Subgroups — #36447642

Order 22784.

434 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020416	C₂	1	1	yes	maximal, characteristic, complemented
2	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018812	C₂	1	2	yes	maximal
4	#26018112	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2931	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	C₂×#15667946	—	2	1	no	—
8	C₂×#15667944	—	2	1	no	—
9	#19151464	C₂²	1	2	yes	—
10	#19151351	—	2	1	no	—
11	#19151349	—	2	1	no	—
12	#19151332	—	2	4	no	—
13	C₂×#131×C₈₉	—	2	1	no	—
14	#532×C₈₉	C₂²	1	1	yes	characteristic
15	#436×C₈₉	—	2	1	no	—
16	#436×C₈₉	C₄	1	2	yes	complemented
17	C₈×D₄×C₈₉	—	2	1	no	—
18	C₈×D₄×C₈₉	C₄	1	2	yes	complemented
19	#423×C₈₉	C₂²	1	1	yes	characteristic
20	#408×C₈₉	—	2	1	no	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₈₉	—	2	2	no	—
23	#15667946	—	4	2	no	—
24	#15667944	—	4	2	no	—
25	C₄⋊#10623494	—	2	2	no	—
26	C₂²⋊#10623494	—	2	2	no	—
27	C₄×#10623494	—	2	2	no	—
28	#15667919	—	2	2	no	—
29	#15667919	—	2	2	no	—
30	C₂×#41×C₈₉	—	2	1	no	—
31	#131×C₈₉	—	4	2	no	—
32	#131×C₈₉	—	4	2	no	—
33	C₂×M₁₆×C₈₉	—	2	1	no	—
34	C₂×M₁₆×C₈₉	—	2	2	no	—
35	C₂²×C₈×C₈₉	—	2	2	no	—
36	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₄⋊Q₈×C₈₉	Q₈	1	1	yes	characteristic
39	#127×C₈₉	Q₈	1	1	yes	characteristic
40	#124×C₈₉	—	2	1	no	—
41	#121×C₈₉	—	2	1	no	—
42	#121×C₈₉	D₄	1	2	yes	—
43	C₄×D₄×C₈₉	—	2	1	no	—
44	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₈₉	—	2	1	no	—
47	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₂²⋊C₈×C₈₉	—	2	2	no	—
49	C₂²⋊C₈×C₈₉	—	2	2	no	—
50	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₂²×Dic₈₉	—	4	2	no	—
55	C₂×#10623494	—	4	2	no	—
56	C₄⋊Dic₈₉	—	4	2	no	—
57	C₄⋊Dic₈₉	—	4	2	no	—
58	#41×C₈₉	—	4	1	no	—
59	#41×C₈₉	—	4	1	no	—
60	#41×C₈₉	—	4	1	no	—
61	#41×C₈₉	—	4	1	no	—
62	C₂×Q₈×C₈₉	—	2	1	no	—
63	C₂×D₄×C₈₉	—	2	2	no	—
64	C₂×D₄×C₈₉	—	2	1	no	—
65	C₂×D₄×C₈₉	—	2	1	no	—
66	C₂×D₄×C₈₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₈₉	—	2	2	no	—
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
71	M₁₆×C₈₉	—	4	1	no	—
72	M₁₆×C₈₉	—	4	1	no	—
73	M₁₆×C₈₉	—	4	4	no	—
74	C₂×C₈×C₈₉	—	2	1	no	—
75	C₂×C₈×C₈₉	—	2	1	no	—
76	C₂×C₈×C₈₉	—	2	2	no	—
77	C₂×C₈×C₈₉	—	4	4	no	—
78	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	D₈	1	2	yes	—
83	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
84	C₄⋊C₄×C₈₉	—	2	1	no	—
85	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
86	C₂²⋊C₄×C₈₉	—	2	2	no	—
87	C₂²⋊C₄×C₈₉	—	2	2	no	—
88	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₈₉	#41	1	1	yes	characteristic
92	C₂³×C₈₉	—	2	2	no	—
93	C₂³×C₈₉	#121	1	1	yes	characteristic
94	Q₈×C₈₉	—	4	1	no	—
95	Q₈×C₈₉	—	4	1	no	—
96	D₄×C₈₉	—	4	2	no	—
97	D₄×C₈₉	—	4	2	no	—
98	D₄×C₈₉	—	4	2	no	—
99	D₄×C₈₉	—	4	2	no	—
100	D₄×C₈₉	—	4	1	no	—
101	D₄×C₈₉	—	4	1	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	4	1	no	—
104	C₂×C₄×C₈₉	—	4	1	no	—
105	C₂×C₄×C₈₉	—	4	2	no	—
106	C₂×C₄×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
117	C₂×Dic₈₉	—	8	2	no	—
118	C₂×Dic₈₉	—	8	2	no	—
119	C₂×Dic₈₉	—	8	2	no	—
120	C₈×C₈₉	—	2	1	no	—
121	C₈×C₈₉	—	2	1	no	—
122	C₈×C₈₉	—	4	2	no	—
123	C₈×C₈₉	#102	1	2	yes	—
124	C₈×C₈₉	#102	1	2	yes	—
125	#10623494	—	8	2	no	—
126	C₂²×C₈₉	—	2	1	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	—	4	2	no	—
129	C₂²×C₈₉	—	4	2	no	—
130	C₂²×C₈₉	—	4	2	no	—
131	C₂²×C₈₉	#380	1	1	yes	characteristic
132	C₂²×C₈₉	#459	1	1	yes	characteristic
133	C₂²×C₈₉	#461	1	1	yes	characteristic
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	4	2	no	—
139	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
140	C₄×C₈₉	#417	1	1	yes	characteristic
141	C₄×C₈₉	#474	1	1	yes	characteristic
142	C₄×C₈₉	#476	1	1	yes	characteristic
143	Dic₈₉	—	16	2	no	—
144	#8676	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	2	1	no	—
146	C₂×C₈₉	—	2	1	no	—
147	C₂×C₈₉	—	8	2	no	—
148	C₂×C₈₉	#1802	1	1	yes	characteristic
149	C₂×C₈₉	#1812	1	1	yes	characteristic
150	C₂×C₈₉	#1814	1	1	yes	characteristic
151	#2931	D₈₉	1	1	yes	characteristic
152	#1846	—	89	1	no	—
153	#1564	—	89	1	no	—
154	#1517	—	89	2	no	—
155	#1286	—	89	2	no	—
156	C₈₉	#8676	1	1	yes	characteristic, complemented, Sylow
157	C₂×#131	—	2	1	no	—
158	#532	C₂×D₈₉	1	1	yes	characteristic
159	#436	—	2	1	no	—
160	#436	Dic₈₉	1	2	yes	complemented
161	C₈×D₄	—	2	1	no	—
162	C₈×D₄	Dic₈₉	1	2	yes	complemented
163	#431	—	89	2	no	—
164	C₂×#107	—	178	1	no	—
165	C₂×#106	—	178	1	no	—
166	#423	C₂×D₈₉	1	1	yes	characteristic
167	#408	—	2	1	no	—
168	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
169	#382	—	89	2	no	—
170	#335	—	178	1	no	—
171	#334	—	178	1	no	—
172	#327	—	178	4	no	—
173	C₂×#41	—	2	1	no	—
174	#131	—	4	2	no	—
175	#131	—	4	2	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
181	C₄⋊Q₈	Dic₁₇₈	1	1	yes	characteristic
182	#127	Dic₁₇₈	1	1	yes	characteristic
183	#124	—	2	1	no	—
184	#121	—	2	1	no	—
185	#121	D₃₅₆	1	2	yes	—
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
188	C₂×C₄⋊C₄	—	178	2	no	—
189	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
190	#107	—	356	2	no	—
191	#106	—	356	2	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	178	2	no	—
194	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	—	178	2	no	—
198	C₄×C₈	—	178	2	no	—
199	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
200	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
201	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
202	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
203	#95	—	178	2	no	—
204	#95	—	178	2	no	—
205	#41	—	4	1	no	—
206	#41	—	4	1	no	—
207	#41	—	4	1	no	—
208	#41	—	4	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	C₄⋊Dic₈₉	1	2	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	356	2	no	—
216	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
218	C₂²×C₄	#13083574	1	1	yes	characteristic
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	356	2	no	—
227	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
231	C₂×C₈	#13083559	1	2	yes	—
232	C₂×C₈	#13083562	1	2	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	356	2	no	—
235	C₄⋊C₄	—	356	2	no	—
236	C₄⋊C₄	C₄⋊Dic₈₉	1	2	yes	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
240	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
241	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
242	C₄²	#13083574	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	#15667986	1	1	yes	characteristic
245	Q₈	—	4	1	no	—
246	Q₈	—	4	1	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	1	no	—
252	D₄	—	4	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	712	2	no	—
259	C₂×C₄	—	712	2	no	—
260	C₂×C₄	—	712	2	no	—
261	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
263	C₂×C₄	#15667971	1	1	yes	characteristic
264	C₂×C₄	#15667986	1	1	yes	characteristic
265	C₂×C₄	#15667991	1	1	yes	characteristic
266	C₂×C₄	#15667991	1	1	yes	characteristic
267	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
268	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
269	C₂×C₄	#15668037	1	1	yes	characteristic
270	C₂×C₄	#15668042	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	2	no	—
274	C₈	—	712	2	no	—
275	C₈	#15667933	1	2	yes	—
276	C₈	#15667933	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#19151458	1	1	yes	characteristic
283	C₂²	#19151793	1	1	yes	characteristic
284	C₂²	#19151803	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	2	no	—
290	C₄	—	1424	2	no	—
291	C₄	C₂×#15667933	1	1	yes	characteristic
292	C₄	#19151599	1	1	yes	characteristic
293	C₄	#19151878	1	1	yes	characteristic
294	C₄	#19151888	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	—	8	2	no	—
298	C₂	#26020252	1	1	yes	characteristic
299	C₂	#26020298	1	1	yes	characteristic
300	C₂	#26020306	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151458
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2931	C₂
socle	C₂²×C₈₉	#380
prime radical	#532×C₈₉	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#2931	D₈₉
hypocenter	C₈₉	#8676
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #26020416	C₂	semidirect
2. #26019000	C₂	semidirect
4. #26018112	C₂	semidirect
16. #436×C₈₉	C₄	semidirect
18. C₈×D₄×C₈₉	C₄	semidirect
156. C₈₉	#8676	semidirect
160. #436	Dic₈₉	semidirect
162. C₈×D₄	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020416	2
#26019000	2
#26018812	2
#26018112	2
C₈₉×#2931	2
#8676	89