Subgroups — #25251098

Order 11008.

650 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	C₂²×#13063271	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583316	C₂	1	1	yes	maximal, characteristic
3	#19144250	C₂	1	1	yes	maximal, characteristic
4	#19136701	C₂	1	4	yes	maximal, complemented
5	#19136684	C₂	1	4	yes	maximal, complemented
6	#19136377	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136315	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136314	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#3059	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
11	#14583470	—	2	1	no	—
12	C₂×#13063271	—	2	4	no	—
13	C₂×#13063271	C₂²	1	2	yes	—
14	C₂×#13063271	C₂²	1	2	yes	—
15	#14583316	—	2	4	no	—
16	C₂×#13063265	C₂²	1	1	yes	characteristic
17	C₂×#13063265	C₂²	1	2	yes	—
18	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
19	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	#14583157	—	2	2	no	—
21	#14583140	C₂²	1	4	yes	—
22	#14582937	C₂²	1	1	yes	characteristic
23	#14582909	—	2	1	no	—
24	#14582726	C₂²	1	2	yes	—
25	#14582724	C₂²	1	2	yes	—
26	#14582703	C₂²	1	4	yes	—
27	#14582702	C₂²	1	4	yes	—
28	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
30	C₄₃×#536	—	2	1	no	—
31	C₄₃×#470	—	2	2	no	—
32	C₄₃×#467	C₂²	1	4	yes	—
33	C₄₃×#427	C₂²	1	1	yes	characteristic
34	C₄₃×#419	—	2	1	no	—
35	C₂²×Dic₈₆	—	2	1	no	—
36	C₂×Dic₁₇₂	—	2	2	no	—
37	C₂×Dic₁₇₂	—	2	2	no	—
38	C₂×Dic₁₇₂	—	2	1	no	—
39	C₂×Dic₁₇₂	C₂³	1	2	yes	—
40	#13063323	—	4	1	no	—
41	#13063312	—	4	1	no	—
42	#13063306	—	4	2	no	—
43	#13063304	—	2	4	no	—
44	#13063284	—	2	1	no	—
45	#13063271	—	2	4	no	—
46	#13063271	—	2	4	no	—
47	#13063265	—	2	2	no	—
48	#13063265	—	2	4	no	—
49	C₂×C₄₃⋊C₁₆	—	2	4	no	—
50	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	—
51	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
54	#13063259	C₂³	1	2	yes	—
55	#13063257	—	2	1	no	—
56	#13063250	—	2	4	no	—
57	C₂×#41×C₄₃	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₄₃	—	2	1	no	—
59	C₂×Q₁₆×C₄₃	—	2	2	no	—
60	C₂×Q₁₆×C₄₃	—	2	1	no	—
61	C₂×Q₁₆×C₄₃	—	2	2	no	—
62	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
63	#131×C₄₃	—	2	2	no	—
64	#131×C₄₃	—	2	2	no	—
65	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
66	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₄₃	—	4	1	no	—
70	#124×C₄₃	—	4	1	no	—
71	#123×C₄₃	—	4	2	no	—
72	C₂²⋊Q₈×C₄₃	—	2	4	no	—
73	#117×C₄₃	—	2	1	no	—
74	#107×C₄₃	C₂³	1	2	yes	—
75	#106×C₄₃	—	2	1	no	—
76	#103×C₄₃	—	2	4	no	—
77	#41×C₄₃	—	2	2	no	—
78	#41×C₄₃	—	2	2	no	—
79	C₂×Q₈×C₄₃	—	2	2	no	—
80	C₂×Q₈×C₄₃	—	4	1	no	—
81	C₂×Q₈×C₄₃	—	4	2	no	—
82	C₂×Q₈×C₄₃	—	4	2	no	—
83	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
84	C₂×D₄×C₄₃	D₈	1	2	yes	—
85	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	—	4	1	no	—
87	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₄₃	—	4	1	no	—
90	C₂×Dic₈₆	—	2	2	no	—
91	C₂×Dic₈₆	—	4	2	no	—
92	C₂×Dic₈₆	—	4	2	no	—
93	C₂×Dic₈₆	—	4	1	no	—
94	Q₁₆×C₄₃	—	2	2	no	—
95	Q₁₆×C₄₃	—	2	2	no	—
96	Q₁₆×C₄₃	—	4	2	no	—
97	M₁₆×C₄₃	—	2	2	no	—
98	M₁₆×C₄₃	—	2	4	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₃	#41	1	2	yes	—
105	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	—	4	4	no	—
108	C₂²⋊C₄×C₄₃	—	4	1	no	—
109	C₂²⋊C₄×C₄₃	—	4	4	no	—
110	C₄²×C₄₃	—	4	1	no	—
111	C₂²⋊Dic₄₃	—	4	1	no	—
112	C₂²⋊Dic₄₃	—	4	4	no	—
113	C₄⋊Dic₄₃	—	2	2	no	—
114	Dic₄₃⋊C₄	—	4	4	no	—
115	C₄×Dic₄₃	—	4	1	no	—
116	Dic₁₇₂	—	2	2	no	—
117	Dic₁₇₂	—	2	2	no	—
118	Dic₁₇₂	—	4	2	no	—
119	C₄₃⋊C₁₆	—	2	2	no	—
120	C₄₃⋊C₁₆	C₂×D₄	1	2	yes	—
121	C₄₃⋊C₁₆	C₂×D₄	1	2	yes	—
122	C₂³×C₄₃	C₂×D₈	1	2	yes	—
123	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
124	Q₈×C₄₃	—	2	2	no	—
125	Q₈×C₄₃	—	4	2	no	—
126	Q₈×C₄₃	—	4	2	no	—
127	Q₈×C₄₃	—	8	2	no	—
128	D₄×C₄₃	—	2	4	no	—
129	D₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	—	8	2	no	—
135	C₂×C₄×C₄₃	#121	1	2	yes	—
136	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#135	1	2	yes	—
139	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	C₂×Dic₄₃	—	8	2	no	—
143	Dic₈₆	—	4	2	no	—
144	Dic₈₆	—	4	2	no	—
145	Dic₈₆	—	8	2	no	—
146	C₈×C₄₃	—	2	2	no	—
147	C₈×C₄₃	#121	1	2	yes	—
148	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
150	#33608	—	43	1	no	maximal, Sylow
151	C₂²×C₄₃	—	2	2	no	—
152	C₂²×C₄₃	—	2	4	no	—
153	C₂²×C₄₃	—	2	2	no	—
154	C₂²×C₄₃	#466	1	2	yes	—
155	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
156	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	8	2	no	—
158	C₄×C₄₃	—	8	2	no	—
159	C₄×C₄₃	#466	1	2	yes	—
160	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
161	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
162	Dic₄₃	—	8	2	no	—
163	Dic₄₃	—	8	2	no	—
164	C₂²×Q₃₂	—	43	1	no	—
165	C₂×#504	—	43	1	no	—
166	#3059	—	43	1	no	—
167	#3059	D₄₃	1	1	yes	characteristic
168	#2219	—	43	4	no	—
169	#2216	—	43	4	no	—
170	#2158	—	43	1	no	—
171	#2141	—	43	1	no	—
172	#2141	—	43	1	no	—
173	C₂×C₄₃	—	2	2	no	—
174	C₂×C₄₃	—	4	2	no	—
175	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
176	C₂×C₄₃	#3417	1	2	yes	—
177	C₂²×Q₁₆	—	43	1	no	—
178	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
179	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
180	#536	—	2	1	no	—
181	#536	—	86	1	no	—
182	C₂×Q₃₂	—	43	2	no	—
183	C₂×Q₃₂	—	43	2	no	—
184	C₂×Q₃₂	—	86	4	no	—
185	#504	—	86	4	no	—
186	C₂×M₃₂	—	43	1	no	—
187	C₂×M₃₂	—	43	2	no	—
188	C₂²×C₁₆	—	43	1	no	—
189	C₂²×C₁₆	—	43	2	no	—
190	#470	—	2	2	no	—
191	#470	—	86	2	no	—
192	#467	—	43	4	no	—
193	#467	C₂×D₄₃	1	4	yes	—
194	#427	—	43	1	no	—
195	#427	C₂×D₄₃	1	1	yes	characteristic
196	#419	—	2	1	no	—
197	#419	—	86	1	no	—
198	#367	—	43	2	no	—
199	#366	—	43	2	no	—
200	#358	—	43	4	no	—
201	#358	—	43	4	no	—
202	C₄₃	#33608	1	1	yes	characteristic, complemented, Sylow
203	C₂×#41	C₂²⋊D₄₃	1	1	yes	characteristic
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×Q₈	—	86	1	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	—	2	2	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	43	2	no	—
210	C₂×Q₁₆	—	86	2	no	—
211	C₂×Q₁₆	—	86	2	no	—
212	C₂×Q₁₆	—	86	1	no	—
213	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
214	#131	—	2	2	no	—
215	#131	—	2	2	no	—
216	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
217	C₂×M₁₆	C₂²⋊D₄₃	1	2	yes	—
218	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
219	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
220	C₄⋊Q₈	—	4	1	no	—
221	C₄⋊Q₈	—	172	1	no	—
222	#124	—	4	1	no	—
223	#124	—	172	1	no	—
224	#123	—	4	2	no	—
225	#123	—	172	2	no	—
226	C₂²⋊Q₈	—	2	4	no	—
227	C₂²⋊Q₈	—	86	4	no	—
228	#117	—	2	1	no	—
229	#117	—	86	1	no	—
230	Q₃₂	—	86	4	no	—
231	Q₃₂	—	86	4	no	—
232	M₃₂	—	86	4	no	—
233	M₃₂	—	86	2	no	—
234	C₂×C₁₆	—	43	4	no	—
235	C₂×C₁₆	—	43	1	no	—
236	C₂×C₁₆	—	43	1	no	—
237	C₂×C₁₆	—	43	2	no	—
238	C₂×C₁₆	—	86	4	no	—
239	#107	—	43	2	no	—
240	#107	C₂²×D₄₃	1	2	yes	—
241	#106	—	2	1	no	—
242	#106	—	86	1	no	—
243	#103	—	2	4	no	—
244	#103	—	86	4	no	—
245	#41	—	2	2	no	—
246	#41	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	86	2	no	—
252	C₂×Q₈	—	172	2	no	—
253	C₂×Q₈	—	172	2	no	—
254	C₂×Q₈	—	172	1	no	—
255	C₂×Q₈	#10621882	1	1	yes	characteristic
256	C₂×D₄	#10621882	1	2	yes	—
257	C₂×D₄	#10621882	1	1	yes	characteristic
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	172	1	no	—
260	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
261	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	4	2	no	—
265	Q₁₆	—	86	2	no	—
266	Q₁₆	—	86	2	no	—
267	Q₁₆	—	172	2	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	2	4	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	D₄×D₄₃	1	2	yes	—
273	C₂×C₈	#10621900	1	2	yes	—
274	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
275	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	86	2	no	—
280	C₄⋊C₄	—	172	4	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	172	1	no	—
284	C₂²⋊C₄	—	172	4	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	172	1	no	—
287	C₁₆	—	43	2	no	—
288	C₁₆	—	43	2	no	—
289	C₁₆	—	86	2	no	—
290	C₂³	C₂×#10621882	1	2	yes	—
291	C₂³	C₂×#10621882	1	1	yes	characteristic
292	Q₈	—	2	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	2	no	—
296	Q₈	—	344	2	no	—
297	Q₈	—	172	2	no	—
298	Q₈	—	172	2	no	—
299	D₄	—	2	4	no	—
300	D₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	344	2	no	—
307	C₂×C₄	—	172	2	no	—
308	C₂×C₄	—	172	2	no	—
309	C₂×C₄	#13063297	1	2	yes	—
310	C₂×C₄	C₂×#10621882	1	2	yes	—
311	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
312	C₂×C₄	#13063345	1	2	yes	—
313	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	#13063297	1	2	yes	—
316	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
317	C₈	C₂×#10621900	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	4	no	—
321	C₂²	#14583133	1	2	yes	—
322	C₂²	C₂²×#10621882	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	344	2	no	—
327	C₄	—	344	2	no	—
328	C₄	#14583133	1	2	yes	—
329	C₄	C₂×#13063297	1	1	yes	characteristic
330	C₄	C₂×#13063345	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	4	2	no	—
333	C₂	C₂×#14583133	1	1	yes	characteristic
334	C₂	#19147746	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3059	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#41×C₄₃	D₄
hypercenter	#3059	D₄₃
hypocenter	C₄₃	#33608
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #3059

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063271	C₂	semidirect
4. #19136701	C₂	semidirect
5. #19136684	C₂	semidirect
6. #19136377	C₂	semidirect
7. #19136315	C₂	semidirect
8. #19136314	C₂	semidirect
202. C₄₃	#33608	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063271	2
C₂×#14583316	2
#19144250	2
#19136701	2
#19136684	2
#19136377	2
#19136315	2
#19136314	2
C₄₃×#3059	2
#33608	43