Subgroups — #36558864

Order 22784.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028903	C₂	1	2	yes	maximal, complemented
2	#26021146	C₂	1	1	yes	maximal, characteristic
3	#26021141	C₂	1	1	yes	maximal, characteristic
4	#26021006	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021001	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020965	C₂	1	2	yes	maximal, complemented
7	#26020958	C₂	1	2	yes	maximal, complemented
8	#26020158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#15667939	C₂	1	2	yes	maximal, complemented
10	#97×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2943	C₂	1	1	yes	maximal, characteristic
12	#19151659	—	2	4	no	—
13	#19151655	—	2	4	no	—
14	C₂×#15667948	C₂²	1	1	yes	characteristic
15	C₂×#15667948	C₂²	1	1	yes	characteristic
16	#19151625	C₂²	1	2	yes	—
17	#19151625	C₂²	1	2	yes	—
18	C₂×#15667939	C₂²	1	2	yes	—
19	C₂×#15667939	C₂²	1	2	yes	—
20	#19151565	C₂²	1	2	yes	—
21	#19151565	C₂²	1	2	yes	—
22	C₄×#13083558	C₂²	1	2	yes	—
23	C₂×#15667923	C₂²	1	1	yes	characteristic
24	C₂×#15667922	C₂²	1	1	yes	characteristic
25	C₂×#15667922	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
28	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
29	#19151363	C₂²	1	2	yes	—
30	#19151357	C₂²	1	2	yes	—
31	#19151357	C₂²	1	2	yes	—
32	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
33	C₄×#41×C₈₉	C₂²	1	2	yes	—
34	#442×C₈₉	C₄	1	2	yes	complemented
35	#441×C₈₉	C₄	1	2	yes	complemented
36	#440×C₈₉	C₄	1	4	yes	complemented
37	#427×C₈₉	C₂²	1	1	yes	characteristic
38	#416×C₈₉	C₂²	1	2	yes	—
39	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×#13083558	C₂³	1	2	yes	—
41	C₂×#13083558	C₂³	1	2	yes	—
42	C₂²×#10623494	C₂³	1	1	yes	characteristic
43	C₂²×#10623494	C₂³	1	1	yes	characteristic
44	C₂×C₄×Dic₈₉	—	2	2	no	—
45	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
47	#15667948	—	2	2	no	—
48	#15667948	—	2	2	no	—
49	C₄⋊#10623494	—	2	4	no	—
50	C₄⋊#10623494	—	2	4	no	—
51	#15667939	—	2	4	no	—
52	#15667939	—	2	4	no	—
53	C₂²⋊#10623494	—	2	4	no	—
54	C₂²⋊#10623494	—	2	4	no	—
55	#15667923	—	2	2	no	—
56	#15667923	D₄	1	4	yes	—
57	#15667922	—	2	2	no	—
58	#15667922	—	2	2	no	—
59	C₄×#10623494	—	2	2	no	—
60	C₈×Dic₈₉	—	2	2	no	—
61	C₈×Dic₈₉	—	2	2	no	—
62	C₂×#41×C₈₉	C₂³	1	2	yes	—
63	#135×C₈₉	—	2	2	no	—
64	#135×C₈₉	—	2	2	no	—
65	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂×SD₁₆×C₈₉	C₂×C₄	1	2	yes	—
67	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
71	C₄×D₄×C₈₉	—	2	4	no	—
72	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
73	#117×C₈₉	—	2	2	no	—
74	#117×C₈₉	C₂³	1	2	yes	—
75	C₂×C₄²×C₈₉	—	2	2	no	—
76	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
77	#107×C₈₉	C₂×C₄	1	2	yes	—
78	#106×C₈₉	C₂×C₄	1	2	yes	—
79	#103×C₈₉	C₂×C₄	1	4	yes	—
80	#102×C₈₉	C₂×C₄	1	4	yes	—
81	#97×C₈₉	C₂×C₄	1	2	yes	—
82	#97×C₈₉	C₂×C₄	1	2	yes	—
83	C₂²×Dic₈₉	—	2	1	no	—
84	C₂²×Dic₈₉	—	2	1	no	—
85	#13083558	—	2	2	no	—
86	#13083558	—	2	2	no	—
87	#13083558	—	2	2	no	—
88	#13083558	—	2	2	no	—
89	C₂×#10623494	—	2	1	no	—
90	C₂×#10623494	—	2	1	no	—
91	C₂×#10623494	—	2	1	no	—
92	C₂×#10623494	—	2	1	no	—
93	C₂×#10623494	—	2	2	no	—
94	C₂×#10623494	—	2	2	no	—
95	C₂×#10623494	C₂×D₄	1	2	yes	—
96	C₂×#10623494	C₂×D₄	1	2	yes	—
97	C₄×Dic₈₉	—	2	1	no	—
98	C₄×Dic₈₉	—	2	1	no	—
99	C₄×Dic₈₉	—	4	4	no	—
100	C₄×Dic₈₉	#41	1	2	yes	—
101	C₄×Dic₈₉	#41	1	2	yes	—
102	#41×C₈₉	—	2	2	no	—
103	#41×C₈₉	—	2	2	no	—
104	#41×C₈₉	—	2	2	no	—
105	#41×C₈₉	—	2	2	no	—
106	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
107	C₂×D₄×C₈₉	—	2	2	no	—
108	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₈₉	—	2	2	no	—
110	C₂²×C₄×C₈₉	—	2	4	no	—
111	C₂²×C₄×C₈₉	#41	1	2	yes	—
112	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
113	Q₁₆×C₈₉	—	2	2	no	—
114	SD₁₆×C₈₉	—	2	4	no	—
115	D₈×C₈₉	—	2	2	no	—
116	C₂×C₈×C₈₉	—	2	1	no	—
117	C₂×C₈×C₈₉	—	2	1	no	—
118	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	#41	1	2	yes	—
122	C₂×C₈×C₈₉	#41	1	2	yes	—
123	C₄⋊C₄×C₈₉	—	2	4	no	—
124	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
125	C₂²⋊C₄×C₈₉	—	2	2	no	—
126	C₂²⋊C₄×C₈₉	—	2	4	no	—
127	C₄²×C₈₉	—	2	4	no	—
128	C₄²×C₈₉	—	2	2	no	—
129	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
130	C₄²×C₈₉	C₂×D₄	1	2	yes	—
131	C₂³×C₈₉	—	2	2	no	—
132	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
133	Q₈×C₈₉	—	2	2	no	—
134	Q₈×C₈₉	—	2	2	no	—
135	D₄×C₈₉	—	2	2	no	—
136	D₄×C₈₉	—	2	2	no	—
137	D₄×C₈₉	—	4	2	no	—
138	D₄×C₈₉	—	4	2	no	—
139	C₂×C₄×C₈₉	—	2	4	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	—	8	4	no	—
145	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
146	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₈₉	—	2	4	no	—
153	C₂×Dic₈₉	—	2	2	no	—
154	C₂×Dic₈₉	—	2	2	no	—
155	C₂×Dic₈₉	—	4	2	no	—
156	C₈×C₈₉	—	2	2	no	—
157	C₈×C₈₉	#117	1	2	yes	—
158	C₈×C₈₉	#117	1	2	yes	—
159	#10623494	—	2	2	no	—
160	#10623494	—	2	2	no	—
161	#10623494	—	4	2	no	—
162	#10623494	—	4	2	no	—
163	C₂²×C₈₉	—	2	2	no	—
164	C₂²×C₈₉	—	4	4	no	—
165	C₂²×C₈₉	—	4	2	no	—
166	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₈₉	#529	1	1	yes	characteristic
168	C₂²×C₈₉	#529	1	1	yes	characteristic
169	C₄×C₈₉	—	2	4	no	—
170	C₄×C₈₉	—	4	4	no	—
171	C₄×C₈₉	—	4	2	no	—
172	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
173	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
174	C₄×C₈₉	#532	1	1	yes	characteristic
175	C₄×C₈₉	#532	1	1	yes	characteristic
176	Dic₈₉	—	2	4	no	—
177	Dic₈₉	—	4	2	no	—
178	#21297	—	89	1	no	maximal, Sylow
179	C₂×C₈₉	—	2	1	no	—
180	C₂×C₈₉	—	2	1	no	—
181	C₂×C₈₉	—	8	2	no	—
182	C₂×C₈₉	#2293	1	1	yes	characteristic
183	C₂×C₈₉	#2956	1	2	yes	—
184	#2943	D₈₉	1	1	yes	characteristic
185	#2923	—	89	2	no	—
186	#1975	—	89	1	no	—
187	#1974	—	89	1	no	—
188	#1946	—	89	1	no	—
189	#1945	—	89	1	no	—
190	#1938	—	89	2	no	—
191	#1937	—	89	2	no	—
192	#1778	—	89	1	no	—
193	C₄×#104	—	89	2	no	—
194	C₄×#97	—	89	1	no	—
195	C₈₉	#21297	1	1	yes	characteristic, complemented, Sylow
196	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
197	C₄×#41	C₂×D₈₉	1	2	yes	—
198	#442	Dic₈₉	1	2	yes	complemented
199	#441	Dic₈₉	1	2	yes	complemented
200	#440	Dic₈₉	1	4	yes	complemented
201	#433	—	178	4	no	—
202	#432	—	178	4	no	—
203	C₂×#108	—	89	1	no	—
204	C₂×#108	—	89	1	no	—
205	#427	C₂×D₈₉	1	1	yes	characteristic
206	#424	—	89	2	no	—
207	#424	—	89	2	no	—
208	C₂×#104	—	89	2	no	—
209	C₂×#104	—	89	2	no	—
210	#416	C₂×D₈₉	1	2	yes	—
211	#408	—	89	2	no	—
212	#408	—	89	2	no	—
213	C₄×M₁₆	—	89	2	no	—
214	C₂×#97	—	89	1	no	—
215	C₂×#97	—	89	1	no	—
216	C₂×#97	—	89	1	no	—
217	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
218	C₂×C₄×C₈	—	89	1	no	—
219	C₂×C₄×C₈	—	89	1	no	—
220	C₄³	—	89	1	no	—
221	#341	—	89	2	no	—
222	#339	—	89	2	no	—
223	#339	—	89	2	no	—
224	C₂×#41	C₂²×D₈₉	1	2	yes	—
225	#135	—	2	2	no	—
226	#135	—	2	2	no	—
227	C₂×Q₁₆	C₂×Dic₈₉	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×Dic₈₉	1	2	yes	—
229	C₂×D₈	C₂×Dic₈₉	1	1	yes	characteristic
230	C₂×M₁₆	—	89	2	no	—
231	C₂×M₁₆	—	89	2	no	—
232	C₂²×C₈	—	89	1	no	—
233	C₂²×C₈	—	89	1	no	—
234	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
237	C₄×D₄	—	2	4	no	—
238	C₄×D₄	C₂×Dic₈₉	1	4	yes	—
239	#117	—	2	2	no	—
240	#117	C₂²×D₈₉	1	2	yes	—
241	C₂×C₄²	—	2	2	no	—
242	C₂×C₄²	—	89	1	no	—
243	C₂×C₄²	—	89	1	no	—
244	C₂×C₄²	—	178	2	no	—
245	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
246	#108	—	178	2	no	—
247	#108	—	178	2	no	—
248	#107	C₂×Dic₈₉	1	2	yes	—
249	#106	C₂×Dic₈₉	1	2	yes	—
250	C₄⋊C₈	—	178	4	no	—
251	C₄⋊C₈	—	178	4	no	—
252	#104	—	178	4	no	—
253	#104	—	178	4	no	—
254	#103	C₂×Dic₈₉	1	4	yes	—
255	#102	C₂×Dic₈₉	1	4	yes	—
256	C₂²⋊C₈	—	178	4	no	—
257	C₂²⋊C₈	—	178	4	no	—
258	#97	—	89	4	no	—
259	#97	—	178	2	no	—
260	#97	—	178	2	no	—
261	#97	—	178	2	no	—
262	#97	C₂×Dic₈₉	1	2	yes	—
263	#97	C₂×Dic₈₉	1	2	yes	—
264	C₄×C₈	—	178	2	no	—
265	C₄×C₈	—	178	2	no	—
266	C₄×C₈	—	178	2	no	—
267	#41	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	—	2	2	no	—
270	#41	—	2	2	no	—
271	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	178	1	no	—
277	C₂²×C₄	—	178	1	no	—
278	C₂²×C₄	#13083573	1	2	yes	—
279	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
280	Q₁₆	—	2	2	no	—
281	SD₁₆	—	2	4	no	—
282	D₈	—	2	2	no	—
283	M₁₆	—	178	2	no	—
284	M₁₆	—	178	2	no	—
285	M₁₆	—	178	2	no	—
286	M₁₆	—	178	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	89	2	no	—
290	C₂×C₈	—	89	2	no	—
291	C₂×C₈	—	178	2	no	—
292	C₂×C₈	—	178	2	no	—
293	C₂×C₈	—	178	1	no	—
294	C₂×C₈	—	178	1	no	—
295	C₂×C₈	—	178	1	no	—
296	C₂×C₈	—	178	1	no	—
297	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
298	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
299	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
300	C₂×C₈	#13083574	1	2	yes	—
301	C₂×C₈	#13083574	1	2	yes	—
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	C₂²×Dic₈₉	1	4	yes	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	4	no	—
306	C₄²	—	2	4	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	89	2	no	—
309	C₄²	—	89	2	no	—
310	C₄²	—	356	4	no	—
311	C₄²	—	178	1	no	—
312	C₄²	—	178	1	no	—
313	C₄²	C₂²×Dic₈₉	1	2	yes	—
314	C₄²	D₄×D₈₉	1	2	yes	—
315	C₂³	—	2	2	no	—
316	C₂³	C₂×#13083573	1	1	yes	characteristic
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	D₄	—	2	2	no	—
320	D₄	—	2	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	4	no	—
329	C₂×C₄	—	356	2	no	—
330	C₂×C₄	—	178	2	no	—
331	C₂×C₄	—	178	4	no	—
332	C₂×C₄	—	178	2	no	—
333	C₂×C₄	D₄×Dic₈₉	1	4	yes	—
334	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
336	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
337	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
338	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
339	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	356	2	no	—
342	C₈	—	356	2	no	—
343	C₈	—	178	2	no	—
344	C₈	—	178	2	no	—
345	C₈	#15667971	1	2	yes	—
346	C₈	#15667971	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	4	4	no	—
350	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
351	C₂²	#19152130	1	1	yes	characteristic
352	C₂²	#19152130	1	1	yes	characteristic
353	C₄	—	2	4	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	4	no	—
356	C₄	—	356	2	no	—
357	C₄	—	178	4	no	—
358	C₄	C₂×#15667971	1	1	yes	characteristic
359	C₄	#41×Dic₈₉	1	1	yes	characteristic
360	C₄	#19152145	1	1	yes	characteristic
361	C₄	#19152145	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	8	2	no	—
365	C₂	#26022717	1	1	yes	characteristic
366	C₂	#26029162	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#2943	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#135×C₈₉	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#2943	D₈₉
hypocenter	C₈₉	#21297
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2943

Complements

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

N	H ≅ G/N	type
1. #26028903	C₂	semidirect
4. #26021006	C₂	semidirect
5. #26021001	C₂	semidirect
6. #26020965	C₂	semidirect
7. #26020958	C₂	semidirect
8. #26020158	C₂	semidirect
9. C₄×#15667939	C₂	semidirect
10. #97×Dic₈₉	C₂	semidirect
34. #442×C₈₉	C₄	semidirect
35. #441×C₈₉	C₄	semidirect
36. #440×C₈₉	C₄	semidirect
195. C₈₉	#21297	semidirect
198. #442	Dic₈₉	semidirect
199. #441	Dic₈₉	semidirect
200. #440	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26028903	2
#26021146	2
#26021141	2
#26021006	2
#26021001	2
#26020965	2
#26020958	2
#26020158	2
C₄×#15667939	2
#97×Dic₈₉	2
C₈₉×#2943	2
#21297	89