Subgroups — #36538440

Order 22784.

766 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029712	C₂	1	2	yes	maximal, complemented
2	C₂×#19151677	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019810	C₂	1	4	yes	maximal, complemented
4	#26019762	C₂	1	2	yes	maximal, complemented
5	#26019728	C₂	1	2	yes	maximal, complemented
6	#26019008	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018929	C₂	1	2	yes	maximal, complemented
8	C₈₉×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668037	—	2	2	no	—
10	C₂²×#13083557	—	2	1	no	—
11	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
12	#19151810	—	2	4	no	—
13	#19151800	C₂²	1	4	yes	—
14	#19151790	C₂²	1	4	yes	—
15	#19151677	—	2	1	no	—
16	#19151677	—	2	1	no	—
17	#19151677	C₂²	1	2	yes	—
18	#19151677	C₂²	1	2	yes	—
19	#19151622	C₂²	1	2	yes	—
20	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
21	#19151599	—	2	2	no	—
22	C₂×#15667924	—	2	1	no	—
23	#19151349	C₂²	1	2	yes	complemented
24	#19151349	C₂²	1	2	yes	complemented
25	#19151344	C₂²	1	4	yes	complemented
26	#19151342	C₂²	1	4	yes	complemented
27	#532×C₈₉	C₂²	1	2	yes	—
28	#500×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	#493×C₈₉	C₂²	1	1	yes	characteristic, complemented
30	#486×C₈₉	C₂²	1	2	yes	complemented
31	#466×C₈₉	—	2	1	no	—
32	#466×C₈₉	—	2	1	no	—
33	C₄×D₈×C₈₉	—	2	1	no	—
34	C₄×D₈×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₈₉	—	2	1	no	—
38	#15668037	—	4	8	no	—
39	C₂×#13083561	—	2	4	no	—
40	C₂×#13083559	—	2	4	no	—
41	C₂×#13083558	—	2	2	no	—
42	C₂×#13083557	—	2	2	no	—
43	C₂×#13083557	—	2	2	no	—
44	C₂×#13083557	—	4	4	no	—
45	C₂²×#10623494	—	2	1	no	—
46	C₄×D₃₅₆	—	2	1	no	—
47	C₄×D₃₅₆	—	2	1	no	—
48	C₄×D₃₅₆	D₄	1	2	yes	—
49	C₄×D₃₅₆	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
51	C₂×#13083551	—	2	1	no	—
52	C₄⋊#10623494	—	2	1	no	—
53	C₄⋊#10623494	—	2	2	no	—
54	C₄⋊#10623494	C₂³	1	2	yes	—
55	C₄⋊#10623494	C₂³	1	4	yes	—
56	#15667936	—	2	4	no	—
57	#15667933	—	2	4	no	—
58	#15667924	—	2	2	no	—
59	#15667924	—	2	2	no	—
60	C₂×#41×C₈₉	—	2	2	no	—
61	C₂×D₈×C₈₉	—	2	1	no	—
62	C₂×D₈×C₈₉	—	2	1	no	—
63	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
65	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
66	#127×C₈₉	C₂³	1	2	yes	—
67	#124×C₈₉	—	2	2	no	—
68	#121×C₈₉	—	2	2	no	—
69	#121×C₈₉	—	2	2	no	—
70	C₄×D₄×C₈₉	—	2	2	no	—
71	C₄×D₄×C₈₉	—	2	2	no	—
72	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
73	#107×C₈₉	—	2	1	no	—
74	#107×C₈₉	—	2	1	no	—
75	#102×C₈₉	—	2	2	no	—
76	#102×C₈₉	—	2	2	no	—
77	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₂³×D₈₉	—	2	1	no	—
82	C₂×D₃₅₆	—	2	4	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
85	C₂²×Dic₈₉	—	2	1	no	—
86	C₂×C₄×D₈₉	—	2	2	no	—
87	C₂×C₄×D₈₉	—	2	2	no	—
88	C₂×C₄×D₈₉	—	4	2	no	—
89	C₂×C₄×D₈₉	—	4	2	no	—
90	#13083561	—	4	8	no	—
91	#13083559	—	4	8	no	—
92	#13083558	—	4	4	no	—
93	#13083557	—	4	4	no	—
94	#13083557	—	4	4	no	—
95	C₂×#10623494	—	2	2	no	—
96	C₂×#10623494	—	2	4	no	—
97	C₂×#10623494	—	4	2	no	—
98	C₄⋊Dic₈₉	—	2	1	no	—
99	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
100	#13083551	—	2	2	no	—
101	#13083551	—	2	2	no	—
102	#41×C₈₉	—	4	4	no	—
103	#41×C₈₉	—	4	4	no	—
104	C₂×Q₈×C₈₉	—	2	2	no	—
105	C₂×D₄×C₈₉	—	2	2	no	—
106	C₂×D₄×C₈₉	—	2	2	no	—
107	C₂×D₄×C₈₉	—	2	2	no	—
108	C₂×D₄×C₈₉	—	4	2	no	—
109	C₂²×C₄×C₈₉	—	4	2	no	—
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
113	D₈×C₈₉	—	4	2	no	—
114	D₈×C₈₉	—	4	2	no	—
115	C₂×C₈×C₈₉	—	2	2	no	—
116	C₂×C₈×C₈₉	—	4	1	no	—
117	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₈₉	—	2	2	no	—
122	C₄⋊C₄×C₈₉	—	2	2	no	—
123	C₂²⋊C₄×C₈₉	—	4	2	no	—
124	C₂²⋊C₄×C₈₉	—	4	2	no	—
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₉	#41	1	1	yes	characteristic
128	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂³×C₈₉	—	4	2	no	—
130	C₂³×C₈₉	#121	1	1	yes	characteristic
131	C₂²×D₈₉	—	2	2	no	—
132	C₂²×D₈₉	—	4	2	no	—
133	C₂²×D₈₉	—	4	2	no	—
134	C₂²×D₈₉	—	4	2	no	—
135	Q₈×C₈₉	—	4	2	no	—
136	Q₈×C₈₉	—	4	2	no	—
137	D₄×C₈₉	—	4	2	no	—
138	D₄×C₈₉	—	4	2	no	—
139	D₄×C₈₉	—	4	2	no	—
140	D₄×C₈₉	—	4	2	no	—
141	D₄×C₈₉	—	8	2	no	—
142	D₄×C₈₉	—	8	4	no	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	—	2	1	no	—
145	C₂×C₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	4	2	no	—
147	C₂×C₄×C₈₉	—	8	4	no	—
148	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
149	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
152	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₈₉	—	2	2	no	—
156	C₂×Dic₈₉	—	4	2	no	—
157	D₃₅₆	—	2	4	no	—
158	D₃₅₆	—	4	2	no	—
159	C₄×D₈₉	—	4	4	no	—
160	C₄×D₈₉	—	4	2	no	—
161	C₄×D₈₉	—	4	2	no	—
162	C₈×C₈₉	—	2	1	no	—
163	C₈×C₈₉	—	2	1	no	—
164	C₈×C₈₉	—	2	1	no	—
165	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	complemented
166	#10623494	—	4	2	no	—
167	#10623494	—	4	4	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	—	2	1	no	—
170	C₂²×C₈₉	—	8	2	no	—
171	C₂²×C₈₉	—	8	2	no	—
172	C₂²×C₈₉	—	8	2	no	—
173	C₂²×C₈₉	#469	1	2	yes	—
174	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
175	C₂×D₈₉	—	4	2	no	—
176	C₂×D₈₉	—	4	2	no	—
177	C₂×D₈₉	—	4	2	no	—
178	C₂×D₈₉	—	8	4	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	—	8	2	no	—
183	C₄×C₈₉	#469	1	2	yes	—
184	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
186	C₄×C₈₉	#575	1	1	yes	characteristic
187	Dic₈₉	—	4	2	no	—
188	#19280	—	89	1	no	maximal, Sylow
189	C₂×C₈₉	—	2	2	no	—
190	C₂×C₈₉	—	16	2	no	—
191	C₂×C₈₉	#3036	1	1	yes	characteristic
192	C₂×C₈₉	C₂×#469	1	1	yes	characteristic
193	C₂×C₈₉	#3067	1	1	yes	characteristic
194	D₈₉	—	8	2	no	—
195	#3169	D₈₉	1	1	yes	characteristic, complemented
196	#3036	—	89	2	no	—
197	C₂×#436	—	89	1	no	—
198	#1701	—	89	4	no	—
199	#1693	—	89	2	no	—
200	#1687	—	89	2	no	—
201	#1566	—	89	1	no	—
202	#1547	—	89	2	no	—
203	C₈₉	#19280	1	1	yes	characteristic, complemented, Sylow
204	C₂×AGL(1,ℤ/8)	—	178	2	no	—
205	C₂²×M₁₆	—	178	1	no	—
206	#532	C₂×D₈₉	1	2	yes	—
207	C₂×C₄×D₄	—	89	1	no	—
208	#500	C₂×D₈₉	1	1	yes	characteristic, complemented
209	#493	C₂×D₈₉	1	1	yes	characteristic, complemented
210	#486	C₂×D₈₉	1	2	yes	complemented
211	#466	—	2	1	no	—
212	#466	—	2	1	no	—
213	#463	—	178	4	no	—
214	#461	—	89	4	no	—
215	#459	—	89	4	no	—
216	C₄×D₈	—	2	1	no	—
217	C₄×D₈	—	2	1	no	—
218	#436	—	89	2	no	—
219	#436	—	89	2	no	—
220	#436	—	178	1	no	—
221	#436	—	178	1	no	—
222	#423	—	89	2	no	—
223	C₂×C₄⋊C₈	—	89	1	no	—
224	#417	—	178	2	no	—
225	C₂×C₂²⋊C₈	—	178	1	no	—
226	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
227	#334	—	89	2	no	—
228	#334	—	89	2	no	—
229	#332	—	89	4	no	—
230	#331	—	89	4	no	—
231	C₂×#41	—	2	2	no	—
232	C₂²×D₄	—	89	1	no	—
233	C₂³×C₄	—	178	1	no	—
234	AGL(1,ℤ/8)	—	356	8	no	—
235	C₂×SD₁₆	—	178	4	no	—
236	C₂×D₈	—	2	1	no	—
237	C₂×D₈	—	2	1	no	—
238	C₂×D₈	—	178	4	no	—
239	C₂×M₁₆	—	356	4	no	—
240	C₂×M₁₆	—	178	2	no	—
241	C₂×M₁₆	—	178	2	no	—
242	C₂×M₁₆	—	178	2	no	—
243	C₂²×C₈	—	178	1	no	—
244	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
245	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
246	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
247	#127	C₂²×D₈₉	1	2	yes	—
248	#124	—	2	2	no	—
249	#121	—	2	2	no	—
250	#121	—	2	2	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	—	2	2	no	—
253	C₄×D₄	—	89	2	no	—
254	C₄×D₄	—	89	2	no	—
255	C₄×D₄	—	178	1	no	—
256	C₄×D₄	—	178	1	no	—
257	C₂×C₄⋊C₄	—	89	1	no	—
258	C₂×C₂²⋊C₄	—	178	1	no	—
259	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	C₄⋊C₈	—	89	2	no	—
263	C₄⋊C₈	—	89	4	no	—
264	C₄⋊C₈	—	178	2	no	—
265	C₄⋊C₈	—	178	1	no	—
266	#103	—	178	4	no	—
267	#102	—	2	2	no	—
268	#102	—	2	2	no	—
269	#102	—	178	4	no	—
270	C₂²⋊C₈	—	178	2	no	—
271	C₂²⋊C₈	—	178	2	no	—
272	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
273	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
276	C₂⁴	—	178	1	no	—
277	#41	—	4	4	no	—
278	#41	—	4	4	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	89	2	no	—
285	C₂×D₄	—	178	1	no	—
286	C₂×D₄	—	178	4	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	356	2	no	—
289	C₂²×C₄	—	356	2	no	—
290	C₂²×C₄	—	178	1	no	—
291	C₂²×C₄	—	178	2	no	—
292	C₂²×C₄	—	178	2	no	—
293	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
295	C₂²×C₄	#13083573	1	1	yes	characteristic
296	SD₁₆	—	356	8	no	—
297	D₈	—	4	2	no	—
298	D₈	—	4	2	no	—
299	D₈	—	356	8	no	—
300	M₁₆	—	356	4	no	—
301	M₁₆	—	356	4	no	—
302	M₁₆	—	356	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	356	2	no	—
306	C₂×C₈	—	178	4	no	—
307	C₂×C₈	—	178	2	no	—
308	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
310	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
311	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	89	2	no	—
315	C₄⋊C₄	—	178	1	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	178	2	no	—
319	C₂²⋊C₄	—	178	2	no	—
320	C₄²	D₄×D₈₉	1	1	yes	characteristic
321	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
322	C₄²	#13083573	1	1	yes	characteristic
323	C₄²	C₂³×D₈₉	1	1	yes	characteristic
324	C₂³	—	4	2	no	—
325	C₂³	—	356	2	no	—
326	C₂³	—	356	2	no	—
327	C₂³	—	356	2	no	—
328	C₂³	—	178	2	no	—
329	C₂³	#15667983	1	1	yes	characteristic
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	2	no	—
336	D₄	—	8	2	no	—
337	D₄	—	8	4	no	—
338	D₄	—	356	2	no	—
339	D₄	—	178	4	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	8	4	no	—
345	C₂×C₄	—	356	4	no	—
346	C₂×C₄	—	356	2	no	—
347	C₂×C₄	—	356	2	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	178	2	no	—
350	C₂×C₄	#15667983	1	1	yes	characteristic
351	C₂×C₄	#15667983	1	1	yes	characteristic
352	C₂×C₄	#15667983	1	1	yes	characteristic
353	C₂×C₄	#15668036	1	4	yes	—
354	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
356	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	356	2	no	—
361	C₈	—	356	4	no	—
362	C₈	#15668037	1	2	yes	complemented
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	356	2	no	—
369	C₂²	—	356	2	no	—
370	C₂²	—	356	2	no	—
371	C₂²	—	712	4	no	—
372	C₂²	#19151848	1	2	yes	—
373	C₂²	C₂×#15667983	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	356	2	no	—
379	C₄	#19151848	1	2	yes	—
380	C₄	C₂×#15668036	1	2	yes	—
381	C₄	C₂×#15668037	1	1	yes	characteristic
382	C₄	#19152396	1	1	yes	characteristic
383	C₂	—	2	2	no	—
384	C₂	—	16	2	no	—
385	C₂	—	712	2	no	—
386	C₂	#26029712	1	1	yes	characteristic
387	C₂	C₂×#19151848	1	1	yes	characteristic
388	C₂	#26029968	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3169	C₂
socle	C₂²×C₈₉	C₂×#121
hypercenter	#3169	D₈₉
hypocenter	C₈₉	#19280
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #26029712	C₂	semidirect
2. C₂×#19151677	C₂	semidirect
3. #26019810	C₂	semidirect
4. #26019762	C₂	semidirect
5. #26019728	C₂	semidirect
6. #26019008	C₂	semidirect
7. #26018929	C₂	semidirect
8. C₈₉×#3169	C₂	semidirect
23. #19151349	C₂²	semidirect
24. #19151349	C₂²	semidirect
25. #19151344	C₂²	semidirect
26. #19151342	C₂²	semidirect
28. #500×C₈₉	C₂²	semidirect
29. #493×C₈₉	C₂²	semidirect
30. #486×C₈₉	C₂²	semidirect
165. C₈×C₈₉	AGL(1,ℤ/8)	semidirect
195. #3169	D₈₉	semidirect
203. C₈₉	#19280	semidirect
208. #500	C₂×D₈₉	semidirect
209. #493	C₂×D₈₉	semidirect
210. #486	C₂×D₈₉	semidirect
362. C₈	#15668037	semidirect

Maximal subgroups

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

subgroup	index
#26029712	2
C₂×#19151677	2
#26019810	2
#26019762	2
#26019728	2
#26019008	2
#26018929	2
C₈₉×#3169	2
#19280	89