Subgroups — #36581082

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	#26020976	C₂	1	2	yes	maximal
2	#26020971	C₂	1	2	yes	maximal
3	#26020818	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	#26020369	C₂	1	1	yes	maximal, characteristic
8	#26020369	C₂	1	1	yes	maximal, characteristic
9	C₄×#125×C₈₉	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
15	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
16	C₂×#15667923	C₂²	1	2	yes	—
17	C₂×C₄×#10623494	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×#125×C₈₉	C₄	1	1	yes	characteristic
27	#389×C₈₉	C₄	1	1	yes	characteristic
28	#389×C₈₉	C₄	1	1	yes	characteristic
29	#387×C₈₉	C₄	1	2	yes	—
30	#387×C₈₉	C₄	1	2	yes	—
31	#382×C₈₉	C₄	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
36	C₄³×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	2	no	—
42	C₄⋊#10623494	—	4	2	no	—
43	C₄⋊#10623494	—	4	1	no	—
44	C₄⋊#10623494	—	4	1	no	—
45	C₄⋊#10623494	—	4	1	no	—
46	C₄⋊#10623494	—	4	1	no	—
47	#15667923	—	4	2	no	—
48	C₄×#10623494	—	4	2	no	—
49	#125×C₈₉	—	2	4	no	—
50	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×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	C₂³	1	1	yes	characteristic
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	2	yes	—
67	#95×C₈₉	C₂×C₄	1	2	yes	—
68	#95×C₈₉	C₂×C₄	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	C₂×#10623494	—	2	2	no	—
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	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	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	—	2	4	no	—
97	C₄⋊C₄×C₈₉	—	2	4	no	—
98	C₄⋊C₄×C₈₉	—	2	2	no	—
99	C₄⋊C₄×C₈₉	—	2	4	no	—
100	C₄⋊C₄×C₈₉	—	2	4	no	—
101	C₄⋊C₄×C₈₉	—	4	1	no	—
102	C₄⋊C₄×C₈₉	—	4	1	no	—
103	C₄⋊C₄×C₈₉	M₁₆	1	4	yes	—
104	C₄²×C₈₉	—	2	2	no	—
105	C₄²×C₈₉	—	2	2	no	—
106	C₄²×C₈₉	—	2	2	no	—
107	C₄²×C₈₉	—	2	2	no	—
108	C₄²×C₈₉	—	2	2	no	—
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	—	2	1	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	2	no	—
116	C₄²×C₈₉	—	2	2	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	2	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	2	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	2	no	—
130	C₂×C₄×C₈₉	—	2	1	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	1	no	—
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	2	1	no	—
138	C₂×C₄×C₈₉	—	2	1	no	—
139	C₂×C₄×C₈₉	—	2	1	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
147	C₂×C₄×C₈₉	#131	1	2	yes	—
148	C₂×C₄×C₈₉	#131	1	2	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	1	yes	characteristic
152	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
153	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
155	#10623494	—	4	2	no	—
156	#10623494	—	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	4	no	—
168	C₄×C₈₉	—	2	2	no	—
169	C₄×C₈₉	—	2	2	no	—
170	C₄×C₈₉	—	4	1	no	—
171	C₄×C₈₉	—	4	1	no	—
172	C₄×C₈₉	—	4	1	no	—
173	C₄×C₈₉	—	4	2	no	—
174	C₄×C₈₉	—	4	2	no	—
175	C₄×C₈₉	—	4	2	no	—
176	C₄×C₈₉	#562	1	1	yes	characteristic
177	C₄×C₈₉	#563	1	1	yes	characteristic
178	C₄×C₈₉	#563	1	1	yes	characteristic
179	C₄×C₈₉	#564	1	1	yes	characteristic
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	D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
197	C₈₉	#23371	1	1	yes	characteristic, complemented, Sylow
198	C₂×#125	Dic₈₉	1	1	yes	characteristic
199	C₂×C₄⋊C₈	—	89	1	no	—
200	C₂×C₄⋊C₈	—	89	1	no	—
201	C₂×C₄⋊C₈	—	89	1	no	—
202	C₂×C₄⋊C₈	—	89	1	no	—
203	C₂×C₄⋊C₈	—	89	2	no	—
204	C₂×C₄⋊C₈	—	89	2	no	—
205	C₂×#97	—	89	2	no	—
206	C₂×C₄×C₈	—	89	2	no	—
207	#389	Dic₈₉	1	1	yes	characteristic
208	#389	Dic₈₉	1	1	yes	characteristic
209	#387	Dic₈₉	1	2	yes	—
210	#387	Dic₈₉	1	2	yes	—
211	#382	Dic₈₉	1	1	yes	characteristic
212	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
213	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
214	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
215	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
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	—	89	2	no	—
224	#336	—	89	2	no	—
225	C₂²×C₈	—	89	2	no	—
226	C₂²×C₈	—	89	2	no	—
227	C₂²×C₈	—	89	2	no	—
228	C₂²×C₈	—	89	2	no	—
229	#125	—	2	4	no	—
230	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
231	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
232	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
234	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
235	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
236	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
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	1	yes	characteristic
245	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
246	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
247	C₄⋊C₈	—	356	2	no	—
248	C₄⋊C₈	—	356	2	no	—
249	C₄⋊C₈	—	356	1	no	—
250	C₄⋊C₈	—	356	1	no	—
251	C₄⋊C₈	—	356	1	no	—
252	C₄⋊C₈	—	356	1	no	—
253	#97	—	356	2	no	—
254	C₄×C₈	—	356	2	no	—
255	#95	C₂×Dic₈₉	1	2	yes	—
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	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
262	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
263	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
264	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
265	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
266	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
267	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
268	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
269	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
270	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
272	C₂×C₈	—	178	2	no	—
273	C₂×C₈	—	178	2	no	—
274	C₂×C₈	—	178	2	no	—
275	C₂×C₈	—	178	2	no	—
276	C₂×C₈	—	178	2	no	—
277	C₂×C₈	—	178	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	—	2	4	no	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	#13083558	1	4	yes	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	2	1	no	—
305	C₄²	—	4	1	no	—
306	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	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	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	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	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	C₂×#13083558	1	2	yes	—
334	C₂×C₄	C₂×#13083558	1	4	yes	—
335	C₂×C₄	#15668019	1	2	yes	—
336	C₂×C₄	#15668019	1	2	yes	—
337	C₂×C₄	#15668058	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	#15668060	1	1	yes	characteristic
342	C₂×C₄	#15668060	1	1	yes	characteristic
343	C₈	—	356	2	no	—
344	C₈	—	356	2	no	—
345	C₈	—	356	2	no	—
346	C₈	—	356	2	no	—
347	C₂²	#19152076	1	1	yes	characteristic
348	C₂²	#19152081	1	1	yes	characteristic
349	C₂²	#19152081	1	1	yes	characteristic
350	C₂²	C₂²×#13083558	1	1	yes	characteristic
351	C₂²	C₂×#15668019	1	1	yes	characteristic
352	C₂²	C₂×#15668019	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	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	#19152346	1	1	yes	characteristic
365	C₄	#19152352	1	1	yes	characteristic
366	C₄	#19152352	1	1	yes	characteristic
367	C₄	C₈₉⋊#564	1	1	yes	characteristic
368	C₂	#26023438	1	1	yes	characteristic
369	C₂	#26029337	1	1	yes	characteristic
370	C₂	#26029354	1	1	yes	characteristic
371	C₂	#26029377	1	1	yes	characteristic
372	C₂	#26029384	1	1	yes	characteristic
373	C₂	#26029391	1	1	yes	characteristic
374	C₂	#26029391	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₄×#125×C₈₉	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#125	D₈₉
hypocenter	C₈₉	#23371
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#125

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
#26020976	2
#26020971	2
#26020818	2
#26020811	2
#26020369	2
C₄×#125×C₈₉	2
#23371	89