Subgroups — #25250883

Order 11008.

706 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583342	C₂	1	2	yes	maximal, complemented
2	#19145023	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136658	C₂	1	4	yes	maximal, complemented
5	#19136646	C₂	1	2	yes	maximal, complemented
6	#19136636	C₂	1	2	yes	maximal, complemented
7	#19136308	C₂	1	2	yes	maximal, complemented
8	C₄₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063347	—	2	1	no	—
10	C₂×#13063347	C₂²	1	2	yes	—
11	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
12	#14583449	—	2	1	no	—
13	C₂×C₄×Dic₈₆	—	2	1	no	—
14	#14583342	—	2	8	no	—
15	C₂×Dic₃₄₄	C₂²	1	4	yes	—
16	C₂×#13063270	C₂²	1	4	yes	—
17	#14583283	—	2	1	no	—
18	#14583283	C₂²	1	2	yes	—
19	#14583279	C₂²	1	1	yes	characteristic
20	#14583276	C₂²	1	1	yes	characteristic
21	#14583272	—	2	1	no	—
22	C₄×Dic₁₇₂	—	2	1	no	—
23	C₄×Dic₁₇₂	—	2	1	no	—
24	C₄×Dic₁₇₂	C₂²	1	2	yes	—
25	C₄×Dic₁₇₂	C₂²	1	2	yes	—
26	C₂×#13063259	C₂²	1	1	yes	characteristic
27	C₂×#13063250	—	2	1	no	—
28	#14582704	C₂²	1	4	yes	—
29	#14582701	C₂²	1	4	yes	—
30	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₂×#10621898	—	2	2	no	—
35	C₂²×Dic₈₆	—	2	1	no	—
36	#13063347	—	2	4	no	—
37	#13063347	—	2	4	no	—
38	#13063347	—	4	4	no	—
39	C₂×Dic₁₇₂	—	2	1	no	—
40	C₂×Dic₁₇₂	—	2	2	no	—
41	C₂×Dic₁₇₂	—	2	1	no	—
42	C₂×Dic₁₇₂	D₄	1	4	yes	complemented
43	C₂×Dic₁₇₂	C₂³	1	2	yes	—
44	C₂×Dic₁₇₂	C₂³	1	2	yes	—
45	C₂×#10621873	—	2	2	no	—
46	C₂×#10621873	—	2	2	no	—
47	C₂×D₃₄₄	—	2	1	no	—
48	C₂×D₃₄₄	C₂³	1	2	yes	—
49	#13063325	—	2	1	no	—
50	#13063322	—	2	1	no	—
51	#13063313	—	2	2	no	—
52	#13063299	—	4	1	no	—
53	#13063299	—	4	1	no	—
54	C₄×Dic₈₆	—	2	2	no	—
55	C₄×Dic₈₆	—	2	2	no	—
56	C₄×Dic₈₆	—	4	1	no	—
57	C₄×Dic₈₆	—	4	1	no	—
58	C₄×D₁₇₂	—	4	1	no	—
59	C₄×D₁₇₂	—	4	1	no	—
60	C₂×C₄⋊Dic₄₃	—	2	1	no	—
61	C₂×Dic₄₃⋊C₄	—	4	1	no	—
62	C₂×C₄×Dic₄₃	—	4	1	no	—
63	Dic₃₄₄	—	2	8	no	—
64	#13063270	—	2	8	no	—
65	#13063259	—	2	1	no	—
66	#13063259	C₂³	1	2	yes	—
67	#13063250	—	2	2	no	—
68	#13063250	—	2	2	no	—
69	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
72	M₃₂×C₄₃	—	2	4	no	—
73	C₂×C₁₆×C₄₃	C₂³	1	4	yes	—
74	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
75	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₄₃	—	4	2	no	—
82	C₂²×Dic₄₃	—	4	1	no	—
83	C₂²×Dic₄₃	—	4	1	no	—
84	#10621898	—	4	4	no	—
85	#10621898	—	4	4	no	—
86	C₂×D₁₇₂	—	2	2	no	—
87	C₂×D₁₇₂	—	4	1	no	—
88	C₂×D₁₇₂	—	4	1	no	—
89	C₂×C₄×D₄₃	—	4	2	no	—
90	C₂×Dic₈₆	—	2	4	no	—
91	C₂×Dic₈₆	—	2	2	no	—
92	C₂×Dic₈₆	—	2	2	no	—
93	C₂×Dic₈₆	—	4	1	no	—
94	C₂×Dic₈₆	—	4	2	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
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	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₄₃	#41	1	2	yes	—
101	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
102	C₄²×C₄₃	D₈	1	1	yes	characteristic
103	C₄²×C₄₃	D₈	1	1	yes	characteristic
104	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	(C₂×D₄₃)⋊C₄	—	4	2	no	—
107	(C₂×D₄₃)⋊C₄	—	4	2	no	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	4	1	no	—
110	C₄⋊Dic₄₃	—	4	1	no	—
111	C₄⋊Dic₄₃	—	4	1	no	—
112	Dic₄₃⋊C₄	—	4	2	no	—
113	Dic₄₃⋊C₄	—	4	2	no	—
114	C₄×Dic₄₃	—	4	2	no	—
115	C₄×Dic₄₃	—	4	2	no	—
116	Dic₁₇₂	—	2	4	no	—
117	Dic₁₇₂	—	2	2	no	—
118	Dic₁₇₂	—	2	2	no	—
119	Dic₁₇₂	—	2	2	no	—
120	Dic₁₇₂	—	4	2	no	—
121	Dic₁₇₂	C₂×D₄	1	4	yes	—
122	D₃₄₄	—	2	2	no	—
123	D₃₄₄	—	2	2	no	—
124	D₃₄₄	—	4	2	no	—
125	#10621873	—	4	4	no	—
126	#10621873	—	4	4	no	—
127	C₁₆×C₄₃	—	2	4	no	—
128	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
129	C₂²×D₄₃	—	4	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	#121	1	2	yes	—
133	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
139	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	C₂²⋊D₄₃	—	8	4	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	C₂×Dic₄₃	—	4	4	no	—
143	C₂×Dic₄₃	—	4	2	no	—
144	C₂×Dic₄₃	—	4	2	no	—
145	C₂×Dic₄₃	—	8	2	no	—
146	D₁₇₂	—	4	2	no	—
147	D₁₇₂	—	4	2	no	—
148	D₁₇₂	—	8	2	no	—
149	D₁₇₂	—	8	2	no	—
150	C₄×D₄₃	—	8	4	no	—
151	Dic₈₆	—	2	4	no	—
152	Dic₈₆	—	4	2	no	—
153	Dic₈₆	—	4	4	no	—
154	Dic₈₆	—	4	2	no	—
155	Dic₈₆	—	4	2	no	—
156	C₈×C₄₃	—	2	1	no	—
157	C₈×C₄₃	—	2	1	no	—
158	C₈×C₄₃	#121	1	2	yes	—
159	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
160	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
161	#33581	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	#459	1	2	yes	—
165	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
166	C₂×D₄₃	—	8	2	no	—
167	C₂×D₄₃	—	8	2	no	—
168	C₂×D₄₃	—	8	2	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	#459	1	2	yes	—
173	C₄×C₄₃	#510	1	2	yes	complemented
174	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
175	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
176	Dic₄₃	—	4	4	no	—
177	Dic₄₃	—	8	2	no	—
178	Dic₄₃	—	8	2	no	—
179	C₂×#510	—	43	2	no	—
180	#3147	—	43	1	no	—
181	C₂×C₄×Q₁₆	—	43	1	no	—
182	#2211	—	43	4	no	—
183	#2209	—	43	2	no	—
184	#2207	—	43	2	no	—
185	#2150	D₄₃	1	1	yes	characteristic, complemented
186	#2139	—	43	2	no	—
187	C₂×C₄₃	—	2	2	no	—
188	C₂×C₄₃	C₂×#459	1	1	yes	characteristic
189	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
190	C₂×C₄₃	#3414	1	1	yes	characteristic
191	D₄₃	—	16	2	no	—
192	C₂×#135	—	43	2	no	—
193	C₂×#135	—	86	1	no	—
194	C₂²×Q₁₆	—	43	1	no	—
195	#532	—	86	1	no	—
196	C₂×C₄×Q₈	—	86	1	no	—
197	#510	—	86	8	no	—
198	C₂×Q₃₂	—	43	4	no	—
199	C₂×SD₃₂	—	43	4	no	—
200	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
201	#495	—	43	2	no	—
202	#495	—	86	1	no	—
203	#494	—	43	1	no	—
204	#493	—	43	1	no	—
205	#492	—	86	1	no	—
206	C₄×Q₁₆	—	43	2	no	—
207	C₄×Q₁₆	—	43	2	no	—
208	C₄×Q₁₆	—	86	1	no	—
209	C₄×Q₁₆	—	86	1	no	—
210	C₂×#107	—	43	1	no	—
211	C₂×#103	—	86	1	no	—
212	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
213	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
214	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
215	#358	—	43	4	no	—
216	#357	—	43	4	no	—
217	C₄₃	#33581	1	1	yes	characteristic, complemented, Sylow
218	C₂×#41	—	86	2	no	—
219	C₂²×Q₈	—	86	1	no	—
220	#135	—	86	4	no	—
221	#135	—	86	4	no	—
222	#135	—	172	4	no	—
223	C₂×Q₁₆	—	43	4	no	—
224	C₂×Q₁₆	—	43	2	no	—
225	C₂×Q₁₆	—	43	2	no	—
226	C₂×Q₁₆	—	86	1	no	—
227	C₂×Q₁₆	—	86	1	no	—
228	C₂×Q₁₆	—	86	2	no	—
229	C₂×SD₁₆	—	86	2	no	—
230	C₂×SD₁₆	—	86	2	no	—
231	C₂×D₈	—	43	2	no	—
232	C₂×D₈	—	86	1	no	—
233	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
235	C₄⋊Q₈	—	86	1	no	—
236	#127	—	86	1	no	—
237	#124	—	86	2	no	—
238	#121	—	172	1	no	—
239	#121	—	172	1	no	—
240	C₄×Q₈	—	86	2	no	—
241	C₄×Q₈	—	86	2	no	—
242	C₄×Q₈	—	172	1	no	—
243	C₄×Q₈	—	172	1	no	—
244	C₄×D₄	—	172	1	no	—
245	C₄×D₄	—	172	1	no	—
246	C₂×C₄⋊C₄	—	86	1	no	—
247	C₂×C₄⋊C₄	—	172	1	no	—
248	C₂×C₄²	—	172	1	no	—
249	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
250	Q₃₂	—	86	8	no	—
251	SD₃₂	—	86	8	no	—
252	M₃₂	—	2	4	no	—
253	C₂×C₁₆	C₂²×D₄₃	1	4	yes	—
254	#107	—	43	2	no	—
255	#107	—	86	1	no	—
256	#103	—	86	2	no	—
257	#103	—	86	2	no	—
258	C₄×C₈	D₁₇₂	1	1	yes	characteristic
259	C₄×C₈	D₁₇₂	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
262	#41	—	172	4	no	—
263	#41	—	172	4	no	—
264	C₂×Q₈	—	86	4	no	—
265	C₂×Q₈	—	86	2	no	—
266	C₂×Q₈	—	86	2	no	—
267	C₂×Q₈	—	172	1	no	—
268	C₂×Q₈	—	172	2	no	—
269	C₂×D₄	—	86	2	no	—
270	C₂×D₄	—	172	2	no	—
271	C₂×D₄	—	172	1	no	—
272	C₂×D₄	—	172	1	no	—
273	C₂²×C₄	—	172	1	no	—
274	C₂²×C₄	—	172	2	no	—
275	C₂²×C₄	—	172	1	no	—
276	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
277	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
279	Q₁₆	—	43	4	no	—
280	Q₁₆	—	86	4	no	—
281	Q₁₆	—	86	2	no	—
282	Q₁₆	—	86	2	no	—
283	Q₁₆	—	86	2	no	—
284	Q₁₆	—	172	2	no	—
285	SD₁₆	—	172	4	no	—
286	SD₁₆	—	172	4	no	—
287	D₈	—	86	2	no	—
288	D₈	—	86	2	no	—
289	D₈	—	172	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
292	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
293	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
294	C₂×C₈	D₄×D₄₃	1	2	yes	—
295	C₂×C₈	#10621902	1	2	yes	—
296	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
297	C₄⋊C₄	—	86	2	no	—
298	C₄⋊C₄	—	172	1	no	—
299	C₄⋊C₄	—	172	2	no	—
300	C₄⋊C₄	—	172	2	no	—
301	C₄⋊C₄	—	172	1	no	—
302	C₄⋊C₄	—	172	1	no	—
303	C₂²⋊C₄	—	172	2	no	—
304	C₂²⋊C₄	—	172	2	no	—
305	C₄²	—	172	2	no	—
306	C₄²	—	172	2	no	—
307	C₄²	D₃₄₄	1	1	yes	characteristic
308	C₄²	D₃₄₄	1	1	yes	characteristic
309	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
310	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
311	C₁₆	—	2	4	no	—
312	C₂³	—	172	2	no	—
313	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
314	Q₈	—	86	4	no	—
315	Q₈	—	172	2	no	—
316	Q₈	—	172	2	no	—
317	Q₈	—	172	4	no	—
318	Q₈	—	172	2	no	—
319	D₄	—	344	2	no	—
320	D₄	—	344	4	no	—
321	D₄	—	344	2	no	—
322	D₄	—	172	2	no	—
323	D₄	—	172	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	344	2	no	—
327	C₂×C₄	—	344	4	no	—
328	C₂×C₄	—	172	2	no	—
329	C₂×C₄	—	172	2	no	—
330	C₂×C₄	—	172	4	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	#13063300	1	2	yes	—
333	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
334	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
335	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
336	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
337	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
338	C₂×C₄	#13063353	1	2	yes	—
339	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	#13063300	1	2	yes	—
343	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
344	C₈	C₂×#10621902	1	1	yes	characteristic
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	344	2	no	—
348	C₂²	—	344	2	no	—
349	C₂²	—	344	2	no	—
350	C₂²	#14583095	1	2	yes	—
351	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	344	2	no	—
356	C₄	—	344	2	no	—
357	C₄	—	172	4	no	—
358	C₄	#14583095	1	2	yes	—
359	C₄	#14583342	1	2	yes	complemented
360	C₄	C₂×#13063300	1	1	yes	characteristic
361	C₄	C₂×#13063353	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	688	2	no	—
364	C₂	C₂×#14583095	1	1	yes	characteristic
365	C₂	C₂×#14583342	1	1	yes	characteristic
366	C₂	#19147731	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2150	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19145023	C₂
hypercenter	#2150	D₄₃
hypocenter	C₄₃	#33581
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. C₂×#14583342	C₂	semidirect
3. C₂×C₄×Dic₁₇₂	C₂	semidirect
4. #19136658	C₂	semidirect
5. #19136646	C₂	semidirect
6. #19136636	C₂	semidirect
7. #19136308	C₂	semidirect
8. C₄₃×#2150	C₂	semidirect
42. C₂×Dic₁₇₂	D₄	semidirect
173. C₄×C₄₃	#510	semidirect
185. #2150	D₄₃	semidirect
217. C₄₃	#33581	semidirect
359. C₄	#14583342	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583342	2
#19145023	2
C₂×C₄×Dic₁₇₂	2
#19136658	2
#19136646	2
#19136636	2
#19136308	2
C₄₃×#2150	2
#33581	43