Subgroups — #36562589

Order 22784.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030575	C₂	1	1	yes	maximal, characteristic
2	#26029189	C₂	1	1	yes	maximal, characteristic, complemented
3	#26028969	C₂	1	1	yes	maximal, characteristic
4	#26021232	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021083	C₂	1	1	yes	maximal, characteristic
6	#26020961	C₂	1	2	yes	maximal, complemented
7	#26020954	C₂	1	2	yes	maximal, complemented
8	#26020095	C₂	1	2	yes	maximal, complemented
9	#26020088	C₂	1	2	yes	maximal, complemented
10	#26019938	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1837	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668020	C₂²	1	2	yes	—
13	#19152155	C₂²	1	1	yes	characteristic
14	#19152148	C₂²	1	1	yes	characteristic
15	#19151953	C₄	1	4	yes	complemented
16	#19151949	C₄	1	2	yes	complemented
17	#19151945	C₄	1	2	yes	complemented
18	#19151727	—	2	4	no	—
19	#19151668	—	2	4	no	—
20	#19151626	C₂²	1	2	yes	—
21	C₂×#15667938	C₂²	1	2	yes	—
22	C₂×#15667938	C₂²	1	2	yes	—
23	#19151595	C₂²	1	2	yes	—
24	#19151595	C₂²	1	2	yes	—
25	#19151566	C₂²	1	2	yes	—
26	#19151556	C₂²	1	2	yes	—
27	C₂×#15667923	C₂²	1	1	yes	characteristic
28	C₂×#15667922	C₂²	1	1	yes	characteristic
29	C₂×#15667922	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
32	#19151444	C₂²	1	1	yes	characteristic
33	#19151359	C₂²	1	2	yes	—
34	#19151359	C₂²	1	2	yes	—
35	#423×C₈₉	C₂²	1	1	yes	characteristic
36	#423×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	#341×C₈₉	C₂²	1	2	yes	—
39	#341×C₈₉	C₂²	1	2	yes	—
40	C₂×#13083574	C₂³	1	2	yes	—
41	#15668020	—	2	4	no	—
42	#15668020	—	2	4	no	—
43	C₂×#13083557	—	2	2	no	—
44	C₂²×#10623494	C₂³	1	1	yes	characteristic
45	C₂²×#10623494	C₂³	1	1	yes	characteristic
46	C₂×C₈×D₈₉	—	2	2	no	—
47	#15668012	C₂×C₄	1	1	yes	characteristic
48	#15668012	C₂×C₄	1	2	yes	—
49	#15668009	C₂×C₄	1	1	yes	characteristic
50	#15668004	C₂×C₄	1	2	yes	—
51	#15668000	C₂×C₄	1	2	yes	—
52	#15667991	—	2	2	no	—
53	#15667986	—	2	2	no	—
54	C₄×Dic₁₇₈	—	2	2	no	—
55	C₄×D₃₅₆	—	2	2	no	—
56	#15667971	C₂³	1	2	yes	—
57	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
59	#15667941	—	2	4	no	—
60	#15667938	—	2	4	no	—
61	#15667938	—	2	4	no	—
62	#15667937	C₂×C₄	1	4	yes	—
63	#15667937	C₂×C₄	1	4	yes	—
64	#15667934	C₂×C₄	1	4	yes	—
65	#15667934	C₂×C₄	1	4	yes	—
66	#15667924	—	2	4	no	—
67	#15667923	D₄	1	4	yes	—
68	#15667922	—	2	2	no	—
69	#15667922	—	2	2	no	—
70	#15667922	—	2	2	no	—
71	C₈×Dic₈₉	—	2	2	no	—
72	C₈×Dic₈₉	—	2	2	no	—
73	C₈×Dic₈₉	—	2	2	no	—
74	#15667919	—	2	2	no	—
75	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
76	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
77	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₈₉	—	2	2	no	—
81	C₄⋊C₈×C₈₉	—	2	2	no	—
82	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
84	#13083574	—	2	2	no	—
85	#13083574	—	2	2	no	—
86	#13083574	—	2	2	no	—
87	#13083574	—	2	2	no	—
88	C₂×Dic₁₇₈	—	2	2	no	—
89	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
90	C₂×D₃₅₆	—	2	2	no	—
91	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
92	C₂×C₂²⋊D₈₉	—	2	2	no	—
93	C₂²×Dic₈₉	—	2	1	no	—
94	C₂²×Dic₈₉	—	2	1	no	—
95	C₂×C₄×D₈₉	—	2	2	no	—
96	#13083557	—	4	4	no	—
97	C₂×#10623494	—	2	1	no	—
98	C₂×#10623494	—	2	1	no	—
99	C₂×#10623494	C₂×D₄	1	2	yes	—
100	C₂×#10623494	C₂×D₄	1	2	yes	—
101	C₂×#10623494	#41	1	2	yes	—
102	C₂×#10623494	#41	1	2	yes	—
103	C₈×D₈₉	—	4	4	no	—
104	C₄⋊Dic₈₉	—	2	2	no	—
105	C₄⋊Dic₈₉	—	2	2	no	—
106	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
107	#13083553	—	2	4	no	—
108	C₂²⋊Dic₈₉	—	2	2	no	—
109	#13083551	—	2	4	no	—
110	C₄×Dic₈₉	—	2	2	no	—
111	C₄×Dic₈₉	—	2	1	no	—
112	C₄×Dic₈₉	—	2	1	no	—
113	C₄×Dic₈₉	#41	1	2	yes	—
114	C₄×Dic₈₉	#41	1	2	yes	—
115	C₂²×C₄×C₈₉	#41	1	2	yes	—
116	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
117	M₁₆×C₈₉	—	2	2	no	—
118	M₁₆×C₈₉	—	2	2	no	—
119	M₁₆×C₈₉	—	2	2	no	—
120	M₁₆×C₈₉	—	2	2	no	—
121	C₂×C₈×C₈₉	—	2	1	no	—
122	C₂×C₈×C₈₉	—	2	1	no	—
123	C₂×C₈×C₈₉	—	2	1	no	—
124	C₂×C₈×C₈₉	—	2	1	no	—
125	C₂×C₈×C₈₉	—	2	2	no	—
126	C₂×C₈×C₈₉	—	2	2	no	—
127	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
129	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₈₉	C₂×D₄	1	2	yes	—
132	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
133	C₂²×D₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
143	C₂²⋊D₈₉	—	4	2	no	—
144	C₂²⋊D₈₉	—	4	2	no	—
145	C₂×Dic₈₉	—	2	4	no	—
146	C₂×Dic₈₉	—	2	2	no	—
147	C₂×Dic₈₉	—	2	2	no	—
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	D₃₅₆	—	2	2	no	—
152	D₃₅₆	—	2	2	no	—
153	D₃₅₆	—	8	2	no	—
154	C₄×D₈₉	—	4	2	no	—
155	C₄×D₈₉	—	4	2	no	—
156	Dic₁₇₈	—	2	2	no	—
157	Dic₁₇₈	—	2	2	no	—
158	Dic₁₇₈	—	8	2	no	—
159	C₈×C₈₉	—	2	2	no	—
160	C₈×C₈₉	—	2	2	no	—
161	C₈×C₈₉	—	4	2	no	—
162	C₈×C₈₉	—	4	2	no	—
163	#10623494	—	2	2	no	—
164	#10623494	#124	1	4	yes	—
165	C₂²×C₈₉	—	2	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	C₂×D₈₉	—	4	4	no	—
170	C₂×D₈₉	—	4	2	no	—
171	C₄×C₈₉	—	2	2	no	—
172	C₄×C₈₉	—	2	2	no	—
173	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
174	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
175	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
176	C₄×C₈₉	#532	1	1	yes	characteristic
177	Dic₈₉	—	4	2	no	—
178	Dic₈₉	—	4	2	no	—
179	Dic₈₉	—	4	4	no	—
180	Dic₈₉	—	4	2	no	—
181	#21659	—	89	1	no	maximal, Sylow
182	C₂×C₈₉	—	2	1	no	—
183	C₂×C₈₉	—	2	1	no	—
184	C₂×C₈₉	C₄×#124	1	1	yes	characteristic
185	C₂×C₈₉	#2957	1	2	yes	—
186	D₈₉	—	8	2	no	—
187	#3131	—	89	1	no	—
188	#2961	—	89	1	no	—
189	#2932	—	89	1	no	—
190	#1995	—	89	1	no	—
191	#1963	—	89	1	no	—
192	#1938	—	89	2	no	—
193	#1937	—	89	2	no	—
194	#1837	D₈₉	1	1	yes	characteristic, complemented
195	#1765	—	89	2	no	—
196	#1764	—	89	2	no	—
197	#1726	—	89	1	no	—
198	C₈₉	#21659	1	1	yes	characteristic, complemented, Sylow
199	C₂×#131	—	89	2	no	—
200	#533	—	89	1	no	—
201	#532	—	89	1	no	—
202	#488	—	89	4	no	—
203	#487	—	89	2	no	—
204	#486	—	89	2	no	—
205	#446	—	178	4	no	—
206	#435	—	178	4	no	—
207	#424	—	89	2	no	—
208	#423	C₂×D₈₉	1	1	yes	characteristic
209	#423	C₂×D₈₉	1	1	yes	characteristic
210	C₂×#104	—	89	2	no	—
211	C₂×#104	—	89	2	no	—
212	#416	—	89	2	no	—
213	#416	—	89	2	no	—
214	#408	—	89	2	no	—
215	#405	—	89	2	no	—
216	C₂×#97	—	89	1	no	—
217	C₂×#97	—	89	1	no	—
218	C₂×#97	—	89	1	no	—
219	C₂×C₄×C₈	—	89	1	no	—
220	C₂×C₄×C₈	—	89	1	no	—
221	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
222	#376	—	89	1	no	—
223	#341	C₂×D₈₉	1	2	yes	—
224	#341	C₂×D₈₉	1	2	yes	—
225	#339	—	89	2	no	—
226	#339	—	89	2	no	—
227	C₂×#41	—	89	2	no	—
228	#131	—	178	4	no	—
229	#131	—	178	4	no	—
230	C₂×M₁₆	—	178	2	no	—
231	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
232	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
233	C₂²×C₈	—	89	1	no	—
234	C₂²×C₈	—	89	1	no	—
235	C₂²×C₈	—	178	2	no	—
236	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
238	C₄⋊Q₈	—	89	2	no	—
239	C₄⋊Q₈	—	89	1	no	—
240	#127	—	89	1	no	—
241	#125	—	89	2	no	—
242	#124	—	89	2	no	—
243	C₂²⋊Q₈	—	178	2	no	—
244	#121	—	178	2	no	—
245	C₄×Q₈	—	178	2	no	—
246	C₄×D₄	—	178	2	no	—
247	#117	—	89	2	no	—
248	C₂×C₄²	—	89	1	no	—
249	C₂×C₄²	—	89	1	no	—
250	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	178	4	no	—
254	#104	—	178	4	no	—
255	#104	—	178	4	no	—
256	#103	—	89	4	no	—
257	#103	—	89	4	no	—
258	#102	—	89	4	no	—
259	#102	—	89	4	no	—
260	C₂²⋊C₈	—	178	4	no	—
261	#97	—	89	4	no	—
262	#97	—	178	2	no	—
263	#97	—	178	2	no	—
264	#97	—	178	2	no	—
265	C₄×C₈	—	178	2	no	—
266	C₄×C₈	—	178	2	no	—
267	C₄×C₈	—	178	2	no	—
268	C₄×C₈	C₄×D₈₉	1	2	yes	—
269	C₄×C₈	C₄×D₈₉	1	2	yes	—
270	#95	—	178	2	no	—
271	#41	—	178	2	no	—
272	#41	—	178	2	no	—
273	#41	—	178	2	no	—
274	#41	—	178	2	no	—
275	C₂×Q₈	—	89	2	no	—
276	C₂×Q₈	—	178	2	no	—
277	C₂×D₄	—	89	2	no	—
278	C₂×D₄	—	178	2	no	—
279	C₂×D₄	—	178	2	no	—
280	C₂²×C₄	—	178	1	no	—
281	C₂²×C₄	—	178	1	no	—
282	C₂²×C₄	—	178	2	no	—
283	C₂²×C₄	#13083575	1	2	yes	—
284	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	356	4	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	89	2	no	—
297	C₂×C₈	—	89	2	no	—
298	C₂×C₈	—	89	2	no	—
299	C₂×C₈	—	89	2	no	—
300	C₂×C₈	—	356	4	no	—
301	C₂×C₈	—	178	1	no	—
302	C₂×C₈	—	178	1	no	—
303	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
304	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
305	C₄⋊C₄	—	89	4	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₄⋊C₄	—	178	4	no	—
308	C₄⋊C₄	—	178	2	no	—
309	C₂²⋊C₄	—	178	2	no	—
310	C₂²⋊C₄	—	178	4	no	—
311	C₄²	—	89	2	no	—
312	C₄²	—	89	2	no	—
313	C₄²	—	178	2	no	—
314	C₄²	—	178	1	no	—
315	C₄²	—	178	1	no	—
316	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
318	C₄²	D₄×D₈₉	1	2	yes	—
319	C₂³	—	178	2	no	—
320	C₂³	C₂×#13083575	1	1	yes	characteristic
321	Q₈	—	178	2	no	—
322	Q₈	—	178	2	no	—
323	Q₈	—	712	2	no	—
324	D₄	—	356	2	no	—
325	D₄	—	356	2	no	—
326	D₄	—	178	2	no	—
327	D₄	—	178	2	no	—
328	D₄	—	712	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	356	2	no	—
331	C₂×C₄	—	356	2	no	—
332	C₂×C₄	—	356	2	no	—
333	C₂×C₄	—	356	2	no	—
334	C₂×C₄	—	178	2	no	—
335	C₂×C₄	—	178	2	no	—
336	C₂×C₄	—	178	4	no	—
337	C₂×C₄	—	178	2	no	—
338	C₂×C₄	#15667974	1	2	yes	—
339	C₂×C₄	#15667974	1	2	yes	—
340	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
342	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
343	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
344	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
345	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	89	4	no	—
351	C₈	—	178	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	356	4	no	—
354	C₂²	—	356	2	no	—
355	C₂²	C₂×#15667974	1	1	yes	characteristic
356	C₂²	#19152133	1	1	yes	characteristic
357	C₂²	#19152133	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	356	2	no	—
361	C₄	—	356	2	no	—
362	C₄	—	356	4	no	—
363	C₄	—	356	2	no	—
364	C₄	#19152066	1	1	yes	characteristic
365	C₄	#19152066	1	1	yes	characteristic
366	C₄	#124×D₈₉	1	1	yes	characteristic
367	C₄	#19152147	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	712	2	no	—
371	C₂	#26022791	1	1	yes	characteristic
372	C₂	#26029168	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×#1837	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#19152148	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1837	D₈₉
hypocenter	C₈₉	#21659
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
2. #26029189	C₂	semidirect
4. #26021232	C₂	semidirect
6. #26020961	C₂	semidirect
7. #26020954	C₂	semidirect
8. #26020095	C₂	semidirect
9. #26020088	C₂	semidirect
10. #26019938	C₂	semidirect
11. C₈₉×#1837	C₂	semidirect
15. #19151953	C₄	semidirect
16. #19151949	C₄	semidirect
17. #19151945	C₄	semidirect
194. #1837	D₈₉	semidirect
198. C₈₉	#21659	semidirect

Maximal subgroups

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

subgroup	index
#26030575	2
#26029189	2
#26028969	2
#26021232	2
#26021083	2
#26020961	2
#26020954	2
#26020095	2
#26020088	2
#26019938	2
C₈₉×#1837	2
#21659	89