Subgroups — #36470649

Order 22784.

440 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151553	C₂	1	1	yes	maximal, characteristic
2	#26021863	C₂	1	1	yes	maximal, characteristic
3	#26018340	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₄	1	1	yes	characteristic
6	C₂×#15667971	C₄	1	1	yes	characteristic
7	C₂×#15667952	C₂²	1	2	yes	—
8	#19151553	—	2	1	no	—
9	#19151553	—	2	1	no	—
10	#19151553	C₄	1	2	yes	—
11	#19151553	C₂²	1	2	yes	—
12	C₂×#15667922	—	2	1	no	—
13	C₂×C₈×Dic₈₉	—	2	1	no	—
14	C₂²⋊#13083549	—	2	1	no	—
15	C₂²⋊#13083549	—	2	1	no	—
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
19	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
20	C₂×#13083558	—	2	4	no	—
21	C₂×#13083558	—	2	1	no	—
22	C₂×#13083558	C₂×C₄	1	2	yes	—
23	C₂²×#10623494	—	2	1	no	—
24	#15667971	—	2	1	no	—
25	#15667971	—	2	1	no	—
26	#15667971	C₈	1	2	yes	—
27	#15667971	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
29	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
31	C₂×C₄×Dic₈₉	—	2	1	no	—
32	#15667952	—	2	4	no	—
33	C₂×#13083549	—	2	2	no	—
34	#15667922	—	2	2	no	—
35	#15667922	—	2	2	no	—
36	C₈×Dic₈₉	—	2	2	no	—
37	C₈×Dic₈₉	—	2	2	no	—
38	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	—	2	4	no	—
40	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₈₉	—	2	2	no	—
47	C₂×C₁₆×C₈₉	—	2	4	no	—
48	C₂²×Dic₈₉	—	2	1	no	—
49	C₂²×Dic₈₉	—	2	1	no	—
50	#13083558	—	2	4	no	—
51	#13083558	—	2	2	no	—
52	#13083558	M₁₆	1	4	yes	—
53	C₂×#10623494	—	2	4	no	—
54	C₂×#10623494	—	2	2	no	—
55	C₄⋊Dic₈₉	—	2	1	no	—
56	C₄⋊Dic₈₉	—	2	1	no	—
57	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
58	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
59	C₂²⋊Dic₈₉	—	2	2	no	—
60	C₂²⋊Dic₈₉	—	2	2	no	—
61	C₄×Dic₈₉	—	2	2	no	—
62	C₄×Dic₈₉	—	2	2	no	—
63	#13083549	—	4	2	no	—
64	C₂⁴×C₈₉	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	—	2	2	no	—
66	C₂²×C₄×C₈₉	—	2	2	no	—
67	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	—	2	2	no	—
74	C₂×C₈×C₈₉	—	2	2	no	—
75	C₂×C₈×C₈₉	—	2	2	no	—
76	C₂×C₈×C₈₉	—	2	2	no	—
77	C₂×C₈×C₈₉	—	2	2	no	—
78	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	#41	1	4	yes	—
83	C₁₆×C₈₉	—	4	2	no	—
84	C₂³×C₈₉	—	2	2	no	—
85	C₂³×C₈₉	—	2	2	no	—
86	C₂³×C₈₉	—	4	1	no	—
87	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
88	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
89	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#117	1	2	yes	—
101	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
103	C₂×Dic₈₉	—	2	2	no	—
104	C₂×Dic₈₉	—	2	2	no	—
105	C₂×Dic₈₉	—	4	2	no	—
106	C₂×Dic₈₉	—	4	2	no	—
107	C₈×C₈₉	—	2	1	no	—
108	C₈×C₈₉	—	2	1	no	—
109	C₈×C₈₉	#123	1	2	yes	—
110	C₈×C₈₉	#123	1	2	yes	—
111	#10623494	—	2	4	no	—
112	C₂²×C₈₉	—	2	1	no	—
113	C₂²×C₈₉	—	2	1	no	—
114	C₂²×C₈₉	—	2	1	no	—
115	C₂²×C₈₉	—	2	1	no	—
116	C₂²×C₈₉	—	2	1	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	—	2	1	no	—
119	C₂²×C₈₉	—	4	4	no	—
120	C₂²×C₈₉	#350	1	2	yes	—
121	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₈₉	#432	1	2	yes	—
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	—	2	1	no	—
125	C₄×C₈₉	#381	1	1	yes	characteristic
126	C₄×C₈₉	#407	1	1	yes	characteristic
127	C₄×C₈₉	#431	1	2	yes	—
128	Dic₈₉	—	4	2	no	—
129	Dic₈₉	—	4	2	no	—
130	#12848	—	89	1	no	maximal, Sylow
131	C₂×C₈₉	—	2	2	no	—
132	C₂×C₈₉	—	2	2	no	—
133	C₂×C₈₉	—	4	1	no	—
134	C₂×C₈₉	#1817	1	1	yes	characteristic
135	C₂×C₈₉	C₂×#350	1	1	yes	characteristic
136	C₂×C₈₉	#2117	1	1	yes	characteristic
137	C₂×#405	—	89	1	no	—
138	#2112	—	89	1	no	—
139	C₂×C₂²⋊C₁₆	D₈₉	1	1	yes	characteristic
140	#1379	—	89	4	no	—
141	C₈₉	#12848	1	1	yes	characteristic, complemented, Sylow
142	C₂²×M₁₆	—	89	1	no	—
143	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
144	C₂×#117	—	89	1	no	—
145	C₂×M₃₂	—	89	2	no	—
146	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
147	#405	—	89	2	no	—
148	#405	—	89	2	no	—
149	#405	—	178	1	no	—
150	#405	—	178	1	no	—
151	C₂×#97	—	178	1	no	—
152	C₂×C₄×C₈	—	178	1	no	—
153	C₂²⋊C₁₆	—	2	1	no	—
154	C₂²⋊C₁₆	—	2	1	no	—
155	C₂²⋊C₁₆	—	178	1	no	—
156	C₂²⋊C₁₆	—	178	1	no	—
157	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
158	C₂×M₁₆	—	89	2	no	—
159	C₂×M₁₆	—	178	4	no	—
160	C₂×M₁₆	—	178	1	no	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	—	178	1	no	—
163	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
164	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
165	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
169	#117	—	89	2	no	—
170	#117	—	89	2	no	—
171	#117	—	178	1	no	—
172	#117	—	178	1	no	—
173	C₂×C₄⋊C₄	—	89	1	no	—
174	C₂×C₄⋊C₄	—	89	1	no	—
175	C₂×C₂²⋊C₄	—	178	1	no	—
176	C₂×C₄²	—	178	1	no	—
177	M₃₂	—	178	4	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	—	178	2	no	—
181	#97	—	178	2	no	—
182	#97	—	178	2	no	—
183	C₄×C₈	—	178	2	no	—
184	C₄×C₈	—	178	2	no	—
185	C₂⁴	#13083557	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	178	1	no	—
189	C₂²×C₄	—	178	1	no	—
190	C₂²×C₄	#13083551	1	1	yes	characteristic
191	C₂²×C₄	#13083551	1	1	yes	characteristic
192	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
193	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
194	C₂²×C₄	#13083557	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
196	M₁₆	—	89	4	no	—
197	M₁₆	—	178	4	no	—
198	M₁₆	—	178	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	178	4	no	—
205	C₂×C₈	—	178	2	no	—
206	C₂×C₈	#13083551	1	1	yes	characteristic
207	C₂×C₈	#13083551	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
210	C₂×C₈	#13083573	1	4	yes	—
211	C₄⋊C₄	—	89	2	no	—
212	C₄⋊C₄	—	89	2	no	—
213	C₄⋊C₄	—	178	1	no	—
214	C₄⋊C₄	—	178	1	no	—
215	C₂²⋊C₄	—	178	2	no	—
216	C₂²⋊C₄	—	178	2	no	—
217	C₄²	—	178	2	no	—
218	C₄²	—	178	2	no	—
219	C₁₆	—	4	2	no	—
220	C₁₆	—	356	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	1	no	—
224	C₂³	#15667924	1	1	yes	characteristic
225	C₂³	#15667924	1	1	yes	characteristic
226	C₂³	C₂×#13083557	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	356	2	no	—
235	C₂×C₄	—	356	2	no	—
236	C₂×C₄	—	178	2	no	—
237	C₂×C₄	—	178	2	no	—
238	C₂×C₄	#15667924	1	1	yes	characteristic
239	C₂×C₄	#15667924	1	1	yes	characteristic
240	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
241	C₂×C₄	#15667968	1	2	yes	—
242	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
243	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	178	4	no	—
247	C₈	#15667993	1	2	yes	—
248	C₈	#15667996	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	#19151381	1	2	yes	—
258	C₂²	C₂×#15667924	1	1	yes	characteristic
259	C₂²	#19151652	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	356	2	no	—
263	C₄	—	356	2	no	—
264	C₄	#19151459	1	1	yes	characteristic
265	C₄	#19151561	1	1	yes	characteristic
266	C₄	#19151648	1	2	yes	—
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	#26020318	1	1	yes	characteristic
271	C₂	C₂×#19151381	1	1	yes	characteristic
272	C₂	#26021879	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₂²⋊C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₈₉	M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₈₉
hypocenter	C₈₉	#12848
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #26018340	C₂	semidirect
141. C₈₉	#12848	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151553	2
#26021863	2
#26018340	2
C₂×C₂²⋊C₁₆×C₈₉	2
#12848	89