Subgroups — #25014827

Order 11008.

590 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583274	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582701	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582700	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582699	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582674	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134758	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
9	C₂²×#10621882	—	2	1	no	—
10	C₂×#13063321	—	2	1	no	—
11	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14583274	—	2	2	no	—
13	#14583274	C₄	1	4	yes	complemented
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063256	—	2	1	no	—
16	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
17	#14582764	—	2	1	no	—
18	#14582701	—	2	2	no	—
19	#14582700	—	2	2	no	—
20	#14582699	—	2	2	no	—
21	#14582674	—	2	2	no	—
22	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#357	—	2	2	no	—
26	C₂²×D₁₇₂	—	2	1	no	—
27	C₂²×C₂²⋊D₄₃	—	4	2	no	—
28	C₂×D₃₄₄	—	2	1	no	—
29	C₂×D₃₄₄	—	2	1	no	—
30	C₂×D₃₄₄	—	2	1	no	—
31	C₂×D₃₄₄	—	2	1	no	—
32	C₂×D₃₄₄	—	2	1	no	—
33	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
34	C₂×#10621882	—	2	4	no	—
35	C₂×#10621882	—	4	2	no	—
36	C₂×#10621882	—	4	2	no	—
37	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
38	#13063321	—	2	4	no	—
39	#13063321	—	4	2	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
42	C₂×C₄₃⋊C₁₆	—	2	1	no	—
43	C₂×C₄₃⋊C₁₆	—	2	1	no	—
44	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	complemented
45	#13063259	—	2	1	no	—
46	#13063259	—	2	1	no	—
47	#13063256	—	4	2	no	—
48	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₄₃	D₄	1	2	yes	—
50	C₄₃⋊#95	—	4	2	no	—
51	C₂²×D₄×C₄₃	—	2	1	no	—
52	C₂×D₈×C₄₃	—	2	1	no	—
53	C₂×D₈×C₄₃	—	2	1	no	—
54	C₂×D₈×C₄₃	—	2	1	no	—
55	C₂×D₈×C₄₃	—	2	1	no	—
56	C₂×D₈×C₄₃	—	2	1	no	—
57	C₂×D₈×C₄₃	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
60	C₂×C₁₆×C₄₃	—	2	1	no	—
61	C₂×C₁₆×C₄₃	—	2	1	no	—
62	C₂×C₁₆×C₄₃	D₄	1	2	yes	complemented
63	#107×C₄₃	—	2	1	no	—
64	#107×C₄₃	—	2	1	no	—
65	C₂⁴×C₄₃	—	4	1	no	—
66	C₂³×D₄₃	—	4	1	no	—
67	C₂×D₄×C₄₃	—	2	2	no	—
68	C₂×D₄×C₄₃	—	4	1	no	—
69	C₂×D₄×C₄₃	—	4	1	no	—
70	C₂×D₄×C₄₃	—	4	1	no	—
71	C₂×D₄×C₄₃	—	4	1	no	—
72	C₂×D₄×C₄₃	—	4	1	no	—
73	C₂²×C₄×C₄₃	—	4	1	no	—
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₄₃	—	4	8	no	—
76	C₂×C₂²⋊D₄₃	—	8	4	no	—
77	C₂×C₂²⋊D₄₃	—	8	4	no	—
78	C₂²×Dic₄₃	—	4	1	no	—
79	C₂²×Dic₄₃	D₈	1	2	yes	—
80	C₂×D₁₇₂	—	2	2	no	—
81	C₂×D₁₇₂	—	4	1	no	—
82	C₂×D₁₇₂	—	4	1	no	—
83	C₂×D₁₇₂	—	4	1	no	—
84	C₂×D₁₇₂	—	4	1	no	—
85	C₂×D₁₇₂	—	4	1	no	—
86	D₈×C₄₃	—	2	2	no	—
87	D₈×C₄₃	—	2	2	no	—
88	D₈×C₄₃	—	4	1	no	—
89	D₈×C₄₃	—	4	1	no	—
90	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
96	C₄⋊C₄×C₄₃	—	4	1	no	—
97	C₄⋊C₄×C₄₃	—	4	1	no	—
98	#10621882	—	4	8	no	—
99	C₄⋊Dic₄₃	—	4	1	no	—
100	C₄⋊Dic₄₃	—	4	1	no	—
101	C₄×Dic₄₃	C₂²⋊C₄	1	2	yes	—
102	C₄×Dic₄₃	SD₁₆	1	2	yes	—
103	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
104	C₂×C₄₃⋊C₈	D₈	1	2	yes	—
105	C₂×C₄₃⋊C₈	SD₁₆	1	2	yes	—
106	D₃₄₄	—	2	2	no	—
107	D₃₄₄	—	2	2	no	—
108	D₃₄₄	—	4	1	no	—
109	D₃₄₄	—	4	1	no	—
110	C₁₆×C₄₃	—	2	2	no	—
111	C₄₃⋊C₁₆	—	2	2	no	—
112	C₂³×C₄₃	—	4	2	no	—
113	C₂³×C₄₃	—	8	1	no	—
114	C₂³×C₄₃	—	8	1	no	—
115	C₂³×C₄₃	—	8	2	no	—
116	C₂³×C₄₃	—	8	2	no	—
117	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
118	C₂²×D₄₃	—	4	2	no	—
119	C₂²×D₄₃	—	8	1	no	—
120	C₂²×D₄₃	—	8	1	no	—
121	C₂²×D₄₃	—	8	2	no	—
122	C₂²×D₄₃	—	8	2	no	—
123	D₄×C₄₃	—	4	2	no	—
124	D₄×C₄₃	—	4	2	no	—
125	D₄×C₄₃	—	8	1	no	—
126	D₄×C₄₃	—	8	1	no	—
127	C₂×C₄×C₄₃	—	8	1	no	—
128	C₂×C₄×C₄₃	—	8	2	no	—
129	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂²⋊D₄₃	—	8	16	no	—
136	C₂×Dic₄₃	—	2	2	no	—
137	C₂×Dic₄₃	—	2	2	no	—
138	C₂×Dic₄₃	—	8	1	no	—
139	C₂×Dic₄₃	—	8	2	no	—
140	C₂×Dic₄₃	#102	1	4	yes	—
141	D₁₇₂	—	4	2	no	—
142	D₁₇₂	—	4	2	no	—
143	D₁₇₂	—	8	1	no	—
144	D₁₇₂	—	8	1	no	—
145	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
146	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
147	C₈×C₄₃	#121	1	1	yes	characteristic
148	C₈×C₄₃	#121	1	1	yes	characteristic
149	C₄₃⋊C₈	#102	1	4	yes	—
150	#13206	—	43	1	no	maximal, Sylow
151	C₂²×C₄₃	—	8	4	no	—
152	C₂²×C₄₃	—	8	2	no	—
153	C₂²×C₄₃	—	16	2	no	—
154	C₂²×C₄₃	—	16	1	no	—
155	C₂²×C₄₃	—	16	1	no	—
156	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
157	C₂²×C₄₃	#486	1	1	yes	characteristic
158	C₂²×C₄₃	#493	1	1	yes	characteristic
159	C₂²×C₄₃	#509	1	2	yes	—
160	C₂²×C₄₃	#509	1	2	yes	—
161	C₂×D₄₃	—	8	4	no	—
162	C₂×D₄₃	—	8	2	no	—
163	C₂×D₄₃	—	16	2	no	—
164	C₂×D₄₃	—	16	1	no	—
165	C₂×D₄₃	—	16	1	no	—
166	C₄×C₄₃	—	16	1	no	—
167	C₄×C₄₃	#390	1	1	yes	characteristic
168	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
169	C₄×C₄₃	#486	1	1	yes	characteristic
170	C₄×C₄₃	#492	1	1	yes	characteristic
171	Dic₄₃	—	2	4	no	—
172	Dic₄₃	—	16	1	no	—
173	C₂×#493	—	43	1	no	—
174	C₂×#357	—	43	1	no	—
175	C₂×#357	—	43	1	no	—
176	C₂×#357	—	43	1	no	—
177	C₂×#357	D₄₃	1	1	yes	characteristic, complemented
178	C₂×#346	—	43	1	no	—
179	#1846	—	43	1	no	—
180	C₂×C₄₃	—	16	2	no	—
181	C₂×C₄₃	#1957	1	1	yes	characteristic
182	C₂×C₄₃	#2140	1	2	yes	—
183	C₂×C₄₃	#2244	1	2	yes	—
184	C₂×C₄₃	#2250	1	1	yes	characteristic
185	C₂×C₄₃	#2250	1	1	yes	characteristic
186	D₄₃	—	16	2	no	—
187	C₂²×D₈	—	43	1	no	—
188	C₂²×D₈	—	86	1	no	—
189	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
190	C₂×#127	—	86	1	no	—
191	C₂²×C₁₆	—	43	1	no	—
192	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
193	#493	—	43	4	no	—
194	#493	—	86	2	no	—
195	C₂×#107	—	43	1	no	—
196	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
197	C₂×#106	—	86	1	no	—
198	C₂×C₄×C₈	—	43	1	no	—
199	#382	—	86	1	no	—
200	#357	—	2	2	no	—
201	#357	—	86	2	no	—
202	#357	—	86	2	no	—
203	#357	—	86	2	no	—
204	#346	—	86	2	no	—
205	C₄₃	#13206	1	1	yes	characteristic, complemented, Sylow
206	C₂²×D₄	—	2	1	no	—
207	C₂²×D₄	—	86	1	no	—
208	C₂²×D₄	—	172	2	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	43	2	no	—
215	C₂×D₈	—	86	4	no	—
216	C₂×D₈	—	86	1	no	—
217	C₂×D₈	—	86	1	no	—
218	C₂×D₈	—	86	1	no	—
219	C₂×D₈	—	86	1	no	—
220	C₂×D₈	—	86	1	no	—
221	C₂×D₈	—	172	2	no	—
222	C₂×D₈	—	172	2	no	—
223	C₂×D₈	C₄×D₄₃	1	2	yes	—
224	C₂²×C₈	—	43	1	no	—
225	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
226	#127	—	86	4	no	—
227	#127	—	172	2	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	86	1	no	—
230	C₂×C₄²	—	43	1	no	—
231	C₂×C₁₆	—	2	1	no	—
232	C₂×C₁₆	—	2	1	no	—
233	C₂×C₁₆	—	43	2	no	—
234	C₂×C₁₆	—	86	1	no	—
235	C₂×C₁₆	—	86	1	no	—
236	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	complemented
237	#107	—	2	1	no	—
238	#107	—	2	1	no	—
239	#107	—	86	1	no	—
240	#107	—	86	1	no	—
241	#106	—	172	2	no	—
242	C₄×C₈	—	43	2	no	—
243	C₄×C₈	—	43	2	no	—
244	#95	—	172	2	no	—
245	C₂⁴	—	4	1	no	—
246	C₂⁴	—	172	1	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	86	2	no	—
254	C₂×D₄	—	344	4	no	—
255	C₂×D₄	—	344	4	no	—
256	C₂×D₄	—	172	8	no	—
257	C₂×D₄	—	172	1	no	—
258	C₂×D₄	—	172	1	no	—
259	C₂×D₄	—	172	1	no	—
260	C₂×D₄	—	172	1	no	—
261	C₂×D₄	—	172	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	43	2	no	—
264	C₂²×C₄	—	172	1	no	—
265	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
266	D₈	—	2	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	4	1	no	—
269	D₈	—	4	1	no	—
270	D₈	—	86	2	no	—
271	D₈	—	86	2	no	—
272	D₈	—	172	8	no	—
273	D₈	—	172	1	no	—
274	D₈	—	172	1	no	—
275	C₂×C₈	—	43	2	no	—
276	C₂×C₈	—	43	2	no	—
277	C₂×C₈	—	43	2	no	—
278	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
279	C₂×C₈	#10621898	1	1	yes	characteristic
280	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
281	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
282	C₂×C₈	#10621900	1	1	yes	characteristic
283	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	172	1	no	—
287	C₄⋊C₄	—	172	1	no	—
288	C₄²	—	43	2	no	—
289	C₄²	—	43	2	no	—
290	C₁₆	—	2	2	no	—
291	C₁₆	—	86	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	2	no	—
295	C₂³	—	8	1	no	—
296	C₂³	—	8	1	no	—
297	C₂³	—	344	2	no	—
298	C₂³	—	344	2	no	—
299	C₂³	—	344	1	no	—
300	C₂³	—	344	1	no	—
301	C₂³	—	172	2	no	—
302	C₂³	D₈×D₄₃	1	1	yes	characteristic
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	1	no	—
306	D₄	—	8	1	no	—
307	D₄	—	344	16	no	—
308	D₄	—	344	1	no	—
309	D₄	—	344	1	no	—
310	D₄	—	172	2	no	—
311	D₄	—	172	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	—	43	4	no	—
315	C₂×C₄	—	86	2	no	—
316	C₂×C₄	—	86	2	no	—
317	C₂×C₄	—	344	2	no	—
318	C₂×C₄	—	344	1	no	—
319	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
320	C₂×C₄	#13063311	1	1	yes	characteristic
321	C₂×C₄	#13063321	1	1	yes	characteristic
322	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
323	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
324	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
325	C₈	—	43	4	no	—
326	C₈	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
327	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
328	C₈	#13063297	1	1	yes	characteristic
329	C₈	#13063298	1	1	yes	characteristic
330	C₂²	—	8	4	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	16	2	no	—
333	C₂²	—	16	1	no	—
334	C₂²	—	16	1	no	—
335	C₂²	—	344	4	no	—
336	C₂²	—	344	2	no	—
337	C₂²	—	688	2	no	—
338	C₂²	—	688	1	no	—
339	C₂²	—	688	1	no	—
340	C₂²	#102×D₄₃	1	1	yes	characteristic
341	C₂²	#14583244	1	1	yes	characteristic
342	C₂²	#14583274	1	1	yes	characteristic
343	C₂²	#14583333	1	2	yes	—
344	C₂²	#14583336	1	2	yes	—
345	C₄	—	16	1	no	—
346	C₄	—	86	4	no	—
347	C₄	—	688	1	no	—
348	C₄	#14582799	1	1	yes	characteristic
349	C₄	#102×D₄₃	1	1	yes	characteristic
350	C₄	#14583244	1	1	yes	characteristic
351	C₄	#14583270	1	1	yes	characteristic
352	C₂	—	16	2	no	—
353	C₂	—	688	2	no	—
354	C₂	#19135399	1	1	yes	characteristic
355	C₂	#19136309	1	2	yes	—
356	C₂	#19136835	1	2	yes	—
357	C₂	#19136858	1	1	yes	characteristic
358	C₂	#19136860	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#357	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂×#14583274	C₂
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#357	D₄₃
hypocenter	C₄₃	#13206
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
2. C₂×#14582701	C₂	semidirect
3. C₂×#14582700	C₂	semidirect
4. C₂×#14582699	C₂	semidirect
5. C₂×#14582674	C₂	semidirect
6. #19134758	C₂	semidirect
7. C₂×C₄₃×#357	C₂	semidirect
13. #14583274	C₄	semidirect
44. C₂×C₄₃⋊C₁₆	D₄	semidirect
62. C₂×C₁₆×C₄₃	D₄	semidirect
177. C₂×#357	D₄₃	semidirect
205. C₄₃	#13206	semidirect
236. C₂×C₁₆	C₂²⋊D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583274	2
C₂×#14582701	2
C₂×#14582700	2
C₂×#14582699	2
C₂×#14582674	2
#19134758	2
C₂×C₄₃×#357	2
#13206	43