Subgroups — #25095197

Order 11008.

998 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583015	C₂	1	2	yes	maximal
2	C₄×#13063347	C₂	1	1	yes	maximal, characteristic
3	#19143024	C₂	1	2	yes	maximal, complemented
4	#19134477	C₂	1	2	yes	maximal, complemented
5	#19134438	C₂	1	4	yes	maximal, complemented
6	#19134408	C₂	1	2	yes	maximal, complemented
7	#19134281	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#108×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063347	C₂²	1	1	yes	characteristic
10	C₂×#13063333	C₂²	1	4	yes	—
11	C₄×#10621898	C₂²	1	2	yes	—
12	#14583015	C₄	1	16	yes	complemented
13	C₄×Dic₁₇₂	C₄	1	2	yes	complemented
14	C₄×#10621873	C₄	1	4	yes	complemented
15	C₄×D₃₄₄	C₄	1	2	yes	complemented
16	#14582937	C₂²	1	1	yes	characteristic
17	C₂×#13063251	C₂²	1	4	yes	—
18	#14582896	C₂²	1	2	yes	—
19	#14582857	C₂²	1	4	yes	—
20	#14582853	—	2	4	no	—
21	#14582851	—	2	2	no	—
22	C₂×#13063236	C₂²	1	1	yes	characteristic
23	C₂×#13063235	C₂²	1	2	yes	—
24	#97×D₄₃	—	2	2	no	—
25	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
26	#14582745	C₂²	1	1	yes	characteristic
27	#14582660	C₂²	1	4	yes	—
28	C₂×#108×C₄₃	C₂²	1	2	yes	—
29	C₄×M₁₆×C₄₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#341	C₂²	1	2	yes	—
32	C₂×#10621898	C₂³	1	2	yes	—
33	#13063347	C₂×C₄	1	8	yes	—
34	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
35	C₂×#10621873	C₂×C₄	1	2	yes	—
36	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
37	#13063333	—	2	8	no	—
38	#13063333	C₂×C₄	1	16	yes	complemented
39	C₂×#10621872	—	2	4	no	—
40	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
41	C₂×C₈×D₄₃	—	2	4	no	—
42	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
43	C₄×D₁₇₂	C₂×C₄	1	4	yes	—
44	C₄×C₂²⋊D₄₃	—	2	4	no	—
45	#13063284	C₂³	1	2	yes	—
46	#13063282	—	2	2	no	—
47	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
48	C₄²×D₄₃	—	2	2	no	—
49	#13063259	C₂×C₄	1	2	yes	—
50	#13063257	C₂×C₄	1	2	yes	—
51	#13063251	C₂×C₄	1	16	yes	complemented
52	#13063250	C₂×C₄	1	4	yes	—
53	#13063247	C₂×C₄	1	4	yes	—
54	#13063236	D₄	1	1	yes	characteristic
55	#13063236	D₄	1	1	yes	characteristic
56	#13063236	D₄	1	2	yes	—
57	#13063235	—	2	4	no	—
58	#13063235	—	2	4	no	—
59	C₈×Dic₄₃	—	2	4	no	—
60	C₈×Dic₄₃	C₂×C₄	1	8	yes	complemented
61	C₄₃⋊#95	—	2	2	no	—
62	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
63	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	#108×C₄₃	C₂×C₄	1	8	yes	complemented
67	#97×C₄₃	—	2	2	no	—
68	C₄×C₈×C₄₃	—	2	2	no	—
69	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
71	C₂²×C₄×C₄₃	#41	1	2	yes	—
72	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₂²⋊D₄₃	—	2	2	no	—
74	C₂²×Dic₄₃	—	2	2	no	—
75	#10621898	—	2	4	no	—
76	#10621898	C₂²×C₄	1	8	yes	—
77	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
78	C₂×C₄×D₄₃	—	2	2	no	—
79	C₂×C₄×D₄₃	—	2	4	no	—
80	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
81	M₁₆×C₄₃	—	2	4	no	—
82	M₁₆×C₄₃	C₂²×C₄	1	8	yes	—
83	C₂×C₈×C₄₃	—	2	4	no	—
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
89	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂²⋊Dic₄₃	—	2	2	no	—
93	(C₂×D₄₃)⋊C₄	—	2	4	no	—
94	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
95	Dic₄₃⋊C₄	—	2	4	no	—
96	C₄×Dic₄₃	—	2	4	no	—
97	C₄×Dic₄₃	—	2	2	no	—
98	C₄×Dic₄₃	—	2	2	no	—
99	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
100	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
103	C₂×C₄₃⋊C₈	#41	1	4	yes	—
104	Dic₁₇₂	C₄²	1	4	yes	—
105	D₃₄₄	C₄²	1	4	yes	—
106	#10621873	C₄²	1	8	yes	—
107	#10621872	—	2	16	no	—
108	C₈×D₄₃	—	2	16	no	—
109	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
110	C₂²×D₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	4	2	no	—
112	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
118	C₂²⋊D₄₃	—	2	8	no	—
119	C₂×Dic₄₃	—	2	2	no	—
120	C₂×Dic₄₃	—	2	4	no	—
121	C₂×Dic₄₃	—	2	4	no	—
122	C₂×Dic₄₃	—	4	4	no	—
123	D₁₇₂	—	2	2	no	—
124	D₁₇₂	C₂×C₄²	1	4	yes	—
125	C₄×D₄₃	—	2	8	no	—
126	C₄×D₄₃	—	8	4	no	—
127	Dic₈₆	—	2	2	no	—
128	Dic₈₆	C₂×C₄²	1	4	yes	—
129	C₈×C₄₃	—	2	8	no	—
130	C₈×C₄₃	—	2	2	no	—
131	C₈×C₄₃	C₂×C₄²	1	2	yes	—
132	C₈×C₄₃	C₂×C₄²	1	2	yes	—
133	C₄₃⋊C₈	C₄×D₄	1	4	yes	—
134	C₄₃⋊C₈	C₄×D₄	1	4	yes	—
135	#20514	—	43	1	no	maximal, Sylow
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₄₃	C₄×#41	1	2	yes	—
139	C₂×D₄₃	—	2	4	no	—
140	C₂×D₄₃	—	4	4	no	—
141	C₄×C₄₃	—	2	2	no	—
142	C₄×C₄₃	—	2	2	no	—
143	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
144	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
145	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
146	Dic₄₃	—	2	4	no	—
147	Dic₄₃	—	4	4	no	—
148	Dic₄₃	—	4	4	no	—
149	C₂×#443	—	43	2	no	—
150	C₄×#135	—	43	1	no	—
151	#2875	—	43	2	no	—
152	C₄×#108	D₄₃	1	1	yes	characteristic, complemented
153	#1772	—	43	2	no	—
154	#1765	—	43	4	no	—
155	#1759	—	43	2	no	—
156	#1726	—	43	1	no	—
157	C₂×C₄₃	—	2	2	no	—
158	C₂×C₄₃	C₄²×D₄	1	1	yes	characteristic
159	C₂×C₄₃	#2957	1	2	yes	—
160	D₄₃	—	4	4	no	—
161	C₂×#135	—	43	1	no	—
162	C₂×#131	—	43	4	no	—
163	C₄×#41	—	43	2	no	—
164	#443	—	43	16	no	—
165	C₄×Q₁₆	—	43	2	no	—
166	C₄×SD₁₆	—	43	4	no	—
167	C₄×D₈	—	43	2	no	—
168	C₂×#108	C₂×D₄₃	1	2	yes	—
169	#427	—	43	1	no	—
170	C₂×#104	—	43	4	no	—
171	#416	—	43	2	no	—
172	#405	—	43	4	no	—
173	#405	—	86	4	no	—
174	C₄×M₁₆	—	86	2	no	—
175	C₄×M₁₆	C₂×D₄₃	1	2	yes	—
176	C₂×#97	—	43	1	no	—
177	C₂×#97	—	43	2	no	—
178	C₂×#97	—	86	2	no	—
179	C₂×C₄×C₈	—	43	2	no	—
180	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
181	#376	—	43	1	no	—
182	#341	C₂×D₄₃	1	2	yes	—
183	#339	—	43	4	no	—
184	C₄₃	#20514	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	43	2	no	—
186	#135	—	43	8	no	—
187	C₂×Q₁₆	—	43	1	no	—
188	C₂×SD₁₆	—	43	2	no	—
189	C₂×D₈	—	43	1	no	—
190	#131	—	43	16	no	—
191	#131	—	86	8	no	—
192	C₂×M₁₆	—	86	4	no	—
193	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
194	C₂²×C₈	—	43	2	no	—
195	C₂²×C₈	—	86	4	no	—
196	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
198	C₄×Q₈	—	43	4	no	—
199	C₄×D₄	—	43	4	no	—
200	C₄×D₄	—	86	4	no	—
201	#117	—	43	2	no	—
202	#117	—	86	2	no	—
203	C₂×C₄²	—	43	2	no	—
204	C₂×C₄²	—	86	2	no	—
205	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
206	#108	C₄×D₄₃	1	8	yes	complemented
207	#107	—	43	2	no	—
208	#106	—	43	2	no	—
209	#104	—	43	16	no	—
210	#103	—	43	4	no	—
211	#102	—	43	4	no	—
212	#97	—	2	2	no	—
213	#97	—	43	1	no	—
214	#97	—	43	1	no	—
215	#97	—	43	2	no	—
216	#97	—	86	4	no	—
217	#97	—	86	4	no	—
218	C₄×C₈	—	2	2	no	—
219	C₄×C₈	—	43	8	no	—
220	C₄×C₈	—	86	4	no	—
221	C₄×C₈	C₄×D₄₃	1	2	yes	—
222	C₄×C₈	C₄×D₄₃	1	2	yes	—
223	#95	—	86	2	no	—
224	#41	—	43	8	no	—
225	#41	—	86	4	no	—
226	C₂×Q₈	—	43	2	no	—
227	C₂×D₄	—	43	2	no	—
228	C₂×D₄	—	86	2	no	—
229	C₂²×C₄	—	86	2	no	—
230	C₂²×C₄	—	86	2	no	—
231	C₂²×C₄	—	86	4	no	—
232	C₂²×C₄	#10621902	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
234	Q₁₆	—	43	4	no	—
235	SD₁₆	—	43	8	no	—
236	D₈	—	43	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	86	16	no	—
239	M₁₆	C₂×C₄×D₄₃	1	8	yes	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	43	4	no	—
243	C₂×C₈	—	43	2	no	—
244	C₂×C₈	—	43	2	no	—
245	C₂×C₈	—	43	4	no	—
246	C₂×C₈	—	86	16	no	—
247	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
249	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
251	C₄⋊C₄	—	43	4	no	—
252	C₄⋊C₄	—	86	4	no	—
253	C₂²⋊C₄	—	86	2	no	—
254	C₂²⋊C₄	—	86	4	no	—
255	C₄²	—	43	4	no	—
256	C₄²	—	86	4	no	—
257	C₄²	—	86	2	no	—
258	C₄²	—	86	2	no	—
259	C₄²	C₂×C₄×D₄₃	1	2	yes	—
260	C₄²	D₄×D₄₃	1	1	yes	characteristic
261	C₄²	D₄×D₄₃	1	1	yes	characteristic
262	C₂³	—	86	2	no	—
263	C₂³	C₂×#10621902	1	1	yes	characteristic
264	Q₈	—	43	4	no	—
265	Q₈	—	86	2	no	—
266	D₄	—	43	4	no	—
267	D₄	—	86	8	no	—
268	D₄	—	86	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	86	2	no	—
271	C₂×C₄	—	86	8	no	—
272	C₂×C₄	—	86	4	no	—
273	C₂×C₄	—	86	4	no	—
274	C₂×C₄	—	344	4	no	—
275	C₂×C₄	—	172	4	no	—
276	C₂×C₄	#13063287	1	2	yes	—
277	C₂×C₄	#13063287	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
279	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
281	C₂×C₄	#41×D₄₃	1	2	yes	—
282	C₈	—	2	8	no	—
283	C₈	—	2	2	no	—
284	C₈	—	43	4	no	—
285	C₈	—	43	4	no	—
286	C₈	C₄²×D₄₃	1	2	yes	—
287	C₈	C₄²×D₄₃	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	86	4	no	—
290	C₂²	—	172	4	no	—
291	C₂²	C₂×#13063287	1	1	yes	characteristic
292	C₂²	C₄×#10621902	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	86	4	no	—
296	C₄	—	172	4	no	—
297	C₄	—	172	4	no	—
298	C₄	C₂×C₄²×D₄₃	1	1	yes	characteristic
299	C₄	C₄×D₄×D₄₃	1	2	yes	—
300	C₄	#14583367	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	172	4	no	—
303	C₂	C₄×#13063287	1	1	yes	characteristic
304	C₂	#19143511	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#108×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂×#13063347	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#108	D₄₃
hypocenter	C₄₃	#20514
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. #19143024	C₂	semidirect
4. #19134477	C₂	semidirect
5. #19134438	C₂	semidirect
6. #19134408	C₂	semidirect
7. #19134281	C₂	semidirect
8. C₄×#108×C₄₃	C₂	semidirect
12. #14583015	C₄	semidirect
13. C₄×Dic₁₇₂	C₄	semidirect
14. C₄×#10621873	C₄	semidirect
15. C₄×D₃₄₄	C₄	semidirect
38. #13063333	C₂×C₄	semidirect
51. #13063251	C₂×C₄	semidirect
60. C₈×Dic₄₃	C₂×C₄	semidirect
66. #108×C₄₃	C₂×C₄	semidirect
152. C₄×#108	D₄₃	semidirect
184. C₄₃	#20514	semidirect
206. #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
C₂×#14583015	2
C₄×#13063347	2
#19143024	2
#19134477	2
#19134438	2
#19134408	2
#19134281	2
C₄×#108×C₄₃	2
#20514	43