Finite Group Databasebeta

← #28364983

Subgroups — #28364983

Order 13952.

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

normal subgroups only

This group has 216 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#20295466C212yesmaximal, complemented
2#20295284C211yesmaximal, characteristic, complemented
3#20294836C212yesmaximal, complemented
4#20294808C211yesmaximal, characteristic, complemented
5#575×C109C211yesmaximal, characteristic
6C2×Dic436C2211yescharacteristic, complemented
7#16812678—21no—
8#16812645—22no—
9#16812643—22no—
10#16812639C2212yescomplemented
11#16812631C2211yescharacteristic, complemented
12#16812624—21no—
13C2×#41×C109—21no—
14#137×C109—22no—
15AGL(1,ℤ/8)×C109—22no—
16#135×C109—21no—
17#135×C109C2212yes—
18C2×M16×C109C2211yescharacteristic
19C2×Dic218—21no—
20Dic436—22no—
21C4⋊Dic109—21no—
22#13182008—41no—
23C22⋊Dic109—41no—
24#13182004—22no—
25#41×C109—22no—
26#41×C109—22no—
27#41×C109—41no—
28#41×C109—41no—
29C2×Q8×C109—21no—
30C2×D4×C109—21no—
31C2×D4×C109—41no—
32C22×C4×C109—41no—
33C22×C4×C109D411yescharacteristic
34Q16×C109—21no—
35Q16×C109D412yes—
36SD16×C109—22no—
37SD16×C109—22no—
38D8×C109—21no—
39D8×C109D412yes—
40M16×C109—22no—
41C2×C8×C109D411yescharacteristic
42C2×C8×C109C2311yescharacteristic
43C23×C109—41no—
44C23×C109D811yescharacteristic
45Q8×C109—22no—
46Q8×C109—41no—
47D4×C109—22no—
48D4×C109—41no—
49D4×C109—42no—
50D4×C109—42no—
51C2×C4×C109—21no—
52C2×C4×C109—21no—
53C2×C4×C109—42no—
54C2×C4×C109—41no—
55C2×C4×C109—42no—
56C2×C4×C109D811yescharacteristic
57C2×C4×C109C2×D411yescharacteristic
58C2×Dic109—41no—
59C2×Dic109—41no—
60Dic218—41no—
61Dic218—41no—
62C8×C109—21no—
63C8×C109C2×D412yes—
64C22×C109—21no—
65C22×C109—41no—
66C22×C109—42no—
67C22×C109—82no—
68C22×C109C2×D811yescharacteristic
69C4×C109—21no—
70C4×C109—42no—
71C4×C109#12011yescharacteristic
72C4×C109AGL(1,ℤ/8)11yescharacteristic
73Dic109—81no—
74Dic109—81no—
75C2×C109—21no—
76C2×C109—41no—
77C2×C109—82no—
78C2×C109#44711yescharacteristic
79#2191—1091nomaximal, Sylow
80C109#219111yescharacteristic, complemented, Sylow
81#575D10911yescharacteristic
82#510—1092no—
83#470—1091no—
84#360—1092no—
85#349—1091no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D10912yes—
91C2×Q16—1091no—
92C2×M16C2×D10911yescharacteristic
93C22⋊Q8—2181no—
94Q32—2182no—
95SD32—2182no—
96M32—1092no—
97#106—1091no—
98#103—2181no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—2181no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D10911yescharacteristic
109Q16—21no—
110Q16—2182no—
111Q16C22⋊D10912yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D10912yes—
116M16—22no—
117C2×C8C22⋊D10911yescharacteristic
118C2×C8C22×D10911yescharacteristic
119C4⋊C4—4361no—
120C4⋊C4—2181no—
121C22⋊C4—4361no—
122C16—2182no—
123C23—41no—
124C23#1318201411yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—4361no—
128Q8—4361no—
129D4—22no—
130D4—41no—
131D4—42no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—4361no—
139C2×C4—4361no—
140C2×C4#1318201411yescharacteristic
141C2×C4C2×C22⋊D10911yescharacteristic
142C8—21no—
143C8C2×C22⋊D10912yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1318201411yescharacteristic
149C4—21no—
150C4—42no—
151C4—8721no—
152C4—8721no—
153C4#1681266811yescharacteristic
154C4#1681272511yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#2029516011yescharacteristic
subgroupG/H
centerC2#20295160
commutator, cosocleC2×C8×C109C23
FrattiniC2×C8C22×D109
Fitting, prime radical#575×C109C2
socleC2×C109#447
hypercenter#575D109
hypocenterC109#2191
LCSG ▷ C2×C8×C109 ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20295466C2semidirect
2. #20295284C2semidirect
3. #20294836C2semidirect
4. #20294808C2semidirect
6. C2×Dic436C22semidirect
10. #16812639C22semidirect
11. #16812631C22semidirect
80. C109#2191semidirect

Maximal subgroups

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

subgroupindex
#202954662
#202952842
#202948362
#202948082
#575×C1092
#2191109