Subgroups — #36537705

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031405	C₂	1	1	yes	maximal, characteristic, complemented
2	#26031195	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151945	C₂	1	1	yes	maximal, characteristic
4	#26029186	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019793	C₂	1	4	yes	maximal, complemented
6	#26019669	C₂	1	4	yes	maximal, complemented
7	#26018932	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018919	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152155	C₂²	1	1	yes	characteristic
11	C₂×#15668012	C₂²	1	1	yes	characteristic
12	C₂×#15668009	C₂²	1	1	yes	characteristic
13	#19152003	—	2	1	no	—
14	#19151971	—	2	1	no	—
15	#19151949	—	2	1	no	—
16	#19151945	—	2	1	no	—
17	#19151945	—	2	1	no	—
18	#19151945	—	2	1	no	—
19	#19151945	C₂²	1	2	yes	—
20	#19151945	C₂²	1	2	yes	—
21	#19151934	—	2	2	no	—
22	#19151894	C₂²	1	4	yes	complemented
23	#19151799	C₂²	1	4	yes	—
24	#19151751	—	2	2	no	—
25	#19151728	—	2	1	no	—
26	#19151728	C₂²	1	2	yes	complemented
27	C₈×Dic₁₇₈	—	2	1	no	—
28	C₈×Dic₁₇₈	C₂²	1	2	yes	complemented
29	#19151624	—	2	1	no	—
30	#19151622	C₂²	1	1	yes	characteristic
31	C₂×#15667934	—	2	1	no	—
32	C₂×#15667934	—	2	1	no	—
33	#19151553	—	2	1	no	—
34	#19151346	C₂²	1	4	yes	complemented
35	#19151341	C₂²	1	4	yes	—
36	#431×C₈₉	C₂²	1	1	yes	characteristic
37	#431×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
39	#335×C₈₉	C₂²	1	2	yes	complemented
40	#334×C₈₉	C₂²	1	2	yes	complemented
41	C₂²×Dic₁₇₈	—	2	1	no	—
42	C₂²×D₃₅₆	—	2	1	no	—
43	C₂²×D₃₅₆	—	2	1	no	—
44	C₂²×D₃₅₆	—	2	1	no	—
45	C₂×#13083563	—	2	4	no	—
46	C₂×#13083558	—	2	1	no	—
47	#15668012	—	2	1	no	—
48	#15668012	—	2	1	no	—
49	#15668012	—	2	1	no	—
50	#15668012	D₄	1	1	yes	characteristic
51	#15668012	D₄	1	1	yes	characteristic
52	#15668012	C₂³	1	2	yes	—
53	#15668009	—	2	1	no	—
54	#15668009	—	2	1	no	—
55	#15668009	—	2	1	no	—
56	#15668009	C₂³	1	2	yes	—
57	#15668004	—	2	1	no	—
58	#15667991	—	2	1	no	—
59	#15667991	D₄	1	2	yes	—
60	C₄×Dic₁₇₈	—	2	1	no	—
61	C₄×Dic₁₇₈	C₂³	1	2	yes	—
62	#15667971	—	2	1	no	—
63	C₂×C₄⋊Dic₈₉	—	2	1	no	—
64	C₂×C₄⋊Dic₈₉	—	2	1	no	—
65	#15667946	—	2	4	no	—
66	C₄⋊#10623494	—	2	1	no	—
67	C₄⋊#10623494	C₂³	1	2	yes	—
68	#15667941	—	2	2	no	—
69	#15667941	—	2	2	no	—
70	#15667937	—	2	4	no	—
71	#15667934	—	2	4	no	—
72	#15667934	—	2	2	no	—
73	#15667934	—	2	2	no	—
74	#15667934	—	2	2	no	—
75	#15667934	—	2	2	no	—
76	#15667922	—	2	2	no	—
77	C₈×Dic₈₉	—	2	2	no	—
78	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
82	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
83	C₂²⋊C₈×C₈₉	—	2	1	no	—
84	C₂²⋊C₈×C₈₉	—	2	1	no	—
85	C₄×C₈×C₈₉	—	2	1	no	—
86	C₄×C₈×C₈₉	—	2	1	no	—
87	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
88	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
89	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₂³×D₈₉	—	4	1	no	—
92	C₂×Dic₁₇₈	—	2	1	no	—
93	C₂×Dic₁₇₈	—	2	2	no	—
94	C₂×Dic₁₇₈	—	4	1	no	—
95	C₂×Dic₁₇₈	—	8	2	no	—
96	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
97	C₂×D₃₅₆	—	2	2	no	—
98	C₂×D₃₅₆	—	2	2	no	—
99	C₂×D₃₅₆	—	2	2	no	—
100	C₂×D₃₅₆	—	4	2	no	—
101	C₂×D₃₅₆	—	4	2	no	—
102	C₂×D₃₅₆	—	4	1	no	—
103	C₂×D₃₅₆	—	4	1	no	—
104	C₂×D₃₅₆	—	4	1	no	—
105	C₂×D₃₅₆	—	8	2	no	—
106	C₂×D₃₅₆	—	8	2	no	—
107	C₂²×Dic₈₉	—	4	1	no	—
108	#13083563	—	8	4	no	—
109	#13083558	—	4	2	no	—
110	C₂×#10623494	—	2	2	no	—
111	C₄⋊Dic₈₉	—	2	1	no	—
112	C₄⋊Dic₈₉	—	2	1	no	—
113	C₄⋊Dic₈₉	—	2	1	no	—
114	C₄⋊Dic₈₉	—	2	2	no	—
115	C₄⋊Dic₈₉	—	2	2	no	—
116	C₄⋊Dic₈₉	—	4	1	no	—
117	C₄⋊Dic₈₉	—	4	1	no	—
118	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
119	#13083553	—	2	2	no	—
120	#13083553	—	2	2	no	—
121	C₂²⋊Dic₈₉	—	2	2	no	—
122	C₄×Dic₈₉	—	2	2	no	—
123	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
126	C₂×C₈×C₈₉	—	2	2	no	—
127	C₂×C₈×C₈₉	—	2	2	no	—
128	C₂×C₈×C₈₉	—	2	2	no	—
129	C₂×C₈×C₈₉	—	2	2	no	—
130	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₈₉	#41	1	1	yes	characteristic
137	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
138	C₂³×C₈₉	#121	1	1	yes	characteristic
139	C₂²×D₈₉	—	4	2	no	—
140	C₂²×D₈₉	—	8	2	no	—
141	C₂²×D₈₉	—	8	2	no	—
142	C₂²×D₈₉	—	8	2	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
148	C₂×C₄×C₈₉	#135	1	2	yes	—
149	C₂×C₄×C₈₉	#137	1	2	yes	—
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
153	C₂×Dic₈₉	—	2	2	no	—
154	C₂×Dic₈₉	—	2	2	no	—
155	C₂×Dic₈₉	—	4	2	no	—
156	C₂×Dic₈₉	—	8	2	no	—
157	D₃₅₆	—	4	2	no	—
158	D₃₅₆	—	4	2	no	—
159	D₃₅₆	—	4	4	no	—
160	D₃₅₆	—	8	2	no	—
161	D₃₅₆	—	8	2	no	—
162	D₃₅₆	—	8	2	no	—
163	D₃₅₆	—	8	2	no	—
164	Dic₁₇₈	—	2	4	no	—
165	Dic₁₇₈	—	4	2	no	—
166	Dic₁₇₈	—	8	2	no	—
167	Dic₁₇₈	—	8	2	no	—
168	C₈×C₈₉	—	2	1	no	—
169	C₈×C₈₉	—	2	1	no	—
170	C₈×C₈₉	—	2	1	no	—
171	C₈×C₈₉	—	2	1	no	—
172	C₈×C₈₉	—	4	2	no	—
173	C₈×C₈₉	—	4	2	no	—
174	#10623494	—	4	2	no	—
175	C₂²×C₈₉	—	2	1	no	—
176	C₂²×C₈₉	—	2	1	no	—
177	C₂²×C₈₉	#464	1	2	yes	—
178	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
179	C₂×D₈₉	—	8	2	no	—
180	C₂×D₈₉	—	8	2	no	—
181	C₂×D₈₉	—	8	2	no	—
182	C₂×D₈₉	—	16	4	no	—
183	C₄×C₈₉	—	2	1	no	—
184	C₄×C₈₉	—	2	1	no	—
185	C₄×C₈₉	#464	1	2	yes	—
186	C₄×C₈₉	#472	1	2	yes	—
187	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
188	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
189	C₄×C₈₉	#577	1	2	yes	—
190	Dic₈₉	—	4	2	no	—
191	Dic₈₉	—	4	2	no	—
192	Dic₈₉	—	8	2	no	—
193	#19212	—	89	1	no	maximal, Sylow
194	C₂×C₈₉	—	2	2	no	—
195	C₂×C₈₉	C₂×#464	1	1	yes	characteristic
196	C₂×C₈₉	#3057	1	1	yes	characteristic
197	C₂×C₈₉	C₂×#472	1	1	yes	characteristic
198	D₈₉	—	16	2	no	—
199	#3230	—	89	1	no	—
200	#3208	—	89	1	no	—
201	C₂×#486	—	89	1	no	—
202	#2960	—	89	1	no	—
203	#1698	—	89	4	no	—
204	#1678	—	89	4	no	—
205	#1564	D₈₉	1	1	yes	characteristic, complemented
206	#1548	—	89	1	no	—
207	#1545	—	89	1	no	—
208	C₈₉	#19212	1	1	yes	characteristic, complemented, Sylow
209	#533	—	89	1	no	—
210	C₂×C₄⋊Q₈	—	89	1	no	—
211	C₂×#127	—	89	1	no	—
212	#500	—	178	1	no	—
213	#492	—	178	1	no	—
214	#487	—	178	1	no	—
215	#486	—	89	2	no	—
216	#486	—	89	2	no	—
217	#486	—	178	1	no	—
218	#486	—	178	1	no	—
219	#486	—	178	1	no	—
220	#484	—	178	2	no	—
221	#477	—	89	4	no	—
222	#460	—	89	4	no	—
223	#450	—	178	2	no	—
224	#446	—	89	2	no	—
225	#446	—	178	1	no	—
226	C₈×Q₈	—	89	2	no	—
227	C₈×Q₈	—	178	1	no	—
228	#431	C₂×D₈₉	1	1	yes	characteristic
229	#431	C₂×D₈₉	1	1	yes	characteristic
230	#424	—	178	1	no	—
231	#423	—	89	1	no	—
232	C₂×#102	—	178	1	no	—
233	C₂×#102	—	178	1	no	—
234	#405	—	178	1	no	—
235	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
236	#335	C₂×D₈₉	1	2	yes	complemented
237	#334	C₂×D₈₉	1	2	yes	complemented
238	#333	—	89	4	no	—
239	#331	—	89	4	no	—
240	C₂²×Q₈	—	178	1	no	—
241	C₂²×D₄	—	178	1	no	—
242	C₂²×D₄	—	178	1	no	—
243	C₂²×D₄	—	178	1	no	—
244	C₂×SD₁₆	—	178	4	no	—
245	C₂×M₁₆	—	178	1	no	—
246	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
247	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
248	C₄⋊Q₈	—	89	1	no	—
249	C₄⋊Q₈	—	89	1	no	—
250	C₄⋊Q₈	—	89	2	no	—
251	C₄⋊Q₈	—	178	1	no	—
252	C₄⋊Q₈	—	178	1	no	—
253	C₄⋊Q₈	—	178	1	no	—
254	#127	—	89	2	no	—
255	#127	—	178	1	no	—
256	#127	—	178	1	no	—
257	#127	—	178	1	no	—
258	#125	—	178	1	no	—
259	C₂²⋊Q₈	—	89	2	no	—
260	C₂²⋊Q₈	—	178	1	no	—
261	C₄×Q₈	—	89	2	no	—
262	C₄×Q₈	—	178	1	no	—
263	#117	—	178	1	no	—
264	C₂×C₄⋊C₄	—	178	1	no	—
265	C₂×C₄⋊C₄	—	178	1	no	—
266	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
267	#107	—	178	4	no	—
268	C₄⋊C₈	—	89	2	no	—
269	C₄⋊C₈	—	178	2	no	—
270	C₄⋊C₈	—	178	2	no	—
271	C₄⋊C₈	—	178	1	no	—
272	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
273	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
274	#103	—	178	4	no	—
275	#102	—	178	4	no	—
276	#102	—	178	2	no	—
277	#102	—	178	2	no	—
278	#102	—	178	2	no	—
279	#102	—	178	2	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	#97	—	178	2	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	—	178	2	no	—
286	C₄×C₈	D₃₅₆	1	1	yes	characteristic
287	C₄×C₈	D₃₅₆	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
290	C₂⁴	—	356	1	no	—
291	C₂×Q₈	—	89	2	no	—
292	C₂×Q₈	—	356	1	no	—
293	C₂×Q₈	—	178	2	no	—
294	C₂×Q₈	—	178	1	no	—
295	C₂×Q₈	—	712	2	no	—
296	C₂×D₄	—	356	1	no	—
297	C₂×D₄	—	356	1	no	—
298	C₂×D₄	—	356	1	no	—
299	C₂×D₄	—	356	2	no	—
300	C₂×D₄	—	356	2	no	—
301	C₂×D₄	—	178	2	no	—
302	C₂×D₄	—	178	2	no	—
303	C₂×D₄	—	178	2	no	—
304	C₂×D₄	—	712	2	no	—
305	C₂×D₄	—	712	2	no	—
306	C₂²×C₄	—	356	1	no	—
307	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
309	C₂²×C₄	#13083575	1	1	yes	characteristic
310	SD₁₆	—	712	4	no	—
311	M₁₆	—	356	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	178	2	no	—
317	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
318	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
319	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
320	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
321	C₄⋊C₄	—	89	2	no	—
322	C₄⋊C₄	—	356	1	no	—
323	C₄⋊C₄	—	356	1	no	—
324	C₄⋊C₄	—	178	2	no	—
325	C₄⋊C₄	—	178	2	no	—
326	C₄⋊C₄	—	178	1	no	—
327	C₄⋊C₄	—	178	1	no	—
328	C₄⋊C₄	—	178	2	no	—
329	C₄⋊C₄	—	178	1	no	—
330	C₄⋊C₄	—	178	2	no	—
331	C₂²⋊C₄	—	178	2	no	—
332	C₄²	—	178	2	no	—
333	C₄²	D₄×D₈₉	1	1	yes	characteristic
334	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
335	C₄²	#13083575	1	1	yes	characteristic
336	C₄²	C₂³×D₈₉	1	1	yes	characteristic
337	C₂³	—	356	2	no	—
338	C₂³	—	712	2	no	—
339	C₂³	—	712	2	no	—
340	C₂³	—	712	2	no	—
341	C₂³	#15667987	1	1	yes	characteristic
342	Q₈	—	356	2	no	—
343	Q₈	—	178	4	no	—
344	Q₈	—	712	2	no	—
345	Q₈	—	712	2	no	—
346	D₄	—	356	2	no	—
347	D₄	—	356	4	no	—
348	D₄	—	356	2	no	—
349	D₄	—	712	2	no	—
350	D₄	—	712	2	no	—
351	D₄	—	712	2	no	—
352	D₄	—	712	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	356	2	no	—
356	C₂×C₄	—	178	2	no	—
357	C₂×C₄	—	178	2	no	—
358	C₂×C₄	—	712	2	no	—
359	C₂×C₄	#15667987	1	1	yes	characteristic
360	C₂×C₄	#15667987	1	1	yes	characteristic
361	C₂×C₄	#15667987	1	1	yes	characteristic
362	C₂×C₄	#15668034	1	2	yes	—
363	C₂×C₄	#15668045	1	2	yes	—
364	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
365	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
366	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₈	—	356	2	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	1424	4	no	—
377	C₂²	—	712	2	no	—
378	C₂²	—	712	2	no	—
379	C₂²	—	712	2	no	—
380	C₂²	#19151821	1	2	yes	—
381	C₂²	C₂×#15667987	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	356	2	no	—
385	C₄	—	356	2	no	—
386	C₄	—	712	2	no	—
387	C₄	#19151821	1	2	yes	—
388	C₄	#19151869	1	2	yes	—
389	C₄	C₂×#15668034	1	1	yes	characteristic
390	C₄	C₂×#15668045	1	1	yes	characteristic
391	C₄	#19152405	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	1424	2	no	—
394	C₂	C₂×#19151821	1	1	yes	characteristic
395	C₂	#26029892	1	1	yes	characteristic
396	C₂	C₂×#19151869	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂×#15668009	C₂²
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#19212
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #26031405	C₂	semidirect
4. #26029186	C₂	semidirect
5. #26019793	C₂	semidirect
6. #26019669	C₂	semidirect
7. #26018932	C₂	semidirect
9. C₈₉×#1564	C₂	semidirect
22. #19151894	C₂²	semidirect
26. #19151728	C₂²	semidirect
28. C₈×Dic₁₇₈	C₂²	semidirect
34. #19151346	C₂²	semidirect
39. #335×C₈₉	C₂²	semidirect
40. #334×C₈₉	C₂²	semidirect
205. #1564	D₈₉	semidirect
208. C₈₉	#19212	semidirect
236. #335	C₂×D₈₉	semidirect
237. #334	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26031405	2
#26031195	2
C₂×#19151945	2
#26029186	2
#26019793	2
#26019669	2
#26018932	2
#26018919	2
C₈₉×#1564	2
#19212	89