Subgroups — #26028813

Order 11392.

484 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13083557	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#117×D₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151626	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151625	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151624	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151623	C₂	1	4	yes	maximal, complemented
8	#19151620	C₂	1	4	yes	maximal, complemented
9	#424×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
11	C₂×#13083558	C₂²	1	1	yes	characteristic
12	C₂×#13083557	—	2	4	no	—
13	C₂×#13083557	C₂²	1	2	yes	—
14	C₂×#13083557	C₂²	1	2	yes	—
15	C₂×#13083557	C₂²	1	2	yes	—
16	M₁₆×D₈₉	—	2	4	no	—
17	C₂²×#10623494	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₉	C₂²	1	2	yes	—
19	#15667971	C₂²	1	1	yes	characteristic
20	#15667970	C₄	1	2	yes	—
21	#15667969	C₂²	1	2	yes	—
22	#15667968	C₄	1	2	yes	—
23	C₄⋊C₄×D₈₉	C₄	1	2	yes	—
24	C₂²⋊C₄×D₈₉	C₄	1	2	yes	—
25	C₄²×D₈₉	C₂²	1	2	yes	—
26	C₄⋊#10623494	C₂²	1	4	yes	complemented
27	#15667941	C₂²	1	4	yes	complemented
28	#15667941	C₂²	1	4	yes	complemented
29	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#117×C₈₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₈₉	C₂²	1	4	yes	complemented
33	C₂³×D₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₈₉	C₂×C₄	1	4	yes	—
36	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₈₉	D₄	1	4	yes	—
39	C₂×C₄×D₈₉	Q₈	1	4	yes	—
40	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
41	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
42	#13083558	—	2	2	no	—
43	#13083557	—	2	4	no	—
44	#13083557	—	2	4	no	—
45	#13083557	—	2	4	no	—
46	C₂×#10623494	—	2	2	no	—
47	C₂×#10623494	C₂³	1	2	yes	—
48	C₂×#10623494	C₂³	1	2	yes	—
49	C₈×D₈₉	—	2	4	no	—
50	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
51	#13083553	C₂×C₄	1	4	yes	—
52	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
53	#13083551	C₂×C₄	1	4	yes	—
54	C₄×Dic₈₉	C₂×C₄	1	4	yes	—
55	C₄×Dic₈₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
57	M₁₆×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	—	2	2	no	—
59	C₂×C₈×C₈₉	C₂³	1	2	yes	—
60	C₂×C₈×C₈₉	C₂³	1	2	yes	—
61	C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
62	C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
63	C₄²×C₈₉	C₂³	1	2	yes	—
64	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₈₉	—	4	2	no	—
66	C₂²×D₈₉	C₄⋊C₄	1	4	yes	—
67	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₈₉	C₄⋊C₄	1	4	yes	—
75	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
76	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₈₉	—	4	4	no	—
79	C₄×D₈₉	C₄⋊C₄	1	4	yes	—
80	C₄×D₈₉	C₄⋊C₄	1	4	yes	—
81	C₄×D₈₉	C₂×D₄	1	2	yes	—
82	C₄×D₈₉	C₂×D₄	1	2	yes	—
83	C₄×D₈₉	C₂×Q₈	1	2	yes	—
84	C₄×D₈₉	C₂×Q₈	1	2	yes	—
85	C₈×C₈₉	—	2	2	no	—
86	C₈×C₈₉	—	2	2	no	—
87	#10623494	—	2	2	no	—
88	#10623494	—	2	2	no	—
89	C₂²×C₈₉	—	2	2	no	—
90	C₂²×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
92	C₂×D₈₉	—	2	2	no	—
93	C₂×D₈₉	—	2	2	no	—
94	C₂×D₈₉	—	4	4	no	—
95	C₂×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
97	C₄×C₈₉	—	2	4	no	—
98	C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
101	Dic₈₉	—	2	4	no	—
102	Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
103	Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₈₉	—	2	2	no	—
105	C₂×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₈₉	#568	1	2	yes	—
107	D₈₉	—	2	2	no	—
108	D₈₉	—	2	2	no	—
109	#2905	—	89	1	no	maximal, Sylow
110	C₈₉	#2905	1	1	yes	characteristic, complemented, Sylow
111	C₂²×M₁₆	—	89	1	no	—
112	C₂²×M₁₆	—	89	1	no	—
113	C₂×#117	—	89	1	no	—
114	#424	—	89	1	no	—
115	#424	—	89	1	no	—
116	#424	—	89	1	no	—
117	#424	—	89	4	no	—
118	#424	D₈₉	1	1	yes	characteristic, complemented
119	#423	—	89	4	no	—
120	C₂³×C₄	—	89	1	no	—
121	C₂×M₁₆	—	89	1	no	—
122	C₂×M₁₆	—	89	2	no	—
123	C₂×M₁₆	—	89	2	no	—
124	C₂×M₁₆	—	89	2	no	—
125	C₂×M₁₆	—	178	4	no	—
126	C₂×M₁₆	—	178	4	no	—
127	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
128	C₂²×C₈	—	89	1	no	—
129	C₂²×C₈	—	89	2	no	—
130	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
131	#117	—	89	1	no	—
132	#117	—	89	2	no	—
133	#117	—	89	2	no	—
134	#117	—	89	2	no	—
135	#117	C₂×D₈₉	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	89	2	no	—
137	C₂×C₂²⋊C₄	—	89	2	no	—
138	C₂×C₄²	—	89	2	no	—
139	C₄⋊C₈	—	89	4	no	—
140	C₄⋊C₈	—	89	4	no	—
141	C₄⋊C₈	—	89	4	no	—
142	C₄⋊C₈	C₂×D₈₉	1	4	yes	complemented
143	C₂⁴	—	89	1	no	—
144	C₂²×C₄	—	89	1	no	—
145	C₂²×C₄	—	89	1	no	—
146	C₂²×C₄	—	89	1	no	—
147	C₂²×C₄	—	89	4	no	—
148	C₂²×C₄	—	89	4	no	—
149	C₂²×C₄	—	89	1	no	—
150	C₂²×C₄	—	89	1	no	—
151	C₂²×C₄	—	89	4	no	—
152	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	178	4	no	—
155	M₁₆	—	178	4	no	—
156	M₁₆	—	178	4	no	—
157	M₁₆	—	178	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	89	2	no	—
160	C₂×C₈	—	89	2	no	—
161	C₂×C₈	—	178	4	no	—
162	C₂×C₈	—	178	2	no	—
163	C₂×C₈	C₂²×D₈₉	1	2	yes	—
164	C₂×C₈	C₂²×D₈₉	1	2	yes	—
165	C₄⋊C₄	—	89	4	no	—
166	C₄⋊C₄	—	89	2	no	—
167	C₄⋊C₄	C₄×D₈₉	1	2	yes	—
168	C₂²⋊C₄	—	89	2	no	—
169	C₂²⋊C₄	—	89	4	no	—
170	C₂²⋊C₄	C₄×D₈₉	1	2	yes	—
171	C₄²	—	89	4	no	—
172	C₄²	—	89	2	no	—
173	C₄²	C₂²×D₈₉	1	2	yes	—
174	C₂³	—	89	4	no	—
175	C₂³	—	89	1	no	—
176	C₂³	—	89	1	no	—
177	C₂³	—	356	2	no	—
178	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
179	C₂×C₄	—	89	1	no	—
180	C₂×C₄	—	89	1	no	—
181	C₂×C₄	—	89	4	no	—
182	C₂×C₄	—	89	2	no	—
183	C₂×C₄	—	89	2	no	—
184	C₂×C₄	—	89	2	no	—
185	C₂×C₄	—	89	2	no	—
186	C₂×C₄	—	89	4	no	—
187	C₂×C₄	—	89	4	no	—
188	C₂×C₄	—	89	4	no	—
189	C₂×C₄	—	356	4	no	—
190	C₂×C₄	C₂×C₄×D₈₉	1	4	yes	—
191	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
192	C₂×C₄	D₄×D₈₉	1	2	yes	—
193	C₂×C₄	Q₈×D₈₉	1	2	yes	—
194	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	2	2	no	—
197	C₈	—	178	2	no	—
198	C₈	—	178	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	89	2	no	—
201	C₂²	—	89	2	no	—
202	C₂²	—	356	4	no	—
203	C₂²	—	178	2	no	—
204	C₂²	—	178	2	no	—
205	C₂²	C₄⋊C₄×D₈₉	1	2	yes	—
206	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	89	2	no	—
209	C₄	—	89	2	no	—
210	C₄	—	178	4	no	—
211	C₄	C₄⋊C₄×D₈₉	1	2	yes	—
212	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
213	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	—	178	2	no	—
216	C₂	—	178	2	no	—
217	C₂	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
218	C₂	#19152363	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#424×C₈₉	C₂
prime radical	C₂³×D₈₉	C₂×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#424	D₈₉
hypocenter	C₈₉	#2905
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. #19151626	C₂	semidirect
5. #19151625	C₂	semidirect
6. #19151624	C₂	semidirect
7. #19151623	C₂	semidirect
8. #19151620	C₂	semidirect
9. #424×C₈₉	C₂	semidirect
26. C₄⋊#10623494	C₂²	semidirect
27. #15667941	C₂²	semidirect
28. #15667941	C₂²	semidirect
32. C₄⋊C₈×C₈₉	C₂²	semidirect
110. C₈₉	#2905	semidirect
118. #424	D₈₉	semidirect
142. C₄⋊C₈	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13083557	2
C₂×M₁₆×D₈₉	2
#117×D₈₉	2
#19151626	2
#19151625	2
#19151624	2
#19151623	2
#19151620	2
#424×C₈₉	2
#2905	89