Subgroups — #25101000

Order 11008.

722 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063347	C₂	1	1	yes	maximal, characteristic
2	#19143247	C₂	1	2	yes	maximal, complemented
3	#19135481	C₂	1	2	yes	maximal
4	#19135341	C₂	1	2	yes	maximal, complemented
5	#19135304	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	C₄×#10621898	C₂²	1	2	yes	—
11	C₄×Dic₁₇₂	C₄	1	2	yes	complemented
12	C₄×#10621873	C₄	1	4	yes	complemented
13	C₄×D₃₄₄	C₄	1	2	yes	complemented
14	#14582958	—	2	4	no	—
15	#14582954	—	2	4	no	—
16	#14582937	C₂²	1	1	yes	characteristic
17	#14582927	C₂²	1	4	yes	—
18	C₂×#13063251	C₂²	1	4	yes	—
19	#14582896	C₂²	1	2	yes	—
20	#14582867	C₂²	1	4	yes	—
21	C₂×#13063236	C₂²	1	1	yes	characteristic
22	C₂×#13063235	C₂²	1	2	yes	—
23	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
24	#14582745	C₂²	1	1	yes	characteristic
25	#14582660	C₂²	1	4	yes	—
26	C₂×#108×C₄₃	C₂²	1	2	yes	—
27	C₄×M₁₆×C₄₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#341	C₂²	1	2	yes	—
30	C₂×#10621898	C₂³	1	2	yes	—
31	#13063347	—	2	4	no	—
32	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
33	C₂×#10621873	C₂×C₄	1	2	yes	—
34	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
36	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
37	C₄×D₁₇₂	C₂×C₄	1	4	yes	—
38	C₄×C₂²⋊D₄₃	—	2	4	no	—
39	#13063284	C₂³	1	2	yes	—
40	#13063282	—	2	2	no	—
41	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
42	C₄²×D₄₃	—	2	2	no	—
43	#13063259	C₂×C₄	1	2	yes	—
44	#13063257	C₂×C₄	1	2	yes	—
45	Dic₄₃⋊C₈	—	2	8	no	—
46	#13063251	—	2	8	no	—
47	#13063250	C₂×C₄	1	4	yes	—
48	#13063247	C₂×C₄	1	4	yes	—
49	(C₂×D₄₃)⋊C₈	—	2	8	no	—
50	#13063236	D₄	1	2	yes	—
51	#13063236	D₄	1	2	yes	—
52	#13063235	—	2	4	no	—
53	C₈×Dic₄₃	—	2	4	no	—
54	C₄₃⋊#95	—	2	2	no	—
55	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	#108×C₄₃	—	2	4	no	—
60	#97×C₄₃	—	2	2	no	—
61	C₄×C₈×C₄₃	—	2	2	no	—
62	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	#41	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₂²⋊D₄₃	—	2	2	no	—
67	C₂²×Dic₄₃	—	2	2	no	—
68	#10621898	—	2	4	no	—
69	#10621898	—	2	4	no	—
70	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
71	C₂×C₄×D₄₃	—	2	2	no	—
72	C₂×C₄×D₄₃	—	2	4	no	—
73	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
74	M₁₆×C₄₃	—	2	4	no	—
75	M₁₆×C₄₃	—	2	4	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	4	no	—
78	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₃	#41	1	4	yes	—
82	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂²⋊Dic₄₃	—	2	2	no	—
86	(C₂×D₄₃)⋊C₄	—	2	4	no	—
87	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
88	Dic₄₃⋊C₄	—	2	4	no	—
89	C₄×Dic₄₃	—	2	4	no	—
90	C₄×Dic₄₃	—	2	2	no	—
91	C₄×Dic₄₃	—	2	2	no	—
92	C₄×Dic₄₃	#41	1	4	yes	—
93	C₂×C₄₃⋊C₈	—	2	2	no	—
94	C₂×C₄₃⋊C₈	—	2	2	no	—
95	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
96	Dic₁₇₂	—	2	2	no	—
97	D₃₄₄	—	2	2	no	—
98	#10621873	—	2	4	no	—
99	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
100	C₂²×D₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	4	2	no	—
102	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
107	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
108	C₂²⋊D₄₃	—	4	4	no	—
109	C₂×Dic₄₃	—	2	2	no	—
110	C₂×Dic₄₃	—	2	4	no	—
111	C₂×Dic₄₃	—	2	4	no	—
112	C₂×Dic₄₃	—	4	4	no	—
113	D₁₇₂	—	2	2	no	—
114	D₁₇₂	—	2	2	no	—
115	C₄×D₄₃	—	4	4	no	—
116	C₄×D₄₃	—	8	4	no	—
117	Dic₈₆	—	2	2	no	—
118	Dic₈₆	—	2	2	no	—
119	C₈×C₄₃	—	2	2	no	—
120	C₈×C₄₃	—	4	4	no	—
121	C₈×C₄₃	#117	1	2	yes	—
122	C₈×C₄₃	#117	1	2	yes	—
123	C₄₃⋊C₈	—	2	4	no	—
124	#21294	—	43	1	no	maximal, Sylow
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂²×C₄₃	#529	1	2	yes	—
128	C₂×D₄₃	—	4	4	no	—
129	C₂×D₄₃	—	4	2	no	—
130	C₄×C₄₃	—	2	2	no	—
131	C₄×C₄₃	—	2	2	no	—
132	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
133	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
134	C₄×C₄₃	#532	1	2	yes	—
135	Dic₄₃	—	4	4	no	—
136	Dic₄₃	—	4	4	no	—
137	Dic₄₃	—	4	2	no	—
138	C₄×#135	—	43	1	no	—
139	#2923	—	43	2	no	—
140	#1974	—	43	2	no	—
141	#1945	—	43	2	no	—
142	#1938	—	43	4	no	—
143	C₄×#108	D₄₃	1	1	yes	characteristic, complemented
144	#1759	—	43	2	no	—
145	#1726	—	43	1	no	—
146	C₂×C₄₃	—	2	2	no	—
147	C₂×C₄₃	#2293	1	1	yes	characteristic
148	C₂×C₄₃	#2957	1	2	yes	—
149	D₄₃	—	8	2	no	—
150	C₂×#135	—	43	1	no	—
151	C₄×#41	—	43	2	no	—
152	C₄×Q₁₆	—	43	2	no	—
153	C₄×SD₁₆	—	43	4	no	—
154	C₄×D₈	—	43	2	no	—
155	#433	—	86	4	no	—
156	#432	—	86	4	no	—
157	C₂×#108	C₂×D₄₃	1	2	yes	—
158	#427	—	43	1	no	—
159	#424	—	43	4	no	—
160	C₂×#104	—	43	4	no	—
161	#416	—	43	2	no	—
162	#408	—	43	4	no	—
163	C₄×M₁₆	C₂×D₄₃	1	2	yes	—
164	C₂×#97	—	43	1	no	—
165	C₂×#97	—	43	2	no	—
166	C₂×C₄×C₈	—	43	2	no	—
167	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
168	#376	—	43	1	no	—
169	#341	C₂×D₄₃	1	2	yes	—
170	#339	—	43	4	no	—
171	C₄₃	#21294	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	43	2	no	—
173	#135	—	86	4	no	—
174	C₂×Q₁₆	—	43	1	no	—
175	C₂×SD₁₆	—	43	2	no	—
176	C₂×D₈	—	43	1	no	—
177	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
178	C₂²×C₈	—	43	2	no	—
179	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
181	C₄×Q₈	—	43	4	no	—
182	C₄×D₄	—	43	4	no	—
183	C₄×D₄	—	86	4	no	—
184	#117	—	43	2	no	—
185	#117	—	86	2	no	—
186	C₂×C₄²	—	43	2	no	—
187	C₂×C₄²	—	86	2	no	—
188	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
189	#108	—	2	4	no	—
190	#107	—	43	2	no	—
191	#106	—	43	2	no	—
192	C₄⋊C₈	—	86	8	no	—
193	#104	—	86	8	no	—
194	#103	—	43	4	no	—
195	#102	—	43	4	no	—
196	C₂²⋊C₈	—	86	8	no	—
197	#97	—	2	2	no	—
198	#97	—	43	2	no	—
199	#97	—	43	2	no	—
200	#97	—	86	4	no	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	—	86	4	no	—
203	C₄×C₈	C₄×D₄₃	1	2	yes	—
204	C₄×C₈	C₄×D₄₃	1	2	yes	—
205	#95	—	86	2	no	—
206	#41	—	86	4	no	—
207	#41	—	86	4	no	—
208	C₂×Q₈	—	43	2	no	—
209	C₂×D₄	—	43	2	no	—
210	C₂×D₄	—	86	2	no	—
211	C₂²×C₄	—	86	2	no	—
212	C₂²×C₄	—	86	4	no	—
213	C₂²×C₄	—	86	2	no	—
214	C₂²×C₄	#10621902	1	2	yes	—
215	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
216	Q₁₆	—	86	2	no	—
217	SD₁₆	—	86	4	no	—
218	D₈	—	86	2	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	43	4	no	—
224	C₂×C₈	—	86	2	no	—
225	C₂×C₈	—	86	2	no	—
226	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
227	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
229	C₂×C₈	#10621898	1	4	yes	—
230	C₄⋊C₄	—	43	4	no	—
231	C₄⋊C₄	—	86	4	no	—
232	C₂²⋊C₄	—	86	2	no	—
233	C₂²⋊C₄	—	86	4	no	—
234	C₄²	—	43	4	no	—
235	C₄²	—	86	4	no	—
236	C₄²	—	86	2	no	—
237	C₄²	—	86	2	no	—
238	C₄²	C₂×C₄×D₄₃	1	2	yes	—
239	C₄²	D₄×D₄₃	1	1	yes	characteristic
240	C₄²	D₄×D₄₃	1	1	yes	characteristic
241	C₂³	—	86	2	no	—
242	C₂³	C₂×#10621902	1	1	yes	characteristic
243	Q₈	—	86	2	no	—
244	Q₈	—	86	2	no	—
245	D₄	—	86	2	no	—
246	D₄	—	86	2	no	—
247	D₄	—	172	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	86	4	no	—
250	C₂×C₄	—	86	2	no	—
251	C₂×C₄	—	86	4	no	—
252	C₂×C₄	—	344	4	no	—
253	C₂×C₄	—	172	4	no	—
254	C₂×C₄	—	172	4	no	—
255	C₂×C₄	#13063287	1	2	yes	—
256	C₂×C₄	#13063287	1	2	yes	—
257	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
259	C₂×C₄	#41×D₄₃	1	2	yes	—
260	C₂×C₄	C₂×#10621898	1	2	yes	—
261	C₈	—	2	2	no	—
262	C₈	—	4	4	no	—
263	C₈	—	86	4	no	—
264	C₈	#13063282	1	2	yes	—
265	C₈	#13063282	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	172	2	no	—
268	C₂²	—	172	4	no	—
269	C₂²	C₂×#13063287	1	1	yes	characteristic
270	C₂²	#14583436	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	172	4	no	—
274	C₄	—	172	2	no	—
275	C₄	—	172	4	no	—
276	C₄	C₂×#13063282	1	1	yes	characteristic
277	C₄	#14583367	1	1	yes	characteristic
278	C₄	#14583446	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	344	2	no	—
281	C₂	#19137056	1	1	yes	characteristic
282	C₂	#19143511	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
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	#21294
LCS	G ▷ C₂×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
2. #19143247	C₂	semidirect
4. #19135341	C₂	semidirect
5. #19135304	C₂	semidirect
6. #19134408	C₂	semidirect
7. #19134281	C₂	semidirect
8. C₄×#108×C₄₃	C₂	semidirect
11. C₄×Dic₁₇₂	C₄	semidirect
12. C₄×#10621873	C₄	semidirect
13. C₄×D₃₄₄	C₄	semidirect
143. C₄×#108	D₄₃	semidirect
171. C₄₃	#21294	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13063347	2
#19143247	2
#19135481	2
#19135341	2
#19135304	2
#19134408	2
#19134281	2
C₄×#108×C₄₃	2
#21294	43