Subgroups — #26029780

Order 11392.

458 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668041	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152168	C₂	1	1	yes	maximal, characteristic
4	#19152074	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151819	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151818	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151817	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151809	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151808	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151623	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151605	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151591	C₂	1	1	yes	maximal, characteristic
15	#464×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
19	#15668041	—	2	4	no	—
20	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
22	C₂×#13083564	C₂²	1	1	yes	characteristic
23	C₂×#13083564	C₂²	1	1	yes	characteristic
24	C₂×#13083564	C₂²	1	1	yes	characteristic
25	Q₁₆×D₈₉	—	2	4	no	—
26	C₂×#13083562	C₂²	1	1	yes	characteristic
27	C₂×#13083557	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
29	#15668012	C₂²	1	1	yes	characteristic
30	#15668010	C₂²	1	1	yes	characteristic
31	#15667999	C₂²	1	1	yes	characteristic
32	#15667994	—	2	1	no	—
33	#15667994	—	2	1	no	—
34	#15667989	—	2	1	no	—
35	#15667989	—	2	1	no	—
36	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₇₈	—	2	1	no	—
38	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
39	#15667977	—	2	1	no	—
40	#15667977	C₂²	1	1	yes	characteristic
41	#15667970	—	2	1	no	—
42	#15667970	C₂²	1	1	yes	characteristic
43	#15667969	—	2	1	no	—
44	#15667969	C₂²	1	1	yes	characteristic
45	C₄⋊#10623494	C₂²	1	1	yes	characteristic
46	#15667941	C₂²	1	1	yes	characteristic
47	#15667941	C₂²	1	1	yes	characteristic
48	#15667937	C₂²	1	1	yes	characteristic
49	#15667937	C₂²	1	1	yes	characteristic
50	#15667936	C₂²	1	1	yes	characteristic
51	#15667935	C₂²	1	1	yes	characteristic
52	#15667935	C₂²	1	1	yes	characteristic
53	#15667933	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
56	#124×C₈₉	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#103×C₈₉	C₂²	1	1	yes	characteristic
60	#102×C₈₉	C₂²	1	1	yes	characteristic
61	#13083573	—	2	2	no	—
62	#13083573	—	2	2	no	—
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	—	2	1	no	—
65	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
68	Q₈×D₈₉	—	2	2	no	—
69	Q₈×D₈₉	—	2	2	no	—
70	Q₈×D₈₉	—	2	2	no	—
71	Q₈×D₈₉	D₄	1	4	yes	—
72	C₂×C₂²⋊D₈₉	—	2	1	no	—
73	C₂²×Dic₈₉	—	2	1	no	—
74	C₂×C₄×D₈₉	—	2	1	no	—
75	C₂×C₄×D₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
77	Dic₃₅₆	—	2	2	no	—
78	Dic₃₅₆	—	2	2	no	—
79	#13083564	—	2	2	no	—
80	#13083564	—	2	2	no	—
81	#13083564	—	2	2	no	—
82	#13083562	—	2	2	no	—
83	#13083557	—	2	2	no	—
84	C₂×#10623494	C₂³	1	1	yes	characteristic
85	C₂×#10623494	C₂³	1	1	yes	characteristic
86	C₈×D₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	2	1	no	—
90	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
91	#13083553	—	2	1	no	—
92	#13083553	—	2	1	no	—
93	#13083553	—	2	1	no	—
94	#13083553	—	2	1	no	—
95	#13083553	D₄	1	1	yes	characteristic
96	#13083553	D₄	1	1	yes	characteristic
97	C₂²⋊Dic₈₉	—	2	1	no	—
98	C₂²⋊Dic₈₉	—	2	1	no	—
99	#13083551	—	2	1	no	—
100	#13083551	D₄	1	1	yes	characteristic
101	#13083551	D₄	1	1	yes	characteristic
102	C₄×Dic₈₉	—	2	1	no	—
103	C₄×Dic₈₉	—	2	1	no	—
104	C₄×Dic₈₉	—	2	1	no	—
105	C₄×Dic₈₉	—	2	1	no	—
106	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
109	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
110	Q₁₆×C₈₉	—	2	2	no	—
111	SD₁₆×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
114	C₄⋊C₄×C₈₉	—	2	1	no	—
115	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
116	C₂²⋊C₄×C₈₉	—	2	1	no	—
117	C₂²⋊C₄×C₈₉	—	2	1	no	—
118	C₄²×C₈₉	—	2	1	no	—
119	C₄²×C₈₉	C₂³	1	1	yes	characteristic
120	C₂³×C₈₉	—	2	1	no	—
121	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₈₉	—	2	1	no	—
123	Q₈×C₈₉	—	2	1	no	—
124	Q₈×C₈₉	—	2	1	no	—
125	Q₈×C₈₉	C₂×D₄	1	2	yes	—
126	D₄×C₈₉	—	2	1	no	—
127	D₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₈₉	—	4	2	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	—	2	1	no	—
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
143	C₄×D₈₉	—	2	4	no	—
144	C₄×D₈₉	—	4	2	no	—
145	C₄×D₈₉	C₂×D₄	1	2	yes	—
146	C₄×D₈₉	#41	1	2	yes	—
147	Dic₁₇₈	—	2	1	no	—
148	Dic₁₇₈	—	2	4	no	—
149	Dic₁₇₈	—	2	1	no	—
150	Dic₁₇₈	—	2	1	no	—
151	Dic₁₇₈	—	2	1	no	—
152	Dic₁₇₈	—	2	1	no	—
153	Dic₁₇₈	—	4	2	no	—
154	Dic₁₇₈	C₂×D₄	1	2	yes	—
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	—	2	1	no	—
157	#10623494	—	2	1	no	—
158	#10623494	—	2	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₈₉	—	2	1	no	—
164	C₂×D₈₉	—	2	1	no	—
165	C₂×D₈₉	#121	1	2	yes	—
166	C₄×C₈₉	—	2	2	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	—	4	1	no	—
171	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
172	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
173	Dic₈₉	—	2	2	no	—
174	Dic₈₉	—	2	1	no	—
175	Dic₈₉	—	2	1	no	—
176	Dic₈₉	—	4	1	no	—
177	Dic₈₉	—	4	1	no	—
178	Dic₈₉	—	4	1	no	—
179	Dic₈₉	#121	1	2	yes	—
180	C₂×C₈₉	—	8	1	no	—
181	C₂×C₈₉	C₂×#121	1	1	yes	characteristic
182	C₂×C₈₉	#578	1	1	yes	characteristic
183	C₂×C₈₉	#578	1	1	yes	characteristic
184	D₈₉	—	2	2	no	—
185	#3044	—	89	1	no	maximal, Sylow
186	C₈₉	#3044	1	1	yes	characteristic, complemented, Sylow
187	C₂×#137	—	89	1	no	—
188	C₂²×Q₁₆	—	89	1	no	—
189	#536	—	89	1	no	—
190	#519	—	89	1	no	—
191	#464	—	89	1	no	—
192	#464	—	89	1	no	—
193	#464	—	89	1	no	—
194	#464	D₈₉	1	1	yes	characteristic, complemented
195	#462	—	89	1	no	—
196	#462	—	89	1	no	—
197	#462	—	89	1	no	—
198	#462	—	89	1	no	—
199	#424	—	89	1	no	—
200	#419	—	89	1	no	—
201	#416	—	89	1	no	—
202	C₂×#41	—	89	1	no	—
203	C₂²×Q₈	—	89	1	no	—
204	C₂²×Q₈	—	89	1	no	—
205	#137	—	178	4	no	—
206	C₂×Q₁₆	—	89	1	no	—
207	C₂×Q₁₆	—	89	1	no	—
208	C₂×Q₁₆	—	89	1	no	—
209	C₂×Q₁₆	—	89	1	no	—
210	C₂×Q₁₆	—	89	1	no	—
211	C₂×Q₁₆	—	178	4	no	—
212	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
213	C₂×SD₁₆	—	89	1	no	—
214	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
215	C₂×M₁₆	—	89	1	no	—
216	C₂²×C₈	—	89	1	no	—
217	C₄⋊Q₈	—	89	1	no	—
218	C₄⋊Q₈	—	89	1	no	—
219	#124	—	89	1	no	—
220	#124	C₂×D₈₉	1	1	yes	characteristic
221	#123	—	178	1	no	—
222	#123	—	178	1	no	—
223	C₂²⋊Q₈	—	178	1	no	—
224	C₂²⋊Q₈	—	178	1	no	—
225	C₄×Q₈	—	89	1	no	—
226	C₄×Q₈	—	89	1	no	—
227	C₄×Q₈	—	89	1	no	—
228	C₄×Q₈	—	178	1	no	—
229	C₄×Q₈	—	178	1	no	—
230	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
231	#117	—	89	1	no	—
232	#117	—	89	1	no	—
233	#117	—	178	1	no	—
234	#117	—	178	1	no	—
235	C₄⋊C₈	—	89	1	no	—
236	C₄⋊C₈	—	89	1	no	—
237	C₄⋊C₈	—	89	1	no	—
238	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
239	#103	—	89	1	no	—
240	#103	—	89	1	no	—
241	#103	—	89	1	no	—
242	#103	—	89	1	no	—
243	#103	—	89	1	no	—
244	#103	C₂×D₈₉	1	1	yes	characteristic
245	#102	—	89	1	no	—
246	#102	C₂×D₈₉	1	1	yes	characteristic
247	#41	—	178	2	no	—
248	#41	—	178	2	no	—
249	C₂×Q₈	—	89	4	no	—
250	C₂×Q₈	—	89	1	no	—
251	C₂×Q₈	—	89	1	no	—
252	C₂×Q₈	—	89	1	no	—
253	C₂×Q₈	—	178	1	no	—
254	C₂×Q₈	—	178	1	no	—
255	C₂×Q₈	—	178	2	no	—
256	C₂×Q₈	—	178	2	no	—
257	C₂×Q₈	—	178	2	no	—
258	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
260	C₂×D₄	—	178	1	no	—
261	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
262	C₂²×C₄	—	89	1	no	—
263	C₂²×C₄	—	178	1	no	—
264	C₂²×C₄	—	178	1	no	—
265	C₂²×C₄	—	178	1	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	178	2	no	—
268	Q₁₆	—	178	2	no	—
269	Q₁₆	—	178	2	no	—
270	Q₁₆	—	178	2	no	—
271	Q₁₆	—	178	2	no	—
272	SD₁₆	—	2	2	no	—
273	SD₁₆	—	178	2	no	—
274	M₁₆	—	178	2	no	—
275	C₂×C₈	—	89	1	no	—
276	C₂×C₈	—	89	1	no	—
277	C₂×C₈	—	178	2	no	—
278	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
279	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	89	1	no	—
282	C₄⋊C₄	—	89	1	no	—
283	C₄⋊C₄	—	89	1	no	—
284	C₄⋊C₄	—	178	1	no	—
285	C₄⋊C₄	—	178	1	no	—
286	C₄⋊C₄	—	178	1	no	—
287	C₄⋊C₄	—	178	1	no	—
288	C₄⋊C₄	—	178	1	no	—
289	C₄⋊C₄	—	178	1	no	—
290	C₄⋊C₄	—	178	1	no	—
291	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	89	1	no	—
295	C₂²⋊C₄	—	89	1	no	—
296	C₂²⋊C₄	—	178	1	no	—
297	C₂²⋊C₄	—	178	1	no	—
298	C₂²⋊C₄	—	178	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	89	1	no	—
301	C₄²	—	178	1	no	—
302	C₄²	—	178	1	no	—
303	C₄²	—	178	1	no	—
304	C₄²	—	178	1	no	—
305	C₄²	C₂²×D₈₉	1	1	yes	characteristic
306	C₂³	—	2	1	no	—
307	C₂³	—	89	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	89	2	no	—
312	Q₈	—	356	2	no	—
313	Q₈	—	178	4	no	—
314	Q₈	—	178	1	no	—
315	Q₈	—	178	1	no	—
316	Q₈	—	178	1	no	—
317	Q₈	—	178	1	no	—
318	Q₈	—	178	1	no	—
319	Q₈	D₄×D₈₉	1	2	yes	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	356	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	89	1	no	—
327	C₂×C₄	—	89	2	no	—
328	C₂×C₄	—	89	2	no	—
329	C₂×C₄	—	89	1	no	—
330	C₂×C₄	—	89	1	no	—
331	C₂×C₄	—	356	2	no	—
332	C₂×C₄	—	356	2	no	—
333	C₂×C₄	—	178	4	no	—
334	C₂×C₄	—	178	1	no	—
335	C₂×C₄	—	178	1	no	—
336	C₂×C₄	—	178	1	no	—
337	C₂×C₄	—	178	1	no	—
338	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
339	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
340	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	178	1	no	—
344	C₈	—	178	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	89	2	no	—
349	C₂²	—	178	1	no	—
350	C₂²	—	178	1	no	—
351	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	89	2	no	—
358	C₄	—	356	1	no	—
359	C₄	—	356	1	no	—
360	C₄	—	356	1	no	—
361	C₄	—	178	1	no	—
362	C₄	—	178	1	no	—
363	C₄	—	178	2	no	—
364	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
365	C₄	#41×D₈₉	1	1	yes	characteristic
366	C₂	—	8	1	no	—
367	C₂	—	178	2	no	—
368	C₂	#121×D₈₉	1	1	yes	characteristic
369	C₂	#19152412	1	1	yes	characteristic
370	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#464×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083573	C₂²
hypercenter	#464	D₈₉
hypocenter	C₈₉	#3044
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×D₈₉	C₂	semidirect
4. #19152074	C₂	semidirect
5. #19151819	C₂	semidirect
6. #19151818	C₂	semidirect
7. #19151817	C₂	semidirect
8. #19151809	C₂	semidirect
9. #19151808	C₂	semidirect
10. #19151807	C₂	semidirect
11. #19151806	C₂	semidirect
12. #19151623	C₂	semidirect
13. #19151605	C₂	semidirect
15. #464×C₈₉	C₂	semidirect
186. C₈₉	#3044	semidirect
194. #464	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668041	2
C₂×Q₁₆×D₈₉	2
#19152168	2
#19152074	2
#19151819	2
#19151818	2
#19151817	2
#19151809	2
#19151808	2
#19151807	2
#19151806	2
#19151623	2
#19151605	2
#19151591	2
#464×C₈₉	2
#3044	89