Subgroups — #36463155

Order 22784.

698 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028816	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151624	C₂	1	2	yes	maximal
3	C₂×#19151370	C₂	1	2	yes	maximal
4	#26019985	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	2	yes	—
7	C₂×#15667971	C₂²	1	2	yes	—
8	#19151625	—	2	4	no	—
9	#19151624	—	2	4	no	—
10	#19151624	—	2	4	no	—
11	#19151622	—	2	2	no	—
12	C₂×#15667941	—	2	4	no	—
13	#19151370	—	2	4	no	—
14	#19151365	C₄	1	4	yes	—
15	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
18	#336×C₈₉	—	2	2	no	—
19	C₂×#13083558	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂²×#10623494	D₄	1	4	yes	—
26	#15667971	—	2	2	no	—
27	#15667971	—	2	2	no	—
28	#15667971	D₄	1	4	yes	—
29	#15667971	Q₈	1	4	yes	—
30	C₂×C₄⋊Dic₈₉	—	2	2	no	—
31	C₂×C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₈₉	—	2	2	no	—
33	C₄⋊#10623494	—	2	8	no	—
34	#15667941	—	2	16	no	—
35	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₈₉	—	2	1	no	—
37	C₂×M₁₆×C₈₉	—	2	2	no	—
38	C₂×M₁₆×C₈₉	—	2	2	no	—
39	C₂×M₁₆×C₈₉	Q₈	1	2	yes	—
40	C₂²×C₈×C₈₉	—	2	2	no	—
41	C₂²×C₈×C₈₉	—	2	2	no	—
42	C₂²×C₈×C₈₉	—	2	1	no	—
43	C₂²×C₈×C₈₉	—	2	1	no	—
44	C₂²×C₈×C₈₉	—	2	1	no	—
45	C₂²×C₈×C₈₉	D₄	1	2	yes	—
46	#117×C₈₉	—	2	2	no	—
47	#117×C₈₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₈₉	Q₈	1	2	yes	—
49	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₈₉	D₄	1	2	yes	—
51	C₂²×Dic₈₉	—	2	4	no	—
52	#13083558	—	4	2	no	—
53	#13083558	—	4	2	no	—
54	#13083558	—	4	2	no	—
55	#13083558	—	4	2	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	—	2	4	no	—
58	C₂×#10623494	—	2	4	no	—
59	C₄⋊Dic₈₉	—	2	4	no	—
60	C₄⋊Dic₈₉	—	2	4	no	—
61	C₂²⋊Dic₈₉	—	2	4	no	—
62	C₂²⋊Dic₈₉	C₄⋊C₄	1	8	yes	—
63	C₄×Dic₈₉	—	2	4	no	—
64	C₄×Dic₈₉	—	2	4	no	—
65	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	#41	1	2	yes	—
76	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
77	M₁₆×C₈₉	—	2	4	no	—
78	M₁₆×C₈₉	—	4	2	no	—
79	C₂×C₈×C₈₉	—	2	4	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	4	1	no	—
83	C₂×C₈×C₈₉	—	4	2	no	—
84	C₂×C₈×C₈₉	—	4	1	no	—
85	C₂×C₈×C₈₉	—	4	1	no	—
86	C₂×C₈×C₈₉	—	4	1	no	—
87	C₄⋊C₄×C₈₉	—	2	4	no	—
88	C₂²⋊C₄×C₈₉	—	2	4	no	—
89	C₄²×C₈₉	—	2	2	no	—
90	C₄²×C₈₉	—	2	2	no	—
91	C₂³×C₈₉	—	4	2	no	—
92	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₈₉	C₄×D₄	1	2	yes	—
95	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
96	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₈₉	—	2	4	no	—
98	C₂×C₄×C₈₉	—	4	4	no	—
99	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#120	1	2	yes	—
101	C₂×C₄×C₈₉	#121	1	2	yes	—
102	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#123	1	2	yes	—
110	C₂×C₄×C₈₉	#125	1	2	yes	—
111	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×Dic₈₉	—	2	8	no	—
118	C₂×Dic₈₉	—	4	8	no	—
119	C₈×C₈₉	—	4	2	no	—
120	C₈×C₈₉	—	4	2	no	—
121	#10623494	—	4	4	no	—
122	C₂²×C₈₉	—	2	4	no	—
123	C₂²×C₈₉	—	2	2	no	—
124	C₂²×C₈₉	—	4	2	no	—
125	C₂²×C₈₉	—	4	1	no	—
126	C₂²×C₈₉	—	4	1	no	—
127	C₂²×C₈₉	#384	1	1	yes	characteristic
128	C₂²×C₈₉	#385	1	2	yes	—
129	C₂²×C₈₉	#385	1	1	yes	characteristic
130	C₂²×C₈₉	#389	1	2	yes	—
131	C₂²×C₈₉	#389	1	1	yes	characteristic
132	C₄×C₈₉	—	4	4	no	—
133	C₄×C₈₉	#392	1	1	yes	characteristic
134	C₄×C₈₉	#395	1	1	yes	characteristic
135	C₄×C₈₉	#395	1	2	yes	—
136	C₄×C₈₉	#398	1	1	yes	characteristic
137	C₄×C₈₉	#398	1	2	yes	—
138	C₄×C₈₉	#399	1	1	yes	characteristic
139	Dic₈₉	—	4	8	no	—
140	#11345	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	—	2	2	no	—
142	C₂×C₈₉	—	2	1	no	—
143	C₂×C₈₉	—	2	2	no	—
144	C₂×C₈₉	—	2	1	no	—
145	C₂×C₈₉	#1444	1	1	yes	characteristic
146	C₂×C₈₉	#1868	1	2	yes	—
147	#2905	—	89	1	no	—
148	C₂×#424	—	89	2	no	—
149	C₂×#344	—	89	2	no	—
150	#1740	—	89	1	no	—
151	#1729	D₈₉	1	1	yes	characteristic
152	C₈₉	#11345	1	1	yes	characteristic, complemented, Sylow
153	C₂²×M₁₆	—	89	2	no	—
154	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
155	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
156	C₂×#117	—	89	2	no	—
157	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
158	#424	—	178	4	no	—
159	#424	—	178	4	no	—
160	#424	—	178	4	no	—
161	#423	—	178	2	no	—
162	C₂×C₄⋊C₈	—	178	4	no	—
163	#344	—	178	4	no	—
164	#342	—	89	4	no	—
165	#336	—	2	2	no	—
166	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	178	2	no	—
171	C₂×M₁₆	—	178	2	no	—
172	C₂×M₁₆	—	178	2	no	—
173	C₂×M₁₆	—	178	2	no	—
174	C₂×M₁₆	—	178	2	no	—
175	C₂×M₁₆	—	178	2	no	—
176	C₂×M₁₆	Dic₁₇₈	1	2	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	89	4	no	—
183	C₂²×C₈	D₃₅₆	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	89	4	no	—
187	#117	—	89	4	no	—
188	#117	—	178	2	no	—
189	#117	—	178	2	no	—
190	C₂×C₄⋊C₄	—	178	2	no	—
191	C₂×C₄⋊C₄	Dic₁₇₈	1	2	yes	—
192	C₂×C₂²⋊C₄	—	89	4	no	—
193	C₂×C₂²⋊C₄	C₄×D₈₉	1	2	yes	—
194	C₂×C₄²	—	178	2	no	—
195	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
196	C₄⋊C₈	—	178	8	no	—
197	C₄⋊C₈	—	178	16	no	—
198	C₂⁴	C₂×C₄×D₈₉	1	1	yes	characteristic
199	C₂²×C₄	—	178	4	no	—
200	C₂²×C₄	#13083553	1	4	yes	—
201	C₂²×C₄	D₄×D₈₉	1	2	yes	—
202	C₂²×C₄	D₄×D₈₉	1	2	yes	—
203	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
205	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
206	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
208	C₂²×C₄	#13083573	1	2	yes	—
209	C₂²×C₄	#13083574	1	1	yes	characteristic
210	C₂²×C₄	#13083574	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	356	2	no	—
214	M₁₆	—	356	2	no	—
215	M₁₆	—	356	2	no	—
216	M₁₆	—	356	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	178	4	no	—
226	C₂×C₈	—	178	4	no	—
227	C₂×C₈	—	178	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	178	4	no	—
230	C₄⋊C₄	—	178	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	89	8	no	—
233	C₂²⋊C₄	—	178	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	178	4	no	—
237	C₄²	—	178	4	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₄⋊C₄×D₈₉	1	2	yes	—
240	C₂³	C₂×#13083553	1	1	yes	characteristic
241	C₂³	#15667972	1	2	yes	—
242	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
243	C₂³	C₄×Dic₁₇₈	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	356	8	no	—
247	C₂×C₄	—	178	8	no	—
248	C₂×C₄	#15667980	1	1	yes	characteristic
249	C₂×C₄	#15667982	1	2	yes	—
250	C₂×C₄	#15667985	1	2	yes	—
251	C₂×C₄	#15667986	1	1	yes	characteristic
252	C₂×C₄	#15667989	1	2	yes	—
253	C₂×C₄	#15667989	1	2	yes	—
254	C₂×C₄	#15667990	1	2	yes	—
255	C₂×C₄	#15667991	1	1	yes	characteristic
256	C₂×C₄	#15667992	1	2	yes	—
257	C₂×C₄	#15667993	1	1	yes	characteristic
258	C₂×C₄	#15667994	1	2	yes	—
259	C₂×C₄	#15668002	1	2	yes	—
260	C₂×C₄	#15668004	1	1	yes	characteristic
261	C₂×C₄	#15668008	1	1	yes	characteristic
262	C₂×C₄	#15668010	1	1	yes	characteristic
263	C₂×C₄	#15668011	1	2	yes	—
264	C₂×C₄	#15668011	1	2	yes	—
265	C₂×C₄	#15668012	1	1	yes	characteristic
266	C₈	—	4	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	356	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	#19151467	1	1	yes	characteristic
275	C₂²	#19151470	1	1	yes	characteristic
276	C₂²	#19151472	1	2	yes	—
277	C₂²	#19151495	1	2	yes	—
278	C₂²	#19151496	1	1	yes	characteristic
279	C₄	—	4	4	no	—
280	C₄	—	356	8	no	—
281	C₄	#19151507	1	1	yes	characteristic
282	C₄	#19151518	1	1	yes	characteristic
283	C₄	#19151519	1	2	yes	—
284	C₄	#19151528	1	1	yes	characteristic
285	C₄	#19151532	1	2	yes	—
286	C₄	#19151534	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	1	no	—
291	C₂	#26018516	1	1	yes	characteristic
292	C₂	#26020537	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667980
commutator, prime radical	C₂⁴×C₈₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₈₉×#1729	C₂
socle	C₂²×C₈₉	#385
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	#1729	D₈₉
hypocenter	C₈₉	#11345
LCS	G ▷ C₂⁴×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
152. C₈₉	#11345	semidirect

Maximal subgroups

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

subgroup	index
#26028816	2
C₂×#19151624	2
C₂×#19151370	2
#26019985	2
C₈₉×#1729	2
#11345	89