Subgroups — #25004686

Order 11008.

682 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063259	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582925	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582667	C₂	1	2	yes	maximal
4	#19134315	C₂	1	2	yes	maximal
5	C₂×C₄₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	1	yes	characteristic
7	C₂×#13063284	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
9	C₂×#13063259	—	2	4	no	—
10	C₂×#13063259	—	2	2	no	—
11	#14582925	—	2	2	no	—
12	#14582925	C₄	1	4	yes	complemented
13	C₂×Dic₄₃⋊C₈	—	2	2	no	—
14	#14582667	—	2	4	no	—
15	#14582662	—	2	4	no	—
16	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
17	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#341	—	2	2	no	—
19	C₂³×Dic₄₃	—	2	2	no	—
20	C₂×#10621877	—	2	1	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	C₂×C₄	1	2	yes	—
24	C₂²×C₄₃⋊C₈	—	2	1	no	—
25	#13063284	—	2	1	no	—
26	#13063284	—	2	1	no	—
27	#13063284	C₂×C₄	1	2	yes	—
28	#13063284	Q₈	1	2	yes	—
29	C₂×C₄⋊Dic₄₃	—	2	2	no	—
30	C₂×C₄⋊Dic₄₃	—	2	2	no	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄⋊Dic₄₃	—	2	2	no	—
34	C₂×C₄⋊Dic₄₃	D₄	1	4	yes	—
35	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
36	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
37	C₂×C₄×Dic₄₃	—	2	1	no	—
38	#13063259	—	2	8	no	—
39	Dic₄₃⋊C₈	—	2	8	no	—
40	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₄₃	—	2	2	no	—
42	C₂×M₁₆×C₄₃	—	2	2	no	—
43	C₂×M₁₆×C₄₃	—	2	2	no	—
44	C₂×M₁₆×C₄₃	—	2	2	no	—
45	C₂×M₁₆×C₄₃	—	2	2	no	—
46	C₂×M₁₆×C₄₃	Q₈	1	4	yes	—
47	C₂²×C₈×C₄₃	—	2	2	no	—
48	C₂²×C₈×C₄₃	—	2	2	no	—
49	C₂²×C₈×C₄₃	—	2	2	no	—
50	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₄₃	D₄	1	2	yes	—
52	C₂²×C₈×C₄₃	D₄	1	2	yes	—
53	C₂⁴×C₄₃	#41	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	#41	1	2	yes	—
63	C₂²×C₄×C₄₃	#41	1	2	yes	—
64	C₂²×Dic₄₃	—	2	2	no	—
65	C₂²×Dic₄₃	—	2	4	no	—
66	C₂²×Dic₄₃	—	4	2	no	—
67	C₂²×Dic₄₃	—	4	2	no	—
68	C₂²×Dic₄₃	—	8	4	no	—
69	M₁₆×C₄₃	—	2	4	no	—
70	M₁₆×C₄₃	—	2	4	no	—
71	M₁₆×C₄₃	—	4	2	no	—
72	M₁₆×C₄₃	—	4	2	no	—
73	C₂×C₈×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	—	2	4	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	4	no	—
78	C₂×C₈×C₄₃	—	4	2	no	—
79	C₂×C₈×C₄₃	—	4	2	no	—
80	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
81	C₂²⋊Dic₄₃	—	2	2	no	—
82	C₂²⋊Dic₄₃	—	2	2	no	—
83	C₄⋊Dic₄₃	—	2	2	no	—
84	C₄⋊Dic₄₃	—	2	4	no	—
85	C₄⋊Dic₄₃	—	2	4	no	—
86	C₄⋊Dic₄₃	—	4	2	no	—
87	C₄⋊Dic₄₃	—	4	2	no	—
88	C₄⋊Dic₄₃	C₄⋊C₄	1	4	yes	—
89	C₄×Dic₄₃	—	2	2	no	—
90	C₄×Dic₄₃	—	2	2	no	—
91	#10621877	—	2	4	no	—
92	#10621877	—	4	2	no	—
93	C₂×C₄₃⋊C₈	—	2	2	no	—
94	C₂×C₄₃⋊C₈	—	4	1	no	—
95	C₂×C₄₃⋊C₈	—	4	1	no	—
96	C₂³×C₄₃	—	2	2	no	—
97	C₂³×C₄₃	—	2	2	no	—
98	C₂³×C₄₃	#117	1	1	yes	characteristic
99	C₂³×C₄₃	#121	1	1	yes	characteristic
100	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂³×C₄₃	#123	1	2	yes	—
102	C₂³×C₄₃	#125	1	2	yes	—
103	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#121	1	2	yes	—
110	C₂×C₄×C₄₃	#121	1	2	yes	—
111	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#125	1	2	yes	—
118	C₂×C₄×C₄₃	#126	1	2	yes	—
119	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×Dic₄₃	—	2	4	no	—
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	—	4	2	no	—
124	C₂×Dic₄₃	—	4	4	no	—
125	C₂×Dic₄₃	—	8	4	no	—
126	C₂×Dic₄₃	—	8	4	no	—
127	C₂×Dic₄₃	—	8	4	no	—
128	C₈×C₄₃	—	2	4	no	—
129	C₈×C₄₃	—	4	4	no	—
130	C₄₃⋊C₈	—	4	2	no	—
131	#11239	—	43	1	no	maximal, Sylow
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	—	2	4	no	—
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	#380	1	1	yes	characteristic
138	C₂²×C₄₃	#387	1	2	yes	—
139	C₂²×C₄₃	#394	1	1	yes	characteristic
140	C₂²×C₄₃	#398	1	1	yes	characteristic
141	C₂²×C₄₃	#398	1	2	yes	—
142	C₂²×C₄₃	#472	1	2	yes	—
143	C₂²×C₄₃	#473	1	2	yes	—
144	C₄×C₄₃	#386	1	1	yes	characteristic
145	C₄×C₄₃	#389	1	2	yes	—
146	C₄×C₄₃	#391	1	1	yes	characteristic
147	C₄×C₄₃	#395	1	1	yes	characteristic
148	C₄×C₄₃	#396	1	2	yes	—
149	C₄×C₄₃	#400	1	1	yes	characteristic
150	Dic₄₃	—	4	4	no	—
151	Dic₄₃	—	8	4	no	—
152	C₂²×#107	—	43	1	no	—
153	C₂×#424	—	43	1	no	—
154	C₂×#343	—	43	2	no	—
155	C₂×#341	D₄₃	1	1	yes	characteristic
156	#1736	—	43	2	no	—
157	C₂×C₄₃	—	2	2	no	—
158	C₂×C₄₃	—	2	2	no	—
159	C₂×C₄₃	#1441	1	1	yes	characteristic
160	C₂×C₄₃	#1816	1	2	yes	—
161	C₂×C₄₃	#2045	1	1	yes	characteristic
162	C₂×C₄₃	#2047	1	1	yes	characteristic
163	C₂×C₄₃	#2089	1	2	yes	—
164	C₂²×M₁₆	—	43	1	no	—
165	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
166	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
167	C₂×#117	—	43	1	no	—
168	C₂²×C₄⋊C₄	—	43	2	no	—
169	C₂×#107	—	86	4	no	—
170	C₂×#107	—	86	2	no	—
171	#424	—	43	4	no	—
172	#424	—	86	2	no	—
173	C₂×C₄⋊C₈	—	86	2	no	—
174	#343	—	86	4	no	—
175	#341	—	2	2	no	—
176	#340	—	86	4	no	—
177	C₄₃	#11239	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	86	2	no	—
179	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	43	2	no	—
186	C₂×M₁₆	—	86	1	no	—
187	C₂×M₁₆	—	86	2	no	—
188	C₂×M₁₆	—	86	2	no	—
189	C₂×M₁₆	Dic₈₆	1	4	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	86	1	no	—
194	C₂²×C₈	C₄×D₄₃	1	2	yes	—
195	C₂²×C₈	D₁₇₂	1	2	yes	—
196	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
197	#117	—	43	2	no	—
198	#117	—	43	2	no	—
199	#117	—	86	1	no	—
200	#117	—	86	1	no	—
201	C₂×C₄⋊C₄	—	43	2	no	—
202	C₂×C₄⋊C₄	—	43	4	no	—
203	C₂×C₄⋊C₄	—	86	2	no	—
204	C₂×C₄⋊C₄	—	86	2	no	—
205	C₂×C₄⋊C₄	—	86	2	no	—
206	C₂×C₄⋊C₄	—	86	2	no	—
207	C₂×C₄⋊C₄	—	86	2	no	—
208	C₂×C₂²⋊C₄	—	86	1	no	—
209	C₂×C₄²	—	86	1	no	—
210	#107	—	86	8	no	—
211	C₄⋊C₈	—	86	8	no	—
212	C₂⁴	#10621902	1	1	yes	characteristic
213	C₂²×C₄	—	86	2	no	—
214	C₂²×C₄	—	86	4	no	—
215	C₂²×C₄	—	344	4	no	—
216	C₂²×C₄	—	172	2	no	—
217	C₂²×C₄	—	172	2	no	—
218	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
221	C₂²×C₄	#10621898	1	2	yes	—
222	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₄₃	1	2	yes	—
224	C₂²×C₄	#10621900	1	2	yes	—
225	C₂²×C₄	#10621900	1	1	yes	characteristic
226	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	86	4	no	—
233	M₁₆	—	172	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	86	2	no	—
242	C₂×C₈	—	172	1	no	—
243	C₂×C₈	—	172	1	no	—
244	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
245	C₄⋊C₄	—	43	4	no	—
246	C₄⋊C₄	—	86	2	no	—
247	C₄⋊C₄	—	86	4	no	—
248	C₄⋊C₄	—	86	4	no	—
249	C₄⋊C₄	—	172	2	no	—
250	C₄⋊C₄	—	172	2	no	—
251	C₂²⋊C₄	—	86	2	no	—
252	C₂²⋊C₄	—	86	2	no	—
253	C₄²	—	86	2	no	—
254	C₄²	—	86	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	#13063283	1	1	yes	characteristic
258	C₂³	#13063300	1	1	yes	characteristic
259	C₂³	#13063301	1	1	yes	characteristic
260	C₂³	#13063310	1	2	yes	—
261	C₂³	#13063316	1	2	yes	—
262	C₂×C₄	—	86	4	no	—
263	C₂×C₄	—	344	4	no	—
264	C₂×C₄	—	344	4	no	—
265	C₂×C₄	—	344	4	no	—
266	C₂×C₄	—	172	4	no	—
267	C₂×C₄	—	172	2	no	—
268	C₂×C₄	—	172	2	no	—
269	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
271	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
272	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
273	C₂×C₄	#13063295	1	1	yes	characteristic
274	C₂×C₄	#13063296	1	1	yes	characteristic
275	C₂×C₄	#13063297	1	2	yes	—
276	C₂×C₄	#13063298	1	2	yes	—
277	C₂×C₄	#13063302	1	2	yes	—
278	C₂×C₄	#13063302	1	2	yes	—
279	C₂×C₄	#13063304	1	2	yes	—
280	C₂×C₄	#13063305	1	1	yes	characteristic
281	C₂×C₄	#13063309	1	1	yes	characteristic
282	C₂×C₄	#13063313	1	1	yes	characteristic
283	C₂×C₄	#13063315	1	2	yes	—
284	C₂×C₄	#13063318	1	2	yes	—
285	C₂×C₄	#13063324	1	2	yes	—
286	C₂×C₄	#13063325	1	1	yes	characteristic
287	C₈	—	2	4	no	—
288	C₈	—	4	4	no	—
289	C₈	—	172	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#14582756	1	1	yes	characteristic
296	C₂²	#14582786	1	2	yes	—
297	C₂²	#14582817	1	1	yes	characteristic
298	C₂²	#14582831	1	1	yes	characteristic
299	C₂²	#14582832	1	2	yes	—
300	C₂²	#14583170	1	2	yes	—
301	C₂²	#14583175	1	2	yes	—
302	C₄	—	344	4	no	—
303	C₄	—	172	4	no	—
304	C₄	#14582776	1	1	yes	characteristic
305	C₄	#14582796	1	2	yes	—
306	C₄	#14582805	1	1	yes	characteristic
307	C₄	#14582820	1	1	yes	characteristic
308	C₄	#14582822	1	2	yes	—
309	C₄	#14582839	1	1	yes	characteristic
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	#19132849	1	1	yes	characteristic
313	C₂	#19134656	1	2	yes	—
314	C₂	#19135852	1	1	yes	characteristic
315	C₂	#19135864	1	1	yes	characteristic
316	C₂	#19136102	1	2	yes	—

	subgroup	G/H
center	C₂³	#13063300
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×#341	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#341	D₄₃
hypocenter	C₄₃	#11239
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
12. #14582925	C₄	semidirect
177. C₄₃	#11239	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063259	2
C₂×#14582925	2
C₂×#14582667	2
#19134315	2
C₂×C₄₃×#341	2
#11239	43