Subgroups — #36447147

Order 22784.

472 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	#26022729	C₂	1	1	yes	maximal, characteristic
2	#26020369	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151330	C₂	1	2	yes	maximal
4	#26018092	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	#19151504	C₄	1	1	yes	characteristic
9	#19151504	C₄	1	1	yes	characteristic
10	#19151496	C₄	1	1	yes	characteristic
11	#19151496	C₄	1	1	yes	characteristic
12	#19151482	—	2	1	no	—
13	#19151482	—	2	1	no	—
14	C₄×#13083553	—	2	1	no	—
15	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
16	#19151444	—	2	1	no	—
17	#19151353	—	2	1	no	—
18	#19151353	—	2	1	no	—
19	#19151330	—	2	4	no	—
20	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
22	C₄³×C₈₉	C₂²	1	1	yes	characteristic
23	#336×C₈₉	—	2	1	no	—
24	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10623494	—	2	2	no	—
26	C₄×Dic₁₇₈	—	2	1	no	—
27	C₄×Dic₁₇₈	—	2	1	no	—
28	C₄×Dic₁₇₈	D₄	1	2	yes	—
29	C₄×Dic₁₇₈	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×#13083553	—	2	1	no	—
34	C₂×#13083553	—	2	1	no	—
35	C₂×#13083553	—	2	1	no	—
36	C₂×C₄×Dic₈₉	—	2	1	no	—
37	C₂×C₄×Dic₈₉	—	2	1	no	—
38	C₄⋊#10623494	—	2	2	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	#15667919	—	2	1	no	—
41	#15667919	—	2	1	no	—
42	#15667919	—	2	1	no	—
43	#15667919	—	2	1	no	—
44	#15667919	—	2	1	no	—
45	#15667919	—	2	1	no	—
46	#15667919	—	2	1	no	—
47	C₂²×C₈×C₈₉	—	2	2	no	—
48	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	—	2	4	no	—
56	C₄×C₈×C₈₉	—	2	2	no	—
57	C₄×C₈×C₈₉	—	2	2	no	—
58	C₂×Dic₁₇₈	—	2	2	no	—
59	C₂×Dic₁₇₈	—	2	2	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₄⋊C₄	1	2	yes	—
62	C₂²×Dic₈₉	—	2	1	no	—
63	C₂²×Dic₈₉	—	2	1	no	—
64	C₂²×Dic₈₉	—	2	1	no	—
65	C₂²×Dic₈₉	—	2	1	no	—
66	C₂×#10623494	—	4	2	no	—
67	C₂×#10623494	—	4	2	no	—
68	C₂×#10623494	—	4	2	no	—
69	C₄⋊Dic₈₉	—	2	1	no	—
70	C₄⋊Dic₈₉	—	2	2	no	—
71	C₄⋊Dic₈₉	—	2	2	no	—
72	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
73	#13083553	—	2	2	no	—
74	#13083553	—	2	2	no	—
75	#13083553	—	4	2	no	—
76	#13083553	—	4	2	no	—
77	C₄×Dic₈₉	—	2	2	no	—
78	C₄×Dic₈₉	—	2	2	no	—
79	C₄×Dic₈₉	—	4	2	no	—
80	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	—	2	4	no	—
88	C₂×C₈×C₈₉	—	4	2	no	—
89	C₂×C₈×C₈₉	—	4	2	no	—
90	C₄²×C₈₉	—	2	2	no	—
91	C₄²×C₈₉	—	2	2	no	—
92	C₄²×C₈₉	—	2	2	no	—
93	C₄²×C₈₉	—	2	2	no	—
94	C₄²×C₈₉	—	2	2	no	—
95	C₄²×C₈₉	—	4	1	no	—
96	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₈₉	#41	1	1	yes	characteristic
99	C₄²×C₈₉	#41	1	1	yes	characteristic
100	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
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	#104	1	2	yes	—
115	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#131	1	2	yes	—
122	C₂×C₄×C₈₉	#131	1	2	yes	—
123	C₂×C₄×C₈₉	#135	1	2	yes	—
124	C₂×C₄×C₈₉	#137	1	2	yes	—
125	C₂×Dic₈₉	—	2	2	no	—
126	C₂×Dic₈₉	—	2	2	no	—
127	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	—	4	1	no	—
129	C₂×Dic₈₉	—	4	1	no	—
130	C₂×Dic₈₉	—	4	1	no	—
131	C₂×Dic₈₉	—	4	1	no	—
132	C₂×Dic₈₉	—	4	1	no	—
133	C₂×Dic₈₉	—	4	1	no	—
134	C₂×Dic₈₉	—	4	1	no	—
135	C₂×Dic₈₉	—	4	1	no	—
136	C₂×Dic₈₉	—	4	1	no	—
137	Dic₁₇₈	—	2	4	no	—
138	Dic₁₇₈	—	4	2	no	—
139	C₈×C₈₉	—	4	4	no	—
140	#10623494	—	8	2	no	—
141	C₂²×C₈₉	#380	1	1	yes	characteristic
142	C₂²×C₈₉	#408	1	1	yes	characteristic
143	C₂²×C₈₉	#408	1	1	yes	characteristic
144	C₂²×C₈₉	#416	1	1	yes	characteristic
145	C₂²×C₈₉	#419	1	1	yes	characteristic
146	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
147	C₂²×C₈₉	#421	1	1	yes	characteristic
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	—	4	1	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	—	4	1	no	—
154	C₄×C₈₉	#424	1	2	yes	—
155	C₄×C₈₉	#433	1	2	yes	—
156	C₄×C₈₉	#464	1	2	yes	—
157	C₄×C₈₉	#479	1	2	yes	—
158	Dic₈₉	—	4	2	no	—
159	Dic₈₉	—	4	2	no	—
160	Dic₈₉	—	8	1	no	—
161	Dic₈₉	—	8	1	no	—
162	#8570	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	#1482	1	1	yes	characteristic
164	C₂×C₈₉	#1485	1	1	yes	characteristic
165	C₂×C₈₉	#1494	1	1	yes	characteristic
166	C₂×C₈₉	#1497	1	1	yes	characteristic
167	C₂×C₈₉	#1798	1	1	yes	characteristic
168	C₂×C₈₉	#1805	1	1	yes	characteristic
169	C₂×C₈₉	#1806	1	1	yes	characteristic
170	#2295	—	89	1	no	—
171	#1831	—	89	1	no	—
172	C₄×C₄⋊C₈	D₈₉	1	1	yes	characteristic
173	C₂×Q₈⋊C₈	—	89	2	no	—
174	#1279	—	89	2	no	—
175	C₈₉	#8570	1	1	yes	characteristic, complemented, Sylow
176	C₂×C₄×Q₈	—	89	1	no	—
177	C₂×C₄⋊C₈	—	2	1	no	—
178	C₂×C₄⋊C₈	—	89	2	no	—
179	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
180	#391	—	89	1	no	—
181	#391	—	89	1	no	—
182	#389	—	89	1	no	—
183	#389	—	89	1	no	—
184	#387	—	178	1	no	—
185	#387	—	178	1	no	—
186	C₄×C₄⋊C₄	—	89	1	no	—
187	C₄×C₄⋊C₄	—	178	1	no	—
188	#376	—	178	1	no	—
189	C₄³	C₂×D₈₉	1	1	yes	characteristic
190	#336	—	2	1	no	—
191	#336	—	178	1	no	—
192	#336	—	178	1	no	—
193	Q₈⋊C₈	—	178	4	no	—
194	C₂²×Q₈	—	89	1	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	178	2	no	—
197	C₄×Q₈	—	89	2	no	—
198	C₄×Q₈	—	89	2	no	—
199	C₄×Q₈	—	178	1	no	—
200	C₄×Q₈	—	178	1	no	—
201	C₂×C₄⋊C₄	—	89	1	no	—
202	C₂×C₄⋊C₄	—	89	1	no	—
203	C₂×C₄⋊C₄	—	89	1	no	—
204	C₂×C₄⋊C₄	—	178	1	no	—
205	C₂×C₄⋊C₄	—	178	1	no	—
206	C₂×C₄⋊C₄	—	178	1	no	—
207	C₂×C₄²	—	178	1	no	—
208	C₂×C₄²	—	178	1	no	—
209	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
211	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
212	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	178	2	no	—
218	C₄⋊C₈	—	178	2	no	—
219	C₄×C₈	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	#95	—	178	1	no	—
222	#95	—	178	1	no	—
223	#95	—	178	1	no	—
224	#95	—	178	1	no	—
225	#95	—	178	1	no	—
226	#95	—	178	1	no	—
227	#95	—	178	1	no	—
228	C₂×Q₈	—	89	2	no	—
229	C₂×Q₈	—	178	2	no	—
230	C₂×Q₈	—	178	2	no	—
231	C₂×Q₈	—	178	1	no	—
232	C₂²×C₄	—	178	1	no	—
233	C₂²×C₄	—	178	1	no	—
234	C₂²×C₄	—	178	1	no	—
235	C₂²×C₄	—	178	1	no	—
236	C₂²×C₄	#13083551	1	1	yes	characteristic
237	C₂²×C₄	#13083551	1	1	yes	characteristic
238	C₂²×C₄	#13083551	1	1	yes	characteristic
239	C₂²×C₄	#13083551	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	356	2	no	—
247	C₂×C₈	—	356	2	no	—
248	C₂×C₈	—	356	2	no	—
249	C₄⋊C₄	—	89	2	no	—
250	C₄⋊C₄	—	356	2	no	—
251	C₄⋊C₄	—	356	2	no	—
252	C₄⋊C₄	—	178	2	no	—
253	C₄⋊C₄	—	178	2	no	—
254	C₄⋊C₄	—	178	1	no	—
255	C₄⋊C₄	—	178	2	no	—
256	C₄⋊C₄	—	178	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	356	2	no	—
264	C₄²	—	178	2	no	—
265	C₄²	—	178	2	no	—
266	C₄²	D₄×D₈₉	1	1	yes	characteristic
267	C₄²	Q₈×D₈₉	1	1	yes	characteristic
268	C₄²	#13083573	1	1	yes	characteristic
269	C₄²	#13083575	1	1	yes	characteristic
270	C₂³	C₂×#13083551	1	1	yes	characteristic
271	Q₈	—	356	2	no	—
272	Q₈	—	178	4	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	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	356	1	no	—
287	C₂×C₄	—	356	1	no	—
288	C₂×C₄	—	356	1	no	—
289	C₂×C₄	—	356	1	no	—
290	C₂×C₄	—	356	1	no	—
291	C₂×C₄	—	356	1	no	—
292	C₂×C₄	—	356	1	no	—
293	C₂×C₄	—	356	1	no	—
294	C₂×C₄	—	356	1	no	—
295	C₂×C₄	—	356	1	no	—
296	C₂×C₄	—	178	2	no	—
297	C₂×C₄	—	178	2	no	—
298	C₂×C₄	#15667940	1	2	yes	—
299	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
300	C₂×C₄	#15667970	1	1	yes	characteristic
301	C₂×C₄	#15667983	1	1	yes	characteristic
302	C₂×C₄	#15667987	1	1	yes	characteristic
303	C₂×C₄	#15667988	1	1	yes	characteristic
304	C₂×C₄	#15667992	1	1	yes	characteristic
305	C₂×C₄	#15668018	1	2	yes	—
306	C₂×C₄	#15668020	1	2	yes	—
307	C₂×C₄	#15668034	1	2	yes	—
308	C₂×C₄	#15668045	1	2	yes	—
309	C₈	—	4	4	no	—
310	C₈	—	712	2	no	—
311	C₂²	#19151455	1	1	yes	characteristic
312	C₂²	#19151564	1	1	yes	characteristic
313	C₂²	#19151566	1	1	yes	characteristic
314	C₂²	#19151595	1	1	yes	characteristic
315	C₂²	#19151608	1	1	yes	characteristic
316	C₂²	C₂×#15667940	1	1	yes	characteristic
317	C₂²	#19151616	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	356	2	no	—
325	C₄	—	356	2	no	—
326	C₄	—	712	1	no	—
327	C₄	—	712	1	no	—
328	C₄	#19151623	1	2	yes	—
329	C₄	#19151658	1	2	yes	—
330	C₄	#19151821	1	2	yes	—
331	C₄	#19151906	1	2	yes	—
332	C₂	#26018661	1	1	yes	characteristic
333	C₂	#26018674	1	1	yes	characteristic
334	C₂	#26018713	1	1	yes	characteristic
335	C₂	#26018726	1	1	yes	characteristic
336	C₂	#26020233	1	1	yes	characteristic
337	C₂	#26020268	1	1	yes	characteristic
338	C₂	#26020273	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#8570
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
175. C₈₉	#8570	semidirect

Maximal subgroups

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

subgroup	index
#26022729	2
#26020369	2
C₂×#19151330	2
#26018092	2
C₄×C₄⋊C₈×C₈₉	2
#8570	89