Subgroups — #36524980

Order 22784.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030811	C₂	1	1	yes	maximal, characteristic
2	#26029186	C₂	1	1	yes	maximal, characteristic, complemented
3	#26028965	C₂	1	1	yes	maximal, characteristic
4	#26019156	C₂	1	4	yes	maximal, complemented
5	#26019066	C₂	1	4	yes	maximal, complemented
6	#26019011	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018708	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018704	C₂	1	1	yes	maximal, characteristic
9	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668018	—	2	1	no	—
11	#19152155	C₂²	1	1	yes	characteristic
12	#19152148	C₂²	1	1	yes	characteristic
13	#19151996	C₄	1	1	yes	characteristic
14	#19151971	C₄	1	1	yes	characteristic
15	#19151949	C₄	1	1	yes	characteristic
16	#19151945	C₄	1	1	yes	characteristic
17	#19151827	C₄	1	4	yes	—
18	#19151728	—	2	1	no	—
19	#19151728	C₂²	1	2	yes	—
20	C₈×Dic₁₇₈	—	2	1	no	—
21	C₈×Dic₁₇₈	C₂²	1	2	yes	—
22	C₄×#13083563	C₂²	1	4	yes	—
23	#19151677	—	2	1	no	—
24	#19151677	C₂²	1	2	yes	—
25	C₈×D₃₅₆	—	2	1	no	—
26	C₈×D₃₅₆	C₂²	1	2	yes	—
27	#19151624	—	2	1	no	—
28	#19151622	C₂²	1	1	yes	characteristic
29	#19151622	C₂²	1	1	yes	characteristic
30	#19151564	—	2	1	no	—
31	#19151553	—	2	1	no	—
32	#19151351	C₂²	1	4	yes	—
33	#19151330	C₂²	1	4	yes	—
34	#19151327	C₂²	1	4	yes	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	#322×C₈₉	C₂²	1	2	yes	—
39	#322×C₈₉	C₂²	1	2	yes	—
40	C₂×#13083574	—	2	1	no	—
41	C₂×#13083563	C₂×C₄	1	4	yes	—
42	#15668018	—	4	4	no	—
43	C₂×#13083558	—	2	1	no	—
44	C₂×#13083558	—	2	1	no	—
45	C₂×#13083557	—	2	2	no	—
46	C₂×C₈×D₈₉	—	2	2	no	—
47	#15668012	C₂×C₄	1	1	yes	characteristic
48	#15668012	C₂×C₄	1	1	yes	characteristic
49	#15668012	C₂×C₄	1	1	yes	characteristic
50	#15668009	C₂×C₄	1	1	yes	characteristic
51	#15668004	—	2	1	no	—
52	#15668000	—	2	1	no	—
53	#15667991	—	2	1	no	—
54	#15667991	C₂×C₄	1	2	yes	—
55	#15667986	—	2	1	no	—
56	#15667986	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₇₈	—	2	1	no	—
58	C₄×Dic₁₇₈	C₂³	1	2	yes	—
59	C₄×D₃₅₆	—	2	1	no	—
60	C₄×D₃₅₆	C₂³	1	2	yes	—
61	#15667971	—	2	1	no	—
62	#15667944	C₂×C₄	1	4	yes	—
63	C₄⋊#10623494	—	2	1	no	—
64	C₄⋊#10623494	—	2	1	no	—
65	C₄⋊#10623494	C₂³	1	2	yes	—
66	C₄⋊#10623494	C₂³	1	2	yes	—
67	#15667941	—	2	2	no	—
68	#15667941	—	2	2	no	—
69	#15667937	C₂×C₄	1	4	yes	—
70	#15667934	C₂×C₄	1	4	yes	—
71	#15667924	—	2	2	no	—
72	#15667924	—	2	2	no	—
73	#15667922	—	2	2	no	—
74	C₈×Dic₈₉	—	2	2	no	—
75	C₂²×C₈×C₈₉	—	2	1	no	—
76	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
77	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₈₉	—	2	1	no	—
81	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
84	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
85	C₄×C₈×C₈₉	C₂³	1	2	yes	—
86	C₄×C₈×C₈₉	C₂³	1	2	yes	—
87	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	#13083574	—	4	2	no	—
90	#13083574	—	4	2	no	—
91	C₂×Dic₁₇₈	—	2	1	no	—
92	C₂×Dic₁₇₈	—	2	1	no	—
93	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
94	C₂×D₃₅₆	—	2	1	no	—
95	C₂×D₃₅₆	—	2	1	no	—
96	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
97	C₂×C₂²⋊D₈₉	—	2	2	no	—
98	C₂×C₄×D₈₉	—	2	2	no	—
99	#13083563	—	2	8	no	—
100	#13083558	—	4	2	no	—
101	#13083558	—	4	2	no	—
102	#13083557	—	4	4	no	—
103	C₂×#10623494	—	2	2	no	—
104	C₂×#10623494	—	2	2	no	—
105	C₈×D₈₉	—	4	4	no	—
106	C₄⋊Dic₈₉	—	2	1	no	—
107	C₄⋊Dic₈₉	—	2	1	no	—
108	C₄⋊Dic₈₉	—	2	1	no	—
109	C₄⋊Dic₈₉	—	2	1	no	—
110	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
111	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
112	#13083553	—	2	2	no	—
113	#13083553	—	2	2	no	—
114	C₂²⋊Dic₈₉	—	2	2	no	—
115	#13083551	—	2	2	no	—
116	#13083551	—	2	2	no	—
117	C₄×Dic₈₉	—	2	2	no	—
118	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
121	C₂×C₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	2	2	no	—
123	C₂×C₈×C₈₉	—	2	4	no	—
124	C₂×C₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	—	4	2	no	—
126	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₈₉	#41	1	1	yes	characteristic
135	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄×C₈₉	#137	1	2	yes	—
145	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₈₉	—	4	4	no	—
149	C₂×Dic₈₉	—	2	2	no	—
150	C₂×Dic₈₉	—	2	2	no	—
151	C₂×Dic₈₉	—	2	2	no	—
152	D₃₅₆	—	2	4	no	—
153	D₃₅₆	—	4	2	no	—
154	D₃₅₆	—	4	2	no	—
155	C₄×D₈₉	—	4	2	no	—
156	C₄×D₈₉	—	4	2	no	—
157	Dic₁₇₈	—	2	4	no	—
158	Dic₁₇₈	—	4	2	no	—
159	Dic₁₇₈	—	4	2	no	—
160	C₈×C₈₉	—	2	2	no	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	—	2	1	no	—
163	C₈×C₈₉	—	2	1	no	—
164	C₈×C₈₉	—	2	1	no	—
165	C₈×C₈₉	—	4	2	no	—
166	C₈×C₈₉	#131	1	4	yes	complemented
167	#10623494	—	4	2	no	—
168	#10623494	—	4	2	no	—
169	C₂²×C₈₉	—	2	1	no	—
170	C₂²×C₈₉	—	2	1	no	—
171	C₂²×C₈₉	#440	1	2	yes	—
172	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₈₉	—	4	2	no	—
174	C₂×D₈₉	—	4	2	no	—
175	C₂×D₈₉	—	4	2	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	#440	1	2	yes	—
179	C₄×C₈₉	#444	1	2	yes	—
180	C₄×C₈₉	C₂×#131	1	2	yes	—
181	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
183	Dic₈₉	—	4	2	no	—
184	Dic₈₉	—	4	2	no	—
185	Dic₈₉	—	4	2	no	—
186	#17960	—	89	1	no	maximal, Sylow
187	C₂×C₈₉	—	2	2	no	—
188	C₂×C₈₉	#2931	1	1	yes	characteristic
189	C₂×C₈₉	C₂×#440	1	1	yes	characteristic
190	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
191	D₈₉	—	8	2	no	—
192	#3167	—	89	1	no	—
193	#2960	—	89	1	no	—
194	#2931	—	89	1	no	—
195	#1592	—	89	4	no	—
196	SD₁₆⋊C₈	—	89	4	no	—
197	#1567	—	89	1	no	—
198	#1493	—	89	1	no	—
199	#1492	—	89	1	no	—
200	C₂×#322	D₈₉	1	1	yes	characteristic, complemented
201	C₈₉	#17960	1	1	yes	characteristic, complemented, Sylow
202	C₂×#131	—	178	1	no	—
203	#533	—	89	1	no	—
204	#532	—	89	1	no	—
205	#498	—	89	1	no	—
206	#492	—	89	1	no	—
207	#487	—	89	1	no	—
208	#486	—	89	1	no	—
209	#465	—	89	4	no	—
210	#446	—	89	2	no	—
211	#446	—	178	1	no	—
212	C₈×Q₈	—	89	2	no	—
213	C₈×Q₈	—	178	1	no	—
214	C₄×SD₁₆	—	89	4	no	—
215	#436	—	89	2	no	—
216	#436	—	178	1	no	—
217	C₈×D₄	—	89	2	no	—
218	C₈×D₄	—	178	1	no	—
219	#424	—	178	1	no	—
220	#423	—	89	1	no	—
221	#423	—	89	1	no	—
222	#408	—	178	1	no	—
223	#405	—	178	1	no	—
224	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
226	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
227	#335	—	89	4	no	—
228	Q₈⋊C₈	—	89	4	no	—
229	D₄⋊C₈	—	89	4	no	—
230	#322	C₂×D₈₉	1	2	yes	—
231	#322	C₂×D₈₉	1	2	yes	—
232	C₂×#41	—	178	1	no	—
233	C₂×SD₁₆	—	89	4	no	—
234	#131	—	356	4	no	—
235	C₂×M₁₆	—	178	1	no	—
236	C₂×M₁₆	—	178	1	no	—
237	C₂×M₁₆	—	178	2	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	178	2	no	—
240	C₂²×C₈	C₄×D₈₉	1	2	yes	—
241	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
242	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
243	C₄⋊Q₈	—	89	1	no	—
244	C₄⋊Q₈	—	89	1	no	—
245	C₄⋊Q₈	—	89	1	no	—
246	#127	—	89	1	no	—
247	#125	—	178	1	no	—
248	#124	—	178	1	no	—
249	C₂²⋊Q₈	—	89	2	no	—
250	C₂²⋊Q₈	—	178	1	no	—
251	#121	—	89	2	no	—
252	#121	—	178	1	no	—
253	C₄×Q₈	—	89	2	no	—
254	C₄×Q₈	—	178	1	no	—
255	C₄×D₄	—	89	2	no	—
256	C₄×D₄	—	178	1	no	—
257	#117	—	178	1	no	—
258	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
259	#106	—	89	4	no	—
260	C₄⋊C₈	—	89	2	no	—
261	C₄⋊C₈	—	89	2	no	—
262	C₄⋊C₈	—	178	2	no	—
263	C₄⋊C₈	—	178	2	no	—
264	C₄⋊C₈	—	178	1	no	—
265	C₄⋊C₈	—	178	1	no	—
266	#103	—	89	4	no	—
267	#102	—	89	4	no	—
268	C₂²⋊C₈	—	178	2	no	—
269	C₂²⋊C₈	—	178	2	no	—
270	#97	—	178	2	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	178	2	no	—
273	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
274	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
275	C₄×C₈	D₃₅₆	1	1	yes	characteristic
276	C₄×C₈	D₃₅₆	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₈₉	1	2	yes	—
278	C₄×C₈	C₂²×D₈₉	1	2	yes	—
279	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
281	#41	—	356	2	no	—
282	#41	—	356	2	no	—
283	C₂×Q₈	—	89	2	no	—
284	C₂×Q₈	—	178	1	no	—
285	C₂×Q₈	—	178	1	no	—
286	C₂×D₄	—	89	2	no	—
287	C₂×D₄	—	178	2	no	—
288	C₂×D₄	—	178	1	no	—
289	C₂×D₄	—	178	1	no	—
290	C₂²×C₄	—	178	2	no	—
291	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
293	C₂²×C₄	#13083574	1	1	yes	characteristic
294	SD₁₆	—	178	8	no	—
295	M₁₆	—	356	4	no	—
296	M₁₆	—	356	2	no	—
297	M₁₆	—	356	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	4	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	2	no	—
303	C₂×C₈	—	356	4	no	—
304	C₂×C₈	—	178	2	no	—
305	C₂×C₈	—	178	2	no	—
306	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
307	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
308	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
309	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
310	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
311	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
312	C₄⋊C₄	—	89	2	no	—
313	C₄⋊C₄	—	89	2	no	—
314	C₄⋊C₄	—	178	1	no	—
315	C₄⋊C₄	—	178	1	no	—
316	C₄⋊C₄	—	178	1	no	—
317	C₄⋊C₄	—	178	2	no	—
318	C₄⋊C₄	—	178	1	no	—
319	C₄⋊C₄	—	178	2	no	—
320	C₂²⋊C₄	—	178	2	no	—
321	C₂²⋊C₄	—	178	2	no	—
322	C₂²⋊C₄	—	178	2	no	—
323	C₄²	—	178	2	no	—
324	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
325	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
326	C₄²	#13083574	1	1	yes	characteristic
327	C₄²	C₂³×D₈₉	1	1	yes	characteristic
328	C₂³	—	178	2	no	—
329	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
330	Q₈	—	356	2	no	—
331	Q₈	—	356	2	no	—
332	Q₈	—	178	4	no	—
333	D₄	—	356	4	no	—
334	D₄	—	356	2	no	—
335	D₄	—	356	2	no	—
336	D₄	—	178	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	356	2	no	—
340	C₂×C₄	—	356	2	no	—
341	C₂×C₄	—	178	2	no	—
342	C₂×C₄	—	178	2	no	—
343	C₂×C₄	—	178	2	no	—
344	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
345	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
346	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
347	C₂×C₄	#15668040	1	2	yes	—
348	C₂×C₄	#15668045	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
350	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
351	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	2	no	—
357	C₈	—	4	2	no	—
358	C₈	—	356	2	no	—
359	C₈	—	356	2	no	—
360	C₈	#15668018	1	4	yes	complemented
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	356	2	no	—
364	C₂²	—	356	2	no	—
365	C₂²	—	356	2	no	—
366	C₂²	#19151700	1	2	yes	—
367	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	356	2	no	—
371	C₄	—	356	2	no	—
372	C₄	—	356	2	no	—
373	C₄	#19151700	1	2	yes	—
374	C₄	#19151721	1	2	yes	—
375	C₄	C₂×#15668018	1	2	yes	—
376	C₄	C₂×#15668040	1	1	yes	characteristic
377	C₄	C₂×#15668045	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	712	2	no	—
380	C₂	#26028965	1	1	yes	characteristic
381	C₂	C₂×#19151700	1	1	yes	characteristic
382	C₂	C₂×#19151721	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#19152148	C₂²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#17960
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #26029186	C₂	semidirect
4. #26019156	C₂	semidirect
5. #26019066	C₂	semidirect
6. #26019011	C₂	semidirect
7. #26018708	C₂	semidirect
9. C₂×#322×C₈₉	C₂	semidirect
166. C₈×C₈₉	#131	semidirect
200. C₂×#322	D₈₉	semidirect
201. C₈₉	#17960	semidirect
360. C₈	#15668018	semidirect

Maximal subgroups

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

subgroup	index
#26030811	2
#26029186	2
#26028965	2
#26019156	2
#26019066	2
#26019011	2
#26018708	2
#26018704	2
C₂×#322×C₈₉	2
#17960	89