes	—
296	C₂×C₄	#13063303	1	2	yes	—
297	C₂×C₄	#13063304	1	1	yes	characteristic
298	C₂×C₄	#13063305	1	2	yes	—
299	C₂×C₄	#13063306	1	1	yes	characteristic
300	C₂×C₄	#13063310	1	2	yes	—
301	C₂×C₄	#13063314	1	1	yes	characteristic
302	C₂×C₄	#13063315	1	2	yes	—
303	C₂×C₄	#13063316	1	2	yes	—
304	C₂×C₄	#13063320	1	2	yes	—
305	C₂×C₄	#13063323	1	1	yes	characteristic
306	C₂×C₄	#13063324	1	2	yes	—
307	C₈	—	172	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	2	no	—
313	C₂²	#14582759	1	1	yes	characteristic
314	C₂²	#14582779	1	1	yes	characteristic
315	C₂²	#14582806	1	1	yes	characteristic
316	C₂²	#14582821	1	2	yes	—
317	C₂²	#14582832	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	2	no	—
322	C₄	—	344	4	no	—
323	C₄	#14582785	1	2	yes	—
324	C₄	#14582798	1	2	yes	—
325	C₄	#14582816	1	1	yes	characteristic
326	C₄	#14582819	1	1	yes	characteristic
327	C₄	#14582829	1	1	yes	characteristic
328	C₄	#14582838	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	2	no	—
333	C₂	#19132848	1	1	yes	characteristic
334	C₂	#19134979	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063293
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#2285	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#2285	D₄₃
hypocenter	C₄₃	#11212
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
16. C₄₃×#384	C₄	semidirect
17. C₄₃×#382	C₄	semidirect
180. #384	Dic₄₃	semidirect
181. #382	Dic₄₃	semidirect
187. C₄₃	#11212	semidirect

Maximal subgroups

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

subgroup	index
#19143159	2
#19134328	2
#19134304	2
C₂×#14582659	2
C₄₃×#2285	2
#11212	43