Subgroups — #36447662

Order 22784.

418 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020383	C₂	1	1	yes	maximal, characteristic, complemented
2	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018812	C₂	1	2	yes	maximal
4	#26018112	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2931	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	#19151464	C₂²	1	2	yes	—
8	#19151361	—	2	1	no	—
9	#19151361	—	2	1	no	—
10	#19151351	—	2	1	no	—
11	#19151349	—	2	1	no	—
12	#19151332	—	2	4	no	—
13	C₂×#131×C₈₉	—	2	1	no	—
14	#532×C₈₉	C₂²	1	1	yes	characteristic
15	#436×C₈₉	—	2	1	no	—
16	#436×C₈₉	C₄	1	2	yes	complemented
17	C₈×D₄×C₈₉	—	2	1	no	—
18	C₈×D₄×C₈₉	C₄	1	2	yes	complemented
19	#423×C₈₉	C₂²	1	1	yes	characteristic
20	#408×C₈₉	—	2	1	no	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₈₉	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₂²⋊#10623494	—	2	2	no	—
25	C₄×#10623494	—	2	2	no	—
26	#15667919	—	2	2	no	—
27	#15667919	—	2	2	no	—
28	C₂×#41×C₈₉	—	2	1	no	—
29	#131×C₈₉	—	4	4	no	—
30	C₂×M₁₆×C₈₉	—	2	1	no	—
31	C₂×M₁₆×C₈₉	—	2	2	no	—
32	C₂²×C₈×C₈₉	—	2	2	no	—
33	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₈₉	Q₈	1	1	yes	characteristic
36	#127×C₈₉	Q₈	1	1	yes	characteristic
37	#124×C₈₉	—	2	1	no	—
38	#121×C₈₉	—	2	1	no	—
39	#121×C₈₉	D₄	1	2	yes	—
40	C₄×D₄×C₈₉	—	2	1	no	—
41	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₈₉	—	2	1	no	—
44	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
45	C₂²⋊C₈×C₈₉	—	2	2	no	—
46	C₂²⋊C₈×C₈₉	—	2	2	no	—
47	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₂²×Dic₈₉	—	4	2	no	—
52	C₂×#10623494	—	4	2	no	—
53	C₄⋊Dic₈₉	—	4	2	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	#41×C₈₉	—	4	2	no	—
56	#41×C₈₉	—	4	2	no	—
57	C₂×Q₈×C₈₉	—	2	1	no	—
58	C₂×D₄×C₈₉	—	2	2	no	—
59	C₂×D₄×C₈₉	—	2	1	no	—
60	C₂×D₄×C₈₉	—	2	1	no	—
61	C₂×D₄×C₈₉	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₈₉	—	2	2	no	—
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
66	M₁₆×C₈₉	—	4	4	no	—
67	M₁₆×C₈₉	—	4	2	no	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	—	2	2	no	—
70	C₂×C₈×C₈₉	—	2	2	no	—
71	C₂×C₈×C₈₉	—	4	4	no	—
72	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₈₉	—	2	1	no	—
77	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
78	C₂²⋊C₄×C₈₉	—	2	2	no	—
79	C₂²⋊C₄×C₈₉	—	2	2	no	—
80	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₈₉	#41	1	1	yes	characteristic
84	C₂³×C₈₉	—	2	2	no	—
85	C₂³×C₈₉	#121	1	1	yes	characteristic
86	Q₈×C₈₉	—	4	2	no	—
87	D₄×C₈₉	—	4	2	no	—
88	D₄×C₈₉	—	4	2	no	—
89	D₄×C₈₉	—	4	4	no	—
90	D₄×C₈₉	—	4	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	4	1	no	—
93	C₂×C₄×C₈₉	—	4	1	no	—
94	C₂×C₄×C₈₉	—	4	2	no	—
95	C₂×C₄×C₈₉	—	4	2	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	#135	1	2	yes	—
103	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×Dic₈₉	—	8	2	no	—
105	C₂×Dic₈₉	—	8	2	no	—
106	C₂×Dic₈₉	—	8	2	no	—
107	C₈×C₈₉	—	2	1	no	—
108	C₈×C₈₉	—	2	1	no	—
109	C₈×C₈₉	—	2	1	no	—
110	C₈×C₈₉	—	2	1	no	—
111	C₈×C₈₉	—	4	2	no	—
112	#10623494	—	8	2	no	—
113	C₂²×C₈₉	—	2	1	no	—
114	C₂²×C₈₉	—	2	1	no	—
115	C₂²×C₈₉	—	4	2	no	—
116	C₂²×C₈₉	—	4	2	no	—
117	C₂²×C₈₉	—	4	2	no	—
118	C₂²×C₈₉	#380	1	1	yes	characteristic
119	C₂²×C₈₉	#463	1	2	yes	—
120	C₄×C₈₉	—	2	1	no	—
121	C₄×C₈₉	—	2	1	no	—
122	C₄×C₈₉	—	2	1	no	—
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	—	4	2	no	—
125	C₄×C₈₉	#416	1	1	yes	characteristic
126	C₄×C₈₉	#418	1	1	yes	characteristic
127	C₄×C₈₉	#478	1	2	yes	—
128	Dic₈₉	—	16	2	no	—
129	#8680	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	—	2	2	no	—
131	C₂×C₈₉	—	8	2	no	—
132	C₂×C₈₉	#1804	1	1	yes	characteristic
133	C₂×C₈₉	#1812	1	1	yes	characteristic
134	C₂×C₈₉	#1814	1	1	yes	characteristic
135	#2931	D₈₉	1	1	yes	characteristic
136	#1835	—	89	1	no	—
137	#1564	—	89	1	no	—
138	#1517	—	89	2	no	—
139	#1286	—	89	2	no	—
140	C₈₉	#8680	1	1	yes	characteristic, complemented, Sylow
141	C₂×#131	—	2	1	no	—
142	#532	C₂×D₈₉	1	1	yes	characteristic
143	#436	—	2	1	no	—
144	#436	Dic₈₉	1	2	yes	complemented
145	C₈×D₄	—	2	1	no	—
146	C₈×D₄	Dic₈₉	1	2	yes	complemented
147	#431	—	89	2	no	—
148	#423	C₂×D₈₉	1	1	yes	characteristic
149	#408	—	2	1	no	—
150	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
151	#382	—	89	2	no	—
152	#340	—	178	1	no	—
153	#340	—	178	1	no	—
154	#335	—	178	1	no	—
155	#334	—	178	1	no	—
156	#327	—	178	4	no	—
157	C₂×#41	—	2	1	no	—
158	#131	—	4	4	no	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
164	C₄⋊Q₈	Dic₁₇₈	1	1	yes	characteristic
165	#127	Dic₁₇₈	1	1	yes	characteristic
166	#124	—	2	1	no	—
167	#121	—	2	1	no	—
168	#121	D₃₅₆	1	2	yes	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
171	C₂×C₄⋊C₄	—	178	2	no	—
172	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	178	2	no	—
175	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₂²⋊C₈	—	178	2	no	—
179	C₄×C₈	—	178	2	no	—
180	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
181	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
184	#95	—	178	2	no	—
185	#95	—	178	2	no	—
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	C₄⋊Dic₈₉	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	356	2	no	—
195	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
197	C₂²×C₄	#13083574	1	1	yes	characteristic
198	M₁₆	—	4	4	no	—
199	M₁₆	—	4	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	—	356	2	no	—
205	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	356	2	no	—
211	C₄⋊C₄	—	356	2	no	—
212	C₄⋊C₄	C₄⋊Dic₈₉	1	2	yes	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
217	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
218	C₄²	#13083574	1	1	yes	characteristic
219	C₂³	—	2	2	no	—
220	C₂³	#15667986	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	712	2	no	—
232	C₂×C₄	—	712	2	no	—
233	C₂×C₄	—	712	2	no	—
234	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
236	C₂×C₄	#15667971	1	1	yes	characteristic
237	C₂×C₄	#15667986	1	1	yes	characteristic
238	C₂×C₄	#15667991	1	1	yes	characteristic
239	C₂×C₄	#15667991	1	1	yes	characteristic
240	C₂×C₄	#15668032	1	2	yes	—
241	C₂×C₄	#15668038	1	2	yes	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	712	2	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	#19151458	1	1	yes	characteristic
254	C₂²	#19151813	1	2	yes	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	1424	2	no	—
261	C₄	#19151594	1	1	yes	characteristic
262	C₄	#19151603	1	1	yes	characteristic
263	C₄	#19151898	1	2	yes	—
264	C₂	—	2	2	no	—
265	C₂	—	8	2	no	—
266	C₂	#26020262	1	1	yes	characteristic
267	C₂	#26020298	1	1	yes	characteristic
268	C₂	#26020306	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151458
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2931	C₂
socle	C₂²×C₈₉	#380
prime radical	#532×C₈₉	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#2931	D₈₉
hypocenter	C₈₉	#8680
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #26020383	C₂	semidirect
2. #26019000	C₂	semidirect
4. #26018112	C₂	semidirect
16. #436×C₈₉	C₄	semidirect
18. C₈×D₄×C₈₉	C₄	semidirect
140. C₈₉	#8680	semidirect
144. #436	Dic₈₉	semidirect
146. C₈×D₄	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020383	2
#26019000	2
#26018812	2
#26018112	2
C₈₉×#2931	2
#8680	89