Subgroups — #36463446

Order 22784.

290 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022049	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈×#13083557	C₂	1	1	yes	maximal, characteristic
4	#26018171	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018169	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018168	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1312	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₈₉	C₄	1	2	yes	—
9	#19151649	C₄	1	1	yes	characteristic
10	C₄×#13083557	C₄	1	1	yes	characteristic
11	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
12	#19151416	C₄	1	2	yes	—
13	#19151377	C₄	1	2	yes	—
14	C₄×#13083549	C₂²	1	1	yes	characteristic
15	C₄×#13083549	C₂²	1	1	yes	characteristic
16	C₁₆×Dic₈₉	C₄	1	2	yes	—
17	#19151320	C₄	1	1	yes	characteristic
18	#19151319	C₄	1	1	yes	characteristic
19	C₈×#10623494	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	C₈²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#13083557	C₈	1	2	yes	—
24	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×#13083549	C₂×C₄	1	2	yes	—
28	C₂×#13083549	C₂×C₄	1	2	yes	—
29	C₁₆×D₈₉	—	2	4	no	—
30	#15667941	C₈	1	2	yes	—
31	#15667924	C₈	1	2	yes	—
32	#15667922	C₈	1	2	yes	—
33	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
34	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
35	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₄×D₈₉	C₄²	1	1	yes	characteristic
44	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
46	#13083557	—	2	4	no	—
47	C₂×#10623494	C₄²	1	1	yes	characteristic
48	C₂×#10623494	C₄²	1	1	yes	characteristic
49	C₂×#10623494	C₂×C₈	1	2	yes	—
50	C₂×#10623494	C₂×C₈	1	2	yes	—
51	C₈×D₈₉	—	2	2	no	—
52	C₈×D₈₉	—	2	2	no	—
53	C₄×Dic₈₉	C₄²	1	1	yes	characteristic
54	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
55	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
56	#13083549	—	2	2	no	—
57	#13083549	M₁₆	1	4	yes	complemented
58	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	C₁₆×C₈₉	—	2	2	no	—
64	C₁₆×C₈₉	—	2	2	no	—
65	C₂²×D₈₉	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
69	C₂×Dic₈₉	C₄×C₈	1	1	yes	characteristic
70	C₂×Dic₈₉	C₄×C₈	1	1	yes	characteristic
71	C₂×Dic₈₉	C₄×C₈	1	1	yes	characteristic
72	C₄×D₈₉	—	2	1	no	—
73	C₄×D₈₉	—	2	1	no	—
74	C₄×D₈₉	—	2	2	no	—
75	C₄×D₈₉	M₃₂	1	4	yes	—
76	C₈×C₈₉	—	2	2	no	—
77	C₈×C₈₉	—	2	2	no	—
78	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
79	C₈×C₈₉	#131	1	2	yes	—
80	#10623494	—	2	1	no	—
81	#10623494	—	2	1	no	—
82	#10623494	—	2	2	no	—
83	#10623494	—	2	2	no	—
84	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
85	C₂×D₈₉	—	2	1	no	—
86	C₂×D₈₉	—	2	1	no	—
87	C₂×D₈₉	#346	1	2	yes	—
88	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
89	C₄×C₈₉	#405	1	1	yes	characteristic
90	C₄×C₈₉	C₂×M₃₂	1	2	yes	—
91	C₄×C₈₉	#504	1	2	yes	—
92	Dic₈₉	—	2	1	no	—
93	Dic₈₉	—	2	1	no	—
94	Dic₈₉	#346	1	2	yes	—
95	#11408	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	C₈×M₁₆	1	1	yes	characteristic
97	C₂×C₈₉	C₂×#346	1	1	yes	characteristic
98	C₂×C₈₉	#2108	1	1	yes	characteristic
99	D₈₉	—	2	2	no	—
100	#2162	—	89	1	no	—
101	C₂×C₄×C₁₆	—	89	1	no	—
102	C₈×M₁₆	—	89	1	no	—
103	#1312	—	89	1	no	—
104	#1312	D₈₉	1	1	yes	characteristic, complemented
105	#1311	—	89	1	no	—
106	#1311	—	89	1	no	—
107	C₈₉	#11408	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₁₆	—	89	2	no	—
109	#431	—	89	1	no	—
110	C₄×M₁₆	—	89	1	no	—
111	C₂×C₄×C₈	—	89	1	no	—
112	C₄⋊C₁₆	—	89	2	no	—
113	C₂²⋊C₁₆	—	89	2	no	—
114	C₄×C₁₆	—	89	2	no	—
115	C₄×C₁₆	—	89	1	no	—
116	C₄×C₁₆	—	89	1	no	—
117	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
118	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
119	#322	—	89	1	no	—
120	#322	—	89	1	no	—
121	C₈²	—	89	1	no	—
122	C₈²	C₂×D₈₉	1	1	yes	characteristic
123	C₂×M₁₆	—	89	2	no	—
124	C₂²×C₈	—	89	1	no	—
125	C₂²×C₈	—	89	1	no	—
126	C₂×C₄²	—	89	1	no	—
127	C₂×C₁₆	—	89	2	no	—
128	C₂×C₁₆	—	89	2	no	—
129	C₂×C₁₆	—	178	4	no	—
130	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
131	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
132	C₄⋊C₈	—	89	2	no	—
133	C₂²⋊C₈	—	89	2	no	—
134	#97	—	89	2	no	—
135	C₄×C₈	—	89	1	no	—
136	C₄×C₈	—	89	1	no	—
137	C₄×C₈	—	89	1	no	—
138	C₄×C₈	—	89	1	no	—
139	C₄×C₈	—	89	1	no	—
140	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
141	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
142	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
143	C₂²×C₄	—	89	1	no	—
144	C₂²×C₄	—	89	1	no	—
145	C₂²×C₄	—	89	1	no	—
146	M₁₆	—	178	4	no	—
147	C₂×C₈	—	89	2	no	—
148	C₂×C₈	—	89	2	no	—
149	C₂×C₈	—	89	1	no	—
150	C₂×C₈	—	89	1	no	—
151	C₂×C₈	—	178	2	no	—
152	C₂×C₈	—	178	2	no	—
153	C₂×C₈	C₈×D₈₉	1	2	yes	—
154	C₂×C₈	C₈×D₈₉	1	2	yes	—
155	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
156	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
157	C₄²	—	89	1	no	—
158	C₄²	—	89	1	no	—
159	C₄²	—	89	1	no	—
160	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
161	C₁₆	—	2	2	no	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	89	4	no	—
164	C₁₆	—	178	2	no	—
165	C₂³	—	89	1	no	—
166	C₂×C₄	—	89	1	no	—
167	C₂×C₄	—	89	4	no	—
168	C₂×C₄	—	89	1	no	—
169	C₂×C₄	—	89	1	no	—
170	C₂×C₄	—	178	2	no	—
171	C₂×C₄	—	178	1	no	—
172	C₂×C₄	—	178	1	no	—
173	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
174	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
175	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	178	2	no	—
179	C₈	—	178	2	no	—
180	C₈	—	178	1	no	—
181	C₈	—	178	1	no	—
182	C₈	M₁₆×D₈₉	1	2	yes	—
183	C₈	#15668020	1	2	yes	—
184	C₂²	—	89	2	no	—
185	C₂²	—	178	1	no	—
186	C₂²	—	178	1	no	—
187	C₂²	C₄×C₈×D₈₉	1	1	yes	characteristic
188	C₄	—	89	2	no	—
189	C₄	—	178	1	no	—
190	C₄	—	178	1	no	—
191	C₄	#19151550	1	1	yes	characteristic
192	C₄	#19151555	1	1	yes	characteristic
193	C₄	M₃₂×D₈₉	1	2	yes	—
194	C₄	#19152014	1	2	yes	—
195	C₂	—	178	2	no	—
196	C₂	#26018531	1	1	yes	characteristic
197	C₂	#346×D₈₉	1	1	yes	characteristic
198	C₂	#26021847	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂×C₄×C₈
Fitting	C₈₉×#1312	C₂
prime radical	C₂²×D₈₉	C₄×C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1312	D₈₉
hypocenter	C₈₉	#11408
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ #1312

Complements

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

N	H ≅ G/N	type
4. #26018171	C₂	semidirect
5. #26018169	C₂	semidirect
6. #26018168	C₂	semidirect
7. C₈₉×#1312	C₂	semidirect
57. #13083549	M₁₆	semidirect
104. #1312	D₈₉	semidirect
107. C₈₉	#11408	semidirect

Maximal subgroups

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

subgroup	index
#26022049	2
C₄×C₁₆×D₈₉	2
C₈×#13083557	2
#26018171	2
#26018169	2
#26018168	2
C₈₉×#1312	2
#11408	89