Subgroups — #36523632

Order 22784.

572 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029366	C₂	1	2	yes	maximal
2	#26028894	C₂	1	1	yes	maximal, characteristic
3	#26019039	C₂	1	2	yes	maximal, complemented
4	#26019034	C₂	1	2	yes	maximal, complemented
5	#26019029	C₂	1	2	yes	maximal, complemented
6	#26019021	C₂	1	2	yes	maximal, complemented
7	#26019017	C₂	1	2	yes	maximal, complemented
8	#26019015	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13083574	C₂²	1	1	yes	characteristic
11	#19151727	—	2	2	no	—
12	#19151674	—	2	2	no	—
13	#19151668	—	2	2	no	—
14	#19151667	—	2	2	no	—
15	#19151657	C₄	1	4	yes	—
16	#19151655	C₂²	1	2	yes	—
17	#19151654	C₂²	1	2	yes	—
18	#19151650	C₂²	1	2	yes	—
19	#19151649	C₂²	1	2	yes	—
20	#19151649	C₂²	1	2	yes	—
21	#19151624	C₄	1	2	yes	—
22	#19151620	C₂²	1	2	yes	—
23	C₄⋊C₈×D₈₉	C₂²	1	2	yes	—
24	#19151564	C₄	1	2	yes	—
25	C₄×#13083558	C₂²	1	1	yes	characteristic
26	#19151351	C₂²	1	2	yes	—
27	#19151350	C₂²	1	2	yes	—
28	#19151350	C₂²	1	2	yes	—
29	#19151349	C₂²	1	1	yes	characteristic
30	#19151349	C₂²	1	1	yes	characteristic
31	#19151348	C₂²	1	2	yes	—
32	#19151348	C₂²	1	2	yes	—
33	#423×C₈₉	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
35	#334×C₈₉	C₂²	1	2	yes	—
36	#334×C₈₉	C₂²	1	1	yes	characteristic
37	#334×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
39	C₂×#13083558	C₂×C₄	1	2	yes	—
40	C₂×#13083557	—	2	2	no	—
41	C₂×C₈×D₈₉	—	2	2	no	—
42	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
43	C₄×Dic₁₇₈	Q₈	1	1	yes	characteristic
44	C₄×D₃₅₆	D₄	1	1	yes	characteristic
45	C₄×D₃₅₆	Q₈	1	1	yes	characteristic
46	C₄×C₂²⋊D₈₉	D₄	1	2	yes	—
47	C₄×C₂²⋊D₈₉	Q₈	1	2	yes	—
48	#15667971	C₂×C₄	1	1	yes	characteristic
49	#15667969	C₂×C₄	1	2	yes	—
50	C₄²×D₈₉	C₂³	1	2	yes	—
51	C₄⋊#10623494	C₂³	1	2	yes	—
52	C₄⋊#10623494	C₂³	1	2	yes	—
53	#15667941	—	2	2	no	—
54	#15667941	—	2	2	no	—
55	#15667941	C₂×C₄	1	4	yes	—
56	#15667941	C₂×C₄	1	4	yes	—
57	C₂²⋊#10623494	—	2	2	no	—
58	#15667924	—	2	2	no	—
59	#15667924	—	2	2	no	—
60	#15667924	C₂×C₄	1	4	yes	—
61	#15667924	C₂×C₄	1	4	yes	—
62	#15667923	—	2	2	no	—
63	#15667923	C₂×C₄	1	2	yes	—
64	#15667922	—	2	2	no	—
65	C₄×#10623494	C₂³	1	2	yes	—
66	C₈×Dic₈₉	—	2	2	no	—
67	C₂×M₁₆×C₈₉	—	2	2	no	—
68	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₈₉	—	2	2	no	—
71	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
72	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
73	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	#13083574	—	4	2	no	—
77	C₂×Dic₁₇₈	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₃₅₆	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₈₉	C₄⋊C₄	1	2	yes	—
80	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
81	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
82	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
83	#13083558	—	4	2	no	—
84	#13083557	—	8	2	no	—
85	C₂×#10623494	—	2	2	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	C₂²×C₄	1	4	yes	—
88	C₈×D₈₉	—	8	2	no	—
89	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
90	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
91	#13083553	C₄⋊C₄	1	2	yes	—
92	#13083553	C₄⋊C₄	1	2	yes	—
93	C₂²⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
94	#13083551	C₄⋊C₄	1	2	yes	—
95	#13083551	C₄⋊C₄	1	2	yes	—
96	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
97	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
98	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
102	M₁₆×C₈₉	—	8	2	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	—	4	2	no	—
106	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
108	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
112	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	4	1	no	—
117	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
123	C₂²⋊D₈₉	—	4	2	no	—
124	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
127	D₃₅₆	—	4	1	no	—
128	C₄×D₈₉	—	2	4	no	—
129	C₄×D₈₉	—	2	2	no	—
130	C₄×D₈₉	—	2	2	no	—
131	C₄×D₈₉	—	4	2	no	—
132	Dic₁₇₈	—	4	1	no	—
133	C₈×C₈₉	—	2	2	no	—
134	C₈×C₈₉	—	2	2	no	—
135	C₈×C₈₉	—	4	2	no	—
136	C₈×C₈₉	—	4	2	no	—
137	#10623494	—	2	4	no	—
138	#10623494	—	4	2	no	—
139	#10623494	—	4	2	no	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂×D₈₉	—	2	2	no	—
145	C₂×D₈₉	—	2	2	no	—
146	C₂×D₈₉	—	2	2	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	#568	1	2	yes	—
152	C₄×C₈₉	#577	1	2	yes	—
153	Dic₈₉	—	2	2	no	—
154	Dic₈₉	—	2	2	no	—
155	Dic₈₉	—	2	2	no	—
156	#17824	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	4	1	no	—
158	C₂×C₈₉	#2905	1	1	yes	characteristic
159	C₂×C₈₉	#2912	1	1	yes	characteristic
160	C₂×C₈₉	#2916	1	1	yes	characteristic
161	D₈₉	—	4	2	no	—
162	#2989	—	89	2	no	—
163	#2922	—	89	1	no	—
164	#1574	—	89	2	no	—
165	#1573	—	89	2	no	—
166	#1572	—	89	2	no	—
167	#1570	—	89	2	no	—
168	#1569	—	89	2	no	—
169	#1568	—	89	1	no	—
170	#1566	D₈₉	1	1	yes	characteristic, complemented
171	C₈₉	#17824	1	1	yes	characteristic, complemented, Sylow
172	C₄×#41	—	89	1	no	—
173	#446	—	178	2	no	—
174	#436	—	178	2	no	—
175	#435	—	178	2	no	—
176	#435	—	178	2	no	—
177	#433	—	89	4	no	—
178	#432	—	89	2	no	—
179	#432	—	89	2	no	—
180	#431	—	89	2	no	—
181	#431	—	89	2	no	—
182	#431	—	89	2	no	—
183	#424	—	89	2	no	—
184	#423	—	89	2	no	—
185	#423	C₂×D₈₉	1	2	yes	—
186	C₂×C₄⋊C₈	—	89	2	no	—
187	#408	—	89	2	no	—
188	C₄×M₁₆	—	89	1	no	—
189	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
190	#335	—	89	2	no	—
191	#335	—	89	2	no	—
192	#335	—	89	2	no	—
193	#334	—	89	2	no	—
194	#334	—	89	2	no	—
195	#334	—	89	1	no	—
196	#334	—	89	1	no	—
197	#334	C₂×D₈₉	1	2	yes	—
198	#334	C₂×D₈₉	1	1	yes	characteristic
199	#334	C₂×D₈₉	1	1	yes	characteristic
200	C₂×#41	—	89	1	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	89	2	no	—
203	C₂×M₁₆	—	178	2	no	—
204	C₂²×C₈	—	178	2	no	—
205	C₂²×C₈	C₄×D₈₉	1	2	yes	—
206	C₄×Q₈	—	89	1	no	—
207	C₄×Q₈	—	89	1	no	—
208	C₄×D₄	—	89	2	no	—
209	C₄×D₄	—	89	2	no	—
210	C₄×D₄	—	89	1	no	—
211	C₄×D₄	—	89	1	no	—
212	#117	—	89	1	no	—
213	#117	—	89	2	no	—
214	C₂×C₄²	—	89	2	no	—
215	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	89	4	no	—
218	C₄⋊C₈	—	89	4	no	—
219	C₄⋊C₈	—	89	2	no	—
220	C₄⋊C₈	—	89	2	no	—
221	C₄⋊C₈	—	178	2	no	—
222	C₄⋊C₈	—	178	2	no	—
223	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
224	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
225	C₂²⋊C₈	—	89	4	no	—
226	C₂²⋊C₈	—	89	4	no	—
227	C₂²⋊C₈	—	178	2	no	—
228	C₂²⋊C₈	—	178	2	no	—
229	C₂²⋊C₈	—	178	2	no	—
230	#97	—	89	2	no	—
231	#97	—	178	2	no	—
232	#97	—	178	2	no	—
233	C₄×C₈	—	89	2	no	—
234	C₄×C₈	—	178	2	no	—
235	C₄×C₈	C₄×D₈₉	1	2	yes	—
236	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
238	#41	—	356	2	no	—
239	C₂×Q₈	—	89	1	no	—
240	C₂×D₄	—	89	2	no	—
241	C₂×D₄	—	89	1	no	—
242	C₂²×C₄	—	89	2	no	—
243	C₂²×C₄	—	89	2	no	—
244	C₂²×C₄	—	89	2	no	—
245	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
247	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
248	M₁₆	—	8	2	no	—
249	M₁₆	—	356	2	no	—
250	M₁₆	—	712	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	89	4	no	—
255	C₂×C₈	—	178	2	no	—
256	C₂×C₈	—	178	2	no	—
257	C₂×C₈	—	712	2	no	—
258	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
259	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
260	C₄⋊C₄	—	89	2	no	—
261	C₄⋊C₄	—	89	2	no	—
262	C₄⋊C₄	—	89	1	no	—
263	C₄⋊C₄	—	89	1	no	—
264	C₂²⋊C₄	—	89	2	no	—
265	C₂²⋊C₄	—	89	2	no	—
266	C₂²⋊C₄	—	89	2	no	—
267	C₄²	—	89	2	no	—
268	C₄²	—	89	2	no	—
269	C₄²	—	89	2	no	—
270	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
271	C₄²	D₄×D₈₉	1	1	yes	characteristic
272	C₄²	Q₈×D₈₉	1	1	yes	characteristic
273	C₄²	C₂³×D₈₉	1	1	yes	characteristic
274	C₂³	—	89	2	no	—
275	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
276	Q₈	—	356	1	no	—
277	D₄	—	356	1	no	—
278	D₄	—	356	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	89	2	no	—
283	C₂×C₄	—	89	2	no	—
284	C₂×C₄	—	89	2	no	—
285	C₂×C₄	—	356	2	no	—
286	C₂×C₄	—	178	2	no	—
287	C₂×C₄	—	178	2	no	—
288	C₂×C₄	—	178	4	no	—
289	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
290	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
291	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₈	—	4	2	no	—
299	C₈	—	356	2	no	—
300	C₈	—	356	2	no	—
301	C₈	—	178	4	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	178	2	no	—
306	C₂²	—	178	2	no	—
307	C₂²	—	178	2	no	—
308	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	178	2	no	—
314	C₄	—	178	2	no	—
315	C₄	—	178	2	no	—
316	C₄	#19152364	1	2	yes	—
317	C₄	#19152404	1	2	yes	—
318	C₂	—	4	1	no	—
319	C₂	—	356	2	no	—
320	C₂	#26028814	1	1	yes	characteristic
321	C₂	#26028843	1	1	yes	characteristic
322	C₂	#26028860	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#17824
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #26019039	C₂	semidirect
4. #26019034	C₂	semidirect
5. #26019029	C₂	semidirect
6. #26019021	C₂	semidirect
7. #26019017	C₂	semidirect
8. #26019015	C₂	semidirect
9. C₈₉×#1566	C₂	semidirect
170. #1566	D₈₉	semidirect
171. C₈₉	#17824	semidirect

Maximal subgroups

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

subgroup	index
#26029366	2
#26028894	2
#26019039	2
#26019034	2
#26019029	2
#26019021	2
#26019017	2
#26019015	2
C₈₉×#1566	2
#17824	89