Subgroups — #25251813

Order 11008.

544 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145245	C₂	1	1	yes	maximal, characteristic
2	#19143464	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136787	C₂	1	2	yes	maximal, complemented
4	#19136760	C₂	1	2	yes	maximal, complemented
5	#19136636	C₂	1	2	yes	maximal, complemented
6	#19136514	C₂	1	2	yes	maximal, complemented
7	#19136461	C₂	1	2	yes	maximal, complemented
8	C₄×#13063270	C₂	1	2	yes	maximal, complemented
9	C₄₃×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
11	#14583504	—	2	1	no	—
12	#14583376	—	2	1	no	—
13	C₂×#13063270	C₂²	1	2	yes	—
14	C₂×#13063270	C₂²	1	2	yes	—
15	#14583301	C₂²	1	1	yes	characteristic
16	#14583276	C₂²	1	1	yes	characteristic
17	#14583264	—	2	1	no	—
18	#14583152	—	2	1	no	—
19	#14583135	C₂²	1	2	yes	—
20	#14583012	—	2	1	no	—
21	#14583007	—	2	1	no	—
22	#14583007	—	2	1	no	—
23	C₄×Dic₁₇₂	C₂²	1	2	yes	—
24	C₄×Dic₁₇₂	C₂²	1	2	yes	—
25	C₄×D₃₄₄	C₂²	1	2	yes	—
26	#14582937	C₂²	1	1	yes	characteristic
27	#14582909	—	2	1	no	—
28	#14582727	C₂²	1	2	yes	—
29	#14582725	C₂²	1	2	yes	—
30	#14582704	C₂²	1	2	yes	—
31	#14582704	C₂²	1	2	yes	—
32	#14582701	C₂²	1	2	yes	—
33	#14582701	C₂²	1	2	yes	—
34	C₄₃×#405	C₂²	1	1	yes	characteristic
35	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
36	C₄₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₄₃×#346	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
39	C₂²×Dic₈₆	—	2	1	no	—
40	C₂×Dic₁₇₂	—	2	1	no	—
41	C₂×Dic₁₇₂	—	2	4	no	—
42	C₂×Dic₁₇₂	C₂³	1	2	yes	—
43	C₂×D₃₄₄	—	2	1	no	—
44	C₂×D₃₄₄	C₂³	1	2	yes	—
45	#13063322	—	2	1	no	—
46	#13063317	—	2	1	no	—
47	#13063310	—	4	2	no	—
48	#13063299	—	2	2	no	—
49	#13063298	—	4	2	no	—
50	C₄×Dic₈₆	—	2	2	no	—
51	C₄×Dic₈₆	—	2	2	no	—
52	#13063290	—	4	2	no	—
53	C₄×D₁₇₂	—	2	2	no	—
54	#13063284	—	2	1	no	—
55	#13063281	—	4	2	no	—
56	#13063270	—	2	4	no	—
57	#13063270	—	2	4	no	—
58	#13063259	C₂³	1	2	yes	—
59	#13063259	C₂³	1	2	yes	—
60	#13063257	—	2	1	no	—
61	#13063257	—	2	1	no	—
62	#13063250	—	2	2	no	—
63	#13063250	—	2	2	no	—
64	#13063247	—	2	2	no	—
65	#13063247	—	2	2	no	—
66	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	#117×C₄₃	D₄	1	1	yes	characteristic
69	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
70	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
71	#97×C₄₃	D₄	1	1	yes	characteristic
72	#97×C₄₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₄₃	—	4	2	no	—
77	C₂²×Dic₄₃	—	4	1	no	—
78	C₂×D₁₇₂	—	2	2	no	—
79	C₂×D₁₇₂	—	4	2	no	—
80	C₂×C₄×D₄₃	—	4	2	no	—
81	C₂×Dic₈₆	—	2	4	no	—
82	C₂×Dic₈₆	—	2	2	no	—
83	C₂×Dic₈₆	—	4	2	no	—
84	C₂×Dic₈₆	—	4	1	no	—
85	M₁₆×C₄₃	—	4	1	no	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
92	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
93	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂²⋊Dic₄₃	—	4	1	no	—
96	(C₂×D₄₃)⋊C₄	—	4	2	no	—
97	(C₂×D₄₃)⋊C₄	—	4	2	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	C₄⋊Dic₄₃	—	2	2	no	—
100	Dic₄₃⋊C₄	—	4	2	no	—
101	Dic₄₃⋊C₄	—	4	2	no	—
102	Dic₄₃⋊C₄	—	4	2	no	—
103	C₄×Dic₄₃	—	4	2	no	—
104	C₄×Dic₄₃	—	4	2	no	—
105	C₄×Dic₄₃	—	4	1	no	—
106	Dic₁₇₂	—	2	4	no	—
107	Dic₁₇₂	—	2	2	no	—
108	Dic₁₇₂	#41	1	4	yes	—
109	D₃₄₄	—	2	2	no	—
110	D₃₄₄	—	2	2	no	—
111	D₃₄₄	—	4	2	no	—
112	C₁₆×C₄₃	—	2	2	no	—
113	C₁₆×C₄₃	—	2	2	no	—
114	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	C₂²⋊D₄₃	—	16	2	no	—
123	C₂×Dic₄₃	—	4	4	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	C₂×Dic₄₃	—	4	2	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	D₁₇₂	—	4	2	no	—
128	D₁₇₂	—	4	2	no	—
129	D₁₇₂	—	16	2	no	—
130	C₄×D₄₃	—	16	2	no	—
131	Dic₈₆	—	2	4	no	—
132	Dic₈₆	—	4	4	no	—
133	Dic₈₆	—	4	2	no	—
134	C₈×C₄₃	—	2	1	no	—
135	C₈×C₄₃	—	2	1	no	—
136	C₈×C₄₃	—	2	1	no	—
137	C₈×C₄₃	C₂×#41	1	2	yes	—
138	#33686	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	—	2	1	no	—
140	C₂²×C₄₃	—	2	2	no	—
141	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₄₃	—	8	2	no	—
143	C₂×D₄₃	—	8	2	no	—
144	C₂×D₄₃	—	8	2	no	—
145	C₄×C₄₃	—	2	1	no	—
146	C₄×C₄₃	—	2	2	no	—
147	C₄×C₄₃	—	2	2	no	—
148	C₄×C₄₃	#525	1	1	yes	characteristic
149	C₄×C₄₃	#575	1	1	yes	characteristic
150	Dic₄₃	—	4	4	no	—
151	Dic₄₃	—	8	2	no	—
152	Dic₄₃	—	8	2	no	—
153	#3176	—	43	1	no	—
154	#2950	—	43	1	no	—
155	#2234	—	43	2	no	—
156	#2229	—	43	2	no	—
157	#2207	—	43	2	no	—
158	#2185	—	43	2	no	—
159	#2175	—	43	2	no	—
160	C₄×SD₃₂	—	43	2	no	—
161	#2164	D₄₃	1	1	yes	characteristic, complemented
162	C₂×C₄₃	—	4	1	no	—
163	C₂×C₄₃	#3100	1	1	yes	characteristic
164	C₂×C₄₃	#3416	1	2	yes	—
165	D₄₃	—	16	2	no	—
166	C₂²×Q₁₆	—	43	1	no	—
167	#540	—	86	1	no	—
168	#519	—	86	1	no	—
169	C₂×SD₃₂	—	43	2	no	—
170	C₂×SD₃₂	—	43	2	no	—
171	#499	—	43	1	no	—
172	#493	—	43	1	no	—
173	#490	—	86	1	no	—
174	#469	—	86	1	no	—
175	#466	—	43	2	no	—
176	#442	—	86	1	no	—
177	#441	—	86	1	no	—
178	#441	—	86	1	no	—
179	C₄×Q₁₆	—	43	2	no	—
180	C₄×Q₁₆	—	43	2	no	—
181	C₄×D₈	—	43	2	no	—
182	#427	—	43	1	no	—
183	#419	—	86	1	no	—
184	#405	C₂×D₄₃	1	1	yes	characteristic
185	#367	—	43	2	no	—
186	#366	—	43	2	no	—
187	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
188	#358	—	43	2	no	—
189	#358	—	43	2	no	—
190	#357	—	43	2	no	—
191	#357	—	43	2	no	—
192	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	—
193	#346	C₂×D₄₃	1	1	yes	characteristic
194	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
195	C₄₃	#33686	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	—	86	1	no	—
197	C₂×Q₁₆	—	43	2	no	—
198	C₂×Q₁₆	—	86	4	no	—
199	C₂×Q₁₆	—	86	1	no	—
200	C₂×D₈	—	43	2	no	—
201	C₂×D₈	—	86	1	no	—
202	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
204	#127	—	86	1	no	—
205	#125	—	86	1	no	—
206	#123	—	172	2	no	—
207	#121	—	86	2	no	—
208	#121	—	172	2	no	—
209	C₄×Q₈	—	86	2	no	—
210	C₄×Q₈	—	86	2	no	—
211	C₄×Q₈	—	172	2	no	—
212	C₄×D₄	—	86	2	no	—
213	#117	—	86	1	no	—
214	#117	—	172	2	no	—
215	#117	D₁₇₂	1	1	yes	characteristic
216	SD₃₂	—	86	4	no	—
217	SD₃₂	—	86	4	no	—
218	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
219	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
220	#107	—	43	2	no	—
221	#107	—	43	2	no	—
222	#106	—	86	1	no	—
223	#106	—	86	1	no	—
224	#103	—	86	2	no	—
225	#103	—	86	2	no	—
226	#102	—	86	2	no	—
227	#102	—	86	2	no	—
228	#97	D₁₇₂	1	1	yes	characteristic
229	#97	D₁₇₂	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₂×Q₈	—	86	4	no	—
233	C₂×Q₈	—	86	2	no	—
234	C₂×Q₈	—	172	2	no	—
235	C₂×Q₈	—	172	1	no	—
236	C₂×D₄	—	86	2	no	—
237	C₂×D₄	—	172	2	no	—
238	C₂×D₄	—	172	2	no	—
239	C₂²×C₄	—	172	1	no	—
240	C₂²×C₄	—	172	2	no	—
241	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
242	Q₁₆	—	43	4	no	—
243	Q₁₆	—	86	4	no	—
244	Q₁₆	—	86	2	no	—
245	D₈	—	86	2	no	—
246	D₈	—	86	2	no	—
247	D₈	—	172	2	no	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
251	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
252	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
254	C₄⋊C₄	—	86	2	no	—
255	C₄⋊C₄	—	86	2	no	—
256	C₄⋊C₄	—	172	2	no	—
257	C₄⋊C₄	—	172	2	no	—
258	C₄⋊C₄	—	172	2	no	—
259	C₄⋊C₄	D₃₄₄	1	2	yes	—
260	C₂²⋊C₄	—	172	1	no	—
261	C₂²⋊C₄	—	172	2	no	—
262	C₂²⋊C₄	—	172	2	no	—
263	C₂²⋊C₄	D₃₄₄	1	2	yes	—
264	C₄²	—	172	1	no	—
265	C₄²	—	172	2	no	—
266	C₄²	—	172	2	no	—
267	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
268	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
269	C₁₆	—	2	2	no	—
270	C₁₆	—	2	2	no	—
271	C₂³	—	172	2	no	—
272	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
273	Q₈	—	86	4	no	—
274	Q₈	—	172	4	no	—
275	Q₈	—	172	2	no	—
276	D₄	—	172	2	no	—
277	D₄	—	172	2	no	—
278	D₄	—	688	2	no	—
279	D₄	—	688	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	172	2	no	—
283	C₂×C₄	—	172	2	no	—
284	C₂×C₄	—	172	4	no	—
285	C₂×C₄	—	172	2	no	—
286	C₂×C₄	—	688	2	no	—
287	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
288	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
289	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	#41×D₄₃	1	2	yes	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	344	2	no	—
298	C₂²	—	344	2	no	—
299	C₂²	—	344	2	no	—
300	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	344	2	no	—
305	C₄	—	344	2	no	—
306	C₄	—	172	4	no	—
307	C₄	#14583411	1	1	yes	characteristic
308	C₄	#14583699	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	—	688	2	no	—
311	C₂	#19144596	1	1	yes	characteristic
312	C₂	#19147743	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2164	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19145245	C₂
hypercenter	#2164	D₄₃
hypocenter	C₄₃	#33686
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2164

Complements

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

N	H ≅ G/N	type
2. #19143464	C₂	semidirect
3. #19136787	C₂	semidirect
4. #19136760	C₂	semidirect
5. #19136636	C₂	semidirect
6. #19136514	C₂	semidirect
7. #19136461	C₂	semidirect
8. C₄×#13063270	C₂	semidirect
9. C₄₃×#2164	C₂	semidirect
161. #2164	D₄₃	semidirect
195. C₄₃	#33686	semidirect

Maximal subgroups

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

subgroup	index
#19145245	2
#19143464	2
#19136787	2
#19136760	2
#19136636	2
#19136514	2
#19136461	2
C₄×#13063270	2
C₄₃×#2164	2
#33686	43