Subgroups — #25011820

Order 11008.

450 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143366	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582707	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582705	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132690	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	1	yes	characteristic
9	#14583367	—	2	1	no	—
10	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14582993	C₄	1	2	yes	—
12	#14582986	—	2	2	no	—
13	#14582981	C₄	1	2	yes	—
14	C₂×#13063261	C₂²	1	1	yes	characteristic
15	#14582893	—	2	1	no	—
16	M₁₆×Dic₄₃	—	2	1	no	—
17	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
18	#14582707	C₄	1	4	yes	complemented
19	#14582705	C₄	1	4	yes	complemented
20	C₁₆×Dic₄₃	—	2	2	no	—
21	#14582663	—	2	1	no	—
22	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#427	C₂²	1	1	yes	characteristic
25	C₂×#10621898	—	2	1	no	—
26	#13063347	—	2	1	no	—
27	#13063347	—	2	1	no	—
28	#13063347	C₂×C₄	1	2	yes	—
29	#13063347	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
31	C₂×#10621873	—	2	1	no	—
32	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
33	C₂×#10621877	—	2	1	no	—
34	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
35	#13063290	—	2	2	no	—
36	#13063287	—	2	2	no	—
37	(C₂²⋊D₄₃)⋊C₄	—	4	2	no	—
38	#13063282	—	4	1	no	—
39	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
40	C₄²×D₄₃	—	4	1	no	—
41	C₂×C₄₃⋊C₁₆	—	2	1	no	—
42	C₂×C₄₃⋊C₁₆	—	2	1	no	—
43	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
44	#13063261	—	2	1	no	—
45	#13063261	C₂×C₄	1	2	yes	complemented
46	#13063248	—	2	2	no	—
47	#13063245	—	2	2	no	—
48	#13063235	—	4	1	no	—
49	C₈×Dic₄₃	—	4	1	no	—
50	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
51	C₈×Dic₄₃	D₄	1	2	yes	—
52	C₂×M₁₆×C₄₃	—	2	1	no	—
53	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
54	#117×C₄₃	—	2	1	no	—
55	C₂×C₁₆×C₄₃	—	2	1	no	—
56	C₂×C₁₆×C₄₃	—	2	1	no	—
57	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	complemented
58	#108×C₄₃	—	2	1	no	—
59	#108×C₄₃	C₂×C₄	1	2	yes	complemented
60	#107×C₄₃	C₂×C₄	1	2	yes	—
61	#106×C₄₃	—	2	1	no	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₄₃	—	4	1	no	—
64	C₂²×Dic₄₃	SD₁₆	1	2	yes	—
65	#10621898	—	2	2	no	—
66	#10621898	—	2	2	no	—
67	#10621898	—	4	1	no	—
68	#10621898	—	4	1	no	—
69	C₂×D₁₇₂	—	2	1	no	—
70	C₂×C₄×D₄₃	—	4	1	no	—
71	C₂×C₄×D₄₃	—	4	2	no	—
72	C₂×Dic₈₆	—	2	1	no	—
73	M₁₆×C₄₃	—	2	2	no	—
74	M₁₆×C₄₃	—	4	1	no	—
75	C₂×C₈×C₄₃	—	4	1	no	—
76	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₄₃	—	2	2	no	—
83	C₂²⋊C₄×C₄₃	—	4	1	no	—
84	C₄²×C₄₃	—	4	1	no	—
85	(C₂×D₄₃)⋊C₄	—	4	2	no	—
86	Dic₄₃⋊C₄	—	4	2	no	—
87	C₄×Dic₄₃	—	4	2	no	—
88	C₄×Dic₄₃	C₂²⋊C₄	1	2	yes	—
89	C₄×Dic₄₃	D₈	1	1	yes	characteristic
90	C₄×Dic₄₃	D₈	1	1	yes	characteristic
91	#10621877	—	2	2	no	—
92	#10621877	—	4	1	no	—
93	C₂×C₄₃⋊C₈	—	2	1	no	—
94	C₂×C₄₃⋊C₈	—	2	1	no	—
95	C₂×C₄₃⋊C₈	—	4	1	no	—
96	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
97	Dic₁₇₂	—	2	1	no	—
98	Dic₁₇₂	C₄²	1	2	yes	—
99	D₃₄₄	—	2	1	no	—
100	D₃₄₄	C₄²	1	2	yes	—
101	#10621873	—	2	2	no	—
102	#10621873	—	2	2	no	—
103	C₁₆×C₄₃	—	2	2	no	—
104	C₄₃⋊C₁₆	—	2	2	no	—
105	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂²×D₄₃	—	4	1	no	—
107	C₂×C₄×C₄₃	—	4	2	no	—
108	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
114	C₂²⋊D₄₃	—	4	2	no	—
115	C₂²⋊D₄₃	—	4	2	no	—
116	C₂×Dic₄₃	—	2	4	no	—
117	C₂×Dic₄₃	—	4	1	no	—
118	C₂×Dic₄₃	#102	1	2	yes	—
119	C₂×Dic₄₃	#102	1	2	yes	—
120	D₁₇₂	—	2	2	no	—
121	D₁₇₂	—	4	1	no	—
122	C₄×D₄₃	—	4	2	no	—
123	C₄×D₄₃	—	4	2	no	—
124	C₄×D₄₃	—	8	4	no	—
125	Dic₈₆	—	2	2	no	—
126	Dic₈₆	—	4	1	no	—
127	C₈×C₄₃	—	4	2	no	—
128	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₄₃	#117	1	1	yes	characteristic
130	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
132	C₄₃⋊C₈	—	2	2	no	—
133	C₄₃⋊C₈	—	4	2	no	—
134	C₄×#359	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
137	C₂²×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
138	C₂²×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
139	C₂×D₄₃	—	4	2	no	—
140	C₂×D₄₃	—	8	2	no	—
141	C₄×C₄₃	—	8	2	no	—
142	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₄₃	#416	1	1	yes	characteristic
144	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
145	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
146	Dic₄₃	—	2	2	no	—
147	Dic₄₃	—	4	2	no	—
148	Dic₄₃	#359	1	4	yes	complemented
149	C₄×#135	—	43	1	no	—
150	C₂×#359	—	43	1	no	—
151	C₂×#359	—	43	1	no	—
152	C₂×C₄×C₁₆	—	43	1	no	—
153	C₄×#108	—	43	1	no	—
154	#1382	—	43	1	no	—
155	#1382	D₄₃	1	1	yes	characteristic, complemented
156	C₂×C₄₃	—	2	1	no	—
157	C₂×C₄₃	—	2	1	no	—
158	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
159	C₂×C₄₃	C₂×#359	1	2	yes	—
160	D₄₃	—	8	2	no	—
161	C₂×#135	—	43	1	no	—
162	C₄×#41	—	86	1	no	—
163	C₂²×C₁₆	—	43	1	no	—
164	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
165	C₄×Q₁₆	—	43	2	no	—
166	C₄×SD₁₆	—	86	2	no	—
167	C₄×D₈	—	43	2	no	—
168	C₂×#108	—	43	1	no	—
169	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
170	#427	C₂×D₄₃	1	1	yes	characteristic
171	#416	—	86	1	no	—
172	C₄×M₁₆	—	86	1	no	—
173	C₂×C₄×C₈	—	43	1	no	—
174	#359	—	43	4	no	—
175	#359	—	43	4	no	—
176	C₄×C₁₆	—	86	2	no	—
177	#341	—	86	1	no	—
178	C₄₃	C₄×#359	1	1	yes	characteristic, complemented, Sylow
179	C₂×#41	—	86	1	no	—
180	#135	—	43	2	no	—
181	#135	—	43	2	no	—
182	#135	—	86	1	no	—
183	#135	—	86	1	no	—
184	C₂×Q₁₆	—	43	1	no	—
185	C₂×SD₁₆	—	86	1	no	—
186	C₂×D₈	—	43	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	86	1	no	—
189	C₂²×C₈	—	43	1	no	—
190	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
191	C₄×Q₈	—	86	2	no	—
192	C₄×D₄	—	86	2	no	—
193	C₄×D₄	—	172	2	no	—
194	#117	—	2	1	no	—
195	#117	—	172	1	no	—
196	C₂×C₄²	—	43	1	no	—
197	C₂×C₄²	—	172	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	43	2	no	—
201	C₂×C₁₆	—	86	1	no	—
202	C₂×C₁₆	—	86	1	no	—
203	C₂×C₁₆	C₄×D₄₃	1	2	yes	complemented
204	#108	—	2	1	no	—
205	#108	—	43	2	no	—
206	#108	—	86	1	no	—
207	#108	C₄×D₄₃	1	2	yes	complemented
208	#107	C₄×D₄₃	1	2	yes	—
209	#106	—	2	1	no	—
210	#103	—	86	2	no	—
211	#102	—	86	2	no	—
212	#97	—	172	1	no	—
213	C₄×C₈	—	43	2	no	—
214	C₄×C₈	—	43	2	no	—
215	C₄×C₈	—	172	1	no	—
216	#41	—	86	2	no	—
217	#41	—	86	2	no	—
218	#41	—	172	1	no	—
219	#41	—	172	1	no	—
220	C₂×Q₈	—	86	1	no	—
221	C₂×D₄	—	86	1	no	—
222	C₂×D₄	—	172	1	no	—
223	C₂²×C₄	—	43	2	no	—
224	C₂²×C₄	—	172	2	no	—
225	C₂²×C₄	—	172	1	no	—
226	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
227	Q₁₆	—	43	2	no	—
228	Q₁₆	—	86	1	no	—
229	SD₁₆	—	86	2	no	—
230	SD₁₆	—	86	2	no	—
231	D₈	—	43	2	no	—
232	D₈	—	86	1	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	86	2	no	—
236	M₁₆	—	172	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	43	2	no	—
239	C₂×C₈	—	86	1	no	—
240	C₂×C₈	—	86	1	no	—
241	C₂×C₈	—	172	1	no	—
242	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
245	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
246	C₂×C₈	#10621900	1	1	yes	characteristic
247	C₂×C₈	#10621902	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	172	2	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₂²⋊C₄	—	172	2	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	43	2	no	—
254	C₄²	—	43	1	no	—
255	C₄²	—	43	1	no	—
256	C₄²	—	172	2	no	—
257	C₁₆	—	2	2	no	—
258	C₁₆	—	86	2	no	—
259	C₂³	—	172	1	no	—
260	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
261	Q₈	—	86	2	no	—
262	Q₈	—	172	1	no	—
263	D₄	—	86	2	no	—
264	D₄	—	172	2	no	—
265	D₄	—	172	2	no	—
266	D₄	—	172	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	43	2	no	—
269	C₂×C₄	—	43	2	no	—
270	C₂×C₄	—	86	4	no	—
271	C₂×C₄	—	344	4	no	—
272	C₂×C₄	—	172	1	no	—
273	C₂×C₄	—	172	2	no	—
274	C₂×C₄	—	172	2	no	—
275	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
276	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	#13063287	1	1	yes	characteristic
278	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
279	C₂×C₄	#13063346	1	1	yes	characteristic
280	C₂×C₄	#13063348	1	1	yes	characteristic
281	C₈	—	4	2	no	—
282	C₈	—	86	2	no	—
283	C₈	—	172	2	no	—
284	C₈	C₄²×D₄₃	1	1	yes	characteristic
285	C₈	#13063283	1	1	yes	characteristic
286	C₈	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
287	C₈	#13063287	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	344	2	no	—
290	C₂²	—	172	2	no	—
291	C₂²	#102×D₄₃	1	1	yes	characteristic
292	C₂²	#14582984	1	1	yes	characteristic
293	C₂²	#14582986	1	1	yes	characteristic
294	C₄	—	8	2	no	—
295	C₄	—	43	4	no	—
296	C₄	—	86	2	no	—
297	C₄	—	172	2	no	—
298	C₄	#14582746	1	1	yes	characteristic
299	C₄	#14582894	1	1	yes	characteristic
300	C₄	#14582979	1	1	yes	characteristic
301	C₄	#14582981	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	344	2	no	—
305	C₂	#19134409	1	1	yes	characteristic
306	C₂	#359×D₄₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1382	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂×#13063347	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1382	D₄₃
hypocenter	C₄₃	C₄×#359
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
4. C₂×C₁₆×Dic₄₃	C₂	semidirect
5. #108×Dic₄₃	C₂	semidirect
6. #19132690	C₂	semidirect
7. C₄₃×#1382	C₂	semidirect
18. #14582707	C₄	semidirect
19. #14582705	C₄	semidirect
43. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
45. #13063261	C₂×C₄	semidirect
57. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
59. #108×C₄₃	C₂×C₄	semidirect
148. Dic₄₃	#359	semidirect
155. #1382	D₄₃	semidirect
178. C₄₃	C₄×#359	semidirect
203. C₂×C₁₆	C₄×D₄₃	semidirect
207. #108	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19143366	2
C₂×#14582707	2
C₂×#14582705	2
C₂×C₁₆×Dic₄₃	2
#108×Dic₄₃	2
#19132690	2
C₄₃×#1382	2
C₄×#359	43