Subgroups — #25013671

Order 11008.

554 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063268	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582703	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582699	C₂	1	1	yes	maximal, characteristic
5	#19132686	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132673	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1969	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₇₂	—	2	1	no	—
10	C₂×#13063268	—	2	2	no	—
11	C₂×#13063268	—	2	2	no	—
12	C₂×#13063268	C₄	1	4	yes	complemented
13	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×D₃₄₄	—	2	2	no	—
16	C₄×D₃₄₄	—	2	2	no	—
17	C₂×#13063259	C₂²	1	1	yes	characteristic
18	C₂×#13063247	—	2	1	no	—
19	#14582703	—	2	2	no	—
20	#14582699	—	2	2	no	—
21	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#103×C₄₃	—	2	1	no	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#391	—	2	1	no	—
26	C₂²×D₁₇₂	—	2	1	no	—
27	C₂²×C₄×D₄₃	—	4	1	no	—
28	C₂×D₃₄₄	—	2	1	no	—
29	C₂×D₃₄₄	—	2	1	no	—
30	C₂×D₃₄₄	—	2	1	no	—
31	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
32	C₂×D₃₄₄	D₄	1	2	yes	—
33	C₂×D₃₄₄	D₄	1	2	yes	—
34	C₄×D₁₇₂	—	2	4	no	—
35	C₄×D₁₇₂	—	4	2	no	—
36	C₂×C₄⋊Dic₄₃	—	2	1	no	—
37	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
38	#13063268	—	2	8	no	—
39	C₂×C₄₃⋊C₁₆	—	2	1	no	—
40	C₂×C₄₃⋊C₁₆	—	2	1	no	—
41	C₂×C₄₃⋊C₁₆	—	2	1	no	—
42	C₂×C₄₃⋊C₁₆	—	2	1	no	—
43	C₂×C₄₃⋊C₁₆	—	2	1	no	—
44	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
45	#13063259	—	2	1	no	—
46	#13063259	D₄	1	2	yes	—
47	#13063247	—	2	4	no	—
48	C₂²×Q₈×C₄₃	—	2	1	no	—
49	C₂×Q₁₆×C₄₃	—	2	1	no	—
50	C₂×Q₁₆×C₄₃	—	2	1	no	—
51	C₂×Q₁₆×C₄₃	—	2	1	no	—
52	C₂×Q₁₆×C₄₃	—	2	1	no	—
53	C₂×Q₁₆×C₄₃	—	2	1	no	—
54	C₂×Q₁₆×C₄₃	C₂×C₄	1	2	yes	complemented
55	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
58	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
59	#107×C₄₃	—	2	1	no	—
60	#107×C₄₃	—	2	1	no	—
61	#103×C₄₃	—	4	2	no	—
62	C₄×C₈×C₄₃	—	2	1	no	—
63	C₄×C₈×C₄₃	—	2	1	no	—
64	#95×C₄₃	—	4	1	no	—
65	#95×C₄₃	—	4	1	no	—
66	C₂³×D₄₃	—	4	1	no	—
67	C₂×Q₈×C₄₃	—	2	2	no	—
68	C₂×Q₈×C₄₃	—	4	1	no	—
69	C₂×Q₈×C₄₃	—	4	1	no	—
70	C₂×Q₈×C₄₃	—	4	1	no	—
71	C₂×Q₈×C₄₃	—	4	1	no	—
72	C₂×Q₈×C₄₃	—	4	1	no	—
73	C₂²×C₄×C₄₃	—	4	1	no	—
74	C₂²×C₄×C₄₃	—	4	1	no	—
75	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₄₃	—	4	1	no	—
79	C₂×D₁₇₂	—	2	2	no	—
80	C₂×D₁₇₂	—	2	2	no	—
81	C₂×D₁₇₂	—	2	2	no	—
82	C₂×D₁₇₂	—	4	1	no	—
83	C₂×D₁₇₂	—	4	1	no	—
84	C₂×D₁₇₂	—	4	1	no	—
85	C₂×C₄×D₄₃	—	4	4	no	—
86	C₂×C₄×D₄₃	—	8	2	no	—
87	C₂×C₄×D₄₃	—	8	2	no	—
88	Q₁₆×C₄₃	—	2	2	no	—
89	Q₁₆×C₄₃	—	2	2	no	—
90	Q₁₆×C₄₃	—	4	1	no	—
91	Q₁₆×C₄₃	—	4	1	no	—
92	C₂×C₈×C₄₃	—	2	1	no	—
93	C₂×C₈×C₄₃	—	2	1	no	—
94	C₂×C₈×C₄₃	—	2	1	no	—
95	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
100	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
101	C₄⋊C₄×C₄₃	—	4	1	no	—
102	C₄⋊C₄×C₄₃	—	4	1	no	—
103	C₄²×C₄₃	—	2	1	no	—
104	C₄²×C₄₃	—	2	1	no	—
105	(C₂×D₄₃)⋊C₄	—	4	4	no	—
106	C₄⋊Dic₄₃	—	2	2	no	—
107	C₄⋊Dic₄₃	—	4	1	no	—
108	D₃₄₄	—	2	2	no	—
109	D₃₄₄	—	2	2	no	—
110	D₃₄₄	—	2	2	no	—
111	D₃₄₄	C₂²⋊C₄	1	4	yes	—
112	C₄₃⋊C₁₆	—	2	2	no	—
113	C₄₃⋊C₁₆	—	4	1	no	—
114	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₄₃	—	4	2	no	—
116	C₂²×D₄₃	—	4	2	no	—
117	C₂²×D₄₃	—	4	2	no	—
118	C₂²×D₄₃	—	8	1	no	—
119	C₂²×D₄₃	—	8	1	no	—
120	C₂²×D₄₃	—	8	1	no	—
121	C₂²×D₄₃	—	8	1	no	—
122	Q₈×C₄₃	—	4	2	no	—
123	Q₈×C₄₃	—	4	2	no	—
124	Q₈×C₄₃	—	8	1	no	—
125	Q₈×C₄₃	—	8	1	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	—	8	1	no	—
133	C₂×C₄×C₄₃	—	8	1	no	—
134	C₂×C₄×C₄₃	—	8	1	no	—
135	C₂×C₄×C₄₃	—	8	1	no	—
136	C₂×C₄×C₄₃	—	8	1	no	—
137	C₂×C₄×C₄₃	SD₃₂	1	2	yes	complemented
138	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
142	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×Dic₄₃	—	4	2	no	—
145	C₂×Dic₄₃	—	8	1	no	—
146	C₂×Dic₄₃	—	8	1	no	—
147	D₁₇₂	—	2	4	no	—
148	D₁₇₂	—	4	2	no	—
149	D₁₇₂	—	4	2	no	—
150	D₁₇₂	—	4	2	no	—
151	C₄×D₄₃	—	8	8	no	—
152	C₈×C₄₃	—	4	1	no	—
153	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
155	C₈×C₄₃	#121	1	1	yes	characteristic
156	C₈×C₄₃	#123	1	1	yes	characteristic
157	#13014	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
159	C₂²×C₄₃	#447	1	1	yes	characteristic
160	C₂²×C₄₃	#459	1	1	yes	characteristic
161	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
162	C₂²×C₄₃	#508	1	1	yes	characteristic
163	C₂²×C₄₃	#509	1	1	yes	characteristic
164	C₂²×C₄₃	#510	1	1	yes	characteristic
165	C₂×D₄₃	—	4	4	no	—
166	C₂×D₄₃	—	8	2	no	—
167	C₂×D₄₃	—	8	2	no	—
168	C₂×D₄₃	—	8	2	no	—
169	C₂×D₄₃	—	8	2	no	—
170	C₂×D₄₃	—	8	2	no	—
171	C₂×D₄₃	—	8	2	no	—
172	C₄×C₄₃	—	2	2	no	—
173	C₄×C₄₃	—	4	1	no	—
174	C₄×C₄₃	—	8	2	no	—
175	C₄×C₄₃	—	16	1	no	—
176	C₄×C₄₃	#386	1	1	yes	characteristic
177	C₄×C₄₃	#442	1	1	yes	characteristic
178	C₄×C₄₃	#449	1	1	yes	characteristic
179	C₄×C₄₃	#463	1	1	yes	characteristic
180	Dic₄₃	—	8	2	no	—
181	C₂²×SD₃₂	—	43	1	no	—
182	C₂×C₄×D₈	—	43	1	no	—
183	C₂×#358	—	43	1	no	—
184	C₂×#357	—	43	1	no	—
185	#1969	D₄₃	1	1	yes	characteristic, complemented
186	#1380	—	43	1	no	—
187	#1374	—	43	1	no	—
188	C₂×C₄₃	#1879	1	1	yes	characteristic
189	C₂×C₄₃	C₄×SD₃₂	1	1	yes	characteristic
190	C₂×C₄₃	#2175	1	1	yes	characteristic
191	C₂×C₄₃	#2187	1	1	yes	characteristic
192	C₂×C₄₃	#2189	1	1	yes	characteristic
193	C₂×C₄₃	#2208	1	1	yes	characteristic
194	C₂×C₄₃	#2210	1	1	yes	characteristic
195	D₄₃	—	8	4	no	—
196	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
197	C₂²×D₈	—	43	1	no	—
198	C₂×C₄×D₄	—	86	1	no	—
199	C₂×SD₃₂	—	43	4	no	—
200	C₂×SD₃₂	—	86	2	no	—
201	C₂×SD₃₂	—	86	2	no	—
202	C₂²×C₁₆	—	43	1	no	—
203	C₂²×C₁₆	—	43	1	no	—
204	C₄×D₈	—	86	2	no	—
205	C₄×D₈	—	86	2	no	—
206	C₂×#107	—	43	1	no	—
207	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
208	C₂×#103	—	2	1	no	—
209	C₂×#102	—	86	1	no	—
210	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
211	#391	—	2	1	no	—
212	#358	—	86	2	no	—
213	#357	—	86	2	no	—
214	C₄₃	#13014	1	1	yes	characteristic, complemented, Sylow
215	C₂²×Q₈	—	2	1	no	—
216	C₂²×D₄	—	86	1	no	—
217	C₂³×C₄	—	172	1	no	—
218	C₂×Q₁₆	—	2	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	—	2	1	no	—
222	C₂×Q₁₆	—	2	1	no	—
223	C₂×Q₁₆	C₄×D₄₃	1	2	yes	complemented
224	C₂×D₈	—	43	2	no	—
225	C₂×D₈	—	43	2	no	—
226	C₂×D₈	—	43	2	no	—
227	C₂×D₈	—	86	1	no	—
228	C₂×D₈	—	86	1	no	—
229	C₂×D₈	—	86	1	no	—
230	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
231	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₄×D₄	—	86	4	no	—
233	C₄×D₄	—	172	2	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	86	1	no	—
236	C₂×C₂²⋊C₄	—	172	1	no	—
237	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
238	SD₃₂	—	86	8	no	—
239	C₂×C₁₆	—	43	2	no	—
240	C₂×C₁₆	—	86	1	no	—
241	C₂×C₁₆	—	86	1	no	—
242	C₂×C₁₆	—	86	1	no	—
243	C₂×C₁₆	—	86	1	no	—
244	C₂×C₁₆	—	86	1	no	—
245	#107	—	2	1	no	—
246	#107	—	2	1	no	—
247	#107	—	43	2	no	—
248	#107	—	86	1	no	—
249	#103	—	4	2	no	—
250	#102	—	86	4	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	#95	—	4	1	no	—
254	#95	—	4	1	no	—
255	C₂⁴	—	172	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×D₄	—	86	2	no	—
263	C₂×D₄	—	86	2	no	—
264	C₂×D₄	—	86	2	no	—
265	C₂×D₄	—	172	1	no	—
266	C₂×D₄	—	172	1	no	—
267	C₂×D₄	—	172	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	344	2	no	—
271	C₂²×C₄	—	344	2	no	—
272	C₂²×C₄	—	172	4	no	—
273	C₂²×C₄	—	172	1	no	—
274	C₂²×C₄	#10621882	1	1	yes	characteristic
275	C₂²×C₄	#10621882	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	4	1	no	—
280	Q₁₆	—	4	1	no	—
281	D₈	—	43	4	no	—
282	D₈	—	86	2	no	—
283	D₈	—	86	2	no	—
284	D₈	—	86	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
289	C₂×C₈	#10621898	1	1	yes	characteristic
290	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
291	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
292	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
293	C₂×C₈	#10621900	1	1	yes	characteristic
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	86	2	no	—
297	C₄⋊C₄	—	172	1	no	—
298	C₂²⋊C₄	—	172	4	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₁₆	—	86	2	no	—
302	C₁₆	—	172	1	no	—
303	C₂³	—	344	1	no	—
304	C₂³	—	344	1	no	—
305	C₂³	—	344	1	no	—
306	C₂³	—	344	1	no	—
307	C₂³	—	172	2	no	—
308	C₂³	—	172	2	no	—
309	C₂³	—	172	2	no	—
310	C₂³	C₂×#10621882	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	1	no	—
314	Q₈	—	8	1	no	—
315	D₄	—	86	4	no	—
316	D₄	—	172	2	no	—
317	D₄	—	172	2	no	—
318	D₄	—	172	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	1	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	—	8	1	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	—	344	8	no	—
331	C₂×C₄	—	344	1	no	—
332	C₂×C₄	—	344	1	no	—
333	C₂×C₄	—	172	2	no	—
334	C₂×C₄	#13063268	1	2	yes	complemented
335	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
336	C₂×C₄	#13063293	1	1	yes	characteristic
337	C₂×C₄	#13063296	1	1	yes	characteristic
338	C₂×C₄	#13063345	1	1	yes	characteristic
339	C₂×C₄	#13063350	1	1	yes	characteristic
340	C₂×C₄	#13063351	1	1	yes	characteristic
341	C₈	—	4	1	no	—
342	C₈	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
343	C₈	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
344	C₈	#13063298	1	1	yes	characteristic
345	C₈	#13063307	1	1	yes	characteristic
346	C₂²	—	344	2	no	—
347	C₂²	—	344	2	no	—
348	C₂²	—	344	2	no	—
349	C₂²	—	344	2	no	—
350	C₂²	—	344	2	no	—
351	C₂²	—	344	2	no	—
352	C₂²	—	172	4	no	—
353	C₂²	C₄×#10621882	1	1	yes	characteristic
354	C₂²	#14583032	1	1	yes	characteristic
355	C₂²	#14583091	1	1	yes	characteristic
356	C₂²	C₂×#13063268	1	1	yes	characteristic
357	C₂²	#14583329	1	1	yes	characteristic
358	C₂²	#14583334	1	1	yes	characteristic
359	C₂²	#14583340	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	4	1	no	—
362	C₄	—	8	2	no	—
363	C₄	—	16	1	no	—
364	C₄	—	344	2	no	—
365	C₄	#14582775	1	1	yes	characteristic
366	C₄	#14583009	1	1	yes	characteristic
367	C₄	#14583042	1	1	yes	characteristic
368	C₄	#14583111	1	1	yes	characteristic
369	C₂	—	344	4	no	—
370	C₂	#19134946	1	1	yes	characteristic
371	C₂	C₄×#13063268	1	1	yes	characteristic
372	C₂	#19136457	1	1	yes	characteristic
373	C₂	#19136523	1	1	yes	characteristic
374	C₂	#19136533	1	1	yes	characteristic
375	C₂	#19136637	1	1	yes	characteristic
376	C₂	#19136647	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1969	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1969	D₄₃
hypocenter	C₄₃	#13014
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

Complements

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

N	H ≅ G/N	type
3. C₂×#14582703	C₂	semidirect
5. #19132686	C₂	semidirect
6. #19132673	C₂	semidirect
7. C₄₃×#1969	C₂	semidirect
12. C₂×#13063268	C₄	semidirect
44. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
54. C₂×Q₁₆×C₄₃	C₂×C₄	semidirect
137. C₂×C₄×C₄₃	SD₃₂	semidirect
185. #1969	D₄₃	semidirect
214. C₄₃	#13014	semidirect
223. C₂×Q₁₆	C₄×D₄₃	semidirect
334. C₂×C₄	#13063268	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063268	2
C₂×C₄×D₃₄₄	2
C₂×#14582703	2
C₂×#14582699	2
#19132686	2
#19132673	2
C₄₃×#1969	2
#13014	43