Subgroups — #36443780

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26019997	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667922	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151320	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
7	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667922	C₄	1	4	yes	—
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₈₉	C₄	1	4	yes	—
13	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
14	#19151352	C₄	1	4	yes	—
15	#19151320	C₄	1	4	yes	complemented
16	C₈×#10623494	C₄	1	4	yes	complemented
17	C₈×#10623494	C₄	1	4	yes	complemented
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	#322×C₈₉	C₄	1	4	yes	complemented
22	C₂²×#10623494	C₂×C₄	1	2	yes	—
23	C₂²×#10623494	C₂×C₄	1	2	yes	—
24	C₂²×#10623494	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
28	#15667922	C₈	1	16	yes	complemented
29	C₄×#10623494	C₂×C₄	1	2	yes	—
30	C₄×#10623494	C₂×C₄	1	2	yes	—
31	C₄×#10623494	C₂×C₄	1	2	yes	—
32	C₄×#10623494	C₂×C₄	1	2	yes	—
33	C₄×#10623494	C₂×C₄	1	2	yes	—
34	C₄×#10623494	C₂×C₄	1	2	yes	—
35	C₈×Dic₈₉	—	2	8	no	—
36	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
46	C₂²×Dic₈₉	C₄²	1	4	yes	—
47	C₂×#10623494	—	2	4	no	—
48	C₂×#10623494	C₄²	1	4	yes	—
49	C₂×#10623494	C₄²	1	4	yes	—
50	C₂×#10623494	C₄²	1	4	yes	—
51	C₂×#10623494	C₂×C₈	1	8	yes	—
52	C₂×#10623494	M₁₆	1	8	yes	—
53	C₄×Dic₈₉	—	2	4	no	—
54	C₄×Dic₈₉	—	2	4	no	—
55	C₄×Dic₈₉	C₂×C₈	1	8	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	—	2	4	no	—
60	C₂×C₈×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	C₄²	1	4	yes	—
62	C₂×C₈×C₈₉	C₄²	1	4	yes	—
63	C₂×C₈×C₈₉	C₄²	1	4	yes	—
64	C₂×C₈×C₈₉	C₂×C₈	1	8	yes	—
65	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
78	C₂×C₄×C₈₉	#131	1	4	yes	—
79	C₂×Dic₈₉	—	2	4	no	—
80	C₂×Dic₈₉	—	2	4	no	—
81	C₂×Dic₈₉	C₄×C₈	1	8	yes	—
82	C₈×C₈₉	—	2	4	no	—
83	C₈×C₈₉	—	2	4	no	—
84	C₈×C₈₉	C₄×C₈	1	8	yes	complemented
85	#10623494	—	2	4	no	—
86	#10623494	C₄×C₈	1	8	yes	complemented
87	#10623494	#97	1	8	yes	complemented
88	C₂²×C₈₉	C₄³	1	1	yes	characteristic
89	C₂²×C₈₉	C₂×C₄×C₈	1	2	yes	—
90	C₂²×C₈₉	C₄×M₁₆	1	2	yes	—
91	C₂²×C₈₉	#405	1	2	yes	—
92	C₄×C₈₉	C₂×C₄×C₈	1	2	yes	—
93	C₄×C₈₉	C₂×C₄×C₈	1	2	yes	—
94	C₄×C₈₉	C₂×#97	1	2	yes	—
95	C₄×C₈₉	C₄×M₁₆	1	2	yes	—
96	C₄×C₈₉	#405	1	2	yes	—
97	C₄×C₈₉	#405	1	2	yes	—
98	Dic₈₉	—	2	8	no	—
99	C₈×#97	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	C₈×M₁₆	1	4	yes	—
101	C₂×C₈₉	C₄²×C₈	1	1	yes	characteristic
102	C₂×C₈₉	C₄×#97	1	1	yes	characteristic
103	C₂×C₈₉	#1726	1	1	yes	characteristic
104	#1744	—	89	1	no	—
105	C₄×#97	—	89	1	no	—
106	C₄²×C₈	—	89	1	no	—
107	C₂×#322	—	89	1	no	—
108	C₂×#322	D₈₉	1	1	yes	characteristic
109	C₂×C₈²	—	89	1	no	—
110	C₂×C₈²	—	89	1	no	—
111	C₈₉	C₈×#97	1	1	yes	characteristic, complemented, Sylow
112	C₂×#97	—	89	4	no	—
113	C₂×C₄×C₈	—	89	1	no	—
114	C₂×C₄×C₈	—	89	1	no	—
115	C₂×C₄×C₈	—	89	1	no	—
116	C₂×C₄×C₈	—	89	4	no	—
117	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
120	C₄³	—	89	1	no	—
121	#336	—	89	4	no	—
122	#322	—	89	4	no	—
123	#322	Dic₈₉	1	4	yes	complemented
124	C₈²	—	89	4	no	—
125	C₈²	—	89	4	no	—
126	C₂²×C₈	—	89	2	no	—
127	C₂²×C₈	—	89	2	no	—
128	C₂²×C₈	—	89	2	no	—
129	C₂²×C₈	C₄×D₈₉	1	2	yes	—
130	C₂²×C₈	C₄×D₈₉	1	2	yes	—
131	C₂²×C₈	C₄×D₈₉	1	2	yes	—
132	C₂×C₄²	—	89	2	no	—
133	C₂×C₄²	—	89	2	no	—
134	C₂×C₄²	—	89	2	no	—
135	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
136	#97	—	89	16	no	—
137	C₄×C₈	—	89	2	no	—
138	C₄×C₈	—	89	2	no	—
139	C₄×C₈	—	89	2	no	—
140	C₄×C₈	—	89	2	no	—
141	C₄×C₈	—	89	2	no	—
142	C₄×C₈	—	89	2	no	—
143	C₄×C₈	—	178	8	no	—
144	C₄×C₈	C₄×D₈₉	1	2	yes	—
145	C₄×C₈	C₄×D₈₉	1	2	yes	—
146	C₄×C₈	C₄×D₈₉	1	2	yes	—
147	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
148	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
149	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
150	C₂²×C₄	—	89	4	no	—
151	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	89	8	no	—
157	C₂×C₈	—	89	8	no	—
158	C₂×C₈	—	89	4	no	—
159	C₂×C₈	—	89	4	no	—
160	C₂×C₈	—	89	4	no	—
161	C₂×C₈	—	178	4	no	—
162	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
163	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
164	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
165	C₂×C₈	C₈×D₈₉	1	8	yes	—
166	C₄²	—	89	8	no	—
167	C₄²	—	178	4	no	—
168	C₄²	—	178	4	no	—
169	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
170	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
171	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
172	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
173	C₂³	C₄²×D₈₉	1	1	yes	characteristic
174	C₂×C₄	—	89	8	no	—
175	C₂×C₄	—	178	4	no	—
176	C₂×C₄	—	178	4	no	—
177	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
178	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
179	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
183	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
184	C₂×C₄	M₁₆×D₈₉	1	4	yes	—
185	C₂×C₄	#15668020	1	4	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	2	4	no	—
188	C₈	—	89	8	no	—
189	C₈	—	89	8	no	—
190	C₈	—	178	4	no	—
191	C₈	C₈×Dic₈₉	1	8	yes	complemented
192	C₂²	C₄²×Dic₈₉	1	1	yes	characteristic
193	C₂²	C₄×C₈×D₈₉	1	2	yes	—
194	C₂²	#19151550	1	2	yes	—
195	C₂²	#19151555	1	2	yes	—
196	C₄	—	178	8	no	—
197	C₄	C₄×C₈×D₈₉	1	2	yes	—
198	C₄	C₂×C₈×Dic₈₉	1	2	yes	—
199	C₄	#97×D₈₉	1	2	yes	—
200	C₄	M₁₆×Dic₈₉	1	2	yes	—
201	C₄	#19151555	1	2	yes	—
202	C₄	#19151556	1	2	yes	—
203	C₂	#26018531	1	4	yes	—
204	C₂	C₄×C₈×Dic₈₉	1	1	yes	characteristic
205	C₂	#97×Dic₈₉	1	1	yes	characteristic
206	C₂	#26019940	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₈₉
commutator	C₂×C₈₉	C₄²×C₈
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	C₈×#97
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
15. #19151320	C₄	semidirect
16. C₈×#10623494	C₄	semidirect
17. C₈×#10623494	C₄	semidirect
21. #322×C₈₉	C₄	semidirect
28. #15667922	C₈	semidirect
84. C₈×C₈₉	C₄×C₈	semidirect
86. #10623494	C₄×C₈	semidirect
87. #10623494	#97	semidirect
111. C₈₉	C₈×#97	semidirect
123. #322	Dic₈₉	semidirect
191. C₈	C₈×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26019997	2
C₄×#15667922	2
C₄×C₈×Dic₈₉	2
C₂×#19151320	2
C₂×C₈×#10623494	2
C₂×#322×C₈₉	2
C₈×#97	89