Subgroups — #36469690

Order 22784.

428 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151714	C₂	1	1	yes	maximal, characteristic
2	#26021999	C₂	1	1	yes	maximal, characteristic
3	#26021989	C₂	1	1	yes	maximal, characteristic
4	M₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#26020134	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	1	yes	characteristic
9	C₂×#15668020	—	2	1	no	—
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	#19151714	—	2	1	no	—
12	#19151714	—	2	1	no	—
13	#19151714	C₄	1	2	yes	complemented
14	#19151714	C₄	1	2	yes	complemented
15	#19151636	C₂²	1	1	yes	characteristic
16	C₂×#15667938	—	2	1	no	—
17	#19151556	—	2	1	no	—
18	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
19	#19151413	C₄	1	2	yes	complemented
20	#19151410	C₄	1	2	yes	complemented
21	#19151409	C₄	1	4	yes	complemented
22	#19151373	—	2	1	no	—
23	C₁₆×Dic₈₉	—	2	1	no	—
24	#19151359	—	2	1	no	—
25	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#13083574	—	2	1	no	—
29	#15668034	—	2	1	no	—
30	#15668034	—	2	1	no	—
31	#15668034	C₂×C₄	1	2	yes	—
32	#15668034	C₂×C₄	1	2	yes	—
33	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
34	C₂×#13083563	—	2	1	no	—
35	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
36	#15668020	—	2	2	no	—
37	#15668020	—	2	2	no	—
38	#15668020	—	4	1	no	—
39	#15668020	—	4	1	no	—
40	C₂×#13083557	—	4	1	no	—
41	C₂²×#10623494	D₄	1	1	yes	characteristic
42	C₂×C₈×D₈₉	—	4	1	no	—
43	#15667971	—	2	1	no	—
44	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
45	#15667952	—	2	1	no	—
46	#15667952	C₂×C₄	1	2	yes	—
47	C₂×#13083549	—	2	1	no	—
48	#15667946	—	2	1	no	—
49	#15667944	C₂×C₄	1	2	yes	—
50	#15667938	—	2	2	no	—
51	#15667938	—	2	2	no	—
52	#15667922	—	4	1	no	—
53	C₈×Dic₈₉	—	4	1	no	—
54	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₈₉	D₄	1	1	yes	characteristic
57	C₈×Dic₈₉	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₈₉	—	2	1	no	—
59	C₂×M₁₆×C₈₉	—	2	1	no	—
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	M₃₂×C₈₉	—	2	1	no	—
62	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
63	C₂×C₁₆×C₈₉	—	2	1	no	—
64	#108×C₈₉	—	2	1	no	—
65	#108×C₈₉	—	2	1	no	—
66	#108×C₈₉	C₂×C₄	1	2	yes	—
67	#108×C₈₉	C₂×C₄	1	2	yes	—
68	#13083574	—	2	2	no	—
69	#13083574	—	2	2	no	—
70	#13083574	—	4	1	no	—
71	#13083574	—	4	1	no	—
72	C₂×Dic₁₇₈	—	2	1	no	—
73	C₂×D₃₅₆	—	2	1	no	—
74	C₂×C₂²⋊D₈₉	—	4	1	no	—
75	C₂²×Dic₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	—	4	1	no	—
77	Dic₃₅₆	—	2	1	no	—
78	Dic₃₅₆	C₄²	1	2	yes	—
79	#13083563	—	2	2	no	—
80	#13083563	—	2	2	no	—
81	D₇₁₂	—	2	1	no	—
82	D₇₁₂	C₄²	1	2	yes	—
83	#13083557	—	4	2	no	—
84	#13083557	—	4	2	no	—
85	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#10623494	D₈	1	1	yes	characteristic
88	C₂×#10623494	D₈	1	1	yes	characteristic
89	C₂×#10623494	SD₁₆	1	1	yes	characteristic
90	C₂×#10623494	SD₁₆	1	1	yes	characteristic
91	C₈×D₈₉	—	4	2	no	—
92	C₈×D₈₉	—	4	2	no	—
93	C₄⋊Dic₈₉	—	2	2	no	—
94	C₂²⋊Dic₈₉	—	4	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	—	4	1	no	—
97	C₄×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
99	#13083549	—	2	2	no	—
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	M₁₆×C₈₉	—	2	2	no	—
102	M₁₆×C₈₉	—	2	2	no	—
103	M₁₆×C₈₉	—	4	1	no	—
104	M₁₆×C₈₉	—	4	1	no	—
105	C₂×C₈×C₈₉	—	4	1	no	—
106	C₂×C₈×C₈₉	—	4	1	no	—
107	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
113	C₁₆×C₈₉	—	2	2	no	—
114	C₂³×C₈₉	#135	1	1	yes	characteristic
115	C₂²×D₈₉	—	4	1	no	—
116	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
122	C₂²⋊D₈₉	—	4	2	no	—
123	C₂²⋊D₈₉	—	4	2	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	—	4	2	no	—
129	D₃₅₆	—	2	2	no	—
130	D₃₅₆	—	4	1	no	—
131	C₄×D₈₉	—	4	2	no	—
132	C₄×D₈₉	—	4	2	no	—
133	Dic₁₇₈	—	2	2	no	—
134	Dic₁₇₈	—	4	1	no	—
135	C₈×C₈₉	—	4	2	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	#102	1	1	yes	characteristic
142	#10623494	#102	1	1	yes	characteristic
143	#10623494	#102	1	1	yes	characteristic
144	#10623494	#102	1	1	yes	characteristic
145	C₂²×C₈₉	—	4	1	no	—
146	C₂²×C₈₉	#416	1	1	yes	characteristic
147	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
148	C₂²×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
149	C₂×D₈₉	—	4	2	no	—
150	C₂×D₈₉	—	8	2	no	—
151	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
153	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
154	C₄×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
155	Dic₈₉	—	2	2	no	—
156	Dic₈₉	—	4	1	no	—
157	Dic₈₉	—	4	2	no	—
158	Dic₈₉	—	8	2	no	—
159	#12646	—	89	1	no	maximal, Sylow
160	C₂×C₈₉	—	2	1	no	—
161	C₂×C₈₉	—	2	1	no	—
162	C₂×C₈₉	—	2	1	no	—
163	C₂×C₈₉	C₄×#102	1	1	yes	characteristic
164	D₈₉	—	8	2	no	—
165	C₂×#443	—	89	1	no	—
166	#2148	—	89	1	no	—
167	#2145	—	89	1	no	—
168	C₄×M₃₂	—	89	1	no	—
169	#1772	—	89	1	no	—
170	#1389	—	89	1	no	—
171	#1388	D₈₉	1	1	yes	characteristic, complemented
172	C₈₉	#12646	1	1	yes	characteristic, complemented, Sylow
173	C₂×#135	—	89	1	no	—
174	C₂×#131	—	178	1	no	—
175	C₂×M₃₂	—	89	1	no	—
176	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
177	#443	—	89	2	no	—
178	#443	—	89	2	no	—
179	#443	—	178	1	no	—
180	#443	—	178	1	no	—
181	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
182	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
183	#427	—	89	1	no	—
184	C₂×#104	—	178	1	no	—
185	#405	—	178	1	no	—
186	C₂×C₄×C₈	—	89	1	no	—
187	#362	—	89	2	no	—
188	#361	—	89	2	no	—
189	#360	—	89	4	no	—
190	#346	—	178	1	no	—
191	C₄×C₁₆	—	178	1	no	—
192	#339	—	178	1	no	—
193	C₂×#41	—	178	1	no	—
194	#135	—	89	2	no	—
195	#135	—	89	2	no	—
196	#135	—	178	1	no	—
197	#135	—	178	1	no	—
198	C₂×Q₁₆	—	89	1	no	—
199	C₂×SD₁₆	—	178	1	no	—
200	C₂×D₈	—	89	1	no	—
201	#131	—	356	1	no	—
202	#131	—	356	1	no	—
203	#131	—	178	2	no	—
204	#131	—	178	2	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	356	1	no	—
208	C₂²×C₈	—	89	1	no	—
209	C₂²×C₈	—	356	1	no	—
210	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
211	#117	—	178	1	no	—
212	C₂×C₄²	—	89	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	89	2	no	—
215	M₃₂	—	178	1	no	—
216	M₃₂	C₄×D₈₉	1	2	yes	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	178	1	no	—
219	#108	—	2	1	no	—
220	#108	—	2	1	no	—
221	#108	C₄×D₈₉	1	2	yes	—
222	#108	C₄×D₈₉	1	2	yes	—
223	#107	—	178	1	no	—
224	#106	—	89	2	no	—
225	#104	—	178	2	no	—
226	#104	—	178	2	no	—
227	#97	—	356	1	no	—
228	C₄×C₈	—	89	1	no	—
229	C₄×C₈	—	89	1	no	—
230	C₄×C₈	—	89	1	no	—
231	C₄×C₈	—	89	1	no	—
232	C₄×C₈	—	356	1	no	—
233	#41	—	356	1	no	—
234	#41	—	356	1	no	—
235	#41	—	178	2	no	—
236	#41	—	178	2	no	—
237	C₂×Q₈	—	178	1	no	—
238	C₂×D₄	—	356	1	no	—
239	C₂×D₄	—	178	1	no	—
240	C₂²×C₄	—	356	1	no	—
241	C₂²×C₄	—	178	1	no	—
242	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
243	Q₁₆	—	89	2	no	—
244	Q₁₆	—	178	1	no	—
245	SD₁₆	—	178	2	no	—
246	SD₁₆	—	178	2	no	—
247	D₈	—	89	2	no	—
248	D₈	—	178	1	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	2	2	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	356	2	no	—
254	M₁₆	—	356	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	89	1	no	—
258	C₂×C₈	—	89	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	—	89	1	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	356	2	no	—
264	C₂×C₈	—	356	2	no	—
265	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
269	C₂×C₈	#13083573	1	1	yes	characteristic
270	C₂×C₈	#13083575	1	1	yes	characteristic
271	C₄⋊C₄	—	178	2	no	—
272	C₂²⋊C₄	—	356	1	no	—
273	C₄²	—	89	1	no	—
274	C₄²	—	89	1	no	—
275	C₄²	—	356	1	no	—
276	C₄²	—	178	1	no	—
277	C₁₆	—	2	2	no	—
278	C₁₆	—	178	2	no	—
279	C₂³	—	356	1	no	—
280	C₂³	#15668035	1	1	yes	characteristic
281	Q₈	—	356	1	no	—
282	Q₈	—	178	2	no	—
283	D₄	—	356	2	no	—
284	D₄	—	356	2	no	—
285	D₄	—	356	1	no	—
286	D₄	—	178	2	no	—
287	C₂×C₄	—	356	1	no	—
288	C₂×C₄	—	356	2	no	—
289	C₂×C₄	—	356	2	no	—
290	C₂×C₄	—	356	2	no	—
291	C₂×C₄	—	356	2	no	—
292	C₂×C₄	—	178	1	no	—
293	C₂×C₄	—	178	1	no	—
294	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
295	C₂×C₄	#15667972	1	1	yes	characteristic
296	C₂×C₄	#15667974	1	1	yes	characteristic
297	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
298	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
299	C₂×C₄	#15668033	1	1	yes	characteristic
300	C₈	—	4	2	no	—
301	C₈	—	4	2	no	—
302	C₈	—	89	1	no	—
303	C₈	—	89	1	no	—
304	C₈	—	89	1	no	—
305	C₈	—	89	1	no	—
306	C₈	C₄²×D₈₉	1	1	yes	characteristic
307	C₈	#15667970	1	1	yes	characteristic
308	C₈	#15667972	1	1	yes	characteristic
309	C₈	#15667974	1	1	yes	characteristic
310	C₂²	—	4	1	no	—
311	C₂²	—	356	2	no	—
312	C₂²	—	712	2	no	—
313	C₂²	#19151593	1	1	yes	characteristic
314	C₂²	#19151680	1	1	yes	characteristic
315	C₂²	#19151683	1	1	yes	characteristic
316	C₄	—	356	1	no	—
317	C₄	—	356	2	no	—
318	C₄	—	178	2	no	—
319	C₄	—	712	2	no	—
320	C₄	#19151445	1	1	yes	characteristic
321	C₄	#102×D₈₉	1	1	yes	characteristic
322	C₄	#19151678	1	1	yes	characteristic
323	C₄	#19151685	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	712	2	no	—
328	C₂	#26020066	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#102×D₈₉
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1388	C₂
socle	C₂×C₈₉	C₄×#102
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1388	D₈₉
hypocenter	C₈₉	#12646
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1388

Complements

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

N	H ≅ G/N	type
4. M₃₂×Dic₈₉	C₂	semidirect
5. #26020134	C₂	semidirect
6. #26018366	C₂	semidirect
7. C₈₉×#1388	C₂	semidirect
13. #19151714	C₄	semidirect
14. #19151714	C₄	semidirect
19. #19151413	C₄	semidirect
20. #19151410	C₄	semidirect
21. #19151409	C₄	semidirect
171. #1388	D₈₉	semidirect
172. C₈₉	#12646	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151714	2
#26021999	2
#26021989	2
M₃₂×Dic₈₉	2
#26020134	2
#26018366	2
C₈₉×#1388	2
#12646	89