Subgroups — #36573360

Order 22784.

712 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151729	C₂	1	2	yes	maximal
2	#26020976	C₂	1	2	yes	maximal
3	#26020976	C₂	1	2	yes	maximal
4	#26020971	C₂	1	2	yes	maximal
5	#26020372	C₂	1	2	yes	maximal
6	#26020369	C₂	1	1	yes	maximal, characteristic
7	#26020109	C₂	1	2	yes	maximal
8	#26020105	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#2295	C₂	1	1	yes	maximal, characteristic
10	#19151729	—	2	8	no	—
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
13	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
14	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
15	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
16	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
17	C₂×#15667923	C₂²	1	2	yes	—
18	C₂×#15667923	C₂²	1	2	yes	—
19	C₂×C₄×#10623494	C₂²	1	2	yes	—
20	#19151353	C₂²	1	1	yes	characteristic
21	#19151353	C₂²	1	1	yes	characteristic
22	#19151353	C₂²	1	2	yes	—
23	#19151353	C₂²	1	1	yes	characteristic
24	#19151353	C₂²	1	1	yes	characteristic
25	#19151353	C₂²	1	2	yes	—
26	#19151353	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
28	#391×C₈₉	C₄	1	1	yes	characteristic
29	#391×C₈₉	C₄	1	1	yes	characteristic
30	#389×C₈₉	C₄	1	1	yes	characteristic
31	#389×C₈₉	C₄	1	1	yes	characteristic
32	#387×C₈₉	C₄	1	2	yes	—
33	#387×C₈₉	C₄	1	2	yes	—
34	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
36	#376×C₈₉	C₂²	1	2	yes	—
37	C₄³×C₈₉	C₂²	1	1	yes	characteristic
38	C₂²×#10623494	C₂³	1	2	yes	—
39	C₂²×#10623494	C₂³	1	2	yes	—
40	C₂²×#10623494	C₂³	1	2	yes	—
41	C₂²×#10623494	C₂³	1	2	yes	—
42	C₄⋊#10623494	—	2	4	no	—
43	C₄⋊#10623494	—	2	4	no	—
44	C₄⋊#10623494	—	2	4	no	—
45	C₄⋊#10623494	—	2	4	no	—
46	C₄⋊#10623494	—	2	4	no	—
47	C₄⋊#10623494	—	2	4	no	—
48	#15667923	—	2	4	no	—
49	#15667923	—	2	4	no	—
50	C₄×#10623494	—	2	4	no	—
51	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₄×Q₈×C₈₉	D₄	1	4	yes	—
53	C₄×Q₈×C₈₉	Q₈	1	4	yes	—
54	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
67	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
69	#95×C₈₉	C₂×C₄	1	2	yes	—
70	#95×C₈₉	C₂×C₄	1	2	yes	—
71	#95×C₈₉	C₂×C₄	1	2	yes	—
72	#95×C₈₉	C₂×C₄	1	2	yes	—
73	#95×C₈₉	C₂×C₄	1	2	yes	—
74	#95×C₈₉	C₂×C₄	1	2	yes	—
75	#95×C₈₉	C₂×C₄	1	2	yes	—
76	C₂×#10623494	—	2	2	no	—
77	C₂×#10623494	—	2	2	no	—
78	C₂×#10623494	—	2	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	—	2	2	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	#41	1	4	yes	—
88	C₂×Q₈×C₈₉	—	4	1	no	—
89	C₂×Q₈×C₈₉	—	4	1	no	—
90	C₂×Q₈×C₈₉	C₄⋊C₄	1	4	yes	—
91	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	—	2	4	no	—
103	C₄⋊C₄×C₈₉	—	4	1	no	—
104	C₄⋊C₄×C₈₉	—	4	1	no	—
105	C₄⋊C₄×C₈₉	—	4	2	no	—
106	C₄⋊C₄×C₈₉	C₄⋊C₄	1	8	yes	—
107	C₄⋊C₄×C₈₉	C₄⋊C₄	1	4	yes	—
108	C₄²×C₈₉	—	2	2	no	—
109	C₄²×C₈₉	—	2	2	no	—
110	C₄²×C₈₉	—	2	2	no	—
111	C₄²×C₈₉	—	2	2	no	—
112	C₄²×C₈₉	—	2	2	no	—
113	C₄²×C₈₉	—	4	1	no	—
114	C₄²×C₈₉	—	4	1	no	—
115	C₄²×C₈₉	—	4	1	no	—
116	C₄²×C₈₉	C₂×D₄	1	4	yes	—
117	C₄²×C₈₉	C₂×D₄	1	2	yes	—
118	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
119	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
120	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
121	Q₈×C₈₉	—	4	4	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	2	1	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	1	no	—
140	C₂×C₄×C₈₉	—	2	1	no	—
141	C₂×C₄×C₈₉	—	2	1	no	—
142	C₂×C₄×C₈₉	—	2	1	no	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	2	2	no	—
147	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
148	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
150	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₈₉	#131	1	2	yes	—
152	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
155	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
156	#10623494	—	2	4	no	—
157	#10623494	—	4	2	no	—
158	#10623494	—	4	2	no	—
159	#10623494	—	4	2	no	—
160	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
162	C₂²×C₈₉	#519	1	1	yes	characteristic
163	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
164	C₂²×C₈₉	#568	1	1	yes	characteristic
165	C₂²×C₈₉	#568	1	1	yes	characteristic
166	C₂²×C₈₉	#568	1	1	yes	characteristic
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	2	4	no	—
169	C₄×C₈₉	—	2	2	no	—
170	C₄×C₈₉	—	2	4	no	—
171	C₄×C₈₉	—	2	2	no	—
172	C₄×C₈₉	—	4	1	no	—
173	C₄×C₈₉	—	4	2	no	—
174	C₄×C₈₉	—	4	2	no	—
175	C₄×C₈₉	—	4	1	no	—
176	C₄×C₈₉	—	4	1	no	—
177	C₄×C₈₉	#548	1	2	yes	—
178	C₄×C₈₉	#557	1	2	yes	—
179	#22737	—	89	1	no	maximal, Sylow
180	C₂×C₈₉	#2351	1	1	yes	characteristic
181	C₂×C₈₉	#2905	1	1	yes	characteristic
182	C₂×C₈₉	#2906	1	1	yes	characteristic
183	C₂×C₈₉	#2967	1	1	yes	characteristic
184	C₂×C₈₉	#2968	1	1	yes	characteristic
185	C₂×C₈₉	#2985	1	1	yes	characteristic
186	C₂×C₈₉	#2987	1	1	yes	characteristic
187	C₂×#446	—	89	2	no	—
188	#2295	D₈₉	1	1	yes	characteristic
189	#1940	—	89	2	no	—
190	#1940	—	89	2	no	—
191	#1939	—	89	2	no	—
192	#1832	—	89	2	no	—
193	#1831	—	89	1	no	—
194	#1768	—	89	2	no	—
195	#1767	—	89	1	no	—
196	C₈₉	#22737	1	1	yes	characteristic, complemented, Sylow
197	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
198	#446	—	178	8	no	—
199	C₂×C₄⋊C₈	—	89	2	no	—
200	C₂×C₄⋊C₈	—	89	2	no	—
201	C₂×C₄⋊C₈	—	89	2	no	—
202	C₂×C₄⋊C₈	—	89	2	no	—
203	C₂×C₄⋊C₈	—	89	2	no	—
204	C₂×C₄⋊C₈	—	89	2	no	—
205	C₂×#97	—	89	2	no	—
206	C₂×#97	—	89	2	no	—
207	C₂×C₄×C₈	—	89	2	no	—
208	#391	Dic₈₉	1	1	yes	characteristic
209	#391	Dic₈₉	1	1	yes	characteristic
210	#389	Dic₈₉	1	1	yes	characteristic
211	#389	Dic₈₉	1	1	yes	characteristic
212	#387	Dic₈₉	1	2	yes	—
213	#387	Dic₈₉	1	2	yes	—
214	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
215	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
216	#376	C₂×D₈₉	1	2	yes	—
217	C₄³	C₂×D₈₉	1	1	yes	characteristic
218	#336	—	89	2	no	—
219	#336	—	89	1	no	—
220	#336	—	89	1	no	—
221	#336	—	89	1	no	—
222	#336	—	89	2	no	—
223	#336	—	89	2	no	—
224	#336	—	89	1	no	—
225	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
226	C₂²×C₈	—	89	2	no	—
227	C₂²×C₈	—	89	2	no	—
228	C₂²×C₈	—	89	2	no	—
229	C₂²×C₈	—	89	2	no	—
230	C₄×Q₈	Dic₁₇₈	1	4	yes	—
231	C₄×Q₈	D₃₅₆	1	4	yes	—
232	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
233	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
234	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
235	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
236	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
237	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
238	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
239	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
240	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
241	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
245	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
247	C₄⋊C₈	—	178	4	no	—
248	C₄⋊C₈	—	178	4	no	—
249	C₄⋊C₈	—	178	4	no	—
250	C₄⋊C₈	—	178	4	no	—
251	C₄⋊C₈	—	178	4	no	—
252	C₄⋊C₈	—	178	4	no	—
253	#97	—	178	4	no	—
254	#97	—	178	4	no	—
255	C₄×C₈	—	178	4	no	—
256	#95	C₂×Dic₈₉	1	2	yes	—
257	#95	C₂×Dic₈₉	1	2	yes	—
258	#95	C₂×Dic₈₉	1	2	yes	—
259	#95	C₂×Dic₈₉	1	2	yes	—
260	#95	C₂×Dic₈₉	1	2	yes	—
261	#95	C₂×Dic₈₉	1	2	yes	—
262	#95	C₂×Dic₈₉	1	2	yes	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	C₄⋊Dic₈₉	1	4	yes	—
266	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
269	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
270	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
271	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
272	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
273	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
274	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
275	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
276	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
277	C₂×C₈	—	89	4	no	—
278	C₂×C₈	—	178	2	no	—
279	C₂×C₈	—	178	2	no	—
280	C₂×C₈	—	178	2	no	—
281	C₂×C₈	—	178	2	no	—
282	C₂×C₈	—	178	2	no	—
283	C₂×C₈	—	178	2	no	—
284	C₂×C₈	—	178	2	no	—
285	C₂×C₈	—	178	2	no	—
286	C₂×C₈	—	178	2	no	—
287	C₂×C₈	—	178	2	no	—
288	C₂×C₈	—	178	2	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	C₄⋊Dic₈₉	1	8	yes	—
294	C₄⋊C₄	C₄⋊Dic₈₉	1	4	yes	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	C₂×D₃₅₆	1	4	yes	—
304	C₄²	C₂×D₃₅₆	1	2	yes	—
305	C₄²	C₂×Dic₁₇₈	1	2	yes	—
306	C₄²	C₂×Dic₁₇₈	1	4	yes	—
307	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
308	Q₈	—	4	4	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
335	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
336	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
337	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
338	C₂×C₄	#15668019	1	2	yes	—
339	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
340	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
341	C₂×C₄	#41×D₈₉	1	2	yes	—
342	C₂×C₄	#15668060	1	2	yes	—
343	C₈	—	356	2	no	—
344	C₈	—	356	2	no	—
345	C₈	—	356	2	no	—
346	C₈	—	178	4	no	—
347	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
348	C₂²	#41×Dic₈₉	1	1	yes	characteristic
349	C₂²	#19152076	1	1	yes	characteristic
350	C₂²	C₂×#15668019	1	1	yes	characteristic
351	C₂²	#19152365	1	1	yes	characteristic
352	C₂²	#19152365	1	1	yes	characteristic
353	C₂²	#19152365	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	4	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	4	no	—
358	C₄	—	2	2	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	#19152260	1	2	yes	—
365	C₄	#19152324	1	2	yes	—
366	C₂	#26023173	1	1	yes	characteristic
367	C₂	#26028816	1	1	yes	characteristic
368	C₂	#26028820	1	1	yes	characteristic
369	C₂	#26029239	1	1	yes	characteristic
370	C₂	#26029248	1	1	yes	characteristic
371	C₂	#26029347	1	1	yes	characteristic
372	C₂	#26029358	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₈₉×#2295	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#2295	D₈₉
hypocenter	C₈₉	#22737
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
196. C₈₉	#22737	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151729	2
#26020976	2
#26020971	2
#26020372	2
#26020369	2
#26020109	2
#26020105	2
C₈₉×#2295	2
#22737	89