Subgroups — #25080527

Order 11008.

702 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	C₂×#14583304	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144083	C₂	1	2	yes	maximal
3	C₈×#10621898	C₂	1	1	yes	maximal, characteristic
4	#19133990	C₂	1	4	yes	maximal, complemented
5	#19133985	C₂	1	4	yes	maximal, complemented
6	#19133268	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063358	—	2	2	no	—
9	C₂×#13063325	C₂²	1	2	yes	—
10	C₄×#10621898	C₂²	1	1	yes	characteristic
11	#14583304	—	2	1	no	—
12	#14583304	—	2	1	no	—
13	#14583304	C₂²	1	4	yes	—
14	#14583110	—	2	2	no	—
15	#14583110	C₂²	1	4	yes	—
16	#14583105	—	2	2	no	—
17	#14583105	C₂²	1	4	yes	—
18	C₈×Dic₈₆	C₂²	1	2	yes	—
19	C₈×D₁₇₂	C₂²	1	2	yes	—
20	C₈×C₂²⋊D₄₃	—	2	2	no	—
21	#14582952	C₂²	1	1	yes	characteristic
22	#14582950	—	2	1	no	—
23	C₂×#13063259	—	2	2	no	—
24	#14582909	—	2	2	no	—
25	C₄×C₈×D₄₃	—	2	1	no	—
26	#14582646	C₂²	1	8	yes	—
27	C₄₃×#423	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#335	C₂²	1	4	yes	—
30	C₂×#10621898	D₄	1	1	yes	characteristic
31	C₂²×Dic₈₆	—	2	2	no	—
32	#13063358	—	4	8	no	—
33	C₂×Dic₁₇₂	—	2	4	no	—
34	C₂×#10621873	—	2	4	no	—
35	C₂×C₈×D₄₃	—	2	2	no	—
36	#13063325	—	2	2	no	—
37	#13063325	—	2	2	no	—
38	#13063325	—	2	2	no	—
39	#13063325	C₂³	1	4	yes	—
40	C₄×Dic₈₆	D₄	1	1	yes	characteristic
41	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
42	C₄×D₁₇₂	D₄	1	1	yes	characteristic
43	C₄×D₁₇₂	C₂³	1	1	yes	characteristic
44	C₄×C₂²⋊D₄₃	—	2	1	no	—
45	C₄×C₂²⋊D₄₃	—	2	1	no	—
46	#13063284	D₄	1	1	yes	characteristic
47	#13063282	—	2	1	no	—
48	C₂×C₄⋊Dic₄₃	—	2	2	no	—
49	C₂×C₄⋊Dic₄₃	—	2	2	no	—
50	C₄²×D₄₃	—	2	1	no	—
51	#13063259	—	2	4	no	—
52	#13063259	—	2	4	no	—
53	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
54	Dic₄₃⋊C₈	—	2	2	no	—
55	#13063250	—	2	4	no	—
56	#13063250	—	2	4	no	—
57	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
58	(C₂×D₄₃)⋊C₈	—	2	2	no	—
59	C₄×C₄₃⋊C₈	—	2	1	no	—
60	C₈×Dic₄₃	—	2	2	no	—
61	C₂×M₁₆×C₄₃	—	2	2	no	—
62	C₂²×C₈×C₄₃	D₄	1	2	yes	—
63	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₄₃	—	2	2	no	—
65	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
66	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₄₃	C₂³	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
72	C₂×C₂²⋊D₄₃	—	2	1	no	—
73	C₂²×Dic₄₃	—	4	2	no	—
74	#10621898	—	2	2	no	—
75	#10621898	D₈	1	4	yes	—
76	C₂×D₁₇₂	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×D₄₃	—	2	1	no	—
78	C₂×C₄×D₄₃	—	2	1	no	—
79	C₂×C₄×D₄₃	—	2	1	no	—
80	C₂×Dic₈₆	—	2	4	no	—
81	C₂×Dic₈₆	—	4	4	no	—
82	C₂×Dic₈₆	—	4	4	no	—
83	C₂×Dic₈₆	—	4	2	no	—
84	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
85	M₁₆×C₄₃	—	4	4	no	—
86	C₂×C₈×C₄₃	—	2	4	no	—
87	C₂×C₈×C₄₃	—	2	4	no	—
88	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
90	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₄₃	#41	1	1	yes	characteristic
93	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊Dic₄₃	—	2	1	no	—
95	(C₂×D₄₃)⋊C₄	—	2	1	no	—
96	(C₂×D₄₃)⋊C₄	—	2	1	no	—
97	C₄⋊Dic₄₃	—	2	4	no	—
98	C₄⋊Dic₄₃	—	2	4	no	—
99	C₄⋊Dic₄₃	—	4	2	no	—
100	C₄⋊Dic₄₃	—	4	2	no	—
101	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
102	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
103	Dic₄₃⋊C₄	—	2	1	no	—
104	Dic₄₃⋊C₄	—	2	1	no	—
105	C₄×Dic₄₃	—	2	1	no	—
106	C₄×Dic₄₃	—	2	1	no	—
107	C₄×Dic₄₃	—	2	1	no	—
108	C₂×C₄₃⋊C₈	—	2	2	no	—
109	Dic₁₇₂	—	4	8	no	—
110	#10621873	—	4	8	no	—
111	C₈×D₄₃	—	4	4	no	—
112	C₂³×C₄₃	#121	1	1	yes	characteristic
113	C₂²×D₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₃	#135	1	2	yes	—
121	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
124	C₂²⋊D₄₃	—	2	4	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	4	4	no	—
129	C₂×Dic₄₃	—	8	4	no	—
130	D₁₇₂	—	2	1	no	—
131	D₁₇₂	C₂×D₈	1	2	yes	—
132	C₄×D₄₃	—	2	4	no	—
133	C₄×D₄₃	—	4	2	no	—
134	C₄×D₄₃	—	4	2	no	—
135	Dic₈₆	—	2	1	no	—
136	Dic₈₆	—	4	4	no	—
137	Dic₈₆	—	4	4	no	—
138	Dic₈₆	—	8	4	no	—
139	Dic₈₆	C₂×D₈	1	2	yes	—
140	C₈×C₄₃	—	2	2	no	—
141	C₈×C₄₃	—	4	4	no	—
142	C₈×C₄₃	#137	1	4	yes	complemented
143	C₄₃⋊C₈	—	4	2	no	—
144	#19271	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	#466	1	2	yes	—
148	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
149	C₂×D₄₃	—	2	2	no	—
150	C₂×D₄₃	—	4	1	no	—
151	C₂×D₄₃	—	4	1	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	#466	1	2	yes	—
155	C₄×C₄₃	#473	1	2	yes	—
156	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
158	C₄×C₄₃	C₂×#137	1	2	yes	—
159	Dic₄₃	—	2	2	no	—
160	Dic₄₃	—	4	1	no	—
161	Dic₄₃	—	4	1	no	—
162	Dic₄₃	—	8	4	no	—
163	C₂×#500	—	43	1	no	—
164	#3039	—	43	2	no	—
165	C₈×#41	—	43	1	no	—
166	#1674	—	43	4	no	—
167	#1673	—	43	4	no	—
168	#1567	D₄₃	1	1	yes	characteristic, complemented
169	#1546	—	43	2	no	—
170	C₂×C₄₃	—	2	2	no	—
171	C₂×C₄₃	#3039	1	1	yes	characteristic
172	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
173	C₂×C₄₃	C₂×#473	1	1	yes	characteristic
174	D₄₃	—	4	2	no	—
175	C₂×#137	—	86	2	no	—
176	C₂×C₄⋊Q₈	—	43	2	no	—
177	C₄×#41	—	43	1	no	—
178	#500	—	43	4	no	—
179	#500	—	86	1	no	—
180	#500	—	86	1	no	—
181	#462	—	43	4	no	—
182	#462	—	86	2	no	—
183	#461	—	43	4	no	—
184	#461	—	86	2	no	—
185	C₈×Q₈	—	43	2	no	—
186	C₈×D₄	—	43	2	no	—
187	C₈×D₄	—	86	2	no	—
188	#431	—	43	1	no	—
189	#431	—	86	1	no	—
190	C₂×#107	—	86	2	no	—
191	#423	C₂×D₄₃	1	2	yes	—
192	#419	—	86	2	no	—
193	C₂×C₄×C₈	—	86	1	no	—
194	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
195	#335	C₂×D₄₃	1	4	yes	—
196	#332	—	43	8	no	—
197	C₄₃	#19271	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	43	1	no	—
199	C₂²×Q₈	—	86	2	no	—
200	#137	—	172	8	no	—
201	C₂×Q₁₆	—	86	4	no	—
202	C₂×SD₁₆	—	86	4	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂²×C₈	—	86	2	no	—
205	C₂²×C₈	D₁₇₂	1	2	yes	—
206	C₄⋊Q₈	—	43	4	no	—
207	C₄⋊Q₈	—	86	2	no	—
208	C₄⋊Q₈	—	86	2	no	—
209	C₄⋊Q₈	—	86	2	no	—
210	C₄×Q₈	—	43	1	no	—
211	C₄×Q₈	—	43	1	no	—
212	C₄×D₄	—	43	1	no	—
213	C₄×D₄	—	43	1	no	—
214	C₄×D₄	—	86	1	no	—
215	C₄×D₄	—	86	1	no	—
216	#117	—	43	1	no	—
217	#117	—	86	1	no	—
218	C₂×C₄⋊C₄	—	86	2	no	—
219	C₂×C₄⋊C₄	—	86	2	no	—
220	C₂×C₄²	—	86	1	no	—
221	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
222	#107	—	86	4	no	—
223	#107	—	86	4	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	43	2	no	—
226	C₄⋊C₈	—	86	2	no	—
227	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
228	#103	—	86	4	no	—
229	#103	—	86	4	no	—
230	C₂²⋊C₈	—	86	1	no	—
231	C₂²⋊C₈	—	86	2	no	—
232	C₄×C₈	—	86	2	no	—
233	C₄×C₈	—	86	1	no	—
234	C₄×C₈	D₁₇₂	1	1	yes	characteristic
235	C₄×C₈	D₁₇₂	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₄₃	1	2	yes	—
237	#41	—	43	4	no	—
238	#41	—	86	2	no	—
239	C₂×Q₈	—	43	1	no	—
240	C₂×Q₈	—	86	4	no	—
241	C₂×Q₈	—	172	4	no	—
242	C₂×Q₈	—	172	4	no	—
243	C₂×Q₈	—	172	2	no	—
244	C₂×D₄	—	43	1	no	—
245	C₂×D₄	—	86	1	no	—
246	C₂²×C₄	—	86	1	no	—
247	C₂²×C₄	—	86	1	no	—
248	C₂²×C₄	—	86	1	no	—
249	C₂²×C₄	—	172	2	no	—
250	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
252	C₂²×C₄	#10621902	1	1	yes	characteristic
253	Q₁₆	—	172	8	no	—
254	SD₁₆	—	172	8	no	—
255	M₁₆	—	4	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	86	2	no	—
259	C₂×C₈	—	172	4	no	—
260	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
261	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
262	C₄⋊C₄	—	43	1	no	—
263	C₄⋊C₄	—	43	1	no	—
264	C₄⋊C₄	—	86	1	no	—
265	C₄⋊C₄	—	86	1	no	—
266	C₄⋊C₄	—	86	4	no	—
267	C₄⋊C₄	—	86	4	no	—
268	C₄⋊C₄	—	172	2	no	—
269	C₄⋊C₄	—	172	2	no	—
270	C₂²⋊C₄	—	86	1	no	—
271	C₂²⋊C₄	—	86	1	no	—
272	C₂²⋊C₄	—	86	1	no	—
273	C₄²	—	86	1	no	—
274	C₄²	—	86	1	no	—
275	C₄²	—	86	1	no	—
276	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
277	C₄²	D₄×D₄₃	1	1	yes	characteristic
278	C₄²	#10621902	1	1	yes	characteristic
279	C₄²	C₂³×D₄₃	1	1	yes	characteristic
280	C₂³	—	86	1	no	—
281	C₂³	#13063300	1	1	yes	characteristic
282	Q₈	—	43	2	no	—
283	Q₈	—	86	1	no	—
284	Q₈	—	344	4	no	—
285	Q₈	—	172	4	no	—
286	Q₈	—	172	4	no	—
287	D₄	—	43	2	no	—
288	D₄	—	86	4	no	—
289	D₄	—	86	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	86	1	no	—
293	C₂×C₄	—	86	4	no	—
294	C₂×C₄	—	86	1	no	—
295	C₂×C₄	—	86	1	no	—
296	C₂×C₄	—	344	4	no	—
297	C₂×C₄	—	172	4	no	—
298	C₂×C₄	—	172	2	no	—
299	C₂×C₄	—	172	2	no	—
300	C₂×C₄	#13063300	1	1	yes	characteristic
301	C₂×C₄	#13063300	1	1	yes	characteristic
302	C₂×C₄	#13063300	1	1	yes	characteristic
303	C₂×C₄	D₈×D₄₃	1	2	yes	—
304	C₂×C₄	#13063348	1	2	yes	—
305	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
307	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	4	no	—
310	C₈	—	172	2	no	—
311	C₈	#13063358	1	4	yes	complemented
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	86	2	no	—
315	C₂²	—	172	1	no	—
316	C₂²	—	172	1	no	—
317	C₂²	#14583136	1	2	yes	—
318	C₂²	C₂×#13063300	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	86	2	no	—
322	C₄	—	344	4	no	—
323	C₄	—	172	1	no	—
324	C₄	—	172	1	no	—
325	C₄	#14583136	1	2	yes	—
326	C₄	#14583175	1	2	yes	—
327	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
328	C₄	C₂×#13063348	1	1	yes	characteristic
329	C₄	C₂×#13063358	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	172	2	no	—
332	C₂	#19144083	1	1	yes	characteristic
333	C₂	C₂×#14583136	1	1	yes	characteristic
334	C₂	C₂×#14583175	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063300
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1567	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂×#10621898	D₄
hypercenter	#1567	D₄₃
hypocenter	C₄₃	#19271
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
1. C₂×#14583304	C₂	semidirect
4. #19133990	C₂	semidirect
5. #19133985	C₂	semidirect
6. #19133268	C₂	semidirect
7. C₄₃×#1567	C₂	semidirect
142. C₈×C₄₃	#137	semidirect
168. #1567	D₄₃	semidirect
197. C₄₃	#19271	semidirect
311. C₈	#13063358	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583304	2
#19144083	2
C₈×#10621898	2
#19133990	2
#19133985	2
#19133268	2
C₄₃×#1567	2
#19271	43