Subgroups — #36581079

Order 22784.

604 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668004	C₂	1	1	yes	maximal, characteristic
2	#26020972	C₂	1	2	yes	maximal
3	#26020969	C₂	1	2	yes	maximal
4	#26020819	C₂	1	2	yes	maximal
5	#26020817	C₂	1	2	yes	maximal
6	#26020808	C₂	1	2	yes	maximal
7	#26020806	C₂	1	2	yes	maximal
8	#26020369	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
10	C₂×#15668004	C₄	1	1	yes	characteristic
11	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	C₂×#15667941	C₂²	1	2	yes	—
14	C₂×#15667941	C₂²	1	2	yes	—
15	C₂×#15667922	C₂²	1	2	yes	—
16	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
17	#19151496	C₄	1	1	yes	characteristic
18	#19151496	C₄	1	1	yes	characteristic
19	#19151482	C₄	1	2	yes	—
20	#19151482	C₄	1	2	yes	—
21	#19151464	C₄	1	1	yes	characteristic
22	C₄×#13083553	C₂²	1	2	yes	—
23	C₄×#13083553	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
26	#19151353	C₂²	1	2	yes	—
27	#19151353	C₂²	1	2	yes	—
28	#19151352	C₂²	1	2	yes	—
29	#19151352	C₂²	1	2	yes	—
30	#19151352	C₂²	1	2	yes	—
31	#19151352	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₄³×C₈₉	C₂²	1	1	yes	characteristic
35	#336×C₈₉	C₂²	1	2	yes	—
36	#336×C₈₉	C₂²	1	2	yes	—
37	C₂²×#10623494	C₂³	1	2	yes	—
38	C₂²×#10623494	C₂³	1	2	yes	—
39	#15668004	—	2	4	no	—
40	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂×#13083553	C₂×C₄	1	2	yes	—
45	C₂×#13083553	C₂×C₄	1	2	yes	—
46	C₂×#13083553	C₂×C₄	1	2	yes	—
47	C₂×#13083553	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
49	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
50	C₄⋊#10623494	—	4	1	no	—
51	C₄⋊#10623494	—	4	1	no	—
52	#15667941	—	4	2	no	—
53	#15667941	—	4	2	no	—
54	#15667922	—	4	2	no	—
55	C₈×Dic₈₉	—	4	2	no	—
56	#15667919	C₂×C₄	1	2	yes	—
57	#15667919	C₂×C₄	1	2	yes	—
58	#15667919	C₂×C₄	1	2	yes	—
59	#15667919	C₂×C₄	1	2	yes	—
60	#15667919	C₂×C₄	1	2	yes	—
61	#15667919	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
63	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
64	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₈₉	—	4	1	no	—
72	C₄⋊C₈×C₈₉	—	4	1	no	—
73	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
74	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
75	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
76	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
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	C₄⋊Dic₈₉	—	2	2	no	—
84	C₄⋊Dic₈₉	—	4	1	no	—
85	C₄⋊Dic₈₉	—	4	1	no	—
86	C₄⋊Dic₈₉	M₁₆	1	4	yes	—
87	#13083553	—	2	4	no	—
88	#13083553	—	2	4	no	—
89	#13083553	—	2	4	no	—
90	#13083553	—	2	4	no	—
91	C₄×Dic₈₉	—	2	2	no	—
92	C₄×Dic₈₉	—	2	2	no	—
93	C₄×Dic₈₉	—	2	2	no	—
94	C₄×Dic₈₉	—	2	2	no	—
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	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
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	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	—	2	2	no	—
107	C₂×C₈×C₈₉	—	2	2	no	—
108	C₄²×C₈₉	—	2	2	no	—
109	C₄²×C₈₉	—	2	2	no	—
110	C₄²×C₈₉	—	2	2	no	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	—	2	1	no	—
113	C₄²×C₈₉	—	2	1	no	—
114	C₄²×C₈₉	—	2	1	no	—
115	C₄²×C₈₉	—	2	1	no	—
116	C₄²×C₈₉	—	2	1	no	—
117	C₄²×C₈₉	—	4	1	no	—
118	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	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	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₈₉	#131	1	2	yes	—
136	C₂×C₄×C₈₉	#131	1	2	yes	—
137	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
141	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	2	2	no	—
147	C₂×Dic₈₉	—	2	2	no	—
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	—	2	2	no	—
150	C₂×Dic₈₉	—	2	2	no	—
151	C₂×Dic₈₉	—	2	2	no	—
152	C₂×Dic₈₉	—	2	2	no	—
153	C₂×Dic₈₉	—	2	2	no	—
154	C₂×Dic₈₉	C₂×M₁₆	1	4	yes	—
155	C₈×C₈₉	—	4	2	no	—
156	C₈×C₈₉	—	4	2	no	—
157	#10623494	—	4	2	no	—
158	#10623494	—	4	2	no	—
159	C₂²×C₈₉	#519	1	1	yes	characteristic
160	C₂²×C₈₉	#520	1	1	yes	characteristic
161	C₂²×C₈₉	#520	1	1	yes	characteristic
162	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
163	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
164	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
165	C₂²×C₈₉	#568	1	1	yes	characteristic
166	C₄×C₈₉	—	2	2	no	—
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	2	2	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	—	4	1	no	—
171	C₄×C₈₉	—	4	1	no	—
172	C₄×C₈₉	#562	1	1	yes	characteristic
173	C₄×C₈₉	#563	1	1	yes	characteristic
174	C₄×C₈₉	#563	1	1	yes	characteristic
175	C₄×C₈₉	#564	1	1	yes	characteristic
176	Dic₈₉	—	2	4	no	—
177	Dic₈₉	—	4	2	no	—
178	Dic₈₉	—	4	2	no	—
179	Dic₈₉	—	4	2	no	—
180	#23371	—	89	1	no	maximal, Sylow
181	C₂×C₈₉	#2382	1	1	yes	characteristic
182	C₂×C₈₉	#2983	1	1	yes	characteristic
183	C₂×C₈₉	#2986	1	1	yes	characteristic
184	C₂×C₈₉	#2990	1	1	yes	characteristic
185	C₂×C₈₉	#2991	1	1	yes	characteristic
186	C₂×C₈₉	#2992	1	1	yes	characteristic
187	C₂×C₈₉	#2992	1	1	yes	characteristic
188	C₄×#125	—	89	1	no	—
189	#1940	—	89	2	no	—
190	#1939	—	89	2	no	—
191	#1917	—	89	2	no	—
192	#1917	—	89	2	no	—
193	#1916	—	89	2	no	—
194	#1916	—	89	2	no	—
195	#1831	—	89	1	no	—
196	#1831	D₈₉	1	1	yes	characteristic
197	C₈₉	#23371	1	1	yes	characteristic, complemented, Sylow
198	C₂×#125	—	89	1	no	—
199	C₂×C₄⋊C₈	—	89	2	no	—
200	C₂×C₄⋊C₈	—	89	2	no	—
201	C₂×C₄⋊C₈	—	89	1	no	—
202	C₂×C₄⋊C₈	—	89	1	no	—
203	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
205	C₂×#97	—	89	2	no	—
206	C₂×C₄×C₈	—	89	2	no	—
207	#389	—	89	1	no	—
208	#389	—	89	1	no	—
209	#387	—	89	2	no	—
210	#387	—	89	2	no	—
211	#382	—	89	1	no	—
212	C₄×C₄⋊C₄	—	89	1	no	—
213	C₄×C₄⋊C₄	—	89	1	no	—
214	C₄×C₄⋊C₄	—	89	2	no	—
215	C₄×C₄⋊C₄	—	89	2	no	—
216	C₄³	C₂×D₈₉	1	1	yes	characteristic
217	#336	—	89	2	no	—
218	#336	—	89	2	no	—
219	#336	—	89	2	no	—
220	#336	—	89	2	no	—
221	#336	—	89	2	no	—
222	#336	—	89	2	no	—
223	#336	C₂×D₈₉	1	2	yes	—
224	#336	C₂×D₈₉	1	2	yes	—
225	C₂²×C₈	—	89	2	no	—
226	C₂²×C₈	—	89	2	no	—
227	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
228	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
229	#125	—	178	4	no	—
230	C₂×C₄⋊C₄	—	89	1	no	—
231	C₂×C₄⋊C₄	—	89	1	no	—
232	C₂×C₄⋊C₄	—	89	1	no	—
233	C₂×C₄⋊C₄	—	89	2	no	—
234	C₂×C₄⋊C₄	—	89	2	no	—
235	C₂×C₄⋊C₄	—	89	1	no	—
236	C₂×C₄⋊C₄	—	89	2	no	—
237	C₂×C₄⋊C₄	—	89	2	no	—
238	C₂×C₄²	—	89	2	no	—
239	C₂×C₄²	—	89	2	no	—
240	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
241	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
242	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
243	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
245	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
247	C₄⋊C₈	—	4	1	no	—
248	C₄⋊C₈	—	4	1	no	—
249	C₄⋊C₈	—	356	1	no	—
250	C₄⋊C₈	—	356	1	no	—
251	C₄⋊C₈	—	356	2	no	—
252	C₄⋊C₈	—	356	2	no	—
253	#97	—	356	2	no	—
254	C₄×C₈	—	356	2	no	—
255	#95	—	89	2	no	—
256	#95	—	89	2	no	—
257	#95	—	89	2	no	—
258	#95	—	89	2	no	—
259	#95	—	89	2	no	—
260	#95	—	89	2	no	—
261	C₂²×C₄	—	89	2	no	—
262	C₂²×C₄	—	89	2	no	—
263	C₂²×C₄	—	89	2	no	—
264	C₂²×C₄	—	89	2	no	—
265	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	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	C₄⋊C₄	—	89	4	no	—
285	C₄⋊C₄	—	356	1	no	—
286	C₄⋊C₄	—	356	1	no	—
287	C₄⋊C₄	—	178	4	no	—
288	C₄⋊C₄	—	178	4	no	—
289	C₄⋊C₄	—	178	2	no	—
290	C₄⋊C₄	—	178	4	no	—
291	C₄⋊C₄	—	178	4	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	178	2	no	—
303	C₄²	—	178	2	no	—
304	C₄²	—	178	2	no	—
305	C₄²	—	178	2	no	—
306	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	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	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	89	4	no	—
323	C₂×C₄	—	178	2	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	—	178	2	no	—
326	C₂×C₄	—	178	2	no	—
327	C₂×C₄	—	178	2	no	—
328	C₂×C₄	—	178	2	no	—
329	C₂×C₄	—	178	2	no	—
330	C₂×C₄	—	178	2	no	—
331	C₂×C₄	—	178	2	no	—
332	C₂×C₄	—	178	2	no	—
333	C₂×C₄	—	178	2	no	—
334	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
335	C₂×C₄	#15668018	1	2	yes	—
336	C₂×C₄	#15668020	1	2	yes	—
337	C₂×C₄	#15668057	1	1	yes	characteristic
338	C₂×C₄	#15668058	1	1	yes	characteristic
339	C₂×C₄	#15668059	1	1	yes	characteristic
340	C₂×C₄	#15668059	1	1	yes	characteristic
341	C₂×C₄	#15668059	1	1	yes	characteristic
342	C₂×C₄	#15668060	1	1	yes	characteristic
343	C₈	—	4	2	no	—
344	C₈	—	4	2	no	—
345	C₈	—	356	2	no	—
346	C₈	—	356	2	no	—
347	C₂²	#19152075	1	1	yes	characteristic
348	C₂²	#19152078	1	1	yes	characteristic
349	C₂²	#19152079	1	1	yes	characteristic
350	C₂²	C₂×M₁₆×D₈₉	1	1	yes	characteristic
351	C₂²	C₂×#15668018	1	1	yes	characteristic
352	C₂²	C₂×#15668020	1	1	yes	characteristic
353	C₂²	#19152364	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	356	2	no	—
361	C₄	—	356	2	no	—
362	C₄	—	356	2	no	—
363	C₄	—	178	4	no	—
364	C₄	#19152342	1	1	yes	characteristic
365	C₄	#19152348	1	1	yes	characteristic
366	C₄	#19152352	1	1	yes	characteristic
367	C₄	C₈₉⋊#564	1	1	yes	characteristic
368	C₂	#26023437	1	1	yes	characteristic
369	C₂	#26029336	1	1	yes	characteristic
370	C₂	#26029353	1	1	yes	characteristic
371	C₂	#26029378	1	1	yes	characteristic
372	C₂	#26029381	1	1	yes	characteristic
373	C₂	#26029388	1	1	yes	characteristic
374	C₂	#26029393	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₈₉×#1831	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#23371
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1831

Complements

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

N	H ≅ G/N	type
197. C₈₉	#23371	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668004	2
#26020972	2
#26020969	2
#26020819	2
#26020817	2
#26020808	2
#26020806	2
#26020369	2
C₈₉×#1831	2
#23371	89