Subgroups — #24990928

Order 11008.

454 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	#19135370	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133354	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133021	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582628	C₂	1	1	yes	maximal, characteristic
6	#19132491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₄₃	—	2	1	no	—
9	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
10	#14582978	—	2	1	no	—
11	#14582972	—	2	2	no	—
12	C₈×D₁₇₂	—	2	1	no	—
13	C₂×#13063260	—	2	1	no	—
14	#14582923	C₂²	1	1	yes	characteristic
15	#14582923	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
17	#14582862	—	2	1	no	—
18	#14582663	—	2	1	no	—
19	#14582652	—	2	2	no	—
20	#14582628	—	2	1	no	—
21	#14582628	—	2	1	no	—
22	#14582628	C₄	1	1	yes	characteristic
23	#14582628	C₄	1	1	yes	characteristic
24	#14582628	C₄	1	1	yes	characteristic
25	#14582628	C₄	1	1	yes	characteristic
26	C₄₃×#431	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×D₁₇₂	Q₈	1	1	yes	characteristic
30	C₂²×C₄×D₄₃	—	2	1	no	—
31	C₂×#10621877	—	2	1	no	—
32	C₂×#10621877	—	2	1	no	—
33	C₂×#10621872	—	2	2	no	—
34	M₁₆×D₄₃	—	4	2	no	—
35	M₁₆×D₄₃	—	4	2	no	—
36	C₂²×C₄₃⋊C₈	—	2	1	no	—
37	C₂×C₈×D₄₃	—	2	2	no	—
38	C₄×D₁₇₂	—	2	1	no	—
39	C₄×D₁₇₂	—	2	1	no	—
40	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
41	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
42	C₄×D₁₇₂	D₄	1	1	yes	characteristic
43	C₄×D₁₇₂	D₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₄₃	Q₈	1	1	yes	characteristic
45	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
46	#13063260	—	4	2	no	—
47	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
50	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
51	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
52	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
53	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
54	(C₂×D₄₃)⋊C₈	—	2	2	no	—
55	(C₂×D₄₃)⋊C₈	—	2	2	no	—
56	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₃	—	2	1	no	—
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₃	—	2	1	no	—
61	C₂²⋊C₈×C₄₃	—	2	1	no	—
62	#97×C₄₃	—	2	1	no	—
63	C₄×C₈×C₄₃	—	2	1	no	—
64	C₄×C₈×C₄₃	—	2	1	no	—
65	C₄×C₈×C₄₃	—	2	1	no	—
66	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
68	C₂³×D₄₃	—	2	1	no	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
72	C₂²×Dic₄₃	—	2	1	no	—
73	C₂×D₁₇₂	—	2	1	no	—
74	C₂×D₁₇₂	—	2	2	no	—
75	C₂×D₁₇₂	—	4	1	no	—
76	C₂×D₁₇₂	C₄⋊C₄	1	1	yes	characteristic
77	C₂×D₁₇₂	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₄₃	—	2	1	no	—
79	C₂×C₄×D₄₃	—	2	1	no	—
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×C₄×D₄₃	—	2	1	no	—
82	C₂×C₄×D₄₃	—	4	1	no	—
83	C₂×C₄×D₄₃	—	4	1	no	—
84	C₂×C₄×D₄₃	—	4	1	no	—
85	C₂×C₄×D₄₃	—	4	1	no	—
86	M₁₆×C₄₃	—	2	1	no	—
87	M₁₆×C₄₃	—	2	1	no	—
88	M₁₆×C₄₃	—	2	1	no	—
89	M₁₆×C₄₃	D₈	1	2	yes	—
90	C₂×C₈×C₄₃	—	2	1	no	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	2	1	no	—
93	C₂×C₈×C₄₃	—	2	1	no	—
94	C₂×C₈×C₄₃	—	4	2	no	—
95	C₂×C₈×C₄₃	—	4	1	no	—
96	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	#41	1	1	yes	characteristic
100	(C₂×D₄₃)⋊C₄	—	2	1	no	—
101	(C₂×D₄₃)⋊C₄	—	2	1	no	—
102	(C₂×D₄₃)⋊C₄	—	2	1	no	—
103	(C₂×D₄₃)⋊C₄	—	2	1	no	—
104	C₄⋊Dic₄₃	—	2	1	no	—
105	C₄⋊Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄⋊Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
107	#10621877	—	4	1	no	—
108	#10621877	—	4	1	no	—
109	#10621877	—	4	1	no	—
110	#10621877	—	4	1	no	—
111	C₂×C₄₃⋊C₈	—	2	1	no	—
112	C₂×C₄₃⋊C₈	—	2	1	no	—
113	C₂×C₄₃⋊C₈	—	2	2	no	—
114	C₂×C₄₃⋊C₈	—	4	1	no	—
115	C₂×C₄₃⋊C₈	—	4	1	no	—
116	C₂×C₄₃⋊C₈	—	4	1	no	—
117	#10621872	—	4	4	no	—
118	C₈×D₄₃	—	4	4	no	—
119	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂²×D₄₃	—	2	1	no	—
121	C₂²×D₄₃	—	2	1	no	—
122	C₂²×D₄₃	—	4	2	no	—
123	C₂²×D₄₃	—	4	1	no	—
124	C₂²×D₄₃	—	4	1	no	—
125	C₂²×D₄₃	—	8	1	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	4	1	no	—
129	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
137	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
139	C₂×Dic₄₃	—	2	1	no	—
140	C₂×Dic₄₃	—	2	1	no	—
141	C₂×Dic₄₃	—	4	1	no	—
142	C₂×Dic₄₃	—	4	1	no	—
143	D₁₇₂	—	2	1	no	—
144	D₁₇₂	—	2	1	no	—
145	D₁₇₂	—	2	1	no	—
146	D₁₇₂	—	4	2	no	—
147	D₁₇₂	#108	1	2	yes	complemented
148	C₄×D₄₃	—	4	1	no	—
149	C₄×D₄₃	—	4	1	no	—
150	C₄×D₄₃	—	4	1	no	—
151	C₄×D₄₃	—	4	1	no	—
152	C₄×D₄₃	—	4	1	no	—
153	C₄×D₄₃	—	4	1	no	—
154	C₄×D₄₃	—	4	1	no	—
155	C₄×D₄₃	—	4	1	no	—
156	C₈×C₄₃	—	2	2	no	—
157	C₈×C₄₃	—	2	2	no	—
158	C₈×C₄₃	—	4	1	no	—
159	C₈×C₄₃	—	4	1	no	—
160	C₈×C₄₃	—	8	1	no	—
161	C₄₃⋊C₈	—	4	1	no	—
162	C₄₃⋊C₈	—	4	1	no	—
163	C₄₃⋊C₈	—	4	2	no	—
164	C₄₃⋊C₈	—	8	1	no	—
165	#8906	—	43	1	no	maximal, Sylow
166	C₂²×C₄₃	—	2	1	no	—
167	C₂²×C₄₃	—	2	1	no	—
168	C₂²×C₄₃	#385	1	1	yes	characteristic
169	C₂²×C₄₃	#474	1	1	yes	characteristic
170	C₂²×C₄₃	#478	1	1	yes	characteristic
171	C₂×D₄₃	—	2	2	no	—
172	C₂×D₄₃	—	4	1	no	—
173	C₂×D₄₃	—	4	1	no	—
174	C₂×D₄₃	—	4	1	no	—
175	C₂×D₄₃	—	4	1	no	—
176	C₂×D₄₃	—	4	1	no	—
177	C₂×D₄₃	—	8	2	no	—
178	C₂×D₄₃	—	8	2	no	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	2	1	no	—
181	C₄×C₄₃	—	2	1	no	—
182	C₄×C₄₃	C₂×#108	1	1	yes	characteristic
183	C₄×C₄₃	#430	1	1	yes	characteristic
184	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
185	C₄×C₄₃	#440	1	1	yes	characteristic
186	C₄×C₄₃	#447	1	1	yes	characteristic
187	C₄×C₄₃	#452	1	1	yes	characteristic
188	Dic₄₃	—	4	1	no	—
189	Dic₄₃	—	4	1	no	—
190	D₄×M₁₆	—	43	1	no	—
191	#1950	—	43	1	no	—
192	#1567	—	43	1	no	—
193	#1486	—	43	1	no	—
194	C₂×D₄⋊C₈	—	43	1	no	—
195	#1301	—	43	1	no	—
196	#1278	D₄₃	1	1	yes	characteristic, complemented
197	C₂×C₄₃	—	2	1	no	—
198	C₂×C₄₃	—	2	1	no	—
199	C₂×C₄₃	#1855	1	1	yes	characteristic
200	C₂×C₄₃	#1861	1	1	yes	characteristic
201	C₂×C₄₃	#1862	1	1	yes	characteristic
202	D₄₃	—	4	2	no	—
203	D₄₃	—	8	1	no	—
204	C₂²×M₁₆	—	86	1	no	—
205	C₂×C₄×D₄	—	43	1	no	—
206	#436	—	86	1	no	—
207	#435	—	86	2	no	—
208	C₈×D₄	—	86	1	no	—
209	#431	C₂×D₄₃	1	1	yes	characteristic
210	C₂×#108	—	86	1	no	—
211	#423	—	43	1	no	—
212	#423	—	43	1	no	—
213	C₂×C₄⋊C₈	—	43	1	no	—
214	#407	—	86	1	no	—
215	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
216	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
217	#341	—	86	1	no	—
218	#335	—	86	2	no	—
219	D₄⋊C₈	—	43	1	no	—
220	D₄⋊C₈	—	43	1	no	—
221	D₄⋊C₈	—	43	1	no	—
222	D₄⋊C₈	—	43	1	no	—
223	D₄⋊C₈	—	86	1	no	—
224	D₄⋊C₈	—	86	1	no	—
225	C₄₃	#8906	1	1	yes	characteristic, complemented, Sylow
226	C₂²×D₄	—	43	1	no	—
227	C₂³×C₄	—	86	1	no	—
228	C₂×M₁₆	—	86	2	no	—
229	C₂×M₁₆	—	86	1	no	—
230	C₂×M₁₆	—	86	1	no	—
231	C₂×M₁₆	—	172	2	no	—
232	C₂×M₁₆	—	172	2	no	—
233	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
234	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	86	2	no	—
237	C₂²×C₈	—	86	1	no	—
238	C₄×D₄	—	43	1	no	—
239	C₄×D₄	—	43	1	no	—
240	C₄×D₄	—	43	1	no	—
241	C₄×D₄	—	43	1	no	—
242	C₄×D₄	—	86	1	no	—
243	C₄×D₄	—	86	1	no	—
244	C₂×C₄⋊C₄	—	43	1	no	—
245	C₂×C₂²⋊C₄	—	86	1	no	—
246	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
247	#108	—	172	2	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	43	1	no	—
250	C₄⋊C₈	—	43	1	no	—
251	C₄⋊C₈	—	43	2	no	—
252	C₄⋊C₈	—	86	1	no	—
253	C₄⋊C₈	—	86	1	no	—
254	C₄⋊C₈	—	86	1	no	—
255	C₄⋊C₈	—	86	1	no	—
256	C₂²⋊C₈	—	2	1	no	—
257	C₂²⋊C₈	—	86	2	no	—
258	C₂²⋊C₈	—	86	2	no	—
259	#97	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	2	1	no	—
262	C₄×C₈	—	2	1	no	—
263	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
264	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
265	C₂⁴	—	86	1	no	—
266	C₂×D₄	—	43	1	no	—
267	C₂×D₄	—	43	1	no	—
268	C₂×D₄	—	86	2	no	—
269	C₂×D₄	—	86	1	no	—
270	C₂×D₄	—	172	1	no	—
271	C₂²×C₄	—	86	1	no	—
272	C₂²×C₄	—	86	1	no	—
273	C₂²×C₄	—	86	1	no	—
274	C₂²×C₄	—	86	1	no	—
275	C₂²×C₄	—	86	1	no	—
276	C₂²×C₄	—	172	1	no	—
277	C₂²×C₄	—	172	1	no	—
278	C₂²×C₄	—	172	1	no	—
279	C₂²×C₄	—	172	1	no	—
280	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
281	C₂²×C₄	#10621900	1	1	yes	characteristic
282	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	2	1	no	—
286	M₁₆	—	172	4	no	—
287	M₁₆	—	172	1	no	—
288	M₁₆	—	172	1	no	—
289	M₁₆	—	172	1	no	—
290	M₁₆	—	172	1	no	—
291	M₁₆	D₃₄₄	1	2	yes	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	86	1	no	—
299	C₂×C₈	—	86	1	no	—
300	C₂×C₈	—	86	2	no	—
301	C₂×C₈	—	172	4	no	—
302	C₂×C₈	—	172	1	no	—
303	C₂×C₈	—	172	1	no	—
304	C₂×C₈	—	172	1	no	—
305	C₄⋊C₄	—	43	1	no	—
306	C₄⋊C₄	—	43	1	no	—
307	C₄⋊C₄	—	86	1	no	—
308	C₂²⋊C₄	—	86	1	no	—
309	C₂²⋊C₄	—	86	1	no	—
310	C₂²⋊C₄	—	86	1	no	—
311	C₂²⋊C₄	—	86	1	no	—
312	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
313	C₄²	#10621898	1	1	yes	characteristic
314	C₄²	D₄×D₄₃	1	1	yes	characteristic
315	C₄²	D₄×D₄₃	1	1	yes	characteristic
316	C₂³	—	86	1	no	—
317	C₂³	—	86	1	no	—
318	C₂³	—	344	1	no	—
319	C₂³	—	172	2	no	—
320	C₂³	—	172	1	no	—
321	C₂³	—	172	1	no	—
322	C₂³	#13063305	1	1	yes	characteristic
323	D₄	—	43	2	no	—
324	D₄	—	86	1	no	—
325	D₄	—	86	1	no	—
326	D₄	—	86	1	no	—
327	D₄	—	172	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	86	1	no	—
332	C₂×C₄	—	86	1	no	—
333	C₂×C₄	—	172	1	no	—
334	C₂×C₄	—	172	1	no	—
335	C₂×C₄	—	172	1	no	—
336	C₂×C₄	—	172	1	no	—
337	C₂×C₄	—	172	1	no	—
338	C₂×C₄	—	172	1	no	—
339	C₂×C₄	—	172	1	no	—
340	C₂×C₄	—	172	1	no	—
341	C₂×C₄	—	172	1	no	—
342	C₂×C₄	—	172	1	no	—
343	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
344	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
345	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
346	C₂×C₄	#13063295	1	1	yes	characteristic
347	C₂×C₄	#13063303	1	1	yes	characteristic
348	C₂×C₄	#13063307	1	1	yes	characteristic
349	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
350	C₂×C₄	#13063347	1	1	yes	characteristic
351	C₂×C₄	#13063353	1	1	yes	characteristic
352	C₂×C₄	#13063358	1	1	yes	characteristic
353	C₈	—	2	2	no	—
354	C₈	—	2	2	no	—
355	C₈	—	4	1	no	—
356	C₈	—	4	1	no	—
357	C₈	—	8	1	no	—
358	C₈	—	344	1	no	—
359	C₈	—	172	1	no	—
360	C₈	—	172	1	no	—
361	C₈	—	172	2	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	86	2	no	—
365	C₂²	—	344	2	no	—
366	C₂²	—	344	2	no	—
367	C₂²	—	172	1	no	—
368	C₂²	—	172	1	no	—
369	C₂²	—	172	1	no	—
370	C₂²	—	172	1	no	—
371	C₂²	—	172	1	no	—
372	C₂²	#14582773	1	1	yes	characteristic
373	C₂²	#14583180	1	1	yes	characteristic
374	C₂²	#14583200	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	172	1	no	—
379	C₄	—	172	1	no	—
380	C₄	#108×D₄₃	1	1	yes	characteristic
381	C₄	#14582946	1	1	yes	characteristic
382	C₄	C₄×D₃₄₄	1	1	yes	characteristic
383	C₄	#14583001	1	1	yes	characteristic
384	C₄	#14583035	1	1	yes	characteristic
385	C₄	#14583063	1	1	yes	characteristic
386	C₂	—	2	1	no	—
387	C₂	—	2	1	no	—
388	C₂	—	344	1	no	—
389	C₂	—	172	2	no	—
390	C₂	#19134801	1	1	yes	characteristic
391	C₂	#19134836	1	1	yes	characteristic
392	C₂	#19134845	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063295
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1278	C₂
socle	C₂²×C₄₃	#385
prime radical	C₂²×D₁₇₂	Q₈
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1278	D₄₃
hypocenter	C₄₃	#8906
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
2. #19135370	C₂	semidirect
3. #19133354	C₂	semidirect
6. #19132491	C₂	semidirect
7. C₄₃×#1278	C₂	semidirect
147. D₁₇₂	#108	semidirect
196. #1278	D₄₃	semidirect
225. C₄₃	#8906	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×D₁₇₂	2
#19135370	2
#19133354	2
#19133021	2
C₂×#14582628	2
#19132491	2
C₄₃×#1278	2
#8906	43