Subgroups — #36525105

Order 22784.

560 conjugacy classes of subgroups.

The subgroup lattice has 360 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	#26029343	C₂	1	2	yes	maximal
3	#26019161	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019160	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019146	C₂	1	2	yes	maximal, complemented
6	#26019072	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#13083564	C₂	1	2	yes	maximal, complemented
8	#26019012	C₂	1	1	yes	maximal, characteristic
9	#26018723	C₂	1	2	yes	maximal
10	#26018537	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1579	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	#19151729	C₂²	1	2	yes	—
20	#19151728	—	2	2	no	—
21	#19151727	—	2	2	no	—
22	#19151725	C₂²	1	2	yes	—
23	C₈×Dic₁₇₈	C₂²	1	2	yes	—
24	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
25	C₄×#13083564	C₂²	1	2	yes	—
26	#19151658	—	2	2	no	—
27	#19151657	—	2	2	no	—
28	#19151654	C₂²	1	2	yes	—
29	#19151649	C₂²	1	1	yes	characteristic
30	#19151649	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#19151350	C₂²	1	2	yes	—
33	#19151349	C₂²	1	1	yes	characteristic
34	#19151330	C₂²	1	2	yes	—
35	Q₈⋊#10623494	C₂²	1	2	yes	—
36	#19151328	C₂²	1	2	yes	—
37	#19151319	C₂²	1	1	yes	characteristic
38	C₈×#10623494	C₂²	1	1	yes	characteristic
39	C₈×#10623494	C₂²	1	1	yes	characteristic
40	C₈×Q₈×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×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
49	#15668011	C₂×C₄	1	2	yes	—
50	#15668010	C₂×C₄	1	2	yes	—
51	#15668002	—	2	2	no	—
52	#15667992	C₂×C₄	1	2	yes	—
53	#15667990	—	2	2	no	—
54	#15667988	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₇₈	—	2	2	no	—
56	C₄×Dic₁₇₈	C₂³	1	2	yes	—
57	#15667969	—	2	2	no	—
58	C₄²×D₈₉	C₂³	1	1	yes	characteristic
59	#15667946	C₂×C₄	1	1	yes	characteristic
60	#15667945	C₂×C₄	1	2	yes	—
61	C₄⋊#10623494	—	2	2	no	—
62	C₄⋊#10623494	C₂³	1	2	yes	—
63	#15667941	—	2	2	no	—
64	#15667941	—	2	1	no	—
65	#15667941	—	2	2	no	—
66	#15667941	D₄	1	2	yes	—
67	#15667937	C₂×C₄	1	2	yes	—
68	#15667936	C₂×C₄	1	2	yes	—
69	#15667935	C₂×C₄	1	2	yes	—
70	#15667924	—	2	2	no	—
71	#15667924	—	2	1	no	—
72	#15667924	D₄	1	2	yes	—
73	#15667923	—	2	2	no	—
74	#15667922	—	2	2	no	—
75	#15667922	—	2	2	no	—
76	C₄×#10623494	C₂³	1	1	yes	characteristic
77	C₄×#10623494	C₂³	1	1	yes	characteristic
78	C₄×#10623494	C₂³	1	1	yes	characteristic
79	C₈×Dic₈₉	—	2	2	no	—
80	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
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	C₄×C₈×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	C₂×#10623494	—	2	2	no	—
100	C₂×#10623494	—	2	1	no	—
101	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
102	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
103	C₂×#10623494	C₂×D₄	1	2	yes	—
104	C₈×D₈₉	—	2	2	no	—
105	C₈×D₈₉	—	2	2	no	—
106	C₄⋊Dic₈₉	—	2	2	no	—
107	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
108	#13083553	—	2	2	no	—
109	#13083553	—	2	2	no	—
110	#13083553	—	2	2	no	—
111	#13083553	—	2	2	no	—
112	#13083553	—	2	2	no	—
113	#13083551	—	2	2	no	—
114	#13083551	—	2	2	no	—
115	C₄×Dic₈₉	—	2	2	no	—
116	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
117	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₈₉	#41	1	1	yes	characteristic
119	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
120	Q₁₆×C₈₉	—	2	2	no	—
121	C₂×C₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	2	1	no	—
123	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₈₉	—	2	2	no	—
127	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
128	C₄²×C₈₉	—	2	2	no	—
129	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
131	Q₈×C₈₉	—	2	2	no	—
132	Q₈×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₈₉	—	2	2	no	—
139	C₂×Dic₈₉	—	2	2	no	—
140	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
141	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
142	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
143	C₄×D₈₉	—	2	1	no	—
144	C₄×D₈₉	—	2	1	no	—
145	C₄×D₈₉	—	2	2	no	—
146	C₄×D₈₉	—	4	1	no	—
147	Dic₁₇₈	—	2	2	no	—
148	Dic₁₇₈	—	2	2	no	—
149	Dic₁₇₈	—	8	2	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	2	2	no	—
152	C₈×C₈₉	—	4	1	no	—
153	C₈×C₈₉	—	4	2	no	—
154	C₈×C₈₉	#131	1	2	yes	—
155	#10623494	—	2	1	no	—
156	#10623494	—	2	2	no	—
157	#10623494	—	4	1	no	—
158	#10623494	—	4	2	no	—
159	#10623494	#131	1	2	yes	—
160	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂×D₈₉	—	2	1	no	—
162	C₂×D₈₉	—	2	1	no	—
163	C₂×D₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	4	2	no	—
166	C₄×C₈₉	—	4	2	no	—
167	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
168	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
169	C₄×C₈₉	#578	1	2	yes	—
170	Dic₈₉	—	2	1	no	—
171	Dic₈₉	—	2	1	no	—
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	4	2	no	—
174	Dic₈₉	—	4	2	no	—
175	#17972	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	#2931	1	1	yes	characteristic
177	C₂×C₈₉	#2950	1	1	yes	characteristic
178	C₂×C₈₉	#2957	1	1	yes	characteristic
179	D₈₉	—	4	1	no	—
180	#3170	—	89	1	no	—
181	#2985	—	89	2	no	—
182	#1593	—	89	1	no	—
183	#1593	—	89	1	no	—
184	#1591	—	89	2	no	—
185	#1579	—	89	1	no	—
186	#1579	D₈₉	1	1	yes	characteristic, complemented
187	C₈×Q₁₆	—	89	2	no	—
188	#1568	—	89	1	no	—
189	#1497	—	89	2	no	—
190	#1450	—	89	1	no	—
191	C₈₉	#17972	1	1	yes	characteristic, complemented, Sylow
192	#533	—	89	2	no	—
193	#500	—	89	1	no	—
194	#494	—	89	1	no	—
195	#487	—	89	2	no	—
196	#467	—	89	1	no	—
197	#467	—	89	1	no	—
198	#467	—	89	2	no	—
199	#446	—	89	2	no	—
200	#446	—	89	2	no	—
201	#446	—	178	2	no	—
202	#446	—	178	2	no	—
203	C₈×Q₈	—	89	2	no	—
204	C₈×Q₈	C₂×D₈₉	1	2	yes	—
205	C₄×Q₁₆	—	89	2	no	—
206	C₄×Q₁₆	—	89	1	no	—
207	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
208	#433	—	178	2	no	—
209	#433	—	178	2	no	—
210	#432	—	89	2	no	—
211	#431	—	89	1	no	—
212	#431	—	89	1	no	—
213	C₂×C₄×C₈	—	89	1	no	—
214	#335	—	89	2	no	—
215	#334	—	89	1	no	—
216	#334	C₂×D₈₉	1	1	yes	characteristic
217	Q₈⋊C₈	—	89	2	no	—
218	Q₈⋊C₈	—	89	2	no	—
219	Q₈⋊C₈	—	89	2	no	—
220	Q₈⋊C₈	C₂×D₈₉	1	2	yes	—
221	#322	—	89	1	no	—
222	#322	C₂×D₈₉	1	1	yes	characteristic
223	C₈²	—	89	1	no	—
224	C₈²	—	89	1	no	—
225	C₂×Q₁₆	—	89	2	no	—
226	C₂×Q₁₆	—	89	1	no	—
227	C₂×Q₁₆	C₄×D₈₉	1	1	yes	characteristic
228	C₂²×C₈	—	89	1	no	—
229	C₂²×C₈	—	89	1	no	—
230	C₄⋊Q₈	—	89	2	no	—
231	C₄⋊Q₈	—	89	2	no	—
232	#125	—	178	2	no	—
233	C₂²⋊Q₈	—	89	2	no	—
234	C₂²⋊Q₈	—	89	2	no	—
235	C₂²⋊Q₈	—	178	2	no	—
236	C₄×Q₈	—	89	2	no	—
237	C₄×Q₈	—	178	2	no	—
238	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
239	#117	—	178	2	no	—
240	C₂×C₄²	—	89	1	no	—
241	#107	—	89	1	no	—
242	#107	—	89	2	no	—
243	#107	C₄×D₈₉	1	1	yes	characteristic
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	89	2	no	—
246	C₄⋊C₈	—	89	2	no	—
247	C₄⋊C₈	—	178	2	no	—
248	C₄⋊C₈	—	178	1	no	—
249	C₄⋊C₈	—	178	2	no	—
250	C₄⋊C₈	—	178	2	no	—
251	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
252	#103	—	89	2	no	—
253	#103	—	89	2	no	—
254	#103	—	89	2	no	—
255	#103	C₄×D₈₉	1	2	yes	—
256	C₂²⋊C₈	—	89	2	no	—
257	C₂²⋊C₈	—	178	1	no	—
258	C₂²⋊C₈	—	178	2	no	—
259	#97	—	178	2	no	—
260	#97	—	178	2	no	—
261	#97	—	178	2	no	—
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	—	89	1	no	—
264	C₄×C₈	—	89	1	no	—
265	C₄×C₈	—	89	1	no	—
266	C₄×C₈	—	89	1	no	—
267	C₄×C₈	—	89	1	no	—
268	C₄×C₈	—	178	2	no	—
269	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
272	C₂×Q₈	—	89	2	no	—
273	C₂×Q₈	—	178	2	no	—
274	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
275	C₂²×C₄	—	89	1	no	—
276	C₂²×C₄	—	89	1	no	—
277	C₂²×C₄	—	89	1	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	178	4	no	—
280	Q₁₆	—	178	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	89	1	no	—
284	C₂×C₈	—	89	1	no	—
285	C₂×C₈	—	89	2	no	—
286	C₂×C₈	—	178	2	no	—
287	C₂×C₈	—	178	2	no	—
288	C₂×C₈	—	178	1	no	—
289	C₂×C₈	—	178	2	no	—
290	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
292	C₂×C₈	D₄×D₈₉	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	89	2	no	—
295	C₄⋊C₄	—	178	2	no	—
296	C₄⋊C₄	—	178	2	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₄⋊C₄	—	178	2	no	—
299	C₄⋊C₄	—	178	2	no	—
300	C₄⋊C₄	—	178	2	no	—
301	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
302	C₂²⋊C₄	—	178	2	no	—
303	C₂²⋊C₄	—	178	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	89	1	no	—
306	C₄²	—	89	1	no	—
307	C₄²	—	89	1	no	—
308	C₄²	—	178	2	no	—
309	C₄²	C₂³×D₈₉	1	1	yes	characteristic
310	C₂³	—	89	1	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	178	2	no	—
314	Q₈	—	178	2	no	—
315	Q₈	—	712	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	89	1	no	—
319	C₂×C₄	—	89	1	no	—
320	C₂×C₄	—	89	1	no	—
321	C₂×C₄	—	356	1	no	—
322	C₂×C₄	—	178	2	no	—
323	C₂×C₄	—	178	2	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	—	178	1	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
329	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	2	1	no	—
332	C₈	—	4	1	no	—
333	C₈	—	4	2	no	—
334	C₈	—	89	2	no	—
335	C₈	—	356	1	no	—
336	C₈	—	356	2	no	—
337	C₈	—	178	2	no	—
338	C₈	—	178	1	no	—
339	C₈	#15668018	1	2	yes	—
340	C₂²	—	178	1	no	—
341	C₂²	—	178	1	no	—
342	C₂²	—	178	1	no	—
343	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	356	2	no	—
348	C₄	—	356	2	no	—
349	C₄	—	178	1	no	—
350	C₄	—	178	1	no	—
351	C₄	—	178	1	no	—
352	C₄	#131×D₈₉	1	1	yes	characteristic
353	C₄	C₂×#15668018	1	1	yes	characteristic
354	C₄	#19152414	1	2	yes	—
355	C₂	—	356	1	no	—
356	C₂	#26028959	1	1	yes	characteristic
357	C₂	#26029117	1	1	yes	characteristic
358	C₂	#26029166	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×#1579	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1579	D₈₉
hypocenter	C₈₉	#17972
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #26019161	C₂	semidirect
4. #26019160	C₂	semidirect
5. #26019146	C₂	semidirect
6. #26019072	C₂	semidirect
7. C₈×#13083564	C₂	semidirect
11. C₈₉×#1579	C₂	semidirect
186. #1579	D₈₉	semidirect
191. C₈₉	#17972	semidirect

Maximal subgroups

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

subgroup	index
#26030835	2
#26029343	2
#26019161	2
#26019160	2
#26019146	2
#26019072	2
C₈×#13083564	2
#26019012	2
#26018723	2
#26018537	2
C₈₉×#1579	2
#17972	89