Subgroups — #36581044

Order 22784.

600 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021193	C₂	1	1	yes	maximal, characteristic
2	#26021189	C₂	1	1	yes	maximal, characteristic
3	#26021189	C₂	1	2	yes	maximal
4	#26020818	C₂	1	2	yes	maximal
5	#26020811	C₂	1	2	yes	maximal
6	#26020811	C₂	1	2	yes	maximal
7	#26020811	C₂	1	2	yes	maximal
8	#26020397	C₂	1	2	yes	maximal
9	C₈₉×#2399	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	C₂×#15667923	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10623494	C₂²	1	2	yes	—
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	#19151353	C₂²	1	2	yes	—
17	#19151353	C₂²	1	2	yes	—
18	#19151353	C₂²	1	2	yes	—
19	#19151353	C₂²	1	2	yes	—
20	#19151353	C₂²	1	2	yes	—
21	#19151353	C₂²	1	2	yes	—
22	#19151353	C₂²	1	2	yes	—
23	#19151353	C₂²	1	2	yes	—
24	#19151353	C₂²	1	2	yes	—
25	#19151353	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
28	#398×C₈₉	C₄	1	2	yes	—
29	#398×C₈₉	C₄	1	1	yes	characteristic
30	#398×C₈₉	C₄	1	1	yes	characteristic
31	#395×C₈₉	C₄	1	1	yes	characteristic
32	#395×C₈₉	C₄	1	1	yes	characteristic
33	#395×C₈₉	C₄	1	2	yes	—
34	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
36	#376×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²×#10623494	C₂³	1	2	yes	—
38	C₂²×#10623494	C₂³	1	2	yes	—
39	C₂²×#10623494	C₂³	1	2	yes	—
40	C₂²×#10623494	C₂³	1	2	yes	—
41	C₄⋊#10623494	—	4	1	no	—
42	C₄⋊#10623494	—	4	1	no	—
43	C₄⋊#10623494	—	4	2	no	—
44	#15667923	—	4	1	no	—
45	C₄×#10623494	—	4	1	no	—
46	C₄×#10623494	—	4	2	no	—
47	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈×C₈₉	—	4	1	no	—
50	C₄×Q₈×C₈₉	—	4	1	no	—
51	C₄×Q₈×C₈₉	—	4	1	no	—
52	C₄×Q₈×C₈₉	—	4	1	no	—
53	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	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	2	yes	—
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×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	#95×C₈₉	C₂×C₄	1	1	yes	characteristic
68	#95×C₈₉	C₂×C₄	1	2	yes	—
69	#95×C₈₉	C₂×C₄	1	1	yes	characteristic
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	C₂×#10623494	—	2	2	no	—
75	C₂×#10623494	—	2	2	no	—
76	C₂×#10623494	—	2	2	no	—
77	C₂×#10623494	—	2	2	no	—
78	C₂×#10623494	—	2	4	no	—
79	C₂×#10623494	—	2	2	no	—
80	C₂×#10623494	—	2	2	no	—
81	C₂×#10623494	—	2	4	no	—
82	C₂×#10623494	—	2	2	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₂×Q₈×C₈₉	—	2	4	no	—
85	C₂×Q₈×C₈₉	—	2	2	no	—
86	C₂×Q₈×C₈₉	—	4	1	no	—
87	C₂×Q₈×C₈₉	—	4	1	no	—
88	C₂×Q₈×C₈₉	—	4	1	no	—
89	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₈₉	C₂²×C₄	1	2	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	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	—	2	4	no	—
101	C₄⋊C₄×C₈₉	—	2	4	no	—
102	C₄⋊C₄×C₈₉	—	2	4	no	—
103	C₄⋊C₄×C₈₉	—	4	2	no	—
104	C₄⋊C₄×C₈₉	—	4	1	no	—
105	C₄⋊C₄×C₈₉	—	4	2	no	—
106	C₄⋊C₄×C₈₉	—	4	1	no	—
107	C₄⋊C₄×C₈₉	—	4	1	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	—	2	2	no	—
113	C₄²×C₈₉	—	2	1	no	—
114	C₄²×C₈₉	—	2	1	no	—
115	C₄²×C₈₉	—	2	2	no	—
116	C₄²×C₈₉	—	2	1	no	—
117	C₄²×C₈₉	—	2	2	no	—
118	C₄²×C₈₉	—	2	2	no	—
119	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
120	Q₈×C₈₉	—	4	4	no	—
121	Q₈×C₈₉	—	4	4	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	2	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	2	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	2	1	no	—
141	C₂×C₄×C₈₉	—	2	1	no	—
142	C₂×C₄×C₈₉	—	2	2	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	1	no	—
147	C₂×C₄×C₈₉	#131	1	2	yes	—
148	C₂×C₄×C₈₉	#131	1	4	yes	—
149	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
152	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
153	#10623494	—	4	2	no	—
154	#10623494	—	4	2	no	—
155	#10623494	—	4	2	no	—
156	#10623494	—	4	2	no	—
157	C₂²×C₈₉	#518	1	1	yes	characteristic
158	C₂²×C₈₉	#519	1	2	yes	—
159	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
160	C₂²×C₈₉	C₂×#131	1	2	yes	—
161	C₂²×C₈₉	#568	1	1	yes	characteristic
162	C₄×C₈₉	—	2	2	no	—
163	C₄×C₈₉	—	2	4	no	—
164	C₄×C₈₉	—	4	2	no	—
165	C₄×C₈₉	—	4	1	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	—	4	1	no	—
168	C₄×C₈₉	—	4	1	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	—	4	2	no	—
171	C₄×C₈₉	—	4	2	no	—
172	C₄×C₈₉	#563	1	1	yes	characteristic
173	C₄×C₈₉	#563	1	1	yes	characteristic
174	C₄×C₈₉	#563	1	2	yes	—
175	#23367	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	#2381	1	1	yes	characteristic
177	C₂×C₈₉	#2976	1	1	yes	characteristic
178	C₂×C₈₉	#2977	1	1	yes	characteristic
179	C₂×C₈₉	#2985	1	2	yes	—
180	C₂×C₈₉	#2986	1	2	yes	—
181	#2399	D₈₉	1	1	yes	characteristic
182	#1985	—	89	1	no	—
183	#1984	—	89	2	no	—
184	#1984	—	89	1	no	—
185	#1917	—	89	2	no	—
186	#1916	—	89	2	no	—
187	#1916	—	89	2	no	—
188	#1916	—	89	2	no	—
189	#1840	—	89	2	no	—
190	C₈₉	#23367	1	1	yes	characteristic, complemented, Sylow
191	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
192	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
193	C₂×C₄⋊C₈	—	89	1	no	—
194	C₂×C₄⋊C₈	—	89	1	no	—
195	C₂×C₄⋊C₈	—	89	2	no	—
196	C₂×#97	—	89	1	no	—
197	C₂×C₄×C₈	—	89	2	no	—
198	C₂×C₄×C₈	—	89	1	no	—
199	#398	Dic₈₉	1	2	yes	—
200	#398	Dic₈₉	1	1	yes	characteristic
201	#398	Dic₈₉	1	1	yes	characteristic
202	#395	Dic₈₉	1	1	yes	characteristic
203	#395	Dic₈₉	1	1	yes	characteristic
204	#395	Dic₈₉	1	2	yes	—
205	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
206	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
207	#376	C₂×D₈₉	1	1	yes	characteristic
208	#336	—	89	2	no	—
209	#336	—	89	2	no	—
210	#336	—	89	2	no	—
211	#336	—	89	2	no	—
212	#336	—	89	2	no	—
213	#336	—	89	2	no	—
214	#336	—	89	2	no	—
215	#336	—	89	2	no	—
216	#336	—	89	2	no	—
217	#336	—	89	2	no	—
218	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
219	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
220	C₂²×C₈	—	89	2	no	—
221	C₂²×C₈	—	89	2	no	—
222	C₂²×C₈	—	89	2	no	—
223	C₂²×C₈	—	89	2	no	—
224	C₄×Q₈	—	4	1	no	—
225	C₄×Q₈	—	4	1	no	—
226	C₄×Q₈	—	4	1	no	—
227	C₄×Q₈	—	4	1	no	—
228	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
231	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
232	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
233	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
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×C₄²	C₂²×D₈₉	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
239	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
242	C₄⋊C₈	—	356	1	no	—
243	C₄⋊C₈	—	356	1	no	—
244	C₄⋊C₈	—	356	2	no	—
245	#97	—	356	1	no	—
246	C₄×C₈	—	356	1	no	—
247	C₄×C₈	—	356	2	no	—
248	#95	C₂×Dic₈₉	1	1	yes	characteristic
249	#95	C₂×Dic₈₉	1	2	yes	—
250	#95	C₂×Dic₈₉	1	1	yes	characteristic
251	#95	C₂×Dic₈₉	1	2	yes	—
252	#95	C₂×Dic₈₉	1	2	yes	—
253	#95	C₂×Dic₈₉	1	2	yes	—
254	#95	C₂×Dic₈₉	1	2	yes	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
262	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
265	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
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	1	yes	characteristic
270	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
271	C₂×C₈	—	178	2	no	—
272	C₂×C₈	—	178	2	no	—
273	C₂×C₈	—	178	2	no	—
274	C₂×C₈	—	178	2	no	—
275	C₂×C₈	—	178	4	no	—
276	C₂×C₈	—	178	2	no	—
277	C₂×C₈	—	178	2	no	—
278	C₂×C₈	—	178	4	no	—
279	C₂×C₈	—	178	2	no	—
280	C₂×C₈	—	178	2	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
301	Q₈	—	4	4	no	—
302	Q₈	—	4	4	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	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	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	#15668019	1	2	yes	—
329	C₂×C₄	#15668019	1	4	yes	—
330	C₂×C₄	#15668057	1	1	yes	characteristic
331	C₂×C₄	#15668057	1	1	yes	characteristic
332	C₂×C₄	#15668060	1	2	yes	—
333	C₂×C₄	#15668060	1	2	yes	—
334	C₈	—	356	2	no	—
335	C₈	—	356	2	no	—
336	C₈	—	356	2	no	—
337	C₈	—	356	2	no	—
338	C₂²	#19152073	1	1	yes	characteristic
339	C₂²	#19152076	1	2	yes	—
340	C₂²	C₂×#15668019	1	2	yes	—
341	C₂²	C₂×#15668019	1	1	yes	characteristic
342	C₂²	#19152365	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	4	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	#19152347	1	1	yes	characteristic
354	C₄	#19152347	1	1	yes	characteristic
355	C₄	#19152347	1	2	yes	—
356	C₂	#26023433	1	1	yes	characteristic
357	C₂	#26029300	1	1	yes	characteristic
358	C₂	#26029308	1	1	yes	characteristic
359	C₂	#26029347	1	2	yes	—
360	C₂	#26029354	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₈₉×#2399	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#2399	D₈₉
hypocenter	C₈₉	#23367
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #2399

Complements

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

N	H ≅ G/N	type
190. C₈₉	#23367	semidirect

Maximal subgroups

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

subgroup	index
#26021193	2
#26021189	2
#26020818	2
#26020811	2
#26020397	2
C₈₉×#2399	2
#23367	89