Subgroups — #25012523

Order 11008.

486 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10621872	C₂	1	1	yes	maximal, characteristic
2	#19136351	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582680	C₂	1	2	yes	maximal
4	#19132678	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10621872	C₄	1	1	yes	characteristic
7	C₂²×#10621872	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₄₃	C₄	1	1	yes	characteristic
9	C₂×#13063265	C₂²	1	2	yes	—
10	C₄×#10621872	—	2	2	no	—
11	C₄×#10621872	—	2	2	no	—
12	C₂×#13063235	C₄	1	1	yes	characteristic
13	C₂×#13063235	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₄₃⋊C₈	C₄	1	1	yes	characteristic
15	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
16	#14582680	—	2	4	no	—
17	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×C₄⋊C₁₆	—	2	2	no	—
20	C₂²×C₄×D₄₃	C₈	1	2	yes	—
21	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#10621872	—	2	2	no	—
23	C₂×#10621872	—	2	2	no	—
24	C₂×#10621872	—	2	1	no	—
25	C₂×#10621872	—	2	1	no	—
26	C₂×#10621872	—	2	1	no	—
27	C₂×#10621872	—	2	1	no	—
28	C₂×#10621872	—	2	2	no	—
29	C₂×#10621872	D₄	1	2	yes	—
30	C₂×#10621872	Q₈	1	2	yes	—
31	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₄₃	C₈	1	2	yes	—
34	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₄₃	—	2	2	no	—
36	C₄²×D₄₃	—	2	1	no	—
37	C₄²×D₄₃	—	2	1	no	—
38	#13063265	—	2	2	no	—
39	#13063265	—	2	2	no	—
40	C₂×C₄₃⋊C₁₆	—	2	2	no	—
41	#13063235	—	2	2	no	—
42	#13063235	—	2	1	no	—
43	#13063235	—	2	1	no	—
44	C₄×C₄₃⋊C₈	—	2	2	no	—
45	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₄₃	—	2	2	no	—
49	M₃₂×C₄₃	—	2	2	no	—
50	C₂×C₁₆×C₄₃	—	2	2	no	—
51	C₄×C₈×C₄₃	D₄	1	2	yes	—
52	C₄×C₈×C₄₃	D₄	1	2	yes	—
53	C₂³×D₄₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₄₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
58	C₂×C₄×D₄₃	—	2	1	no	—
59	C₂×C₄×D₄₃	—	2	1	no	—
60	C₂×C₄×D₄₃	—	2	1	no	—
61	C₂×C₄×D₄₃	—	2	1	no	—
62	C₂×C₄×D₄₃	—	2	2	no	—
63	C₂×C₄×D₄₃	—	2	2	no	—
64	C₂×C₄×D₄₃	—	4	4	no	—
65	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₄₃	M₁₆	1	2	yes	—
67	C₂×C₈×C₄₃	—	2	1	no	—
68	C₂×C₈×C₄₃	—	2	1	no	—
69	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
75	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
76	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₃	M₁₆	1	1	yes	characteristic
78	C₄²×C₄₃	M₁₆	1	1	yes	characteristic
79	C₄×Dic₄₃	—	2	2	no	—
80	C₄×Dic₄₃	—	2	2	no	—
81	C₄×Dic₄₃	—	2	1	no	—
82	C₄×Dic₄₃	—	2	1	no	—
83	C₂×C₄₃⋊C₈	—	2	1	no	—
84	C₂×C₄₃⋊C₈	—	2	1	no	—
85	C₂×C₄₃⋊C₈	—	2	1	no	—
86	C₂×C₄₃⋊C₈	—	2	1	no	—
87	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
88	C₂×C₄₃⋊C₈	M₁₆	1	2	yes	—
89	#10621872	—	2	8	no	—
90	#10621872	—	2	2	no	—
91	#10621872	—	2	2	no	—
92	#10621872	—	2	2	no	—
93	#10621872	—	2	2	no	—
94	C₁₆×C₄₃	—	4	2	no	—
95	C₄₃⋊C₁₆	—	4	2	no	—
96	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
97	C₂²×D₄₃	—	2	1	no	—
98	C₂²×D₄₃	—	2	1	no	—
99	C₂²×D₄₃	—	4	2	no	—
100	C₂²×D₄₃	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×Dic₄₃	—	2	1	no	—
111	C₂×Dic₄₃	—	2	1	no	—
112	C₂×Dic₄₃	—	2	2	no	—
113	C₂×Dic₄₃	—	4	2	no	—
114	C₂×Dic₄₃	C₄⋊C₈	1	2	yes	—
115	C₄×D₄₃	—	2	2	no	—
116	C₄×D₄₃	—	2	2	no	—
117	C₄×D₄₃	—	2	2	no	—
118	C₄×D₄₃	—	2	2	no	—
119	C₄×D₄₃	—	4	4	no	—
120	C₄×D₄₃	—	4	4	no	—
121	C₈×C₄₃	—	2	2	no	—
122	C₈×C₄₃	#121	1	1	yes	characteristic
123	C₈×C₄₃	#121	1	1	yes	characteristic
124	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₄₃⋊C₈	—	2	2	no	—
127	C₄₃⋊C₈	—	2	2	no	—
128	#12785	—	43	1	no	maximal, Sylow
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
132	C₂²×C₄₃	#431	1	1	yes	characteristic
133	C₂×D₄₃	—	2	2	no	—
134	C₂×D₄₃	—	4	2	no	—
135	C₂×D₄₃	—	4	2	no	—
136	C₂×D₄₃	—	4	2	no	—
137	C₄×C₄₃	—	2	2	no	—
138	C₄×C₄₃	#349	1	4	yes	complemented
139	C₄×C₄₃	#380	1	1	yes	characteristic
140	C₄×C₄₃	#407	1	1	yes	characteristic
141	C₄×C₄₃	#423	1	1	yes	characteristic
142	C₄×C₄₃	#432	1	1	yes	characteristic
143	Dic₄₃	—	2	2	no	—
144	Dic₄₃	—	4	2	no	—
145	C₂×C₄×M₁₆	—	43	1	no	—
146	#2150	—	43	1	no	—
147	#2150	D₄₃	1	1	yes	characteristic, complemented
148	C₂×#349	—	43	2	no	—
149	#1377	—	43	2	no	—
150	C₂×C₄₃	—	2	1	no	—
151	C₂×C₄₃	—	2	1	no	—
152	C₂×C₄₃	#1797	1	1	yes	characteristic
153	C₂×C₄₃	C₂×#349	1	2	yes	—
154	D₄₃	—	4	2	no	—
155	C₂²×M₁₆	—	43	1	no	—
156	C₂²×M₁₆	—	43	1	no	—
157	C₂²×C₄²	—	43	1	no	—
158	C₂×M₃₂	—	43	2	no	—
159	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
160	C₄×M₁₆	—	86	2	no	—
161	C₄×M₁₆	—	86	2	no	—
162	C₂×#97	—	43	1	no	—
163	C₂×#97	—	43	1	no	—
164	C₂×C₄×C₈	—	43	1	no	—
165	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
166	C₄⋊C₁₆	—	2	2	no	—
167	C₄⋊C₁₆	—	86	2	no	—
168	#349	—	86	4	no	—
169	C₄₃	#12785	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	43	1	no	—
171	C₂³×C₄	—	43	2	no	—
172	C₂×M₁₆	—	43	2	no	—
173	C₂×M₁₆	—	43	2	no	—
174	C₂×M₁₆	—	86	2	no	—
175	C₂×M₁₆	—	86	2	no	—
176	C₂×M₁₆	—	86	2	no	—
177	C₂×M₁₆	—	86	1	no	—
178	C₂×M₁₆	—	86	1	no	—
179	C₂×M₁₆	—	86	1	no	—
180	C₂×M₁₆	—	86	1	no	—
181	C₂²×C₈	—	43	1	no	—
182	C₂²×C₈	—	43	1	no	—
183	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
185	C₂×C₄²	—	43	2	no	—
186	C₂×C₄²	—	43	1	no	—
187	C₂×C₄²	—	86	1	no	—
188	C₂×C₄²	—	86	1	no	—
189	C₂×C₄²	—	86	2	no	—
190	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
191	M₃₂	—	2	2	no	—
192	M₃₂	—	2	2	no	—
193	M₃₂	—	86	2	no	—
194	M₃₂	—	86	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	86	2	no	—
197	#97	—	86	2	no	—
198	#97	—	86	1	no	—
199	#97	—	86	1	no	—
200	C₄×C₈	—	86	2	no	—
201	C₄×C₈	D₁₇₂	1	2	yes	—
202	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
203	C₂⁴	—	43	1	no	—
204	C₂²×C₄	—	43	1	no	—
205	C₂²×C₄	—	43	2	no	—
206	C₂²×C₄	—	43	2	no	—
207	C₂²×C₄	—	43	2	no	—
208	C₂²×C₄	—	86	1	no	—
209	C₂²×C₄	—	86	1	no	—
210	C₂²×C₄	—	86	1	no	—
211	C₂²×C₄	—	86	1	no	—
212	C₂²×C₄	—	86	2	no	—
213	C₂²×C₄	—	86	2	no	—
214	C₂²×C₄	—	172	4	no	—
215	C₂²×C₄	C₈×D₄₃	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
217	M₁₆	—	86	8	no	—
218	M₁₆	—	86	2	no	—
219	M₁₆	—	86	2	no	—
220	M₁₆	—	86	2	no	—
221	M₁₆	—	86	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	43	2	no	—
225	C₂×C₈	—	43	2	no	—
226	C₂×C₈	—	86	1	no	—
227	C₂×C₈	—	86	1	no	—
228	C₂×C₈	—	86	1	no	—
229	C₂×C₈	—	86	1	no	—
230	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
231	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
232	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
233	C₂×C₈	#10621900	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
235	C₂×C₈	#10621902	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
237	C₄²	—	86	2	no	—
238	C₄²	—	86	2	no	—
239	C₄²	—	86	1	no	—
240	C₄²	—	86	1	no	—
241	C₄²	#10621872	1	1	yes	characteristic
242	C₄²	#10621872	1	1	yes	characteristic
243	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
244	C₁₆	—	4	2	no	—
245	C₁₆	—	172	2	no	—
246	C₂³	—	43	2	no	—
247	C₂³	—	86	1	no	—
248	C₂³	—	86	1	no	—
249	C₂³	—	172	2	no	—
250	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	43	2	no	—
253	C₂×C₄	—	86	1	no	—
254	C₂×C₄	—	86	1	no	—
255	C₂×C₄	—	86	2	no	—
256	C₂×C₄	—	86	2	no	—
257	C₂×C₄	—	86	2	no	—
258	C₂×C₄	—	86	2	no	—
259	C₂×C₄	—	86	2	no	—
260	C₂×C₄	—	172	4	no	—
261	C₂×C₄	—	172	4	no	—
262	C₂×C₄	—	172	2	no	—
263	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
264	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
265	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
267	C₂×C₄	#13063283	1	1	yes	characteristic
268	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
269	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
270	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	86	2	no	—
273	C₈	—	86	2	no	—
274	C₈	#13063296	1	1	yes	characteristic
275	C₈	#13063300	1	1	yes	characteristic
276	C₈	#13063301	1	1	yes	characteristic
277	C₈	#13063305	1	1	yes	characteristic
278	C₂²	—	2	2	no	—
279	C₂²	—	86	2	no	—
280	C₂²	—	172	2	no	—
281	C₂²	—	172	2	no	—
282	C₂²	—	172	2	no	—
283	C₂²	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
284	C₂²	C₄⋊C₈×D₄₃	1	1	yes	characteristic
285	C₂²	#14582951	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	86	2	no	—
288	C₄	—	172	2	no	—
289	C₄	#14582680	1	4	yes	complemented
290	C₄	#14582756	1	1	yes	characteristic
291	C₄	#14582862	1	1	yes	characteristic
292	C₄	#14582921	1	1	yes	characteristic
293	C₄	#14582955	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	172	2	no	—
297	C₂	#19134572	1	1	yes	characteristic
298	C₂	C₂×#14582680	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂³×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#2150	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₄₃	C₂×C₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#2150	D₄₃
hypocenter	C₄₃	#12785
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #19136351	C₂	semidirect
4. #19132678	C₂	semidirect
5. C₄₃×#2150	C₂	semidirect
138. C₄×C₄₃	#349	semidirect
147. #2150	D₄₃	semidirect
169. C₄₃	#12785	semidirect
289. C₄	#14582680	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10621872	2
#19136351	2
C₂×#14582680	2
#19132678	2
C₄₃×#2150	2
#12785	43