Subgroups — #36444795

Order 22784.

562 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	#26028885	C₂	1	1	yes	maximal, characteristic
2	#26028885	C₂	1	1	yes	maximal, characteristic
3	#26018160	C₂	1	4	yes	maximal
4	C₈₉×#2283	C₂	1	1	yes	maximal, characteristic
5	#19151655	—	2	1	no	—
6	#19151655	—	2	1	no	—
7	#19151655	C₄	1	2	yes	complemented
8	#19151655	C₄	1	2	yes	complemented
9	#19151651	—	2	1	no	—
10	#19151651	—	2	1	no	—
11	#19151651	C₄	1	2	yes	complemented
12	#19151651	C₄	1	2	yes	complemented
13	#19151622	—	2	1	no	—
14	#19151622	—	2	1	no	—
15	#19151562	C₂²	1	2	yes	—
16	#19151562	C₂²	1	2	yes	—
17	C₄×#13083558	—	2	1	no	—
18	C₄×#13083558	—	2	1	no	—
19	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	#383×C₈₉	—	2	1	no	—
21	#382×C₈₉	—	2	1	no	—
22	#381×C₈₉	C₂²	1	2	yes	—
23	#380×C₈₉	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
25	C₄×C₂²⋊C₄×C₈₉	C₄	1	2	yes	—
26	#376×C₈₉	—	2	1	no	—
27	C₂×#13083558	—	2	2	no	—
28	C₂×#13083558	—	2	2	no	—
29	C₄⋊#10623494	—	2	2	no	—
30	C₄⋊#10623494	—	2	2	no	—
31	C₄⋊#10623494	—	2	2	no	—
32	C₄⋊#10623494	—	2	2	no	—
33	C₂²⋊#10623494	—	2	2	no	—
34	C₂²⋊#10623494	—	2	2	no	—
35	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
36	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
37	#15667923	—	2	2	no	—
38	#15667923	—	2	2	no	—
39	C₄×#10623494	—	2	2	no	—
40	C₄×#10623494	—	2	2	no	—
41	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₈₉	—	2	2	no	—
47	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
59	#95×C₈₉	—	2	2	no	—
60	#95×C₈₉	—	2	2	no	—
61	#95×C₈₉	—	2	2	no	—
62	#95×C₈₉	—	2	2	no	—
63	#13083558	—	4	4	no	—
64	#13083558	—	4	4	no	—
65	C₂×#10623494	—	2	4	no	—
66	C₂×#10623494	—	2	4	no	—
67	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	—	2	4	no	—
69	C₂²×C₄×C₈₉	—	4	2	no	—
70	C₂²×C₄×C₈₉	—	4	2	no	—
71	C₂²×C₄×C₈₉	—	4	2	no	—
72	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
90	C₄⋊C₄×C₈₉	—	4	4	no	—
91	C₂²⋊C₄×C₈₉	—	4	4	no	—
92	C₄²×C₈₉	—	4	4	no	—
93	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₈₉	—	2	2	no	—
110	C₂³×C₈₉	—	4	1	no	—
111	C₂³×C₈₉	—	4	1	no	—
112	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
113	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	4	1	no	—
124	C₂×C₄×C₈₉	—	4	1	no	—
125	C₂×C₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	—	4	4	no	—
134	C₂×C₄×C₈₉	—	4	4	no	—
135	C₂×C₄×C₈₉	—	4	4	no	—
136	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
137	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
139	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
140	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
142	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
143	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
148	#10623494	—	4	4	no	—
149	#10623494	—	4	4	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	—	2	1	no	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	4	4	no	—
159	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
160	C₂²×C₈₉	#410	1	2	yes	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	—	8	4	no	—
172	C₄×C₈₉	#344	1	2	yes	—
173	C₄×C₈₉	#344	1	2	yes	—
174	#8113	—	89	1	no	maximal, Sylow
175	C₂×C₈₉	—	2	2	no	—
176	C₂×C₈₉	—	4	1	no	—
177	C₂×C₈₉	—	4	1	no	—
178	C₂×C₈₉	#1740	1	1	yes	characteristic
179	C₂×C₈₉	C₂×#344	1	1	yes	characteristic
180	C₂×C₈₉	C₂×#344	1	1	yes	characteristic
181	#2920	—	89	1	no	—
182	#2920	—	89	1	no	—
183	#2283	D₈₉	1	1	yes	characteristic
184	#1306	—	89	4	no	—
185	C₈₉	#8113	1	1	yes	characteristic, complemented, Sylow
186	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
187	#432	—	89	2	no	—
188	#432	—	89	2	no	—
189	#432	—	178	1	no	—
190	#432	—	178	1	no	—
191	#431	—	89	2	no	—
192	#431	—	89	2	no	—
193	#431	—	178	1	no	—
194	#431	—	178	1	no	—
195	#423	—	178	1	no	—
196	#423	—	178	1	no	—
197	#407	—	89	2	no	—
198	#407	—	89	2	no	—
199	C₄×M₁₆	—	178	1	no	—
200	C₄×M₁₆	—	178	1	no	—
201	#383	—	2	1	no	—
202	#382	—	2	1	no	—
203	#381	C₂×D₈₉	1	2	yes	—
204	#380	Dic₈₉	1	2	yes	—
205	C₄×C₄⋊C₄	—	2	1	no	—
206	C₄×C₂²⋊C₄	Dic₈₉	1	2	yes	—
207	#376	—	2	1	no	—
208	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
209	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
210	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
211	C₂×M₁₆	—	178	2	no	—
212	C₂×M₁₆	—	178	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	C₂×Dic₈₉	1	4	yes	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
217	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
218	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
219	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
220	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
221	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
222	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
223	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
227	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
228	C₄⋊C₈	—	178	2	no	—
229	C₄⋊C₈	—	178	2	no	—
230	C₄⋊C₈	—	178	2	no	—
231	C₄⋊C₈	—	178	2	no	—
232	C₂²⋊C₈	—	89	4	no	—
233	C₂²⋊C₈	—	89	4	no	—
234	C₂²⋊C₈	—	178	2	no	—
235	C₂²⋊C₈	—	178	2	no	—
236	#97	—	178	2	no	—
237	#97	—	178	2	no	—
238	C₄×C₈	—	178	2	no	—
239	C₄×C₈	—	178	2	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
250	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
251	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
252	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
257	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
258	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
259	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
260	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
266	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
267	M₁₆	—	356	4	no	—
268	M₁₆	—	356	4	no	—
269	C₂×C₈	—	178	4	no	—
270	C₂×C₈	—	178	4	no	—
271	C₄⋊C₄	—	4	4	no	—
272	C₂²⋊C₄	—	4	4	no	—
273	C₄²	—	4	4	no	—
274	C₄²	#13083551	1	1	yes	characteristic
275	C₄²	#13083551	1	1	yes	characteristic
276	C₄²	#13083551	1	1	yes	characteristic
277	C₄²	#13083551	1	1	yes	characteristic
278	C₄²	#13083551	1	1	yes	characteristic
279	C₄²	#13083551	1	1	yes	characteristic
280	C₄²	#13083551	1	1	yes	characteristic
281	C₄²	#13083551	1	1	yes	characteristic
282	C₄²	#13083553	1	1	yes	characteristic
283	C₄²	#13083553	1	1	yes	characteristic
284	C₄²	#13083553	1	1	yes	characteristic
285	C₄²	#13083553	1	1	yes	characteristic
286	C₄²	#13083553	1	1	yes	characteristic
287	C₄²	#13083553	1	1	yes	characteristic
288	C₄²	#13083553	1	1	yes	characteristic
289	C₄²	#13083553	1	1	yes	characteristic
290	C₂³	—	2	2	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
294	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
295	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	#15667919	1	1	yes	characteristic
318	C₂×C₄	#15667919	1	1	yes	characteristic
319	C₂×C₄	#15667919	1	1	yes	characteristic
320	C₂×C₄	#15667919	1	1	yes	characteristic
321	C₂×C₄	#15667919	1	1	yes	characteristic
322	C₂×C₄	#15667919	1	1	yes	characteristic
323	C₂×C₄	#15667919	1	1	yes	characteristic
324	C₂×C₄	#15667919	1	1	yes	characteristic
325	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
326	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
327	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
328	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
329	C₈	—	356	4	no	—
330	C₈	—	356	4	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	4	no	—
340	C₂²	C₂×#15667919	1	1	yes	characteristic
341	C₂²	#19151572	1	2	yes	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	8	4	no	—
353	C₄	#19151368	1	2	yes	—
354	C₄	#19151368	1	2	yes	—
355	C₂	—	2	2	no	—
356	C₂	—	4	1	no	—
357	C₂	—	4	1	no	—
358	C₂	#26019985	1	1	yes	characteristic
359	C₂	C₂×#19151368	1	1	yes	characteristic
360	C₂	C₂×#19151368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667919
commutator	C₂³×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₈₉×#2283	C₂
socle	C₂²×C₈₉	C₂×#95
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	#2283	D₈₉
hypocenter	C₈₉	#8113
LCS	G ▷ C₂³×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2283

Complements

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

N	H ≅ G/N	type
7. #19151655	C₄	semidirect
8. #19151655	C₄	semidirect
11. #19151651	C₄	semidirect
12. #19151651	C₄	semidirect
185. C₈₉	#8113	semidirect

Maximal subgroups

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

subgroup	index
#26028885	2
#26018160	2
C₈₉×#2283	2
#8113	89