Subgroups — #36526457

Order 22784.

564 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030838	C₂	1	1	yes	maximal, characteristic
2	#26029353	C₂	1	2	yes	maximal
3	#26019160	C₂	1	2	yes	maximal, complemented
4	#26019101	C₂	1	4	yes	maximal, complemented
5	#26019070	C₂	1	2	yes	maximal, complemented
6	#26018672	C₂	1	2	yes	maximal
7	#26018541	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic
9	#19152155	C₂²	1	2	yes	—
10	#19152003	C₄	1	1	yes	characteristic
11	#19151978	C₄	1	1	yes	characteristic
12	#19151949	C₄	1	2	yes	—
13	#19151839	C₄	1	2	yes	—
14	#19151839	C₄	1	2	yes	—
15	#19151728	—	2	2	no	—
16	#19151725	C₂²	1	4	yes	—
17	C₈×Dic₁₇₈	—	2	2	no	—
18	#19151722	C₂²	1	4	yes	—
19	C₄×Dic₃₅₆	C₂²	1	2	yes	—
20	C₄×Dic₃₅₆	C₂²	1	2	yes	—
21	#19151656	—	2	4	no	—
22	#19151651	C₂²	1	1	yes	characteristic
23	C₄×#13083558	C₂²	1	1	yes	characteristic
24	#19151328	C₂²	1	4	yes	—
25	#19151328	C₂²	1	4	yes	—
26	#19151320	C₂²	1	2	yes	—
27	#19151319	C₂²	1	2	yes	—
28	#431×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	#335×C₈₉	C₂²	1	2	yes	—
31	#334×C₈₉	C₂²	1	1	yes	characteristic
32	#334×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
35	C₂×#13083558	—	2	1	no	—
36	#15668012	C₂×C₄	1	2	yes	—
37	#15668012	C₂×C₄	1	2	yes	—
38	#15668004	—	2	2	no	—
39	#15667991	—	2	2	no	—
40	#15667991	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₇₈	—	2	2	no	—
42	C₄×Dic₁₇₈	C₂³	1	4	yes	—
43	#15667971	—	2	2	no	—
44	#15667946	C₂×C₄	1	2	yes	—
45	#15667946	C₂×C₄	1	2	yes	—
46	C₄⋊#10623494	D₄	1	2	yes	—
47	#15667941	—	2	4	no	—
48	#15667941	—	2	4	no	—
49	#15667937	C₂×C₄	1	4	yes	—
50	#15667937	C₂×C₄	1	4	yes	—
51	C₂²⋊#10623494	D₄	1	2	yes	—
52	#15667923	—	2	1	no	—
53	#15667922	—	2	4	no	—
54	C₄×#10623494	C₂³	1	2	yes	—
55	C₄×#10623494	C₂³	1	2	yes	—
56	C₈×Dic₈₉	—	2	4	no	—
57	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
61	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
62	C₂²⋊C₈×C₈₉	—	2	2	no	—
63	C₄×C₈×C₈₉	—	2	2	no	—
64	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₇₈	—	2	2	no	—
69	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
70	Dic₃₅₆	—	2	4	no	—
71	Dic₃₅₆	—	4	2	no	—
72	#13083558	—	4	2	no	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	C₂×D₄	1	4	yes	—
75	C₄⋊Dic₈₉	—	2	2	no	—
76	C₄⋊Dic₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
79	#13083553	—	2	4	no	—
80	#13083553	—	2	4	no	—
81	C₂²⋊Dic₈₉	—	2	4	no	—
82	C₄×Dic₈₉	—	2	4	no	—
83	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	4	1	no	—
90	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₈₉	#41	1	1	yes	characteristic
97	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
98	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
109	C₂×Dic₈₉	—	2	4	no	—
110	C₂×Dic₈₉	—	2	4	no	—
111	Dic₁₇₈	—	2	4	no	—
112	Dic₁₇₈	—	2	4	no	—
113	Dic₁₇₈	—	4	4	no	—
114	C₈×C₈₉	—	2	1	no	—
115	C₈×C₈₉	—	2	1	no	—
116	C₈×C₈₉	—	2	1	no	—
117	C₈×C₈₉	—	4	2	no	—
118	C₈×C₈₉	—	4	2	no	—
119	C₈×C₈₉	#131	1	2	yes	—
120	#10623494	—	2	4	no	—
121	#10623494	—	4	2	no	—
122	C₂²×C₈₉	—	2	1	no	—
123	C₂²×C₈₉	—	2	1	no	—
124	C₂²×C₈₉	—	2	1	no	—
125	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₈₉	—	2	1	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	#444	1	2	yes	—
130	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
131	C₄×C₈₉	#568	1	1	yes	characteristic
132	C₄×C₈₉	#578	1	2	yes	—
133	Dic₈₉	—	4	4	no	—
134	Dic₈₉	—	4	4	no	—
135	#18090	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	—	4	1	no	—
137	C₂×C₈₉	#2935	1	1	yes	characteristic
138	C₂×C₈₉	#2950	1	1	yes	characteristic
139	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
140	#3170	—	89	1	no	—
141	#2986	—	89	2	no	—
142	#1593	—	89	2	no	—
143	#1584	—	89	4	no	—
144	#1579	—	89	2	no	—
145	#1564	D₈₉	1	1	yes	characteristic
146	#1485	—	89	2	no	—
147	#1451	—	89	1	no	—
148	C₈₉	#18090	1	1	yes	characteristic, complemented, Sylow
149	#533	—	89	2	no	—
150	#500	—	89	1	no	—
151	#494	—	89	1	no	—
152	#487	—	89	2	no	—
153	#467	—	89	2	no	—
154	#467	—	89	2	no	—
155	#446	—	89	4	no	—
156	#446	—	178	2	no	—
157	C₈×Q₈	—	89	4	no	—
158	C₈×Q₈	—	178	2	no	—
159	C₄×Q₁₆	—	89	2	no	—
160	C₄×Q₁₆	—	89	2	no	—
161	#433	—	178	4	no	—
162	#431	—	89	1	no	—
163	#431	C₂×D₈₉	1	2	yes	—
164	C₄×M₁₆	—	89	1	no	—
165	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
166	#335	C₂×D₈₉	1	2	yes	—
167	#334	C₂×D₈₉	1	1	yes	characteristic
168	#334	C₂×D₈₉	1	1	yes	characteristic
169	Q₈⋊C₈	—	89	4	no	—
170	Q₈⋊C₈	—	89	4	no	—
171	#322	—	89	2	no	—
172	#322	—	89	2	no	—
173	C₂×Q₁₆	—	89	2	no	—
174	C₂×Q₁₆	—	89	2	no	—
175	C₂×M₁₆	—	178	1	no	—
176	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
177	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
178	C₄⋊Q₈	—	89	2	no	—
179	C₄⋊Q₈	—	89	2	no	—
180	#125	—	178	2	no	—
181	C₂²⋊Q₈	—	89	4	no	—
182	C₂²⋊Q₈	—	178	2	no	—
183	C₄×Q₈	—	89	4	no	—
184	C₄×Q₈	—	178	2	no	—
185	#117	—	178	2	no	—
186	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
187	#107	—	89	2	no	—
188	#107	—	89	2	no	—
189	C₄⋊C₈	—	89	2	no	—
190	C₄⋊C₈	—	178	4	no	—
191	C₄⋊C₈	—	178	4	no	—
192	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
193	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
194	#103	—	89	4	no	—
195	#103	—	89	4	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	—	89	2	no	—
198	#97	—	178	4	no	—
199	#97	—	178	1	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	89	2	no	—
202	C₄×C₈	—	89	2	no	—
203	C₄×C₈	—	178	4	no	—
204	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
205	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
208	C₂×Q₈	—	89	4	no	—
209	C₂×Q₈	—	178	2	no	—
210	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
212	C₂²×C₄	#13083575	1	1	yes	characteristic
213	Q₁₆	—	356	2	no	—
214	Q₁₆	—	178	4	no	—
215	M₁₆	—	356	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	89	4	no	—
221	C₂×C₈	—	178	2	no	—
222	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
226	C₄⋊C₄	—	89	4	no	—
227	C₄⋊C₄	—	178	2	no	—
228	C₄⋊C₄	—	178	2	no	—
229	C₄⋊C₄	—	178	4	no	—
230	C₄⋊C₄	—	178	2	no	—
231	C₄⋊C₄	—	178	4	no	—
232	C₂²⋊C₄	—	178	4	no	—
233	C₄²	—	178	4	no	—
234	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
235	C₄²	D₄×D₈₉	1	1	yes	characteristic
236	C₄²	#13083575	1	1	yes	characteristic
237	C₄²	C₂³×D₈₉	1	1	yes	characteristic
238	C₂³	#15667974	1	1	yes	characteristic
239	Q₈	—	356	4	no	—
240	Q₈	—	178	4	no	—
241	Q₈	—	178	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	178	4	no	—
247	C₂×C₄	—	178	4	no	—
248	C₂×C₄	#15667974	1	1	yes	characteristic
249	C₂×C₄	#15667974	1	1	yes	characteristic
250	C₂×C₄	#15667974	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
253	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	356	2	no	—
260	C₈	—	178	4	no	—
261	C₈	#15668018	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×#15667974	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	356	4	no	—
270	C₄	—	356	4	no	—
271	C₄	#19151719	1	2	yes	—
272	C₄	C₂×#15668018	1	1	yes	characteristic
273	C₄	#19152364	1	1	yes	characteristic
274	C₄	#19152412	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	#26028995	1	1	yes	characteristic
277	C₂	#26029119	1	1	yes	characteristic
278	C₂	C₂×#19151719	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#18090
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #26019160	C₂	semidirect
4. #26019101	C₂	semidirect
5. #26019070	C₂	semidirect
148. C₈₉	#18090	semidirect

Maximal subgroups

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

subgroup	index
#26030838	2
#26029353	2
#26019160	2
#26019101	2
#26019070	2
#26018672	2
#26018541	2
C₈₉×#1564	2
#18090	89