Subgroups — #36537499

Order 22784.

842 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031167	C₂	1	2	yes	maximal, complemented
2	C₂×#19151975	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019731	C₂	1	4	yes	maximal, complemented
4	#26019607	C₂	1	4	yes	maximal, complemented
5	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018921	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2931	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₁₂	—	2	1	no	—
9	C₂²×D₇₁₂	—	2	1	no	—
10	C₂×#15668009	C₂²	1	2	yes	—
11	#19151975	—	2	1	no	—
12	#19151975	—	2	1	no	—
13	#19151975	C₂²	1	2	yes	—
14	#19151975	C₂²	1	2	yes	—
15	#19151974	—	2	2	no	—
16	#19151946	—	2	2	no	—
17	#19151793	C₂²	1	8	yes	complemented
18	#19151731	—	2	4	no	—
19	#19151651	C₂²	1	2	yes	—
20	#19151351	C₂²	1	2	yes	complemented
21	#19151349	C₂²	1	2	yes	complemented
22	#19151341	C₂²	1	8	yes	complemented
23	C₂×#131×C₈₉	—	2	1	no	—
24	#532×C₈₉	C₂²	1	1	yes	characteristic
25	#436×C₈₉	—	2	1	no	—
26	#436×C₈₉	C₂²	1	2	yes	complemented
27	C₈×D₄×C₈₉	—	2	1	no	—
28	C₈×D₄×C₈₉	C₂²	1	2	yes	complemented
29	#423×C₈₉	C₂²	1	1	yes	characteristic
30	#408×C₈₉	—	2	1	no	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²×D₃₅₆	—	2	2	no	—
33	C₂²×D₃₅₆	—	2	2	no	—
34	C₂²×D₃₅₆	—	2	2	no	—
35	C₂×D₇₁₂	—	2	2	no	—
36	C₂×D₇₁₂	—	2	2	no	—
37	C₂×D₇₁₂	—	2	2	no	—
38	C₂×D₇₁₂	—	2	2	no	—
39	C₂×D₇₁₂	—	4	4	no	—
40	C₂×D₇₁₂	—	4	4	no	—
41	C₂×#13083559	—	2	8	no	—
42	#15668009	—	2	2	no	—
43	#15668009	—	2	2	no	—
44	#15668009	—	2	2	no	—
45	#15668009	C₂³	1	4	yes	—
46	C₄⋊#10623494	C₂³	1	4	yes	—
47	#15667932	—	2	8	no	—
48	C₂²⋊#10623494	—	2	2	no	—
49	C₄×#10623494	—	2	2	no	—
50	C₂×#41×C₈₉	—	2	1	no	—
51	#131×C₈₉	—	4	4	no	—
52	C₂×M₁₆×C₈₉	—	2	1	no	—
53	C₂×M₁₆×C₈₉	—	2	2	no	—
54	C₂²×C₈×C₈₉	—	2	2	no	—
55	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
58	#127×C₈₉	D₄	1	1	yes	characteristic
59	#124×C₈₉	—	2	1	no	—
60	#121×C₈₉	—	2	1	no	—
61	#121×C₈₉	D₄	1	2	yes	—
62	C₄×D₄×C₈₉	—	2	1	no	—
63	C₄×D₄×C₈₉	C₂³	1	2	yes	—
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₈₉	—	2	1	no	—
66	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
67	C₂²⋊C₈×C₈₉	—	2	2	no	—
68	C₂²⋊C₈×C₈₉	—	2	2	no	—
69	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
70	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂³×D₈₉	—	4	2	no	—
74	C₂×D₃₅₆	—	2	4	no	—
75	C₂×D₃₅₆	—	2	4	no	—
76	C₂×D₃₅₆	—	2	4	no	—
77	C₂×D₃₅₆	—	4	4	no	—
78	C₂×D₃₅₆	—	4	4	no	—
79	C₂×D₃₅₆	—	4	2	no	—
80	C₂×D₃₅₆	—	4	2	no	—
81	C₂×D₃₅₆	—	4	2	no	—
82	C₂×D₃₅₆	—	8	4	no	—
83	C₂×D₃₅₆	—	8	4	no	—
84	D₇₁₂	—	4	4	no	—
85	D₇₁₂	—	4	4	no	—
86	D₇₁₂	—	4	4	no	—
87	D₇₁₂	—	4	4	no	—
88	#13083559	—	8	8	no	—
89	C₂×#10623494	—	2	4	no	—
90	#41×C₈₉	—	4	2	no	—
91	#41×C₈₉	—	4	2	no	—
92	C₂×Q₈×C₈₉	—	2	1	no	—
93	C₂×D₄×C₈₉	—	2	2	no	—
94	C₂×D₄×C₈₉	—	2	1	no	—
95	C₂×D₄×C₈₉	—	2	1	no	—
96	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₈₉	—	2	2	no	—
98	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
101	M₁₆×C₈₉	—	4	2	no	—
102	M₁₆×C₈₉	—	4	4	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	—	4	4	no	—
106	C₂×C₈×C₈₉	D₈	1	4	yes	complemented
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₈₉	—	2	2	no	—
114	C₂²⋊C₄×C₈₉	—	2	2	no	—
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₈₉	#41	1	1	yes	characteristic
118	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₂³×C₈₉	—	2	2	no	—
120	C₂³×C₈₉	#121	1	1	yes	characteristic
121	C₂²×D₈₉	—	4	4	no	—
122	C₂²×D₈₉	—	8	4	no	—
123	C₂²×D₈₉	—	8	4	no	—
124	C₂²×D₈₉	—	8	4	no	—
125	Q₈×C₈₉	—	4	2	no	—
126	D₄×C₈₉	—	2	4	no	—
127	D₄×C₈₉	—	4	4	no	—
128	D₄×C₈₉	—	4	2	no	—
129	D₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
143	D₃₅₆	—	4	4	no	—
144	D₃₅₆	—	4	4	no	—
145	D₃₅₆	—	4	8	no	—
146	D₃₅₆	—	8	4	no	—
147	D₃₅₆	—	8	4	no	—
148	D₃₅₆	—	8	4	no	—
149	D₃₅₆	—	8	4	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	2	1	no	—
152	C₈×C₈₉	—	4	2	no	—
153	C₈×C₈₉	C₂×D₈	1	2	yes	—
154	C₈×C₈₉	C₂×D₈	1	2	yes	—
155	#10623494	—	4	4	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	4	2	no	—
159	C₂²×C₈₉	—	4	2	no	—
160	C₂²×C₈₉	—	4	2	no	—
161	C₂²×C₈₉	#459	1	2	yes	—
162	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
163	C₂×D₈₉	—	8	4	no	—
164	C₂×D₈₉	—	8	4	no	—
165	C₂×D₈₉	—	8	4	no	—
166	C₂×D₈₉	—	16	8	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	4	2	no	—
170	C₄×C₈₉	#459	1	2	yes	—
171	C₄×C₈₉	#472	1	2	yes	—
172	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₄×C₈₉	#576	1	2	yes	—
175	#19187	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	2	2	no	—
177	C₂×C₈₉	—	8	2	no	—
178	C₂×C₈₉	C₂×#459	1	1	yes	characteristic
179	C₂×C₈₉	#3058	1	1	yes	characteristic
180	C₂×C₈₉	C₂×#472	1	1	yes	characteristic
181	D₈₉	—	16	4	no	—
182	#3205	—	89	2	no	—
183	C₂×#493	—	89	1	no	—
184	#2931	D₈₉	1	1	yes	characteristic, complemented
185	#1687	—	89	4	no	—
186	#1667	—	89	4	no	—
187	#1564	—	89	1	no	—
188	#1545	—	89	2	no	—
189	C₈₉	#19187	1	1	yes	characteristic, complemented, Sylow
190	C₂²×D₈	—	178	1	no	—
191	C₂²×D₈	—	178	1	no	—
192	C₂×#131	—	2	1	no	—
193	#532	C₂×D₈₉	1	1	yes	characteristic
194	C₂×#127	—	89	2	no	—
195	#493	—	89	2	no	—
196	#493	—	89	2	no	—
197	#493	—	178	2	no	—
198	#493	—	178	1	no	—
199	#493	—	178	1	no	—
200	#486	—	178	2	no	—
201	#459	—	89	8	no	—
202	#447	—	178	4	no	—
203	#436	—	2	1	no	—
204	#436	C₂×D₈₉	1	2	yes	complemented
205	C₈×D₄	—	2	1	no	—
206	C₈×D₄	C₂×D₈₉	1	2	yes	complemented
207	#431	—	89	2	no	—
208	#423	C₂×D₈₉	1	1	yes	characteristic
209	#408	—	2	1	no	—
210	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
211	#335	—	89	2	no	—
212	#334	—	89	2	no	—
213	#331	—	89	8	no	—
214	C₂×#41	—	2	1	no	—
215	C₂²×D₄	—	178	2	no	—
216	C₂²×D₄	—	178	2	no	—
217	C₂²×D₄	—	178	2	no	—
218	C₂×D₈	—	356	4	no	—
219	C₂×D₈	—	356	4	no	—
220	C₂×D₈	—	178	8	no	—
221	C₂×D₈	—	178	2	no	—
222	C₂×D₈	—	178	2	no	—
223	C₂×D₈	—	178	2	no	—
224	C₂×D₈	—	178	2	no	—
225	#131	—	4	4	no	—
226	C₂×M₁₆	—	2	1	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂²×C₈	—	2	2	no	—
229	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
230	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
231	C₄⋊Q₈	D₃₅₆	1	1	yes	characteristic
232	#127	—	89	4	no	—
233	#127	—	178	2	no	—
234	#127	—	178	2	no	—
235	#127	—	178	2	no	—
236	#127	D₃₅₆	1	1	yes	characteristic
237	#124	—	2	1	no	—
238	#121	—	2	1	no	—
239	#121	D₃₅₆	1	2	yes	—
240	C₄×D₄	—	2	1	no	—
241	C₄×D₄	C₂²×D₈₉	1	2	yes	—
242	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	89	4	no	—
245	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
246	#102	—	178	8	no	—
247	C₂²⋊C₈	—	2	2	no	—
248	C₂²⋊C₈	—	2	2	no	—
249	C₂²⋊C₈	—	178	2	no	—
250	C₄×C₈	—	178	2	no	—
251	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
252	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
255	C₂⁴	—	356	2	no	—
256	#41	—	4	2	no	—
257	#41	—	4	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	356	2	no	—
263	C₂×D₄	—	356	2	no	—
264	C₂×D₄	—	356	2	no	—
265	C₂×D₄	—	356	4	no	—
266	C₂×D₄	—	356	4	no	—
267	C₂×D₄	—	178	4	no	—
268	C₂×D₄	—	178	4	no	—
269	C₂×D₄	—	178	4	no	—
270	C₂×D₄	—	712	4	no	—
271	C₂×D₄	—	712	4	no	—
272	C₂×D₄	C₂×D₃₅₆	1	2	yes	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
276	C₂²×C₄	#13083574	1	1	yes	characteristic
277	D₈	—	356	4	no	—
278	D₈	—	356	4	no	—
279	D₈	—	356	4	no	—
280	D₈	—	356	4	no	—
281	D₈	—	712	8	no	—
282	M₁₆	—	4	2	no	—
283	M₁₆	—	4	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	4	4	no	—
287	C₂×C₈	—	178	4	no	—
288	C₂×C₈	#13083559	1	4	yes	complemented
289	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
290	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	C₂×D₃₅₆	1	2	yes	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
298	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
299	C₄²	#13083574	1	1	yes	characteristic
300	C₄²	C₂³×D₈₉	1	1	yes	characteristic
301	C₂³	—	2	2	no	—
302	C₂³	—	356	4	no	—
303	C₂³	—	712	4	no	—
304	C₂³	—	712	4	no	—
305	C₂³	—	712	4	no	—
306	C₂³	#15667986	1	1	yes	characteristic
307	Q₈	—	4	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	356	4	no	—
313	D₄	—	356	8	no	—
314	D₄	—	356	4	no	—
315	D₄	—	712	4	no	—
316	D₄	—	712	4	no	—
317	D₄	—	712	4	no	—
318	D₄	—	712	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	#15667986	1	1	yes	characteristic
325	C₂×C₄	#15667986	1	1	yes	characteristic
326	C₂×C₄	#15667986	1	1	yes	characteristic
327	C₂×C₄	C₂×#13083559	1	2	yes	—
328	C₂×C₄	#15668037	1	2	yes	—
329	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
331	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	4	2	no	—
335	C₈	—	356	4	no	—
336	C₈	C₂×#13083559	1	2	yes	—
337	C₈	C₂×#13083559	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	1424	8	no	—
344	C₂²	—	712	4	no	—
345	C₂²	—	712	4	no	—
346	C₂²	—	712	4	no	—
347	C₂²	#19151793	1	2	yes	—
348	C₂²	C₂×#15667986	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	#19151793	1	2	yes	—
353	C₄	#19151868	1	2	yes	—
354	C₄	C₂²×#13083559	1	1	yes	characteristic
355	C₄	C₂×#15668037	1	1	yes	characteristic
356	C₄	#19152403	1	2	yes	—
357	C₂	—	2	2	no	—
358	C₂	—	8	2	no	—
359	C₂	—	1424	4	no	—
360	C₂	C₂×#19151793	1	1	yes	characteristic
361	C₂	#26029900	1	1	yes	characteristic
362	C₂	C₂×#19151868	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667986
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#2931	C₂
socle	C₂²×C₈₉	C₂×#121
hypercenter	#2931	D₈₉
hypocenter	C₈₉	#19187
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #26031167	C₂	semidirect
2. C₂×#19151975	C₂	semidirect
3. #26019731	C₂	semidirect
4. #26019607	C₂	semidirect
5. #26019000	C₂	semidirect
6. #26018921	C₂	semidirect
7. C₈₉×#2931	C₂	semidirect
17. #19151793	C₂²	semidirect
20. #19151351	C₂²	semidirect
21. #19151349	C₂²	semidirect
22. #19151341	C₂²	semidirect
26. #436×C₈₉	C₂²	semidirect
28. C₈×D₄×C₈₉	C₂²	semidirect
106. C₂×C₈×C₈₉	D₈	semidirect
184. #2931	D₈₉	semidirect
189. C₈₉	#19187	semidirect
204. #436	C₂×D₈₉	semidirect
206. C₈×D₄	C₂×D₈₉	semidirect
288. C₂×C₈	#13083559	semidirect

Maximal subgroups

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

subgroup	index
#26031167	2
C₂×#19151975	2
#26019731	2
#26019607	2
#26019000	2
#26018921	2
C₈₉×#2931	2
#19187	89