Subgroups — #36447481

Order 22784.

542 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151663	C₂	1	1	yes	maximal, characteristic
2	#26020411	C₂	1	1	yes	maximal, characteristic, complemented
3	#26020253	C₂	1	1	yes	maximal, characteristic
4	#26020249	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151350	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018136	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₈₉	—	2	1	no	—
9	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
10	#19151663	—	2	2	no	—
11	#19151663	C₄	1	4	yes	complemented
12	C₂×#15667945	—	2	1	no	—
13	C₂×#15667945	—	2	1	no	—
14	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	C₂×#15667934	—	2	1	no	—
16	C₂×#15667932	—	2	1	no	—
17	C₂×#15667924	—	2	1	no	—
18	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
19	#19151468	C₂²	1	1	yes	characteristic
20	#19151361	—	2	1	no	—
21	#19151360	—	2	1	no	—
22	#19151350	—	2	2	no	—
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#384×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₈₉	—	2	1	no	—
28	C₂²×#10623494	—	2	1	no	—
29	C₂²×#10623494	D₄	1	1	yes	characteristic
30	C₂²×#10623494	D₄	1	1	yes	characteristic
31	C₂×C₈×D₈₉	—	2	4	no	—
32	C₂×C₈×D₈₉	—	4	2	no	—
33	C₂×C₈×D₈₉	—	4	2	no	—
34	C₄×D₃₅₆	—	2	1	no	—
35	C₄×D₃₅₆	—	2	1	no	—
36	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
37	C₄×D₃₅₆	Q₈	1	2	yes	—
38	C₂×C₄⋊Dic₈₉	—	2	1	no	—
39	C₂×C₄⋊Dic₈₉	—	2	1	no	—
40	C₂×C₄⋊Dic₈₉	—	2	1	no	—
41	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
42	C₂×#13083551	—	2	1	no	—
43	#15667945	—	4	2	no	—
44	#15667945	—	4	2	no	—
45	C₄⋊#10623494	—	2	1	no	—
46	C₄⋊#10623494	—	2	1	no	—
47	#15667934	—	4	2	no	—
48	#15667932	—	4	2	no	—
49	#15667924	—	2	4	no	—
50	C₄×#10623494	C₂×C₄	1	2	yes	—
51	C₄×#10623494	D₄	1	2	yes	—
52	C₂²×C₈×C₈₉	—	2	1	no	—
53	C₂²×C₈×C₈₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₈₉	—	2	1	no	—
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₄⋊C₈×C₈₉	—	2	1	no	—
61	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
62	C₂³×D₈₉	—	2	1	no	—
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	—	2	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
67	C₂×D₃₅₆	D₈	1	2	yes	—
68	C₂×D₃₅₆	Q₁₆	1	2	yes	—
69	C₂²×Dic₈₉	—	2	1	no	—
70	C₂²×Dic₈₉	—	4	1	no	—
71	C₂×C₄×D₈₉	—	2	2	no	—
72	C₂×C₄×D₈₉	—	2	2	no	—
73	C₂×C₄×D₈₉	—	4	1	no	—
74	C₂×C₄×D₈₉	—	4	1	no	—
75	C₂×C₄×D₈₉	—	4	1	no	—
76	C₂×C₄×D₈₉	—	4	1	no	—
77	C₂×#10623494	—	2	1	no	—
78	C₂×#10623494	—	2	1	no	—
79	C₂×#10623494	—	4	1	no	—
80	C₂×#10623494	—	4	1	no	—
81	C₂×#10623494	—	4	1	no	—
82	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
83	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
84	C₂×#10623494	D₈	1	2	yes	—
85	C₂×#10623494	SD₁₆	1	2	yes	—
86	C₈×D₈₉	—	4	8	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	4	1	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	C₄⋊Dic₈₉	—	4	1	no	—
92	C₄⋊Dic₈₉	—	8	1	no	—
93	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
94	#13083551	—	2	2	no	—
95	#13083551	—	2	2	no	—
96	C₂²×C₄×C₈₉	—	4	1	no	—
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
100	C₂×C₈×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	—	4	1	no	—
102	C₂×C₈×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	—	4	1	no	—
104	C₂×C₈×C₈₉	—	4	1	no	—
105	C₂×C₈×C₈₉	—	4	1	no	—
106	C₄⋊C₄×C₈₉	—	4	1	no	—
107	C₄⋊C₄×C₈₉	—	4	1	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄⋊C₄×C₈₉	—	4	1	no	—
110	C₄⋊C₄×C₈₉	—	8	1	no	—
111	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
114	C₄²×C₈₉	#41	1	1	yes	characteristic
115	C₂³×C₈₉	#121	1	1	yes	characteristic
116	C₂²×D₈₉	—	2	2	no	—
117	C₂²×D₈₉	—	2	2	no	—
118	C₂²×D₈₉	—	2	2	no	—
119	C₂²×D₈₉	—	4	1	no	—
120	C₂²×D₈₉	—	4	1	no	—
121	C₂²×D₈₉	—	4	1	no	—
122	C₂²×D₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	8	1	no	—
126	C₂×C₄×C₈₉	—	8	1	no	—
127	C₂×C₄×C₈₉	—	8	1	no	—
128	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
139	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
140	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	—	4	1	no	—
144	C₂×Dic₈₉	—	4	1	no	—
145	C₂×Dic₈₉	—	8	1	no	—
146	C₂×Dic₈₉	—	8	1	no	—
147	C₂×Dic₈₉	—	8	1	no	—
148	D₃₅₆	—	2	2	no	—
149	D₃₅₆	—	2	2	no	—
150	D₃₅₆	—	2	2	no	—
151	D₃₅₆	#107	1	4	yes	complemented
152	C₄×D₈₉	—	4	2	no	—
153	C₄×D₈₉	—	4	2	no	—
154	C₄×D₈₉	—	4	2	no	—
155	C₄×D₈₉	—	4	2	no	—
156	C₈×C₈₉	—	4	2	no	—
157	C₈×C₈₉	—	8	1	no	—
158	#10623494	—	2	2	no	—
159	#10623494	—	8	1	no	—
160	#10623494	#102	1	4	yes	complemented
161	C₂²×C₈₉	#380	1	1	yes	characteristic
162	C₂²×C₈₉	#459	1	1	yes	characteristic
163	C₂²×C₈₉	#461	1	1	yes	characteristic
164	C₂²×C₈₉	#466	1	1	yes	characteristic
165	C₂²×C₈₉	#467	1	1	yes	characteristic
166	C₂²×C₈₉	#472	1	1	yes	characteristic
167	C₂²×C₈₉	#473	1	1	yes	characteristic
168	C₂×D₈₉	—	2	4	no	—
169	C₂×D₈₉	—	4	2	no	—
170	C₂×D₈₉	—	4	2	no	—
171	C₂×D₈₉	—	4	2	no	—
172	C₂×D₈₉	—	4	2	no	—
173	C₂×D₈₉	—	4	2	no	—
174	C₂×D₈₉	—	4	2	no	—
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	16	1	no	—
178	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
179	C₄×C₈₉	#417	1	1	yes	characteristic
180	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
181	C₄×C₈₉	#427	1	1	yes	characteristic
182	C₄×C₈₉	#466	1	1	yes	characteristic
183	C₄×C₈₉	#467	1	1	yes	characteristic
184	C₄×C₈₉	#474	1	1	yes	characteristic
185	C₄×C₈₉	#476	1	1	yes	characteristic
186	Dic₈₉	—	4	2	no	—
187	Dic₈₉	—	16	1	no	—
188	#8644	—	89	1	no	maximal, Sylow
189	C₂×C₈₉	#1667	1	1	yes	characteristic
190	C₂×C₈₉	#1669	1	1	yes	characteristic
191	C₂×C₈₉	#1673	1	1	yes	characteristic
192	C₂×C₈₉	#1675	1	1	yes	characteristic
193	C₂×C₈₉	#1802	1	1	yes	characteristic
194	C₂×C₈₉	#1808	1	1	yes	characteristic
195	C₂×C₈₉	#1816	1	1	yes	characteristic
196	D₈₉	—	4	4	no	—
197	C₂×C₈×D₄	—	89	1	no	—
198	#1845	—	89	1	no	—
199	#1802	—	89	1	no	—
200	#1802	—	89	1	no	—
201	C₂×#335	—	89	1	no	—
202	#1296	—	89	1	no	—
203	#1296	D₈₉	1	1	yes	characteristic, complemented
204	C₈₉	#8644	1	1	yes	characteristic, complemented, Sylow
205	C₂³×C₈	—	178	1	no	—
206	C₂×C₄×D₄	—	89	1	no	—
207	C₈×D₄	—	89	4	no	—
208	C₈×D₄	—	178	2	no	—
209	C₂×#107	—	178	1	no	—
210	C₂×#107	—	178	1	no	—
211	C₂×C₄⋊C₈	—	89	1	no	—
212	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
213	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
214	C₂×#102	—	178	1	no	—
215	C₂×#102	—	178	1	no	—
216	C₂×C₂²⋊C₈	—	178	1	no	—
217	C₂×C₄×C₈	—	89	1	no	—
218	#384	—	89	1	no	—
219	#384	C₂×D₈₉	1	1	yes	characteristic
220	#340	—	178	1	no	—
221	#340	—	178	1	no	—
222	#335	—	178	2	no	—
223	C₂²×D₄	—	89	1	no	—
224	C₂³×C₄	—	178	1	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	—	89	1	no	—
228	C₂²×C₈	—	89	1	no	—
229	C₂²×C₈	—	356	2	no	—
230	C₂²×C₈	—	356	2	no	—
231	C₂²×C₈	—	178	4	no	—
232	C₂²×C₈	—	178	1	no	—
233	C₄×D₄	—	89	2	no	—
234	C₄×D₄	—	89	2	no	—
235	C₄×D₄	—	178	1	no	—
236	C₄×D₄	—	178	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	89	1	no	—
241	C₂×C₄⋊C₄	—	178	1	no	—
242	C₂×C₄⋊C₄	—	178	1	no	—
243	C₂×C₄⋊C₄	—	178	1	no	—
244	C₂×C₂²⋊C₄	—	178	1	no	—
245	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
246	#107	—	356	2	no	—
247	#107	—	356	2	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	178	1	no	—
252	C₄⋊C₈	—	178	1	no	—
253	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
254	#102	—	356	2	no	—
255	#102	—	356	2	no	—
256	C₂²⋊C₈	—	178	4	no	—
257	C₄×C₈	—	89	2	no	—
258	C₄×C₈	—	89	2	no	—
259	C₂⁴	—	178	1	no	—
260	C₂×D₄	—	89	2	no	—
261	C₂×D₄	—	89	2	no	—
262	C₂×D₄	—	89	2	no	—
263	C₂×D₄	—	178	1	no	—
264	C₂×D₄	—	178	1	no	—
265	C₂×D₄	—	178	1	no	—
266	C₂²×C₄	—	4	1	no	—
267	C₂²×C₄	—	356	1	no	—
268	C₂²×C₄	—	356	1	no	—
269	C₂²×C₄	—	356	1	no	—
270	C₂²×C₄	—	356	1	no	—
271	C₂²×C₄	—	356	1	no	—
272	C₂²×C₄	—	178	1	no	—
273	C₂²×C₄	—	178	2	no	—
274	C₂²×C₄	—	178	2	no	—
275	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
276	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
277	C₂²×C₄	#13083574	1	1	yes	characteristic
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	89	2	no	—
285	C₂×C₈	—	89	2	no	—
286	C₂×C₈	—	89	2	no	—
287	C₂×C₈	—	89	2	no	—
288	C₂×C₈	—	356	8	no	—
289	C₂×C₈	—	356	1	no	—
290	C₂×C₈	—	356	1	no	—
291	C₂×C₈	—	356	1	no	—
292	C₂×C₈	—	178	1	no	—
293	C₂×C₈	—	178	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	8	1	no	—
299	C₄⋊C₄	—	89	2	no	—
300	C₄⋊C₄	—	356	1	no	—
301	C₄⋊C₄	—	356	1	no	—
302	C₄⋊C₄	—	356	1	no	—
303	C₄⋊C₄	—	356	1	no	—
304	C₄⋊C₄	—	178	1	no	—
305	C₄⋊C₄	—	712	1	no	—
306	C₂²⋊C₄	—	178	2	no	—
307	C₂²⋊C₄	—	178	2	no	—
308	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
309	C₄²	D₄×D₈₉	1	1	yes	characteristic
310	C₄²	Q₈×D₈₉	1	1	yes	characteristic
311	C₄²	#13083574	1	1	yes	characteristic
312	C₂³	—	356	1	no	—
313	C₂³	—	356	1	no	—
314	C₂³	—	356	1	no	—
315	C₂³	—	356	1	no	—
316	C₂³	—	178	2	no	—
317	C₂³	—	178	2	no	—
318	C₂³	—	178	2	no	—
319	C₂³	#15667985	1	1	yes	characteristic
320	D₄	—	89	4	no	—
321	D₄	—	178	2	no	—
322	D₄	—	178	2	no	—
323	D₄	—	178	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	—	8	1	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	—	356	1	no	—
330	C₂×C₄	—	356	1	no	—
331	C₂×C₄	—	356	2	no	—
332	C₂×C₄	—	356	2	no	—
333	C₂×C₄	—	356	2	no	—
334	C₂×C₄	—	356	2	no	—
335	C₂×C₄	—	178	2	no	—
336	C₂×C₄	—	712	1	no	—
337	C₂×C₄	—	712	1	no	—
338	C₂×C₄	—	712	1	no	—
339	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
340	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
341	C₂×C₄	#15667969	1	1	yes	characteristic
342	C₂×C₄	#15667985	1	1	yes	characteristic
343	C₂×C₄	#15667990	1	1	yes	characteristic
344	C₂×C₄	#15667990	1	1	yes	characteristic
345	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
346	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
347	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
348	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
349	C₂×C₄	#15668033	1	1	yes	characteristic
350	C₂×C₄	#15668033	1	1	yes	characteristic
351	C₂×C₄	#15668039	1	1	yes	characteristic
352	C₂×C₄	#15668044	1	1	yes	characteristic
353	C₈	—	4	2	no	—
354	C₈	—	8	1	no	—
355	C₈	—	89	4	no	—
356	C₈	—	178	2	no	—
357	C₈	—	712	1	no	—
358	C₂²	—	356	2	no	—
359	C₂²	—	356	2	no	—
360	C₂²	—	356	2	no	—
361	C₂²	—	356	2	no	—
362	C₂²	—	356	2	no	—
363	C₂²	—	356	2	no	—
364	C₂²	—	178	4	no	—
365	C₂²	#19151457	1	1	yes	characteristic
366	C₂²	#19151792	1	1	yes	characteristic
367	C₂²	#19151802	1	1	yes	characteristic
368	C₂²	#19151833	1	1	yes	characteristic
369	C₂²	#19151838	1	1	yes	characteristic
370	C₂²	#19151867	1	1	yes	characteristic
371	C₂²	#19151872	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	16	1	no	—
375	C₄	—	356	2	no	—
376	C₄	—	1424	1	no	—
377	C₄	#102×D₈₉	1	1	yes	characteristic
378	C₄	#19151598	1	1	yes	characteristic
379	C₄	#107×D₈₉	1	1	yes	characteristic
380	C₄	#19151633	1	1	yes	characteristic
381	C₄	#19151833	1	1	yes	characteristic
382	C₄	#19151838	1	1	yes	characteristic
383	C₄	#19151877	1	1	yes	characteristic
384	C₄	#19151887	1	1	yes	characteristic
385	C₂	—	356	4	no	—
386	C₂	#26019606	1	1	yes	characteristic
387	C₂	#26019616	1	1	yes	characteristic
388	C₂	#26019640	1	1	yes	characteristic
389	C₂	#26019650	1	1	yes	characteristic
390	C₂	#26020251	1	1	yes	characteristic
391	C₂	#26020280	1	1	yes	characteristic
392	C₂	#26020315	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667985
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1296	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₃₅₆	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1296	D₈₉
hypocenter	C₈₉	#8644
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #26020411	C₂	semidirect
5. C₂×#19151350	C₂	semidirect
6. #26018136	C₂	semidirect
7. C₈₉×#1296	C₂	semidirect
11. #19151663	C₄	semidirect
151. D₃₅₆	#107	semidirect
160. #10623494	#102	semidirect
203. #1296	D₈₉	semidirect
204. C₈₉	#8644	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151663	2
#26020411	2
#26020253	2
#26020249	2
C₂×#19151350	2
#26018136	2
C₈₉×#1296	2
#8644	89