Subgroups — #25002338

Order 11008.

722 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	#19134334	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582658	C₂	1	2	yes	maximal
4	C₂×#14582655	C₂	1	2	yes	maximal
5	C₄₃×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂×#13063284	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×Dic₄₃	—	2	2	no	—
10	C₂²⋊C₄×Dic₄₃	—	2	2	no	—
11	#14582745	C₄	1	2	yes	complemented
12	C₄²×Dic₄₃	C₄	1	2	yes	complemented
13	#14582668	—	2	2	no	—
14	#14582658	C₄	1	8	yes	complemented
15	#14582655	—	2	4	no	—
16	C₂×#117×C₄₃	C₂²	1	2	yes	—
17	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#342	—	2	2	no	—
19	C₂³×Dic₄₃	—	2	1	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	C₂×C₄	1	4	yes	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	#13063284	—	2	1	no	—
28	#13063284	—	2	1	no	—
29	#13063284	D₄	1	2	yes	—
30	#13063284	Q₈	1	2	yes	—
31	C₂×C₄⋊Dic₄₃	—	2	1	no	—
32	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
33	C₂×C₄×Dic₄₃	—	2	1	no	—
34	C₂×C₄×Dic₄₃	—	2	2	no	—
35	C₂×C₄×Dic₄₃	—	2	4	no	—
36	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
40	C₄₃⋊#95	—	2	4	no	—
41	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	#117×C₄₃	—	2	2	no	—
43	#117×C₄₃	—	2	2	no	—
44	#117×C₄₃	—	2	2	no	—
45	#117×C₄₃	—	2	2	no	—
46	#117×C₄₃	—	2	2	no	—
47	#117×C₄₃	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
50	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
52	C₂×C₄²×C₄₃	—	2	2	no	—
53	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
54	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	—	4	2	no	—
56	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	#41	1	2	yes	—
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	#41	1	2	yes	—
66	C₂²×C₄×C₄₃	#41	1	2	yes	—
67	C₂²×Dic₄₃	—	2	1	no	—
68	C₂²×Dic₄₃	—	2	1	no	—
69	C₂²×Dic₄₃	—	2	2	no	—
70	C₂²×Dic₄₃	—	2	4	no	—
71	C₂²×Dic₄₃	—	4	4	no	—
72	C₄⋊C₄×C₄₃	—	2	4	no	—
73	C₄⋊C₄×C₄₃	—	2	4	no	—
74	C₄⋊C₄×C₄₃	—	4	2	no	—
75	C₂²⋊C₄×C₄₃	—	2	4	no	—
76	C₂²⋊C₄×C₄₃	—	2	4	no	—
77	C₂²⋊C₄×C₄₃	—	4	2	no	—
78	C₄²×C₄₃	—	2	2	no	—
79	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	2	4	no	—
81	C₄²×C₄₃	—	4	2	no	—
82	C₂²⋊Dic₄₃	—	2	2	no	—
83	C₂²⋊Dic₄₃	—	2	2	no	—
84	C₄⋊Dic₄₃	—	2	2	no	—
85	C₄⋊Dic₄₃	—	2	2	no	—
86	C₄×Dic₄₃	—	2	1	no	—
87	C₄×Dic₄₃	—	2	1	no	—
88	C₄×Dic₄₃	—	2	2	no	—
89	C₄×Dic₄₃	—	2	4	no	—
90	C₄×Dic₄₃	—	4	8	no	—
91	C₄×Dic₄₃	C₄²	1	2	yes	—
92	C₄×Dic₄₃	C₄²	1	2	yes	—
93	C₄×Dic₄₃	C₄⋊C₄	1	4	yes	—
94	#10621877	—	2	4	no	—
95	#10621877	—	2	4	no	—
96	#10621877	—	4	2	no	—
97	#10621877	—	4	2	no	—
98	C₂×C₄₃⋊C₈	—	2	4	no	—
99	C₂×C₄₃⋊C₈	—	4	2	no	—
100	C₂×C₄₃⋊C₈	—	4	2	no	—
101	C₂³×C₄₃	—	4	2	no	—
102	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₄₃	C₄×D₄	1	2	yes	—
104	C₂³×C₄₃	#121	1	1	yes	characteristic
105	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₄₃	#123	1	2	yes	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	—	4	4	no	—
109	C₂×C₄×C₄₃	—	4	4	no	—
110	C₂×C₄×C₄₃	—	8	4	no	—
111	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#117	1	2	yes	—
115	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#117	1	2	yes	—
119	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#123	1	2	yes	—
125	C₂×C₄×C₄₃	#124	1	2	yes	—
126	C₂×C₄×C₄₃	#125	1	2	yes	—
127	C₂×C₄×C₄₃	#126	1	2	yes	—
128	C₂×C₄×C₄₃	#126	1	2	yes	—
129	C₂×Dic₄₃	—	2	4	no	—
130	C₂×Dic₄₃	—	2	2	no	—
131	C₂×Dic₄₃	—	2	2	no	—
132	C₂×Dic₄₃	—	2	4	no	—
133	C₂×Dic₄₃	—	4	8	no	—
134	C₂×Dic₄₃	—	4	4	no	—
135	C₂×Dic₄₃	—	4	2	no	—
136	C₄₃⋊C₈	—	4	4	no	—
137	#10824	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	—	2	4	no	—
140	C₂²×C₄₃	—	4	2	no	—
141	C₂²×C₄₃	—	4	1	no	—
142	C₂²×C₄₃	—	4	1	no	—
143	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄₃	#380	1	1	yes	characteristic
145	C₂²×C₄₃	#381	1	1	yes	characteristic
146	C₂²×C₄₃	#387	1	2	yes	—
147	C₂²×C₄₃	#388	1	2	yes	—
148	C₄×C₄₃	—	4	4	no	—
149	C₄×C₄₃	—	8	4	no	—
150	C₄×C₄₃	#376	1	1	yes	characteristic
151	C₄×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₄₃	#382	1	1	yes	characteristic
153	C₄×C₄₃	#383	1	1	yes	characteristic
154	C₄×C₄₃	#387	1	2	yes	—
155	C₄×C₄₃	#388	1	2	yes	—
156	Dic₄₃	—	2	4	no	—
157	Dic₄₃	—	4	4	no	—
158	Dic₄₃	—	4	2	no	—
159	C₄×#117	—	43	1	no	—
160	#1742	—	43	1	no	—
161	#1740	D₄₃	1	1	yes	characteristic
162	C₂×#339	—	43	2	no	—
163	C₂×H(3,ℤ/4)	—	43	2	no	—
164	C₂×C₄₃	—	2	1	no	—
165	C₂×C₄₃	—	2	1	no	—
166	C₂×C₄₃	—	2	2	no	—
167	C₂×C₄₃	—	2	2	no	—
168	C₂×C₄₃	#1434	1	1	yes	characteristic
169	C₂×C₄₃	#1827	1	2	yes	—
170	C₂²×M₁₆	—	43	2	no	—
171	C₂×#117	—	43	1	no	—
172	C₂×#117	C₂×D₄₃	1	2	yes	—
173	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
174	C₂²×C₄²	—	43	1	no	—
175	C₄×C₄⋊C₄	—	86	2	no	—
176	C₄×C₂²⋊C₄	—	86	2	no	—
177	#376	—	43	2	no	—
178	C₄³	—	43	2	no	—
179	#343	—	86	2	no	—
180	#342	—	2	2	no	—
181	#339	—	43	8	no	—
182	H(3,ℤ/4)	—	86	4	no	—
183	C₄₃	#10824	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	86	1	no	—
185	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
186	C₂×M₁₆	—	43	4	no	—
187	C₂×M₁₆	—	86	2	no	—
188	C₂×M₁₆	—	86	2	no	—
189	C₂×M₁₆	—	86	2	no	—
190	C₂×M₁₆	—	86	2	no	—
191	C₂×M₁₆	—	86	2	no	—
192	C₂²×C₈	—	86	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	43	2	no	—
199	#117	—	43	2	no	—
200	#117	—	86	1	no	—
201	#117	—	86	1	no	—
202	#117	C₄×D₄₃	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	86	1	no	—
205	C₂×C₄⋊C₄	D₁₇₂	1	2	yes	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	—	86	1	no	—
208	C₂×C₂²⋊C₄	C₂²⋊D₄₃	1	2	yes	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	43	2	no	—
211	C₂×C₄²	—	43	4	no	—
212	C₂×C₄²	—	43	1	no	—
213	C₂×C₄²	—	43	1	no	—
214	C₂×C₄²	—	86	1	no	—
215	C₂×C₄²	—	86	2	no	—
216	C₂×C₄²	—	86	4	no	—
217	C₂×C₄²	C₄×D₄₃	1	2	yes	—
218	#95	—	86	4	no	—
219	C₂⁴	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	86	1	no	—
222	C₂²×C₄	—	86	1	no	—
223	C₂²×C₄	—	86	2	no	—
224	C₂²×C₄	—	86	4	no	—
225	C₂²×C₄	—	172	4	no	—
226	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
227	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
229	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
230	C₂²×C₄	#10621898	1	2	yes	—
231	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
232	C₂²×C₄	#10621900	1	1	yes	characteristic
233	C₂²×C₄	#10621900	1	2	yes	—
234	C₂²×C₄	#10621900	1	2	yes	—
235	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
236	C₂²×C₄	#10621902	1	1	yes	characteristic
237	M₁₆	—	86	4	no	—
238	M₁₆	—	86	4	no	—
239	M₁₆	—	172	2	no	—
240	M₁₆	—	172	2	no	—
241	C₂×C₈	—	86	4	no	—
242	C₂×C₈	—	172	2	no	—
243	C₂×C₈	—	172	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	86	2	no	—
248	C₄⋊C₄	—	86	2	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	86	2	no	—
253	C₂²⋊C₄	—	86	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	43	2	no	—
259	C₄²	—	43	2	no	—
260	C₄²	—	43	4	no	—
261	C₄²	—	86	1	no	—
262	C₄²	—	86	1	no	—
263	C₄²	—	86	2	no	—
264	C₄²	—	86	4	no	—
265	C₄²	—	172	8	no	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
268	C₂³	C₄×C₂²⋊D₄₃	1	2	yes	—
269	C₂³	#13063296	1	1	yes	characteristic
270	C₂³	#13063301	1	1	yes	characteristic
271	C₂³	#13063306	1	2	yes	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	—	86	4	no	—
277	C₂×C₄	—	86	2	no	—
278	C₂×C₄	—	86	2	no	—
279	C₂×C₄	—	86	4	no	—
280	C₂×C₄	—	172	8	no	—
281	C₂×C₄	—	172	4	no	—
282	C₂×C₄	—	172	2	no	—
283	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
284	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
285	C₂×C₄	#13063281	1	2	yes	—
286	C₂×C₄	#13063281	1	1	yes	characteristic
287	C₂×C₄	#13063281	1	1	yes	characteristic
288	C₂×C₄	#13063282	1	1	yes	characteristic
289	C₂×C₄	#13063283	1	2	yes	—
290	C₂×C₄	#13063283	1	1	yes	characteristic
291	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
292	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
293	C₂×C₄	#13063290	1	2	yes	—
294	C₂×C₄	#13063300	1	1	yes	characteristic
295	C₂×C₄	#13063305	1	1	yes	characteristic
296	C₂×C₄	#13063309	1	2	yes	—
297	C₂×C₄	#13063311	1	2	yes	—
298	C₂×C₄	#13063315	1	2	yes	—
299	C₂×C₄	#13063318	1	2	yes	—
300	C₂×C₄	#13063320	1	2	yes	—
301	C₈	—	172	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
308	C₂²	#14582756	1	1	yes	characteristic
309	C₂²	#14582760	1	1	yes	characteristic
310	C₂²	#14582780	1	2	yes	—
311	C₂²	#14582787	1	2	yes	—
312	C₄	—	4	4	no	—
313	C₄	—	8	4	no	—
314	C₄	—	86	4	no	—
315	C₄	—	172	2	no	—
316	C₄	—	172	4	no	—
317	C₄	#14582744	1	1	yes	characteristic
318	C₄	#14582750	1	1	yes	characteristic
319	C₄	#14582763	1	1	yes	characteristic
320	C₄	#14582766	1	1	yes	characteristic
321	C₄	#14582784	1	2	yes	—
322	C₄	#14582791	1	2	yes	—
323	C₂	—	2	2	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	2	no	—
327	C₂	#19132819	1	1	yes	characteristic
328	C₂	#19134696	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063281
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1740	C₂
socle	C₂²×C₄₃	#381
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1740	D₄₃
hypocenter	C₄₃	#10824
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
11. #14582745	C₄	semidirect
12. C₄²×Dic₄₃	C₄	semidirect
14. #14582658	C₄	semidirect
183. C₄₃	#10824	semidirect

Maximal subgroups

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

subgroup	index
#117×Dic₄₃	2
#19134334	2
C₂×#14582658	2
C₂×#14582655	2
C₄₃×#1740	2
#10824	43