Subgroups — #36524984

Order 22784.

598 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030835	C₂	1	1	yes	maximal, characteristic
2	#26029183	C₂	1	2	yes	maximal
3	#26019162	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019159	C₂	1	2	yes	maximal, complemented
5	#26019071	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019070	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019068	C₂	1	2	yes	maximal, complemented
8	#26019006	C₂	1	1	yes	maximal, characteristic
9	#26018705	C₂	1	2	yes	maximal
10	#322×D₈₉	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152152	C₂²	1	2	yes	—
13	#19152002	C₄	1	1	yes	characteristic
14	#19151976	C₄	1	1	yes	characteristic
15	#19151947	C₄	1	2	yes	—
16	#19151840	C₄	1	1	yes	characteristic
17	#19151838	C₄	1	2	yes	—
18	#19151836	C₄	1	1	yes	characteristic
19	#19151728	C₂²	1	2	yes	—
20	#19151726	—	2	2	no	—
21	Q₈×#10623494	C₂²	1	2	yes	—
22	#19151722	—	2	2	no	—
23	#19151722	C₂²	1	2	yes	—
24	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
25	C₄×#13083564	C₂²	1	2	yes	—
26	#19151623	—	2	2	no	—
27	#19151620	C₂²	1	2	yes	—
28	#19151555	—	2	2	no	—
29	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#19151349	C₂²	1	1	yes	characteristic
33	#19151348	C₂²	1	2	yes	—
34	#19151330	C₂²	1	2	yes	—
35	Q₈⋊#10623494	C₂²	1	2	yes	—
36	#19151328	C₂²	1	2	yes	—
37	#19151320	C₂²	1	1	yes	characteristic
38	#19151320	C₂²	1	1	yes	characteristic
39	#19151319	C₂²	1	1	yes	characteristic
40	#446×C₈₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#334×C₈₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	—
44	#322×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#13083564	C₂×C₄	1	2	yes	—
47	C₂×#13083557	—	2	2	no	—
48	C₂×C₈×D₈₉	—	2	1	no	—
49	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₈₉	D₄	1	2	yes	—
52	#15668011	C₂×C₄	1	2	yes	—
53	#15668010	C₂×C₄	1	2	yes	—
54	#15668002	—	2	2	no	—
55	#15667992	C₂×C₄	1	2	yes	—
56	#15667990	—	2	2	no	—
57	#15667988	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₇₈	—	2	2	no	—
59	C₄×Dic₁₇₈	C₂³	1	2	yes	—
60	#15667969	—	2	2	no	—
61	C₄²×D₈₉	C₂³	1	1	yes	characteristic
62	#15667946	C₂×C₄	1	1	yes	characteristic
63	#15667945	C₂×C₄	1	2	yes	—
64	C₄⋊#10623494	—	2	2	no	—
65	C₄⋊#10623494	C₂³	1	2	yes	—
66	#15667941	—	2	2	no	—
67	#15667941	—	2	2	no	—
68	#15667941	—	2	2	no	—
69	#15667937	C₂×C₄	1	2	yes	—
70	#15667936	C₂×C₄	1	2	yes	—
71	#15667935	C₂×C₄	1	2	yes	—
72	#15667922	—	2	2	no	—
73	C₄×#10623494	—	2	2	no	—
74	C₄×#10623494	C₂³	1	1	yes	characteristic
75	C₄×#10623494	C₂³	1	1	yes	characteristic
76	C₄×#10623494	C₂³	1	1	yes	characteristic
77	C₈×Dic₈₉	—	2	2	no	—
78	C₈×Dic₈₉	—	2	1	no	—
79	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₈₉	D₄	1	2	yes	—
82	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
84	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₈₉	—	2	2	no	—
86	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
87	#103×C₈₉	C₂×C₄	1	2	yes	—
88	#97×C₈₉	—	2	2	no	—
89	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₇₈	—	2	2	no	—
93	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
97	Dic₃₅₆	—	2	2	no	—
98	#13083564	—	2	4	no	—
99	#13083557	—	4	4	no	—
100	C₂×#10623494	—	2	2	no	—
101	C₂×#10623494	—	2	1	no	—
102	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
103	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
104	C₂×#10623494	C₂×D₄	1	2	yes	—
105	C₈×D₈₉	—	2	4	no	—
106	C₈×D₈₉	—	2	2	no	—
107	C₈×D₈₉	—	2	2	no	—
108	C₈×D₈₉	—	4	2	no	—
109	C₄⋊Dic₈₉	—	2	2	no	—
110	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
111	#13083553	—	2	2	no	—
112	#13083553	—	2	2	no	—
113	#13083553	—	2	2	no	—
114	#13083553	—	2	2	no	—
115	#13083553	—	2	2	no	—
116	#13083551	—	2	2	no	—
117	#13083551	—	2	2	no	—
118	C₄×Dic₈₉	—	2	2	no	—
119	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₈₉	#41	1	1	yes	characteristic
122	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
123	Q₁₆×C₈₉	—	2	2	no	—
124	C₂×C₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	—	2	1	no	—
126	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
129	C₄⋊C₄×C₈₉	—	2	2	no	—
130	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
131	C₄²×C₈₉	—	2	2	no	—
132	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₈₉	—	2	2	no	—
135	Q₈×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₈₉	—	2	2	no	—
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
146	C₄×D₈₉	—	2	1	no	—
147	C₄×D₈₉	—	2	1	no	—
148	C₄×D₈₉	—	2	2	no	—
149	C₄×D₈₉	#137	1	4	yes	—
150	Dic₁₇₈	—	2	2	no	—
151	Dic₁₇₈	—	2	2	no	—
152	Dic₁₇₈	—	4	4	no	—
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	—	2	2	no	—
155	C₈×C₈₉	—	4	1	no	—
156	C₈×C₈₉	—	4	2	no	—
157	C₈×C₈₉	#131	1	2	yes	—
158	#10623494	—	2	1	no	—
159	#10623494	—	2	2	no	—
160	#10623494	—	4	1	no	—
161	#10623494	—	4	2	no	—
162	#10623494	#131	1	2	yes	—
163	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×D₈₉	—	2	1	no	—
165	C₂×D₈₉	—	2	1	no	—
166	C₂×D₈₉	#441	1	2	yes	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	4	2	no	—
169	C₄×C₈₉	—	4	2	no	—
170	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
171	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
172	C₄×C₈₉	C₂×#137	1	2	yes	—
173	Dic₈₉	—	2	1	no	—
174	Dic₈₉	—	4	2	no	—
175	Dic₈₉	—	4	2	no	—
176	Dic₈₉	#441	1	2	yes	—
177	Dic₈₉	#444	1	2	yes	—
178	#17961	—	89	1	no	maximal, Sylow
179	C₂×C₈₉	#2931	1	1	yes	characteristic
180	C₂×C₈₉	C₂×#441	1	1	yes	characteristic
181	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
182	D₈₉	—	2	2	no	—
183	#3170	—	89	1	no	—
184	#2960	—	89	2	no	—
185	#1593	—	89	2	no	—
186	#1593	—	89	1	no	—
187	#1593	D₈₉	1	1	yes	characteristic, complemented
188	#1579	—	89	1	no	—
189	#1579	—	89	1	no	—
190	#1579	—	89	2	no	—
191	#1566	—	89	1	no	—
192	#1493	—	89	2	no	—
193	C₂×#322	—	89	1	no	—
194	C₈₉	#17961	1	1	yes	characteristic, complemented, Sylow
195	#533	—	89	2	no	—
196	#500	—	89	1	no	—
197	#494	—	89	1	no	—
198	#487	—	89	2	no	—
199	#467	—	89	1	no	—
200	#467	—	89	1	no	—
201	#467	—	89	2	no	—
202	#446	—	89	2	no	—
203	#446	—	178	2	no	—
204	#446	C₂×D₈₉	1	2	yes	—
205	C₈×Q₈	—	89	2	no	—
206	C₈×Q₈	—	89	2	no	—
207	C₈×Q₈	—	178	2	no	—
208	C₄×Q₁₆	—	89	2	no	—
209	C₄×Q₁₆	—	89	1	no	—
210	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
211	#424	—	178	2	no	—
212	#423	—	89	2	no	—
213	#405	—	178	2	no	—
214	C₂×C₄×C₈	—	89	1	no	—
215	C₂×C₄×C₈	—	89	1	no	—
216	C₂×C₄×C₈	—	89	1	no	—
217	#334	—	89	2	no	—
218	#334	—	89	1	no	—
219	#334	C₂×D₈₉	1	1	yes	characteristic
220	Q₈⋊C₈	—	89	2	no	—
221	Q₈⋊C₈	—	89	2	no	—
222	Q₈⋊C₈	—	89	2	no	—
223	Q₈⋊C₈	C₂×D₈₉	1	2	yes	—
224	#322	—	89	1	no	—
225	#322	—	89	1	no	—
226	#322	—	89	1	no	—
227	#322	C₂×D₈₉	1	1	yes	characteristic
228	C₂×Q₁₆	—	89	2	no	—
229	C₂×Q₁₆	—	89	1	no	—
230	C₂×Q₁₆	C₄×D₈₉	1	1	yes	characteristic
231	C₂×M₁₆	—	178	2	no	—
232	C₂²×C₈	—	89	1	no	—
233	C₂²×C₈	—	89	1	no	—
234	C₂²×C₈	—	89	2	no	—
235	C₂²×C₈	—	178	1	no	—
236	C₄⋊Q₈	—	89	2	no	—
237	C₄⋊Q₈	—	89	2	no	—
238	#125	—	178	2	no	—
239	C₂²⋊Q₈	—	89	2	no	—
240	C₂²⋊Q₈	—	89	2	no	—
241	C₂²⋊Q₈	—	178	2	no	—
242	C₄×Q₈	—	89	2	no	—
243	C₄×Q₈	—	178	2	no	—
244	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
245	#117	—	178	2	no	—
246	C₂×C₄²	—	89	1	no	—
247	#107	—	89	1	no	—
248	#107	—	89	2	no	—
249	#107	C₄×D₈₉	1	1	yes	characteristic
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	89	2	no	—
252	C₄⋊C₈	—	178	2	no	—
253	C₄⋊C₈	—	178	2	no	—
254	C₄⋊C₈	—	178	2	no	—
255	C₄⋊C₈	—	178	2	no	—
256	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
257	#103	—	89	2	no	—
258	#103	—	89	2	no	—
259	#103	—	89	2	no	—
260	#103	C₄×D₈₉	1	2	yes	—
261	#97	—	2	2	no	—
262	#97	—	178	2	no	—
263	C₄×C₈	—	89	1	no	—
264	C₄×C₈	—	89	1	no	—
265	C₄×C₈	—	89	2	no	—
266	C₄×C₈	—	89	1	no	—
267	C₄×C₈	—	89	1	no	—
268	C₄×C₈	—	89	1	no	—
269	C₄×C₈	—	178	2	no	—
270	C₄×C₈	—	178	1	no	—
271	C₄×C₈	—	178	2	no	—
272	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
275	C₂×Q₈	—	89	2	no	—
276	C₂×Q₈	—	178	2	no	—
277	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
278	C₂²×C₄	—	89	1	no	—
279	C₂²×C₄	—	89	1	no	—
280	C₂²×C₄	—	89	1	no	—
281	Q₁₆	—	2	2	no	—
282	Q₁₆	—	178	4	no	—
283	Q₁₆	—	178	2	no	—
284	M₁₆	—	356	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	89	1	no	—
288	C₂×C₈	—	89	1	no	—
289	C₂×C₈	—	89	2	no	—
290	C₂×C₈	—	356	2	no	—
291	C₂×C₈	—	178	2	no	—
292	C₂×C₈	—	178	4	no	—
293	C₂×C₈	—	178	2	no	—
294	C₂×C₈	—	178	1	no	—
295	C₂×C₈	—	178	2	no	—
296	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
298	C₂×C₈	D₄×D₈₉	1	2	yes	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	89	2	no	—
301	C₄⋊C₄	—	178	2	no	—
302	C₄⋊C₄	—	178	2	no	—
303	C₄⋊C₄	—	178	2	no	—
304	C₄⋊C₄	—	178	2	no	—
305	C₄⋊C₄	—	178	2	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
308	C₂²⋊C₄	—	178	2	no	—
309	C₂²⋊C₄	—	178	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	89	1	no	—
312	C₄²	—	89	1	no	—
313	C₄²	—	89	1	no	—
314	C₄²	—	178	2	no	—
315	C₄²	C₂³×D₈₉	1	1	yes	characteristic
316	C₂³	—	89	1	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	356	4	no	—
320	Q₈	—	178	2	no	—
321	Q₈	—	178	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	89	1	no	—
325	C₂×C₄	—	89	1	no	—
326	C₂×C₄	—	89	4	no	—
327	C₂×C₄	—	89	1	no	—
328	C₂×C₄	—	178	2	no	—
329	C₂×C₄	—	178	2	no	—
330	C₂×C₄	—	178	2	no	—
331	C₂×C₄	—	178	1	no	—
332	C₂×C₄	—	178	1	no	—
333	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
335	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	—	2	1	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	89	2	no	—
341	C₈	—	356	1	no	—
342	C₈	—	356	2	no	—
343	C₈	—	178	2	no	—
344	C₈	—	178	1	no	—
345	C₈	#15668020	1	2	yes	—
346	C₂²	—	89	2	no	—
347	C₂²	—	178	1	no	—
348	C₂²	—	178	1	no	—
349	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	89	2	no	—
354	C₄	—	89	2	no	—
355	C₄	—	356	2	no	—
356	C₄	—	356	2	no	—
357	C₄	—	178	1	no	—
358	C₄	#131×D₈₉	1	1	yes	characteristic
359	C₄	C₂×#15668020	1	1	yes	characteristic
360	C₄	#137×D₈₉	1	2	yes	—
361	C₂	—	178	2	no	—
362	C₂	#26028961	1	1	yes	characteristic
363	C₂	#441×D₈₉	1	1	yes	characteristic
364	C₂	#444×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1593	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1593	D₈₉
hypocenter	C₈₉	#17961
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
3. #26019162	C₂	semidirect
4. #26019159	C₂	semidirect
5. #26019071	C₂	semidirect
6. #26019070	C₂	semidirect
7. #26019068	C₂	semidirect
11. C₈₉×#1593	C₂	semidirect
187. #1593	D₈₉	semidirect
194. C₈₉	#17961	semidirect

Maximal subgroups

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

subgroup	index
#26030835	2
#26029183	2
#26019162	2
#26019159	2
#26019071	2
#26019070	2
#26019068	2
#26019006	2
#26018705	2
#322×D₈₉	2
C₈₉×#1593	2
#17961	89