Subgroups — #25002236

Order 11008.

650 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582855	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582669	C₂	1	2	yes	maximal
3	#19134334	C₂	1	1	yes	maximal, characteristic
4	#19134292	C₂	1	2	yes	maximal
5	C₄₃×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂²×#10621877	C₂²	1	1	yes	characteristic
8	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	#14582855	—	2	4	no	—
10	C₂×#13063236	C₄	1	2	yes	complemented
11	C₂×C₄×C₄₃⋊C₈	C₄	1	2	yes	complemented
12	#14582669	—	2	4	no	—
13	#14582668	C₄	1	4	yes	complemented
14	#14582654	—	2	4	no	—
15	C₂×#117×C₄₃	C₂²	1	2	yes	—
16	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#342	C₄	1	4	yes	—
18	C₂×#10621877	—	2	2	no	—
19	C₂×#10621877	—	2	2	no	—
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	C₂×C₄	1	4	yes	—
26	C₂×#10621877	D₄	1	2	yes	—
27	C₂×#10621877	Q₈	1	2	yes	—
28	C₂²×C₄₃⋊C₈	—	2	2	no	—
29	C₂²×C₄₃⋊C₈	—	2	1	no	—
30	C₂²×C₄₃⋊C₈	—	2	1	no	—
31	C₂²×C₄₃⋊C₈	—	2	2	no	—
32	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
36	#13063236	—	2	4	no	—
37	C₄×C₄₃⋊C₈	—	2	4	no	—
38	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	#117×C₄₃	—	2	2	no	—
40	#117×C₄₃	—	2	2	no	—
41	#117×C₄₃	—	2	2	no	—
42	#117×C₄₃	—	2	2	no	—
43	#117×C₄₃	—	2	2	no	—
44	#117×C₄₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₃	—	2	2	no	—
50	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
51	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	—	2	4	no	—
53	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
54	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	#41	1	2	yes	—
60	C₂²×C₄×C₄₃	#41	1	2	yes	—
61	C₂²×C₄×C₄₃	#41	1	2	yes	—
62	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₄⋊C₄×C₄₃	—	2	4	no	—
65	C₄⋊C₄×C₄₃	—	4	2	no	—
66	C₄⋊C₄×C₄₃	—	4	2	no	—
67	C₂²⋊C₄×C₄₃	—	2	4	no	—
68	C₂²⋊C₄×C₄₃	—	2	4	no	—
69	C₂²⋊C₄×C₄₃	—	2	4	no	—
70	C₄²×C₄₃	—	2	2	no	—
71	C₄²×C₄₃	—	2	2	no	—
72	C₄²×C₄₃	—	4	2	no	—
73	C₄²×C₄₃	—	4	2	no	—
74	#10621877	—	2	4	no	—
75	#10621877	—	2	4	no	—
76	#10621877	—	2	4	no	—
77	#10621877	—	2	4	no	—
78	#10621877	—	4	2	no	—
79	#10621877	—	4	2	no	—
80	C₂×C₄₃⋊C₈	—	2	2	no	—
81	C₂×C₄₃⋊C₈	—	2	1	no	—
82	C₂×C₄₃⋊C₈	—	2	1	no	—
83	C₂×C₄₃⋊C₈	—	2	1	no	—
84	C₂×C₄₃⋊C₈	—	2	1	no	—
85	C₂×C₄₃⋊C₈	—	2	2	no	—
86	C₂×C₄₃⋊C₈	—	2	4	no	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	4	2	no	—
90	C₂×C₄₃⋊C₈	—	4	2	no	—
91	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
92	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
93	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
94	C₂³×C₄₃	—	4	2	no	—
95	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
96	C₂³×C₄₃	#117	1	2	yes	—
97	C₂³×C₄₃	C₄×D₄	1	2	yes	—
98	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
99	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₄₃	—	2	4	no	—
101	C₂×C₄×C₄₃	—	4	4	no	—
102	C₂×C₄×C₄₃	—	4	8	no	—
103	C₂×C₄×C₄₃	—	4	4	no	—
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	#117	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#117	1	2	yes	—
110	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#121	1	2	yes	—
115	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₃	#123	1	2	yes	—
117	C₂×C₄×C₄₃	#123	1	2	yes	—
118	C₂×C₄×C₄₃	#124	1	2	yes	—
119	C₂×C₄×C₄₃	#125	1	2	yes	—
120	C₂×C₄×C₄₃	#126	1	2	yes	—
121	C₂×C₄×C₄₃	#126	1	2	yes	—
122	C₄₃⋊C₈	—	2	4	no	—
123	C₄₃⋊C₈	—	2	2	no	—
124	C₄₃⋊C₈	—	2	2	no	—
125	C₄₃⋊C₈	—	4	4	no	—
126	#10803	—	43	1	no	maximal, Sylow
127	C₂²×C₄₃	—	2	2	no	—
128	C₂²×C₄₃	—	2	4	no	—
129	C₂²×C₄₃	—	4	2	no	—
130	C₂²×C₄₃	—	4	1	no	—
131	C₂²×C₄₃	—	4	1	no	—
132	C₂²×C₄₃	#376	1	1	yes	characteristic
133	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₃	#387	1	2	yes	—
136	C₂²×C₄₃	#388	1	2	yes	—
137	C₄×C₄₃	—	4	4	no	—
138	C₄×C₄₃	—	8	4	no	—
139	C₄×C₄₃	#380	1	1	yes	characteristic
140	C₄×C₄₃	#381	1	1	yes	characteristic
141	C₄×C₄₃	#382	1	1	yes	characteristic
142	C₄×C₄₃	#383	1	1	yes	characteristic
143	C₄×C₄₃	#387	1	2	yes	—
144	C₄×C₄₃	#388	1	2	yes	—
145	C₂×#405	—	43	1	no	—
146	C₂×#344	—	43	2	no	—
147	#1742	—	43	1	no	—
148	#1740	D₄₃	1	1	yes	characteristic
149	#1729	—	43	2	no	—
150	C₂×C₄₃	—	2	2	no	—
151	C₂×C₄₃	—	2	1	no	—
152	C₂×C₄₃	—	2	1	no	—
153	C₂×C₄₃	—	2	2	no	—
154	C₂×C₄₃	#1434	1	1	yes	characteristic
155	C₂×C₄₃	#1757	1	2	yes	—
156	C₂²×M₁₆	—	43	1	no	—
157	C₂²×M₁₆	—	43	2	no	—
158	C₂³×C₈	—	43	1	no	—
159	C₂×#117	C₂×D₄₃	1	2	yes	—
160	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
161	#405	—	86	4	no	—
162	C₂×#97	—	43	2	no	—
163	C₂×C₄×C₈	—	43	2	no	—
164	#344	—	86	4	no	—
165	#343	—	43	4	no	—
166	#342	Dic₄₃	1	4	yes	—
167	#336	—	86	4	no	—
168	C₄₃	#10803	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
170	C₂×M₁₆	—	43	2	no	—
171	C₂×M₁₆	—	43	2	no	—
172	C₂×M₁₆	—	43	4	no	—
173	C₂×M₁₆	—	86	2	no	—
174	C₂×M₁₆	—	86	2	no	—
175	C₂×M₁₆	—	86	2	no	—
176	C₂×M₁₆	—	86	2	no	—
177	C₂×M₁₆	—	86	2	no	—
178	C₂×M₁₆	—	86	2	no	—
179	C₂×M₁₆	—	86	2	no	—
180	C₂²×C₈	—	43	1	no	—
181	C₂²×C₈	—	43	1	no	—
182	C₂²×C₈	—	43	2	no	—
183	C₂²×C₈	—	43	4	no	—
184	C₂²×C₈	—	86	2	no	—
185	C₂²×C₈	—	86	1	no	—
186	C₂²×C₈	—	86	1	no	—
187	C₂²×C₈	—	86	2	no	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
196	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
197	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₄×D₄₃	1	2	yes	—
200	#97	—	86	4	no	—
201	C₄×C₈	—	86	4	no	—
202	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
205	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
207	C₂²×C₄	#10621898	1	2	yes	—
208	C₂²×C₄	#10621898	1	2	yes	—
209	C₂²×C₄	D₄×D₄₃	1	2	yes	—
210	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
211	C₂²×C₄	#10621900	1	2	yes	—
212	C₂²×C₄	#10621900	1	1	yes	characteristic
213	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
214	C₂²×C₄	#10621902	1	1	yes	characteristic
215	M₁₆	—	86	4	no	—
216	M₁₆	—	86	4	no	—
217	M₁₆	—	86	4	no	—
218	M₁₆	—	86	4	no	—
219	M₁₆	—	172	2	no	—
220	M₁₆	—	172	2	no	—
221	C₂×C₈	—	43	2	no	—
222	C₂×C₈	—	43	2	no	—
223	C₂×C₈	—	43	4	no	—
224	C₂×C₈	—	86	2	no	—
225	C₂×C₈	—	86	1	no	—
226	C₂×C₈	—	86	1	no	—
227	C₂×C₈	—	86	2	no	—
228	C₂×C₈	—	86	1	no	—
229	C₂×C₈	—	86	1	no	—
230	C₂×C₈	—	86	4	no	—
231	C₂×C₈	—	86	4	no	—
232	C₂×C₈	—	86	4	no	—
233	C₂×C₈	—	172	2	no	—
234	C₂×C₈	—	172	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
247	C₂³	#13063284	1	2	yes	—
248	C₂³	D₄×Dic₄₃	1	2	yes	—
249	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
250	C₂³	Q₈×Dic₄₃	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
257	C₂×C₄	#13063281	1	1	yes	characteristic
258	C₂×C₄	#13063282	1	2	yes	—
259	C₂×C₄	#13063282	1	1	yes	characteristic
260	C₂×C₄	#13063282	1	1	yes	characteristic
261	C₂×C₄	#13063283	1	1	yes	characteristic
262	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
263	C₂×C₄	#13063287	1	1	yes	characteristic
264	C₂×C₄	#13063290	1	1	yes	characteristic
265	C₂×C₄	#13063298	1	2	yes	—
266	C₂×C₄	#13063303	1	2	yes	—
267	C₂×C₄	#13063307	1	2	yes	—
268	C₂×C₄	#13063310	1	2	yes	—
269	C₂×C₄	#13063311	1	2	yes	—
270	C₂×C₄	#13063315	1	2	yes	—
271	C₂×C₄	#13063318	1	2	yes	—
272	C₂×C₄	#13063320	1	2	yes	—
273	C₈	—	86	4	no	—
274	C₈	—	86	2	no	—
275	C₈	—	86	2	no	—
276	C₈	—	172	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	#14582745	1	1	yes	characteristic
283	C₂²	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
284	C₂²	C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
285	C₂²	#14582785	1	2	yes	—
286	C₂²	#14582792	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	8	4	no	—
289	C₄	#14582758	1	1	yes	characteristic
290	C₄	#14582761	1	1	yes	characteristic
291	C₄	#14582763	1	1	yes	characteristic
292	C₄	#14582766	1	1	yes	characteristic
293	C₄	#14582781	1	2	yes	—
294	C₄	#14582788	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	#19132820	1	1	yes	characteristic
300	C₂	#19134397	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₄₃
commutator	C₂³×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1740	C₂
socle	C₂²×C₄₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1740	D₄₃
hypocenter	C₄₃	#10803
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
10. C₂×#13063236	C₄	semidirect
11. C₂×C₄×C₄₃⋊C₈	C₄	semidirect
13. #14582668	C₄	semidirect
168. C₄₃	#10803	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582855	2
C₂×#14582669	2
#19134334	2
#19134292	2
C₄₃×#1740	2
#10803	43