Subgroups — #36470945

Order 22784.

404 conjugacy classes of subgroups.

The subgroup lattice has 336 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×#19151428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151377	C₂	1	1	yes	maximal, characteristic
5	#26020287	C₂	1	1	yes	maximal, characteristic
6	#26018345	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₈₉	—	2	1	no	—
9	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	C₂²×#13083549	C₂²	1	1	yes	characteristic
11	C₂×#15667948	C₂²	1	1	yes	characteristic
12	C₂×#15667948	C₂²	1	1	yes	characteristic
13	C₂×#15667947	—	2	1	no	—
14	#108×D₈₉	—	2	2	no	—
15	#108×D₈₉	—	2	2	no	—
16	#19151561	—	2	1	no	—
17	#19151428	—	2	2	no	—
18	#19151427	—	2	2	no	—
19	#19151377	C₄	1	2	yes	—
20	#19151377	C₄	1	2	yes	—
21	#19151362	—	2	1	no	—
22	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
26	C₂×#13083558	—	2	1	no	—
27	C₂×#13083558	—	2	1	no	—
28	C₂×#13083557	—	4	1	no	—
29	M₁₆×D₈₉	—	2	4	no	—
30	M₁₆×D₈₉	—	4	2	no	—
31	C₂²×#10623494	D₄	1	1	yes	characteristic
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	—	2	1	no	—
36	C₂×C₈×D₈₉	—	4	1	no	—
37	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
40	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
41	C₂×#13083549	—	2	1	no	—
42	C₂×#13083549	—	2	1	no	—
43	C₂×#13083549	C₂×C₄	1	2	yes	—
44	#15667948	—	2	1	no	—
45	#15667948	—	2	1	no	—
46	#15667948	—	2	1	no	—
47	#15667948	D₄	1	2	yes	—
48	#15667947	—	2	4	no	—
49	#15667924	—	4	1	no	—
50	#15667924	—	4	1	no	—
51	C₂×M₁₆×C₈₉	—	2	1	no	—
52	C₂×M₁₆×C₈₉	—	2	1	no	—
53	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₈₉	—	2	1	no	—
55	C₂×C₁₆×C₈₉	—	2	1	no	—
56	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
57	#108×C₈₉	—	2	1	no	—
58	#108×C₈₉	—	2	1	no	—
59	#108×C₈₉	—	2	1	no	—
60	#108×C₈₉	D₄	1	2	yes	—
61	C₂³×D₈₉	Q₁₆	1	1	yes	characteristic
62	C₂²×Dic₈₉	Q₁₆	1	1	yes	characteristic
63	C₂×C₄×D₈₉	—	2	1	no	—
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	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
70	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
71	#13083558	—	2	2	no	—
72	#13083558	—	4	1	no	—
73	#13083558	—	4	1	no	—
74	#13083558	—	4	1	no	—
75	#13083557	—	4	4	no	—
76	C₂×#10623494	—	2	1	no	—
77	C₂×#10623494	—	2	1	no	—
78	C₂×#10623494	—	4	1	no	—
79	C₂×#10623494	—	4	1	no	—
80	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
81	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
89	C₈×D₈₉	—	2	1	no	—
90	C₈×D₈₉	—	4	4	no	—
91	#13083549	—	2	2	no	—
92	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
93	M₁₆×C₈₉	—	2	2	no	—
94	M₁₆×C₈₉	—	4	1	no	—
95	M₁₆×C₈₉	—	4	1	no	—
96	M₁₆×C₈₉	—	4	1	no	—
97	C₂×C₈×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	—	4	1	no	—
99	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
104	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
105	C₁₆×C₈₉	—	2	2	no	—
106	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
107	C₂²×D₈₉	—	2	1	no	—
108	C₂²×D₈₉	—	2	1	no	—
109	C₂²×D₈₉	—	2	2	no	—
110	C₂²×D₈₉	—	4	1	no	—
111	C₂²×D₈₉	#107	1	1	yes	characteristic
112	C₂²×D₈₉	#107	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	#107	1	1	yes	characteristic
122	C₂×Dic₈₉	#107	1	1	yes	characteristic
123	C₄×D₈₉	—	2	1	no	—
124	C₄×D₈₉	—	2	1	no	—
125	C₄×D₈₉	—	2	1	no	—
126	C₄×D₈₉	—	2	1	no	—
127	C₄×D₈₉	—	2	1	no	—
128	C₄×D₈₉	—	2	1	no	—
129	C₄×D₈₉	—	2	1	no	—
130	C₄×D₈₉	—	2	1	no	—
131	C₈×C₈₉	—	4	2	no	—
132	C₈×C₈₉	—	8	1	no	—
133	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
134	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
135	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
137	#10623494	—	2	1	no	—
138	#10623494	—	2	1	no	—
139	#10623494	—	4	2	no	—
140	#10623494	—	8	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
144	C₂²×C₈₉	#451	1	1	yes	characteristic
145	C₂²×C₈₉	#484	1	1	yes	characteristic
146	C₂×D₈₉	—	2	1	no	—
147	C₂×D₈₉	—	2	1	no	—
148	C₂×D₈₉	—	2	1	no	—
149	C₂×D₈₉	—	2	1	no	—
150	C₂×D₈₉	—	2	1	no	—
151	C₂×D₈₉	—	4	2	no	—
152	C₂×D₈₉	—	4	2	no	—
153	C₂×D₈₉	#368	1	2	yes	complemented
154	C₄×C₈₉	#385	1	1	yes	characteristic
155	C₄×C₈₉	#428	1	1	yes	characteristic
156	C₄×C₈₉	#447	1	1	yes	characteristic
157	C₄×C₈₉	#480	1	1	yes	characteristic
158	Dic₈₉	—	2	1	no	—
159	Dic₈₉	—	2	1	no	—
160	#12922	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	—	2	1	no	—
162	C₂×C₈₉	—	2	1	no	—
163	C₂×C₈₉	#1852	1	1	yes	characteristic
164	C₂×C₈₉	C₂×#368	1	1	yes	characteristic
165	C₂×C₈₉	#2161	1	1	yes	characteristic
166	D₈₉	—	2	2	no	—
167	D₈₉	—	4	1	no	—
168	C₂²×#108	—	89	1	no	—
169	C₂×#368	—	89	1	no	—
170	C₂×#368	—	89	1	no	—
171	C₂×C₂²⋊C₁₆	—	89	1	no	—
172	#1810	—	89	1	no	—
173	#1381	—	89	1	no	—
174	#1381	D₈₉	1	1	yes	characteristic, complemented
175	C₈₉	#12922	1	1	yes	characteristic, complemented, Sylow
176	C₂²×M₁₆	—	178	1	no	—
177	C₂³×C₈	—	89	1	no	—
178	C₂²×C₁₆	—	89	1	no	—
179	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
180	C₂×#108	—	89	1	no	—
181	C₂×#108	—	89	1	no	—
182	C₂×#108	—	178	2	no	—
183	C₂×#108	—	178	1	no	—
184	C₂×#108	—	178	2	no	—
185	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
186	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
187	#407	—	178	1	no	—
188	#368	—	178	2	no	—
189	#368	—	178	2	no	—
190	C₂²⋊C₁₆	—	89	2	no	—
191	C₂²⋊C₁₆	—	89	2	no	—
192	#341	—	178	1	no	—
193	C₂³×C₄	—	89	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	356	1	no	—
197	C₂×M₁₆	—	356	2	no	—
198	C₂×M₁₆	—	178	4	no	—
199	C₂×M₁₆	—	178	1	no	—
200	C₂×M₁₆	—	178	1	no	—
201	C₂²×C₈	—	89	1	no	—
202	C₂²×C₈	—	89	1	no	—
203	C₂²×C₈	—	89	1	no	—
204	C₂²×C₈	—	89	1	no	—
205	C₂²×C₈	—	89	1	no	—
206	C₂²×C₈	—	356	1	no	—
207	C₂²×C₈	—	178	1	no	—
208	C₂²×C₈	—	178	1	no	—
209	C₂²×C₈	—	178	1	no	—
210	C₂²×C₈	—	178	1	no	—
211	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	89	2	no	—
215	C₂×C₁₆	—	178	1	no	—
216	C₂×C₁₆	—	178	1	no	—
217	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	—	2	1	no	—
221	#108	—	89	2	no	—
222	#108	—	178	4	no	—
223	#108	—	178	1	no	—
224	#108	—	178	1	no	—
225	#108	—	178	1	no	—
226	#108	D₃₅₆	1	2	yes	—
227	C₂²⋊C₈	—	356	1	no	—
228	C₂²⋊C₈	—	356	1	no	—
229	C₂⁴	—	89	1	no	—
230	C₂²×C₄	—	89	1	no	—
231	C₂²×C₄	—	89	1	no	—
232	C₂²×C₄	—	89	1	no	—
233	C₂²×C₄	—	89	1	no	—
234	C₂²×C₄	—	89	1	no	—
235	C₂²×C₄	—	178	1	no	—
236	C₂²×C₄	—	178	1	no	—
237	C₂²×C₄	—	178	1	no	—
238	C₂²×C₄	—	178	1	no	—
239	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	4	1	no	—
244	M₁₆	—	356	4	no	—
245	M₁₆	—	356	1	no	—
246	M₁₆	—	356	1	no	—
247	M₁₆	—	356	1	no	—
248	M₁₆	—	178	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	89	1	no	—
252	C₂×C₈	—	89	1	no	—
253	C₂×C₈	—	356	4	no	—
254	C₂×C₈	—	356	1	no	—
255	C₂×C₈	—	356	1	no	—
256	C₂×C₈	—	178	1	no	—
257	C₂×C₈	—	178	1	no	—
258	C₂×C₈	—	178	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	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
269	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
270	C₂×C₈	#13083573	1	1	yes	characteristic
271	C₂×C₈	#13083574	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₁₆	—	178	2	no	—
274	C₂³	—	89	1	no	—
275	C₂³	—	89	1	no	—
276	C₂³	—	356	1	no	—
277	C₂³	—	178	2	no	—
278	C₂³	—	178	1	no	—
279	C₂³	—	178	1	no	—
280	C₂³	Q₁₆×D₈₉	1	1	yes	characteristic
281	C₂×C₄	—	89	1	no	—
282	C₂×C₄	—	89	1	no	—
283	C₂×C₄	—	178	1	no	—
284	C₂×C₄	—	178	1	no	—
285	C₂×C₄	—	178	1	no	—
286	C₂×C₄	—	178	1	no	—
287	C₂×C₄	—	178	1	no	—
288	C₂×C₄	—	178	1	no	—
289	C₂×C₄	—	178	1	no	—
290	C₂×C₄	—	178	1	no	—
291	C₂×C₄	—	178	1	no	—
292	C₂×C₄	—	178	1	no	—
293	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
294	C₂×C₄	#15667982	1	1	yes	characteristic
295	C₂×C₄	#15667994	1	1	yes	characteristic
296	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
297	C₂×C₄	#15668039	1	1	yes	characteristic
298	C₂×C₄	#15668041	1	1	yes	characteristic
299	C₈	—	4	2	no	—
300	C₈	—	8	1	no	—
301	C₈	—	356	2	no	—
302	C₈	—	178	1	no	—
303	C₈	—	178	1	no	—
304	C₈	—	712	1	no	—
305	C₈	#15667972	1	1	yes	characteristic
306	C₈	C₄×D₃₅₆	1	1	yes	characteristic
307	C₈	#15667990	1	1	yes	characteristic
308	C₈	#15667992	1	1	yes	characteristic
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	89	2	no	—
312	C₂²	—	356	2	no	—
313	C₂²	—	356	2	no	—
314	C₂²	—	178	1	no	—
315	C₂²	—	178	1	no	—
316	C₂²	—	178	1	no	—
317	C₂²	—	178	1	no	—
318	C₂²	—	178	1	no	—
319	C₂²	#107×D₈₉	1	1	yes	characteristic
320	C₂²	#19151755	1	1	yes	characteristic
321	C₂²	#19151932	1	1	yes	characteristic
322	C₄	—	178	1	no	—
323	C₄	—	178	1	no	—
324	C₄	#19151472	1	1	yes	characteristic
325	C₄	#19151637	1	1	yes	characteristic
326	C₄	#19151733	1	1	yes	characteristic
327	C₄	#19151910	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	356	1	no	—
331	C₂	—	178	2	no	—
332	C₂	#26020441	1	1	yes	characteristic
333	C₂	#368×D₈₉	1	1	yes	characteristic
334	C₂	#26022045	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1381	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×D₈₉	Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1381	D₈₉
hypocenter	C₈₉	#12922
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

Complements

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

N	H ≅ G/N	type
2. C₂×#19151428	C₂	semidirect
3. C₂×#19151427	C₂	semidirect
6. #26018345	C₂	semidirect
7. C₈₉×#1381	C₂	semidirect
153. C₂×D₈₉	#368	semidirect
174. #1381	D₈₉	semidirect
175. C₈₉	#12922	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×#19151428	2
C₂×#19151427	2
C₂×#19151377	2
#26020287	2
#26018345	2
C₈₉×#1381	2
#12922	89