Subgroups — #36470661

Order 22784.

416 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028670	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151380	C₂	1	1	yes	maximal, characteristic
3	#26018339	C₂	1	2	yes	maximal, complemented
4	#26018337	C₂	1	2	yes	maximal, complemented
5	C₂×#349×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₄	1	1	yes	characteristic
7	C₂×#15667971	C₄	1	1	yes	characteristic
8	C₂×#15667952	C₂²	1	2	yes	—
9	#19151556	—	2	1	no	—
10	#19151556	—	2	1	no	—
11	#19151553	C₄	1	1	yes	characteristic
12	#19151553	C₄	1	1	yes	characteristic
13	#19151553	C₂²	1	1	yes	characteristic
14	#19151553	C₂²	1	1	yes	characteristic
15	M₁₆×Dic₈₉	—	2	1	no	—
16	M₁₆×Dic₈₉	—	2	1	no	—
17	#19151380	—	2	1	no	—
18	#19151380	—	2	1	no	—
19	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
21	#349×C₈₉	—	2	1	no	—
22	#349×C₈₉	—	2	1	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	2	1	no	—
26	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
27	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
28	C₂²×#10623494	—	2	1	no	—
29	#15667971	—	2	1	no	—
30	#15667971	—	2	1	no	—
31	#15667971	C₈	1	2	yes	—
32	#15667971	C₂×C₄	1	1	yes	characteristic
33	#15667971	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₈₉	—	2	1	no	—
35	C₂×C₂²⋊Dic₈₉	C₈	1	2	yes	—
36	C₂×C₄×Dic₈₉	—	2	1	no	—
37	#15667952	—	2	2	no	—
38	#15667952	—	2	2	no	—
39	C₂×#13083549	—	2	2	no	—
40	#15667922	—	2	1	no	—
41	#15667922	—	2	1	no	—
42	#15667922	—	2	1	no	—
43	#15667922	—	2	1	no	—
44	C₈×Dic₈₉	—	2	1	no	—
45	C₈×Dic₈₉	—	2	1	no	—
46	C₈×Dic₈₉	—	2	1	no	—
47	C₈×Dic₈₉	—	2	1	no	—
48	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂×M₁₆×C₈₉	—	2	2	no	—
50	C₂×M₁₆×C₈₉	—	2	2	no	—
51	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
52	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	M₃₂×C₈₉	—	2	2	no	—
58	M₃₂×C₈₉	—	2	2	no	—
59	C₂×C₁₆×C₈₉	—	2	2	no	—
60	C₂²×Dic₈₉	—	2	2	no	—
61	#13083558	—	2	4	no	—
62	#13083558	—	2	1	no	—
63	#13083558	—	2	1	no	—
64	#13083558	M₁₆	1	2	yes	—
65	#13083558	M₁₆	1	2	yes	—
66	C₂×#10623494	—	2	4	no	—
67	C₂×#10623494	—	2	1	no	—
68	C₂×#10623494	—	2	1	no	—
69	C₄⋊Dic₈₉	—	2	2	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₂²⋊Dic₈₉	—	2	2	no	—
73	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
74	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
75	C₄×Dic₈₉	—	2	2	no	—
76	C₄×Dic₈₉	—	2	1	no	—
77	C₄×Dic₈₉	—	2	1	no	—
78	#13083549	—	4	2	no	—
79	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	—	2	1	no	—
81	C₂²×C₄×C₈₉	—	2	1	no	—
82	C₂²×C₄×C₈₉	—	2	1	no	—
83	C₂²×C₄×C₈₉	—	2	1	no	—
84	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	M₁₆×C₈₉	—	2	2	no	—
91	M₁₆×C₈₉	—	2	2	no	—
92	M₁₆×C₈₉	—	2	2	no	—
93	M₁₆×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	#41	1	2	yes	—
101	C₂×C₈×C₈₉	#41	1	2	yes	—
102	C₁₆×C₈₉	—	4	2	no	—
103	C₂³×C₈₉	—	2	1	no	—
104	C₂³×C₈₉	—	2	1	no	—
105	C₂³×C₈₉	—	8	1	no	—
106	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
107	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
127	C₂×Dic₈₉	—	2	2	no	—
128	C₂×Dic₈₉	—	2	2	no	—
129	C₂×Dic₈₉	—	8	2	no	—
130	C₈×C₈₉	—	2	1	no	—
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	#123	1	2	yes	—
133	C₈×C₈₉	#123	1	2	yes	—
134	#10623494	—	2	2	no	—
135	#10623494	—	2	2	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	—	4	1	no	—
139	C₂²×C₈₉	—	4	1	no	—
140	C₂²×C₈₉	—	4	1	no	—
141	C₂²×C₈₉	—	8	1	no	—
142	C₂²×C₈₉	—	8	1	no	—
143	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₂²×C₈₉	#431	1	1	yes	characteristic
145	C₂²×C₈₉	#431	1	1	yes	characteristic
146	C₄×C₈₉	—	2	1	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	#381	1	1	yes	characteristic
149	C₄×C₈₉	#407	1	1	yes	characteristic
150	C₄×C₈₉	#432	1	1	yes	characteristic
151	C₄×C₈₉	#432	1	1	yes	characteristic
152	Dic₈₉	—	4	2	no	—
153	Dic₈₉	—	4	2	no	—
154	#12851	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	—	2	1	no	—
156	C₂×C₈₉	—	2	1	no	—
157	C₂×C₈₉	—	2	1	no	—
158	C₂×C₈₉	—	4	1	no	—
159	C₂×C₈₉	—	4	1	no	—
160	C₂×C₈₉	#1817	1	1	yes	characteristic
161	#2873	—	89	1	no	—
162	C₂×#349	—	89	1	no	—
163	C₂×#349	D₈₉	1	1	yes	characteristic
164	#1378	—	89	2	no	—
165	#1378	—	89	2	no	—
166	C₈₉	#12851	1	1	yes	characteristic, complemented, Sylow
167	C₂²×M₁₆	—	89	1	no	—
168	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
169	C₂×#117	—	89	1	no	—
170	C₂×M₃₂	—	89	2	no	—
171	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
172	#405	—	89	1	no	—
173	#405	—	89	1	no	—
174	#405	—	89	1	no	—
175	#405	—	89	1	no	—
176	#405	—	178	1	no	—
177	#405	—	178	1	no	—
178	C₄×M₁₆	—	178	1	no	—
179	C₄×M₁₆	—	178	1	no	—
180	#349	—	2	1	no	—
181	#349	—	2	1	no	—
182	#349	—	178	1	no	—
183	#349	—	178	1	no	—
184	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	89	1	no	—
188	C₂×M₁₆	—	89	1	no	—
189	C₂×M₁₆	—	178	2	no	—
190	C₂×M₁₆	—	178	2	no	—
191	C₂×M₁₆	—	178	1	no	—
192	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
193	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
194	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
195	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
196	C₂²×C₈	—	178	1	no	—
197	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
199	#117	—	89	2	no	—
200	#117	—	89	1	no	—
201	#117	—	89	1	no	—
202	#117	—	178	1	no	—
203	#117	—	178	1	no	—
204	C₂×C₄⋊C₄	—	178	1	no	—
205	C₂×C₂²⋊C₄	—	89	2	no	—
206	C₂×C₄²	—	178	1	no	—
207	M₃₂	—	2	2	no	—
208	M₃₂	—	2	2	no	—
209	M₃₂	—	178	2	no	—
210	M₃₂	—	178	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	—	178	2	no	—
213	#97	—	178	1	no	—
214	#97	—	178	1	no	—
215	#97	—	178	1	no	—
216	#97	—	178	1	no	—
217	C₄×C₈	—	178	1	no	—
218	C₄×C₈	—	178	1	no	—
219	C₄×C₈	—	178	1	no	—
220	C₄×C₈	—	178	1	no	—
221	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	178	2	no	—
227	C₂²×C₄	#13083551	1	1	yes	characteristic
228	C₂²×C₄	#13083551	1	1	yes	characteristic
229	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
230	C₂²×C₄	#13083557	1	1	yes	characteristic
231	C₂²×C₄	#13083557	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	89	2	no	—
238	M₁₆	—	89	2	no	—
239	M₁₆	—	178	4	no	—
240	M₁₆	—	178	1	no	—
241	M₁₆	—	178	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	178	4	no	—
245	C₂×C₈	—	178	1	no	—
246	C₂×C₈	—	178	1	no	—
247	C₂×C₈	#13083551	1	1	yes	characteristic
248	C₂×C₈	#13083551	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
250	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
251	C₂×C₈	#13083573	1	2	yes	—
252	C₂×C₈	#13083573	1	2	yes	—
253	C₄⋊C₄	—	178	1	no	—
254	C₄⋊C₄	—	178	1	no	—
255	C₄⋊C₄	—	178	2	no	—
256	C₂²⋊C₄	—	89	2	no	—
257	C₂²⋊C₄	—	89	2	no	—
258	C₂²⋊C₄	—	178	2	no	—
259	C₄²	—	178	2	no	—
260	C₄²	—	178	1	no	—
261	C₄²	—	178	1	no	—
262	C₁₆	—	4	2	no	—
263	C₁₆	—	356	2	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	8	1	no	—
267	C₂³	#15667924	1	1	yes	characteristic
268	C₂³	#15667924	1	1	yes	characteristic
269	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	178	2	no	—
281	C₂×C₄	—	178	2	no	—
282	C₂×C₄	—	712	2	no	—
283	C₂×C₄	#15667924	1	1	yes	characteristic
284	C₂×C₄	#15667924	1	1	yes	characteristic
285	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
286	C₂×C₄	#15667968	1	1	yes	characteristic
287	C₂×C₄	#15667968	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
289	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
290	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	178	2	no	—
294	C₈	—	178	2	no	—
295	C₈	#15667993	1	2	yes	—
296	C₈	#15667996	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	8	1	no	—
303	C₂²	—	8	1	no	—
304	C₂²	C₂×#15667924	1	1	yes	characteristic
305	C₂²	#19151648	1	1	yes	characteristic
306	C₂²	#19151648	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	356	2	no	—
310	C₄	—	356	2	no	—
311	C₄	#19151459	1	1	yes	characteristic
312	C₄	#19151561	1	1	yes	characteristic
313	C₄	#19151652	1	1	yes	characteristic
314	C₄	#19151652	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	4	1	no	—
319	C₂	—	4	1	no	—
320	C₂	#26020318	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151459
commutator	C₂³×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#349×C₈₉	C₂
socle	C₂×C₈₉	#1817
prime radical	C₂⁴×C₈₉	C₂×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#349	D₈₉
hypocenter	C₈₉	#12851
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
3. #26018339	C₂	semidirect
4. #26018337	C₂	semidirect
166. C₈₉	#12851	semidirect

Maximal subgroups

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

subgroup	index
#26028670	2
C₂×#19151380	2
#26018339	2
#26018337	2
C₂×#349×C₈₉	2
#12851	89