Finite Group Databasebeta

← #27197098

Subgroups — #27197098

Order 13184.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#20268593C211yesmaximal, characteristic, complemented
2C2×#15692962C211yesmaximal, characteristic, complemented
3#20268058C211yesmaximal, characteristic, complemented
4#20267605C211yesmaximal, characteristic, complemented
5#20267598C211yesmaximal, characteristic, complemented
6#20267576C211yesmaximal, characteristic, complemented
7#509×C103C211yesmaximal, characteristic, complemented
8#15693067—21no—
9#15693048—22no—
10#15693047—21no—
11#15693046—21no—
12#15693044—21no—
13C2×D824C2211yescharacteristic
14D8×D103—21no—
15#15693027C2211yescharacteristic
16#15692962—22no—
17C2×C103⋊C16C2211yescharacteristic, complemented
18#15692957C2211yescharacteristic, complemented
19#15692938—21no—
20#135×C103C2211yescharacteristic
21C2×D8×C103C2211yescharacteristic, complemented
22SD32×C103—21no—
23D16×C103—21no—
24M32×C103C2211yescharacteristic, complemented
25#13110106—22no—
26#13110105—41no—
27#13110105D411yescharacteristic
28C2×D412—21no—
29C2×D412—41no—
30D4×D103—22no—
31D4×D103—42no—
32#13110093—41no—
33#13110092—22no—
34D824—21no—
35D824D412yes—
36#13110090—22no—
37#13110090—41no—
38#13110089—21no—
39#13110089D411yescharacteristic
40#13110088—21no—
41C8×D103—21no—
42C103⋊C16—21no—
43#41×C103—21no—
44C2×D4×C103—21no—
45Q16×C103D411yescharacteristic
46SD16×C103—21no—
47D8×C103—21no—
48D8×C103—21no—
49D8×C103D411yescharacteristic
50C2×C8×C103C2311yescharacteristic
51C16×C103—21no—
52C23×C103—41no—
53C22×D103—42no—
54C22×D103—41no—
55Q8×C103—21no—
56D4×C103—21no—
57D4×C103—41no—
58D4×C103—41no—
59D4×C103—41no—
60C2×C4×C103—41no—
61C2×C4×C103C2×D411yescharacteristic
62C22⋊D103—21no—
63C22⋊D103—42no—
64D412—22no—
65D412—42no—
66D412—41no—
67D412D811yescharacteristic
68C4×D103—21no—
69C4×D103—42no—
70Dic206D811yescharacteristic
71C8×C103C2×D411yescharacteristic
72C8×C103C2×D411yescharacteristic
73C103⋊C8—21no—
74C103⋊C8—41no—
75C22×C103—41no—
76C22×C103—81no—
77C22×C103—161no—
78C22×C103AGL(1,ℤ/8)11yescharacteristic
79C2×D103—21no—
80C2×D103—42no—
81C2×D103—162no—
82C2×D103—161no—
83C4×C103—41no—
84C4×C103#12011yescharacteristic
85C4×C103C2×D811yescharacteristic
86Dic103—21no—
87C2×C103—21no—
88C2×C103—81no—
89C2×C103—161no—
90C2×C103#44711yescharacteristic
91D103—41no—
92D103—82no—
93#2192—1031nomaximal, Sylow
94C103#219211yescharacteristic, complemented, Sylow
95#576—1031no—
96#509D10311yescharacteristic, complemented
97C2×D16—1031no—
98#472—1031no—
99#361—1031no—
100#359—1031no—
101#350—1031no—
102H(5,𝔽2)—2061no—
103AGL(1,ℤ/8)—2061no—
104AGL(1,ℤ/8)—2062no—
105AGL(1,ℤ/8)—2061no—
106#135—2061no—
107#135C2×D10311yescharacteristic
108C2×D8—1031no—
109C2×D8—2061no—
110C2×D8C2×D10311yescharacteristic, complemented
111#131—1031no—
112SD32—21no—
113D16—21no—
114D16—2062no—
115M32C2×D10311yescharacteristic, complemented
116C2×C16—1031no—
117#108—1031no—
118#100—2061no—
119#41—21no—
120#41—1031no—
121#41—4121no—
122#41—2062no—
123C2×D4—21no—
124C2×D4—4121no—
125C2×D4—4122no—
126C2×D4—2062no—
127C2×D4—2061no—
128Q16C22⋊D10311yescharacteristic
129SD16—21no—
130SD16—4121no—
131SD16—2062no—
132D8—21no—
133D8—21no—
134D8—1032no—
135D8—4121no—
136D8—2062no—
137D8—2061no—
138D8C22⋊D10311yescharacteristic
139M16—1031no—
140M16—2061no—
141M16—2061no—
142C2×C8—2061no—
143C2×C8C22×D10311yescharacteristic
144C16—21no—
145C16—2061no—
146C23—41no—
147C23—4122no—
148C23—4121no—
149Q8—21no—
150Q8—1031no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—1031no—
156D4—4122no—
157D4—4122no—
158D4—4121no—
159D4—2061no—
160D4—2062no—
161C2×C4—41no—
162C2×C4—4122no—
163C2×C4—2061no—
164C2×C4D4×D10311yescharacteristic
165C8—4121no—
166C8—2061no—
167C8D4×D10311yescharacteristic
168C8C2×C22⋊D10311yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—16481no—
173C22—16482no—
174C22—4122no—
175C22—2061no—
176C22#1569304511yescharacteristic
177C4—41no—
178C4—2061no—
179C4#1569298911yescharacteristic
180C4D8×D10311yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—8242no—
185C2—4121no—
186C2#2026792311yescharacteristic
subgroupG/H
centerC2#20267923
commutator, cosocleC2×C8×C103C23
FrattiniC2×C8C22×D103
Fitting#509×C103C2
socleC2×C103#447
hypercenter#509D103
hypocenterC103#2192
LCSG ▷ C2×C8×C103 ▷ C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20268593C2semidirect
2. C2×#15692962C2semidirect
3. #20268058C2semidirect
4. #20267605C2semidirect
5. #20267598C2semidirect
6. #20267576C2semidirect
7. #509×C103C2semidirect
17. C2×C103⋊C16C22semidirect
18. #15692957C22semidirect
21. C2×D8×C103C22semidirect
24. M32×C103C22semidirect
94. C103#2192semidirect
96. #509D103semidirect
110. C2×D8C2×D103semidirect
115. M32C2×D103semidirect

Maximal subgroups

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

subgroupindex
#202685932
C2×#156929622
#202680582
#202676052
#202675982
#202675762
#509×C1032
#2192103