Subgroups — #36447643

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	#26028965	C₂	1	1	yes	maximal, characteristic
2	#26020416	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018813	C₂	1	2	yes	maximal
4	#26018111	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668018	—	2	1	no	—
7	#19152148	C₂²	1	1	yes	characteristic
8	#19151677	—	2	1	no	—
9	#19151677	C₄	1	2	yes	complemented
10	C₈×D₃₅₆	—	2	1	no	—
11	C₈×D₃₅₆	C₄	1	2	yes	complemented
12	C₂×#15667946	—	2	1	no	—
13	C₂×#15667944	—	2	1	no	—
14	#19151622	C₂²	1	1	yes	characteristic
15	#19151564	—	2	1	no	—
16	#19151464	C₂²	1	2	yes	—
17	#19151333	—	2	4	no	—
18	#431×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#335×C₈₉	—	2	1	no	—
21	#334×C₈₉	—	2	1	no	—
22	C₂×#13083574	—	2	1	no	—
23	#15668018	—	4	2	no	—
24	#15668018	—	4	2	no	—
25	C₂×#13083558	—	2	1	no	—
26	C₂×#13083557	—	2	2	no	—
27	C₂×C₈×D₈₉	—	2	2	no	—
28	#15668012	Q₈	1	1	yes	characteristic
29	#15668009	Q₈	1	1	yes	characteristic
30	#15668000	—	2	1	no	—
31	#15667986	—	2	1	no	—
32	#15667986	D₄	1	2	yes	—
33	C₄×D₃₅₆	—	2	1	no	—
34	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₈₉	—	2	2	no	—
36	#15667946	—	4	2	no	—
37	#15667944	—	4	2	no	—
38	C₄⋊#10623494	—	2	1	no	—
39	C₄⋊#10623494	C₂×C₄	1	2	yes	—
40	#15667924	—	2	2	no	—
41	#15667924	—	2	2	no	—
42	#15667919	—	2	2	no	—
43	#15667919	—	2	2	no	—
44	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₈₉	—	2	2	no	—
48	C₂²⋊C₈×C₈₉	—	2	2	no	—
49	C₄×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	#13083574	—	4	1	no	—
55	#13083574	—	4	1	no	—
56	#13083574	—	4	1	no	—
57	#13083574	—	4	1	no	—
58	C₂×Dic₁₇₈	—	2	1	no	—
59	C₂×D₃₅₆	—	2	1	no	—
60	C₂×D₃₅₆	—	2	1	no	—
61	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
62	C₂×C₂²⋊D₈₉	—	2	2	no	—
63	C₂²×Dic₈₉	—	4	2	no	—
64	C₂×C₄×D₈₉	—	2	2	no	—
65	#13083558	—	4	1	no	—
66	#13083558	—	4	1	no	—
67	#13083557	—	4	4	no	—
68	C₂×#10623494	—	2	2	no	—
69	C₈×D₈₉	—	4	4	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	4	2	no	—
72	C₄⋊Dic₈₉	—	4	2	no	—
73	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
74	#13083551	—	2	2	no	—
75	#13083551	—	2	2	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
79	C₂×C₈×C₈₉	—	2	1	no	—
80	C₂×C₈×C₈₉	—	2	1	no	—
81	C₂×C₈×C₈₉	—	4	2	no	—
82	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	D₈	1	2	yes	—
87	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
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	#121	1	1	yes	characteristic
93	C₂²×D₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	4	1	no	—
95	C₂×C₄×C₈₉	—	4	1	no	—
96	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
106	C₂²⋊D₈₉	—	4	2	no	—
107	C₂²⋊D₈₉	—	4	2	no	—
108	C₂×Dic₈₉	—	2	2	no	—
109	C₂×Dic₈₉	—	8	2	no	—
110	C₂×Dic₈₉	—	8	2	no	—
111	C₂×Dic₈₉	—	8	2	no	—
112	D₃₅₆	—	4	2	no	—
113	D₃₅₆	—	4	2	no	—
114	D₃₅₆	—	4	1	no	—
115	D₃₅₆	—	4	1	no	—
116	C₄×D₈₉	—	4	2	no	—
117	C₄×D₈₉	—	4	2	no	—
118	Dic₁₇₈	—	4	1	no	—
119	Dic₁₇₈	—	4	1	no	—
120	C₈×C₈₉	—	2	1	no	—
121	C₈×C₈₉	—	2	1	no	—
122	C₈×C₈₉	—	8	2	no	—
123	C₈×C₈₉	#102	1	2	yes	—
124	C₈×C₈₉	#102	1	2	yes	—
125	#10623494	—	4	2	no	—
126	C₂²×C₈₉	—	2	1	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	#380	1	1	yes	characteristic
129	C₂²×C₈₉	#459	1	1	yes	characteristic
130	C₂²×C₈₉	#461	1	1	yes	characteristic
131	C₂×D₈₉	—	4	2	no	—
132	C₂×D₈₉	—	4	2	no	—
133	C₂×D₈₉	—	4	2	no	—
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	C₂×#102	1	1	yes	characteristic
139	C₄×C₈₉	#417	1	1	yes	characteristic
140	C₄×C₈₉	#474	1	1	yes	characteristic
141	C₄×C₈₉	#476	1	1	yes	characteristic
142	Dic₈₉	—	4	2	no	—
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	#1802	1	1	yes	characteristic
148	C₂×C₈₉	#1812	1	1	yes	characteristic
149	C₂×C₈₉	#1814	1	1	yes	characteristic
150	D₈₉	—	8	2	no	—
151	#2931	—	89	1	no	—
152	#1846	—	89	1	no	—
153	#1564	D₈₉	1	1	yes	characteristic, complemented
154	#1517	—	89	2	no	—
155	#1286	—	89	2	no	—
156	C₈₉	#8676	1	1	yes	characteristic, complemented, Sylow
157	C₂×#131	—	178	1	no	—
158	#532	—	89	1	no	—
159	#436	—	89	2	no	—
160	#436	—	178	1	no	—
161	C₈×D₄	—	89	2	no	—
162	C₈×D₄	—	178	1	no	—
163	#431	C₂×D₈₉	1	2	yes	—
164	C₂×#107	—	178	1	no	—
165	C₂×#106	—	178	1	no	—
166	#423	—	89	1	no	—
167	#408	—	178	1	no	—
168	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
169	#382	—	89	2	no	—
170	#335	—	2	1	no	—
171	#334	—	2	1	no	—
172	#327	—	178	4	no	—
173	C₂×#41	—	178	1	no	—
174	#131	—	356	2	no	—
175	#131	—	356	2	no	—
176	C₂×M₁₆	—	178	1	no	—
177	C₂×M₁₆	—	178	2	no	—
178	C₂²×C₈	—	178	2	no	—
179	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
180	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
181	C₄⋊Q₈	—	89	1	no	—
182	#127	—	89	1	no	—
183	#124	—	178	1	no	—
184	#121	—	89	2	no	—
185	#121	—	178	1	no	—
186	C₄×D₄	—	89	2	no	—
187	C₄×D₄	—	178	1	no	—
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	2	no	—
193	C₄⋊C₈	—	89	2	no	—
194	C₄⋊C₈	—	178	1	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	—	178	2	no	—
197	C₂²⋊C₈	—	178	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
200	C₄×C₈	C₄×D₈₉	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	—	356	1	no	—
206	#41	—	356	1	no	—
207	#41	—	356	1	no	—
208	#41	—	356	1	no	—
209	C₂×Q₈	—	178	1	no	—
210	C₂×D₄	—	89	2	no	—
211	C₂×D₄	—	178	2	no	—
212	C₂×D₄	—	178	1	no	—
213	C₂×D₄	—	178	1	no	—
214	C₂²×C₄	—	356	2	no	—
215	C₂²×C₄	—	178	2	no	—
216	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
218	C₂²×C₄	#13083575	1	1	yes	characteristic
219	M₁₆	—	356	1	no	—
220	M₁₆	—	356	1	no	—
221	M₁₆	—	356	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	356	4	no	—
226	C₂×C₈	—	178	2	no	—
227	C₂×C₈	#13083551	1	1	yes	characteristic
228	C₂×C₈	#13083551	1	1	yes	characteristic
229	C₂×C₈	#13083551	1	1	yes	characteristic
230	C₂×C₈	#13083551	1	1	yes	characteristic
231	C₂×C₈	D₇₁₂	1	2	yes	—
232	C₂×C₈	#13083563	1	2	yes	—
233	C₄⋊C₄	—	89	2	no	—
234	C₄⋊C₄	—	356	2	no	—
235	C₄⋊C₄	—	356	2	no	—
236	C₄⋊C₄	—	178	1	no	—
237	C₂²⋊C₄	—	178	2	no	—
238	C₂²⋊C₄	—	178	2	no	—
239	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
240	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
241	C₄²	Q₈×D₈₉	1	1	yes	characteristic
242	C₄²	#13083573	1	1	yes	characteristic
243	C₂³	—	178	2	no	—
244	C₂³	#15667987	1	1	yes	characteristic
245	Q₈	—	356	1	no	—
246	Q₈	—	356	1	no	—
247	D₄	—	356	2	no	—
248	D₄	—	356	2	no	—
249	D₄	—	356	2	no	—
250	D₄	—	356	2	no	—
251	D₄	—	356	1	no	—
252	D₄	—	356	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	356	2	no	—
256	C₂×C₄	—	356	2	no	—
257	C₂×C₄	—	178	2	no	—
258	C₂×C₄	—	712	2	no	—
259	C₂×C₄	—	712	2	no	—
260	C₂×C₄	—	712	2	no	—
261	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
262	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
263	C₂×C₄	#15667970	1	1	yes	characteristic
264	C₂×C₄	#15667983	1	1	yes	characteristic
265	C₂×C₄	#15667988	1	1	yes	characteristic
266	C₂×C₄	#15667992	1	1	yes	characteristic
267	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
268	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
269	C₂×C₄	#15668040	1	1	yes	characteristic
270	C₂×C₄	#15668045	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	8	2	no	—
274	C₈	—	356	2	no	—
275	C₈	#15667934	1	2	yes	—
276	C₈	#15667934	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	356	2	no	—
280	C₂²	—	356	2	no	—
281	C₂²	—	356	2	no	—
282	C₂²	#19151455	1	1	yes	characteristic
283	C₂²	#19151794	1	1	yes	characteristic
284	C₂²	#19151804	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	—	356	2	no	—
290	C₄	—	1424	2	no	—
291	C₄	C₂×#15667934	1	1	yes	characteristic
292	C₄	#19151600	1	1	yes	characteristic
293	C₄	#19151879	1	1	yes	characteristic
294	C₄	#19151889	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	—	712	2	no	—
298	C₂	#26020253	1	1	yes	characteristic
299	C₂	#26020295	1	1	yes	characteristic
300	C₂	#26020307	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151455
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	#380
prime radical	#19152148	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#8676
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. #26020416	C₂	semidirect
4. #26018111	C₂	semidirect
5. C₈₉×#1564	C₂	semidirect
9. #19151677	C₄	semidirect
11. C₈×D₃₅₆	C₄	semidirect
153. #1564	D₈₉	semidirect
156. C₈₉	#8676	semidirect

Maximal subgroups

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

subgroup	index
#26028965	2
#26020416	2
#26018813	2
#26018111	2
C₈₉×#1564	2
#8676	89