Subgroups — #36469680

Order 22784.

498 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151680	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151412	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151410	C₂	1	1	yes	maximal, characteristic
4	M₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018360	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
9	C₂×#15667974	—	2	1	no	—
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	#19151680	—	2	1	no	—
12	#19151680	—	2	1	no	—
13	#19151680	C₄	1	2	yes	—
14	#19151680	C₄	1	2	yes	—
15	C₂×#15667948	C₂²	1	1	yes	characteristic
16	C₂×#15667932	—	2	1	no	—
17	M₁₆×Dic₈₉	—	2	1	no	—
18	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
19	#19151412	C₄	1	4	yes	complemented
20	#19151410	C₄	1	4	yes	complemented
21	#19151373	—	2	1	no	—
22	C₁₆×Dic₈₉	—	2	1	no	—
23	#19151362	—	2	1	no	—
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×D₃₅₆	—	2	1	no	—
28	C₂²×C₄×D₈₉	—	4	1	no	—
29	C₂×D₇₁₂	—	2	1	no	—
30	C₂×D₇₁₂	—	2	1	no	—
31	C₂×D₇₁₂	—	2	1	no	—
32	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
33	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
34	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
35	C₂×#13083558	—	2	1	no	—
36	C₂²×#10623494	D₄	1	1	yes	characteristic
37	#15667974	—	2	2	no	—
38	#15667974	—	2	2	no	—
39	#15667974	—	4	1	no	—
40	#15667974	—	4	1	no	—
41	C₂×#13083551	—	4	1	no	—
42	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
43	#15667952	—	2	1	no	—
44	#15667952	C₂×C₄	1	2	yes	complemented
45	C₂×#13083549	—	2	1	no	—
46	#15667948	—	2	1	no	—
47	#15667948	C₂×C₄	1	2	yes	complemented
48	#15667932	—	2	2	no	—
49	#15667932	—	2	2	no	—
50	#15667922	—	4	1	no	—
51	C₈×Dic₈₉	—	4	1	no	—
52	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₈₉	D₄	1	1	yes	characteristic
55	C₈×Dic₈₉	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₈₉	—	2	1	no	—
57	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
59	M₃₂×C₈₉	—	2	1	no	—
60	M₃₂×C₈₉	C₂×C₄	1	2	yes	complemented
61	C₂×C₁₆×C₈₉	—	2	1	no	—
62	#108×C₈₉	—	2	1	no	—
63	#108×C₈₉	C₂×C₄	1	2	yes	complemented
64	#107×C₈₉	—	2	1	no	—
65	#107×C₈₉	C₂×C₄	1	2	yes	—
66	C₂³×D₈₉	—	4	1	no	—
67	C₂×D₃₅₆	—	2	2	no	—
68	C₂×D₃₅₆	—	2	2	no	—
69	C₂×D₃₅₆	—	2	2	no	—
70	C₂×D₃₅₆	—	4	1	no	—
71	C₂×D₃₅₆	—	4	1	no	—
72	C₂×D₃₅₆	—	4	1	no	—
73	C₂²×Dic₈₉	SD₁₆	1	1	yes	characteristic
74	C₂²×Dic₈₉	SD₁₆	1	1	yes	characteristic
75	C₂×C₄×D₈₉	—	4	2	no	—
76	C₂×C₄×D₈₉	—	4	2	no	—
77	C₂×C₄×D₈₉	—	8	4	no	—
78	D₇₁₂	—	2	2	no	—
79	D₇₁₂	—	2	2	no	—
80	D₇₁₂	—	2	2	no	—
81	D₇₁₂	C₄²	1	4	yes	—
82	#13083558	—	2	2	no	—
83	#13083558	—	4	1	no	—
84	C₂×#10623494	—	2	1	no	—
85	C₂×#10623494	—	2	1	no	—
86	C₂×#10623494	—	4	1	no	—
87	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
89	#13083551	—	4	2	no	—
90	#13083551	—	4	2	no	—
91	C₄×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×Dic₈₉	D₈	1	1	yes	characteristic
94	C₄×Dic₈₉	D₈	1	1	yes	characteristic
95	#13083549	—	2	2	no	—
96	C₂²×C₄×C₈₉	—	4	1	no	—
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	M₁₆×C₈₉	—	2	2	no	—
99	M₁₆×C₈₉	—	4	1	no	—
100	C₂×C₈×C₈₉	—	4	1	no	—
101	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
106	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₁₆×C₈₉	—	2	2	no	—
110	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
111	C₂²×D₈₉	—	4	2	no	—
112	C₂²×D₈₉	—	4	2	no	—
113	C₂²×D₈₉	—	4	2	no	—
114	C₂²×D₈₉	—	8	2	no	—
115	C₂²×D₈₉	—	8	2	no	—
116	C₂×C₄×C₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	8	2	no	—
118	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
124	C₂×Dic₈₉	—	2	2	no	—
125	C₂×Dic₈₉	—	4	1	no	—
126	C₂×Dic₈₉	#102	1	1	yes	characteristic
127	C₂×Dic₈₉	#102	1	1	yes	characteristic
128	C₂×Dic₈₉	#102	1	1	yes	characteristic
129	C₂×Dic₈₉	#102	1	1	yes	characteristic
130	D₃₅₆	—	2	4	no	—
131	D₃₅₆	—	4	2	no	—
132	D₃₅₆	—	4	2	no	—
133	D₃₅₆	—	4	2	no	—
134	C₄×D₈₉	—	8	4	no	—
135	C₄×D₈₉	—	8	4	no	—
136	C₈×C₈₉	—	4	2	no	—
137	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
138	C₈×C₈₉	#117	1	1	yes	characteristic
139	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
140	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
141	#10623494	—	2	1	no	—
142	#10623494	—	2	1	no	—
143	#10623494	—	4	2	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
147	C₂²×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
148	C₂²×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
149	C₂×D₈₉	—	4	4	no	—
150	C₂×D₈₉	—	8	4	no	—
151	C₂×D₈₉	—	8	4	no	—
152	C₂×D₈₉	—	8	2	no	—
153	C₂×D₈₉	—	8	2	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₈₉	#416	1	1	yes	characteristic
157	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
158	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
159	Dic₈₉	—	2	1	no	—
160	Dic₈₉	—	2	1	no	—
161	Dic₈₉	—	2	1	no	—
162	Dic₈₉	#361	1	2	yes	complemented
163	C₄×#361	—	89	1	no	maximal, Sylow
164	C₂×C₈₉	—	2	1	no	—
165	C₂×C₈₉	—	2	1	no	—
166	C₂×C₈₉	C₄×#102	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#361	1	1	yes	characteristic
168	C₂×C₈₉	#2148	1	1	yes	characteristic
169	D₈₉	—	8	4	no	—
170	C₂×C₄×D₈	—	89	1	no	—
171	C₂×#361	—	89	1	no	—
172	C₂×#361	—	89	1	no	—
173	C₄×M₃₂	—	89	1	no	—
174	C₄×#108	—	89	1	no	—
175	#1387	—	89	1	no	—
176	#1387	D₈₉	1	1	yes	characteristic, complemented
177	C₈₉	C₄×#361	1	1	yes	characteristic, complemented, Sylow
178	C₂²×D₈	—	89	1	no	—
179	C₂×C₄×D₄	—	178	1	no	—
180	C₂×M₃₂	—	89	1	no	—
181	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
182	C₄×D₈	—	89	2	no	—
183	C₄×D₈	—	89	2	no	—
184	C₄×D₈	—	178	1	no	—
185	C₄×D₈	—	178	1	no	—
186	C₂×#108	—	89	1	no	—
187	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
188	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
189	C₂×#102	—	178	1	no	—
190	C₄×M₁₆	—	178	1	no	—
191	C₂×C₄×C₈	—	89	1	no	—
192	#361	—	89	4	no	—
193	#361	—	89	4	no	—
194	#346	—	178	1	no	—
195	C₄×C₁₆	—	178	1	no	—
196	#341	—	178	1	no	—
197	C₂²×D₄	—	178	1	no	—
198	C₂³×C₄	—	356	1	no	—
199	C₂×D₈	—	89	2	no	—
200	C₂×D₈	—	89	2	no	—
201	C₂×D₈	—	89	2	no	—
202	C₂×D₈	—	178	1	no	—
203	C₂×D₈	—	178	1	no	—
204	C₂×D₈	—	178	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	178	1	no	—
207	C₂²×C₈	—	89	1	no	—
208	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
209	C₄×D₄	—	356	1	no	—
210	C₄×D₄	—	356	1	no	—
211	C₄×D₄	—	178	2	no	—
212	C₄×D₄	—	178	2	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₂²⋊C₄	—	356	1	no	—
215	C₂×C₄²	—	89	1	no	—
216	M₃₂	—	2	1	no	—
217	M₃₂	—	89	2	no	—
218	M₃₂	—	178	1	no	—
219	M₃₂	C₄×D₈₉	1	2	yes	complemented
220	C₂×C₁₆	—	2	1	no	—
221	C₂×C₁₆	—	178	1	no	—
222	#108	—	2	1	no	—
223	#108	—	89	2	no	—
224	#108	—	178	1	no	—
225	#108	C₄×D₈₉	1	2	yes	complemented
226	#107	—	2	1	no	—
227	#107	C₄×D₈₉	1	2	yes	—
228	#102	—	178	2	no	—
229	#102	—	178	2	no	—
230	#97	—	356	1	no	—
231	C₄×C₈	—	89	1	no	—
232	C₄×C₈	—	89	1	no	—
233	C₄×C₈	—	89	1	no	—
234	C₄×C₈	—	89	1	no	—
235	C₄×C₈	—	356	1	no	—
236	C₂⁴	—	356	1	no	—
237	C₂×D₄	—	356	1	no	—
238	C₂×D₄	—	356	1	no	—
239	C₂×D₄	—	356	1	no	—
240	C₂×D₄	—	178	2	no	—
241	C₂×D₄	—	178	2	no	—
242	C₂×D₄	—	178	2	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	89	1	no	—
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	356	2	no	—
247	C₂²×C₄	—	356	2	no	—
248	C₂²×C₄	—	712	4	no	—
249	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
250	D₈	—	89	4	no	—
251	D₈	—	178	2	no	—
252	D₈	—	178	2	no	—
253	D₈	—	178	2	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	356	1	no	—
257	M₁₆	—	178	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	—	356	1	no	—
262	C₂×C₈	—	178	1	no	—
263	C₂×C₈	—	178	1	no	—
264	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	#13083573	1	1	yes	characteristic
269	C₂×C₈	#13083575	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	356	2	no	—
273	C₂²⋊C₄	—	356	2	no	—
274	C₄²	—	89	1	no	—
275	C₄²	—	89	1	no	—
276	C₄²	—	89	1	no	—
277	C₄²	—	89	1	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	178	2	no	—
280	C₂³	—	356	2	no	—
281	C₂³	—	356	2	no	—
282	C₂³	—	356	2	no	—
283	C₂³	—	712	2	no	—
284	C₂³	—	712	2	no	—
285	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
286	D₄	—	356	2	no	—
287	D₄	—	356	2	no	—
288	D₄	—	356	2	no	—
289	D₄	—	178	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	89	1	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	89	1	no	—
295	C₂×C₄	—	89	1	no	—
296	C₂×C₄	—	356	1	no	—
297	C₂×C₄	—	178	2	no	—
298	C₂×C₄	—	712	4	no	—
299	C₂×C₄	—	712	4	no	—
300	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
301	C₂×C₄	#15667972	1	1	yes	characteristic
302	C₂×C₄	#15667974	1	1	yes	characteristic
303	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
304	C₂×C₄	#15668033	1	1	yes	characteristic
305	C₂×C₄	#15668035	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	—	356	2	no	—
308	C₈	—	178	1	no	—
309	C₈	—	178	1	no	—
310	C₈	C₄²×D₈₉	1	1	yes	characteristic
311	C₈	#15667970	1	1	yes	characteristic
312	C₈	#15667972	1	1	yes	characteristic
313	C₈	#15667974	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	356	4	no	—
317	C₂²	—	712	4	no	—
318	C₂²	—	712	4	no	—
319	C₂²	—	712	2	no	—
320	C₂²	—	712	2	no	—
321	C₂²	#102×D₈₉	1	1	yes	characteristic
322	C₂²	#19151683	1	1	yes	characteristic
323	C₂²	#19151685	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	89	2	no	—
326	C₄	—	178	1	no	—
327	C₄	—	178	1	no	—
328	C₄	—	178	1	no	—
329	C₄	#19151445	1	1	yes	characteristic
330	C₄	#19151593	1	1	yes	characteristic
331	C₄	#19151678	1	1	yes	characteristic
332	C₄	#19151680	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	712	4	no	—
336	C₂	#26020066	1	1	yes	characteristic
337	C₂	#361×D₈₉	1	1	yes	characteristic
338	C₂	#26022000	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#102×D₈₉
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1387	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1387	D₈₉
hypocenter	C₈₉	C₄×#361
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

Complements

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

N	H ≅ G/N	type
4. M₃₂×Dic₈₉	C₂	semidirect
5. #108×Dic₈₉	C₂	semidirect
6. #26018360	C₂	semidirect
7. C₈₉×#1387	C₂	semidirect
19. #19151412	C₄	semidirect
20. #19151410	C₄	semidirect
44. #15667952	C₂×C₄	semidirect
47. #15667948	C₂×C₄	semidirect
60. M₃₂×C₈₉	C₂×C₄	semidirect
63. #108×C₈₉	C₂×C₄	semidirect
162. Dic₈₉	#361	semidirect
176. #1387	D₈₉	semidirect
177. C₈₉	C₄×#361	semidirect
219. M₃₂	C₄×D₈₉	semidirect
225. #108	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151680	2
C₂×#19151412	2
C₂×#19151410	2
M₃₂×Dic₈₉	2
#108×Dic₈₉	2
#26018360	2
C₈₉×#1387	2
C₄×#361	89