Subgroups — #36581025

Order 22784.

636 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021193	C₂	1	2	yes	maximal
2	#26021189	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	#26020395	C₂	1	1	yes	maximal, characteristic
8	#26020395	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂×#15667923	C₂²	1	2	yes	—
13	C₂×C₄×#10623494	C₂²	1	2	yes	—
14	#19151353	C₂²	1	1	yes	characteristic
15	#19151353	C₂²	1	2	yes	—
16	#19151353	C₂²	1	1	yes	characteristic
17	#19151353	C₂²	1	1	yes	characteristic
18	#19151353	C₂²	1	2	yes	—
19	#19151353	C₂²	1	2	yes	—
20	#19151353	C₂²	1	2	yes	—
21	#19151353	C₂²	1	1	yes	characteristic
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	C₄	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
28	#391×C₈₉	C₄	1	2	yes	—
29	#389×C₈₉	C₄	1	2	yes	—
30	#389×C₈₉	C₄	1	1	yes	characteristic
31	#389×C₈₉	C₄	1	1	yes	characteristic
32	#384×C₈₉	C₄	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
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	#15667923	—	4	2	no	—
44	C₄×#10623494	—	4	2	no	—
45	C₂²×Q₈×C₈₉	C₂×C₄	1	2	yes	—
46	C₄⋊Q₈×C₈₉	—	2	4	no	—
47	C₄×Q₈×C₈₉	—	4	2	no	—
48	C₄×Q₈×C₈₉	—	4	2	no	—
49	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
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×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	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	—	2	2	no	—
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×Q₈×C₈₉	—	2	4	no	—
84	C₂×Q₈×C₈₉	—	2	4	no	—
85	C₂×Q₈×C₈₉	—	4	2	no	—
86	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
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	2	yes	—
93	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
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	—	4	1	no	—
101	C₄⋊C₄×C₈₉	—	4	1	no	—
102	C₄⋊C₄×C₈₉	—	4	2	no	—
103	C₄⋊C₄×C₈₉	—	4	2	no	—
104	C₄⋊C₄×C₈₉	—	4	2	no	—
105	C₄⋊C₄×C₈₉	M₁₆	1	4	yes	—
106	C₄²×C₈₉	—	2	2	no	—
107	C₄²×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	2	no	—
112	C₄²×C₈₉	—	2	2	no	—
113	C₄²×C₈₉	—	2	2	no	—
114	C₄²×C₈₉	—	2	1	no	—
115	C₄²×C₈₉	—	2	1	no	—
116	C₄²×C₈₉	—	2	2	no	—
117	C₄²×C₈₉	—	4	1	no	—
118	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
119	Q₈×C₈₉	—	4	8	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	2	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	2	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	1	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	1	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	2	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	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
148	C₂×C₄×C₈₉	#131	1	2	yes	—
149	C₂×C₄×C₈₉	#131	1	2	yes	—
150	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
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	2	yes	—
154	#10623494	—	4	2	no	—
155	#10623494	—	4	2	no	—
156	#10623494	—	4	2	no	—
157	#10623494	—	4	2	no	—
158	C₂²×C₈₉	#518	1	1	yes	characteristic
159	C₂²×C₈₉	#519	1	1	yes	characteristic
160	C₂²×C₈₉	#519	1	1	yes	characteristic
161	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
162	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
163	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
164	C₂²×C₈₉	#568	1	1	yes	characteristic
165	C₄×C₈₉	—	2	4	no	—
166	C₄×C₈₉	—	2	2	no	—
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	2	2	no	—
169	C₄×C₈₉	—	4	2	no	—
170	C₄×C₈₉	—	4	1	no	—
171	C₄×C₈₉	—	4	2	no	—
172	C₄×C₈₉	—	4	2	no	—
173	C₄×C₈₉	—	4	1	no	—
174	C₄×C₈₉	—	4	1	no	—
175	C₄×C₈₉	#563	1	2	yes	—
176	C₄×C₈₉	#563	1	2	yes	—
177	#23364	—	89	1	no	maximal, Sylow
178	C₂×C₈₉	#2381	1	1	yes	characteristic
179	C₂×C₈₉	#2974	1	1	yes	characteristic
180	C₂×C₈₉	#2981	1	1	yes	characteristic
181	C₂×C₈₉	#2983	1	1	yes	characteristic
182	C₂×C₈₉	#2984	1	1	yes	characteristic
183	C₂×C₈₉	#2985	1	1	yes	characteristic
184	C₂×C₈₉	#2986	1	1	yes	characteristic
185	C₄×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
186	#1985	—	89	2	no	—
187	#1984	—	89	2	no	—
188	#1917	—	89	2	no	—
189	#1917	—	89	2	no	—
190	#1916	—	89	2	no	—
191	#1916	—	89	2	no	—
192	#1839	—	89	1	no	—
193	#1839	—	89	1	no	—
194	C₈₉	#23364	1	1	yes	characteristic, complemented, Sylow
195	C₂×C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
196	C₂×C₄×Q₈	C₂×D₈₉	1	2	yes	—
197	C₂×C₄⋊C₈	—	89	2	no	—
198	C₂×C₄⋊C₈	—	89	2	no	—
199	C₂×#97	—	89	2	no	—
200	C₂×C₄×C₈	—	89	2	no	—
201	#391	Dic₈₉	1	2	yes	—
202	#389	Dic₈₉	1	2	yes	—
203	#389	Dic₈₉	1	1	yes	characteristic
204	#389	Dic₈₉	1	1	yes	characteristic
205	#384	Dic₈₉	1	1	yes	characteristic
206	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
207	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
208	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
209	C₄³	C₂×D₈₉	1	1	yes	characteristic
210	#336	—	89	1	no	—
211	#336	—	89	2	no	—
212	#336	—	89	2	no	—
213	#336	—	89	2	no	—
214	#336	—	89	1	no	—
215	#336	—	89	1	no	—
216	#336	—	89	2	no	—
217	#336	—	89	1	no	—
218	#336	—	89	2	no	—
219	#336	—	89	2	no	—
220	#336	—	89	2	no	—
221	#336	—	89	2	no	—
222	C₂²×Q₈	C₂×Dic₈₉	1	2	yes	—
223	C₂²×C₈	—	89	2	no	—
224	C₂²×C₈	—	89	2	no	—
225	C₂²×C₈	—	89	2	no	—
226	C₂²×C₈	—	89	2	no	—
227	C₄⋊Q₈	—	2	4	no	—
228	C₄×Q₈	—	4	2	no	—
229	C₄×Q₈	—	4	2	no	—
230	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
231	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	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	2	yes	—
235	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
236	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
237	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
238	C₂×C₄⋊C₄	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	C₂×Dic₈₉	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	2	yes	—
246	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
248	C₄⋊C₈	—	356	2	no	—
249	C₄⋊C₈	—	356	2	no	—
250	#97	—	356	2	no	—
251	C₄×C₈	—	356	2	no	—
252	#95	C₂×Dic₈₉	1	2	yes	—
253	#95	C₂×Dic₈₉	1	2	yes	—
254	#95	C₂×Dic₈₉	1	2	yes	—
255	#95	C₂×Dic₈₉	1	2	yes	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	2	4	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
261	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
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	2	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	1	yes	characteristic
269	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
270	C₂×C₈	—	178	2	no	—
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	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	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	#13083558	1	4	yes	—
291	C₄²	—	2	2	no	—
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	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	4	1	no	—
303	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
304	Q₈	—	4	8	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
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	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	C₂×#13083558	1	2	yes	—
332	C₂×C₄	C₂×#13083558	1	4	yes	—
333	C₂×C₄	#15668019	1	2	yes	—
334	C₂×C₄	#15668019	1	2	yes	—
335	C₂×C₄	#15668057	1	2	yes	—
336	C₂×C₄	#15668060	1	1	yes	characteristic
337	C₂×C₄	#15668060	1	1	yes	characteristic
338	C₂×C₄	#15668060	1	2	yes	—
339	C₈	—	356	2	no	—
340	C₈	—	356	2	no	—
341	C₈	—	356	2	no	—
342	C₈	—	356	2	no	—
343	C₂²	#19152073	1	1	yes	characteristic
344	C₂²	#19152076	1	1	yes	characteristic
345	C₂²	#19152076	1	1	yes	characteristic
346	C₂²	C₂²×#13083558	1	1	yes	characteristic
347	C₂²	C₂×#15668019	1	1	yes	characteristic
348	C₂²	C₂×#15668019	1	1	yes	characteristic
349	C₂²	#19152365	1	1	yes	characteristic
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	4	no	—
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	2	no	—
360	C₄	#19152347	1	2	yes	—
361	C₄	#19152347	1	2	yes	—
362	C₂	#26023433	1	1	yes	characteristic
363	C₂	#26029287	1	1	yes	characteristic
364	C₂	#26029328	1	1	yes	characteristic
365	C₂	#26029337	1	1	yes	characteristic
366	C₂	#26029341	1	1	yes	characteristic
367	C₂	#26029347	1	1	yes	characteristic
368	C₂	#26029354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₄×C₄⋊Q₈×C₈₉	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#23364
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
194. C₈₉	#23364	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
#26020395	2
C₄×C₄⋊Q₈×C₈₉	2
#23364	89