Subgroups — #25004706

Order 11008.

670 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143185	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582922	C₂	1	1	yes	maximal, characteristic
3	#19134333	C₂	1	2	yes	maximal
4	C₂×#14582660	C₂	1	2	yes	maximal
5	C₂×C₄₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	1	yes	characteristic
7	C₂×#13063284	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
9	#14582941	—	2	4	no	—
10	#14582937	—	2	2	no	—
11	#14582922	—	2	2	no	—
12	#14582922	C₄	1	4	yes	complemented
13	C₂×Dic₄₃⋊C₈	—	2	2	no	—
14	#14582667	—	2	4	no	—
15	#14582660	—	2	2	no	—
16	#14582660	—	2	2	no	—
17	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
18	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#338	—	2	2	no	—
20	C₂³×Dic₄₃	—	2	1	no	—
21	C₂×#10621877	—	2	1	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	C₂×C₄	1	2	yes	—
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	#13063284	—	2	2	no	—
27	#13063284	—	2	2	no	—
28	#13063284	—	2	2	no	—
29	#13063284	—	2	2	no	—
30	C₂×C₄⋊Dic₄₃	D₄	1	4	yes	—
31	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄×Dic₄₃	—	2	1	no	—
33	C₂×C₄×Dic₄₃	—	2	1	no	—
34	C₂×C₄×Dic₄₃	—	2	1	no	—
35	C₂×C₄×Dic₄₃	—	2	2	no	—
36	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₄₃	Q₈	1	2	yes	—
40	#13063259	—	2	4	no	—
41	#13063257	—	2	4	no	—
42	Dic₄₃⋊C₈	—	2	8	no	—
43	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₄₃	—	2	2	no	—
45	C₂×M₁₆×C₄₃	—	2	2	no	—
46	C₂×M₁₆×C₄₃	—	2	2	no	—
47	C₂×M₁₆×C₄₃	—	2	2	no	—
48	C₂×M₁₆×C₄₃	—	2	2	no	—
49	C₂×M₁₆×C₄₃	—	2	2	no	—
50	C₂×M₁₆×C₄₃	—	2	2	no	—
51	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
53	C₂×M₁₆×C₄₃	Q₈	1	4	yes	—
54	C₂²×C₈×C₄₃	—	2	2	no	—
55	C₂²×C₈×C₄₃	D₄	1	2	yes	—
56	C₂⁴×C₄₃	#41	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	#41	1	2	yes	—
66	C₂²×C₄×C₄₃	#41	1	2	yes	—
67	C₂²×Dic₄₃	—	2	2	no	—
68	C₂²×Dic₄₃	—	2	1	no	—
69	C₂²×Dic₄₃	—	2	1	no	—
70	C₂²×Dic₄₃	—	2	2	no	—
71	C₂²×Dic₄₃	—	2	4	no	—
72	C₂²×Dic₄₃	—	4	4	no	—
73	M₁₆×C₄₃	—	2	4	no	—
74	M₁₆×C₄₃	—	2	4	no	—
75	M₁₆×C₄₃	—	2	4	no	—
76	M₁₆×C₄₃	—	2	4	no	—
77	M₁₆×C₄₃	—	4	2	no	—
78	M₁₆×C₄₃	—	4	2	no	—
79	C₂×C₈×C₄₃	—	2	2	no	—
80	C₂×C₈×C₄₃	—	2	2	no	—
81	C₂×C₈×C₄₃	—	2	4	no	—
82	C₂×C₈×C₄₃	—	4	2	no	—
83	C₂×C₈×C₄₃	—	4	2	no	—
84	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
85	C₂²⋊Dic₄₃	—	4	2	no	—
86	C₂²⋊Dic₄₃	—	4	2	no	—
87	C₄⋊Dic₄₃	—	2	4	no	—
88	C₄⋊Dic₄₃	—	2	4	no	—
89	C₄×Dic₄₃	—	2	2	no	—
90	C₄×Dic₄₃	—	2	1	no	—
91	C₄×Dic₄₃	—	2	1	no	—
92	C₄×Dic₄₃	—	2	2	no	—
93	C₄×Dic₄₃	—	4	2	no	—
94	C₄×Dic₄₃	—	4	2	no	—
95	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
96	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
97	#10621877	—	2	4	no	—
98	#10621877	—	4	2	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	4	1	no	—
101	C₂×C₄₃⋊C₈	—	4	1	no	—
102	C₂³×C₄₃	—	4	2	no	—
103	C₂³×C₄₃	#117	1	1	yes	characteristic
104	C₂³×C₄₃	#121	1	1	yes	characteristic
105	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₄₃	#123	1	2	yes	—
107	C₂³×C₄₃	#125	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#121	1	2	yes	—
115	C₂×C₄×C₄₃	#121	1	2	yes	—
116	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#125	1	2	yes	—
123	C₂×C₄×C₄₃	#126	1	2	yes	—
124	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×Dic₄₃	—	2	4	no	—
127	C₂×Dic₄₃	—	2	2	no	—
128	C₂×Dic₄₃	—	2	2	no	—
129	C₂×Dic₄₃	—	4	4	no	—
130	C₂×Dic₄₃	—	4	4	no	—
131	C₂×Dic₄₃	—	4	2	no	—
132	C₂×Dic₄₃	—	4	4	no	—
133	C₂×Dic₄₃	—	8	4	no	—
134	C₈×C₄₃	—	2	4	no	—
135	C₈×C₄₃	—	4	4	no	—
136	C₄₃⋊C₈	—	4	2	no	—
137	#11243	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	—	2	4	no	—
140	C₂²×C₄₃	—	4	1	no	—
141	C₂²×C₄₃	—	4	1	no	—
142	C₂²×C₄₃	—	4	2	no	—
143	C₂²×C₄₃	#380	1	1	yes	characteristic
144	C₂²×C₄₃	#387	1	2	yes	—
145	C₂²×C₄₃	#394	1	1	yes	characteristic
146	C₂²×C₄₃	#398	1	1	yes	characteristic
147	C₂²×C₄₃	#398	1	2	yes	—
148	C₄×C₄₃	#386	1	1	yes	characteristic
149	C₄×C₄₃	#389	1	2	yes	—
150	C₄×C₄₃	#391	1	1	yes	characteristic
151	C₄×C₄₃	#395	1	1	yes	characteristic
152	C₄×C₄₃	#396	1	2	yes	—
153	C₄×C₄₃	#400	1	1	yes	characteristic
154	Dic₄₃	—	2	4	no	—
155	Dic₄₃	—	4	2	no	—
156	Dic₄₃	—	8	4	no	—
157	#2911	—	43	1	no	—
158	C₂×#423	—	43	1	no	—
159	#1742	—	43	2	no	—
160	C₂×#339	—	43	2	no	—
161	C₂×#338	D₄₃	1	1	yes	characteristic
162	C₂×C₄₃	—	2	1	no	—
163	C₂×C₄₃	—	2	2	no	—
164	C₂×C₄₃	—	2	1	no	—
165	C₂×C₄₃	—	2	2	no	—
166	C₂×C₄₃	#1441	1	1	yes	characteristic
167	C₂×C₄₃	#2050	1	2	yes	—
168	C₂²×M₁₆	—	43	1	no	—
169	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
170	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
171	C₂×#117	—	43	2	no	—
172	C₂²×C₄²	—	43	1	no	—
173	#428	—	86	4	no	—
174	#427	—	86	2	no	—
175	#423	—	43	4	no	—
176	#423	—	86	2	no	—
177	C₂×C₄⋊C₈	—	86	2	no	—
178	#343	—	86	4	no	—
179	#339	—	86	2	no	—
180	#339	—	86	2	no	—
181	#338	—	2	2	no	—
182	C₄₃	#11243	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₄	—	86	1	no	—
184	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	43	2	no	—
193	C₂×M₁₆	—	86	1	no	—
194	C₂×M₁₆	—	86	2	no	—
195	C₂×M₁₆	—	86	2	no	—
196	C₂×M₁₆	Dic₈₆	1	4	yes	—
197	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
198	C₂×M₁₆	C₂²⋊D₄₃	1	2	yes	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	86	1	no	—
201	C₂²×C₈	D₁₇₂	1	2	yes	—
202	#117	—	86	2	no	—
203	#117	—	86	2	no	—
204	#117	—	86	2	no	—
205	#117	—	86	2	no	—
206	C₂×C₄⋊C₄	—	43	4	no	—
207	C₂×C₂²⋊C₄	—	86	2	no	—
208	C₂×C₄²	—	43	2	no	—
209	C₂×C₄²	—	43	1	no	—
210	C₂×C₄²	—	43	1	no	—
211	C₂×C₄²	—	43	2	no	—
212	C₂×C₄²	—	86	1	no	—
213	C₂×C₄²	—	86	1	no	—
214	C₂×C₄²	—	86	1	no	—
215	C₂×C₄²	—	86	2	no	—
216	#107	—	86	4	no	—
217	#106	—	86	4	no	—
218	C₄⋊C₈	—	86	8	no	—
219	C₂⁴	#10621902	1	1	yes	characteristic
220	C₂²×C₄	—	86	1	no	—
221	C₂²×C₄	—	86	2	no	—
222	C₂²×C₄	—	86	1	no	—
223	C₂²×C₄	—	86	2	no	—
224	C₂²×C₄	—	86	4	no	—
225	C₂²×C₄	—	172	4	no	—
226	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
229	C₂²×C₄	#10621898	1	2	yes	—
230	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₄₃	1	2	yes	—
232	C₂²×C₄	#10621900	1	2	yes	—
233	C₂²×C₄	#10621900	1	1	yes	characteristic
234	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	86	4	no	—
243	M₁₆	—	172	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	86	2	no	—
250	C₂×C₈	—	172	1	no	—
251	C₂×C₈	—	172	1	no	—
252	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
253	C₄⋊C₄	—	86	4	no	—
254	C₄⋊C₄	—	86	4	no	—
255	C₂²⋊C₄	—	172	2	no	—
256	C₂²⋊C₄	—	172	2	no	—
257	C₄²	—	43	2	no	—
258	C₄²	—	43	2	no	—
259	C₄²	—	86	2	no	—
260	C₄²	—	86	1	no	—
261	C₄²	—	86	1	no	—
262	C₄²	—	86	2	no	—
263	C₄²	—	172	2	no	—
264	C₄²	—	172	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	#13063283	1	1	yes	characteristic
267	C₂³	#13063300	1	1	yes	characteristic
268	C₂³	#13063301	1	1	yes	characteristic
269	C₂³	#13063310	1	2	yes	—
270	C₂³	#13063316	1	2	yes	—
271	C₂×C₄	—	86	4	no	—
272	C₂×C₄	—	86	2	no	—
273	C₂×C₄	—	86	2	no	—
274	C₂×C₄	—	344	4	no	—
275	C₂×C₄	—	172	4	no	—
276	C₂×C₄	—	172	4	no	—
277	C₂×C₄	—	172	4	no	—
278	C₂×C₄	—	172	2	no	—
279	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
281	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
282	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
283	C₂×C₄	#13063295	1	1	yes	characteristic
284	C₂×C₄	#13063296	1	1	yes	characteristic
285	C₂×C₄	#13063297	1	2	yes	—
286	C₂×C₄	#13063298	1	2	yes	—
287	C₂×C₄	#13063302	1	2	yes	—
288	C₂×C₄	#13063302	1	2	yes	—
289	C₂×C₄	#13063304	1	2	yes	—
290	C₂×C₄	#13063305	1	1	yes	characteristic
291	C₂×C₄	#13063309	1	1	yes	characteristic
292	C₂×C₄	#13063313	1	1	yes	characteristic
293	C₂×C₄	#13063315	1	2	yes	—
294	C₂×C₄	#13063318	1	2	yes	—
295	C₂×C₄	#13063324	1	2	yes	—
296	C₂×C₄	#13063325	1	1	yes	characteristic
297	C₈	—	2	4	no	—
298	C₈	—	4	4	no	—
299	C₈	—	172	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	#14582756	1	1	yes	characteristic
306	C₂²	#14582786	1	2	yes	—
307	C₂²	#14582817	1	1	yes	characteristic
308	C₂²	#14582831	1	1	yes	characteristic
309	C₂²	#14582832	1	2	yes	—
310	C₄	—	86	4	no	—
311	C₄	—	344	4	no	—
312	C₄	—	172	2	no	—
313	C₄	#14582776	1	1	yes	characteristic
314	C₄	#14582796	1	2	yes	—
315	C₄	#14582805	1	1	yes	characteristic
316	C₄	#14582820	1	1	yes	characteristic
317	C₄	#14582822	1	2	yes	—
318	C₄	#14582839	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	#19132849	1	1	yes	characteristic
324	C₂	#19135883	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063313
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×#338	C₂
socle	C₂²×C₄₃	#394
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#338	D₄₃
hypocenter	C₄₃	#11243
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
12. #14582922	C₄	semidirect
182. C₄₃	#11243	semidirect

Maximal subgroups

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

subgroup	index
#19143185	2
C₂×#14582922	2
#19134333	2
C₂×#14582660	2
C₂×C₄₃×#338	2
#11243	43