Subgroups — #36470978

Order 22784.

420 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151420	C₂	1	1	yes	maximal, characteristic, complemented
3	#26021862	C₂	1	1	yes	maximal, characteristic
4	#26020274	C₂	1	1	yes	maximal, characteristic
5	#26018352	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018351	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#365×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₈₉	—	2	1	no	—
9	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	C₂×#15667948	C₂²	1	1	yes	characteristic
12	C₂×#15667947	—	2	1	no	—
13	#108×D₈₉	—	2	2	no	—
14	#108×D₈₉	—	2	2	no	—
15	C₂×#15667944	C₂²	1	1	yes	characteristic
16	#19151455	—	2	1	no	—
17	#19151420	—	2	2	no	—
18	#19151377	C₄	1	2	yes	complemented
19	#19151377	C₄	1	2	yes	complemented
20	#19151360	—	2	1	no	—
21	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
24	#365×C₈₉	—	2	2	no	—
25	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
26	C₂×#13083558	—	2	1	no	—
27	C₂×#13083557	—	4	1	no	—
28	M₁₆×D₈₉	—	2	4	no	—
29	M₁₆×D₈₉	—	4	2	no	—
30	C₂²×#10623494	D₄	1	1	yes	characteristic
31	C₂×C₈×D₈₉	—	2	1	no	—
32	C₂×C₈×D₈₉	—	2	1	no	—
33	C₂×C₈×D₈₉	—	2	1	no	—
34	C₂×C₈×D₈₉	—	2	1	no	—
35	C₂×C₈×D₈₉	—	4	1	no	—
36	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
39	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
40	C₂×C₄⋊Dic₈₉	—	2	1	no	—
41	C₂×#13083551	—	4	1	no	—
42	#15667952	—	2	1	no	—
43	#15667952	—	2	1	no	—
44	C₂×#13083549	C₂×C₄	1	2	yes	—
45	#15667948	—	2	1	no	—
46	#15667948	D₄	1	2	yes	—
47	#15667947	—	2	4	no	—
48	#15667944	—	2	1	no	—
49	#15667944	—	2	1	no	—
50	#15667919	—	4	1	no	—
51	C₂×M₁₆×C₈₉	—	2	1	no	—
52	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
54	M₃₂×C₈₉	—	2	1	no	—
55	M₃₂×C₈₉	—	2	1	no	—
56	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
57	#108×C₈₉	—	2	1	no	—
58	#108×C₈₉	D₄	1	2	yes	—
59	#106×C₈₉	—	2	1	no	—
60	#106×C₈₉	—	2	1	no	—
61	C₂³×D₈₉	SD₁₆	1	1	yes	characteristic
62	C₂²×Dic₈₉	—	4	1	no	—
63	C₂²×Dic₈₉	SD₁₆	1	1	yes	characteristic
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×D₈₉	SD₁₆	1	1	yes	characteristic
71	C₂×C₄×D₈₉	SD₁₆	1	1	yes	characteristic
72	#13083558	—	2	2	no	—
73	#13083558	—	4	1	no	—
74	#13083557	—	4	4	no	—
75	C₂×#10623494	—	2	1	no	—
76	C₂×#10623494	—	2	1	no	—
77	C₂×#10623494	—	4	1	no	—
78	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
79	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
80	C₈×D₈₉	—	2	1	no	—
81	C₈×D₈₉	—	2	1	no	—
82	C₈×D₈₉	—	2	1	no	—
83	C₈×D₈₉	—	2	1	no	—
84	C₈×D₈₉	—	2	1	no	—
85	C₈×D₈₉	—	2	1	no	—
86	C₈×D₈₉	—	2	1	no	—
87	C₈×D₈₉	—	2	1	no	—
88	C₈×D₈₉	—	4	4	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	#13083551	—	8	2	no	—
92	#13083549	—	2	2	no	—
93	C₂²×C₄×C₈₉	—	4	1	no	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	M₁₆×C₈₉	—	2	2	no	—
96	M₁₆×C₈₉	—	4	1	no	—
97	C₂×C₈×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
102	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
103	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	—	4	1	no	—
105	C₄⋊C₄×C₈₉	—	4	1	no	—
106	C₁₆×C₈₉	—	2	2	no	—
107	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
108	C₂²×D₈₉	—	2	1	no	—
109	C₂²×D₈₉	—	2	1	no	—
110	C₂²×D₈₉	—	2	2	no	—
111	C₂²×D₈₉	—	4	1	no	—
112	C₂²×D₈₉	#106	1	1	yes	characteristic
113	C₂²×D₈₉	#106	1	1	yes	characteristic
114	C₂×C₄×C₈₉	—	8	1	no	—
115	C₂×C₄×C₈₉	—	8	1	no	—
116	C₂×C₄×C₈₉	—	8	1	no	—
117	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
119	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	8	1	no	—
126	C₂×Dic₈₉	—	8	1	no	—
127	C₂×Dic₈₉	—	8	1	no	—
128	C₂×Dic₈₉	#106	1	1	yes	characteristic
129	C₂×Dic₈₉	#106	1	1	yes	characteristic
130	C₄×D₈₉	—	2	1	no	—
131	C₄×D₈₉	—	2	1	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	2	1	no	—
134	C₄×D₈₉	—	2	1	no	—
135	C₄×D₈₉	—	2	1	no	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	2	1	no	—
138	C₈×C₈₉	—	4	2	no	—
139	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
140	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
141	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
142	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
143	#10623494	—	2	1	no	—
144	#10623494	—	2	1	no	—
145	#10623494	—	4	2	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
149	C₂²×C₈₉	#448	1	1	yes	characteristic
150	C₂²×C₈₉	#481	1	1	yes	characteristic
151	C₂×D₈₉	—	2	1	no	—
152	C₂×D₈₉	—	2	1	no	—
153	C₂×D₈₉	—	2	1	no	—
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	4	2	no	—
157	C₂×D₈₉	—	4	2	no	—
158	C₂×D₈₉	#365	1	2	yes	complemented
159	C₄×C₈₉	—	16	1	no	—
160	C₄×C₈₉	#385	1	1	yes	characteristic
161	C₄×C₈₉	#428	1	1	yes	characteristic
162	C₄×C₈₉	#450	1	1	yes	characteristic
163	C₄×C₈₉	#483	1	1	yes	characteristic
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	2	1	no	—
166	Dic₈₉	—	16	1	no	—
167	#12928	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	2	1	no	—
169	C₂×C₈₉	—	2	1	no	—
170	C₂×C₈₉	#1853	1	1	yes	characteristic
171	C₂×C₈₉	C₂×#365	1	1	yes	characteristic
172	C₂×C₈₉	#2155	1	1	yes	characteristic
173	D₈₉	—	2	2	no	—
174	D₈₉	—	4	1	no	—
175	C₂²×#108	—	89	1	no	—
176	C₂×#365	—	89	1	no	—
177	C₂×#365	D₈₉	1	1	yes	characteristic, complemented
178	#2112	—	89	1	no	—
179	#1807	—	89	1	no	—
180	#1383	—	89	1	no	—
181	#1383	—	89	1	no	—
182	C₈₉	#12928	1	1	yes	characteristic, complemented, Sylow
183	C₂²×M₁₆	—	178	1	no	—
184	C₂³×C₈	—	89	1	no	—
185	C₂×M₃₂	—	89	1	no	—
186	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
187	C₂×#108	—	89	1	no	—
188	C₂×#108	—	178	2	no	—
189	C₂×#108	—	178	1	no	—
190	C₂×#108	—	178	2	no	—
191	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
192	C₂×#106	—	89	1	no	—
193	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
194	#380	—	178	1	no	—
195	#365	—	2	2	no	—
196	#365	—	178	2	no	—
197	C₂²⋊C₁₆	—	89	2	no	—
198	C₂²⋊C₁₆	—	89	2	no	—
199	#340	—	178	1	no	—
200	C₂³×C₄	—	89	1	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	356	1	no	—
203	C₂×M₁₆	—	356	2	no	—
204	C₂×M₁₆	—	178	4	no	—
205	C₂×M₁₆	—	178	1	no	—
206	C₂²×C₈	—	89	1	no	—
207	C₂²×C₈	—	89	1	no	—
208	C₂²×C₈	—	89	1	no	—
209	C₂²×C₈	—	89	1	no	—
210	C₂²×C₈	—	89	1	no	—
211	C₂²×C₈	—	356	1	no	—
212	C₂²×C₈	—	178	1	no	—
213	C₂²×C₈	—	178	1	no	—
214	C₂²×C₈	—	178	1	no	—
215	C₂²×C₈	—	178	1	no	—
216	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	178	1	no	—
219	C₂×C₂²⋊C₄	—	356	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	M₃₂	—	178	1	no	—
223	M₃₂	—	178	1	no	—
224	C₂×C₁₆	—	89	2	no	—
225	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
226	#108	—	2	1	no	—
227	#108	—	89	2	no	—
228	#108	—	178	4	no	—
229	#108	—	178	1	no	—
230	#108	C₂²⋊D₈₉	1	2	yes	—
231	#106	—	2	1	no	—
232	#106	—	2	1	no	—
233	#106	—	178	1	no	—
234	#106	—	178	1	no	—
235	#95	—	356	1	no	—
236	C₂⁴	—	89	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	89	1	no	—
239	C₂²×C₄	—	89	1	no	—
240	C₂²×C₄	—	89	1	no	—
241	C₂²×C₄	—	89	1	no	—
242	C₂²×C₄	—	89	1	no	—
243	C₂²×C₄	—	356	1	no	—
244	C₂²×C₄	—	178	1	no	—
245	C₂²×C₄	—	178	1	no	—
246	C₂²×C₄	—	178	1	no	—
247	C₂²×C₄	—	178	1	no	—
248	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
249	M₁₆	—	2	2	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	356	4	no	—
252	M₁₆	—	356	1	no	—
253	M₁₆	—	178	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	89	1	no	—
256	C₂×C₈	—	89	1	no	—
257	C₂×C₈	—	356	4	no	—
258	C₂×C₈	—	356	1	no	—
259	C₂×C₈	—	178	1	no	—
260	C₂×C₈	—	178	1	no	—
261	C₂×C₈	—	178	1	no	—
262	C₂×C₈	—	178	1	no	—
263	C₂×C₈	—	178	1	no	—
264	C₂×C₈	—	178	1	no	—
265	C₂×C₈	—	178	1	no	—
266	C₂×C₈	—	178	1	no	—
267	C₂×C₈	—	178	1	no	—
268	C₂×C₈	—	178	1	no	—
269	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
270	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
272	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
273	C₂×C₈	#13083574	1	1	yes	characteristic
274	C₂×C₈	#13083575	1	1	yes	characteristic
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	356	1	no	—
278	C₄⋊C₄	—	356	1	no	—
279	C₂²⋊C₄	—	712	2	no	—
280	C₁₆	—	2	2	no	—
281	C₁₆	—	178	2	no	—
282	C₂³	—	89	1	no	—
283	C₂³	—	89	1	no	—
284	C₂³	—	356	1	no	—
285	C₂³	—	178	2	no	—
286	C₂³	—	178	1	no	—
287	C₂³	—	178	1	no	—
288	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	89	1	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	178	1	no	—
295	C₂×C₄	—	178	1	no	—
296	C₂×C₄	—	178	1	no	—
297	C₂×C₄	—	178	1	no	—
298	C₂×C₄	—	178	1	no	—
299	C₂×C₄	—	178	1	no	—
300	C₂×C₄	—	178	1	no	—
301	C₂×C₄	—	178	1	no	—
302	C₂×C₄	—	178	1	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	712	1	no	—
305	C₂×C₄	—	712	1	no	—
306	C₂×C₄	—	712	1	no	—
307	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
308	C₂×C₄	#15667980	1	1	yes	characteristic
309	C₂×C₄	#15667997	1	1	yes	characteristic
310	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
311	C₂×C₄	#15668039	1	1	yes	characteristic
312	C₂×C₄	#15668041	1	1	yes	characteristic
313	C₈	—	4	2	no	—
314	C₈	—	356	2	no	—
315	C₈	—	178	1	no	—
316	C₈	—	178	1	no	—
317	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
318	C₈	#15667974	1	1	yes	characteristic
319	C₈	#15667988	1	1	yes	characteristic
320	C₈	#15667990	1	1	yes	characteristic
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	89	2	no	—
324	C₂²	—	356	2	no	—
325	C₂²	—	356	2	no	—
326	C₂²	—	178	1	no	—
327	C₂²	—	178	1	no	—
328	C₂²	—	178	1	no	—
329	C₂²	—	178	1	no	—
330	C₂²	—	178	1	no	—
331	C₂²	#106×D₈₉	1	1	yes	characteristic
332	C₂²	#19151735	1	1	yes	characteristic
333	C₂²	#19151918	1	1	yes	characteristic
334	C₄	—	16	1	no	—
335	C₄	—	1424	1	no	—
336	C₄	—	178	1	no	—
337	C₄	—	178	1	no	—
338	C₄	#19151469	1	1	yes	characteristic
339	C₄	#19151637	1	1	yes	characteristic
340	C₄	#19151747	1	1	yes	characteristic
341	C₄	#19151930	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	356	1	no	—
345	C₂	—	178	2	no	—
346	C₂	#26020443	1	1	yes	characteristic
347	C₂	#365×D₈₉	1	1	yes	characteristic
348	C₂	#26022023	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#106×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#365×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#106
prime radical	C₂³×D₈₉	SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#365	D₈₉
hypocenter	C₈₉	#12928
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
2. C₂×#19151420	C₂	semidirect
5. #26018352	C₂	semidirect
6. #26018351	C₂	semidirect
7. C₂×#365×C₈₉	C₂	semidirect
18. #19151377	C₄	semidirect
19. #19151377	C₄	semidirect
158. C₂×D₈₉	#365	semidirect
177. C₂×#365	D₈₉	semidirect
182. C₈₉	#12928	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108×D₈₉	2
C₂×#19151420	2
#26021862	2
#26020274	2
#26018352	2
#26018351	2
C₂×#365×C₈₉	2
#12928	89