Subgroups — #36581012

Order 22784.

706 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020748	C₂	1	2	yes	maximal, complemented
2	#26020741	C₂	1	2	yes	maximal, complemented
3	#26020734	C₂	1	2	yes	maximal, complemented
4	#26020727	C₂	1	2	yes	maximal, complemented
5	#26020656	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020641	C₂	1	1	yes	maximal, characteristic, complemented
7	#26020636	C₂	1	2	yes	maximal, complemented
8	#26020401	C₂	1	2	yes	maximal, complemented
9	C₈₉×#3153	C₂	1	1	yes	maximal, characteristic
10	#19152365	C₂²	1	2	yes	—
11	#19152076	C₂²	1	1	yes	characteristic
12	#19152073	C₂²	1	1	yes	characteristic
13	#19151615	C₂²	1	2	yes	—
14	#19151615	C₂²	1	2	yes	—
15	#19151607	C₂²	1	2	yes	—
16	#19151603	C₂²	1	2	yes	—
17	C₂×#15667931	C₂²	1	1	yes	characteristic
18	C₂×#15667929	C₂²	1	1	yes	characteristic
19	#19151572	C₂²	1	2	yes	—
20	#19151369	C₂²	1	2	yes	—
21	#19151369	C₂²	1	2	yes	—
22	#19151358	C₂²	1	2	yes	—
23	#19151358	C₂²	1	2	yes	—
24	#19151356	C₂²	1	2	yes	—
25	#19151355	C₂²	1	2	yes	—
26	C₂×#137×C₈₉	C₂²	1	2	yes	—
27	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	—
28	#532×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
30	#497×C₈₉	C₄	1	2	yes	complemented
31	#496×C₈₉	C₄	1	2	yes	complemented
32	#495×C₈₉	C₄	1	4	yes	complemented
33	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
34	#15668019	—	4	2	no	—
35	#15668019	—	4	2	no	—
36	C₂×#13083558	—	4	2	no	—
37	C₂×#13083558	C₂³	1	2	yes	—
38	C₂×#13083558	C₂³	1	2	yes	—
39	Q₈×Dic₈₉	—	4	2	no	—
40	D₄×Dic₈₉	—	4	2	no	—
41	#15667971	—	4	1	no	—
42	#15667971	C₂³	1	2	yes	—
43	#15667971	C₂³	1	2	yes	—
44	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
45	#15667939	—	4	2	no	—
46	#15667939	—	4	2	no	—
47	#15667936	—	4	2	no	—
48	#15667933	—	4	2	no	—
49	#15667931	—	4	1	no	—
50	#15667929	—	4	1	no	—
51	#15667927	—	4	2	no	—
52	C₂×#41×C₈₉	C₂³	1	2	yes	—
53	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
55	#137×C₈₉	—	4	2	no	—
56	#137×C₈₉	—	4	2	no	—
57	AGL(1,ℤ/8)×C₈₉	—	4	2	no	—
58	AGL(1,ℤ/8)×C₈₉	—	4	2	no	—
59	C₂×Q₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
60	C₂×SD₁₆×C₈₉	C₂×C₄	1	4	yes	—
61	C₂×SD₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
62	C₂×D₈×C₈₉	C₂×C₄	1	4	yes	—
63	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic, complemented
65	#127×C₈₉	C₂×C₄	1	1	yes	characteristic, complemented
66	#124×C₈₉	—	4	1	no	—
67	#124×C₈₉	C₂×C₄	1	2	yes	—
68	#124×C₈₉	C₂×C₄	1	2	yes	—
69	#124×C₈₉	C₂×C₄	1	2	yes	complemented
70	#121×C₈₉	—	4	1	no	—
71	C₄×D₄×C₈₉	—	4	1	no	—
72	C₂×C₂²⋊C₄×C₈₉	—	4	1	no	—
73	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
74	#97×C₈₉	C₂×C₄	1	2	yes	complemented
75	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	complemented
76	C₂²×Dic₈₉	—	4	2	no	—
77	#13083558	—	2	2	no	—
78	#13083558	—	2	2	no	—
79	#13083558	—	2	2	no	—
80	#13083558	—	2	2	no	—
81	#13083558	—	4	4	no	—
82	C₂×#10623494	—	2	2	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₂×#10623494	—	4	4	no	—
85	C₄⋊Dic₈₉	—	2	2	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	4	2	no	—
88	C₂²⋊Dic₈₉	—	2	2	no	—
89	C₂²⋊Dic₈₉	—	2	2	no	—
90	C₂²⋊Dic₈₉	—	4	2	no	—
91	C₄×Dic₈₉	—	2	2	no	—
92	C₄×Dic₈₉	—	2	2	no	—
93	C₄×Dic₈₉	—	4	2	no	—
94	C₂⁴×C₈₉	—	2	1	no	—
95	#41×C₈₉	—	2	2	no	—
96	#41×C₈₉	—	2	2	no	—
97	#41×C₈₉	—	2	2	no	—
98	#41×C₈₉	—	2	2	no	—
99	C₂×Q₈×C₈₉	—	2	1	no	—
100	C₂×Q₈×C₈₉	—	2	2	no	—
101	C₂×Q₈×C₈₉	—	2	2	no	—
102	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
104	C₂×D₄×C₈₉	—	2	2	no	—
105	C₂×D₄×C₈₉	—	2	2	no	—
106	C₂×D₄×C₈₉	—	2	2	no	—
107	C₂×D₄×C₈₉	—	2	1	no	—
108	C₂×D₄×C₈₉	—	4	1	no	—
109	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
110	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₈₉	—	2	1	no	—
112	C₂²×C₄×C₈₉	—	2	1	no	—
113	C₂²×C₄×C₈₉	—	2	2	no	—
114	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
115	Q₁₆×C₈₉	—	4	4	no	—
116	SD₁₆×C₈₉	—	4	4	no	—
117	SD₁₆×C₈₉	—	4	4	no	—
118	D₈×C₈₉	—	4	4	no	—
119	M₁₆×C₈₉	—	2	2	no	—
120	M₁₆×C₈₉	—	2	2	no	—
121	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₂²⋊C₄×C₈₉	—	4	2	no	—
124	C₂²⋊C₄×C₈₉	—	4	2	no	—
125	C₂²⋊C₄×C₈₉	—	4	2	no	—
126	C₄²×C₈₉	—	2	1	no	—
127	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₂³×C₈₉	—	2	2	no	—
130	C₂³×C₈₉	—	2	2	no	—
131	C₂³×C₈₉	—	4	1	no	—
132	C₂³×C₈₉	—	4	1	no	—
133	C₂³×C₈₉	—	8	2	no	—
134	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
135	Q₈×C₈₉	—	2	2	no	—
136	Q₈×C₈₉	—	2	4	no	—
137	Q₈×C₈₉	—	2	2	no	—
138	Q₈×C₈₉	—	4	2	no	—
139	D₄×C₈₉	—	2	2	no	—
140	D₄×C₈₉	—	2	4	no	—
141	D₄×C₈₉	—	2	2	no	—
142	D₄×C₈₉	—	4	2	no	—
143	D₄×C₈₉	—	4	2	no	—
144	D₄×C₈₉	—	4	2	no	—
145	D₄×C₈₉	—	8	2	no	—
146	C₂×C₄×C₈₉	—	2	1	no	—
147	C₂×C₄×C₈₉	—	2	1	no	—
148	C₂×C₄×C₈₉	—	2	2	no	—
149	C₂×C₄×C₈₉	—	2	2	no	—
150	C₂×C₄×C₈₉	—	4	1	no	—
151	C₂×C₄×C₈₉	—	4	1	no	—
152	C₂×C₄×C₈₉	—	4	2	no	—
153	C₂×C₄×C₈₉	—	4	2	no	—
154	C₂×C₄×C₈₉	—	8	1	no	—
155	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
157	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
158	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×Dic₈₉	—	4	2	no	—
160	C₂×Dic₈₉	—	4	2	no	—
161	C₂×Dic₈₉	—	16	2	no	—
162	C₈×C₈₉	—	2	4	no	—
163	#10623494	—	4	2	no	—
164	#10623494	—	4	2	no	—
165	C₂²×C₈₉	—	4	1	no	—
166	C₂²×C₈₉	—	4	2	no	—
167	C₂²×C₈₉	—	4	2	no	—
168	C₂²×C₈₉	—	8	2	no	—
169	C₂²×C₈₉	—	8	2	no	—
170	C₂²×C₈₉	—	8	2	no	—
171	C₂²×C₈₉	—	16	1	no	—
172	C₂²×C₈₉	#518	1	1	yes	characteristic
173	C₂²×C₈₉	#560	1	1	yes	characteristic
174	C₂²×C₈₉	#560	1	1	yes	characteristic
175	C₄×C₈₉	—	4	1	no	—
176	C₄×C₈₉	—	4	1	no	—
177	C₄×C₈₉	—	4	2	no	—
178	C₄×C₈₉	—	4	2	no	—
179	C₄×C₈₉	#518	1	2	yes	—
180	C₄×C₈₉	#560	1	2	yes	—
181	Dic₈₉	—	8	2	no	—
182	Dic₈₉	—	8	2	no	—
183	#23362	—	89	1	no	maximal, Sylow
184	C₂×C₈₉	—	2	1	no	—
185	C₂×C₈₉	—	2	1	no	—
186	C₂×C₈₉	—	2	1	no	—
187	C₂×C₈₉	—	8	2	no	—
188	C₂×C₈₉	—	8	2	no	—
189	C₂×C₈₉	#2380	1	1	yes	characteristic
190	#3153	D₈₉	1	1	yes	characteristic
191	#1905	—	89	2	no	—
192	#1904	—	89	2	no	—
193	#1903	—	89	2	no	—
194	#1902	—	89	2	no	—
195	#1888	—	89	1	no	—
196	#1885	—	89	1	no	—
197	#1884	—	89	2	no	—
198	#1841	—	89	2	no	—
199	C₈₉	#23362	1	1	yes	characteristic, complemented, Sylow
200	C₂×#137	C₂×D₈₉	1	2	yes	—
201	C₂×AGL(1,ℤ/8)	C₂×D₈₉	1	2	yes	—
202	#568	—	89	2	no	—
203	#532	C₂×D₈₉	1	1	yes	characteristic
204	C₂×#124	C₂×D₈₉	1	1	yes	characteristic
205	#519	—	89	1	no	—
206	#518	—	89	1	no	—
207	#497	Dic₈₉	1	2	yes	complemented
208	#496	Dic₈₉	1	2	yes	complemented
209	#495	Dic₈₉	1	4	yes	complemented
210	#421	—	89	2	no	—
211	#421	—	89	2	no	—
212	#419	—	89	2	no	—
213	#418	—	89	2	no	—
214	C₂×#101	—	89	1	no	—
215	C₂×#100	—	89	1	no	—
216	#410	—	89	2	no	—
217	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
218	#344	—	89	2	no	—
219	#344	—	89	2	no	—
220	#339	—	89	2	no	—
221	#339	—	89	2	no	—
222	#338	—	89	2	no	—
223	H(3,ℤ/4)	—	89	2	no	—
224	C₂×#41	C₂²×D₈₉	1	2	yes	—
225	C₂²×Q₈	C₂²×D₈₉	1	1	yes	characteristic
226	C₂²×D₄	C₂²×D₈₉	1	1	yes	characteristic
227	#137	—	4	2	no	—
228	#137	—	4	2	no	—
229	AGL(1,ℤ/8)	—	4	2	no	—
230	AGL(1,ℤ/8)	—	4	2	no	—
231	C₂×Q₁₆	C₂×Dic₈₉	1	4	yes	complemented
232	C₂×SD₁₆	C₂×Dic₈₉	1	4	yes	—
233	C₂×SD₁₆	C₂×Dic₈₉	1	4	yes	complemented
234	C₂×D₈	C₂×Dic₈₉	1	4	yes	—
235	#131	—	356	2	no	—
236	#131	—	356	2	no	—
237	C₂×M₁₆	—	89	2	no	—
238	C₂×M₁₆	—	89	2	no	—
239	C₂×M₁₆	—	356	2	no	—
240	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
241	C₄⋊Q₈	C₂×Dic₈₉	1	1	yes	characteristic, complemented
242	#127	C₂×Dic₈₉	1	1	yes	characteristic, complemented
243	#124	—	4	1	no	—
244	#124	C₂×Dic₈₉	1	2	yes	—
245	#124	C₂×Dic₈₉	1	2	yes	—
246	#124	C₂×Dic₈₉	1	2	yes	complemented
247	#121	—	4	1	no	—
248	C₄×Q₈	—	356	2	no	—
249	C₄×D₄	—	4	1	no	—
250	C₄×D₄	—	356	2	no	—
251	#117	—	89	2	no	—
252	#117	—	89	2	no	—
253	#117	—	356	1	no	—
254	C₂×C₂²⋊C₄	—	4	1	no	—
255	C₂×C₂²⋊C₄	—	356	1	no	—
256	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
257	#104	—	356	2	no	—
258	#104	—	356	2	no	—
259	#103	—	356	2	no	—
260	#102	—	356	2	no	—
261	#101	—	356	1	no	—
262	#100	—	356	1	no	—
263	#99	—	356	2	no	—
264	#97	C₂×Dic₈₉	1	2	yes	complemented
265	C₄×C₈	C₂×Dic₈₉	1	2	yes	complemented
266	C₂⁴	—	2	1	no	—
267	#41	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	—	2	2	no	—
270	#41	—	2	2	no	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
275	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
282	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	356	2	no	—
287	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
288	Q₁₆	—	4	4	no	—
289	SD₁₆	—	4	4	no	—
290	SD₁₆	—	4	4	no	—
291	D₈	—	4	4	no	—
292	M₁₆	—	2	2	no	—
293	M₁₆	—	2	2	no	—
294	M₁₆	—	356	4	no	—
295	M₁₆	—	178	2	no	—
296	M₁₆	—	178	2	no	—
297	M₁₆	—	178	2	no	—
298	M₁₆	—	178	2	no	—
299	C₂×C₈	—	356	4	no	—
300	C₂×C₈	—	178	2	no	—
301	C₂×C₈	—	178	2	no	—
302	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	356	2	no	—
305	C₄⋊C₄	—	178	2	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₂²⋊C₄	—	4	2	no	—
309	C₂²⋊C₄	—	4	2	no	—
310	C₂²⋊C₄	—	356	2	no	—
311	C₂²⋊C₄	—	178	2	no	—
312	C₂²⋊C₄	—	178	2	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	356	2	no	—
315	C₄²	—	178	2	no	—
316	C₄²	—	178	2	no	—
317	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
318	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
319	C₂³	—	2	2	no	—
320	C₂³	—	2	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	4	1	no	—
323	C₂³	—	8	2	no	—
324	C₂³	#15668057	1	1	yes	characteristic
325	Q₈	—	2	2	no	—
326	Q₈	—	2	4	no	—
327	Q₈	—	2	2	no	—
328	Q₈	—	4	2	no	—
329	D₄	—	2	2	no	—
330	D₄	—	2	4	no	—
331	D₄	—	2	2	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	2	no	—
335	D₄	—	8	2	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	8	1	no	—
345	C₂×C₄	—	356	2	no	—
346	C₂×C₄	—	356	2	no	—
347	C₂×C₄	—	1424	2	no	—
348	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
349	C₂×C₄	#15668057	1	2	yes	—
350	C₂×C₄	#15668057	1	2	yes	—
351	C₂×C₄	#15668057	1	1	yes	characteristic
352	C₈	—	2	4	no	—
353	C₈	—	356	2	no	—
354	C₈	—	356	2	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	8	2	no	—
361	C₂²	—	16	1	no	—
362	C₂²	#19152073	1	1	yes	characteristic
363	C₂²	#19152335	1	1	yes	characteristic
364	C₂²	#19152335	1	1	yes	characteristic
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	2	no	—
369	C₄	—	712	2	no	—
370	C₄	—	712	2	no	—
371	C₄	#19152073	1	2	yes	—
372	C₄	#19152335	1	2	yes	—
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	8	2	no	—
377	C₂	—	8	2	no	—
378	C₂	#26023429	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26023429
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3153	C₂
socle	C₂×C₈₉	#2380
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#3153	D₈₉
hypocenter	C₈₉	#23362
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3153

Complements

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

N	H ≅ G/N	type
1. #26020748	C₂	semidirect
2. #26020741	C₂	semidirect
3. #26020734	C₂	semidirect
4. #26020727	C₂	semidirect
5. #26020656	C₂	semidirect
6. #26020641	C₂	semidirect
7. #26020636	C₂	semidirect
8. #26020401	C₂	semidirect
30. #497×C₈₉	C₄	semidirect
31. #496×C₈₉	C₄	semidirect
32. #495×C₈₉	C₄	semidirect
59. C₂×Q₁₆×C₈₉	C₂×C₄	semidirect
61. C₂×SD₁₆×C₈₉	C₂×C₄	semidirect
64. C₄⋊Q₈×C₈₉	C₂×C₄	semidirect
65. #127×C₈₉	C₂×C₄	semidirect
69. #124×C₈₉	C₂×C₄	semidirect
74. #97×C₈₉	C₂×C₄	semidirect
75. C₄×C₈×C₈₉	C₂×C₄	semidirect
199. C₈₉	#23362	semidirect
207. #497	Dic₈₉	semidirect
208. #496	Dic₈₉	semidirect
209. #495	Dic₈₉	semidirect
231. C₂×Q₁₆	C₂×Dic₈₉	semidirect
233. C₂×SD₁₆	C₂×Dic₈₉	semidirect
241. C₄⋊Q₈	C₂×Dic₈₉	semidirect
242. #127	C₂×Dic₈₉	semidirect
246. #124	C₂×Dic₈₉	semidirect
264. #97	C₂×Dic₈₉	semidirect
265. C₄×C₈	C₂×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020748	2
#26020741	2
#26020734	2
#26020727	2
#26020656	2
#26020641	2
#26020636	2
#26020401	2
C₈₉×#3153	2
#23362	89