Subgroups — #36707676

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033355	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	C₂×#19151382	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	4	no	—
9	#19152015	C₂²	1	8	yes	complemented
10	C₂×#15667952	—	2	4	no	—
11	C₂×#15667952	C₂²	1	2	yes	—
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂²×#13083549	C₂²	1	1	yes	characteristic
14	#19151419	C₂²	1	4	yes	complemented
15	#19151382	C₂²	1	4	yes	complemented
16	#19151382	C₂²	1	4	yes	complemented
17	#19151374	C₂²	1	2	yes	complemented
18	C₄×#13083549	C₂²	1	2	yes	complemented
19	#575×C₈₉	C₄	1	1	yes	characteristic
20	C₂×#131×C₈₉	C₂²	1	2	yes	—
21	#444×C₈₉	C₄	1	4	yes	—
22	#443×C₈₉	C₂²	1	4	yes	—
23	#430×C₈₉	C₄	1	1	yes	characteristic
24	#421×C₈₉	C₄	1	2	yes	—
25	#405×C₈₉	C₂²	1	1	yes	characteristic
26	#15667952	—	2	8	no	—
27	#15667952	—	2	2	no	—
28	#15667952	D₄	1	4	yes	—
29	#15667952	C₂³	1	4	yes	—
30	C₂×#13083549	—	2	8	no	—
31	C₂×#13083549	—	2	2	no	—
32	C₂×#13083549	—	2	2	no	—
33	C₂×#13083549	C₂³	1	1	yes	characteristic
34	C₂×#13083549	C₂³	1	1	yes	characteristic
35	C₂×#13083549	C₂³	1	2	yes	—
36	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
37	#137×C₈₉	C₈	1	4	yes	—
38	AGL(1,ℤ/8)×C₈₉	C₈	1	4	yes	—
39	#135×C₈₉	C₂×C₄	1	4	yes	—
40	#131×C₈₉	—	2	2	no	—
41	#131×C₈₉	—	2	2	no	—
42	#131×C₈₉	C₂×C₄	1	4	yes	—
43	#131×C₈₉	C₂³	1	4	yes	—
44	C₂×M₁₆×C₈₉	—	2	2	no	—
45	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	—	2	2	no	—
48	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
50	#108×C₈₉	C₂×C₄	1	4	yes	—
51	#104×C₈₉	C₂×C₄	1	8	yes	—
52	#97×C₈₉	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₈₉	C₂³	1	2	yes	—
54	#13083549	—	2	4	no	—
55	#13083549	—	2	1	no	—
56	#13083549	—	2	1	no	—
57	#13083549	C₂×D₄	1	4	yes	—
58	#41×C₈₉	—	2	2	no	—
59	#41×C₈₉	—	2	2	no	—
60	#41×C₈₉	C₂×C₈	1	4	yes	—
61	#41×C₈₉	C₂²×C₄	1	4	yes	—
62	C₂×Q₈×C₈₉	C₂×C₈	1	2	yes	—
63	C₂×D₄×C₈₉	—	2	2	no	—
64	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₈₉	—	2	2	no	—
66	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	Q₁₆×C₈₉	C₂×C₈	1	4	yes	—
68	SD₁₆×C₈₉	C₂×C₈	1	8	yes	—
69	D₈×C₈₉	C₂×C₈	1	4	yes	—
70	M₁₆×C₈₉	—	2	2	no	—
71	M₁₆×C₈₉	—	2	4	no	—
72	M₁₆×C₈₉	—	2	4	no	—
73	M₁₆×C₈₉	C₂×C₈	1	4	yes	—
74	M₁₆×C₈₉	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₈₉	—	2	2	no	—
76	C₂×C₈×C₈₉	—	2	4	no	—
77	C₂×C₈×C₈₉	—	2	4	no	—
78	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₈₉	#41	1	2	yes	—
82	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₈₉	—	2	1	no	—
84	C₂²⋊C₄×C₈₉	—	2	1	no	—
85	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
86	C₂³×C₈₉	—	2	2	no	—
87	C₂³×C₈₉	#131	1	1	yes	characteristic
88	Q₈×C₈₉	—	2	2	no	—
89	Q₈×C₈₉	C₂²×C₈	1	4	yes	—
90	D₄×C₈₉	—	2	4	no	—
91	D₄×C₈₉	—	2	4	no	—
92	D₄×C₈₉	—	2	2	no	—
93	D₄×C₈₉	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	4	no	—
96	C₂×C₄×C₈₉	—	2	4	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
102	C₈×C₈₉	—	2	4	no	—
103	C₈×C₈₉	C₄×D₄	1	2	yes	—
104	C₈×C₈₉	C₂²×C₈	1	4	yes	—
105	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
107	C₂²×C₈₉	—	2	4	no	—
108	C₂²×C₈₉	—	4	1	no	—
109	C₂²×C₈₉	—	8	2	no	—
110	C₂²×C₈₉	C₈×D₄	1	2	yes	—
111	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
112	C₄×C₈₉	—	2	4	no	—
113	C₄×C₈₉	—	4	2	no	—
114	C₄×C₈₉	C₈×D₄	1	2	yes	—
115	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
116	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
117	#33442	—	89	1	no	maximal, Sylow
118	C₂×C₈₉	—	2	1	no	—
119	C₂×C₈₉	—	2	2	no	—
120	C₂×C₈₉	—	4	4	no	—
121	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
122	#3407	—	89	2	no	—
123	#2955	D₈₉	1	1	yes	characteristic
124	#2171	—	89	4	no	—
125	#2170	—	89	4	no	—
126	#2153	—	89	1	no	—
127	C₂×#350	—	89	2	no	—
128	#2108	—	89	1	no	—
129	C₈₉	#33442	1	1	yes	characteristic, complemented, Sylow
130	#575	Dic₈₉	1	1	yes	characteristic
131	C₂×#131	C₂×D₈₉	1	2	yes	—
132	#504	—	89	8	no	—
133	#504	—	178	4	no	—
134	C₂×M₃₂	—	89	1	no	—
135	C₂×M₃₂	—	89	2	no	—
136	C₂×M₃₂	—	178	4	no	—
137	C₂²×C₁₆	—	89	1	no	—
138	#444	Dic₈₉	1	4	yes	—
139	#443	C₂×D₈₉	1	4	yes	—
140	#430	Dic₈₉	1	1	yes	characteristic
141	#421	Dic₈₉	1	2	yes	—
142	#405	C₂×D₈₉	1	1	yes	characteristic
143	#364	—	89	4	no	—
144	#350	—	89	4	no	—
145	#350	—	89	4	no	—
146	#346	—	89	2	no	—
147	C₄×C₁₆	—	89	2	no	—
148	C₂×#41	C₂×Dic₈₉	1	2	yes	—
149	#137	#10623494	1	4	yes	—
150	AGL(1,ℤ/8)	#10623494	1	4	yes	—
151	#135	C₂×Dic₈₉	1	4	yes	—
152	#131	—	2	2	no	—
153	#131	—	2	2	no	—
154	#131	C₂×Dic₈₉	1	4	yes	—
155	#131	C₂²×D₈₉	1	4	yes	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
158	C₂×M₁₆	C₂×Dic₈₉	1	1	yes	characteristic
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
161	#117	C₂×Dic₈₉	1	1	yes	characteristic
162	M₃₂	—	89	4	no	—
163	M₃₂	—	89	4	no	—
164	M₃₂	—	178	8	no	—
165	M₃₂	—	178	2	no	—
166	C₂×C₁₆	—	89	2	no	—
167	C₂×C₁₆	—	89	1	no	—
168	C₂×C₁₆	—	89	1	no	—
169	C₂×C₁₆	—	178	8	no	—
170	C₂×C₁₆	—	178	2	no	—
171	C₂×C₁₆	—	178	2	no	—
172	#108	C₂×Dic₈₉	1	4	yes	—
173	#104	C₂×Dic₈₉	1	8	yes	—
174	#97	C₂×Dic₈₉	1	2	yes	—
175	C₄×C₈	C₂²×D₈₉	1	2	yes	—
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	#41	C₂×#10623494	1	4	yes	—
179	#41	C₂²×Dic₈₉	1	4	yes	—
180	C₂×Q₈	C₂×#10623494	1	2	yes	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	C₂×#10623494	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
185	Q₁₆	C₂×#10623494	1	4	yes	—
186	SD₁₆	C₂×#10623494	1	8	yes	—
187	D₈	C₂×#10623494	1	4	yes	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	C₂×#10623494	1	4	yes	—
192	M₁₆	C₂²×Dic₈₉	1	4	yes	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
197	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
198	C₂×C₈	D₄×D₈₉	1	2	yes	—
199	C₂×C₈	#13083573	1	2	yes	—
200	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	C₂²×Dic₈₉	1	2	yes	—
204	C₁₆	—	89	4	no	—
205	C₁₆	—	178	4	no	—
206	C₁₆	—	178	1	no	—
207	C₁₆	—	178	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	#15668019	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	Q₈	C₂²×#10623494	1	4	yes	—
212	D₄	—	2	4	no	—
213	D₄	—	2	4	no	—
214	D₄	—	2	2	no	—
215	D₄	C₂²×#10623494	1	4	yes	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
221	C₂×C₄	C₂²×#10623494	1	2	yes	—
222	C₂×C₄	#15668019	1	1	yes	characteristic
223	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	D₄×Dic₈₉	1	2	yes	—
226	C₈	C₂²×#10623494	1	4	yes	—
227	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
228	C₈	C₂×#13083573	1	1	yes	characteristic
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	8	2	no	—
232	C₂²	D₄×#10623494	1	2	yes	—
233	C₂²	C₂×#15668019	1	1	yes	characteristic
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	D₄×#10623494	1	2	yes	—
237	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
238	C₄	C₂³×#10623494	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	4	no	—
242	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	#33442
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. #26033355	C₂	semidirect
2. #26022094	C₂	semidirect
3. #26022089	C₂	semidirect
4. #26022018	C₂	semidirect
5. C₂×#19151382	C₂	semidirect
6. #26021849	C₂	semidirect
9. #19152015	C₂²	semidirect
14. #19151419	C₂²	semidirect
15. #19151382	C₂²	semidirect
16. #19151382	C₂²	semidirect
17. #19151374	C₂²	semidirect
18. C₄×#13083549	C₂²	semidirect
129. C₈₉	#33442	semidirect

Maximal subgroups

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

subgroup	index
#26033355	2
#26022094	2
#26022089	2
#26022018	2
C₂×#19151382	2
#26021849	2
C₈₉×#2955	2
#33442	89