Subgroups — #36707840

Order 22784.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033388	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151714	C₂	1	2	yes	maximal, complemented
3	#26022137	C₂	1	4	yes	maximal, complemented
4	#26022132	C₂	1	4	yes	maximal, complemented
5	#26021979	C₂	1	2	yes	maximal, complemented
6	M₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2161	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	2	yes	—
9	C₂×#15668020	—	2	2	no	—
10	#19152041	C₄	1	4	yes	complemented
11	#19152036	C₄	1	4	yes	complemented
12	#19152030	C₂²	1	4	yes	—
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	#19151714	—	2	2	no	—
15	#19151714	—	2	2	no	—
16	#19151714	C₂²	1	4	yes	complemented
17	#19151714	C₂²	1	4	yes	complemented
18	C₂×#15667938	—	2	2	no	—
19	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
20	#19151404	C₂²	1	8	yes	complemented
21	#19151373	C₂²	1	2	yes	complemented
22	C₁₆×Dic₈₉	C₂²	1	2	yes	complemented
23	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#108×C₈₉	C₂²	1	2	yes	—
25	#368×C₈₉	C₂²	1	4	yes	complemented
26	C₂×#13083574	—	2	2	no	—
27	#15668034	—	2	2	no	—
28	#15668034	—	2	2	no	—
29	#15668034	C₂×C₄	1	4	yes	—
30	#15668034	C₂³	1	4	yes	—
31	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
32	C₂×#13083563	—	2	2	no	—
33	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
34	#15668020	—	2	4	no	—
35	#15668020	—	2	4	no	—
36	#15668020	—	4	2	no	—
37	#15668020	—	4	2	no	—
38	C₂×#13083557	—	4	2	no	—
39	C₂²×#10623494	D₄	1	1	yes	characteristic
40	C₂×C₈×D₈₉	—	4	2	no	—
41	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
42	Dic₇₁₂	C₂×C₄	1	4	yes	complemented
43	#15667957	C₂×C₄	1	8	yes	complemented
44	D₁₄₂₄	C₂×C₄	1	4	yes	complemented
45	#15667952	—	2	2	no	—
46	C₂×#13083549	C₂³	1	2	yes	—
47	#15667938	—	2	4	no	—
48	#15667938	—	2	4	no	—
49	C₈×Dic₈₉	D₄	1	1	yes	characteristic
50	C₈×Dic₈₉	D₄	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
52	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₈₉	—	2	2	no	—
54	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	M₃₂×C₈₉	C₂×C₄	1	4	yes	—
56	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
57	#108×C₈₉	—	2	2	no	—
58	#108×C₈₉	C₂³	1	4	yes	—
59	#13083574	—	2	4	no	—
60	#13083574	—	2	4	no	—
61	#13083574	—	4	2	no	—
62	#13083574	—	4	2	no	—
63	C₂×Dic₁₇₈	—	2	2	no	—
64	C₂×D₃₅₆	—	2	2	no	—
65	C₂×C₂²⋊D₈₉	—	4	2	no	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	4	2	no	—
68	Dic₃₅₆	—	2	2	no	—
69	Dic₃₅₆	C₂²×C₄	1	4	yes	—
70	#13083563	—	2	4	no	—
71	#13083563	—	2	4	no	—
72	D₇₁₂	—	2	2	no	—
73	D₇₁₂	C₂²×C₄	1	4	yes	—
74	#13083557	—	4	4	no	—
75	#13083557	—	4	4	no	—
76	C₂×#10623494	D₈	1	2	yes	—
77	C₂×#10623494	D₈	1	2	yes	—
78	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
79	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
80	C₈×D₈₉	—	4	4	no	—
81	C₈×D₈₉	—	4	4	no	—
82	C₄×Dic₈₉	—	2	1	no	—
83	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
84	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
85	#13083549	—	2	2	no	—
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	M₁₆×C₈₉	—	2	4	no	—
88	M₁₆×C₈₉	—	4	2	no	—
89	C₂×C₈×C₈₉	—	4	2	no	—
90	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	#41	1	2	yes	—
93	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
94	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
95	C₂³×C₈₉	#135	1	1	yes	characteristic
96	C₂²×D₈₉	—	4	2	no	—
97	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₂²⋊D₈₉	—	4	4	no	—
102	C₂²⋊D₈₉	—	4	4	no	—
103	C₂×Dic₈₉	—	2	1	no	—
104	C₂×Dic₈₉	—	2	1	no	—
105	C₂×Dic₈₉	—	4	2	no	—
106	C₂×Dic₈₉	—	4	2	no	—
107	D₃₅₆	—	2	4	no	—
108	D₃₅₆	—	4	2	no	—
109	C₄×D₈₉	—	4	4	no	—
110	C₄×D₈₉	—	4	4	no	—
111	Dic₁₇₈	—	2	4	no	—
112	Dic₁₇₈	—	4	2	no	—
113	C₈×C₈₉	—	4	4	no	—
114	C₈×C₈₉	C₄×D₄	1	2	yes	—
115	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
116	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
117	#10623494	C₂×D₈	1	1	yes	characteristic
118	#10623494	C₂×D₈	1	2	yes	—
119	#10623494	C₂×D₈	1	1	yes	characteristic
120	C₂²×C₈₉	—	4	1	no	—
121	C₂²×C₈₉	C₄×D₈	1	2	yes	—
122	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
123	C₂×D₈₉	—	4	4	no	—
124	C₂×D₈₉	—	8	4	no	—
125	C₄×C₈₉	C₄×D₈	1	2	yes	—
126	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
128	Dic₈₉	—	2	2	no	—
129	Dic₈₉	—	4	1	no	—
130	Dic₈₉	—	4	4	no	—
131	#33463	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	—	2	1	no	—
133	C₂×C₈₉	—	2	2	no	—
134	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
135	D₈₉	—	8	4	no	—
136	#3414	—	89	1	no	—
137	C₂×#443	—	89	2	no	—
138	#2179	—	89	4	no	—
139	#2178	—	89	4	no	—
140	#2161	D₈₉	1	1	yes	characteristic, complemented
141	#2143	—	89	2	no	—
142	C₄×M₃₂	—	89	1	no	—
143	C₈₉	#33463	1	1	yes	characteristic, complemented, Sylow
144	C₂×#135	—	89	2	no	—
145	C₂×#131	—	178	2	no	—
146	#510	—	89	4	no	—
147	#509	—	89	4	no	—
148	#508	—	89	4	no	—
149	C₂×M₃₂	—	89	1	no	—
150	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
151	#443	—	89	4	no	—
152	#443	—	89	4	no	—
153	#443	—	178	2	no	—
154	#443	—	178	2	no	—
155	C₂×#108	C₂×D₈₉	1	2	yes	—
156	C₂×#104	—	178	2	no	—
157	C₂×C₄×C₈	—	89	1	no	—
158	#368	C₂×D₈₉	1	4	yes	complemented
159	#359	—	89	8	no	—
160	#346	—	89	2	no	—
161	C₄×C₁₆	—	89	2	no	—
162	C₂×#41	—	178	2	no	—
163	#135	—	89	4	no	—
164	#135	—	89	4	no	—
165	#135	—	178	2	no	—
166	#135	—	178	2	no	—
167	C₂×Q₁₆	—	89	2	no	—
168	C₂×SD₁₆	—	178	2	no	—
169	C₂×D₈	—	89	2	no	—
170	#131	—	356	2	no	—
171	#131	—	356	2	no	—
172	#131	—	178	4	no	—
173	#131	—	178	4	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	356	2	no	—
176	C₂²×C₈	—	89	1	no	—
177	C₂²×C₈	—	356	2	no	—
178	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
179	C₂×C₄²	—	89	1	no	—
180	Q₃₂	—	89	4	no	—
181	SD₃₂	—	89	8	no	—
182	D₁₆	—	89	4	no	—
183	M₃₂	—	178	2	no	—
184	M₃₂	C₄×D₈₉	1	4	yes	—
185	C₂×C₁₆	—	89	2	no	—
186	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
187	#108	—	2	2	no	—
188	#108	C₂²×D₈₉	1	4	yes	—
189	#104	—	178	4	no	—
190	#104	—	178	4	no	—
191	C₄×C₈	—	89	1	no	—
192	C₄×C₈	—	89	1	no	—
193	C₄×C₈	—	89	1	no	—
194	C₄×C₈	—	89	1	no	—
195	#41	—	356	2	no	—
196	#41	—	356	2	no	—
197	#41	—	178	4	no	—
198	#41	—	178	4	no	—
199	C₂×Q₈	—	178	2	no	—
200	C₂×D₄	—	356	2	no	—
201	C₂×D₄	—	178	2	no	—
202	C₂²×C₄	—	356	2	no	—
203	C₂²×C₄	—	178	1	no	—
204	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
205	Q₁₆	—	89	4	no	—
206	Q₁₆	—	178	2	no	—
207	SD₁₆	—	178	4	no	—
208	SD₁₆	—	178	4	no	—
209	D₈	—	89	4	no	—
210	D₈	—	178	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	356	4	no	—
214	M₁₆	—	356	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	89	2	no	—
217	C₂×C₈	—	89	2	no	—
218	C₂×C₈	—	89	1	no	—
219	C₂×C₈	—	89	1	no	—
220	C₂×C₈	—	356	4	no	—
221	C₂×C₈	—	356	4	no	—
222	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
223	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
224	C₂×C₈	#13083575	1	2	yes	—
225	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
226	C₄²	—	89	1	no	—
227	C₄²	—	89	1	no	—
228	C₄²	—	178	1	no	—
229	C₁₆	—	178	2	no	—
230	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
231	C₂³	—	356	2	no	—
232	C₂³	#15668035	1	1	yes	characteristic
233	Q₈	—	356	2	no	—
234	Q₈	—	178	4	no	—
235	D₄	—	356	4	no	—
236	D₄	—	356	4	no	—
237	D₄	—	356	2	no	—
238	D₄	—	178	4	no	—
239	C₂×C₄	—	356	2	no	—
240	C₂×C₄	—	356	2	no	—
241	C₂×C₄	—	356	4	no	—
242	C₂×C₄	—	356	4	no	—
243	C₂×C₄	—	178	1	no	—
244	C₂×C₄	—	178	1	no	—
245	C₂×C₄	#15667974	1	2	yes	—
246	C₂×C₄	D₈×D₈₉	1	2	yes	—
247	C₂×C₄	#15668035	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	—	89	1	no	—
251	C₈	—	89	2	no	—
252	C₈	—	89	1	no	—
253	C₈	#15667974	1	2	yes	—
254	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
255	C₈	C₂×#13083575	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	356	4	no	—
258	C₂²	—	712	4	no	—
259	C₂²	#19151680	1	2	yes	—
260	C₂²	C₂×#15668035	1	1	yes	characteristic
261	C₄	—	356	1	no	—
262	C₄	—	356	4	no	—
263	C₄	—	178	2	no	—
264	C₄	#19151680	1	2	yes	—
265	C₄	C₂×#15667974	1	1	yes	characteristic
266	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	712	4	no	—
270	C₂	C₂×#19151680	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2161	C₂
socle	C₂×C₈₉	C₂×C₄×D₈
prime radical	#26033388	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2161	D₈₉
hypocenter	C₈₉	#33463
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

Complements

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

N	H ≅ G/N	type
2. C₂×#19151714	C₂	semidirect
3. #26022137	C₂	semidirect
4. #26022132	C₂	semidirect
5. #26021979	C₂	semidirect
6. M₃₂×Dic₈₉	C₂	semidirect
7. C₈₉×#2161	C₂	semidirect
10. #19152041	C₄	semidirect
11. #19152036	C₄	semidirect
16. #19151714	C₂²	semidirect
17. #19151714	C₂²	semidirect
20. #19151404	C₂²	semidirect
21. #19151373	C₂²	semidirect
22. C₁₆×Dic₈₉	C₂²	semidirect
25. #368×C₈₉	C₂²	semidirect
42. Dic₇₁₂	C₂×C₄	semidirect
43. #15667957	C₂×C₄	semidirect
44. D₁₄₂₄	C₂×C₄	semidirect
140. #2161	D₈₉	semidirect
143. C₈₉	#33463	semidirect
158. #368	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033388	2
C₂×#19151714	2
#26022137	2
#26022132	2
#26021979	2
M₃₂×Dic₈₉	2
C₈₉×#2161	2
#33463	89