Subgroups — #36448090

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020368	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26018087	C₂	1	1	yes	maximal, characteristic
4	#26018086	C₂	1	2	yes	maximal
5	C₈₉×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
12	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
13	#19151352	—	2	2	no	—
14	#19151352	—	2	2	no	—
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×#10623494	—	2	2	no	—
18	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₈₉	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₈₉	Q₈	1	1	yes	characteristic
26	C₄⋊#10623494	—	2	2	no	—
27	C₄⋊#10623494	—	2	2	no	—
28	C₄×#10623494	—	2	2	no	—
29	C₈×Dic₈₉	—	2	2	no	—
30	C₈×Dic₈₉	—	2	2	no	—
31	C₈×Dic₈₉	—	2	2	no	—
32	C₈×Dic₈₉	C₈	1	4	yes	complemented
33	C₂²×C₈×C₈₉	—	2	2	no	—
34	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₈₉	—	2	2	no	—
38	C₄⋊C₈×C₈₉	—	2	2	no	—
39	C₄×C₈×C₈₉	—	2	2	no	—
40	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#10623494	—	2	1	no	—
45	C₂×#10623494	—	2	1	no	—
46	C₂×#10623494	—	2	1	no	—
47	C₂×#10623494	—	2	1	no	—
48	C₂×#10623494	—	4	2	no	—
49	C₂×#10623494	—	4	2	no	—
50	C₂×#10623494	—	4	2	no	—
51	C₂×#10623494	C₂×C₈	1	2	yes	—
52	C₂×#10623494	M₁₆	1	2	yes	—
53	C₄×Dic₈₉	—	2	1	no	—
54	C₄×Dic₈₉	—	2	1	no	—
55	C₄×Dic₈₉	—	2	1	no	—
56	C₄×Dic₈₉	—	2	2	no	—
57	C₄×Dic₈₉	—	2	1	no	—
58	C₄×Dic₈₉	—	2	2	no	—
59	C₄×Dic₈₉	—	2	2	no	—
60	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
61	C₄×Dic₈₉	M₁₆	1	2	yes	—
62	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₈₉	—	2	1	no	—
66	C₂×C₈×C₈₉	—	2	1	no	—
67	C₂×C₈×C₈₉	—	2	1	no	—
68	C₂×C₈×C₈₉	—	2	1	no	—
69	C₂×C₈×C₈₉	—	4	2	no	—
70	C₂×C₈×C₈₉	—	4	2	no	—
71	C₂×C₈×C₈₉	—	4	2	no	—
72	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
74	C₄²×C₈₉	#41	1	2	yes	—
75	C₄²×C₈₉	#41	1	2	yes	—
76	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	—	2	1	no	—
78	C₂×C₄×C₈₉	—	2	1	no	—
79	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#124	1	2	yes	—
82	C₂×C₄×C₈₉	#125	1	2	yes	—
83	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#135	1	2	yes	—
89	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
90	C₂×Dic₈₉	—	2	1	no	—
91	C₂×Dic₈₉	—	2	1	no	—
92	C₂×Dic₈₉	—	2	1	no	—
93	C₂×Dic₈₉	—	2	1	no	—
94	C₂×Dic₈₉	—	2	1	no	—
95	C₂×Dic₈₉	—	2	1	no	—
96	C₂×Dic₈₉	—	2	1	no	—
97	C₂×Dic₈₉	—	2	1	no	—
98	C₂×Dic₈₉	C₄⋊C₈	1	2	yes	—
99	C₂×Dic₈₉	C₄⋊C₈	1	2	yes	—
100	C₂×Dic₈₉	#108	1	2	yes	—
101	C₂×Dic₈₉	#108	1	2	yes	—
102	C₈×C₈₉	—	2	2	no	—
103	C₈×C₈₉	—	2	2	no	—
104	C₈×C₈₉	—	8	2	no	—
105	#10623494	—	2	2	no	—
106	#10623494	—	2	2	no	—
107	#10623494	—	8	2	no	—
108	C₂²×C₈₉	#382	1	1	yes	characteristic
109	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂²×C₈₉	#423	1	1	yes	characteristic
111	C₂²×C₈₉	#427	1	1	yes	characteristic
112	C₂²×C₈₉	#427	1	1	yes	characteristic
113	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
114	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
115	C₄×C₈₉	—	2	2	no	—
116	C₄×C₈₉	#431	1	1	yes	characteristic
117	C₄×C₈₉	#431	1	1	yes	characteristic
118	C₄×C₈₉	#432	1	1	yes	characteristic
119	C₄×C₈₉	#432	1	1	yes	characteristic
120	C₄×C₈₉	#488	1	2	yes	—
121	C₄×C₈₉	#488	1	2	yes	—
122	Dic₈₉	—	2	2	no	—
123	Dic₈₉	—	2	2	no	—
124	Dic₈₉	—	2	2	no	—
125	Dic₈₉	—	2	2	no	—
126	#8785	—	89	1	no	maximal, Sylow
127	C₂×C₈₉	#1564	1	1	yes	characteristic
128	C₂×C₈₉	#1564	1	1	yes	characteristic
129	C₂×C₈₉	#1568	1	1	yes	characteristic
130	C₂×C₈₉	#1568	1	1	yes	characteristic
131	C₂×C₈₉	#1831	1	1	yes	characteristic
132	C₂×C₈₉	#1835	1	1	yes	characteristic
133	C₂×C₈₉	#1837	1	1	yes	characteristic
134	#1831	—	89	2	no	—
135	C₄²×C₈	—	89	1	no	—
136	#1277	—	89	2	no	—
137	#1277	—	89	1	no	—
138	#1277	D₈₉	1	1	yes	characteristic
139	C₈₉	#8785	1	1	yes	characteristic, complemented, Sylow
140	C₂×C₄⋊C₈	—	89	2	no	—
141	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
142	C₂×C₄×C₈	—	89	1	no	—
143	C₂×C₄×C₈	—	89	1	no	—
144	C₂×C₄×C₈	—	89	1	no	—
145	C₂×C₄×C₈	—	89	1	no	—
146	C₂×C₄×C₈	—	89	1	no	—
147	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
148	C₄³	—	89	1	no	—
149	#336	—	178	2	no	—
150	#336	—	178	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	89	1	no	—
153	C₂²×C₈	—	89	1	no	—
154	C₂²×C₈	—	178	2	no	—
155	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
156	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
157	C₂×C₄²	—	89	1	no	—
158	C₂×C₄²	—	89	1	no	—
159	C₂×C₄²	—	89	1	no	—
160	C₂×C₄²	—	89	1	no	—
161	C₂×C₄²	—	89	1	no	—
162	C₂×C₄²	—	89	1	no	—
163	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	178	2	no	—
167	C₄⋊C₈	—	178	2	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	—	89	4	no	—
170	C₄×C₈	—	178	2	no	—
171	C₄×C₈	—	178	2	no	—
172	C₄×C₈	—	178	2	no	—
173	C₄×C₈	—	178	2	no	—
174	C₂²×C₄	—	89	1	no	—
175	C₂²×C₄	—	89	1	no	—
176	C₂²×C₄	—	89	1	no	—
177	C₂²×C₄	—	89	1	no	—
178	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
179	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
180	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	89	2	no	—
189	C₂×C₈	—	89	2	no	—
190	C₂×C₈	—	356	2	no	—
191	C₂×C₈	—	356	2	no	—
192	C₂×C₈	—	356	2	no	—
193	C₂×C₈	—	178	1	no	—
194	C₂×C₈	—	178	1	no	—
195	C₂×C₈	—	178	1	no	—
196	C₂×C₈	—	178	1	no	—
197	C₂×C₈	C₈×D₈₉	1	2	yes	—
198	C₂×C₈	#13083557	1	2	yes	—
199	C₄²	—	89	2	no	—
200	C₄²	—	89	2	no	—
201	C₄²	—	178	2	no	—
202	C₄²	—	178	2	no	—
203	C₄²	—	178	1	no	—
204	C₄²	—	178	1	no	—
205	C₄²	—	178	1	no	—
206	C₄²	—	178	2	no	—
207	C₄²	—	178	1	no	—
208	C₄²	#13083573	1	2	yes	—
209	C₄²	#13083574	1	2	yes	—
210	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	89	2	no	—
214	C₂×C₄	—	89	2	no	—
215	C₂×C₄	—	89	2	no	—
216	C₂×C₄	—	89	2	no	—
217	C₂×C₄	—	178	1	no	—
218	C₂×C₄	—	178	1	no	—
219	C₂×C₄	—	178	1	no	—
220	C₂×C₄	—	178	1	no	—
221	C₂×C₄	—	178	1	no	—
222	C₂×C₄	—	178	1	no	—
223	C₂×C₄	—	178	1	no	—
224	C₂×C₄	—	178	1	no	—
225	C₂×C₄	#15667968	1	1	yes	characteristic
226	C₂×C₄	#15667969	1	1	yes	characteristic
227	C₂×C₄	#15667998	1	2	yes	—
228	C₂×C₄	#15668002	1	2	yes	—
229	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
230	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
231	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
232	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
233	C₂×C₄	#15668031	1	1	yes	characteristic
234	C₂×C₄	#15668033	1	2	yes	—
235	C₂×C₄	#15668035	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	8	2	no	—
239	C₈	—	178	2	no	—
240	C₈	—	178	2	no	—
241	C₈	—	712	2	no	—
242	C₂²	#19151462	1	1	yes	characteristic
243	C₂²	C₄⋊C₈×D₈₉	1	1	yes	characteristic
244	C₂²	#19151620	1	1	yes	characteristic
245	C₂²	#19151633	1	1	yes	characteristic
246	C₂²	#19151635	1	1	yes	characteristic
247	C₂²	#108×D₈₉	1	1	yes	characteristic
248	C₂²	#108×D₈₉	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	178	2	no	—
251	C₄	—	178	2	no	—
252	C₄	—	178	2	no	—
253	C₄	—	178	2	no	—
254	C₄	#19151648	1	1	yes	characteristic
255	C₄	#19151649	1	1	yes	characteristic
256	C₄	#19151652	1	1	yes	characteristic
257	C₄	#19151653	1	1	yes	characteristic
258	C₄	#19151951	1	2	yes	—
259	C₄	#19151954	1	2	yes	—
260	C₂	#26018997	1	1	yes	characteristic
261	C₂	#26018999	1	1	yes	characteristic
262	C₂	#26019012	1	1	yes	characteristic
263	C₂	#26019014	1	1	yes	characteristic
264	C₂	#26020367	1	1	yes	characteristic
265	C₂	#26020381	1	1	yes	characteristic
266	C₂	#26020388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1277	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1277	D₈₉
hypocenter	C₈₉	#8785
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
32. C₈×Dic₈₉	C₈	semidirect
139. C₈₉	#8785	semidirect

Maximal subgroups

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

subgroup	index
#26020368	2
C₄×C₈×Dic₈₉	2
#26018087	2
#26018086	2
C₈₉×#1277	2
#8785	89