Subgroups — #25250313

Order 11008.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143570	C₂	1	1	yes	maximal, characteristic
2	#19136500	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136497	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136426	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136420	C₂	1	2	yes	maximal, complemented
6	#19136414	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₆×C₂²⋊D₄₃	C₂	1	2	yes	maximal, complemented
8	#19136401	C₂	1	2	yes	maximal, complemented
9	#19136394	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136352	C₂	1	2	yes	maximal, complemented
11	C₂×C₄₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583433	C₄	1	1	yes	characteristic
13	C₂×#13063264	C₂²	1	2	yes	—
14	C₂×C₁₆×D₄₃	C₂²	1	2	yes	—
15	#14583029	C₄	1	1	yes	characteristic
16	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
17	#14582978	C₄	1	1	yes	characteristic
18	#14582970	C₄	1	2	yes	—
19	C₈×D₁₇₂	C₂²	1	1	yes	characteristic
20	C₈×C₂²⋊D₄₃	C₂²	1	2	yes	—
21	#14582958	C₄	1	2	yes	—
22	#14582956	C₄	1	1	yes	characteristic
23	#14582853	C₂²	1	2	yes	—
24	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
26	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
27	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
28	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
29	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
30	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
31	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
32	#14582674	C₂²	1	2	yes	—
33	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₄₃	C₂²	1	2	yes	—
36	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₂×#10621872	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₄₃	C₂³	1	2	yes	—
43	#13063319	C₈	1	2	yes	—
44	#13063317	C₈	1	1	yes	characteristic
45	#13063313	C₈	1	1	yes	characteristic
46	#13063308	C₈	1	2	yes	—
47	#13063304	C₈	1	1	yes	characteristic
48	#13063299	C₈	1	1	yes	characteristic
49	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
50	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
52	#13063282	C₂×C₄	1	2	yes	—
53	#13063264	—	4	2	no	—
54	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₁₆×D₄₃	—	4	2	no	—
59	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
60	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
61	Dic₄₃⋊C₈	C₂×C₄	1	2	yes	—
62	Dic₄₃⋊C₈	C₂×C₄	1	2	yes	—
63	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
64	(C₂×D₄₃)⋊C₈	C₂×C₄	1	2	yes	—
65	(C₂×D₄₃)⋊C₈	C₂×C₄	1	2	yes	—
66	#13063236	C₂×C₄	1	2	yes	—
67	#13063235	C₂×C₄	1	2	yes	—
68	C₈×Dic₄₃	C₂³	1	2	yes	—
69	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₄₃	—	4	2	no	—
73	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
75	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₂²⋊D₄₃	C₂×C₈	1	2	yes	—
83	C₂×D₁₇₂	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
86	C₂×C₈×C₄₃	—	2	1	no	—
87	C₂×C₈×C₄₃	—	2	1	no	—
88	C₂×C₈×C₄₃	—	4	1	no	—
89	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²⋊Dic₄₃	C₂×C₈	1	2	yes	—
98	(C₂×D₄₃)⋊C₄	C₂×C₈	1	2	yes	—
99	(C₂×D₄₃)⋊C₄	C₂×C₈	1	2	yes	—
100	(C₂×D₄₃)⋊C₄	C₂×C₈	1	2	yes	—
101	(C₂×D₄₃)⋊C₄	C₂×C₈	1	2	yes	—
102	C₄⋊Dic₄₃	C₂×C₈	1	1	yes	characteristic
103	C₄⋊Dic₄₃	C₂×C₈	1	1	yes	characteristic
104	Dic₄₃⋊C₄	C₂×C₈	1	2	yes	—
105	Dic₄₃⋊C₄	C₂×C₈	1	2	yes	—
106	Dic₄₃⋊C₄	C₂×C₈	1	2	yes	—
107	Dic₄₃⋊C₄	C₂×C₈	1	2	yes	—
108	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
109	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
111	#10621872	—	4	2	no	—
112	C₈×D₄₃	—	2	2	no	—
113	C₈×D₄₃	—	2	2	no	—
114	C₁₆×C₄₃	—	2	2	no	—
115	C₁₆×C₄₃	—	2	2	no	—
116	C₄₃⋊C₁₆	—	2	1	no	—
117	C₄₃⋊C₁₆	—	2	1	no	—
118	C₄₃⋊C₁₆	—	2	1	no	—
119	C₄₃⋊C₁₆	—	2	1	no	—
120	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
121	C₂²×D₄₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
132	C₂²⋊D₄₃	—	2	4	no	—
133	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
134	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
135	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
136	D₁₇₂	—	2	2	no	—
137	C₄×D₄₃	—	2	2	no	—
138	C₄×D₄₃	—	2	2	no	—
139	Dic₈₆	—	2	2	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	—	2	1	no	—
143	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
145	C₄₃⋊C₈	—	2	2	no	—
146	C₄₃⋊C₈	—	2	2	no	—
147	#33508	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	2	1	no	—
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	#504	1	2	yes	—
151	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
152	C₂×D₄₃	—	2	2	no	—
153	C₂×D₄₃	—	2	2	no	—
154	C₂×D₄₃	—	2	2	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	#504	1	2	yes	—
160	C₄×C₄₃	#520	1	1	yes	characteristic
161	C₄×C₄₃	#568	1	1	yes	characteristic
162	Dic₄₃	—	2	2	no	—
163	Dic₄₃	—	2	2	no	—
164	Dic₄₃	—	2	2	no	—
165	#2966	—	43	1	no	—
166	#2183	—	43	1	no	—
167	Q₈×C₁₆	—	43	1	no	—
168	#2169	—	43	1	no	—
169	#2168	—	43	2	no	—
170	D₄×C₁₆	—	43	2	no	—
171	D₄×C₁₆	—	43	1	no	—
172	#2164	—	43	2	no	—
173	#2162	—	43	1	no	—
174	#2151	—	43	2	no	—
175	C₂×C₄⋊C₁₆	D₄₃	1	1	yes	characteristic, complemented
176	C₂×C₄₃	—	2	2	no	—
177	C₂×C₄₃	#2988	1	1	yes	characteristic
178	C₂×C₄₃	C₂×#504	1	1	yes	characteristic
179	C₂×C₄₃	C₂×#504	1	1	yes	characteristic
180	D₄₃	—	4	2	no	—
181	#529	—	43	1	no	—
182	C₂×M₃₂	—	43	2	no	—
183	C₂²×C₁₆	—	43	2	no	—
184	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
185	#446	—	43	1	no	—
186	C₈×Q₈	—	43	1	no	—
187	#436	—	43	1	no	—
188	#435	—	43	2	no	—
189	C₈×D₄	—	43	2	no	—
190	C₈×D₄	—	43	1	no	—
191	#433	—	43	2	no	—
192	#432	—	43	1	no	—
193	#405	—	43	2	no	—
194	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
195	C₄⋊C₁₆	—	43	2	no	—
196	C₄⋊C₁₆	—	43	2	no	—
197	C₄⋊C₁₆	—	43	1	no	—
198	C₄⋊C₁₆	—	43	1	no	—
199	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
200	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
201	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
202	C₂²⋊C₁₆	—	43	1	no	—
203	C₂²⋊C₁₆	—	43	1	no	—
204	C₂²⋊C₁₆	—	43	2	no	—
205	C₂²⋊C₁₆	—	43	2	no	—
206	#346	—	43	2	no	—
207	C₄×C₁₆	—	43	2	no	—
208	C₄×C₁₆	—	43	1	no	—
209	C₄×C₁₆	—	43	1	no	—
210	C₄₃	#33508	1	1	yes	characteristic, complemented, Sylow
211	C₂×M₁₆	—	43	2	no	—
212	C₂²×C₈	—	43	2	no	—
213	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
215	#126	—	43	2	no	—
216	#125	—	43	1	no	—
217	#124	—	43	1	no	—
218	#123	—	43	2	no	—
219	C₂²⋊Q₈	—	43	1	no	—
220	#121	—	43	1	no	—
221	C₄×Q₈	—	43	1	no	—
222	C₄×D₄	—	43	2	no	—
223	C₄×D₄	—	43	1	no	—
224	#117	—	43	2	no	—
225	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
226	M₃₂	—	172	2	no	—
227	C₂×C₁₆	—	4	2	no	—
228	C₂×C₁₆	—	43	1	no	—
229	C₂×C₁₆	—	43	1	no	—
230	C₂×C₁₆	—	43	1	no	—
231	C₂×C₁₆	—	43	1	no	—
232	C₂×C₁₆	—	172	2	no	—
233	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
234	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
235	C₄⋊C₈	—	43	2	no	—
236	C₄⋊C₈	—	43	2	no	—
237	C₄⋊C₈	—	43	1	no	—
238	C₄⋊C₈	—	43	1	no	—
239	C₂²⋊C₈	—	43	2	no	—
240	C₂²⋊C₈	—	43	2	no	—
241	C₂²⋊C₈	—	43	2	no	—
242	#97	—	43	2	no	—
243	#97	—	43	2	no	—
244	C₄×C₈	—	43	2	no	—
245	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
246	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
249	C₂×Q₈	—	43	1	no	—
250	C₂×D₄	—	43	2	no	—
251	C₂×D₄	—	43	1	no	—
252	C₂²×C₄	—	43	2	no	—
253	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
254	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
256	M₁₆	—	172	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	43	2	no	—
261	C₂×C₈	—	43	2	no	—
262	C₂×C₈	—	86	2	no	—
263	C₂×C₈	—	86	2	no	—
264	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
267	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
268	C₄⋊C₄	—	43	2	no	—
269	C₄⋊C₄	—	43	2	no	—
270	C₄⋊C₄	—	43	1	no	—
271	C₄⋊C₄	—	43	2	no	—
272	C₄⋊C₄	—	43	1	no	—
273	C₄⋊C₄	—	43	2	no	—
274	C₂²⋊C₄	—	43	2	no	—
275	C₂²⋊C₄	—	43	2	no	—
276	C₂²⋊C₄	—	43	2	no	—
277	C₂²⋊C₄	—	43	2	no	—
278	C₂²⋊C₄	—	43	2	no	—
279	C₄²	—	43	2	no	—
280	C₄²	C₈×D₄₃	1	1	yes	characteristic
281	C₄²	C₈×D₄₃	1	1	yes	characteristic
282	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
283	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	2	no	—
286	C₁₆	—	86	1	no	—
287	C₁₆	—	86	1	no	—
288	C₁₆	—	86	1	no	—
289	C₁₆	—	86	1	no	—
290	C₂³	—	43	2	no	—
291	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
292	Q₈	—	86	2	no	—
293	D₄	—	86	4	no	—
294	D₄	—	86	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	43	2	no	—
300	C₂×C₄	—	43	2	no	—
301	C₂×C₄	—	43	2	no	—
302	C₂×C₄	—	86	2	no	—
303	C₂×C₄	—	86	2	no	—
304	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	86	2	no	—
314	C₈	—	86	2	no	—
315	C₈	#13063370	1	1	yes	characteristic
316	C₈	#13063373	1	1	yes	characteristic
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	86	2	no	—
320	C₂²	—	86	2	no	—
321	C₂²	—	86	2	no	—
322	C₂²	#14583317	1	2	yes	—
323	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	86	2	no	—
329	C₄	—	86	2	no	—
330	C₄	—	86	2	no	—
331	C₄	#14583315	1	2	yes	—
332	C₄	#14583380	1	1	yes	characteristic
333	C₄	#14583665	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	172	2	no	—
336	C₂	#19143704	1	1	yes	characteristic
337	C₂	C₂×#14583315	1	1	yes	characteristic
338	C₂	C₂×#14583317	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	C₂×C₄₃×C₄⋊C₁₆	C₂
prime radical	#13063299	C₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₄₃
hypocenter	C₄₃	#33508
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #19136500	C₂	semidirect
3. #19136497	C₂	semidirect
4. #19136426	C₂	semidirect
5. #19136420	C₂	semidirect
6. #19136414	C₂	semidirect
7. C₁₆×C₂²⋊D₄₃	C₂	semidirect
8. #19136401	C₂	semidirect
9. #19136394	C₂	semidirect
10. #19136352	C₂	semidirect
11. C₂×C₄₃×C₄⋊C₁₆	C₂	semidirect
175. C₂×C₄⋊C₁₆	D₄₃	semidirect
210. C₄₃	#33508	semidirect

Maximal subgroups

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

subgroup	index
#19143570	2
#19136500	2
#19136497	2
#19136426	2
#19136420	2
#19136414	2
C₁₆×C₂²⋊D₄₃	2
#19136401	2
#19136394	2
#19136352	2
C₂×C₄₃×C₄⋊C₁₆	2
#33508	43