Subgroups — #36707669

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152015	C₂	1	2	yes	maximal, complemented
2	#26022094	C₂	1	4	yes	maximal, complemented
3	#26022089	C₂	1	4	yes	maximal, complemented
4	#26022018	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021880	C₂	1	2	yes	maximal, complemented
6	#26021849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2955	C₂	1	1	yes	maximal, characteristic
8	#19152015	—	2	2	no	—
9	#19152015	—	2	2	no	—
10	#19152015	C₂²	1	4	yes	complemented
11	#19152015	C₂²	1	4	yes	complemented
12	C₂×#15667952	—	2	2	no	—
13	C₂×#15667952	C₂²	1	2	yes	—
14	C₂×#15667952	C₂²	1	1	yes	characteristic
15	C₂²×#13083549	—	2	2	no	—
16	C₂²×#13083549	C₂²	1	1	yes	characteristic
17	#19151419	C₂²	1	4	yes	complemented
18	#19151382	C₂²	1	8	yes	complemented
19	#19151374	C₂²	1	2	yes	complemented
20	C₄×#13083549	C₂²	1	2	yes	complemented
21	#575×C₈₉	C₄	1	1	yes	characteristic
22	C₂×#131×C₈₉	C₂²	1	2	yes	—
23	#444×C₈₉	C₄	1	4	yes	—
24	#443×C₈₉	C₂²	1	4	yes	—
25	#430×C₈₉	C₄	1	1	yes	characteristic
26	#421×C₈₉	C₄	1	2	yes	—
27	#405×C₈₉	C₂²	1	1	yes	characteristic
28	#15667952	—	2	4	no	—
29	#15667952	—	2	4	no	—
30	#15667952	—	2	2	no	—
31	#15667952	—	2	2	no	—
32	#15667952	C₂³	1	4	yes	—
33	C₂×#13083549	—	2	4	no	—
34	C₂×#13083549	—	2	4	no	—
35	C₂×#13083549	—	2	2	no	—
36	C₂×#13083549	D₄	1	2	yes	—
37	C₂×#13083549	D₄	1	2	yes	—
38	C₂×#13083549	C₂³	1	2	yes	—
39	C₂×#13083549	C₂³	1	1	yes	characteristic
40	C₂×#13083549	C₂³	1	1	yes	characteristic
41	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
42	#137×C₈₉	C₈	1	4	yes	—
43	AGL(1,ℤ/8)×C₈₉	C₈	1	4	yes	—
44	#135×C₈₉	C₂×C₄	1	4	yes	—
45	#131×C₈₉	—	2	2	no	—
46	#131×C₈₉	—	2	2	no	—
47	#131×C₈₉	C₂×C₄	1	4	yes	—
48	#131×C₈₉	C₂³	1	4	yes	—
49	C₂×M₁₆×C₈₉	—	2	2	no	—
50	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
51	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	—	2	2	no	—
53	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
55	#108×C₈₉	C₂×C₄	1	4	yes	—
56	#104×C₈₉	C₂×C₄	1	8	yes	—
57	#97×C₈₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₈₉	C₂³	1	2	yes	—
59	#13083549	—	2	4	no	—
60	#13083549	—	2	2	no	—
61	#13083549	C₂×D₄	1	1	yes	characteristic
62	#13083549	C₂×D₄	1	2	yes	—
63	#13083549	C₂×D₄	1	1	yes	characteristic
64	#41×C₈₉	—	2	2	no	—
65	#41×C₈₉	—	2	2	no	—
66	#41×C₈₉	C₂×C₈	1	4	yes	—
67	#41×C₈₉	C₂²×C₄	1	4	yes	—
68	C₂×Q₈×C₈₉	C₂×C₈	1	2	yes	—
69	C₂×D₄×C₈₉	—	2	2	no	—
70	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₈₉	—	2	2	no	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	Q₁₆×C₈₉	C₂×C₈	1	4	yes	—
74	SD₁₆×C₈₉	C₂×C₈	1	8	yes	—
75	D₈×C₈₉	C₂×C₈	1	4	yes	—
76	M₁₆×C₈₉	—	2	2	no	—
77	M₁₆×C₈₉	—	2	4	no	—
78	M₁₆×C₈₉	—	2	4	no	—
79	M₁₆×C₈₉	C₂×C₈	1	4	yes	—
80	M₁₆×C₈₉	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	2	4	no	—
83	C₂×C₈×C₈₉	—	2	4	no	—
84	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₈₉	#41	1	2	yes	—
88	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₈₉	—	2	1	no	—
90	C₂²⋊C₄×C₈₉	—	2	1	no	—
91	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
92	C₂³×C₈₉	—	2	2	no	—
93	C₂³×C₈₉	#131	1	1	yes	characteristic
94	Q₈×C₈₉	—	2	2	no	—
95	Q₈×C₈₉	C₂²×C₈	1	4	yes	—
96	D₄×C₈₉	—	2	4	no	—
97	D₄×C₈₉	—	2	4	no	—
98	D₄×C₈₉	—	2	2	no	—
99	D₄×C₈₉	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₈₉	—	2	2	no	—
101	C₂×C₄×C₈₉	—	2	4	no	—
102	C₂×C₄×C₈₉	—	2	4	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₈₉	—	2	4	no	—
109	C₈×C₈₉	C₄×D₄	1	2	yes	—
110	C₈×C₈₉	C₂²×C₈	1	4	yes	—
111	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
113	C₂²×C₈₉	—	2	4	no	—
114	C₂²×C₈₉	—	4	1	no	—
115	C₂²×C₈₉	—	4	4	no	—
116	C₂²×C₈₉	C₈×D₄	1	2	yes	—
117	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
118	C₄×C₈₉	—	2	4	no	—
119	C₄×C₈₉	—	4	2	no	—
120	C₄×C₈₉	C₈×D₄	1	2	yes	—
121	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
123	#33441	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	—	2	1	no	—
125	C₂×C₈₉	—	2	2	no	—
126	C₂×C₈₉	—	4	4	no	—
127	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
128	C₂×#504	—	89	2	no	—
129	#2955	D₈₉	1	1	yes	characteristic
130	#2171	—	89	4	no	—
131	#2170	—	89	4	no	—
132	#2153	—	89	1	no	—
133	#2117	—	89	2	no	—
134	#2108	—	89	1	no	—
135	C₈₉	#33441	1	1	yes	characteristic, complemented, Sylow
136	#575	Dic₈₉	1	1	yes	characteristic
137	C₂×#131	C₂×D₈₉	1	2	yes	—
138	#504	—	89	4	no	—
139	#504	—	89	4	no	—
140	#504	—	178	2	no	—
141	#504	—	178	2	no	—
142	C₂×M₃₂	—	89	1	no	—
143	C₂×M₃₂	—	89	2	no	—
144	C₂×M₃₂	—	178	2	no	—
145	C₂²×C₁₆	—	89	1	no	—
146	C₂²×C₁₆	—	178	2	no	—
147	#444	Dic₈₉	1	4	yes	—
148	#443	C₂×D₈₉	1	4	yes	—
149	#430	Dic₈₉	1	1	yes	characteristic
150	#421	Dic₈₉	1	2	yes	—
151	#405	C₂×D₈₉	1	1	yes	characteristic
152	#364	—	89	4	no	—
153	#350	—	89	8	no	—
154	#346	—	89	2	no	—
155	C₄×C₁₆	—	89	2	no	—
156	C₂×#41	C₂×Dic₈₉	1	2	yes	—
157	#137	#10623494	1	4	yes	—
158	AGL(1,ℤ/8)	#10623494	1	4	yes	—
159	#135	C₂×Dic₈₉	1	4	yes	—
160	#131	—	2	2	no	—
161	#131	—	2	2	no	—
162	#131	C₂×Dic₈₉	1	4	yes	—
163	#131	C₂²×D₈₉	1	4	yes	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
166	C₂×M₁₆	C₂×Dic₈₉	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
169	#117	C₂×Dic₈₉	1	1	yes	characteristic
170	M₃₂	—	89	4	no	—
171	M₃₂	—	178	4	no	—
172	M₃₂	—	178	4	no	—
173	M₃₂	—	178	2	no	—
174	M₃₂	—	178	2	no	—
175	C₂×C₁₆	—	89	2	no	—
176	C₂×C₁₆	—	89	2	no	—
177	C₂×C₁₆	—	89	1	no	—
178	C₂×C₁₆	—	89	1	no	—
179	C₂×C₁₆	—	89	2	no	—
180	C₂×C₁₆	—	178	4	no	—
181	C₂×C₁₆	—	178	4	no	—
182	C₂×C₁₆	—	178	2	no	—
183	#108	C₂×Dic₈₉	1	4	yes	—
184	#104	C₂×Dic₈₉	1	8	yes	—
185	#97	C₂×Dic₈₉	1	2	yes	—
186	C₄×C₈	C₂²×D₈₉	1	2	yes	—
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	C₂×#10623494	1	4	yes	—
190	#41	C₂²×Dic₈₉	1	4	yes	—
191	C₂×Q₈	C₂×#10623494	1	2	yes	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₂×#10623494	1	2	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
196	Q₁₆	C₂×#10623494	1	4	yes	—
197	SD₁₆	C₂×#10623494	1	8	yes	—
198	D₈	C₂×#10623494	1	4	yes	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	C₂×#10623494	1	4	yes	—
203	M₁₆	C₂²×Dic₈₉	1	4	yes	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
209	C₂×C₈	D₄×D₈₉	1	2	yes	—
210	C₂×C₈	#13083573	1	2	yes	—
211	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂²×Dic₈₉	1	2	yes	—
215	C₁₆	—	89	1	no	—
216	C₁₆	—	89	2	no	—
217	C₁₆	—	89	1	no	—
218	C₁₆	—	178	4	no	—
219	C₁₆	—	178	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#15668019	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	C₂²×#10623494	1	4	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	2	no	—
227	D₄	C₂²×#10623494	1	4	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
233	C₂×C₄	C₂²×#10623494	1	2	yes	—
234	C₂×C₄	#15668019	1	1	yes	characteristic
235	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	D₄×Dic₈₉	1	2	yes	—
238	C₈	C₂²×#10623494	1	4	yes	—
239	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
240	C₈	C₂×#13083573	1	1	yes	characteristic
241	C₂²	—	2	4	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	D₄×#10623494	1	2	yes	—
245	C₂²	C₂×#15668019	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	D₄×#10623494	1	2	yes	—
249	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
250	C₄	C₂³×#10623494	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₂×D₄×#10623494	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2955	C₂
socle	C₂×C₈₉	C₂×C₈×D₄
prime radical	#575×C₈₉	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2955	D₈₉
hypocenter	C₈₉	#33441
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2955

Complements

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

N	H ≅ G/N	type
1. C₂×#19152015	C₂	semidirect
2. #26022094	C₂	semidirect
3. #26022089	C₂	semidirect
4. #26022018	C₂	semidirect
5. #26021880	C₂	semidirect
6. #26021849	C₂	semidirect
10. #19152015	C₂²	semidirect
11. #19152015	C₂²	semidirect
17. #19151419	C₂²	semidirect
18. #19151382	C₂²	semidirect
19. #19151374	C₂²	semidirect
20. C₄×#13083549	C₂²	semidirect
135. C₈₉	#33441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152015	2
#26022094	2
#26022089	2
#26022018	2
#26021880	2
#26021849	2
C₈₉×#2955	2
#33441	89