Subgroups — #24999200

Order 11008.

562 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583249	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583190	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583176	C₂	1	1	yes	maximal, characteristic
4	#19135388	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582647	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132429	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063325	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
10	#14583249	—	2	2	no	—
11	#14583249	—	2	2	no	—
12	#14583190	—	2	2	no	—
13	#14583190	—	2	2	no	—
14	#14583176	—	2	2	no	—
15	#14583176	—	2	2	no	—
16	C₂×#13063258	—	2	1	no	—
17	C₂×#13063257	—	2	1	no	—
18	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×#13063249	—	2	2	no	—
20	C₂×#13063247	—	2	1	no	—
21	C₂×#13063245	—	2	1	no	—
22	C₂×(C₂×D₄₃)⋊C₈	—	2	2	no	—
23	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
24	#14582647	—	2	2	no	—
25	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#332	—	2	2	no	—
29	C₂²×Dic₈₆	—	2	1	no	—
30	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₃	—	2	1	no	—
32	C₂²×C₄₃⋊C₈	—	2	1	no	—
33	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
34	#13063325	—	2	1	no	—
35	#13063325	—	2	1	no	—
36	#13063325	—	2	1	no	—
37	#13063325	D₄	1	2	yes	—
38	C₄×D₁₇₂	—	2	1	no	—
39	C₄×D₁₇₂	—	2	1	no	—
40	C₄×D₁₇₂	—	2	1	no	—
41	C₄×D₁₇₂	—	2	1	no	—
42	C₂×C₄⋊Dic₄₃	—	2	1	no	—
43	C₂×C₄⋊Dic₄₃	—	2	1	no	—
44	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
45	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
46	#13063258	—	2	4	no	—
47	#13063257	—	2	4	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
50	#13063249	—	2	8	no	—
51	#13063247	—	2	4	no	—
52	#13063245	—	2	4	no	—
53	(C₂×D₄₃)⋊C₈	—	4	4	no	—
54	C₄×C₄₃⋊C₈	—	2	1	no	—
55	C₄×C₄₃⋊C₈	—	2	1	no	—
56	C₂²×Q₈×C₄₃	—	2	1	no	—
57	C₂²×C₈×C₄₃	—	2	1	no	—
58	C₂²×C₈×C₄₃	—	2	1	no	—
59	C₄⋊Q₈×C₄₃	—	2	1	no	—
60	C₄⋊Q₈×C₄₃	—	2	1	no	—
61	C₄⋊Q₈×C₄₃	—	2	1	no	—
62	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₃	—	2	1	no	—
67	C₄⋊C₈×C₄₃	—	2	1	no	—
68	C₄⋊C₈×C₄₃	—	2	1	no	—
69	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
70	C₂³×D₄₃	—	2	1	no	—
71	C₂×Q₈×C₄₃	—	2	2	no	—
72	C₂×Q₈×C₄₃	—	4	1	no	—
73	C₂×Q₈×C₄₃	—	4	2	no	—
74	C₂×Q₈×C₄₃	—	4	2	no	—
75	C₂²×C₄×C₄₃	—	4	1	no	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
79	C₂²×Dic₄₃	—	2	1	no	—
80	C₂²×Dic₄₃	—	4	1	no	—
81	C₂×D₁₇₂	—	2	1	no	—
82	C₂×D₁₇₂	—	2	1	no	—
83	C₂×D₁₇₂	—	2	1	no	—
84	C₂×D₁₇₂	—	2	1	no	—
85	C₂×D₁₇₂	—	2	1	no	—
86	C₂×D₁₇₂	Q₁₆	1	2	yes	—
87	C₂×C₄×D₄₃	—	4	1	no	—
88	C₂×C₄×D₄₃	—	4	1	no	—
89	C₂×C₄×D₄₃	—	4	2	no	—
90	C₂×C₄×D₄₃	—	4	2	no	—
91	C₂×Dic₈₆	—	2	2	no	—
92	C₂×Dic₈₆	—	4	1	no	—
93	C₂×Dic₈₆	—	8	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	4	1	no	—
96	C₂×C₈×C₄₃	—	4	1	no	—
97	C₂×C₈×C₄₃	—	4	1	no	—
98	C₂×C₈×C₄₃	—	4	2	no	—
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	2	2	no	—
101	C₄⋊C₄×C₄₃	—	4	1	no	—
102	C₄⋊C₄×C₄₃	—	4	1	no	—
103	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
107	(C₂×D₄₃)⋊C₄	—	4	1	no	—
108	(C₂×D₄₃)⋊C₄	—	4	1	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	2	1	no	—
112	C₄⋊Dic₄₃	—	2	1	no	—
113	C₄⋊Dic₄₃	—	4	1	no	—
114	C₄⋊Dic₄₃	—	4	1	no	—
115	C₂×C₄₃⋊C₈	—	2	2	no	—
116	C₂×C₄₃⋊C₈	—	2	2	no	—
117	C₂×C₄₃⋊C₈	—	2	2	no	—
118	C₂×C₄₃⋊C₈	—	2	2	no	—
119	C₂×C₄₃⋊C₈	—	4	1	no	—
120	C₂×C₄₃⋊C₈	—	4	1	no	—
121	C₂³×C₄₃	#121	1	1	yes	characteristic
122	C₂²×D₄₃	—	2	2	no	—
123	C₂²×D₄₃	—	4	1	no	—
124	C₂²×D₄₃	—	4	2	no	—
125	C₂²×D₄₃	—	4	1	no	—
126	C₂²×D₄₃	—	4	2	no	—
127	Q₈×C₄₃	—	4	4	no	—
128	Q₈×C₄₃	—	8	1	no	—
129	Q₈×C₄₃	—	8	1	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	—	8	2	no	—
133	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#135	1	2	yes	—
141	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
142	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
145	C₂×Dic₄₃	—	4	2	no	—
146	C₂×Dic₄₃	—	4	1	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	8	2	no	—
149	D₁₇₂	—	2	2	no	—
150	D₁₇₂	—	2	2	no	—
151	D₁₇₂	—	4	1	no	—
152	D₁₇₂	—	4	1	no	—
153	C₄×D₄₃	—	8	2	no	—
154	C₄×D₄₃	—	8	1	no	—
155	C₄×D₄₃	—	8	1	no	—
156	Dic₈₆	—	8	2	no	—
157	Dic₈₆	—	8	2	no	—
158	C₈×C₄₃	—	4	2	no	—
159	C₈×C₄₃	—	8	1	no	—
160	C₄₃⋊C₈	—	4	2	no	—
161	C₄₃⋊C₈	—	4	2	no	—
162	#10275	—	43	1	no	maximal, Sylow
163	C₂²×C₄₃	#392	1	1	yes	characteristic
164	C₂²×C₄₃	#463	1	2	yes	—
165	C₂²×C₄₃	#467	1	1	yes	characteristic
166	C₂²×C₄₃	#470	1	1	yes	characteristic
167	C₂²×C₄₃	#471	1	2	yes	—
168	C₂×D₄₃	—	4	4	no	—
169	C₂×D₄₃	—	4	2	no	—
170	C₂×D₄₃	—	8	2	no	—
171	C₂×D₄₃	—	8	1	no	—
172	C₂×D₄₃	—	8	1	no	—
173	C₄×C₄₃	—	2	1	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	8	2	no	—
176	C₄×C₄₃	#449	1	2	yes	—
177	C₄×C₄₃	#451	1	1	yes	characteristic
178	C₄×C₄₃	#451	1	1	yes	characteristic
179	C₄×C₄₃	#452	1	1	yes	characteristic
180	C₄×C₄₃	#452	1	1	yes	characteristic
181	C₄×C₄₃	#495	1	1	yes	characteristic
182	C₄×C₄₃	#496	1	1	yes	characteristic
183	Dic₄₃	—	8	1	no	—
184	Dic₄₃	—	8	2	no	—
185	C₂×#487	—	43	1	no	—
186	C₂×#476	—	43	1	no	—
187	C₂×#474	—	43	1	no	—
188	#1954	—	43	1	no	—
189	C₂×#333	—	43	1	no	—
190	C₂×#332	D₄₃	1	1	yes	characteristic, complemented
191	#1277	—	43	1	no	—
192	C₂×C₄₃	#1683	1	2	yes	—
193	C₂×C₄₃	#1703	1	2	yes	—
194	C₂×C₄₃	#2001	1	1	yes	characteristic
195	C₂×C₄₃	#2012	1	1	yes	characteristic
196	C₂×C₄₃	#2019	1	1	yes	characteristic
197	D₄₃	—	8	2	no	—
198	C₂×C₄⋊Q₈	—	43	1	no	—
199	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
200	C₂×C₄×D₄	—	43	1	no	—
201	#487	—	86	2	no	—
202	#487	—	86	2	no	—
203	#476	—	86	2	no	—
204	#476	—	86	2	no	—
205	#474	—	86	2	no	—
206	#474	—	86	2	no	—
207	C₂×#107	—	86	1	no	—
208	C₂×#106	—	86	1	no	—
209	C₂×C₄⋊C₈	—	43	1	no	—
210	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
211	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
212	C₂×#103	—	86	2	no	—
213	C₂×#102	—	86	1	no	—
214	C₂×#102	—	86	1	no	—
215	C₂×C₂²⋊C₈	—	86	2	no	—
216	C₂×C₄×C₈	—	43	1	no	—
217	#333	—	86	2	no	—
218	#332	—	2	2	no	—
219	C₄₃	#10275	1	1	yes	characteristic, complemented, Sylow
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	—	86	1	no	—
222	C₂²×D₄	—	43	1	no	—
223	C₂³×C₄	—	86	1	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	43	2	no	—
227	C₂²×C₈	—	86	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	43	2	no	—
232	C₄⋊Q₈	—	86	1	no	—
233	C₄⋊Q₈	—	86	1	no	—
234	C₄⋊Q₈	—	86	1	no	—
235	C₄⋊Q₈	D₁₇₂	1	2	yes	—
236	C₄×D₄	—	86	1	no	—
237	C₄×D₄	—	86	1	no	—
238	C₄×D₄	—	86	1	no	—
239	C₄×D₄	—	86	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	43	1	no	—
243	C₂×C₄⋊C₄	—	86	1	no	—
244	C₂×C₄⋊C₄	—	86	1	no	—
245	C₂×C₂²⋊C₄	—	86	1	no	—
246	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
247	#107	—	86	4	no	—
248	#106	—	86	4	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	—	43	2	no	—
253	C₄⋊C₈	—	86	1	no	—
254	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
255	#103	—	86	8	no	—
256	#102	—	86	4	no	—
257	#102	—	86	4	no	—
258	C₂²⋊C₈	—	172	4	no	—
259	C₄×C₈	—	86	1	no	—
260	C₄×C₈	—	86	1	no	—
261	C₂⁴	—	86	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	—	4	2	no	—
266	C₂×Q₈	—	86	2	no	—
267	C₂×Q₈	—	344	2	no	—
268	C₂×Q₈	—	172	1	no	—
269	C₂×D₄	—	43	2	no	—
270	C₂×D₄	—	86	1	no	—
271	C₂×D₄	—	86	1	no	—
272	C₂×D₄	—	86	1	no	—
273	C₂×D₄	—	86	1	no	—
274	C₂×D₄	—	86	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	86	1	no	—
277	C₂²×C₄	—	172	1	no	—
278	C₂²×C₄	—	172	1	no	—
279	C₂²×C₄	—	172	1	no	—
280	C₂²×C₄	—	172	2	no	—
281	C₂²×C₄	—	172	2	no	—
282	C₂²×C₄	#10621898	1	1	yes	characteristic
283	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	—	86	2	no	—
291	C₂×C₈	—	86	2	no	—
292	C₂×C₈	—	86	2	no	—
293	C₂×C₈	—	86	2	no	—
294	C₂×C₈	—	172	1	no	—
295	C₂×C₈	—	172	1	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	86	2	no	—
301	C₄⋊C₄	—	86	1	no	—
302	C₄⋊C₄	—	86	2	no	—
303	C₄⋊C₄	—	86	1	no	—
304	C₄⋊C₄	—	172	1	no	—
305	C₄⋊C₄	—	172	1	no	—
306	C₂²⋊C₄	—	172	1	no	—
307	C₂²⋊C₄	—	172	1	no	—
308	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
309	C₄²	D₄×D₄₃	1	1	yes	characteristic
310	C₄²	D₄×D₄₃	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
312	C₂³	—	86	2	no	—
313	C₂³	—	172	1	no	—
314	C₂³	—	172	2	no	—
315	C₂³	—	172	1	no	—
316	C₂³	—	172	2	no	—
317	C₂³	#13063298	1	1	yes	characteristic
318	Q₈	—	4	4	no	—
319	Q₈	—	8	1	no	—
320	Q₈	—	8	1	no	—
321	Q₈	—	344	2	no	—
322	Q₈	—	344	2	no	—
323	D₄	—	86	2	no	—
324	D₄	—	86	2	no	—
325	D₄	—	172	1	no	—
326	D₄	—	172	1	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	2	no	—
330	C₂×C₄	—	344	2	no	—
331	C₂×C₄	—	344	2	no	—
332	C₂×C₄	—	344	1	no	—
333	C₂×C₄	—	344	1	no	—
334	C₂×C₄	—	172	1	no	—
335	C₂×C₄	—	172	2	no	—
336	C₂×C₄	—	172	2	no	—
337	C₂×C₄	#13063293	1	1	yes	characteristic
338	C₂×C₄	#13063295	1	1	yes	characteristic
339	C₂×C₄	#13063295	1	1	yes	characteristic
340	C₂×C₄	#13063298	1	1	yes	characteristic
341	C₂×C₄	#13063299	1	1	yes	characteristic
342	C₂×C₄	#13063321	1	1	yes	characteristic
343	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
344	C₂×C₄	#13063346	1	2	yes	—
345	C₂×C₄	#13063346	1	1	yes	characteristic
346	C₂×C₄	#13063349	1	2	yes	—
347	C₂×C₄	#13063354	1	1	yes	characteristic
348	C₂×C₄	#13063357	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	8	1	no	—
351	C₈	—	172	2	no	—
352	C₈	—	172	2	no	—
353	C₂²	—	344	1	no	—
354	C₂²	—	344	2	no	—
355	C₂²	—	344	1	no	—
356	C₂²	—	172	4	no	—
357	C₂²	—	172	2	no	—
358	C₂²	#14582808	1	1	yes	characteristic
359	C₂²	#14583113	1	2	yes	—
360	C₂²	#14583139	1	1	yes	characteristic
361	C₂²	#14583156	1	1	yes	characteristic
362	C₂²	#14583165	1	2	yes	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	344	1	no	—
367	C₄	—	344	2	no	—
368	C₄	#14583045	1	2	yes	—
369	C₄	#14583053	1	1	yes	characteristic
370	C₄	#14583056	1	1	yes	characteristic
371	C₄	#14583058	1	1	yes	characteristic
372	C₄	#14583064	1	1	yes	characteristic
373	C₄	#14583280	1	1	yes	characteristic
374	C₄	#14583287	1	1	yes	characteristic
375	C₂	—	344	2	no	—
376	C₂	#19134045	1	2	yes	—
377	C₂	#19134169	1	2	yes	—
378	C₂	#19135604	1	1	yes	characteristic
379	C₂	#19135674	1	1	yes	characteristic
380	C₂	#19135708	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063298
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#332	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂²×D₁₇₂	D₄
hypercenter	C₂×#332	D₄₃
hypocenter	C₄₃	#10275
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
1. C₂×#14583249	C₂	semidirect
5. C₂×#14582647	C₂	semidirect
6. #19132429	C₂	semidirect
7. C₂×C₄₃×#332	C₂	semidirect
190. C₂×#332	D₄₃	semidirect
219. C₄₃	#10275	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583249	2
C₂×#14583190	2
C₂×#14583176	2
#19135388	2
C₂×#14582647	2
#19132429	2
C₂×C₄₃×#332	2
#10275	43