Subgroups — #36707597

Order 22784.

710 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#26028944	C₂	1	1	yes	maximal, characteristic
3	#26022075	C₂	1	4	yes	maximal, complemented
4	C₁₆×C₂²⋊D₈₉	C₂	1	4	yes	maximal, complemented
5	#26022012	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021866	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021862	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021848	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
11	#19152105	C₄	1	1	yes	characteristic
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂²×#13083549	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₈₉	—	2	4	no	—
15	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
16	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
17	#19151667	C₄	1	4	yes	—
18	C₈×C₂²⋊D₈₉	C₂²	1	4	yes	—
19	#19151626	C₄	1	1	yes	characteristic
20	#19151565	C₄	1	1	yes	characteristic
21	#19151561	C₄	1	1	yes	characteristic
22	#19151553	C₂²	1	1	yes	characteristic
23	#19151416	C₂²	1	4	yes	complemented
24	C₂²⋊#13083549	C₂²	1	4	yes	complemented
25	#19151377	C₂²	1	4	yes	—
26	#19151373	C₂²	1	2	yes	complemented
27	C₁₆×Dic₈₉	C₂²	1	2	yes	complemented
28	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
29	#504×C₈₉	—	2	4	no	—
30	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₈₉	C₂²	1	2	yes	complemented
32	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	complemented
34	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
36	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₈₉	—	2	2	no	—
38	C₂×C₈×D₈₉	—	2	2	no	—
39	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₈₉	C₂³	1	2	yes	—
41	#15667993	C₈	1	2	yes	—
42	#15667988	C₈	1	2	yes	—
43	#15667983	C₈	1	2	yes	—
44	#15667980	C₈	1	2	yes	—
45	C₄×C₂²⋊D₈₉	C₂×C₄	1	4	yes	—
46	#15667971	C₂×C₄	1	1	yes	characteristic
47	#15667952	—	2	2	no	—
48	C₂×#13083549	—	2	2	no	—
49	C₂×#13083549	C₂³	1	2	yes	—
50	C₂×#13083549	C₂³	1	2	yes	—
51	C₁₆×D₈₉	—	2	4	no	—
52	C₁₆×D₈₉	—	2	4	no	—
53	#15667941	C₂×C₄	1	4	yes	—
54	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
55	#15667924	C₂×C₄	1	4	yes	—
56	#15667922	C₂×C₄	1	2	yes	—
57	C₈×Dic₈₉	C₂³	1	2	yes	—
58	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
59	#131×C₈₉	—	2	2	no	—
60	#131×C₈₉	—	2	2	no	—
61	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	M₃₂×C₈₉	—	2	2	no	—
66	M₃₂×C₈₉	—	2	4	no	—
67	C₂×C₁₆×C₈₉	—	2	4	no	—
68	C₂×C₁₆×C₈₉	D₄	1	4	yes	complemented
69	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
70	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
72	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
73	C₂×C₂²⋊D₈₉	C₂×C₈	1	4	yes	—
74	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₈₉	—	2	2	no	—
76	C₂×C₄×D₈₉	—	2	2	no	—
77	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
79	#13083558	—	2	2	no	—
80	C₂×#10623494	—	2	2	no	—
81	C₂×#10623494	C₂²×C₄	1	2	yes	—
82	C₂×#10623494	C₂²×C₄	1	2	yes	—
83	C₈×D₈₉	—	2	4	no	—
84	C₈×D₈₉	—	2	2	no	—
85	C₈×D₈₉	—	2	2	no	—
86	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
87	#13083553	C₂×C₈	1	4	yes	—
88	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
89	C₂²⋊Dic₈₉	C₂×C₈	1	4	yes	—
90	#13083551	C₂×C₈	1	4	yes	—
91	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
92	#13083549	—	2	2	no	—
93	#13083549	—	2	2	no	—
94	#41×C₈₉	—	2	2	no	—
95	#41×C₈₉	—	2	2	no	—
96	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
97	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
98	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	M₁₆×C₈₉	—	2	2	no	—
102	M₁₆×C₈₉	—	2	4	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₈₉	#41	1	2	yes	—
109	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₁₆×C₈₉	—	2	2	no	—
111	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
112	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
113	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
114	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
115	C₂²×D₈₉	—	2	2	no	—
116	C₂²×D₈₉	—	2	4	no	—
117	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
118	Q₈×C₈₉	—	2	2	no	—
119	D₄×C₈₉	—	2	4	no	—
120	D₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	#131	1	2	yes	—
127	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	2	8	no	—
129	C₂×Dic₈₉	—	2	2	no	—
130	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
131	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
132	C₄×D₈₉	—	2	2	no	—
133	C₄×D₈₉	—	2	2	no	—
134	C₄×D₈₉	—	2	4	no	—
135	C₈×C₈₉	—	2	2	no	—
136	C₈×C₈₉	C₄×D₄	1	2	yes	—
137	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
139	#10623494	—	2	2	no	—
140	#10623494	—	2	2	no	—
141	C₂²×C₈₉	—	2	2	no	—
142	C₂²×C₈₉	—	2	4	no	—
143	C₂²×C₈₉	—	2	2	no	—
144	C₂²×C₈₉	C₈×D₄	1	2	yes	—
145	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
146	C₂×D₈₉	—	2	8	no	—
147	C₂×D₈₉	—	2	2	no	—
148	C₂×D₈₉	—	2	2	no	—
149	C₂×D₈₉	#504	1	4	yes	complemented
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	C₈×D₄	1	2	yes	—
152	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
154	Dic₈₉	—	2	4	no	—
155	Dic₈₉	—	2	2	no	—
156	#33432	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	2	2	no	—
158	C₂×C₈₉	—	4	2	no	—
159	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#504	1	2	yes	—
161	D₈₉	—	2	4	no	—
162	C₂×#504	D₈₉	1	1	yes	characteristic, complemented
163	C₂³×C₁₆	—	89	1	no	—
164	#2929	—	89	1	no	—
165	#2168	—	89	4	no	—
166	D₄×C₁₆	—	89	4	no	—
167	#2151	—	89	1	no	—
168	#2113	—	89	1	no	—
169	#2112	—	89	1	no	—
170	#2108	—	89	1	no	—
171	C₈₉	#33432	1	1	yes	characteristic, complemented, Sylow
172	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
173	C₂³×C₈	—	89	1	no	—
174	#525	—	89	1	no	—
175	#504	—	2	4	no	—
176	C₂×M₃₂	—	89	1	no	—
177	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
178	C₂×M₃₂	C₂×D₈₉	1	2	yes	complemented
179	C₂²×C₁₆	—	89	1	no	—
180	C₂²×C₁₆	—	89	2	no	—
181	C₂²×C₁₆	—	89	2	no	—
182	C₂²×C₁₆	—	178	4	no	—
183	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
184	C₂²×C₁₆	C₂×D₈₉	1	2	yes	complemented
185	#435	—	89	4	no	—
186	C₈×D₄	—	89	4	no	—
187	#424	—	89	1	no	—
188	#408	—	89	1	no	—
189	#407	—	89	1	no	—
190	#405	—	89	1	no	—
191	C₄⋊C₁₆	—	89	4	no	—
192	C₂²⋊C₁₆	—	89	4	no	—
193	C₂²⋊C₁₆	—	89	4	no	—
194	#346	—	89	2	no	—
195	C₄×C₁₆	—	89	2	no	—
196	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
197	C₂³×C₄	—	89	1	no	—
198	#131	—	2	2	no	—
199	#131	—	2	2	no	—
200	C₂×M₁₆	—	89	1	no	—
201	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
202	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
203	C₂²×C₈	—	89	1	no	—
204	C₂²×C₈	—	89	2	no	—
205	C₂²×C₈	—	89	2	no	—
206	C₂²×C₈	—	178	2	no	—
207	C₂²×C₈	—	178	2	no	—
208	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
209	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
210	#123	—	89	2	no	—
211	C₂²⋊Q₈	—	89	2	no	—
212	#121	—	89	2	no	—
213	#120	—	89	2	no	—
214	C₄×D₄	—	89	4	no	—
215	#117	—	89	1	no	—
216	M₃₂	—	2	2	no	—
217	M₃₂	—	2	4	no	—
218	M₃₂	—	178	2	no	—
219	C₂×C₁₆	—	2	4	no	—
220	C₂×C₁₆	—	89	2	no	—
221	C₂×C₁₆	—	89	2	no	—
222	C₂×C₁₆	—	178	2	no	—
223	C₂×C₁₆	—	178	4	no	—
224	C₂×C₁₆	—	178	4	no	—
225	C₂×C₁₆	C₂²⋊D₈₉	1	4	yes	complemented
226	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
227	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
228	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
229	C₄⋊C₈	—	89	4	no	—
230	C₂²⋊C₈	—	89	4	no	—
231	C₂²⋊C₈	—	89	4	no	—
232	#97	—	89	2	no	—
233	C₄×C₈	—	89	2	no	—
234	C₂⁴	—	89	1	no	—
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
238	C₂×D₄	—	89	4	no	—
239	C₂×D₄	C₈×D₈₉	1	2	yes	—
240	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
241	C₂²×C₄	—	89	1	no	—
242	C₂²×C₄	—	89	2	no	—
243	C₂²×C₄	—	89	2	no	—
244	C₂²×C₄	—	178	2	no	—
245	C₂²×C₄	—	178	2	no	—
246	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	178	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	89	2	no	—
254	C₂×C₈	—	89	2	no	—
255	C₂×C₈	—	178	2	no	—
256	C₂×C₈	—	178	4	no	—
257	C₂×C₈	—	178	2	no	—
258	C₂×C₈	—	178	2	no	—
259	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
260	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
262	C₂×C₈	#13083574	1	2	yes	—
263	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	—	89	2	no	—
265	C₄⋊C₄	—	89	4	no	—
266	C₂²⋊C₄	—	89	2	no	—
267	C₂²⋊C₄	—	89	4	no	—
268	C₂²⋊C₄	—	89	4	no	—
269	C₄²	—	89	2	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	178	2	no	—
272	C₁₆	—	178	2	no	—
273	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
274	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
275	C₂³	—	89	2	no	—
276	C₂³	—	178	4	no	—
277	C₂³	—	178	2	no	—
278	C₂³	C₂×C₈×D₈₉	1	2	yes	—
279	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	D₄	—	2	4	no	—
282	D₄	—	2	2	no	—
283	D₄	—	178	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	89	4	no	—
287	C₂×C₄	—	89	2	no	—
288	C₂×C₄	—	178	2	no	—
289	C₂×C₄	—	178	4	no	—
290	C₂×C₄	—	178	2	no	—
291	C₂×C₄	—	178	2	no	—
292	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
293	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
294	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
295	C₂×C₄	#15668018	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	178	2	no	—
299	C₈	—	178	2	no	—
300	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
301	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
302	C₈	C₂×#13083574	1	1	yes	characteristic
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	89	4	no	—
307	C₂²	—	178	8	no	—
308	C₂²	—	178	2	no	—
309	C₂²	—	178	2	no	—
310	C₂²	C₈×C₂²⋊D₈₉	1	2	yes	—
311	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	178	2	no	—
314	C₄	—	178	4	no	—
315	C₄	C₈×C₂²⋊D₈₉	1	2	yes	—
316	C₄	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
317	C₄	C₂×#15668018	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	178	4	no	—
321	C₂	C₂×C₈×C₂²⋊D₈₉	1	1	yes	characteristic
322	C₂	#504×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#504×C₈₉	C₂
prime radical	#19152105	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#504	D₈₉
hypocenter	C₈₉	#33432
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆×D₈₉	C₂	semidirect
3. #26022075	C₂	semidirect
4. C₁₆×C₂²⋊D₈₉	C₂	semidirect
5. #26022012	C₂	semidirect
6. #26021866	C₂	semidirect
7. #26021862	C₂	semidirect
8. #26021848	C₂	semidirect
9. C₂×#504×C₈₉	C₂	semidirect
23. #19151416	C₂²	semidirect
24. C₂²⋊#13083549	C₂²	semidirect
26. #19151373	C₂²	semidirect
27. C₁₆×Dic₈₉	C₂²	semidirect
31. C₂×M₃₂×C₈₉	C₂²	semidirect
33. C₂²×C₁₆×C₈₉	C₂²	semidirect
68. C₂×C₁₆×C₈₉	D₄	semidirect
149. C₂×D₈₉	#504	semidirect
162. C₂×#504	D₈₉	semidirect
171. C₈₉	#33432	semidirect
178. C₂×M₃₂	C₂×D₈₉	semidirect
184. C₂²×C₁₆	C₂×D₈₉	semidirect
225. C₂×C₁₆	C₂²⋊D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₈₉	2
#26028944	2
#26022075	2
C₁₆×C₂²⋊D₈₉	2
#26022012	2
#26021866	2
#26021862	2
#26021848	2
C₂×#504×C₈₉	2
#33432	89