Subgroups — #36445464

Order 22784.

444 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029221	C₂	1	1	yes	maximal, characteristic
2	#26019939	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018841	C₂	1	1	yes	maximal, characteristic
4	#26018736	C₂	1	1	yes	maximal, characteristic
5	M₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1300	C₂	1	1	yes	maximal, characteristic, complemented
8	#19152132	C₂²	1	1	yes	characteristic
9	#19151725	C₄	1	1	yes	characteristic, complemented
10	#19151722	C₄	1	1	yes	characteristic, complemented
11	#19151675	C₄	1	1	yes	characteristic, complemented
12	#19151666	—	2	1	no	—
13	#19151663	C₄	1	1	yes	characteristic, complemented
14	#19151661	—	2	1	no	—
15	#19151660	—	2	1	no	—
16	#19151651	C₂²	1	1	yes	characteristic
17	#19151555	—	2	1	no	—
18	C₂×#15667923	—	2	1	no	—
19	C₂×#15667923	C₂²	1	1	yes	characteristic
20	C₂×C₄×#10623494	—	2	1	no	—
21	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
22	#19151339	C₄	1	2	yes	complemented
23	#19151337	C₄	1	2	yes	complemented
24	#19151334	C₄	1	2	yes	—
25	#19151331	C₄	1	2	yes	—
26	#19151319	—	2	1	no	—
27	C₈×#10623494	—	2	1	no	—
28	#432×C₈₉	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	#376×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#13083557	—	2	1	no	—
32	C₂²×#10623494	—	2	1	no	—
33	C₂²×#10623494	D₄	1	1	yes	characteristic
34	C₂²×#10623494	D₄	1	1	yes	characteristic
35	C₂×C₈×D₈₉	—	2	1	no	—
36	#15668006	—	2	1	no	—
37	#15668004	C₂×C₄	1	1	yes	characteristic
38	#15668000	C₂×C₄	1	1	yes	characteristic
39	#15667995	—	2	1	no	—
40	#15667991	C₂×C₄	1	1	yes	characteristic
41	#15667986	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
43	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
44	C₄×C₂²⋊D₈₉	—	2	1	no	—
45	#15667969	—	2	1	no	—
46	C₄⋊#10623494	—	2	1	no	—
47	#15667941	—	2	1	no	—
48	#15667941	—	2	1	no	—
49	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
50	#15667924	—	2	1	no	—
51	#15667924	—	2	1	no	—
52	#15667923	—	2	4	no	—
53	#15667923	—	2	1	no	—
54	#15667923	C₂×C₄	1	2	yes	—
55	#15667922	—	2	1	no	—
56	C₄×#10623494	—	2	4	no	—
57	C₄×#10623494	—	2	1	no	—
58	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
59	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
60	C₄×#10623494	D₄	1	1	yes	characteristic
61	C₄×#10623494	D₄	1	1	yes	characteristic
62	C₈×Dic₈₉	—	2	1	no	—
63	C₂×M₁₆×C₈₉	—	2	1	no	—
64	C₂×C₄²×C₈₉	—	2	1	no	—
65	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊C₈×C₈₉	—	2	1	no	—
69	#97×C₈₉	—	2	1	no	—
70	#97×C₈₉	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₈₉	—	2	1	no	—
72	#95×C₈₉	—	2	1	no	—
73	#95×C₈₉	C₂×C₄	1	2	yes	—
74	C₂×Dic₁₇₈	C₄²	1	1	yes	characteristic
75	C₂×D₃₅₆	C₄²	1	1	yes	characteristic
76	C₂×C₂²⋊D₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	—	2	1	no	—
78	#13083557	—	4	2	no	—
79	C₂×#10623494	—	2	2	no	—
80	C₂×#10623494	—	2	2	no	—
81	C₂×#10623494	—	2	2	no	—
82	C₂×#10623494	—	2	2	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₂×#10623494	—	2	2	no	—
85	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
89	C₈×D₈₉	—	4	2	no	—
90	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
91	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
92	#13083553	—	2	1	no	—
93	#13083553	—	2	1	no	—
94	#13083553	—	2	1	no	—
95	#13083553	—	2	1	no	—
96	C₂²⋊Dic₈₉	—	2	1	no	—
97	#13083551	—	2	1	no	—
98	#13083551	—	2	1	no	—
99	#13083551	—	2	1	no	—
100	#13083551	—	2	1	no	—
101	C₄×Dic₈₉	—	2	1	no	—
102	C₂²×C₄×C₈₉	—	2	2	no	—
103	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
106	M₁₆×C₈₉	—	2	4	no	—
107	C₂×C₈×C₈₉	—	2	2	no	—
108	C₂×C₈×C₈₉	—	2	1	no	—
109	C₂×C₈×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	—	2	2	no	—
111	C₄²×C₈₉	—	2	2	no	—
112	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	#41	1	1	yes	characteristic
116	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	4	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
131	C₂²⋊D₈₉	—	4	2	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	D₃₅₆	—	4	1	no	—
136	C₄×D₈₉	—	4	1	no	—
137	C₄×D₈₉	—	4	1	no	—
138	Dic₁₇₈	—	4	1	no	—
139	C₈×C₈₉	—	2	4	no	—
140	C₈×C₈₉	—	4	1	no	—
141	C₈×C₈₉	—	4	1	no	—
142	#10623494	—	2	4	no	—
143	#10623494	—	2	1	no	—
144	#10623494	—	2	1	no	—
145	#10623494	—	4	2	no	—
146	#10623494	#104	1	2	yes	—
147	#10623494	#104	1	2	yes	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₈₉	#443	1	2	yes	—
152	C₂×D₈₉	—	4	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₂×D₈₉	—	4	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	4	4	no	—
161	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
162	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
163	Dic₈₉	—	4	1	no	—
164	Dic₈₉	—	4	1	no	—
165	Dic₈₉	—	4	1	no	—
166	#8247	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	—	2	2	no	—
168	C₂×C₈₉	#1757	1	1	yes	characteristic
169	C₂×C₈₉	C₄×#104	1	1	yes	characteristic
170	C₂×C₈₉	#1764	1	1	yes	characteristic
171	D₈₉	—	8	1	no	—
172	#2966	—	89	1	no	—
173	#1726	—	89	1	no	—
174	#1524	—	89	1	no	—
175	#1500	—	89	1	no	—
176	C₈×M₁₆	—	89	1	no	—
177	#1302	—	89	1	no	—
178	#1300	D₈₉	1	1	yes	characteristic, complemented
179	C₈₉	#8247	1	1	yes	characteristic, complemented, Sylow
180	#529	—	89	1	no	—
181	#446	—	89	1	no	—
182	C₈×Q₈	—	89	1	no	—
183	#436	—	89	1	no	—
184	#435	—	178	1	no	—
185	C₈×D₄	—	89	1	no	—
186	C₈×D₄	—	178	1	no	—
187	#433	—	178	1	no	—
188	#432	C₂×D₈₉	1	1	yes	characteristic
189	#431	—	89	1	no	—
190	#405	—	178	1	no	—
191	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
192	C₂×#97	—	89	1	no	—
193	C₂×#97	—	178	1	no	—
194	C₂×C₄×C₈	—	89	1	no	—
195	C₂×C₄×C₈	—	178	1	no	—
196	#376	C₂×D₈₉	1	1	yes	characteristic
197	#330	—	89	2	no	—
198	#329	—	89	2	no	—
199	#328	—	89	2	no	—
200	#327	—	89	2	no	—
201	#322	—	178	1	no	—
202	C₈²	—	178	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	178	1	no	—
205	C₂²×C₈	—	89	1	no	—
206	C₂²×C₈	—	89	1	no	—
207	C₂²×C₈	—	178	1	no	—
208	C₂²×C₈	—	178	1	no	—
209	#126	—	178	1	no	—
210	#125	—	89	1	no	—
211	#124	—	89	1	no	—
212	#123	—	178	1	no	—
213	C₂²⋊Q₈	—	89	1	no	—
214	#121	—	89	1	no	—
215	C₄×Q₈	—	89	1	no	—
216	C₄×D₄	—	89	1	no	—
217	C₄×D₄	—	178	1	no	—
218	#117	—	178	1	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
221	C₄⋊C₈	—	178	1	no	—
222	C₄⋊C₈	—	178	1	no	—
223	C₄⋊C₈	—	178	1	no	—
224	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
225	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	89	2	no	—
228	C₂²⋊C₈	—	178	1	no	—
229	C₂²⋊C₈	—	178	1	no	—
230	#97	—	2	1	no	—
231	#97	—	89	2	no	—
232	#97	—	178	4	no	—
233	#97	—	178	1	no	—
234	#97	—	178	1	no	—
235	#97	C₄×D₈₉	1	2	yes	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	89	1	no	—
238	C₄×C₈	—	89	1	no	—
239	C₄×C₈	—	89	1	no	—
240	C₄×C₈	—	89	1	no	—
241	C₄×C₈	—	178	4	no	—
242	C₄×C₈	—	178	1	no	—
243	C₄×C₈	—	178	1	no	—
244	#95	—	2	1	no	—
245	#95	C₄×D₈₉	1	2	yes	—
246	C₂×Q₈	—	89	1	no	—
247	C₂×D₄	—	89	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	178	1	no	—
251	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	#13083573	1	1	yes	characteristic
254	M₁₆	—	2	4	no	—
255	M₁₆	—	356	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	—	89	1	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	356	2	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	—	178	2	no	—
266	C₂×C₈	—	178	2	no	—
267	C₂×C₈	—	178	2	no	—
268	C₂×C₈	—	178	2	no	—
269	C₂×C₈	—	178	2	no	—
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	89	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₂²⋊C₄	—	178	1	no	—
277	C₂²⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	178	1	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	178	1	no	—
284	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
286	C₄²	D₄×D₈₉	1	1	yes	characteristic
287	C₄²	#13083575	1	1	yes	characteristic
288	C₂³	—	178	1	no	—
289	C₂³	#15667972	1	1	yes	characteristic
290	Q₈	—	356	1	no	—
291	D₄	—	356	1	no	—
292	D₄	—	356	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	1	no	—
302	C₂×C₄	—	178	1	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	#15667970	1	1	yes	characteristic
308	C₂×C₄	#15667972	1	1	yes	characteristic
309	C₂×C₄	#15667974	1	1	yes	characteristic
310	C₂×C₄	#15667974	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	4	1	no	—
313	C₈	—	4	1	no	—
314	C₈	—	89	2	no	—
315	C₈	—	89	2	no	—
316	C₈	—	356	2	no	—
317	C₈	—	178	4	no	—
318	C₈	—	178	1	no	—
319	C₈	—	178	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	356	1	no	—
323	C₂²	—	356	1	no	—
324	C₂²	—	356	1	no	—
325	C₂²	#19151445	1	1	yes	characteristic
326	C₂²	#19151712	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	356	1	no	—
334	C₄	—	356	1	no	—
335	C₄	—	356	1	no	—
336	C₄	#104×D₈₉	1	1	yes	characteristic
337	C₄	#104×D₈₉	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	712	1	no	—
340	C₂	#26020051	1	1	yes	characteristic
341	C₂	#26020056	1	1	yes	characteristic
342	C₂	#26020086	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1300	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	#15667986	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1300	D₈₉
hypocenter	C₈₉	#8247
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1300

Complements

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

N	H ≅ G/N	type
2. #26019939	C₂	semidirect
5. M₁₆×#10623494	C₂	semidirect
6. #26018150	C₂	semidirect
7. C₈₉×#1300	C₂	semidirect
9. #19151725	C₄	semidirect
10. #19151722	C₄	semidirect
11. #19151675	C₄	semidirect
13. #19151663	C₄	semidirect
22. #19151339	C₄	semidirect
23. #19151337	C₄	semidirect
178. #1300	D₈₉	semidirect
179. C₈₉	#8247	semidirect

Maximal subgroups

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

subgroup	index
#26029221	2
#26019939	2
#26018841	2
#26018736	2
M₁₆×#10623494	2
#26018150	2
C₈₉×#1300	2
#8247	89