Finite Group Databasebeta

← #26019203

Subgroups — #26019203

Order 11392.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19152160C211yesmaximal, characteristic
2#19151737C212yesmaximal, complemented
3#19151331C212yesmaximal, complemented
4#19151322C211yesmaximal, characteristic, complemented
5#393×C89C211yesmaximal, characteristic, complemented
6C22×C22⋊D89—21no—
7C2×#13083561—22no—
8#15667996—22no—
9#15667986—21no—
10#15667986C2212yes—
11#15667982—22no—
12#15667936—22no—
13C22⋊#10623494C2212yescomplemented
14C22×Q8×C89C2211yescharacteristic, complemented
15C2×C22⋊C4×C89—21no—
16C2×C22⋊C4×C89C2211yescharacteristic
17#95×C89—21no—
18#95×C89C2211yescharacteristic, complemented
19C23×D89—21no—
20C2×D356—21no—
21C2×D356D412yes—
22C2×C22⋊D89—22no—
23C2×C22⋊D89—22no—
24C2×C22⋊D89—42no—
25C2×C22⋊D89—42no—
26C22×Dic89—21no—
27#13083561—44no—
28C2×#10623494—22no—
29C4⋊Dic89—21no—
30C4⋊Dic89D412yes—
31#13083551—22no—
32#13083551—22no—
33C24×C89D411yescharacteristic
34C2×Q8×C89—22no—
35C2×Q8×C89—42no—
36C22×C4×C89—21no—
37C22×C4×C89—21no—
38C22×C4×C89D411yescharacteristic
39C22×C4×C89C2311yescharacteristic
40C22⋊C4×C89—22no—
41C22⋊C4×C89—42no—
42C23×C89—21no—
43C23×C89—21no—
44C23×C89—41no—
45C23×C89—41no—
46C23×C89SD1612yescomplemented
47C23×C89C2×D411yescharacteristic
48C22×D89—22no—
49C22×D89—42no—
50C22×D89—81no—
51C22×D89—81no—
52Q8×C89—24no—
53Q8×C89—42no—
54C2×C4×C89—22no—
55C2×C4×C89—21no—
56C2×C4×C89—41no—
57C2×C4×C89—41no—
58C2×C4×C89—41no—
59C2×C4×C89—41no—
60C2×C4×C89—41no—
61C2×C4×C89—41no—
62C2×C4×C89—41no—
63C2×C4×C89C2×D412yes—
64C22⋊D89—44no—
65C22⋊D89—42no—
66C22⋊D89—42no—
67C2×Dic89—22no—
68C2×Dic89—42no—
69D356—42no—
70D356—81no—
71#10623494—42no—
72C22×C89—21no—
73C22×C89—21no—
74C22×C89—22no—
75C22×C89—42no—
76C22×C89—41no—
77C22×C89—41no—
78C22×C89—81no—
79C22×C89#12011yescharacteristic
80C22×C89C2×SD1611yescharacteristic
81C22×C89AGL(1,ℤ/8)11yescharacteristic
82C2×D89—42no—
83C2×D89—42no—
84C2×D89—42no—
85C2×D89—84no—
86C4×C89—22no—
87C4×C89—41no—
88C4×C89—42no—
89C4×C89—81no—
90Dic89—42no—
91C2×C89—21no—
92C2×C89—21no—
93C2×C89—42no—
94C2×C89#45011yescharacteristic
95C2×C89#45511yescharacteristic
96C2×C89B(4,𝔽2)11yescharacteristic
97D89—82no—
98#1601—891nomaximal, Sylow
99C89#160111yescharacteristic, complemented, Sylow
100#534—891no—
101#448—892no—
102#393D8911yescharacteristic, complemented
103#327—892no—
104#323—891no—
105C22×Q8C2×D8911yescharacteristic, complemented
106C22×D4—1781no—
107C2×SD16—1782no—
108#123—1782no—
109#121—892no—
110#121—1781no—
111#120—1782no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4C2×D8911yescharacteristic
114#103—1782no—
115C22⋊C8—892no—
116#95—21no—
117#95C2×D8911yescharacteristic, complemented
118C24—1781no—
119C24C22⋊D8911yescharacteristic
120C2×Q8—22no—
121C2×Q8—42no—
122C2×D4—892no—
123C2×D4—3562no—
124C2×D4—3562no—
125C2×D4—1781no—
126C2×D4—1782no—
127C2×D4—1782no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—1781no—
131C22×C4C22⋊D8911yescharacteristic
132C22×C4C22×D8911yescharacteristic
133SD16—3564no—
134C2×C8—1782no—
135C4⋊C4—892no—
136C4⋊C4—1781no—
137C22⋊C4—22no—
138C22⋊C4—42no—
139C22⋊C4—1782no—
140C22⋊C4—1782no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—3562no—
146C23—1782no—
147C23—7121no—
148C23—7121no—
149C23#1308356112yescomplemented
150C23C2×C22⋊D8911yescharacteristic
151Q8—24no—
152Q8—42no—
153D4—3562no—
154D4—3564no—
155D4—3562no—
156D4—3562no—
157D4—7121no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—3562no—
168C2×C4—1782no—
169C2×C4D4×D8912yes—
170C8—3562no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—3562no—
179C22—3562no—
180C22—3562no—
181C22—7124no—
182C22#1566798011yescharacteristic
183C22C2×#1308356111yescharacteristic
184C22#1566803711yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—81no—
189C4—3562no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—7122no—
194C2#1915174811yescharacteristic
195C2#1915177511yescharacteristic
196C2#1915178411yescharacteristic
subgroupG/H
centerC22#15667980
commutator, cosocleC22×C4×C89C23
FrattiniC22×C4C22×D89
Fitting#393×C89C2
socleC22×C89#120
prime radical#19152160C2
hypercenter#393D89
hypocenterC89#1601
LCSG ▷ C22×C4×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #19151737C2semidirect
3. #19151331C2semidirect
4. #19151322C2semidirect
5. #393×C89C2semidirect
13. C22⋊#10623494C22semidirect
14. C22×Q8×C89C22semidirect
18. #95×C89C22semidirect
46. C23×C89SD16semidirect
99. C89#1601semidirect
102. #393D89semidirect
105. C22×Q8C2×D89semidirect
117. #95C2×D89semidirect
149. C23#13083561semidirect

Maximal subgroups

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

subgroupindex
#191521602
#191517372
#191513312
#191513222
#393×C892
#160189