Subgroups — #36470906

Order 22784.

370 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033354	C₂	1	1	yes	maximal, characteristic
2	#26022017	C₂	1	1	yes	maximal, characteristic, complemented
3	#26021867	C₂	1	1	yes	maximal, characteristic
4	#26020053	C₂	1	1	yes	maximal, characteristic
5	#26018342	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	#19152016	—	2	2	no	—
10	#19152014	—	2	2	no	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	M₃₂×D₈₉	—	2	2	no	—
13	C₂²×#13083549	C₂²	1	1	yes	characteristic
14	#19151580	C₄	1	1	yes	characteristic
15	#19151571	C₄	1	1	yes	characteristic
16	#19151553	C₂²	1	1	yes	characteristic
17	#19151418	—	2	2	no	—
18	#19151377	—	2	2	no	—
19	#19151370	C₄	1	1	yes	characteristic
20	#19151368	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
23	#405×C₈₉	C₂²	1	1	yes	characteristic
24	#364×C₈₉	—	2	2	no	—
25	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
26	#15668018	—	2	1	no	—
27	#15668018	—	2	1	no	—
28	#15668018	—	2	1	no	—
29	#15668018	—	2	1	no	—
30	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
32	C₂×#13083557	Q₈	1	2	yes	—
33	C₂×C₈×D₈₉	D₄	1	2	yes	—
34	#15667971	C₂×C₄	1	1	yes	characteristic
35	#15667952	—	2	1	no	—
36	#15667952	D₄	1	2	yes	—
37	#15667951	—	2	4	no	—
38	C₂×#13083549	—	2	1	no	—
39	C₂×#13083549	—	2	1	no	—
40	C₂×#13083549	—	2	1	no	—
41	C₂×#13083549	—	2	1	no	—
42	C₁₆×D₈₉	—	2	4	no	—
43	#15667930	C₈	1	2	yes	—
44	#15667928	C₈	1	2	yes	—
45	#15667926	—	2	2	no	—
46	#15667922	—	2	1	no	—
47	C₈×Dic₈₉	—	2	1	no	—
48	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₈₉	—	2	1	no	—
52	M₃₂×C₈₉	—	2	1	no	—
53	M₃₂×C₈₉	—	2	1	no	—
54	M₃₂×C₈₉	D₄	1	2	yes	—
55	C₂×C₁₆×C₈₉	—	2	1	no	—
56	C₂×C₁₆×C₈₉	—	2	1	no	—
57	#97×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	#13083574	—	2	1	no	—
60	#13083574	—	2	1	no	—
61	#13083574	—	2	1	no	—
62	#13083574	—	2	1	no	—
63	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
64	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₉	M₁₆	1	2	yes	—
66	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
67	#13083558	—	2	1	no	—
68	#13083558	—	2	1	no	—
69	#13083558	—	2	1	no	—
70	#13083558	C₂×C₈	1	2	yes	—
71	#13083557	—	2	2	no	—
72	#13083557	—	2	2	no	—
73	C₂×#10623494	—	2	1	no	—
74	C₂×#10623494	C₄⋊C₄	1	2	yes	—
75	C₈×D₈₉	—	2	2	no	—
76	C₈×D₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	1	no	—
78	C₂²⋊Dic₈₉	—	2	1	no	—
79	C₄×Dic₈₉	—	2	1	no	—
80	#13083549	—	2	2	no	—
81	#13083549	—	4	1	no	—
82	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	M₁₆×C₈₉	—	2	1	no	—
84	M₁₆×C₈₉	C₂×C₈	1	2	yes	—
85	C₂×C₈×C₈₉	—	2	1	no	—
86	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
91	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
92	C₄⋊C₄×C₈₉	—	2	1	no	—
93	C₂²⋊C₄×C₈₉	—	2	1	no	—
94	C₄²×C₈₉	—	2	1	no	—
95	C₁₆×C₈₉	—	2	2	no	—
96	C₁₆×C₈₉	—	4	1	no	—
97	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×D₈₉	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₈₉	—	4	1	no	—
100	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
106	C₂²⋊D₈₉	—	2	2	no	—
107	C₂²⋊D₈₉	—	2	2	no	—
108	C₂×Dic₈₉	—	4	1	no	—
109	C₂×Dic₈₉	C₄⋊C₈	1	2	yes	—
110	D₃₅₆	—	2	1	no	—
111	D₃₅₆	—	2	1	no	—
112	C₄×D₈₉	—	2	2	no	—
113	C₄×D₈₉	—	2	2	no	—
114	Dic₁₇₈	—	2	1	no	—
115	Dic₁₇₈	—	2	1	no	—
116	C₈×C₈₉	—	2	2	no	—
117	C₈×C₈₉	#120	1	1	yes	characteristic
118	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₈₉	#123	1	1	yes	characteristic
121	#10623494	—	2	2	no	—
122	#10623494	—	2	2	no	—
123	C₂²×C₈₉	—	4	1	no	—
124	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₈₉	#435	1	1	yes	characteristic
126	C₂²×C₈₉	#435	1	1	yes	characteristic
127	C₂×D₈₉	—	2	2	no	—
128	C₂×D₈₉	—	4	2	no	—
129	C₄×C₈₉	—	8	1	no	—
130	C₄×C₈₉	#385	1	1	yes	characteristic
131	C₄×C₈₉	#424	1	1	yes	characteristic
132	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
133	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
134	Dic₈₉	—	2	2	no	—
135	Dic₈₉	—	8	1	no	—
136	#12916	—	89	1	no	maximal, Sylow
137	C₂×C₈₉	—	2	1	no	—
138	C₂×C₈₉	—	2	1	no	—
139	C₂×C₈₉	—	2	1	no	—
140	C₂×C₈₉	#1850	1	1	yes	characteristic
141	D₈₉	—	4	2	no	—
142	#3407	—	89	1	no	—
143	#2153	—	89	1	no	—
144	#2153	D₈₉	1	1	yes	characteristic, complemented
145	#2113	—	89	1	no	—
146	#1757	—	89	1	no	—
147	#1379	—	89	1	no	—
148	#1379	—	89	1	no	—
149	C₈₉	#12916	1	1	yes	characteristic, complemented, Sylow
150	C₂×#131	—	89	1	no	—
151	#504	—	178	2	no	—
152	#504	—	178	2	no	—
153	C₂×M₃₂	—	89	1	no	—
154	C₂×M₃₂	—	178	2	no	—
155	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
156	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
157	C₂²×C₁₆	—	89	1	no	—
158	#413	—	89	1	no	—
159	#410	—	89	1	no	—
160	#405	—	89	1	no	—
161	#405	C₂×D₈₉	1	1	yes	characteristic
162	#364	—	2	2	no	—
163	#364	—	178	2	no	—
164	C₂²⋊C₁₆	—	178	2	no	—
165	#344	—	89	1	no	—
166	#344	—	89	1	no	—
167	C₂×#41	—	89	1	no	—
168	#131	—	178	1	no	—
169	#131	—	178	1	no	—
170	#131	—	178	1	no	—
171	#131	—	178	1	no	—
172	C₂×M₁₆	—	89	1	no	—
173	C₂×M₁₆	—	89	2	no	—
174	C₂×M₁₆	—	89	1	no	—
175	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
176	C₂²×C₈	—	89	2	no	—
177	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
178	#117	—	89	1	no	—
179	#117	C₄×D₈₉	1	1	yes	characteristic
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	89	2	no	—
184	M₃₂	—	178	4	no	—
185	M₃₂	—	178	1	no	—
186	M₃₂	C₂²⋊D₈₉	1	2	yes	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	178	4	no	—
190	C₂×C₁₆	—	178	1	no	—
191	C₂×C₁₆	—	178	1	no	—
192	C₂×C₁₆	—	178	1	no	—
193	C₂×C₁₆	—	178	1	no	—
194	#101	—	89	2	no	—
195	#100	—	89	2	no	—
196	#99	—	178	2	no	—
197	#97	—	2	1	no	—
198	#97	—	178	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	178	1	no	—
201	#41	—	178	1	no	—
202	#41	—	178	1	no	—
203	#41	—	178	1	no	—
204	#41	—	178	1	no	—
205	C₂×Q₈	—	89	1	no	—
206	C₂×D₄	—	89	2	no	—
207	C₂×D₄	—	89	1	no	—
208	C₂²×C₄	—	89	2	no	—
209	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
210	M₁₆	—	2	1	no	—
211	M₁₆	—	89	2	no	—
212	M₁₆	—	178	1	no	—
213	M₁₆	—	178	1	no	—
214	M₁₆	—	178	1	no	—
215	M₁₆	—	178	2	no	—
216	M₁₆	—	178	2	no	—
217	M₁₆	C₈×D₈₉	1	2	yes	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	89	2	no	—
220	C₂×C₈	—	178	1	no	—
221	C₂×C₈	—	178	2	no	—
222	C₂×C₈	—	178	2	no	—
223	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
224	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
227	C₂×C₈	#13083574	1	1	yes	characteristic
228	C₂×C₈	#13083575	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	178	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	178	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	178	1	no	—
235	C₁₆	—	2	2	no	—
236	C₁₆	—	4	1	no	—
237	C₁₆	—	356	1	no	—
238	C₁₆	—	178	2	no	—
239	C₂³	—	89	2	no	—
240	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
241	Q₈	—	178	1	no	—
242	Q₈	—	178	1	no	—
243	D₄	—	178	2	no	—
244	D₄	—	178	2	no	—
245	D₄	—	178	1	no	—
246	D₄	—	178	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	89	2	no	—
249	C₂×C₄	—	356	1	no	—
250	C₂×C₄	—	178	2	no	—
251	C₂×C₄	—	178	2	no	—
252	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
253	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
254	C₂×C₄	#15667974	1	1	yes	characteristic
255	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
256	C₂×C₄	#15668018	1	1	yes	characteristic
257	C₂×C₄	#15668020	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	178	2	no	—
260	C₈	—	178	2	no	—
261	C₈	#15667980	1	1	yes	characteristic
262	C₈	#15667988	1	1	yes	characteristic
263	C₈	#15667990	1	1	yes	characteristic
264	C₈	#15667997	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	356	2	no	—
267	C₂²	—	178	2	no	—
268	C₂²	C₄⋊C₈×D₈₉	1	1	yes	characteristic
269	C₂²	#19151667	1	1	yes	characteristic
270	C₂²	#19151668	1	1	yes	characteristic
271	C₄	—	8	1	no	—
272	C₄	—	178	2	no	—
273	C₄	—	712	1	no	—
274	C₄	#19151469	1	1	yes	characteristic
275	C₄	#19151623	1	1	yes	characteristic
276	C₄	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
277	C₄	#19151663	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	356	2	no	—
282	C₂	#26020428	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15667980
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2153	C₂
socle	C₂×C₈₉	#1850
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2153	D₈₉
hypocenter	C₈₉	#12916
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
2. #26022017	C₂	semidirect
5. #26018342	C₂	semidirect
6. #26018341	C₂	semidirect
7. C₈₉×#2153	C₂	semidirect
144. #2153	D₈₉	semidirect
149. C₈₉	#12916	semidirect

Maximal subgroups

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

subgroup	index
#26033354	2
#26022017	2
#26021867	2
#26020053	2
#26018342	2
#26018341	2
C₈₉×#2153	2
#12916	89