Subgroups — #36462815

Order 22784.

602 conjugacy classes of subgroups.

The subgroup lattice has 288 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×#19151370	C₂	1	2	yes	maximal
3	#26019985	C₂	1	1	yes	maximal, characteristic
4	#26019948	C₂	1	2	yes	maximal
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	#19151622	—	2	2	no	—
10	#19151370	—	2	4	no	—
11	#19151365	C₄	1	4	yes	—
12	#19151352	—	2	4	no	—
13	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
16	#336×C₈₉	—	2	2	no	—
17	C₂×#13083558	—	2	2	no	—
18	C₂×#13083558	—	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	C₂²×#10623494	D₄	1	4	yes	—
24	#15667971	—	2	2	no	—
25	#15667971	—	2	2	no	—
26	#15667971	—	2	2	no	—
27	#15667971	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₈₉	—	2	2	no	—
31	C₄⋊#10623494	—	2	8	no	—
32	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₈₉	—	2	1	no	—
34	C₂×M₁₆×C₈₉	—	2	2	no	—
35	C₂×M₁₆×C₈₉	—	2	2	no	—
36	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
37	C₂²×C₈×C₈₉	—	2	1	no	—
38	C₂²×C₈×C₈₉	—	2	2	no	—
39	C₂²×C₈×C₈₉	—	2	1	no	—
40	C₂²×C₈×C₈₉	—	2	2	no	—
41	C₂²×C₈×C₈₉	—	2	1	no	—
42	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
43	#117×C₈₉	—	2	2	no	—
44	#117×C₈₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₈₉	Q₈	1	2	yes	—
46	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₈₉	D₄	1	2	yes	—
48	C₂²×Dic₈₉	—	2	4	no	—
49	#13083558	—	4	2	no	—
50	#13083558	—	4	2	no	—
51	#13083558	—	4	2	no	—
52	#13083558	—	4	2	no	—
53	C₂×#10623494	—	2	4	no	—
54	C₂×#10623494	—	2	4	no	—
55	C₂×#10623494	—	2	4	no	—
56	C₄⋊Dic₈₉	—	4	2	no	—
57	C₄⋊Dic₈₉	—	4	2	no	—
58	C₂²⋊Dic₈₉	—	2	4	no	—
59	C₂²⋊Dic₈₉	—	2	4	no	—
60	C₄×Dic₈₉	—	4	2	no	—
61	C₄×Dic₈₉	—	4	2	no	—
62	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	#41	1	2	yes	—
71	C₂²×C₄×C₈₉	#41	1	2	yes	—
72	C₂²×C₄×C₈₉	#41	1	2	yes	—
73	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
74	M₁₆×C₈₉	—	2	4	no	—
75	M₁₆×C₈₉	—	4	2	no	—
76	C₂×C₈×C₈₉	—	2	2	no	—
77	C₂×C₈×C₈₉	—	2	4	no	—
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	—	4	1	no	—
80	C₂×C₈×C₈₉	—	4	1	no	—
81	C₂×C₈×C₈₉	—	4	2	no	—
82	C₂×C₈×C₈₉	—	4	1	no	—
83	C₂×C₈×C₈₉	—	4	1	no	—
84	C₄⋊C₄×C₈₉	—	2	4	no	—
85	C₂²⋊C₄×C₈₉	—	2	4	no	—
86	C₄²×C₈₉	—	2	2	no	—
87	C₄²×C₈₉	—	2	2	no	—
88	C₂³×C₈₉	—	4	2	no	—
89	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₈₉	#117	1	2	yes	—
91	C₂³×C₈₉	C₄×D₄	1	2	yes	—
92	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	—	2	4	no	—
95	C₂×C₄×C₈₉	—	4	4	no	—
96	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#121	1	2	yes	—
99	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#123	1	2	yes	—
104	C₂×C₄×C₈₉	#123	1	2	yes	—
105	C₂×C₄×C₈₉	#124	1	2	yes	—
106	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#125	1	2	yes	—
109	C₂×C₄×C₈₉	#125	1	2	yes	—
110	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#126	1	2	yes	—
112	C₂×C₄×C₈₉	#126	1	2	yes	—
113	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×Dic₈₉	—	4	8	no	—
115	C₂×Dic₈₉	—	4	4	no	—
116	C₈×C₈₉	—	4	2	no	—
117	C₈×C₈₉	—	4	2	no	—
118	#10623494	—	4	4	no	—
119	C₂²×C₈₉	—	2	4	no	—
120	C₂²×C₈₉	—	2	2	no	—
121	C₂²×C₈₉	—	4	2	no	—
122	C₂²×C₈₉	—	4	1	no	—
123	C₂²×C₈₉	—	4	1	no	—
124	C₂²×C₈₉	#382	1	1	yes	characteristic
125	C₂²×C₈₉	#385	1	1	yes	characteristic
126	C₂²×C₈₉	#387	1	2	yes	—
127	C₂²×C₈₉	#388	1	2	yes	—
128	C₂²×C₈₉	#389	1	1	yes	characteristic
129	C₄×C₈₉	—	4	4	no	—
130	C₄×C₈₉	#393	1	1	yes	characteristic
131	C₄×C₈₉	#394	1	1	yes	characteristic
132	C₄×C₈₉	#397	1	1	yes	characteristic
133	C₄×C₈₉	#398	1	2	yes	—
134	C₄×C₈₉	#400	1	1	yes	characteristic
135	C₄×C₈₉	#400	1	2	yes	—
136	Dic₈₉	—	8	4	no	—
137	#11288	—	89	1	no	maximal, Sylow
138	C₂×C₈₉	—	2	2	no	—
139	C₂×C₈₉	—	2	1	no	—
140	C₂×C₈₉	—	2	2	no	—
141	C₂×C₈₉	—	2	1	no	—
142	C₂×C₈₉	#1442	1	1	yes	characteristic
143	C₂×C₈₉	#1868	1	2	yes	—
144	#2905	—	89	1	no	—
145	C₂×#344	—	89	2	no	—
146	#1740	—	89	1	no	—
147	#1729	—	89	2	no	—
148	#1729	D₈₉	1	1	yes	characteristic
149	C₈₉	#11288	1	1	yes	characteristic, complemented, Sylow
150	C₂²×M₁₆	—	89	2	no	—
151	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
152	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
153	C₂×#117	—	89	2	no	—
154	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
155	#424	—	178	4	no	—
156	#423	—	178	2	no	—
157	#344	—	178	4	no	—
158	#342	—	89	4	no	—
159	#336	—	2	2	no	—
160	#336	—	178	4	no	—
161	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
162	C₂×M₁₆	—	2	1	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	178	2	no	—
166	C₂×M₁₆	—	178	2	no	—
167	C₂×M₁₆	—	178	2	no	—
168	C₂×M₁₆	—	178	2	no	—
169	C₂×M₁₆	—	178	2	no	—
170	C₂×M₁₆	—	178	2	no	—
171	C₂×M₁₆	D₃₅₆	1	2	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	89	4	no	—
178	C₂²×C₈	Dic₁₇₈	1	2	yes	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	178	2	no	—
182	#117	—	178	2	no	—
183	#117	—	178	2	no	—
184	#117	—	178	2	no	—
185	C₂×C₄⋊C₄	—	178	2	no	—
186	C₂×C₄⋊C₄	Dic₁₇₈	1	2	yes	—
187	C₂×C₂²⋊C₄	—	89	4	no	—
188	C₂×C₂²⋊C₄	C₄×D₈₉	1	2	yes	—
189	C₂×C₄²	—	178	2	no	—
190	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
191	C₄⋊C₈	—	178	8	no	—
192	C₂⁴	C₂×C₄×D₈₉	1	1	yes	characteristic
193	C₂²×C₄	—	178	4	no	—
194	C₂²×C₄	#13083553	1	4	yes	—
195	C₂²×C₄	D₄×D₈₉	1	2	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
200	C₂²×C₄	#13083573	1	2	yes	—
201	C₂²×C₄	#13083573	1	2	yes	—
202	C₂²×C₄	#13083574	1	1	yes	characteristic
203	C₂²×C₄	#13083574	1	1	yes	characteristic
204	C₂²×C₄	#13083575	1	2	yes	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	356	2	no	—
208	M₁₆	—	356	2	no	—
209	M₁₆	—	356	2	no	—
210	M₁₆	—	356	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	178	4	no	—
220	C₂×C₈	—	178	4	no	—
221	C₂×C₈	—	178	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	356	2	no	—
224	C₄⋊C₄	—	356	2	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	178	4	no	—
227	C₂²⋊C₄	—	178	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	356	2	no	—
231	C₄²	—	356	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×#13083553	1	1	yes	characteristic
234	C₂³	#15667970	1	2	yes	—
235	C₂³	#15667972	1	2	yes	—
236	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
237	C₂³	C₄×Dic₁₇₈	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	356	8	no	—
241	C₂×C₄	—	356	4	no	—
242	C₂×C₄	#15667980	1	1	yes	characteristic
243	C₂×C₄	#15667986	1	1	yes	characteristic
244	C₂×C₄	#15667987	1	2	yes	—
245	C₂×C₄	#15667989	1	2	yes	—
246	C₂×C₄	#15667989	1	2	yes	—
247	C₂×C₄	#15667991	1	1	yes	characteristic
248	C₂×C₄	#15667993	1	1	yes	characteristic
249	C₂×C₄	#15667996	1	2	yes	—
250	C₂×C₄	#15667997	1	2	yes	—
251	C₂×C₄	#15667998	1	2	yes	—
252	C₂×C₄	#15667999	1	1	yes	characteristic
253	C₂×C₄	#15668001	1	1	yes	characteristic
254	C₂×C₄	#15668003	1	2	yes	—
255	C₂×C₄	#15668003	1	2	yes	—
256	C₂×C₄	#15668004	1	1	yes	characteristic
257	C₂×C₄	#15668005	1	2	yes	—
258	C₂×C₄	#15668007	1	2	yes	—
259	C₂×C₄	#15668012	1	1	yes	characteristic
260	C₈	—	4	2	no	—
261	C₈	—	4	2	no	—
262	C₈	—	356	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#19151463	1	1	yes	characteristic
269	C₂²	#19151470	1	1	yes	characteristic
270	C₂²	#19151485	1	2	yes	—
271	C₂²	#19151490	1	2	yes	—
272	C₂²	#19151496	1	1	yes	characteristic
273	C₄	—	4	4	no	—
274	C₄	—	712	4	no	—
275	C₄	#19151510	1	1	yes	characteristic
276	C₄	#19151515	1	1	yes	characteristic
277	C₄	#19151523	1	1	yes	characteristic
278	C₄	#19151530	1	2	yes	—
279	C₄	#19151537	1	1	yes	characteristic
280	C₄	#19151540	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	#26018508	1	1	yes	characteristic
286	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	#11288
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
149. C₈₉	#11288	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×#19151370	2
#26019985	2
#26019948	2
C₈₉×#1729	2
#11288	89