Subgroups — #25251505

Order 11008.

534 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	#19144246	C₂	1	1	yes	maximal, characteristic
2	#19143203	C₂	1	1	yes	maximal, characteristic
3	#19136813	C₂	1	8	yes	maximal, complemented
4	#19136369	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582715	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136215	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583665	—	2	1	no	—
10	C₂×#13063331	C₂²	1	1	yes	characteristic
11	#14583473	—	2	1	no	—
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	#14583154	C₂²	1	4	yes	—
14	#14583137	—	2	2	no	—
15	#14582947	—	2	1	no	—
16	#14582946	C₂²	1	4	yes	—
17	C₂×#13063261	C₂²	1	1	yes	characteristic
18	#108×D₄₃	—	2	2	no	—
19	#14582937	C₂²	1	1	yes	characteristic
20	#14582908	—	2	1	no	—
21	#14582723	C₂²	1	4	yes	—
22	#14582722	C₂²	1	4	yes	—
23	#14582715	C₂²	1	4	yes	—
24	#14582680	C₂²	1	4	yes	—
25	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#362	C₂²	1	4	yes	—
29	C₂×#10621898	D₄	1	1	yes	characteristic
30	#13063333	—	4	2	no	—
31	#13063331	—	2	1	no	—
32	#13063331	—	2	1	no	—
33	#13063331	—	2	1	no	—
34	#13063331	—	2	1	no	—
35	#13063331	—	4	2	no	—
36	C₂×#10621877	—	2	1	no	—
37	C₂×#10621877	D₄	1	1	yes	characteristic
38	C₂×#10621872	C₂³	1	2	yes	—
39	M₁₆×D₄₃	—	2	4	no	—
40	C₂×C₈×D₄₃	D₄	1	2	yes	—
41	#13063323	—	4	1	no	—
42	#13063314	—	4	1	no	—
43	#13063308	—	4	2	no	—
44	#13063301	—	2	4	no	—
45	#13063284	—	2	1	no	—
46	#13063265	—	2	1	no	—
47	#13063265	C₂³	1	2	yes	—
48	C₂×C₄₃⋊C₁₆	—	2	1	no	—
49	#13063261	—	2	1	no	—
50	#13063261	C₂³	1	2	yes	—
51	#13063260	—	2	4	no	—
52	#13063259	—	2	1	no	—
53	#13063257	C₂³	1	2	yes	—
54	#13063249	—	2	4	no	—
55	C₂²×Q₈×C₄₃	—	2	1	no	—
56	C₂×Q₁₆×C₄₃	—	2	1	no	—
57	C₂×Q₁₆×C₄₃	—	2	1	no	—
58	C₂×Q₁₆×C₄₃	—	2	1	no	—
59	C₂×Q₁₆×C₄₃	—	2	1	no	—
60	C₂×Q₁₆×C₄₃	—	2	1	no	—
61	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
62	C₂×M₁₆×C₄₃	—	2	1	no	—
63	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	M₃₂×C₄₃	—	2	1	no	—
65	M₃₂×C₄₃	C₂³	1	2	yes	—
66	C₂×C₁₆×C₄₃	—	2	1	no	—
67	#108×C₄₃	—	2	1	no	—
68	#108×C₄₃	C₂³	1	2	yes	—
69	C₂×Q₈×C₄₃	—	2	2	no	—
70	C₂×Q₈×C₄₃	—	4	1	no	—
71	C₂×Q₈×C₄₃	—	4	1	no	—
72	C₂×Q₈×C₄₃	—	4	1	no	—
73	C₂×Q₈×C₄₃	—	4	1	no	—
74	C₂×Q₈×C₄₃	—	4	1	no	—
75	C₂²×C₄×C₄₃	—	4	1	no	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₄₃	SD₁₆	1	2	yes	—
78	#10621898	—	2	1	no	—
79	#10621898	—	2	1	no	—
80	#10621898	—	2	1	no	—
81	#10621898	—	2	1	no	—
82	C₂×D₁₇₂	SD₁₆	1	1	yes	characteristic
83	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
84	C₂×Dic₈₆	SD₁₆	1	1	yes	characteristic
85	Q₁₆×C₄₃	—	2	2	no	—
86	Q₁₆×C₄₃	—	2	2	no	—
87	Q₁₆×C₄₃	—	4	1	no	—
88	Q₁₆×C₄₃	—	4	1	no	—
89	M₁₆×C₄₃	—	2	2	no	—
90	M₁₆×C₄₃	—	4	1	no	—
91	C₂×C₈×C₄₃	—	4	1	no	—
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₄₃	—	4	1	no	—
99	(C₂×D₄₃)⋊C₄	—	4	4	no	—
100	C₄⋊Dic₄₃	—	2	2	no	—
101	Dic₄₃⋊C₄	—	4	4	no	—
102	C₄×Dic₄₃	—	4	1	no	—
103	#10621877	—	2	1	no	—
104	#10621877	—	2	1	no	—
105	#10621877	—	2	2	no	—
106	#10621877	—	4	1	no	—
107	C₂×C₄₃⋊C₈	—	4	1	no	—
108	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
109	#10621872	—	2	2	no	—
110	#10621872	—	2	2	no	—
111	#10621872	—	4	4	no	—
112	C₈×D₄₃	—	2	2	no	—
113	C₈×D₄₃	—	2	2	no	—
114	C₈×D₄₃	—	4	4	no	—
115	C₁₆×C₄₃	—	2	2	no	—
116	C₄₃⋊C₁₆	—	2	2	no	—
117	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂²×D₄₃	C₂×SD₁₆	1	2	yes	—
119	Q₈×C₄₃	—	4	2	no	—
120	Q₈×C₄₃	—	4	2	no	—
121	Q₈×C₄₃	—	8	1	no	—
122	Q₈×C₄₃	—	8	1	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	8	1	no	—
125	C₂×C₄×C₄₃	—	8	1	no	—
126	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₄₃	—	2	2	no	—
133	C₂²⋊D₄₃	—	2	2	no	—
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	C₂×SD₁₆	1	2	yes	—
136	D₁₇₂	—	2	1	no	—
137	D₁₇₂	—	2	1	no	—
138	C₄×D₄₃	—	2	2	no	—
139	C₄×D₄₃	—	2	2	no	—
140	Dic₈₆	—	2	1	no	—
141	Dic₈₆	—	2	1	no	—
142	C₈×C₄₃	—	4	2	no	—
143	C₈×C₄₃	#123	1	1	yes	characteristic
144	C₈×C₄₃	#123	1	1	yes	characteristic
145	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
147	C₄₃⋊C₈	—	2	2	no	—
148	C₄₃⋊C₈	—	4	2	no	—
149	#33658	—	43	1	no	maximal, Sylow
150	C₂²×C₄₃	—	4	1	no	—
151	C₂²×C₄₃	#481	1	1	yes	characteristic
152	C₂²×C₄₃	#481	1	1	yes	characteristic
153	C₂²×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
154	C₂×D₄₃	—	2	2	no	—
155	C₂×D₄₃	—	4	2	no	—
156	C₄×C₄₃	—	8	2	no	—
157	C₄×C₄₃	#481	1	1	yes	characteristic
158	C₄×C₄₃	#481	1	1	yes	characteristic
159	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
160	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	Dic₄₃	—	2	2	no	—
162	Dic₄₃	—	8	2	no	—
163	#3059	—	43	1	no	—
164	#2916	—	43	1	no	—
165	#2239	—	43	8	no	—
166	#2155	—	43	1	no	—
167	#2155	—	43	1	no	—
168	C₂×#362	—	43	1	no	—
169	C₂×#362	D₄₃	1	1	yes	characteristic, complemented
170	#2115	—	43	1	no	—
171	C₂×C₄₃	—	2	1	no	—
172	C₂×C₄₃	—	2	1	no	—
173	C₂×C₄₃	—	2	1	no	—
174	C₂×C₄₃	C₂×#481	1	1	yes	characteristic
175	D₄₃	—	4	2	no	—
176	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
177	#568	—	86	1	no	—
178	C₂×#131	—	43	1	no	—
179	#536	—	86	1	no	—
180	C₂×M₃₂	—	43	1	no	—
181	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
182	#470	—	43	4	no	—
183	#467	—	86	2	no	—
184	#430	—	43	4	no	—
185	#430	—	86	1	no	—
186	C₂×#108	—	43	1	no	—
187	C₂×#108	—	86	2	no	—
188	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
189	#427	—	43	1	no	—
190	#419	—	86	1	no	—
191	#365	—	43	4	no	—
192	#365	—	43	4	no	—
193	#362	—	43	4	no	—
194	#362	C₂×D₄₃	1	4	yes	—
195	#349	—	43	4	no	—
196	C₄₃	#33658	1	1	yes	characteristic, complemented, Sylow
197	C₂×#41	—	43	1	no	—
198	C₂²×Q₈	—	2	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
205	#131	—	86	1	no	—
206	#131	—	86	1	no	—
207	#131	—	86	1	no	—
208	#131	—	86	1	no	—
209	#131	—	172	2	no	—
210	#131	—	172	2	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	43	1	no	—
213	C₂×M₁₆	—	43	2	no	—
214	C₂×M₁₆	—	86	4	no	—
215	C₂×M₁₆	—	86	1	no	—
216	C₂²×C₈	—	43	2	no	—
217	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
218	C₄⋊Q₈	—	172	1	no	—
219	#124	—	172	1	no	—
220	#123	—	172	2	no	—
221	C₂²⋊Q₈	—	86	4	no	—
222	#117	—	86	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	43	2	no	—
225	M₃₂	—	86	1	no	—
226	M₃₂	C₂²×D₄₃	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	86	1	no	—
229	#108	—	2	1	no	—
230	#108	—	43	2	no	—
231	#108	—	86	4	no	—
232	#108	—	86	1	no	—
233	#108	C₂²×D₄₃	1	2	yes	—
234	#107	—	86	1	no	—
235	#106	—	43	2	no	—
236	#103	—	86	4	no	—
237	#41	—	86	1	no	—
238	#41	—	86	1	no	—
239	#41	—	86	1	no	—
240	#41	—	86	1	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	43	1	no	—
248	C₂×D₄	—	43	2	no	—
249	C₂×D₄	—	43	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	43	2	no	—
252	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	4	1	no	—
256	Q₁₆	—	4	1	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	86	1	no	—
260	M₁₆	—	86	1	no	—
261	M₁₆	—	86	2	no	—
262	M₁₆	—	86	2	no	—
263	M₁₆	—	86	2	no	—
264	M₁₆	—	172	4	no	—
265	M₁₆	—	172	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	43	2	no	—
268	C₂×C₈	—	86	2	no	—
269	C₂×C₈	—	86	2	no	—
270	C₂×C₈	—	172	4	no	—
271	C₂×C₈	—	172	1	no	—
272	C₂×C₈	#10621898	1	1	yes	characteristic
273	C₂×C₈	#10621898	1	1	yes	characteristic
274	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
275	C₂×C₈	#10621900	1	1	yes	characteristic
276	C₂×C₈	#10621900	1	1	yes	characteristic
277	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
278	C₄⋊C₄	—	86	2	no	—
279	C₄⋊C₄	—	172	4	no	—
280	C₂²⋊C₄	—	172	4	no	—
281	C₂²⋊C₄	—	172	1	no	—
282	C₄²	—	172	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	86	2	no	—
285	C₂³	—	43	2	no	—
286	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	8	1	no	—
290	Q₈	—	8	1	no	—
291	Q₈	—	86	1	no	—
292	Q₈	—	86	1	no	—
293	D₄	—	86	2	no	—
294	D₄	—	86	2	no	—
295	D₄	—	86	1	no	—
296	D₄	—	86	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	43	2	no	—
301	C₂×C₄	—	86	2	no	—
302	C₂×C₄	—	86	2	no	—
303	C₂×C₄	—	172	2	no	—
304	C₂×C₄	#13063307	1	1	yes	characteristic
305	C₂×C₄	#13063307	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
307	C₂×C₄	#13063349	1	1	yes	characteristic
308	C₂×C₄	#13063349	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	86	2	no	—
312	C₈	—	172	2	no	—
313	C₈	#13063307	1	1	yes	characteristic
314	C₈	#13063307	1	1	yes	characteristic
315	C₈	C₂×#10621900	1	1	yes	characteristic
316	C₈	C₂×#10621898	1	1	yes	characteristic
317	C₂²	—	4	1	no	—
318	C₂²	—	86	2	no	—
319	C₂²	—	172	2	no	—
320	C₂²	#14583216	1	1	yes	characteristic
321	C₂²	#14583216	1	1	yes	characteristic
322	C₂²	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
323	C₄	—	8	2	no	—
324	C₄	—	86	2	no	—
325	C₄	—	344	2	no	—
326	C₄	#14583216	1	1	yes	characteristic
327	C₄	#14583216	1	1	yes	characteristic
328	C₄	C₂×#13063307	1	1	yes	characteristic
329	C₄	C₂×#13063349	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	172	2	no	—
334	C₂	C₂×#14583216	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#14583216
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#362	C₂
socle	C₂×C₄₃	C₂×#481
prime radical	C₂×#10621898	D₄
hypercenter	C₂×#362	D₄₃
hypocenter	C₄₃	#33658
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

Complements

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

N	H ≅ G/N	type
3. #19136813	C₂	semidirect
4. #19136369	C₂	semidirect
5. #19136368	C₂	semidirect
6. C₂×#14582715	C₂	semidirect
8. C₂×C₄₃×#362	C₂	semidirect
169. C₂×#362	D₄₃	semidirect
196. C₄₃	#33658	semidirect

Maximal subgroups

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

subgroup	index
#19144246	2
#19143203	2
#19136813	2
#19136369	2
#19136368	2
C₂×#14582715	2
#19136215	2
C₂×C₄₃×#362	2
#33658	43