Subgroups — #36537539

Order 22784.

674 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	#26031179	C₂	1	2	yes	maximal
2	C₂×#19152003	C₂	1	1	yes	maximal, characteristic, complemented
3	#26028965	C₂	1	1	yes	maximal, characteristic
4	#26019766	C₂	1	4	yes	maximal, complemented
5	#26019642	C₂	1	4	yes	maximal, complemented
6	#26018936	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	—	2	1	no	—
9	#19152148	C₂²	1	1	yes	characteristic
10	C₂×#15668012	C₂²	1	2	yes	—
11	#19152003	—	2	1	no	—
12	#19152003	—	2	1	no	—
13	#19152003	C₂²	1	2	yes	—
14	#19152003	C₂²	1	2	yes	—
15	#19151971	—	2	2	no	—
16	#19151949	—	2	2	no	—
17	#19151804	C₂²	1	8	yes	—
18	#19151739	—	2	4	no	—
19	#19151677	—	2	1	no	—
20	#19151677	C₂²	1	2	yes	—
21	C₈×D₃₅₆	—	2	1	no	—
22	C₈×D₃₅₆	C₂²	1	2	yes	—
23	C₂×#15667946	—	2	1	no	—
24	C₂×#15667946	—	2	1	no	—
25	#19151622	C₂²	1	1	yes	characteristic
26	#19151564	—	2	1	no	—
27	#19151345	C₂²	1	8	yes	—
28	#431×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	#335×C₈₉	C₂²	1	2	yes	—
31	#334×C₈₉	C₂²	1	2	yes	—
32	C₂×#13083574	—	2	1	no	—
33	C₂²×Dic₁₇₈	—	2	2	no	—
34	C₂×#13083563	—	2	8	no	—
35	#15668018	—	4	4	no	—
36	C₂×#13083558	—	2	1	no	—
37	C₂×#13083557	—	2	2	no	—
38	C₂×C₈×D₈₉	—	2	2	no	—
39	#15668012	—	2	2	no	—
40	#15668012	—	2	2	no	—
41	#15668012	—	2	2	no	—
42	#15668012	D₄	1	1	yes	characteristic
43	#15668012	C₂³	1	4	yes	—
44	#15668009	D₄	1	1	yes	characteristic
45	#15668000	—	2	1	no	—
46	#15667986	—	2	1	no	—
47	#15667986	D₄	1	2	yes	—
48	C₄×D₃₅₆	—	2	1	no	—
49	C₄×D₃₅₆	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₈₉	—	2	2	no	—
51	C₂×C₄⋊Dic₈₉	—	2	2	no	—
52	#15667946	—	2	2	no	—
53	#15667946	—	2	2	no	—
54	#15667946	—	2	2	no	—
55	#15667946	—	2	2	no	—
56	C₄⋊#10623494	—	2	1	no	—
57	C₄⋊#10623494	C₂³	1	2	yes	—
58	#15667937	—	2	8	no	—
59	#15667924	—	2	2	no	—
60	#15667924	—	2	2	no	—
61	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
65	C₂²⋊C₈×C₈₉	—	2	2	no	—
66	C₄×C₈×C₈₉	—	2	2	no	—
67	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	#13083574	—	4	2	no	—
72	#13083574	—	4	2	no	—
73	C₂×Dic₁₇₈	—	2	1	no	—
74	C₂×Dic₁₇₈	—	2	4	no	—
75	C₂×Dic₁₇₈	—	4	2	no	—
76	C₂×Dic₁₇₈	—	8	4	no	—
77	C₂×D₃₅₆	—	2	1	no	—
78	C₂×D₃₅₆	—	2	1	no	—
79	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
80	C₂×C₂²⋊D₈₉	—	2	2	no	—
81	C₂²×Dic₈₉	—	4	2	no	—
82	C₂×C₄×D₈₉	—	2	2	no	—
83	#13083563	—	8	8	no	—
84	#13083558	—	4	2	no	—
85	#13083557	—	4	4	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₈×D₈₉	—	4	4	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	2	4	no	—
90	C₄⋊Dic₈₉	—	2	4	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	C₄⋊Dic₈₉	—	4	2	no	—
93	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
94	#13083551	—	2	2	no	—
95	#13083551	—	2	2	no	—
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
99	C₂×C₈×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	—	2	4	no	—
101	C₂×C₈×C₈₉	SD₁₆	1	4	yes	complemented
102	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₈₉	#41	1	1	yes	characteristic
109	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₂³×C₈₉	#121	1	1	yes	characteristic
111	C₂²×D₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₈₉	#137	1	2	yes	—
119	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
122	C₂²⋊D₈₉	—	4	4	no	—
123	C₂×Dic₈₉	—	2	2	no	—
124	C₂×Dic₈₉	—	4	4	no	—
125	C₂×Dic₈₉	—	8	4	no	—
126	D₃₅₆	—	2	4	no	—
127	D₃₅₆	—	4	2	no	—
128	D₃₅₆	—	4	2	no	—
129	C₄×D₈₉	—	4	2	no	—
130	C₄×D₈₉	—	4	2	no	—
131	Dic₁₇₈	—	4	2	no	—
132	Dic₁₇₈	—	4	8	no	—
133	Dic₁₇₈	—	8	4	no	—
134	C₈×C₈₉	—	2	1	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	—	4	4	no	—
137	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
138	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
139	#10623494	—	4	2	no	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	#461	1	2	yes	—
143	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
144	C₂×D₈₉	—	4	2	no	—
145	C₂×D₈₉	—	4	2	no	—
146	C₂×D₈₉	—	4	2	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	#461	1	2	yes	—
150	C₄×C₈₉	#473	1	2	yes	—
151	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
153	C₄×C₈₉	#576	1	2	yes	—
154	Dic₈₉	—	4	2	no	—
155	Dic₈₉	—	8	4	no	—
156	#19192	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	2	2	no	—
158	C₂×C₈₉	C₂×#461	1	1	yes	characteristic
159	C₂×C₈₉	#3058	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
161	D₈₉	—	8	2	no	—
162	#3206	—	89	2	no	—
163	C₂×#500	—	89	1	no	—
164	#2931	—	89	1	no	—
165	#1693	—	89	4	no	—
166	#1673	—	89	4	no	—
167	#1564	D₈₉	1	1	yes	characteristic, complemented
168	#1549	—	89	2	no	—
169	C₈₉	#19192	1	1	yes	characteristic, complemented, Sylow
170	C₂×#131	—	178	1	no	—
171	#532	—	89	1	no	—
172	C₂×C₄⋊Q₈	—	89	2	no	—
173	#500	—	89	2	no	—
174	#500	—	89	2	no	—
175	#500	—	178	1	no	—
176	#500	—	178	1	no	—
177	#492	—	178	2	no	—
178	#487	—	178	2	no	—
179	#461	—	89	8	no	—
180	#448	—	178	4	no	—
181	#436	—	89	2	no	—
182	#436	—	178	1	no	—
183	C₈×D₄	—	89	2	no	—
184	C₈×D₄	—	178	1	no	—
185	#431	C₂×D₈₉	1	2	yes	—
186	C₂×#107	—	178	1	no	—
187	C₂×#107	—	178	1	no	—
188	#423	—	89	1	no	—
189	#408	—	178	1	no	—
190	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
191	#335	C₂×D₈₉	1	2	yes	—
192	#334	C₂×D₈₉	1	2	yes	—
193	#332	—	89	8	no	—
194	C₂×#41	—	178	1	no	—
195	C₂²×Q₈	—	178	2	no	—
196	C₂×SD₁₆	—	178	8	no	—
197	#131	—	356	4	no	—
198	C₂×M₁₆	—	178	1	no	—
199	C₂×M₁₆	—	178	2	no	—
200	C₂²×C₈	—	178	2	no	—
201	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
202	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
203	C₄⋊Q₈	—	89	1	no	—
204	C₄⋊Q₈	—	89	4	no	—
205	C₄⋊Q₈	—	178	2	no	—
206	C₄⋊Q₈	—	178	2	no	—
207	C₄⋊Q₈	—	178	2	no	—
208	#127	—	89	1	no	—
209	#124	—	178	1	no	—
210	#121	—	89	2	no	—
211	#121	—	178	1	no	—
212	C₄×D₄	—	89	2	no	—
213	C₄×D₄	—	178	1	no	—
214	C₂×C₄⋊C₄	—	178	2	no	—
215	C₂×C₄⋊C₄	—	178	2	no	—
216	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
217	#107	—	178	2	no	—
218	#107	—	178	2	no	—
219	#107	—	178	2	no	—
220	#107	—	178	2	no	—
221	C₄⋊C₈	—	89	2	no	—
222	C₄⋊C₈	—	178	1	no	—
223	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
224	#103	—	178	8	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	178	2	no	—
227	C₂²⋊C₈	—	178	2	no	—
228	C₄×C₈	—	2	2	no	—
229	C₄×C₈	D₃₅₆	1	1	yes	characteristic
230	C₄×C₈	D₃₅₆	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
233	#41	—	356	2	no	—
234	#41	—	356	2	no	—
235	C₂×Q₈	—	356	2	no	—
236	C₂×Q₈	—	178	4	no	—
237	C₂×Q₈	—	178	1	no	—
238	C₂×Q₈	—	712	4	no	—
239	C₂×D₄	—	89	2	no	—
240	C₂×D₄	—	178	2	no	—
241	C₂×D₄	—	178	1	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂²×C₄	—	356	2	no	—
244	C₂²×C₄	—	178	2	no	—
245	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
247	C₂²×C₄	#13083575	1	1	yes	characteristic
248	SD₁₆	—	712	8	no	—
249	M₁₆	—	356	2	no	—
250	M₁₆	—	356	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	356	4	no	—
254	C₂×C₈	—	178	2	no	—
255	C₂×C₈	#13083563	1	4	yes	complemented
256	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
257	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
258	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
259	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
260	C₄⋊C₄	—	89	2	no	—
261	C₄⋊C₄	—	356	2	no	—
262	C₄⋊C₄	—	356	2	no	—
263	C₄⋊C₄	—	178	4	no	—
264	C₄⋊C₄	—	178	4	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₂²⋊C₄	—	178	2	no	—
267	C₂²⋊C₄	—	178	2	no	—
268	C₄²	D₄×D₈₉	1	1	yes	characteristic
269	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
270	C₄²	#13083575	1	1	yes	characteristic
271	C₄²	C₂³×D₈₉	1	1	yes	characteristic
272	C₂³	—	178	2	no	—
273	C₂³	#15667987	1	1	yes	characteristic
274	Q₈	—	356	2	no	—
275	Q₈	—	356	8	no	—
276	Q₈	—	712	4	no	—
277	D₄	—	356	4	no	—
278	D₄	—	356	2	no	—
279	D₄	—	356	2	no	—
280	D₄	—	178	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	356	2	no	—
284	C₂×C₄	—	356	2	no	—
285	C₂×C₄	—	356	4	no	—
286	C₂×C₄	—	178	2	no	—
287	C₂×C₄	—	712	4	no	—
288	C₂×C₄	#15667987	1	1	yes	characteristic
289	C₂×C₄	#15667987	1	1	yes	characteristic
290	C₂×C₄	#15667987	1	1	yes	characteristic
291	C₂×C₄	C₂×#13083563	1	2	yes	—
292	C₂×C₄	#15668045	1	2	yes	—
293	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
295	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	4	4	no	—
299	C₈	—	356	2	no	—
300	C₈	C₂×#13083563	1	2	yes	—
301	C₈	C₂×#13083563	1	2	yes	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	356	2	no	—
305	C₂²	—	356	2	no	—
306	C₂²	—	356	2	no	—
307	C₂²	#19151804	1	2	yes	—
308	C₂²	C₂×#15667987	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	356	2	no	—
312	C₄	—	712	4	no	—
313	C₄	#19151804	1	2	yes	—
314	C₄	#19151874	1	2	yes	—
315	C₄	C₂²×#13083563	1	1	yes	characteristic
316	C₄	C₂×#15668045	1	1	yes	characteristic
317	C₄	#19152400	1	2	yes	—
318	C₂	—	2	2	no	—
319	C₂	—	712	2	no	—
320	C₂	C₂×#19151804	1	1	yes	characteristic
321	C₂	#26029901	1	1	yes	characteristic
322	C₂	C₂×#19151874	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	#19152148	C₂²
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#19192
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. C₂×#19152003	C₂	semidirect
4. #26019766	C₂	semidirect
5. #26019642	C₂	semidirect
6. #26018936	C₂	semidirect
7. C₈₉×#1564	C₂	semidirect
101. C₂×C₈×C₈₉	SD₁₆	semidirect
167. #1564	D₈₉	semidirect
169. C₈₉	#19192	semidirect
255. C₂×C₈	#13083563	semidirect

Maximal subgroups

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

subgroup	index
#26031179	2
C₂×#19152003	2
#26028965	2
#26019766	2
#26019642	2
#26018936	2
C₈₉×#1564	2
#19192	89